Advances in Heat Transfer, Volume 28: Transport Phenomena in Materials Processing

ADVANCES IN HEAT TRANSFER Volume 28 This Page Intentionally Left Blank Advances in HEAT TRANSFER Guest Editor Dimo...

Author: James P. Hartnett

15 downloads 561 Views 23MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

ADVANCES IN HEAT TRANSFER Volume 28

This Page Intentionally Left Blank

Advances in

HEAT TRANSFER Guest Editor Dimos Poulikakos* Department of Mechanical Engineering University of Illinois at Chicago Chicago, Illinoh

*PresentAddress: Department of Mechanical and Process Engineering, Institute of Energy Technology, Swiss Federal Institute of Technology, ETH Center, Zurich, Switzerland.

Serial Editors James P. Hartnett

Thomas F. Irvine

Energy Resources Center Unwersity of Illinois at Chicago Chicago, Illinois

Department of Mechanical Engineering State Universiry of New York at Stony Brook Stony Brook, New York

Serial Associate Editors Young I. Cho

George A. Greene

Department of Mechanical Engineering Drewel University Philadelphia, Pennsylvania

Department of Advanced Technology Brookhaven National Laboratoly Upton, New York

Volume 28

ACADEMIC PRESS San Diego Boston New York London Sydney Tokyo Toronto

This book is printed on acid-free paper.

@

Copyright 0 1996 by ACADEMIC PRESS, INC.

All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Academic Press, Inc. A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495 United Kingdom Edition published by Academic Press Limited 24-28 Oval Road, London NW 1 7DX

International Standard Serial Number: 0065-27 17 International Standard Book Number: 0-12-020028-7 PRINTED IN THE UNITED STATES OF AMERICA

CONTENTS Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix xi

Heat Transfer and Fluid Dynamics in the Process of Spray Deposition DIMOS POULIKAKOS AND JOHNM. WALDVOGEL

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. TheSprayRegion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 A. Convective Cooling of a Single Liquid Metal Droplet . . . . . . . . . . . . . . . 3 B. In-Flight Solidification of a Liquid Metal Droplet . . . . . . . . . . . . . . . . 11 C. Studies of Sprays in Spray Deposition . . . . . . . . . . . . . . . . . . . . . . . 18 I11. The Impact Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A. Splat Cooling of a Single Liquid Metal Droplet . . . . . . . . . . . . . . . . . 23 B. Impact and Solidification of Multiple Liquid Metal Droplets and Sprays . . . 54 IV . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Heat and Mass Transfer in Pulsed-Laser-Induced Phase Transformations P . GRIGOROPOULOS. TEDD . BENWEIT.JENG-RONG Ho. COSTAS XIANFAN XU.AND XIANG ZHANG

I . Pulsed Laser Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..................................... B. ThermalModeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Experimental Verification of the Melting Process . . . . . . . . . . . . . . . . D . Ultrashallow p+-Junction Formation in Silicon by Excimer Laser Doping . .

A . Background

E . Topography Formation

..............................

I1. Pulsed Laser Sputtering of Metals

..................... .................................... B. Time-of-Flight Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . C. Considering Thermal and Electronic Effects . . . . . . . . . . . . . . . . . . I11. Computational Modeling of Pulsed Laser Vaporization . . . . . . A. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Modeling Description-Transparent Vapor Assumption . . . . . . . . . . . A. Background

V

75 75 76 80 96 101 109 109 112 116 123 123 125

vi

CONTENTS

IV. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135 136 138

Heat and Mass Transfer in the Extrusion of Non-Newtonian Materials YOGESH JALURIA

I . Introduction

.................................... .................................... ................................. I1. Material Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . I11. Single-Screw Extruder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Tapered Screw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Residence-Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Mixing Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Twin-Screw Extruder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Background B . Literature Review

V . FlowinDies

....................................

.......................... ............................

A. Coupling of Extruder with Die B. Transport in Complex Dies

VI . Combined Heat and Mass Transfer . . . . . . . . . . . . . . . . . . . .

................................

A . Moisture Transport B. Chemical Reaction and Conversion

.......................

VII . Additional Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . VIII . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 146 149 152 155 155 176 176 180 184 187 190 197 201 201 205 213 213 218 220 225 226 227

Convection Heat and Mass Transfer in Alloy Solidification PATRICK J . PRESCOTT AND FRANK P . INCROPERA

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Physical Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

................. .............................. ............................... .....................................

A . Historical Perspective of Solidification Models B. Single Domain Models C. Micro/Macro Models D . Submodels

231 238 249 250 253 261 264

vii

CONTENTS

.......... ......................... .................................... V. Strategies for Intelligent Process Control . . . . . . . . . . . . . . . . VI. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IV. Theoretical Results and Experimental Validation A . Semitransparent Analog Alloys B. MetalAlloys

269 270 288 308 326 328 329

Transport Phenomena in Chemical Vapor-Deposition Systems Roop L. W A N

1. Introduction

..................................... ........................................ ..................... I1. Transport Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Rate-Limiting Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Some Basic Transport Considerations . . . . . . . . . . . . . . . . . . . . . . I11. Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.Scope B. Common CVD Reactor Configurations

A . Equations for a Multicomponent Mixture . . . . . . . . . . . . . . . . . . . . B. Simplified Governing Equations

C. Transport Properties

. . . . . . . . . .. .. .. .. .. .. .. .. ...............................

IV. Solutions for Selected Reactor Configurations

............

................................ ................................... ..................................

A . Horizontal Reactors B. Barrel Reactor C. Pancake Reactor D . &symmetric Rotating.Disk. Impinging.Jet, and Planar Stagnation-Flow Reactors E. Hot-Wall LPCVD Reactors

339 339 344 346 346 351 353 353 356 362 365 365 375 386

...................................... ...........................

389 399 408 413 414 415

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

427

V . Artificial Neural Network Models for CVD Processes . . . . . . . VI. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


CONTRIBUTORS

Numbers in parentheses indicate the pages on which the authors' contributionsbegin.

TEDD. BENNETT (75), Department of Mechanical Engineering, University of California, Berkeley, California 94720. COSTASP. GRIGOROPOULOS (75), Department of Mechanical Engineering, University of California, Berkeley, California 94720. JENG-RONG Ho (751, Department of Mechanical Engineering, University of California, Berkeley, California 94720. FRANK P. INCROPERA (230, School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907. YOGESHJALURIA (1451, Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, Piscataway, New Jersey, 08903. Roop L. W A (3391, N Department of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309. DIMOSPOULIKAKOS' (l), Mechanical Engineering Department, University of Illinois at Chicago, Chicago, Illinois 60607. PATRICK J. PRESCOTT (231), Department of Mechanical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802. JOHNM. WALDVOGEL (11, Motorola Inc., Schaumburg, Illinois 60196. XIANFANXu2 (751, Department of Mechanical Engineering, University of California, Berkeley, California 94720. XYWG ZHANG(75), Department of Mechanical Engineering, University of California, Berkeley, California 94720.

Present Address: Department of Mechanical and Process Engineering, Institute of Energy Technology, Swiss Federal Institute of Technology, ETH Center, Zurich, Switzerland. Present Address: School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907.


PREFACE

The area of materials processing has progressively become one of the focal points of university and industrial research in the 1990s. Transport phenomena play a central role in a plethora of applications in materials processing and manufacturing. However, the flow of information and collaboration among the scientific communities in the areas of transport phenomena and materials processing are not yet optimal. The purpose of this volume is to present a representative sample of existing research efforts in the area of transport phenomena, directly related to materials processing. This task is accomplished through five review papers (which compose the present volume) selected to cover a wide spectrum of applications. Naturally, due to space limitations the volume is not all-inclusive. However, I feel that it will provide the reader with a good flavor of the many exciting research areas in materials processing and manufacturing in which the transport phenomena scientific commumnity can contribute significantly. I thank the series editors of Advances in Heat Transfer for sharing my viewpoint that there is a pressing need for this special volume and for giving me the opportunity to put it together. Finally, thanks are due to all the contributors who made this volume possible in a timely fashion. D. Poulikakos, Guest Editor

xi


ADVANCES IN HEAT TRANSFER,VOLUME 28

Heat Transfer and Fluid Dynamics in the Process of Spray Deposition

DIMOS POULIKAKOS* Institute of Energy Technology, Swiss Federal Institute of Technology (ETH), ETH Centec Zurich, Switzerland

JOHN M. WALDVOGEL Motorola, Inc. Schaumbutg, Illinois

I. Introduction

Spray deposition (or spray casting) is a novel rapid solidification technol-

ogy for the creation of advanced metals and metal composites. This technology is particularIy attractive to manufacturing because it shows promise to provide materials and products that combine superior properties and near net shape. With reference to the former, the extremely high cooling rates present in the process of spray deposition (especially at the early stages) capture nonequilibrium states that cannot be captured by more conventional casting methods (foundry solidification, for example) because the atomic mobility in the liquid phase of a metal is far greater than that in the solid phase. To this end, the cooling rates at the early stages of the spray deposition process are of the order of (106-108YC/s. With reference to the latter, the spray deposition process has been shown to produce near net shape products which eliminate the need for additional finishing steps in the manufacturing process. Moreover, the fine and homogeneous grain microstructure that appears to result from the spray deposition process may eliminate the need for additional mechanical working [l, 21. In this paper, a review is presented of the existing knowledge base of the process of spray deposition, focusing on issues in which transport phenomena are relevant. ‘Present address: Department of Mechanical and Process Engineering, Institute of Energy Technology, Swiss Federal Institute of Technology, ETH Center, Zurich, Switzerland. 1

Copyright 0 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

2

DIMOS POULIKAKOS AND JOHN M. WALDVOGEL

The defining features of the process of spray deposition are shown in Fig. 1. This process involves four distinct regions. In the first region, the metal or metal alloy under processing is melted inside a crucible (often utilizing induction heating) and subsequently heated to a desired superheat temperature that ensures good fluidity. The melting occurs in an inert environment (e.g., argon or nitrogen) to limit oxidation. A stream of molten metal exits through the bottom of the crucible and enters the second of the four regions mentioned above, the atomization region, in which the liquid metal stream is blasted with an inert atomizing gas and disintegrates into a spray. In the third region of the spray deposition process, the spray region, liquid metal elements disintegrate further into droplets. Droplet coalescence also takes place in the spray region. The liquid metal droplets constituting the spray travel in an inert environment

FIG.1. Schematic illustrating the spray deposition process.

HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION

3

(to limit oxidation). Cooling occurs during this travel, which often results in the partial solidification of the droplets prior to impact. The impact and solidification of the droplets on the substrate constitutes the fourth region of the spray deposition process. From the preceding description it is clear that transport phenomena play a pivotal role on the spray deposition process. To exemplify, heat transfer and fluid dynamics phenomena take place in all four regions outlined above: the melting in the crucible, the breakup of the liquid metal stream into the spray, the transportation of the droplet in the form of a spray, and the splashing and solidification of the droplet on the substrate. Despite this fact, out knowledge base on the effect of transport phenomena in the spray deposition process is very limited. Most of the existing studies have been performed by materials scientists and focus on metallurgical aspects of the process, which are also of great relevance and importance. The mission of this work is to review the existing studies in the open literature that focus on the effect of transport phenomena on the spray deposition process. In doing so, the existing base of knowledge and the state of the art of this process from the standpoint of transport phenomena will be defined. In addition, the research needs in these areas will be identified. The presentation will be centered around the most challenging fluid dynamics and heat transfer aspects of the spray deposition process that occur in the spray region and the impact region. It is worth noting that the melting process in the crucible is rather well understood. For each of these regions both basic studies involving single droplets and droplet arrays as well as more applied studies involving sprays will be presented.

11. The Spray Region

A.

CONVECTIVE COOLING OF A SINGLE LIQUID

METALDROPLET

Basic studies on the convective cooling of a liquid metal droplet placed in a gas stream are the first step toward the investigation of the heat and fluid flow phenomena in sprays. In addition, these studies are relevant to dilute regions in the spray where the effect of interaction between droplets is not important. A significant base of knowledge in the general area of convective cooling of droplets already exists because of the wide use of sprays in many engineering applications exemplified by spray combustion. Most of this knowledge is summarized in a recent review paper by Sirignano [3]. It was not until recently, however, that a complete dedicated

4


numerical study focusing on the presolidification fluid dynamics and convection phenomena of the problem of a superheated liquid metal droplet placed in a uniform gas stream was published by Megaridis [4].This study is pertinent to the laminar flow regime and assumes axisymmetric flow conditions. In what follows immediately, the mathematical model and the main findings in Megaridis [4]are highlighted. A schematic of a liquid metal droplet in flight under laminar axisymmetric flow conditions is shown in Fig. 2 The study in Megaridis [41 simulates the acceleration and simultaneous cooling of a liquid metal droplet suddenly placed in a uniform stream of an inert gas. To this end, the flow field in the gaseous stream and the shear-induced flow field in the liquid metal droplet are considered simultaneously. The model relies on experience of droplet transport phenomena gathered from earlier combustion-related studies. The conservation equations in the gas phase are [4]. Continuity:

Radial momentum:

Liquid-Metal Droplet

FIG. 2. Schematic of axisymmetric flow inside and around a liquid metal droplet from Megaridis [4].


5

Axial momentum:

Energy:

In the preceding equations, ur and u, are the radial and axial velocity components, respectively; p is the pressure; a, and QZ contain the relevant viscous terms in the momentum equations; T is the temperature; and pg, k,, and cp, are the gas density, thermal conductivity, and specific heat at constant pressure, respectively. The conservation equations in the liquid phase modeling the flow in the liquid metal droplet are cast in the stream function-vorticity formulation after introducing the stream function in the usual manner: 1 r dz

1 a*

u l , z = - - -. r dr Thus, the conservation equations in the liquid metal region (phase) are Ul,r =

- -,

Vorticity equation: dt = --

Stream-function equation:

Energy equation:

6


In these equations the subscript 1 denotes the liquid metal region; +, the stream function; w , the vorticity; T , the temperature; t, the time; and c p , , p , , k , , and p I , the specific heat at constant pressure, viscosity, thermal conductivity, and density of the liquid metal, respectively. To complete the model formulation, the initial and boundary conditions utilized in Megaridis [4]are postulated. 1. Initial conditions a. Gas phase: Att=O:

u,=O,

u , = U ~ , ~p ,= p m , T = T m . (10)

The subscript m denotes incoming free-stream conditions, and Urn. is the initial relative velocity between the droplet and the free stream. b. Liquid phase +=w=O, T=To. (11),(12) Att=O: where To is the injection temperature of the droplet. c. Droplet surface u,=u,=O, p = p m , T = T o . (13)-(15) Att=O: 2. Boundary conditions a. Gas-liquid interface: The conditions at the gas-liquid in spherical coordinates and with n denoting the direction perpendicular to the interface are as follows: Shear stress continuity:

pg

[---

du,,,

=PI

a UI.0

-1

+ -1 d o ,

a

a

0

, (16)

&I

where a is the droplet radius and subscripts 1 and g denote the liquid and gas phases, respectively. Tangential velocity continuity: u I , Ols = ug,O l s .

(17)

Temperature and heat flux continuity:

b. Inflow and outflow boundaries of the computational domain Inflow boundary U, =

0,

U, =

U,, p = p a , T

=

T,.

(20)-(23)


7

Note that the free-stream velocity U, is time-dependent as a result of the relative deceleration of the droplet with respect to the free stream. Outflow boundary Du, Dt

Du, - _ _ _ = -DT Dt

Dt

- -Dp Dt

=

0,

where D/Dt denotes the total derivative with respect to time. c. Axis of symmetry a v , _ap= _dT - - u, = 0 , Gas phase: -(25) dr dr dr Liquid phase:

dT $ = o = - = 0. dr

Equations (1)-(26) constitute the theoretical model solved numerically in Megaridis [4] to study the presolidification convection phenomena in a single liquid metal droplet placed in a quiescent stream of inert gas. The base case of the numerical solution simulated the cooling of a superheated liquid aluminum droplet initially at 1000 K, suddenly injected in a stream of nitrogen at 400 K. The ambient pressure was 1 atm and the initial value of the Reynolds number [Re = d & / k ) ] was 100. The symbol d in the definition of the Reynolds number is the droplet diameter. The remaining symbols were defined earlier. The variable properties of the gas phase were obtained from standard correlations [5]. The thermophysical properties of the liquid metal phase were assumed constant [61. The first result of interest in Megaridis [4] was a temporal comparison of the drag coefficient to the well-known correlation for laminar flow over a solid sphere [7]:

c

-

24

-(I Re,

+ 0.1935Re:.6305);

20 I Re, I 260.

(27)

This equation utilizes the film-adjusted Reynolds number, which is based on the relative velocity between the droplet and the free stream, the droplet diameter, the free-stream density, and the gas velocity evaluated at the film temperature (average between the free stream and the droplet surface temperatures). The result of this comparison is shown in Fig. 3. Since the Reynolds number is based on the relative velocity between the free stream and the droplet, high values of the Reynolds number in this graph correspond to early times in the cooling process. Clearly, except for

8


1.56 1.52 1.48 -

75.0 77.5 80.0 82.5 85.0 87.5 90.0 92.5 95.0 97.5 100.0

Instantaneous Reynolds Number FIG.3. C, vs. Re from Megaridis [4].

the early stages of the process when the droplet is introduced into the liquid stream, Eq. (27) predicts well the drag coefficient and its use is recommended. A characteristic map of liquid isotherms obtained in Megaridis [4] is shown in Fig. 4a. The nondimensionalization of time was based on the , tU,/a2). Clearly, the viscous diffusion time scale in the gas phase ( T ~ = coldest location in the droplet is in the vicinity of the forward stagnation point. It is in this vicinity where the solidification process will be initiated. Hence, despite the fact that the maximum temperature difference in the droplet is small (1 K in Fig. 4a), the solidification will not be radially symmetric and should not be modeled as such. Note that solidification has been observed to take place under conditions of severe undercooling and in the presense of recalescence [8-101, as will be discussed later herein, which also renders the radically symmetric modeling of the process inappropriate.

a

LIQUID-PHASE ISOTHERMS

Contour Interval: 8.65E-02 K, Min: 990 K , Max: 991 K

Reynolds Number = 97.32 Ambient Temperature= 400 K Initial Droplet Temperahre= 1000 Initial Reynolds Number = 100

K

A. Liquid-Aluminum Dro let B. Ranr-Marshall, Film d3eynolds Number C. Ram-Marshall, Free-Stream Reynolds Number

10.0 9.0 8.0

7.0

6.0 5.0 4.0

3.0 2.0 1

.o

0.0

0.0 3.0

6.0

9.0

12.0 15.0

18.0 21.0 24.0 27.0

30.0

Gas Hydrodynamic Diffusion Time Scale FIG.4. (a) Map of isotherms in liquid metal droplet from Megaridis [4]. The gas flow is from left to right. The arrow indicates the direction of increasing temperature. (b) Nu vs. dimensionless time T from Megaridis [4].

10

DIMOS POULIKAKOS A N D JOHN M. WALDVOGEL

The final main result in [4] was to test whether the popular Ranz-Marshall correlation for laminar convection from a solid sphere is appropriate for the problem of interest. This correlation reads [ l l ] Nu

=

2

+ 0.6Re1/’ Pr1I3.

(28) The average Nusselt number (Nu) is defined on the basis of the droplet diameter, the surface-averaged heat transfer coefficient between the gas and the droplet, and the free-stream thermal conductivity. The Prandtl number (Pr) is that of the gas at free-stream conditions. The Reynolds number is based on the relative velocity between the droplet and the free stream. Figure 4b shows that comparison between Eq. (28) and the numerical predictions of the numerical model outlined earlier [4]. Curve A in Fig. 4b shows the results predicted by the model, curve B indicates the results of Eq. (28) utilizing the film-adjusted Reynolds number (Re,) defined earlier in connection with Eq. (27), and curve C shows the results of Eq. (28) utilizing a Reynolds number based on the droplet diameter, the relative velocity between the gas and the free stream, and the gas properties at free-stream conditions. As shown in Fig. 4b, curve C agrees better with the numerical results than does curve B, which implies that the free-stream properties should be used in Eq. (28) to estimate Nu in liquid metal droplets. Furthermore, the agreement between the numerical results and curve C can be described as fair (within 15%)with the Ranz-Marshall correlation underpredicting Nu. This may result in significant errors, especially if solid nucleation and partial solidification with recalescence occur during the droplet flight. A need for improved correlations exists in this area. Although the laminar flow results of Megaridis [4] improve our knowledge of the basic mechanisms in the cooling of a liquid metal droplet, they cannot be applied directly to the real spray deposition process because the relevant heat and fluid flow phenomena often are in the turbulent regime. Our literature review indicated that no study analogous to Megaridis [4] for liquid metal droplets in the turbulent regime exists in the open literature. Instead, lumped models combined with empirical correlations are utilized. A description of a typical model of this kind is given in Gutierrez-Miravete et al. [12]. The droplet velocity is obtained from a simple force balance on the droplet (Newton’s second law)

where pd, ud, 6 , A , , pg, u g , and g are the droplet density, velocity, volume, surface area, gas density, velocity, and gravitational acceleration, respectively. The drag coefficient is denoted by C,. The value of the drag

11


coefficient recommended in Lavernia et al. 112, 131 is the one obtained in the 1960s [14]: 6 21 C , = 0.28 + -+ -. 0.1 IRe I4000. (30) Re ' In this equation the Reynolds number is based on the gas properties, the droplet diameter, and the relative velocity between the droplet and the gas. When compressibility effects are important (the Mach number of the gas flow in the atomizer may be quite high [15]), they should be accounted for in the drag coefficient expression. To this end, drag coefficient relations for high-speed flow past a small sphere can be utilized. A correlation of this kind developed from research related to rocket nozzle design is [16, 171

CD

[:

r

(1

=

+ 0.15 Re0.687)[1 ( 1 + explexp ~ ~ 0 . 8 8 p- *

(31)

Re In this equation M is the Mach number and Re the Reynolds number, both of which are based on the relative velocity between the particle and the gaseous stream and the gaseous stream properties. Regarding cooling of the droplet in the transitional and turbulent regimes, simple, lumped models are commonly used [12-141:

where A is the droplet surface area, V the volume, h the heat transfer coefficient, T the temperature, and To the initial temperature. The radiative cooling of the droplet (which may or may not be important) is accounted for by the term Qrad.The thermophysical properties are those of the liquid metal. In order to use this equation to obtain the temperature history of the droplet, information on the heat transfer coefficient is needed. Despite its limited validity, the Ranz-Marshall correlation mentioned earlier [Eq. (28)] is commonly used to provide this information. A better alternative would perhaps be the correlation proposed by Whitaker [18, 191 for heat transfer from an isothermal spherical surface: Nu

hD k

= -=

2

+ (0.4Re1/*+ 0.06Re2/3)Pr0.4 3.5 < Re < 7.6

X

lo4. (33)

12


This equation has been tested for 0.71 < Pr < 380 and 1 < pJpS < 3.2. It should be used with caution outside these ranges. All the thermophysical properties in Eq. (33) are of the gaseous stream evaluated at free-stream conditions, except for the viscosity p,, which is evaluated at the sphere surface temperature. At this point it is worth stressing that Eqs. (28) and (33) are both applicable for solid spheres. Obviously, similar correlations must be developed for liquid metal droplets in the high-Reynolds-number regime.

B. IN-FLIGHTSOLIDIFICATIONOF

A

LIQUIDMETALDROPLET

The liquid metal droplet size in the spray deposition process varies between typically 10 and 300 pm. In addition, the flow of the inert gas causing the atomization process depends on location in the domain of the spray. Therefore, several scenarios are possible for each droplet during its flight. Some droplets (usually the smaller droplets) completely solidify during their flight and impact the substrate in solid form. Some droplets solidify only partially, with the degree of solidification depending on the droplet size. Finally, some droplets do not solidify at all during their flight (usually the largest droplets) and impact the substrate in the liquid state. As mentioned earlier, when solidification ensues during the flight of liquid metal droplets in the process of spray deposition, it does so in the presence of severe undercooling. From these observations, it is obvious that the study of solidification of a single liquid metal droplet flying in an inert-gas environment is very relevant to the process of spray deposition. Our literature search showed that such study has not been performed at a level of sophistication that would involve the solution of the Navier-Stokes and energy equations in the gas and liquid metal regions. This is true even for the case of axisymmetric laminar flow in the gas region. Instead, approximate (usually lumped) models are used to estimate the in-flight solidification process. Representatives of such models will be discussed next. The basic elements of a simple, spatially isothermal solidification model for a liquid metal droplet of initial temperature T , flying in a gas environment are contained in Dubroff [8] and Lavernia et al. [131. As discussed earlier, solidification takes place under severe undercooling. Four distinct regimes descriptive of the process can be defined as summarized in Fig. 5. In the first regime convective and radiative cooling takes place until a solid nucleation temperature (TN) is reached. Note that the solid nucleation temperature is lower than the equilibrium solid nucleation temperature (TL), which in the case of an alloy is the liquidus temperature. The difference TN - TL is the undercooling present at the initial


13

t

*'

TL

h

6

8

F

Time

FIG. 5. Illustration of the four distinct regimes of in-flight cooling and solidification: convective and radiative cooling, recalescence, slow solidification up to Ts, and cooling of solid sphere.

stage of freezing. The second regime is recalescence. Here the solidification progresses extremely fast and the latent heat released raises the droplet temperature to the recalescence temperature T R, which is often very close to the equilibrium liquidus temperature, TL. After recalescence, the third regime starts (Fig. 51, where solidification proceeds at a much slower rate and the droplet temperature continues to decrease (the heat removal from the droplet surface is larger than the latent heat of fusion in this regime) until the solidification process is completed and the droplet is at the solidus temperature, T,. In the last (fourth) regime the droplet is a solid sphere cooled convectively and radiatively by its gaseous environment. As discussed earlier, depending on droplet size and flow and temperature conditions, a liquid metal droplet can impact the substrate while in any of these four regimes.

1. Modeling of Convective Cooling and Radiatwe Cooling Regimes Since the regimes do not involve solidification they can be modeled in an identical manner [8, 131. An energy balance in a spatially isothermal

14

DlMOS POULIKAKOS AND JOHN M. WALDVOGEL

control volume defined by the outer surface of the droplet yields dT ~ VC dt

+ h,A( T - T,) + UEA(T 4 - T:)

=

0.

(34)

This equation is similar to Eq. (32) and accounts for graybody radiative cooling. The droplet volume is denoted by V , the surface area by A , the specific heat by c, the temperature by T , the time by t , the density by p, the gas free-stream temperature by T,, the convective heat transfer coefficient by h,, the emissivity by E , and the Stefan-Boltzmann constant by u.If the dependence of the liquid metal thermophysical properties on temperature is known (or if these properties are assumed for simplicity independent of temperature) and if T, is a known function of t or constant, Eq. (34) can be integrated forward in time, starting from an initial condition, to yield the temperature history in the droplet in regimes 1 and 4. Note that an equation like Eq. (28) or (33) needs to be used to calculate the convective heat transfer coefficient in the model. Utilizing Eq. (28), for example, yields

h

k -(2

,-D + 0.6Re'/2Pr'/3), -

(35)

where the droplet thermal conductivity is denoted by k and its diameter, by D. Since the Reynolds number is based on the relative velocity between the droplet and the free stream, the definition of this velocity in the present lumped model needs to be discussed. To this end, the gas velocity can either be assumed to be a function of the droplet velocity, or it can be approximately estimated as a function of the distance from the nozzle exit. With reference to the former, an example of a simple (albeit arbitrary) assumption is that the gas velocity is a constant percentage of the droplet velocity

where 4 is a number between zero and unity (e.g., 0.5). Using Eq. (361, we can obtain the absolute droplet velocity as a function of time. Subsequently, we can determine the relative droplet velocity (and the relevant Reynolds number) as a function of time. Aided by knowledge gained from the preceding observations, we can use Eqs. (34) and (35) to determine the temperature history of the droplet during its flight. With reference to the latter, Lavernia et al. [13], without offering a rigorous proof, assumed the gas velocity to decay from its initial nozzle exit


15

value according to

where z is the distance from the nozzle exit, zref is a reference distance, and uge represents the gas velocity at the nozzle exit. As discussed earlier, after the gas velocity is obtained from Eq. (37), the absolute droplet velocity results from Eq. (29) and the temperature history of the droplet, from Eq. (34). Another issue relevant to the preceding discussion is the value of gas density used in the velocity and temperature calculations. Veistinen et al. [20, 211 showed that in the case of argon as the atomization gas, a density value at the nozzle exit equal 2.7 times the gas density at room conditions yielded excellent agreement between calculated and measured [221 gas velocities. For the case of argon as the atomization gas again, regression analysis of experimental data in Beattle and Julien [231 for the variation of argon density with pressure yielded 1131 p g = 1.6317 x lO-’p

+ 1.0585,

(38)

where the gas density is expressed in kilograms per cubic meters (kg/m3) and the pressure is expressed in newtons per square meter (N/m2).

2. Modeling of Recalescence Regime The initiation of this regime is marked by the solid nucleation (Fig. 5). The exact value of the nucleation temperature (T,) for the various metals and alloys depends on a host of parameters exemplified by the metal purity, the cooling rates, and the flow conditions. Therefore, this value is seldom known and is assigned arbitrarily in models of the process. There is significant need to create a reliable database of nucleation temperatures for flying liquid metal droplets for a variety of materials. A simple lumped model of the recalescence regime can be constructed if it is assumed that solid nucleates at the outer surface and advances concentrically inward (radial symmetry). This assumption facilitates the analysis but as discussed earlier in connection with Megaridis [41 it is unlikely that radial symmetry exists in the freezing process. Solid nucleation is likely to occur first at the forward stagnation point of the droplet. This fact was recognized by Levi and Mehrabian [24], who presented a solidification model in which a solid nucleated at a single point of the outer surface and not the entire surface. The simple model presented below does assume radial symmetry of the

16


freezing front and should be viewed in the context of this limitation. If the solid shell after nucleation and the liquid region are concentric, the solid fraction on the droplet is given by r

3

X=l-(&

(39)

where R is the droplet radius and r is the radial coordinate. The speed of propagation of the solid front [U = (dr/dt)]is obtained by taking the time derivative of Eq. (39): dX 3(1 - x ) ~ / ~ dt R During recalescence heat transfer at the phase-change interface occurs under nonequilibrium conditions. To this end, a crystallization kinetics relationship for the freezing velocity is required at the freezing front. Assuming that curvature effects do not dominate the freezing process the following relation can be used at the freezing front [25]: - =

where dm is the molecular diameter taken as the molecular jump distance, DL is the diffusivity in the liquid phase, A H , is the latent heat of fusion per molecule, T is the temperature, AT is the undercooling, Tf is the fusion temperature, and K is the Boltzmann constant [expressed in joules per kelvin per molecule (JK-*molecule-')]. for low and up to moderate undercooling, a Taylor series expansion of Eq. (41) accurate to the first order yields the Wilson-Frenkel relation [26, 271:

U =KAT,

( 42)

where the kinetic coefficient is

Note the introduction of a correction factor P [25, 281. This correction factor was suggested [28] to account for the fact that the molecular jump distance across the solid/liquid interface may be smaller than that for diffusion in the bulk liquid. In addition it corrects for reorientation effects necessary with asymmetric molecules. Determining the value of the correction factor P is obscure and is commonly taken to be equal to unity in the literature. The value of the kinetic coefficient can be determined from Eq. (43) utilizing the relevant thermophysical properties. Errors will, of course,

17


arise from approximations in property variations (e.g., the dependence of

D, on AT [251). These errors usually do not drastically affect the nucleation estimates, which are much more sensitive to the effect of nucleants. In the model presented here, Eq. (42) wiIl be assumed to describe adequately the crystallization kinetics [8, 121. Combining Eqs. (40) and (42) yields

The volumetric heat generation rate in the droplet due to the solidification process is

and the energy equation [Eq. (34)] modified to account for this heat generation becomes dT dX ~ VC + pVAH, - h c A ( T - T,) u & A ( T 4- T,) = 0. (46) dt dt All the symbols in Eq. (46) have been defined earlier. Radiation cooling is taken into account for the sake of generality, unlike in Grant et al. [8, 12, 131, where radiation effects were justifiably neglected. Equations (44) and (46) can be solved simultaneously with simple numerical means to yield the temperature history in the droplet in the recalescence regime. The initial conditions for the numerical solution are that at t = 0 the solid fraction is equal to zero ( x = 0) and the temperature is equal to the nucleation temperature ( T = T N ) .The solution should be advanced in time until the recalescence temperature is reached, marking the end of the recalescence regime.

+

+

3. Modeling of Solidification Regime This regime involves conventional solidification starting at the recalescence temperature, T R .The solid fraction at the beginning of this regime is that obtained at the end of the recalescence ( x R ) .It is again assumed that the heat of fusion is released uniformly within the droplet as the temperature decreases from T R to T,. Therefore, the energy equation for this regime becomes pVc

+ pVAH, TR--XT,R

IdT

-

dt

+ h c A ( T - T,) + v & A ( T 4- T,)

=

0.

18


Equation (47) can be integrated in time numerically until the solidification is complete (the droplet temperature becomes the solidus temperature). Regime 4, discussed earlier, follows. At this point the description of the solidification process of a single liquid metal droplet in flight based on a simple, lumped model is complete. It should be reiterated that the assumption of radial symmetry in the solidification process inherent in the studies that formed the basis for construction of the model presented in this section [8, 12, 131 was relaxed in the work of Levi and Mehrabian [24]. These authors developed a mathematical model for the freezing process in the undercooled droplet from a single (point) nucleation site at its surface neglecting radiative cooling from the surface. They also discuss the implications of single versus multiple nucleation sites. Their results on the freezing process indicate the presence of two distinct solidification regimes. In the first regime the solidification interface velocities are high, the droplet absorbs most of the released latent heat of fusion, and the surface cooling plays a minor role. In the second regime the solid growth is much slower and depends greatly on the cooling of the droplet surface. The extent of rapid solidification was determined to be a function of the nucleation temperature, the particle size, the kinematic coefficient, and the heat transfer coefficient. In a recent theoretical study of solidification of metal drops, Bayazitoglu and Cerny [29] analyzed the process of conduction freezing using a lumped model as well as a radially symmetric nonisothermal model. In addition to the radial symmetry imposed in the model (the shortcomings of this convenient assumption were discussed earlier), the presence of recalescence resulting from the severe undercooling and the associated nonequilibrium phenomena was also neglected. It was found that for relatively slow cooling rates up to lo4 K/s, which the authors claimed to be relevant to powder production, the lumped model was sufficiently accurate and the assumption of constant temperature inside the droplet justified. Numerical results for nonisothermal freezing showed that the proper choice of convective heat transfer coefficient and the accurate determination of the thermal emissivity are important in the determination of the temperature field and the freezing velocity of the interface.

c. STUDIES

OF SPRAYS IN SPRAY

DEPOSITION

Not many studies have been performed on real liquid metal sprays focusing strictly on the process of spray deposition. Most of the studies of this kind are pertinent to powder production. In addition, the main goal in the majority of published investigations on powder production is the


19

determination of particle size and distribution and not the relevant complex heat and fluid flow phenomena in the liquid metal spray. Representatives of these studies will be reviewed next. The discussion will center around studies utilizing the popular inert-gas metal atomization. Biancaniello et al. [301 performed an experimental study of metal atomization utilizing a supersonic inert gas. Using Schlieren, shadowgraph, and flash photography methods, they studied qualitively the flow fields in the gas and liquid regions. Emphasis was placed on aspiration conditions and nozzle designs that promoted the atomization process. The aspiration condition is the variation of the nozzle exit pressure with the die plenum pressure. It was found that a characteristic wave structure dominated the gas flow field at the maximum aspiration operating conditions. Using water as the operating fluid, it was also shown that the shape on the inner nozzle bore was instrumental in producing a liquid sheet. In a subsequent paper, Biancaniello et al. [31] analyzed metal powders produced by supersonic gas metal atomization. They discussed the droplet fragmentation mechanisms in the spray leading to the powder size distribution and offered three possible mechanisms of secondary droplet formation. In the first scenario, the primary droplet is pulled into a ligament that pinches down to the classic dumbbell shape and eventually breaks into two droplets. In the second mechanism the gas flow distorts the primary droplet into an umbrella shape, and this droplet further divides into many small droplets. The third mechanism is a variation of the first mechanism, in which satellite droplets form from the ligament in addition to the two secondary droplets mentioned above. In a companion paper, Biancaniello et al. [32] performed a real-time particle size analysis during inert-gas atomization based on the principle of Fraunhofer diffraction. They found their method to be a suitable candidate for process feedback control. An important parameter in liquid metal sprays is the mean diameter of the droplets in the spray. To this end, for lack of a better alternative, results from the literature on metal powders are commonly used to characterize gas-atomized liquid metal sprays in casting processes [ 121. A popular correlation of this kind is the Lubanska correlation for the mass mean droplet diameter [33, 341:

where dm0,5is the mass droplet diameter. According to its definition [351, 50% (fraction 0.5) of the total mass of the spray contains droplets of diameters smaller than that given by Eq. (48). In other words, Eq. (48) defines the diameter of the holes of a screen that would allow only 50% of

20


the total mass of the spray to pass through it. The parameters K , v, , vg, M,, M g ,d , , and We denote an empirical constant, the liquid metal kinematic viscosity, the gas kinematic viscosity, the melt mass flow rate, the gas mass flow rate, the melt stream diameter prior to atomization, and the Weber number of the liquid metal, respectively. The liquid metal and gas flow rates mentioned above are given by

where A , A , , pm , pg , g, F , 1, y , and Po are the cross-sectional area of the metal jet, the effective area of the gas nozzle, the metal density, the gas density, the gravitational acceleration, the discharge coefficient from the crucible, the height of the melt in the crucible, the ratio of the specific heat for constant pressure to that for constant volume of the atomizing gas, and the plenum pressure of the atomizing gas, respectively. The ratio M g / M , is a major factor on gas atomization of liquid metals, since it appears explicitly in Eq. (48). The gas flow rate can be kept practically constant for a fixed atomizer design [36, 371 by maintaining a constant gas atomization pressure. This is not true for the metal flow rate. Conventionally, the molten metal stream is allowed to fall freely by gravity to enter the region where the atomization process takes place. Hence, the metal mass flow rate depends on the metal static pressure head [denoted by 1 in Eq. (49)l. Controlling this pressure head controls M,. Ando et al. [36] performed a study to determine the pressure at the exit of the metal delivery tube during gas atomization. Knowledge of this pressure is necessary for the determination of the metal mass flow rate. On the basis of Bernoulli's theorem, they developed a method for the determination of the above-mentioned exit pressure, They found it to differ considerably from the ambient pressure in confined gas atomization where the melt stream is confined in the close vicinity of the atomizing gas jets. They used an ultrasonic gas atomizer for their work and water as the atomized fluid. The main theoretical result of their work (based on Bernoulli's theorem) is the following equation on the pressure difference between the ambient pressure ( P , ) and the pressure at the delivery tube exit ( P 2 ) A P = P 2 - P , = ~2 ( 2 g l -

(5:fj;i.

All the symbols in this equation are defined either in Fig. 6 or earlier in


i, i I

21

0 0 0 0 0 0 0

FIG.6. Illustration showing details of device to generate liquid metal spray.

connection with Eqs. (481450). The discharge coefficient is defined as

where E, is the mechanical energy per liquid mass of the liquid metal lost as frictional heating, V, is the velocity at the exit of the metal delivery tube (position 2 in Fig. 61, and p2 is a factor correlating the velocity to the kinetic energy at the same location [37]. The value of F is usually defined experimentally [361. Ando eta[. also propose the following equation for the metal mass flow rate as a function of the metal static head and pressure difference

By moving the delivery slit up and down relative to the gas jet impingement point, the authors [36] also found that if the gas jets are deflected on the delivery slit (Fig. 7a), the pressure is lowered, resulting in aspiration of the metal stream. If the gas jets just miss the delivery slit, the pressure increases resulting in undesirable backpressure effects in the metal stream (Fig. 7b). Finally, if the gas jets impinge well below the delivery slit (Fig. 7c), only a slight decrease or no effect was observed in the pressure and the metal flows freely by free fall. Earlier works on pressure determination

22


FIG.7. Delivery slit and gas jet effect on spray Schetch prepared following the discussion in Ando et al. [36].

at the exit tube of ultrasonic gas atomizers (USGAs) using a circular gas atomizer without metal flow [21, 38, 391 have also shown that significant variations of the delivery slit pressure compared to the atmospheric pressure may result depending on the relative position of the delivery slit and the gas jets. Before closing this section it is worth reiterating that a significant need for heat transfer and fluid mechanics research in real liquid metal sprays in the process of spray deposition exists. The existing base of knowledge relies on findings more pertinent to the related process of powder metallurgy and does not address numerous issues unique to the spray deposition process.

111. The Impact Region

The impact region in the process of spray deposition is perhaps the most challenging from the standpoint of transport phenomena. Splashing of liquid metal droplets initially on the substrate at high speeds (up to 100 m/s) and, later, on the completely or partially solidified layer of the already deposited material in the presence of rapid heat transfer and solidification under nonequilibrium conditions are features of this region. A good description of the various mechanisms responsible for the evolution of the microstructure in the deposited layer in the duration of the process given in Annavarapu et af. [2] is as follows:


23

Splat solidification. Here, discrete droplets that impact on the substrate spread extremely fast (the spreading time scales can be as low as nanoseconds [40]) and solidify primarily by conduction through the substrate. A droplet is completely solidified before another droplet impinges on it. Solidification rates are very high. This mechanism is characteristic of early stages of the spray deposition process and of thin deposit layers in particular. It yields fine-grain structures. Growth ofnuclei. This mechanism is present when low heat extraction from the bottom of the deposited layer results in a buildup of energy within it. The top region of the deposit (on top of which the incoming droplets fall) is only partially solidified. The cooling rates are low [12, 411. Studies have indicated [41,42] that the growth and coarsening of solid-phase nuclei in the partially solidified layer is directly related to the production of fine equiaxed grains in the final microstructure of the solid. Incremental solidification. Incoming droplets impact on a thin, completely liquid layer at the top of the deposit. The solidified material underneath the liquid layer acts as a chill, causing the advancement of the freezing front and the growth of the deposited layer. Incremental solidification also occurs when the thickness of partially solidified layer mentioned above becomes constant because the energy input equals the energy extraction in this layer. The solidification rates in the incremental solidification mechanism are low.

On the basis of this discussion it is clear that studies of the heat transfer and fluid mechanisms of both single as well as groups of liquid metal droplets impacting and solidifying on a substrate are directly related to the process of spray deposition. Single-droplet studies are particularly relevant to the splat solidification mechanism of the process. In the following sections single as well as multiple droplet studies will be reviewed sequentially. A.

SPLAT COOLING OF A SINGLE LIQUID

METALDROPLET

The heat and fluid flow phenomena occurring during the impact of a single liquid metal droplet on a cold substrate are neither conventional nor easy to study. There are several reasons for this fact. The fluid dynamics of high-speed droplet spreading is a free-surface problem with dramatic domain deformations in the presence of surface tension, and with possible droplet breakup phenomena and three dimensional effects. The heat transfer process involves rapid solidification, possibly under nonequilibrium conditions, in the presence of convection in a severely deforming

24


domain, coupled with conduction in the substrate and, whenever important, radiation to the environment. It is for these reasons that theoretical models of splat cooling were (and still are) constructed on the basis of educated assumptions. The first generation of such models does not involve sophisticated modeling of the complex impact fluid dynamics. Such studies will be reviewed first. 1. Studies without Sophisticated Fluid Dynamics Modeling Attempts have been made to facilitate the study of the various mechanisms of the fluid dynamics of the splashing process based on order-ofmagnitude (scaling) arguments. Bennett and Poulikakos [43] reviewed the state of the art of these attempts and proposed appropriate criteria that define the effect of surface tension and viscous forces on the maximum spreading of a droplet impacting a solid surface in connection with the process of splat-quench solidification. They defined two domains: the viscous dissipation domain and the surface tension domain, which are characterized by the Reynolds number and the Weber number and are discriminated by the principal mechanism responsible for arresting the splat. It was found that correctly determining the equilibrium contact angle was important to the prediction of the maximum spreading. Conditions under which the solidification process should not be expected to affect the maximum spreading were also determined. Utilizing a combination of arguments published earlier in Madejski [44], Collings et al. [45], and Chandra and. Avedisian [46], Bennett and Poulikakos [43] proposed the following equation for the spread factor 6 (the ratio of the splat diameter to the droplet diameter)

( 6/1.2941)5 + 3[(1 - cosWe8 ) e 2 - 41 = 1, (54) Re where 8 is the equilibrium contact angle, Re is the Reynolds number (based on the droplet diameter, the impact velocity, and the liquid metal viscosity and density, p u d / p ) , and We is the Weber number based on the liquid metal surface tension, density, droplet diameter, and impact velocity, pu2d/a. For the purpose of illustrating the relative contribution of the viscous energy dissipation and surface tension in terminating the splat spreading, a typical value of the equilibrium contact angle 8 = 7r/2 was assigned in Eq. (54) in Bennett and Poulikakos [43] to yield ( 5/1.294q5 Re

2- 41 + 3[ tWe

=

1.

(55)

25


In the extreme when We (55) becomes

+

~0

(surface tension effects are negligible), Eq.

5, = 1.2941Re'/'.

(56) On the other hand, when the viscous energy dissipation is negligible, Re + m: '/2

5,=(?+4)

.

(57)

The subscripts v and s in Eqs. (56) and (57) represent viscous and surface tension effects, respectively. If Eq. (56) were used to estimate the spread factor, it would yield a value larger than the actual value because it completely neglects surface tension effects. To improve the predictions of Eq. (561, a correction factor C , was introduced in Bennett and Poulikakos [43] such that

6 = C,& = C,1.2941Re'l5. Eliminating the spread factor 6 between Eqs. (55) and (58) yields Re=

[

(58)

[(We/3)(1 - C:) + 41'" 1.2941C,

(59)

Similarly, introducing a correction factor C , to improve the predictions of Eq. (57), which neglects viscous effects, we obtain

Combining Eqs. (55) and (60) to eliminate the spread factor, we obtain Re

=

(Cs[We/3 + 4]1/2/1.2941)5 * (1 - C;)[I + 12/We]

Equations (59) and (61) were plotted together in Fig. 8 [431, which defines graphically the viscous dissipation and surface tension domains in the splat-quenching process. The border between the two domains is marked by the boldface curve C , = C, = 0.816. Observing Fig. 8, we see that even well into the viscous dissipation domain the surface tension effects are significant. On the other hand, the viscous dissipation effects disappear more rapidly in the surface tension domain. By curve-fitting the boldface border curve of Fig. 8, the following condition was proposed in Bennett and Poulikakos [43], under which surface tension effects dominate the

26


Cv=0.95

500

.-Cs=0.60

_ _ - - -_ _ - - Surface tension domain

.... '

0

0

*

*

;

'

2000

'

'

;

'

c

*

;

'

'

*

6000

4000

:

'

n

'

8000

10000

Re FIG.8. We vs. Re from Bennett and Poulikakos [431 showing the viscous and surface tension domains.

termination of the splat spreading We < 2.8

(62)

In the scaling arguments of Bennett and Poulikakos ([431, and references therein), the effect of solidification in the droplet spreading was neglected. In the great majority of existing heat transfer studies on splat-quench solidification the fluid dynamics aspects of the process are neglected for simplicity. It is assumed that the droplet after impact spreads first, cools down, and subsequently solidifies. A comparison of the relevant time scales indicates that such an assumption is perhaps justified, especially at a first attempt to study the complex problem of splat-quenching of liquid metal droplets. Investigations in this category (neglecting the fluid dynamics) will be reviewed first. Research efforts in the area of splat-quench solidification first became noticeable in the 1960s, after it was observed that certain alloys could yield new metastable crystalline phases and amorphous solid phases. These results were attributed to the very high. cooling rates (in excess of lo5 K/s). Most of the research in the 1960s and early 1970s focused primarily on metallurgical aspects of the process and it has been reviewed by Jones [47] and by Anantharaman and Suryanarayana [48]. Studies of


27

heat transfer aspects of splat-quenching solidification are more recent. Madejski [44, 491 performed analytical and experimental studies of heat transfer during splat cooling of metal droplets. Based on a unidirectional solution of the Stefan type that assumes that both the splat and the substrate are semiinfinite bodies he determined the dependence of the spread factor 6 (defined earlier) on the Weber, Reynolds, and Peclet numbers and a parameter, k, defined as

where the subscripts s and 1 denote density in the solid and liquid regions, U is a freezing constant, and E = R,/D is the ratio of the liquid metal disk radius at the time of initial contact to the droplet diameter. Note that in the analysis of Madejski 1441 it was assumed that a droplet of diameter D deforms to a disk of radius R, when it contacts the substrate. Madejski [49] postulated that:

0

In the limit k

=

(1/Re)

In the limit k

=

(1/We)

6,

=

=

=

0, We > 100:

0:

1.2941(Re

+ 0.9517)”5.

(65) In these equations 6, is the maximum spread factor. In the general case concerning the flattening of a droplet without freezing ( k = 0) if We > 100 and Re > 100, the following equation was recommended for the maximum spread ratio [44]:

Results for the case where freezing was present in the spreading process

( k > 0) were obtained numerically and showed an insensitivity of the value of the maximum spread factor on k for small values of the Weber number. In the opposite extreme (Weber and Reynolds numbers approach infinity) the dependence of the maximum spread ratio on k was given by

6, = 1.5344 k-0.395. (67) The agreement between theoretical predictions and experiments in [49] were deemed to be “not bad” by Madejski. Numerical investigations of heat transfer and solidification aspects of a splat-cooled liquid metal droplet without accounting for the associated fluid dynamics phenomena and by modeling the heat transfer in the

28


substrate and the splat as unidirectional were performed by Wang and Matthys [50-521. They presented results of the interface velocity as a function of propagation distance with and without undercooling of the melt. With undercooling, the freezing interface velocity was shown to decrease rapidly as the freezing front advances. Without melt undercooling, the freezing interface velocity is heat-transfer-limited. This resulted in less drastic changes in the interface velocity with the propagation of the freezing front. The quality of the thermal contact between splat and substrate was influential is sustaining the interface velocity in an undercooling melt and critical in the absence of undercooling. Relevant to the splat-quenching process is the work of Shingu and Ozaki 1531, who investigated numerically rapid solidification occurring by conduction cooling. Rosner and Epstein [54] studied theoretically the simultaneous kinetic and heat transfer limitations in the crystallization of highly supercooled melts. Evans and Greer [55] developed a one-dimensional numerical solution to the rapid solidification of an alloy melt in order to study the solute trapping. They employed a two-equation model relating the interface velocity and solid composition to the temperature and liquid composition at the interface. In a recent paper, Bennett and Poulikakos [56] presented a combined theoretical and experimental study of the splat-quenching process. Although they did not consider the fluid mechanics of the process, they presented an extensive conduction-based model of the freezing process accounting for axisymmetric conduction in the substrate. In this respect, their model is more general than what has been presented in previous studies of similar nature [44, 49-52] and will be discussed in detail below. On the basis of order-of-magnitude arguments Bennett and Poulikakos [56] stated that as a first approximation it is sometimes reasonable to assume that in the process of splat-quenching the liquid metal droplet spreads first and solidifies subsequently. To this end, the splat was modeled as a thin liquid metal disk initially at uniform temperature, T,, which was suddenly brought into contact with a large (by comparison) substrate of initial temperature, To considerably lower than the freezing temperature of the splat material, Tf (Fig. 9). Heat was conducted away from the splat into the substrate. Solidification ensued and progressed until the entire splat was solidified. The heat conduction cooling of the splat continued after solidification was completed, until the splat temperature reached the substrate temperature. The heat conduction process was modeled as two-dimensional in both the splat and the substrate. To this end, the range of validity of previous one-dimensional models was explored in Bennett and Poulikakos [561. The conduction equation describing the transport of heat in the splat with


29

Liquid

i Splat

*

1 2

FIG.9. Schematic of the “disk” model in Bennett and Poulikakos [43].

respect to the cylindrical coordinate system (r, z ) of Fig. 9 is [561 j

dt

=

l,s,

(68)

where the subscript j takes on the values 1 or s when Eq. (68) is applied to the liquid or to the solid portion of the solidifying splat, respectively. The temperature is denoted by T,, the time by t , and the thermal conductivity, density, and specific heat of the splat by k j , p j , and c j , respectively. Equation (68) reflects the independence of the heat transport on the angular position from symmetry considerations. The heat conduction equation in the substrate is dT [d2T 1 dT p c -at= k : + d- r- + - ,r d r

]:lt (69)

in which the notation is analogous to that defined earlier, following Eq. (68). To complete the model formulation, the relevant initial, boundary, and matching conditions need to be discussed [56].The initial conditions of the problem are that both the substrate and the splat were isothermal prior to making contact with one another: At t = 0: T, = T,, T = To. (7017(71) The boundary conditions at the top and at the lateral surface of the splat are

’

dT,

Atz=0:

k .= h,(T, dz

Atr=R:

- k .I = ha(? dr

-

Ta), j

=

~,1,

(72)

-

Ta), j

=

s,1,

(73)

d q

30


where ha is the heat transfer coefficient between the splat surface and the ambient, T, is the ambient temperature, and the symbols s and 1 respectively, denote solid and liquid. These boundary conditions account for the convective removal of heat from the splat surface. A similar boundary condition to Eq. (73) accounts for convection of heat from the top surface of the substrate. Since freezing takes place in the splat, the solid and the liquid regions are separated by a freezing interface. The matching conditions for the temperature field at this interface are

where the subscript i denotes the position of the freezing interface and U, its velocity. Conditions (74)and (75) stand for the temperature continuity and for heat flux discontinuity because of the heat released on solidification. The negative sign in the left-hand side of Eq. (75) reflects the fact that the freezing interface velocity is pointing to the negative z direction. Note that in writing Eq. (75), the radial conduction was neglected for simplicity. This approximation is appropriate within the context of this model since the splat thickness is at least two orders of magnitude smaller than the splat diameter. The matching conditions at the splat-substrate interface are as follows:

This condition can alternatively be written for the substrate side of the interface: dT Atz=H: -- k - = h , AT, (77) dz These matching conditions account for the presence of a contact thermal resistance at the splat-substrate interface. To this end, matching condition (76) states the fact that the heat flux leaving the splat at the interface equals the product of a heat transfer (resistance) coefficient descriptive of the imperfect thermal contact at the interface, multiplied by the temperature jump across the interface (AT,) defined as the difference between the interface temperatures at the splat and the substrate sides. Condition (77) is analogous to Eq. (76) written for the substrate side of the interface. The temperature of the substrate far away from the interface is not affected by the presence of the splat

As z

+

a:

T + To.

(78)


31

The last issue to be discussed before completing the description of the heat conduction model is the undercooling present in the splat at the initiation and subsequent development of the solidification process. In the classic treatment of a freezing front, the front is defined by the freezing temperature of the material and its propagation velocity is limited by the rate at which heat can be conducted away from this front into the liquid and solid regions [Eq. (731. However, this treatment does not account for the presence of undercooling in the melt prior to the initiation of solidification. Such undercooling is a common occurrence in the splat-quenching process and other rapid solidification processes and results in the freezing front being at a temperature below the equilibrium freezing temperature. As discussed earlier in connection with Eqs. (41) and (42) to account for this fact, a freezing kinetics relationship between the amount of undercooling and the velocity of propagation of the freezing interface is needed. The equation used in Bennett and Poulikakos [56] is identical to Eq. (42):

U

= K(Tf-

Ti),

(79)

where K is the freezing kinetics coefficient, T, is the equilibrium freezing temperature of the solid-liquid interface, and Ti is the actual temperature of this interface. No details of the finite-difference method used for the numerical solution of the model in Bennett and Poulikakos [56] are given herein for brevity. Figures 10 and 11 present comparisons of theoretical and experimental results for the temperature and quenching rate histories, respectively, at the splat-substrate interface. A copper substrate was used in Fig. 10 and a Pyrex substrate was employed in Fig. 11. The initial temperature of both substrates was 25°C. The numerical model for the results in Fig. 10 duplicated the experimental condition of a 3-mm lead droplet released from 30 cm above the substrate with temperature at a release time of 468°C. The temperature at impact time was estimated to be 460°C. The experimentally measured spread factor was 4.1. The heat transfer coefficient at the splat-substrate interface that defines the contact resistance was assumed to be h , = 15 kW/m2 K. This value is within the ranges reported in the literature and was chosen so as to yield the best agreement between the theoretical model and the experiment. The conditions for the results of Fig. 11 were similar. This time the lead droplet diameter was measured to be 2.7 mm, the droplet impact temperature 486"C, and the spread factor 4.4. The contact heat transfer coefficient used was h , = 100 kW/m2 K. Examining Figs. 10 and 11, we conclude that the predictions of the model are satisfactory, especially if one takes into account the relative simplicity of the model. The temperature of the splat-substrate interface

a

500 400

300

200

100

o t 0

0.02

I

I

0.04

0.06

0.08

0.1

Time (s)

40,000 35,000

-9

30,000

v)

25,000 Numerical

Y

a

I

m c& 20,000

.-C

r 0

5

15,000

0 10,000

5,000

0 0

0.01

0.02

0.03

0.04

Time (s)

0.05

0.06

0.07


33

decreases monotonically in both Figs. 10a and l l a , except when solidification is initiated in the splat, resulting in temporary increase in temperature. The rate of cooling is very rapid initially, and slower after the completion of the freezing process. Figures 10b and l l b present a more critical comparison between experimental and numerical results in the form of quenching rate/time curves. It can be seen from these figures that after the quenching rates have become relatively small, the numerical and experimental results compare very favorably. At early times qualitative agreement is present but the sluggish response of thermocouples prohibits good quantitative agreement between theory and experiment. The numerical model was used to further explore and establish quantitatively the effect of the contact thermal resistance. Figure 12 shows the numerical results on the thermal history of splats quenched with varying degrees of thermal contact quality. The effect is substantial. The thermal histories of the bottom-center surface node of the splat for interface heat transfer coefficients ( h , ) ranging from 10 to 100 kW/m2 K are shown. It can be seen that as the interface heat transfer coefficient becomes small (poor thermal contact), its effect dominates the cooling rate of the splat (Newtonian cooling). Conversely, as the heat transfer coefficient becomes large, its effect becomes less influential to the cooling rate of the splat (ideal cooling). It is also clear that the interface heat transfer coefficient has significant influence over the length of time required to initiate freezing as well as over the duration of the freezing process. A characteristic feature of the numerical results of Figs. 10-12 is the undercooling disappearing just prior to the onset of freezing. This is an interesting detail that experimental results were unable to detect clearly. Referring back to Fig. 10a, it is apparent that the heat extraction is insufficient to sustain the undercooling achieved prior to solidification; that is, the kinetics of crystalline formation is so rapid that the rate of latent heat released is sufficient to substantially reheat the splat. This is an observation worth further consideration, because it indicates that despite the original undercooling, the interface freezing temperature may quickly

FIG.10. Comparison of experimental (circles) and numerical results of splat quenching of lead on a copper substrate from Bennett and Poulikakos [43]: (a) splat-substrate interface temperature vs. time; (b) splat-quenching rate vs. time. Experimental results: initial droplet temperature 468"C, initial substrate temperature 2 5 T , droplet diameter 3.0 mm, free-fall distance 30 cm, spread factor 4.1. Numerical results: initial splat temperature 460°C, initial substrate temperature 25"C, droplet diameter 3.0 mm, free-fall distance 30 cm, spread factor 4.1, heat transfer coefficient 15 kW m-' K-',undercooling, 40°C.

a

5001

Numerical

0

b

0.02

0.04

0.06

0.08

Time (s)

- Numerical

Time (s)

0.1


35

rise to the melting temperature. Note that microstructural features of the resulting solid are determined largely by the rate of solidification, which is dictated by thermal conditions at the freezing interface. Details of the transient thermal conditions within the splat during its solidification obtained numerically are presented in Fig. 13. For the sake of clarity, there are relatively few time steps presented. It can be seen that the freezing interface temperature rises to within 1°C of the melting temperature after this interface has propagated only 10 p m into the splat ( t = 5.13 ms). This result informs us that recalescence is confined to a relative small region adjacent to the contact surface. Hence, the enhanced solidification speed afforded by undercooling the melt is largely ineffectual at influencing the gross properties of the splat when thermal contact with the substrate is poor ( h , = 15 kW/rn2 K in Fig. 13) and the splat thickness is relatively large (156 p m in Fig. 13). One distinctive attribute of the numerical solution to the splatquenching problem in Bennett and Poulikakos [56] is that it accounts for two-dimensional conduction into the substrate. Previous investigations into this system have assumed that the conduction of heat into the substrate can be treated as one-dimensional. Figure 14 shows the dissipation of heat into a copper substrate assuming close to ideal thermal contact between the splat and substrate ( h , = 100 kW/m2 K). The evolution-decay of thermal gradients in the substrate is very rapid. The substrate surface temperature reaches a peak of approximately 95"C, during the period in which latent heat is released while the splat freezes. The elevation of temperature much beyond the outside radius of the splat (5.37 mm) is marginal, especially in the initial period of quenching. For most of the very early period, in which heat is being removed from the splat, isotherms developing beneath the splat are very flat, indicating one-dimensional conduction in the substrate material. The long-term transfer of heat away from the vicinity of splat, however, becomes significantly two-dimensional as demonstrated by the curvature of the isotherms.

FIG. 11. Comparison of experimental (circles) and numerical results of splat quenching of lead on a Pyrex substrate [43] (a) splat-substrate interface temperature vs. time; (b) splat-quenching rate vs. time. Experimental results: initial droplet temperature 494°C initial substrate temperature 2 5 T , droplet diameter 2.7 mm, free-fall distance 30 cm, spread factor 4.4. Numerical results: initial splat temperature 486"C, initial substrate temperature 2 5 T , droplet diameter 2.7 mm, free-fall distance 30 cm, spread factor 4.4, heat transfer coefficient 100 kW m-* K - ' , undercooling, 40°C.

36


450 400 350 300 250

200

150

100

50

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Time (s) FIG.12. Effect of thermal contact resistance between lead splat and copper substrate on the thermal history of the bottom-center location of the splat. Initial splat temperature 460°C, initial substrate temperature 25”C, droplet diameter 3.0 mm, spread factor 4.1, heat transfer coefficient 15 kW m-’ K - ’ [431.

2. Studies with Sophisticated Fluid L$narnics Modeling The work reported by Madejski and others [44, 49-561 represents a “fist generation” of studies of transport phenomena in the process of impact and solidification of a liquid metal droplet on a substrate. In all these studies the fluid mechanics of the process, for all practical purposes, was not taken into account in order to circumvent associated difficulties. Very recently, studies have been performed to explore the fluid dynamics of a liquid metal droplet impact. Examples of such studies are the works of Fukai et al. 157, 581, Trapaga and Szekely 1591, Lui et al. [60, 611, Marchi et al. [62], and Tsurutani et al. [63]. These and other relevant works will be reviewed in this section,


-

37

= 4.74ms

-t

= 5.13111s

-t

- - = 5.31111s - - - - - = 5.65ms - - - I = 6.60ms - - - .- = 9.08ms 1-1

t

t

-..--t = 13.08ms .....t = 14.54111s

t

\y, ’

0

20

295

60

40

Distance

from

80

Splat/Substrate

100

120

Interface

140

160

( pm)

FIG.13. Temperature variation with distance from the splat-substrate interface for a host of times I431.

Fukai et al. [57] published a theoretical study of the deformation of a spherical liquid metal droplet impinging on a flat surface. The study accounts for the presence of surface tension during the spreading process. The theoretical model is solved numerically utilizing deforming finite elements and grid generation to simulate accurately the large deformations, as well as the domain nonuniformities characteristic of the spreading process. The results document the effects of impact velocity, droplet diameter, surface tension, and material properties on the fluid dynamics of the deforming droplet. Two liquids with markedly different thermophysical properties, water and liquid tin, are utilized in the numerical simulations. The occurrence of droplet recoiling and mass accumulation around the splat periphery are standout features of the numerical simulations and

38


Time = 5 . 0 ~ 1 0 -(s).~

IO'C

.

0

pcr Isolhcrm. Ouler moat Isotherm = 30.C.

b ? l\

-..- T

3uu

/

I

1,000 lull 1,500 2.000 2,500

6000

8000

4000

2000

"

urn

;

'

l 6000

"

' 4000

6000

4000

8000

I O T per Isotherm. Outer most Isotherm = 30%.

Time = 2 . 0 ~ 1 0 .(s~).

8000

2000

0 wm

l

'

'

(

l 0

2000

~

~ 2000

'

l

~

4000

~

~

6000

~

'

8000

wm 10.C pcr Isotherm. Outer most Isotherm = 30.C.

Time = 5 . 0 ~ 1 0 -(~ s).

2.000 2,500

8000

6000

4000

0

2000

2000

4000

6000

8000

P Time = I.OXIO-~(s).

IO'C

per Isotherm. Outer most Isotherm = 30%.

0 500 1,000

wm 1.soo 2,000 2.500 8000

6000

4000

2000

0

2000

4000

6000

8000

P

FIG.14. Transient isotherms in the copper substrate. Initial splat temperature 46o"C, initial substrate temperature 2 5 T , droplet diameter 3.0 mm, spread factor 4.1, heat transfer coefficient 100 kW m - 2 K - ' undercooling, 40°C [43].

~

'

~


a

t"

39

bZ

0

X

FIG.15. Deforming droplet coordinate system in Fukai et al. [57].

yielded a nonmonotonic dependence of the maximum splat radius on time. The work in Fukai et al. [57] is a departure from earlier attempts at similar problems [59, 631 that were based on finite differences (marker and cell and control volume methods) with fixed grids. The mathematical model for the droplet spreading in Fukai et al. [57] is presented next. The model was formulated to simulate the impact of a liquid droplet on a solid substrate, starting at the instant that the droplet comes into contact with the substrate and proceeding until the droplet comes to rest after the splashing process is completed. A Lagrangian approach was adopted [64] because it facilitated the accurate simulation of the motion of the deforming free surface. In the Lagrangian axisymmetric conservation equations within an initially spherical droplet impacting on a solid surface (Fig. 151, r, z , and 6 are respectively the radial, axial, and azimuthal coordinates (Fig 19, p is the density, u is the radial velocity component, u is the axial velocity component, t is the time, p is the pressure, p is the viscosity, Y is the kinematic viscosity, g is the gravitational acceleration, c is the speed of sound, and y is the surface tension. The stresses are denoted by ui,, the uniform droplet velocity at the time of impact by u,-, and the radius of the droplet at impact by r,,. The conservation equations in dimensionless form are

dV

1

d

1

40


The dimensionless initial and boundary conditions of the problem are [57]

U=O, V = - l ,

AtT=O:

At 2

=

2 P = - We .

(83)

U = V = 0.

0:

(85)

At the free surface

-

Grrnr

+

H CrZnz= - 2 -n,, We -

C,,n,

+ q z n z= - 2

H

-n,. We

The nondimensionalization was carried out according to the following definitions: r R=z = -Z u = -U T / = - U r0 ro UO uo

The mean curvature of the free surface was defined as

+ [(rrl2+ (zt)2]nr 2 r ~ [ ( r ‘+) (~z r ) 2 ] 3 ’ 2

r2(r’.zr! - z’rrr)

H=

(89)

In these equations primes denote differentiation with respect to the arc length along the free surface s (Fig. 15). The nondimensionalization process created the following dimensionless groups (Reynolds, Weber, Froude, and Mach numbers, respectively): R e = - ,uoro U

We=--pr,u,2 Y

F r = - ,4 ro g

M = -UO. C

(90)

Note that the time derivative has been maintained in the continuity equation to facilitate the numerical solution of the model, as explained in


41

detail in Fukai et al. [57]. This solution was based on deforming finite elements and grid generation in connection with the artificial compressibility method and the Galerkin method [571. Background on the several issues involved in the artificial compressibility method is contained elsewhere [65-761. Herein, key results from Fukai et al. [57] will be discussed next. A numerical simulation was conducted involving a liquid tin droplet of radius r = 12 p m impinging on a stationary flat plate with a velocity ug = 25 m/s. These conditions were chosen as typical to spray coating applications. It is important to note, however, that solidification was not modeled in Fukai et al. [57]. For liquid tin, the following values were used: surface tension coefficient y = 0.554 N/m, density p = 7000 kg/m3, and kinematic viscosity v = 2.6 x lo-’ m2/s. These above conditions resulted in the following values of the relevant dimensionless numbers: Re = 1200, We = 100, and Fr = 5.6 X lo4. Figure 16 depicts a sequence of frames corresponding to different instances of the metal droplet impact process. Immediately after contact, a thin film is formed at the periphery of the splat, which propagates radially at velocities substantially higher than the impact velocity. These velocities varied as a function of time, but in general, their temporal peaks occurred at early stages, and were 2-3 times larger than the droplet impact velocities. The formation of a ring-shaped tip of the laterally propagating sheet is present in the tin simulation. However, it appears that the impeding effects of surface tension to the overall spreading process take longer to dominate compared to numerical simulations for water droplets [57]. This trend is expected, due to the higher value of We for the tin droplet compared to the water droplet (100 vs. 10 or 1.4). As seen in Fig. 16, the droplet stretches to a significant degree before its spreading is halted by the dominance of the surface tension mechanisms ( T > 4). It is important to note the significant mass accumulation around the periphery of the liquid tin splat, which is most pronounced after T = 2 (see Fig. 16). Its existence was verified with simple laboratory experiments by Fukai et al. [57]. The lateral flow direction of the edge of the splat is eventually reversed, as clearly shown in the late stages of the simulation. The tin simulation was carried to rather long times as a result of properly resolved splat thicknesses in the vicinity of the axis. The longer simulation times resulted in the capturing of the splashing event occurring when the reversed flow reached the axis of symmetry. The computation was terminated at T = 12.4 (5 ps after impact) due to grid generation limitations to adequately follow the upward motion of the apex formed in the center of the splat and the subsequent possible breakup of the flow into ligaments and/or droplets.

42


time=2.000

t

time=4.001

t

rs

time=l.200

time=6.801

tirne=l1.601

FIG.16. Splashing sequence of a liquid tin droplet with Re

=

1200 from Fukai et al. [57].

Substantial insight may be gained on the splashing dynamics by monitoring representative splat parameters as a function of time. For example, the splat radius as well as the splat thickness on the axis of symmetry are two parameters whose temporal variation can elucidate the relevant mass and momentum transport processes. Several additional simulations were performed in Fukai et af.[57], and selected results are discussed below.


43

Figure 17 displays the effect of Reynolds number on the nondimensional splat thickness Z,,,,, at the axis of symmetry of the flow domain. Four different runs are compared, characterized by the same Weber number (We = 801, and Reynolds numbers of 120, 1200, 6000, and 12,000, respectively. It is immediately apparent that the initial stage of impact (up to T = 2) is almost identical for all four cases, when the splat thickness at the axis of symmetry is employed as a measure of the deforming droplet dynamics. This result, which is in agreement with the predictions of Fukai et al. [58], suggests that the initial rate of change of the splat thickness at r = 0 is directly proportional to the impact velocity (u,,). The slope of the linear portion of the graph displayed in Fig. 17 is approximately equal to 0.88, a value slightly different from the values 0.7 to 0.84 reported in Trapaga and Szekely [591. Subsequent stages of the simulations are markedly different, displaying thicker splats corresponding to lower values of Re. Figure 17 also demonstrates that the splat thickness approaches an asymptotic value that increases with decreasing Re. In all four cases

Tin

(

We = 80 )

[a] - Re = 12,000 [b] - Re = 6,000 [c] - Re = 1,200 [d] - Re = 120

0

2

4

Dimensionless time

6

8 T

10

(tv I ro)

FIG. 17. Effect of the Reynolds number on the dimensionless splat thickness from Fukai et al. [57].

44

DIMOS POULlKAKOS AND JOHN M. WALDVOGEL

depicted in this figure, the final splat thickness at r = 0 is smaller than 10% of the preimpact diameter of the droplet. In fact, almost all tin droplet simulations completed showed that the final thickness at the center of the splat represents only a small fraction of initial droplet diameter (typical value around 5%). Figure 18 shows the effect of Reynolds number on the time variation of the dimensionless splat radius R,,, (the distance of the outermost liquid element from the axis of symmetry). Three simulations considered previously (We = 80 and Re = 120, 1200, and 6000) are compared in this figure, which demonstrates some important features of the spreading process. Initially, the splat radius increases with nondimensional time up to a maximum value. The splat radius subsequently decreases, and for the lowest Reynolds number approaches an asymptotic value. As expected, the maximum splat radius depends on the Reynolds number, since larger values of Re would naturally result in larger spreading. In addition, the simulations showed that the time at which the maximum occurs shifts

3.5 Tin

(

We = 8 0 )

3 .O

2.5

1

/, ‘4

\

K\ 1’ \ .A.

\ \

\

2.0

[a] - Re = 6,000 [b] - Re =1,200 [c] - Re = 120

1.5

1 .o 0

2

4

Dimensionless time

8

6 7

10

(tv / r ) 0

0

FIG. 18. Effect of the Reynolds number on the dimensionless splat radius from Fukai et al. [57].


45

toward later stages as the Reynolds number increases. The reason for this finding is as follows: The maximum of the curves shown in Fig. 18 corresponds to the time at which the outward spreading of the fluid has terminated. After this time, flow reversal and recoiling of the droplet occur. As the Reynolds number (flow inertia) increases, it takes more time for the combined action of viscosity and surface tension to slow down and eventually arrest the flow, thus explaining the shift in the maximum of the curves mentioned above. Figure 19 documents the effect of Weber number on the dimensionless splat radius R,,,. Three different runs are compared, characterized by the same Reynolds number (Re = 1200) and Weber numbers of 500,1000, and 5000, respectively. In all cases, the splat radius initially increases with nondimensional time, then achieves a maximum, and eventually decreases before it approaches an asymptotic value. Since larger Weber numbers correspond to lower surface tension forces, the maximum splat radius at high We values is expected to be higher (a trend clearly displayed in Fig.

4.0

I

I

3.5

3.0

x

A

I2

2.5

2.0

1.5

1 .o 0

2

4

6

Dimensionless time r

8 (tvo/

10

ro)

FIG. 19. Effect of the Weber number on the dimensionless splat radius from Fukai et al. [571.

46


19). In addition, the instant where the maximum splat radius is achieved is shifted toward later times with increasing values of We. The current study clearly shows that the final splat spread is affected by the corresponding Weber number. This trend does not agree with the results reported in Trapaga and Szekely [59], where a relative insensitivity of the rate of spreading on Weber number was found for the range of flow conditions investigated therein; We = 200-2000 and Re = 100-105. The effect of droplet size on the splat spreading rate was investigated by performing two additional simulations, each of a liquid tin droplet impinging on a stationary flat plate with velocity uo = 4 m/s. The two droplets, however, were characterized by different radii: ro = 375 and 37.5 pm, respectively. These conditions resulted in the following values of dimensionless numbers: Re = 6000, We = 80, Fr = 4600 for the ro = 375 p m droplet, and Re = 600, We = 8, Fr = 46,000 for the ro = 37.5 pm droplet. Figure 20 shows the temporal variation of splat radius for the two splatting events. The lower values of Re and We for the smaller droplets suggest that surface tension effects are more important than in the larger

3.5

j /

2.5

[a]

- ro =

I

I

&

vo = 4 m/s

Tin

.375 mm

Re=6,000 We=80 Fr=4.6E+3

i I

2.0

[b] - ro = .0375 mm

/

Re=600 Fr=.6E+4

1.5

1 .o

0

2 Dimensionless time

4 T

6 (tvJ ro)

FIG.20. Effect of droplet size on the splat spreading rate from Fukai el al. [57].


47

droplet case; therefore, the corresponding splat radius curve remains consistently lower than that of the larger droplet throughout the simulation. The modeling of the splat cooling of a liquid metal droplet accounting for the associated fluid mechanics phenomena as described in Fukai et al. [57] but in the absence of solidification was completed recently by Zhao et al. [77]. These authors used the Lagrangian formulation and extended the fluid dynamics model of Fukai et al. [57] to account for the heat transfer process in the droplet and the substrate. Following the notation in Fig. 15, the dimensionless heat transfer model accompanying the fluid dynamics model of Fukai et al. [57] in Zhao et al. [77] is

$1 2)

Energy equation in the splat: dT =

dt

-(

- [dR R e P r R dR 1

1

+

Energy equation in the substrate: dT, --

dt

-(-

A

1

d

- [ R g ]

R e P r R dR

+

z). d=T,

(92)

Initial and boundary conditions: T=l,

Art=0: At the droplet surface:

dT

T,=O. dT

-n ,

dR

(93)Y (94)

+n, = 0 dZ

(95)

dT

At the substrate surface prior to the splat arrival: In the substrate far from the interface ( 2 + - m) : At the splat-substrate interface ( Z

=

- = 0 (96) dZ

T, = 0 (97)

0) :

The nondimensionalization was carried out according to the following definitions:

The nondimensionalization of all additional quantities was defined earlier [in Eq. (SS)].

48


In addition to the usual Prandtl number (Pr = v / a ) , the following dimensionless groups appeared in the heat transfer model:

In the preceding equations T is the temperature, k is the thermal conductivity, and CY is the thermal diffusivity. The subscripts s, *, and 0 denote substrate, dimensional quantity, and initial state, respectively. The remaining quantities were defined earlier in connection with the fluid dynamics model of Fukai et al. [571. In addition to the theoretical model outlined above, an experimental study was presented in Zhao et al. [78]. The experimental findings verified the numerical predictions adequately. The intricate details of both the numerical and the experimental procedures are contained in Zhao et al. [77, 781 and will not be repeated here for brevity. Representative results, shown in Figs. 21 and 22, will be discussed next. The first set of results examines the effect of the substrate material on the cooling of a molten metal droplet in low-speed spray coating applications. The temperature distribution is represented by the contour lines denoting the isotherms (Fig. 21). Instantaneous streamlines and the velocity vectors are also plotted in the left half of the droplet region to better illustrate the fluid flow effects on the thermal development history of the droplet. A tin droplet of radius ro = 9 p m was considered to impinge on three different substrates at a velocity uo = 29.4 m/s. For molten tin, the following property values were used: surface tension coefficient y = 0.544 N/m, density p = 7000 kg/m3, kinematic viscosity v = 2.64 X m2/s, thermal diffusivity CY = 1.714 X m2/s, and thermal conductivity k = 30 W/m * K. The values of the relevant dimensionless groups are Re = 1000, We = 100, and Fr = lo7. The substrate materials examined are copper, steel, and glass, respectively, to cover a large portion of the spectrum of the thermal diffusivity and the thermal conductivity. The thermal diffusivity and thermal conductivities for the substrate materim2/s and als used in the simulations are correspondingly 1.17 x 401 W/m K for copper, 3.95 X m2/s and 14.9 W/m . K for steel [American Iron and Steel Institute (AISI) 304 stainless steel], and 7.47 X lo-’ m2/s and 1.4 W/m * K for glass. It is immediately apparent from Fig. 21a-c that the cooling of the impinging droplets occurs practically simultaneously with the spreading. In the entire droplet spreading process the droplet temperature field demonstrates convective and two-dimensional features. In all cases, the fluid temperature is higher in the center region and lower around the spreading front. This is because high-temperature fluid is continuously supplied to

-


49

FIG.2Na). Tin droplet spreading on different substrates from Zhao et al. [77]: (a) T = 1.0.

50


FIG.21(b). Tin droplet spreading on different substrates from Zhao el al. [77]: (b) T

=

2.0.


FIG.2").

51

Tin droplet spreading on different substrates from Zhao el al. [77]: (c) T = 4.0.

52


10

:

9 -

Ink Droplet

ink experimental - - - - - - - ink numerical - - - - - . solder experimental solder numerical

r, = 1.48 mm v, = 1.945 m/s Re = 2860 We = 76 Fr = 262

8 7 -

5 h

6 -

Solder Droplet (TinnRad 50/50) ro = 1.6215 mm v, = 1.7 m/s Re = 10894 We = 83 Fr = 183

v

J

5: 4 -

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Time (nu)

FIG.22. Comparison of theoretical and experimental results on the maximum splat radius from Zhao et af. [78].

the center region, and the splat periphery is continuously cooled down by contacting the low-temperature surface of the substrate as it spreads outward. In the cases of steel and glass substrates, at late stages of the droplet spreading (7 = 4.0, Fig. 2 1 4 the temperature distribution within the splat is largely one-dimensional. The temperature gradients within the splat occur in the radial rather than the axial direction despite the fact that the splat thickness is much smaller than the splat diameter. It appears that at this time the center of the splat is cooled by the splat periphery rather than the substrate. This is a result of the strongly convective nature of the cooling process at the earlier stages and implies that approximate modeling attempts using an axial heat conduction model for a thin disk to simulate the heat transfer within the fully spread droplet are not accurate when the thermal conductivity of the substrate is low. The premise that “the droplet spreads first and cools down later” associated with the early analyses of splat cooling and thermal spray deposition may not always be valid. As shown in Fig. 21a-c, the copper substrate temperature changes very little during the entire process of spreading, whereas the glass substrate temperature increases rapidly in the neighborhood of the

HEAT TRANSFER AND FIUID DYNAMICS IN SPRAY DEPOSITION

53

splat-substrate interface. The droplet impacting on the copper substrate cools down the fastest compared to the droplets impacting on the steel and glass substrates under identical conditions. The flow structure is dominated by the inertial force at the initial stages of spreading. The flow structure experiences a drastic change at the late stages of spreading when the inertial force decreases as the fluid spreading slows down by the action of surface tension forces. Noteworthy is the mass accumulation around the periphery of the splat. A secondary flow vortex emerges near the contact line (magnified in the detail of Fig. 21c). The vortex grows larger and moves up to the top of the free surface in the splat periphery region. The theoretical modeling was quantitatively validated by performing numerical simulations with the same conditions present in experimental investigations performed using a photoelectric technique [78]. The temporal variation of spread ratios measured with the photoelectric method were compared with the numerical predictions [781. Representative results of such comparisons are shown in Fig. 22. The agreement between theory and experiment is good in the case of a liquid solder droplet. This is due to the fact that wetting effects are not as important in this case because liquid solder does not significantly wet the glass surface. The agreement deteriorates after the initiation of recoiling primarily because the effect of contact angle hysterisis is not modeled in Zhao et al. [77]. In a recent paper Liu et al. [60] presented a numerical investigation of micropore formation during the impact of molten droplets on a substrate in a plasma deposition process. Their work extends the earlier paper of Marchi et al. [62]. They utilized a fixed-grid finite-difference model for the fluid mechanics component of the problem, and they adopted the model of Madejski [44], which follows the unidirectional conduction (Stefan) approach for the heat transfer and freezing components of the problem. Although they dealt with velocities as high as 400 m/s, their model was axisymmetric. Note that earlier experiments on plasma-sprayed niobium particles on a substrate [79] showed severe droplet fragmentation (particle crushing) and a splat configuration that is far from axisymmetric. Subject to these approximations and to experimental verification, Liu et al. [601 postulate that following flattening and solidification the splat edge may separate from the solid-liquid interface, causing the formation of micropores in the fringe region of the splat. In the limits of high and low impact velocity and substrate temperature, they report low microporosity. If the impact velocity or substrate temperature is between these two extremes, the deformation and solidification velocities become comparable and most of the voids formed due to separation are fixed in the solidification layer, yielding large microporosity. Liu et al. [60] summarize that the combination of a liquid or mushy droplet condition at high impact velocity with a

54


solid or mushy surface condition improves contact and adhesion and reduces microporosity. Liu et al. [61] reported a numerical study of molten droplet impingement on a nonflat surface. They conclude that during the impingement onto an axisymmetric wavy surface, a single droplet spreads and eventually forms a thin, nonflat splat. If the surface wavelength is larger than the droplet diameter, the spreading process features an acceleration-deceleration cycle, which results in a violent breakup of the liquid. If the surface wavelength is smaller than the droplet diameter, the surface hinders the spreading process. The normal stress introduced by the curved surface affects the spreading process. As expected, decreasing the roughness size of the deposition surface and increasing the roughness spacing improved the splat flattening and reduced the liquid breakup. Waldvogel and Poulikakos [85] recently performed numerical simulations of the impact and solidification of picoliter size solder droplets on a substrate. They extended the numerical and theoretical tools described in Fukai et al. [57, 581 by improving the grid generation methodology and by modeling the solidification processing. At the time this present chapter was written, the results produced in Waldvogel and Poulikakos [SO] were typical of low impact velocities not common in the process of spray deposition. Despite this fact, it was clear that the final shape of the solidified splat depended greatly on the process parameters. Figure 23 shows three timeframes of the process of impact and solidification of a 50-pm-diameter solder (63/37 Sn/Pb) droplet impacting with 2-m/s velocity on an FR4 substrate (FR4 is a composite material commonly used in the electronics manufacturing industry). The droplet temperature at impact is 200°C and the initial substrate temperature 25°C. The droplet is completely solidified in the last frame (after 50 p s ) . In this case, the solidified splat has a doughnut shape. The velocity vectors in Fig. 23 populate the liquid region, which has not solidified yet, and visualize the fluid motion. The solidified portion of the droplet lies below the liquid region. A host of splat shapes were produced in Waldvogel and Poulikakos [80] depending on the operating conditions.

B. IMPACTAND SOLIDIFICATION OF MULTIPLELIQUIDMETAL DROPLETSAND SPRAYS 1. Studies without Sophisticated Fluid Dynamics Modeling The problem of heat transfer in the splat cooling of two liquid metal droplets impacting sequentially on a substrate was studied recently by Kang et al. [Sl]. The theoretical part of the study was focused on the heat

0.0

I .II

FIG.23. Representative stages of solidification of the 63/37-Sn/Pb droplet impacting on an FR4 substrate (from Waldvogel and Poulikakos [80]).

56


transfer aspects of the solidification process and the difference in the behavior of the solidification of the first and second droplet. The experimental part of the study aimed at the characterization of the structure of solidified splats composed of one or two droplets. It was found that the solidification of the second droplet exhibited drastically slower cooling rates compared to the first droplet. As a result, the grain structure of the top of a two-droplet splat was considerably coarser than the structure of the top of a single-droplet splat. The findings of Kang et al. [811 implied that in splat cooling severe limitations need to be imposed on the thickness of the resulting solid layer to ensure rapid solidification and fine-grain structure. In addition, it was shown that the temperature field in the substrate is two-dimensional and radial conduction in the substrate should not be neglected in the modeling of the process. The theoretical model of Kang et al. [81] is presented below. Following the simple approach used for single splats (discussed earlier), the difficulties associated with the fluid dynamics of the process were circumvented in Kang et al. [81]. In the resulting simple and easy-to-use model of the solidification process, each droplet in the splat was viewed as a thin metal disk initially in the liquid phase and finally, after the solidification was completed, in the solid phase. It was assumed that the second disk (splat) contacted the first disk (splat) a short time after the solidification process of the first splat was completed. Figure 24 shows a schematic of the model of a single-droplet splat, and Fig. 24 shows a schematic of the model of a two-droplet splat according to the aforementioned simplifications. Owing to the fact that the thickness to diameter ratio of the splat is very small (typically on the order of 0.05-0.15) the heat conduction process in the splat is modeled as unidirectional [44, 49-561. However, the present model will account for both radial and axial conduction in the substrate following the recommendations of Bennett and Poulikakos [56] for single splats. The first droplet in the splat is assumed to be a thin metal disk at uniform temperature that is suddenly brought into contact with a large (by comparison) substrate of initial temperature considerably lower than the freezing temperature of the disk metal, Tf.Heat is conducted away from the splat to the substrate. Solidification ensues and progresses until the entire splat is solidified. After a short time, the second droplet of the splat is deposited on top of the first droplet. The modeling and the initial conditions of the second droplet are identical to those of the first droplet. Heat is conducted from the second splat to the first splat and eventually to the substrate. Solidification of the second splat ensures and progresses until the entire region solidifies. Following the preceding discussion, the


57

b

1 s t Splat

FIG. 24. Schematic of single and double droplet impact (from Kang ef al. [Sl], with permission from ASME).

heat conduction equation in both the first and the second droplet in the splat reads

where the subscript j takes on the values 1 or 2 corresponding to the first (bottom) or the second (top) disk and the superscript m takes on the values s or 1 denoting solid or liquid region. The temperature is denoted by T ; the time, by t ; the axial coordinate, by z (Fig. 24); and the density, specific heat, and the thermal conductivity of the solidifying material, by r, c, and k , respectively. The heat conduction equation in the substrate is dT dt

where the notation is analogous to what was defined above with the added clarification that r stands for the radial coordinate (Fig. 24).

58


The initial conditions for the splat and the substrate are

To, Ti( z , t' = 0) = T o , R ( r , z , t = 0) = T, . T : ( z , t = 0)

(104)

=

(105)

( 106) It is worth clarifying that t is the time from the contact of the first splat with the substrate and t ' is the time from the contact of the second splat with the first splat. The boundary and matching condition accompanying the preceding equations are T ( r -+ m, z T ( r , z -+

-k;"

-kT -k

=

0,t)

-m,t)

-

dz

=

( 107) ( 108)

T,, T,,

=h,[T;"(z=H,t) -T,];

-

")

=

=

h , [ T , " ( z = 2 H , t ) - T,];

m=sorI,

(109)

m = s or I, (110)

= h , [ T ( r > R , z = O , t ) -T,] r > R , r=0,1

Boundary conditions (107) and (108) simply state the fact that the temperature inside the substrate far away from the surface as well as the temperature of the substrate surface remote from the splat region remain unaffected by the solidification process and retain their initial value. Boundary conditions (109)-(111) account for convective losses through the top of the first splat, the top of the second splat, and the top of the substrate, respectively. Clearly, boundary condition (109) is in effect prior to the deposition of the second splat. After the second splat is deposited on top of the first splat boundary condition (109) is replaced by

m = s or I, (112) where h, is the contact heat transfer coefficient (the reciprocal of the contact resistance) between the two splats. A matching condition analogous to Eq. (112) is utilized at the interface to couple the substrate with the first splat. -k

fl) dz

=

h,[T(r < R,z=O,t) - T;"(z=O,t)];

r> B , using a simple scaling analysis, and is borne out by experimental data. The temperature gradient in the y direction is expected to be much greater than that in the down-channel direction. The diffusion term in the y

158

YOGESH JALURIA

direction is, therefore, retained. Thus, the energy equation becomes

where T is the local temperature, p the density, C p the specific heat at constant pressure, and k the thermal conductivity of the fluid. The first term on the right side is due to thermal diffusion in the y direction, and the last two terms are due to viscous dissipation. The shear stress T , , ~ and are given for this 2D flow by

where p is the molecular viscosity of the fluid. The shear rate for this 2D circumstance is given by the expression

with the viscosity p given by one of the various models, such as Eq. (l), whichever is appropriate for a given fluid. If the power-law model is used, the singularity at zero shear rate must be avoided, often by simply putting n = 1 as shear rate approaches zero, i.e., Newtonian behavior at small shear rates. The preceding energy equation, Eq. (51, is parabolic in the z direction and marching may be used to obtain the solution. A restriction to the flow is imposed by the presence of a die at the end of the extruder. The flow in the extruder is strongly coupled with that inside the die. For very narrow dies and large extruder speeds, a backflow may arise in the extruder channel in terms of the coordinate system described. This makes the problem elliptic, requiring a different approach for the solution. These considerations are discussed later in the chapter. The boundary conditions are also shown in Fig. 5. The temperature distribution at the barrel is specified as Tb(z),which, in many cases is a constant or has different values in different sections. The screw is taken as adiabatic. We may write these conditions as

159

EXTRUSION OF NON-NEWTONIAN MATERIALS

Since the energy equation is parabolic in z , boundary conditions are necessary only at z = 0 to allow marching in the z direction and, thus, obtain the solution in the entire domain. The boundary conditions at z = 0 are provided in terms of the developed velocity profiles, denoted by subscript dev, at T = T i . These are obtained by solving the momentum equations, keeping the temperature constant at T i ,by means of an implicit finite-difference scheme [331. These equations may be nondimensionalized with channel height H , barrel velocity component in the z direction Vb, , barrel and inlet temperatures Tb and T i ,and p o as characteristic quantities. The resulting dimensionless variables are

W

w * = -, 'b

fj=

T - Ti

-

p * = =P ,

Tb - Ti '

I

jj=j+

'b z

P

Pe

'bz

= -, CY

G=

k ( Tb - Ti) '

where Pe is the Peclet number, G is the Griffith number, and parameter p represents the dependence of viscosity on temperature. The dimensionless equations thus obtained are

The continuity equation for the conservation of mass is dU*

-d +X * -

dW* dZ*

=

0.

(13)

However, the constraints on the flow are generally written in integral form,

160

YOGESH JALURIA

given in dimensionless terms as

[u* dy*

=

0,

[w*dy*

=

Q/B qv = -, Hvb,

where the first condition ensures that the net flow across the channel is zero, if the leakage flow across the screw flights is negligible, and the second condition gives the down-channel flow rate. Therefore, the parameter qv is the dimensionless volumetric flow rate, generally called the throughput, emerging from the extruder. If the screw channel is not rectangular, qv is defined as Q/AVbz,where A is the channel cross section. The integral form of the continuity equation is generally used since the throughput can be specified as an operating condition. A similar nondimensionalization is used for an Arrehenius temperature dependence of viscosity, i.e., p varying as exp(b/T). For this case, two parameters, p1 and &, are defined as

p1 = Tb/Ti

and

Pz = b/Ti,

as employed by Gopalakrishna et al. [18]. Thus, the governing equations are nondimensionalized for different circumstances. Similarly, the boundary conditions are also obtained in dimensionless form. However, results are often presented in physical terms because of complicated property variation characteristics. The governing dimensionless equations are conveniently solved by means of finite-difference techniques for simple geometries. The computational domain is the rectangular channel shown. The iterative Newton-Raphson method [27] may be used to satisfy the conditions on the flow rates. The iteration is terminated when the pressure gradients satisfy a chosen convergence criterion. Using the boundary conditions in terms of u , w,and T at any upstream z location, the energy equation is solved to obtain the temperature distribution at the next downstream z location. The same approach is extended to solve the mass transfer problem. The numerical scheme is validated by comparisons with experimental data on actual, full-size, extruders, using both Newtonian and non-Newtonian fluids, as discussed later. For further details on the numerical scheme, see Karwe and Jaluria [33] and Gopalakrishna et al. [HI. 2. Fulb Developed Flow The simplest solution is that of the fully developed circumstance for which the temperature and the velocity fields are assumed not to vary downstream. The barrel is taken as isothermal and at the inlet tempera-


161

ture of the fluid. Convective transport of heat is neglected, but the viscous dissipation effects are considered, so that the energy generated is lost to the barrel. Although analytic solutions can be obtained for certain channel flows driven by pressure or shear, the present circumstance, with combined pressure and shear effects, screw helix and non-Newtonian fluids, requires a numerical solution of the governing equations. Figure 6a shows the calculated w* velocity profiles for different values of q, . The characteristic curves, in terms of the throughput and the dimensionless pressure gradient, are shown for n = 0.5 and different Griffith numbers in Fig. 6b. When the pressure gradient is zero, the flow is due only to the viscous effect of the moving barrel and is termed as drugflow. For Newtonian flow, the velocity profile is linear for this circumstance and q v = 0.5. This situation is similar to the Couette flow between two parallel plates in the absence of a pressure gradient. For a favorable pressure gradient, the throughput exceeds 0.5, and for an adverse pressure gradient, it is less. The velocity profile bulges outward, with the velocity exceeding the linear variation, for the favorable case. The opposite occurs for the adverse pressure case. Similarly, for non-Newtonian flow, the profiles are seen to be strongly dependent on the throughput, although drag flow does not arise at q, = 0.5 but at a value that depends on the fluid, temperature, and other conditions. The screw channel is assumed to be completely filled with the non-Newtonian fluid. Therefore, a decrease in the throughput at a given screw speed implies a smaller-diameter die, with greater obstruction to the flow. This results in a greater pressure rise downstream and an increased adverse pressure gradient, which is reflected in decreased velocity levels. Reverse flow may also arise at very small throughputs in this coordinate system. Larger throughputs are obtained with a favorable pressure gradient, specifically, pressure decreasing downstream. In an extruder, the obstruction provided by the die and by a tapered screw in many cases increases the pressure, resulting in an opposing pressure circumstance, as seen for small values of q,. A higher Griffith number implies greater viscous heating. This results in higher temperatures and lower viscosity. This also gives rise to a smaller pressure gradient at a given throughput, in the positive-pressure-gradient range, which is of interest in extrusion. 3. Developing Flow

The results presented here are based on the coordinate axes fixed to the rotating screw. The ratio of axial screw length L to channel height H is taken as about 70, corresponding to practical extruders. This results in a dimensionless down-channel distance of around 200. Once the numerical

YOGESH JALURIA

/ / 5

-0.3

-0.1

I

I

I

I

0.1

0.3

0.5

0.7

I

I

1.1

0.9

I

I

1.3

1.5

W*

Screw helix angle = 16.0

LEGEND *=:G=0.0 a =:G=2.5 a = : G = 5.0

x( -5.0

, -4.0

1

-3.0

I

-20

I

-1.0

r

1

0.0

1.0

20

dimensionless pressure gradient

I

3.0

g aZ

I

4.0

I

5.0

FIG.6. Computed results for the fully developed case at n = 0.5 and p = 2.0: (a) profiles of the w * velocity component at G = 2.5 and different values of the throughput q v ; (b) characteristic curves showing variation of pressure gradient with throughput at various values of the Griffith number G (adapted from Kwon et al. [38]).

163


results for the velocity and temperature fields are obtained, various other quantities of interest, such as the heat input to and from the barrel, local Nusselt number Nu,, bulk temperature, shear stress, and pressure at various downstream locations, including that at the die, are calculated. Only a few typical results are presented here for conciseness. Figure 7 shows the results in terms of isotherms and constant velocity lines along the unraveled screw channel. The temperature and velocity profiles at four downstream locations are also known. The temperature of the fluid far downstream is seen to increase above the barrel temperature. Therefore, beyond a certain downstream location, heat transfer occurs from the flow to the barrel if the barrel is maintained at a fixed temperature. This

0

SCREW ( adiabatic )

0

40.0

20.0

60.0

80.0

100.0

120.0

1eo.o

140.0

1~0.0

z.= z/H

.; 2

0

-0

a 0 0 l O A O 8 0 m LO 1s

-OD 00 10 4 0 so a I o ia

W.

40.0

200

60.0+

80.0

-0s 0 0 1 0 4 0 S O 8 1 0 1 I

-0800I0~08001011

W*

W*

=.

= 1000 z/H

1200

140.0

W*

160.0

180.0

0

00

00

10

9

I6

I0

0

o e 01 i e 9

10

10

00 06

I0

e

I0

LO

FIG.7. Calculated velocity and temperature fields in the extruder at n Pe = 3427, p = 1.61, and 4, = 0.3 (adapted from Kwon et al. [38]).

00

00

I0

I0

9

=

0.5, G

=

10.0,

LO

164

YOGESH JALURIA

implies initial heat transfer to the fluid by the barrel followed by heat removal from the fluid further downstream. This effect is due to the viscous heating of the fluid and varies strongly with the Griffith number. At larger Griffith numbers, the fluid temperature may be much higher than the barrel temperature. It is seen from these results that the constant velocity lines are almost parallel to the barrel, indicating very little convective mixing in the fluid, which is largely stretched as it goes through the extruder. To enhance mixing, reverse screw elements and breaks in the screws are often employed in practical single-screw extruders. Mixing is also substantially increased in twin-screw extruders, as discussed later in the chapter. It is also seen that large temperature differences exist in the fluid, from the barrel to the screw. This is due largely to the low thermal conductivity of typical polymeric materials. At relatively high values of the dimensionless throughput q v , the backflow is small and much of the fluid near the screw root remains unheated. Viscous dissipation is important and affects the thermal field substantially. Figure 8a shows the variation of the dimensionless pressure p* and the pressure gradient dp*/dz* along the screw channel. Figure 8b shows the corresponding variation of the bulk temperature 6 b u l k and the local Nusselt number Nu, at the inside surface of the barrel. These trends agree with the physical behavior in actual systems. The pressure rises toward the outlet, as does the bulk temperature. The actual values attained vary strongly with q v , as discussed earlier. The bulk temperature and the Nusselt number Nu, are defined as

where qin is the heat flux Input into the fluid at the barrel. Far downstream, the value approaches zero, indicating the small amount of energy transfer needed to maintain the barrel temperature at a given value. It may also be negative if the fluid loses energy to the barrel. 4. Three-Dimensional Transport

The basic transport processes in the extruder channel are threedimensional, although two-dimensional models, as outlined above, have been used extensively to model the flow and generate data needed for design. The main problem with such 2D models is that the effect of the flights is brought in by mass conservation considerations only, approximating the flights as being very far apart. Thus the recirculating flow gener-

. !

-1

/

3 34

3' > am

-0 d-

4 xY

SfE a 2-

i2-

4.8 a-

$1-

2 2I

0-

x7a.0

0.0

40.0

M.O

110.0

100.0

im.0

Dimensionless distance along m

110.0

1w.o

IM.O

:

w helix ZC

1' I-

Mrnenaionlen distance along arer helix

+

FIG.8. Variation of the pressure, pressure gradient, bulk temperature, and local Nusselt number in the down-channel direction for the conditions of Fig. 7 (adapted from Kwon et al. [381).

166

YOGESH JALURIA

ated in a screw channel, between two flights separated by a finite distance, is not simulated. Although this is applicable for shallow channels, 3D models are needed for deep channels and for a more realistic modeling of practical extruders. However, 3D modeling is fairly involved and not much work has been done on it [21]. A relatively simple model to simulate 3D flows has been developed by Sastrohartono et al. [59] and is outlined here. In the unwrapped extruder channel, it is reasonable to assume that the velocity vector V does not change significantly along the screw channel direction, i.e., z-coordinate direction, d V / d z O

-1k(vk )k) -

'

(< Pk+k+k>) ( 16)

*Volume-averaged quantities are enclosed in brackets, ( ), and the su erscript k is used to denote volume-averaged quantities for phase k. Thus, (V,) = c k ( V kP), where &k is the volume fraction of phase k within the REV h e . , ck = g,).

258

PATRICK J. PRESCO’IT AND FRANK P. INCROPERA

Energy:

Species:

=

-v

*

(&))

-

v

*

( 1 or c k < 0) and converges quickly.

CONVECTION IN ALLOY SOLIDIFICATION

261

0 liquid

- interface

b

interface

0 solid

FIG. 14. Illustration of interfacial diffusion length scales for (a) heat transfer and (b) species transfer (reprinted with permission from Ni and Beckermann [49], 1991, ASM International).

C. MICRO/MACRO MODELS As discussed in Sections I1 and III.B.2, microscopic and macroscopic phenomena are coupled, and a research area of considerable importance relates to the development of solidification models that simultaneously predict the evolution of macroscopic and microscopic phenomena. However, the current understanding of micro/macro interactions is primarily qualitative, and macroscopic convection models that also predict features such as microsegregation, grain size and orientation, columnar-to-equiaxed

262

PATRICK J. PRESCO'R AND FRANK P. INCROPERA

transition (CET), primary and secondary arm spacings, and porosity are not yet well developed. A shortcoming of the original single-domain mixture models [12, 42, 431 is the manner in which solute is partitioned between solid and liquid phases. At a given temperature, the solid phase is assumed to be fully saturated with solute according to the equilibrium phase diagram. That is, the h e r law is applied under the assumption that freezing occurs through a series of equilibrium states. In order for equilibrium freezing to be a reasonable assumption, solidification must occur very slowly or the primary dendrites must be pure (there is complete rejection of solute by the dendrites). In most casting processes, the distribution of solute in single dendrite arm is nonuniform, due to the changing concentration of precipitated solid (Fig. 3) and the very low rate of diffusion in the solid phase. The condition is termed microsegregation. The manner in which solute is partitioned microscopically affects the calculation of solid fraction and liquid concentration, which, in turn, affect permeability and buoyancy forces, respectively. It also affects the amount of eutectic material formed in the final casting and the properties of the casting. Microsegregation has been considered by invoking the Scheil assumption [68], which presumes interdendritic liquid to be locally well mixed and negligible diffusion within solid dendrites. However, use of the assumption is not a matter of applying the Scheil equation [68] for partitioning the solute and determining the local solid fraction, since the Scheil equation does not account for the advection of solute into and out of a volume element. Rappaz and Voller 1691 discuss implementation of the Scheil assumption in modeling microsegregation in a macroscopic solidification model. The average concentration of solute in the solid phase is represented as

where the integral, which accounts for the solidification history, is incrementally evaluated over each time step and the integrand over a time step is simply the solid composition taken from the equilibrium phase diagram. A difficulty with this method is that it requires maintaining a complete record of the solidification history at each spatial grid node in order to property account for microsegregation during remelting [69]. For problems with no remelting, the equation works well without special consideration 147, 701. Remelting can be considered with Eq. (29) by approximating the composition of remelted solid to be [71], which is an underestimation for hypoeutectic alloys. Felicelli et al. [72] realized, however, that the Scheil

c,


263

assumption and Eq. (29) can be applied and used to accurately account for remelting without requiring excessive memory to record the Cs -fs history by discretizing solid fraction. That is, Eq. (29) is expressed as

Acs,l

where the increments are calculated, recorded, and adjusted (if remelting occurs). The detailed solute accounting procedure of Eqs. (29) and (30) becomes significantly more complicated when solid grains are no longer stationary, and the Scheil assumption has not been applied in simulations that account for advection of equiaxed grains. The lever law and Scheil assumption represent two limiting dendrite conditions, namely, those of infinite and zero diffusion, respectively. In some cases, diffusion occurs in dendrites during solidification at a rate that renders neither the lever law nor the Scheil assumption valid, and the magnitude of uncertainty associated with invoking either one of the two limiting assumptions can be assessed by comparing results of simulations based on each limiting condition [711. A model for microscopic diffusion would have to be incorporated for greater precision, and Sundarraj and Voller 1731 developed a dual-scale model, which combined a onedimensional microsegregation model [74, 751 with a macrosegregation model, to treat solute transport during solidification of an AI-Cu alloy. Mo [76] proposed a model utilizing internal variables to account for microsegregation in a macrosegregation model. This model agrees with predictions made using the Scheil and lever law (equilibrium) assumptions for short and long solidification times, respectively. It is also able to simulate conditions between the two limiting cases and accommodates remelting in a straightforward manner [761. Another microscopic feature of importance is the actual microstructure, which affects terms like S, and 29 in Eq. (26) and ultimately determines the mechanical properties of casting. Phenomena such as nucleation, growth, impingement, and coarsening of grains [681 are influenced by thermal, solutal, and flow conditions, and models relating microstructural features to microscopic and macroscopic transport phenomena were reviewed by Rappaz [50]. However, many of the recently developed micro/macro models that incorporate nucleation, growth, and impingement of grains neglect convection in the melt [77-801. Nucleation and grain growth models [50, 681 were included in a volumeaveraged two-phase solidification model [491 for predicting final-grain-size distributions in solidified alloys, as well as convective transport phenomena during solidification [81]. Similar extensions have been proposed for continuum models [52, 531, although calculated results are not yet available.

264

PATRICK J. PRESCO'IT AND FRANK P. INCROPERA

Although two-phase models [48, 491 are less restrictive than mixture models [12, 43, 471, they are still limited in their ability to resolve microscopic phenomena, for which three different length scales can be identified [82]: (1) the overall radius of a grain, (2) the dendrite tip radius, and (3) the secondary arm spacing. In view of these considerations and with the goal of predicting microstructural details in a macroscopic transport model, including convection, Wang and Beckermann have proposed a volume-averaged multiphase model [51, 83, 841, which distinguishes interdendritic liquid from extradendritic liquid by modeling them as different phases (Fig. 15). The multiphase model is an extension of the two-phase model [49], and it considers two interfaces, one between solid dendrites and interdendritic liquid and another between the interdendritic and extradendritic liquids (i.e, the grain envelope). Each phase interface has a characteristic length scale, requiring two sets of interfacial species diffusion length scales (Fig. 16). The model has been used to predict columnar to equiaxed transition [85, 861, but without convection. Micro/macro modeling of alloy solidification is a relatively new approach to simulating casting processes, and it is only beginning to be included in models that account for convection. Micro/macro considerations of convective transport during alloy solidification are complex and computationally intensive, and much additional model development is needed. However, as computational power continues to increase, verified micro/macro solidification models are likely to become useful engineering tools for process design and analysis. D. SUBMODELS The single-domain models described in Section 1II.B require the introduction of several submodels. For example, a model based on experimental data for permeability must be used in the mixture momentum Eq. (31, rendering it semiempirical. Similarly, drag coefficients and solid- and liquid-phase viscosities are required for the two-phase momentum Eq. (161, and models must be used for transport coefficients such as thermal conductivity. The purpose of this section is to review various models that have been proposed for use in single-domain formulations of convection during alloy solidification. 1. Permeability and Dray Coeficients

Darcy's law is used in momentum equations to account for momentum exchange between interdendritic liquid and solid dendrites. The model assumes that the rate of momentum exchange between phases is propor-


265

dendrite envelope

phase s

phase d

phase 1

a dendrite envelope phase I

phase: d

phase S

b FIG.15. Schematic illustration of physical conditions considered by the multiphase model: (a) columnar dendritic growth; (b) equiaxed dendritic growth (reprinted with permission from Wang and Beckermann 1831, 1993, Elsevier Science Inc.).

tional to the difference in their respective velocities and inversely proportional to permeability, which represents the square of an appropriate microscopic, viscous length scale. With dendrite arm spacings being on the order of l o w 5rn, permeabilities on the order of lo-" to lop9 m2 would seem appropriate.

266


s-d interfnce

% d-I

interface

Liquid

FIG.16. Illustration of diffusion length scales associated with interfacial species transfer in the multiphase model (reprinted with permission from Wang and Beckermann [841, 1993, ASM International).

The permeability of a dendritic array depends on several factors, including the local volume fraction solid and its structure. Regions within the mushy zone near the liquidus interface have relatively small local volume fractions of solid and a relatively large permeability. Conversely, the permeability of the dendritic array is relatively small near the solidus interface. Furthermore, the permeability of a dendritic mushy zone may depend on flow direction (anisotropic), as in the case with a columnar structure, or it may be isotropic when grains are equiaxed and small. The original macrosegregation models [5, 36, 371, which considered transport only within the mushy zone, used the following isotropic model to account for the variation of permeability with volume fraction of solid:

K = KO(1 - g , ) 2 = Keg:, (31) where KO is a model constant chosen to fit permeability data [21]. Although this model is appropriate within regions of the mushy zone for which g, > i, it is not valid near the liquidus interface. Furthermore, Eq. (31) is not suitable for use in the mixture momentum Eq. (31, because Eq. (31) yields K = KO when g, = 0 (gl = 1). That is, Eq. (31) fails to eliminate the third term on the right side of Eq. (3) when g, = 1 and, hence, would impose a non-physical damping on flow in the fully melted region. A model more suitable for the continuum momentum equation is the Kozeny-Carman (or Blake-Koseny) equation for permeability [12, 42,461:

267


where the permeability constant KO is an empirical constant that depends on dendrite arm spacings [87]. This model assures that the permeability is infinite in the fully melted region (g, = 0, g , = l), and K varies continuously with g, to a value of zero in the fully solid region where g, = 1 ( g , = 0). However, the Blake-Kozeny equation is not intended for use with g, less than about 0.50 [88]. West [89] proposed a piecewise continuous permeability model that differentiates regions in the mushy zone according to their proximity to the liquidus interface. The model has been adopted by Amberg and coworkers in their simulations of binary metal alloy solidification [57, 701. West's model assumes capillary behavior in regions far from the liquidus, and it provides a transition to dispersed particle behavior in regions with large liquid volume fractions. The model is of the form

(8,

> 4).

(35b)

m2 and K 2 = 8.8 X The suggested model constants of K , = 6.4 X lo-'' m2 were chosen to fit experimental data [21]. The reliability of permeability models is significantly limited by a lack of permeability data. Of the limited data that have been published [21, 90-921, none report values of permeability for solid fractions less than approximately 30% [93]. However, the corresponding region of the mushy strongly influences macrosegregation, since the relative motion between solid and liquid is largest for small solid volume fractions. Such data are not available because of difficulties associated with their measurement, and numerical simulations of flow through tortuous paths have recently been performed in efforts to provide permeability data in the low-g, regions of mushy zones [94, 951. The two-phase solidification models described in Section III.B.2 require submodels for interfacial momentum exchange, and for flow through a coherent mushy zone, such as columnar dendritic regions or regions with packed equiaxed crystals, Darcy's law may be applied [49]. In regions with

268

PATRICK I. PRESCOTT AND FRANK P. INCROPERA

free-floating crystals, the rate of momentum exchange between solid and liquid phases can be expressed as [491 (V,)'),

(36)

where Ad/Vo represents the total projected area of the solid phase per unit volume and C, is a drag coefficient that depends on a two-phase Reynolds number. Experimental data are required to correlate C , with two-phase flow conditions and microstructural features, and correlations based on a limited amount of data have been used [81, 96-981. In a proposed extension of the continuum model [52] to account for the motion of free-floating dendrites in the melt or slurry region, it is argued that the only significant relative motion between free-floating dendrites and liquid is due to buoyancy. The difference between solid and liquid velocities is based on a balance between local (microscopic) viscous and buoyancy forces

where d, is a characteristic grain diameter and pmis the effective viscosity of the slurry [53]. Instead of calculating the solid velocity from a separate solid momentum equation [e.g., Eq. (lo)], V, is related to the liquid velocity and used in the energy and species transport equations.

2. Transport Properties Transport properties, such as thermal conductivity, viscosity, and mass diffusivity, must be prescribed or modeled in order to solve the governing equations for either a mixture or a two-phase model. The most common modeling technique is to use independent sets of constant transport properties for the liquid and solid phases, with the assumption that neither the coexistence of phases nor the morphology of the interface affects the macroscopic properties. Hence, conventional, single-phase property data are used, and the transport properties of the mixture (continuum) are allowed to vary according to the local volume fractions of phases. The thermal conductivity of the mixture is expressed as


269

The mixture mass diffusivity is modeled in a similar fashion, and with Ds = 0 and g, p , = f,p , the mass diffusion coefficient appearing in Eq. ( 5 ) is

D = filll. (39) As discussed by Beckermann and Viskanfa [12] and Ni and Beckermann [491, the volume-averaging technique suggests that conventional singlephase values for properties such as k , and k , can be used only as approximations in lieu of available data that suggest a relationship between microstructural features and individual phase transport coefficients. In general, the macroscopic diffusion transport coefficients for a phase within a two- or multiphase system can be expected to be different from their microscopic versions and even anisotropic, despite isotropic diffusion at the microscale [49]. However, sufficient data are not yet available to justify detailed models that account for tortuosity. Liquid, solid, and mixture (slurry) viscosities are also important parameters in solidification models with convection. The solid phase within a coherent mushy zone may be regarded as having an infinite viscosity, meaning that it can withstand any stress without deforming. Because of the absence of experimental data from dendritic flows or a general theory [82], the liquid-phase viscosity in the mushy (or slurry) zone is typically prescribed to be the actual (microscopic) liquid viscosity, despite the presence of the dendritic structure. In a slurry region, the solid viscosity can be modeled in the following manner [49, 821:

where ps0is the solid viscosity in the limit when E, = 0, E , , ~is a critical value of solid fraction at which the solid becomes rigid, and a is an empirical constant (typically a = 2.5). Alternative models for solid viscosity can be adopted [49, 821.

IV. Theoretical Results and Experimental Validation Continuum and volume-averaged solidification models discussed in Section I11 have been used to predict convective phenomena for several different conditions. Many studies have focused on solidification of aqueous salt solutions, since these systems can readily be used in the laboratory and the convection patterns can be visualized. Metal alloys are receiving increased attention, but with fewer comparisons of predictions with experi-

270


mental results. Representative model predictions are reviewed in the following subsections, along with assessments based on comparing predictions with experimental results. A. SEMITRANSPARENT ~INALOG ALLOYS

Semitransparent analog alloys are used because they permit flow visualization. The most common analog alloy used in laboratory experiments is aqueous ammonium chloride (NH,Cl-H,O), with NH,Cl concentrations between 25 and 32 wt%. This system is popular because its freezing range corresponds to readily controllable conditions (7'' = - 15°C) and NH,Cl crystallizes with a dendritic morphology (i.e., with primary, secondary, and higher-order branches), which is similar to that of most metals [99, 1001. A disadvantage of using aqueous ammonium chloride is that it is highly corrosive and, thus, requires that appropriate measures be taken in the laboratory. Another analog alloy used by experimenters is aqueous sodium carbonate (NaC0,-H,O), which is less caustic than aqueous ammonium chloride. Its freezing range is also convenient for laboratory studies, but the NaCO, crystals which form in the mushy zone are devoid of secondary dendrite branches. Hence, the mushy zone is microstructurally different than those associated with most metal alloys. 1. Solidijication from the Sidewall of a Rectangular Cavity

The continuum model of binary solid-liquid phase change 1431 was first used to simulate convection heat, mass, and momentum transfer during solidification of a NH,Cl-H,O solution from one sidewall of a rectangular cavity of aspect ratio (height/width) A = 4 [42]. Two cases were considered, both of which involved opposing thermal and solutal buoyancy forces. In one case, a sidewall was cooled and maintained at a temperature below the solidus temperature, while the opposite wall was maintained at the initial temperature of the melt (above the liquidus temperature). Plots of the velocity, stream function, temperature, and liquid composition fields were presented at various times during solidification. Conditions were conduction dominated at the beginning of the process, and the simulation showed the transient development of convection patterns. Convection in the melt was thermally driven by heating from the hot sidewall and cooling through the solid and mushy zones, and solutally driven flow was predicted in the mushy zone. The coupled flows induced significant deviation from conduction dominated conditions, and the thicknesses of the fully solidified and mushy regions varied with vertical position. Furthermore, as a result of variations in thermal and solutal conditions near the liquidus

271


interface separating the mushy and melted zones, the shape of the liquidus interface was highly irregular. Figure 17 shows conditions that were predicted after 360 s of cooling. A thin layer of fully solid material covers the left sidewall, and mushy (s + I), and liquid (1) zones are indicated on the velocity plot (Fig. 17a). The streamlines in Fig. 17b reveal more clearly the thermally driven (counterclockwise) cell in the melt and the solutal (clockwise) cell in the mushy zone, and Figs. 17c,d show the temperature and liquid compositions fields, which have been distorted significantly by advection effects. Eventually, as the solidification rate diminished to near-zero, thermal-convection-dominated conditions in the melt, whereas solutal convection was confined to a relatively small region near the bottom of the mushy zone.

a

b

C

d

+ 8.1 mm/s

FIG.17. Predicted conditions at f = 360 s for solidification of aqueous ammonium chloride in a differentially heated rectangular cavity: (a) velocity vectors, (b) streamlines, (c) isotherms, and (d) liquid isocomps. (Reprinted from Bennon and Incropera [42]. Copyright 1987, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, U.K.).

272


A second simulation case involved an adiabatic sidewall opposite the chilled surface. Following the initial conduction-dominated period, thermal and solutal convection cells again developed in the melted and mushy zones, respectively. However, in this case, thermal convection in the melt was not sustained by a heated sidewall. As solidification progressed, temperature gradients in the melt became small and flow conditions in both the melt and mushy zones were increasingly influenced by solutal buoyancy. The evolution of macrosegregation patterns was also studied [loll. Figures 18a,b show, respectively, velocity vectors and liquid compositions predicted at t = 390 s, with solid (s), mushy (s + I), and liquid (1) zones identified in Fig. 18a. In addition, four regions in the mushy zone are outlined with dashed lines. Within these regions, interdendritic fluid is channeled, and the subsequent effects are revealed by the mixture composition plot of Fig. 18c, which corresponds to t = 870 s. Channels are created when water-rich interdendritic fluid flows upward into a warmer region of the mushy zone, displacing saltier fluid and partially remelting or dissolving the existing dendrites. Hence, a high-permeability region is formed, providing a preferred path for buoyant fluid. As interdendritic fluid is drawn into a channel, it precipitates NH,Cl in a region to the right and below the channel, leading to the A-segregate pattern (Fig. 18c) of alternating positive and negative segregation layers that slant upward and to the right from the chilled (left) wall. The aforementioned simulations were the first to account for coupled heat, mass, and momentum transport in solid, mushy, and melted regions and to predict features such as an irregular liquidus interface, the channeling of interdendritic fluid, and the formation of distinct A segregates in the final casting. The effects of microsegregation (coring) [69] and solid movement on macrosegregation were studied by Voller et al. [47]. It was found that when the solid was assumed to move with the liquid (no relative motion), macrosegregation was small. This result is expected because, without relative motion between solid and liquid, macrosegregation can occur only by species diffusion in the liquid, which occurs at a very low rate relative to advection. Although microsegregation does not physically occur in NH,CI dendrites, which are pure, it was simulated [47] and found to increase the severity of macrosegregation slightly, without changing the overall pattern of segregation. Continuum model predictions were compared with experimental observations in another study [102], in which a H,O-31 wt% NH,CI solution was solidified in a rectangular cavity. The model successfully predicted several qualitative features of the process, including liquidus interface irregularities, channel formation, and the formation and subsequent ero-

273


a

b

C ( t = 870s)

( t = 390s)

Ve Ioc it y

- 0.55

Macrosegregation

Liquid Composition

1 mm/s

f, HzO = 0.70

f

HzO

00.47- 0.56 0.56- 0.64 B 0.64- 0.72 C_ 0.72- 0.80

FIG. 18. Solidification of aqueous ammonium chloride in a side-chilled rectangular mold: (a) velocity vectors at t = 390 s, (b) liquid isocomps at t = 390 s, and (c) macrosegregation at t = 870 s (reprinted with permission from Bennon and Incropera [loll, 1987, ASM International).

sion of a double-diffusive interface. Figure 19 shows predicted convection conditions after 480 s of cooling through the left sidewall. A layer of cold but solutally buoyant (water-rich) fluid has formed at the top of the melted region (Figs. 19a,b) and is separated from the bulk melt by a doublediffusive interface (Figs. 19c,d). Thermal buoyancy forces act to recirculate

274

PATRICK J. PRESCOTT AND FRANK P. INCROPERA

C

-

5.69mm/s

FIG. 19. Predicted conditions after 480 s of solidification of 31 wt% NH,CI-H,O in a differentially heated rectangular cavity: (a) velocity vectors; (b) streamlines; (c) isotherms; (d) liquid isocomps. (Reprinted from Christenson et ul. [102]. Copyright 1989, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB,

U.K.)

the fluid in this layer, as well as the bulk liquid beneath it (Fig. 19b). Interdendritic fluid ascends within the mushy zone as a result of solutal buoyancy forces (Figs. 19b, d), and completely remelted channels were predicted in the upper part of the mushy zone (Fig. 19a). With time, the rate of solidification decreases, as does the flow of interdendritic fluid from the mushy zone to the top layer. Hence, the double-diffusive interface could not be sustained, and at t = 720 s, the interface was eroded. Although the predicted transport phenomena were in qualitative agreement with experiments, predictions did not compare well quantitatively [1021. An example of the discrepancy between predicted and measured results is shown in Fig. 20, where the solidus and liquidus interface

CONVECIION IN ALLOY SOLIDIFICATION

a

275

--- Measured

-

Predicted

b

7 \

I

I I FIG.20. Measured and predicted interface locations at (a) t = 180 s, (b) t = 660 s, and (c) t = 1200 s. (Reprinted from Christenson et at. [102]. Copyright 1989, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 IGB, U.K.)

locations are plotted at three different times during solidification. The model significantly underpredicted the rate of advancement of the solidification fronts, with much of the discrepancy attributable to uncertainties in prescribed model parameters. It was subsequently determined [1031 that the value prescribed for the solid thermal conductivity in the early singledomain model simulations [12, 42, 101, 1021 was incorrect (due to a typographical error in the source from which it was obtained [38]) and that the actual value was approximately six times larger. Other sources of disagreement include uncertainties in the permeability of the mushy zone and the effects of temperature and concentration on all properties. A model consisting of equations for momentum and species transport in the liquid phase (assuming a stationary solid phase with no diffusion), along with an equation accounting for heat transfer in a solid-liquid

276


mixture, was used to simulate experimental conditions for which an aqueous ammonium chloride solution was solidified from one side of a square cavity [12,104]. Although the model was developed from volume-averaging theorems and its form is slightly different than that of the continuum model [43, 451, the two models contain the same essential features. Predicted results were in qualitative agreement with measurements, but quantitative agreement was only fair [12, 1041. The prescribed value of the solid thermal conductivity was taken from Szekely and Jassal [38], which, as discussed previously, was much too small. Because the simulation did not consider a layer of fully solidified material, the effect of this erroneous input was, perhaps, less noticeable than in the study of Christenson et al. [1021. Discrepancies between predicted and measured results were also attributed to uncertainties in other model inputs and assumptions, including that of an isotropic permeability for the mushy zone. The solidification of aqueous ammonium chloride in square and tall rectangular ( A = 4) cavities was simulated recently by Zeng and Faghri [lo51 using a continuum model with a temperature-based energy equation [651. Thermal and solutal buoyancy effects were examined for solidification in the square cavity by considering (1) only thermal buoyancy, (2) only solutal buoyancy, and (3) thermosolutal convection. The conditions considered by Christenson et al. [13, 1021 were simulated in a tall rectangular cavity, with similar results. It appears as though Zeng and Faghri [lo51 also used an unrealistically low value for the solid thermal conductivity. The effect of anisotropic permeability was considered by Yo0 and Viskanta [106]. A continuum model was used with an anisotropic permeability model to simulate some of the experimental conditions reported by Beckermann [104]. Permeability in the mushy zone depended on whether the flow was parallel, perpendicular, or oblique to the growth direction of primary dendrites, which was assumed to be aligned with the local temperature gradient. The ratio of the principal permeabilities (parallel or perpendicular to temperature gradient) was varied from 0.5 to 2, and it was found that predicted convection conditions were very sensitive to this ratio. Figures 21a-c show predicted streamlines for R = 0.5, 1.0, and 2.0, respectively, after 20 min of solidification. Since the temperature gradients are primarily horizontal, R represents the ratio of permeability for flow in the horizontal direction to that in the vertical direction (KJK,,). The geometric mean of K x and K , was kept constant. When the permeability to flow in the vertical direction is relatively small (R = 2.0) (Fig. 21c), the flow of water-rich interdendritic fluid to the top of the cavity is impeded, thereby retarding development of a double-diffusive layer along the top of the liquid region. In contrast, when the permeability to flow in the vertical direction is relatively large ( R = 0.5) (Fig. 21a), the growth of a

(dw)


a

277

b

FIG. 21. Streamlines predicted at t = 20 min for solidification of aqueous ammonium chloride in a square cavity with anisotropic permeability in the mushy zone: (a) R = 0.5; (b) R = 1.0; (c) R = 2.0. (Reprinted from Yo0 and Viskanta [1061. Copyright 1992, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 IGB,

U.K.)

water-rich layer along the top of the liquid pool is accelerated. The significant influence of anisotropy on solidification suggests that the effect should be considered to achieve accurate simulations.

2. Solidificationfrom the Bottom Wall-Unidirectional Solidification Double-diffusive convection leading to freckle formation in unidirectionally solidified ingots has been observed experimentally in aqueous ammonium chloride solutions [16, 107-1111 but had not been predicted

278


until a continuum model was applied [112, 1131. In one of these investigations [113], the continuum model was used to study the effects of permeability and cooling rate on the convective transport phenomena and macrosegregation during unidirectional solidification in a vertically aligned cylindrical mold. The melt was assumed to be initially quiescent, homogeneous (fONHbC' = 0.32), and isothermal with an 8°C superheat (To = 50°C). The axisymmetric simulation began with a sudden reduction in temperature of the bottom circular surface of the mold (T,,), and three cases were considered. Case 1 used a permeability constant of K O = 5.556 X m2 in Darcy's law and a chill plate temperature of T, = -30°C. Conditions for cases 2 and 3 corresponded to KO = 5.556 X lo-.'' m2, T, = - 30°C and K O = 5.556 x lo-'' m2, T, = - 100°C, respectively. For case 1, conditions during the first 6 min of solidification were conduction-dominated, with the advancing solidification front, isotherms, and liquid isocomps remaining planar and horizontal. However, the density gradient within the mushy zone is unstable because the NH,Cl concentration increases with temperature, according to the equilibrium phase diagram. Figure 22a shows velocity vectors, streamlines, and isotherms at t = 6 min, when fluid motion is initiated. The attendant disturbances amplify, and by t = 7 min (Fig. 22b), the effect of fluid motion induces

FIG. 22. Predicted flow field and isotherms for unidirectional solidification from below (case 1): (a) t = 6 min; (b) 1 = 7 min; (c) t = 8 min. (Reprinted from Neilson and Incropera [113].Copyright 1992, with permission from Springer-Verlag.)


279

small perturbations in the isotherms near the liquidus interface. At t

=

8 min (Fig. 2 2 4 fluid velocities have become significantly larger and the

array of recirculation cells along the liquidus interface is exchanging fluid between the melted and mushy zones. Cold fluid ascending from the mushy zone is solutally buoyant and extracts heat from the surrounding, warmer fluid of nominal composition, which is descending to the mushy zone. As solidification continues, convection becomes stronger and continues to distort otherwise planar isotherms and liquid isocomps. Fluid motion extends throughout the melt, and the complex flow pattern changes continuously. The perturbed temperature and liquid composition fields along the liquidus interface affect the growth of the interface, and because the rate of species diffusion is much smaller than the rate of heat diffusion, channels were eventually predicted to originate at the liquidus interface [113]. Cold, water-enriched (solutally buoyant) liquid that is generated in the mushy zone by the precipitation of solid ammonium chloride, ascend to warmer, water-deficient regions of the mushy zone, acquiring energy far more effectively than ammonium chloride. The net effect is an increasing potential to remelt or dissolve dendrites and to thereby induce the downward growth of fully melted channels [112]. The upward flow of fluid through a channel and into the bulk melt is augmented by the fact that it is heated (gains thermal buoyancy) without significant loss in solutal buoyancy. For the conditions of case 1 [113], two prominent channels develop and provide for the discharge of interdendritic fluid into the bulk melt, while bulk fluid seeps across the liquidus interface and into the mushy zone around the channel openings. When the permeability of the mushy zone is increased by one order of magnitude (case 21, the aforementioned transition from conduction- to convection-dominated conditions occurs much sooner, indicating that Darcy damping reduces the rate at which perturbations can grow. Also, because of the lower resistance to interdendritic fluid flow, the increased permeability enhances exchange of fluid between the melt and mushy zones, with a corresponding increase in the number of fully melted channels that are predicted to form following the onset of convection [113]. Figure 23 corresponds to f = 25 min for case 2 and reveals five channels, one of which is at the outer radius, or left edge, of the plot. When the solidification rate was increased by lowering the bottom surface temperature to -100°C (case 3), the onset of convection was delayed and the amount of fluid exchanged between the melted and mushy zones decreased significantly. The effects of permeability and cooling rate on macrosegregation are shown in Fig. 24. Liquidus and solidus lines are superimposed on the

280

PATRICK J. PRESCOIT AND FRANK P. INCROPERA

FIG. 23. Predicted flow field for unidirectional solidification from below (case 2) at t = 25 min: (a) velocity vectors; (b) streamlines. (Reprinted from Neilson and Incropera [113]. Copyright 1992, with permission from Springer-Verlag.)

macrosegregation plots. Macrosegregation in fully solidified regions is permanent, whereas macrosegregation in the mushy zones changes slightly, as solidification continues past 50 min. Case 2 provides the most severe macrosegregation (Fig. 24b), since it corresponds to the earliest onset of convection and the largest fluid velocities. Since solidification was conduction-dominated for the longest period of time in case 3, macrosegregation is least severe. The foregoing simulations of unidirectional solidification [112, 1131 were performed under the assumption of axial symmetry. Therefore, the channels and freckles were predicted to be rings, rather than the discrete, pencil-shaped regions observed experimentally [ 16, 107-1 111. However, despite this obvious difference, the axisymmetric calculations are consistent with experimental observations in several important respects. Specifically, channels have been observed to initiate at the liquidus interface and to grow downward [107,108].Furthermore, the number of active channels decreases with time, and terminated channels have been revealed in etched sections of unidirectional solidified metal castings [15, 1141. Transient, three-dimensional calculations in cylindrical coordinates were performed [1151, and the predictions revealed discrete channels, while confirming the conclusions drawn from axisymmetric simulations [112, 1131.

N 00

Y

FIG.24. Macrosegregation patterns predicted at r = 50 min for unidirectional solidification: (a) case 1; (b) case 2; (c) case 3. (Reprinted from Neilson and Incropera [113]. Copyright 1992, with permission from Springer-Verlag.)

282


The problem of unidirectional solidification of a binary alloy from below continues to be the subject of many experimental [17, 116-1191 and theoretical [ 120, 1221 studies. Numerical simulations of unidirectional solidified metal alloys are discussed in Section 1V.B.

3. Multidirectional Solidification Despite being a common casting configuration, multidirectional solidification has received much less attention than unidirectional solidification from a chilled bottom wall or sidewall. Solidification of aqueous ammonium chloride in a square cavity chilled from the bottom wall and vertical sidewalls was recently studied both numerically and experimentally [18, 1231. The numerical calculations were based on a continuum model for momentum, heat and species transport during binary solid-liquid phase change [43] and simulated experimental conditions in a square test cell (101.6 x 101.6 mm). The nominal composition of the alloy was 27 wt% NH,Cl, and the initial temperature of the melt was uniform at 25°C which is 6.4"C above the nominal liquidus temperature of 18.6"C. At t = 0, the left and right vertical and bottom walls of the mold were chilled to - 30"C, which is 14.6"C below the eutectic temperature of -154°C. Details regarding the properties of the aqueous NH,CI system, experimental methods, and numerical procedures can be found elsewhere [18, 1231. Predictions during the early stages of solidification indicated a strong influence of thermal convection in the melt, causing the mushy zone thickness along the vertical sidewalls to increase with increasing distance from the top. The mushy zone thickness along the bottom wall was uniform and relatively large. At t = 10 min (Fig. 2 9 , conditions within the melt are asymmetric (despite the symmetry of the imposed boundary conditions). This result is attributed to the somewhat random locations at which two channels formed in the mushy zone over the bottom wall (Fig. 25b). Thermally driven downflows along the vertical liquidus interfaces turn inward at the liquidus interface along the bottom wall, causing solutally buoyant fluid emerging from the channels to coalesce near, but slightly to the right of, the cavity midplane (Figs. 25a, b). Interdendritic liquid in the bottom mushy zone is drawn into the two channels, in a manner similar to that for unidirectional solidification from below [ 17, 1121, while interdendritic fluid in the sidewall mushy regions ascends and is discharged into the melt along the top of the cavity. Figures 25c, d indicate that cold-water-enriched fluid is accumulating at the top of the mold, and, due to the depression of the liquidus temperature with water enrichment, dendrites are remelted (Fig. 25b).


a

2-D (102x66) t=lO.O min

-

b

Wmar = 1.51 x 1 0 2

t= 600.0 sec

3.40 mm/s

283

ymin= -1.31 x 10.2

Y

f

I 1 1 W

d

+ x

X

t= 600.0 sec

W

X

FIG.25. Predicted conditions at t = 10 min for multidirectional solidification of aqueous ammonium chloride in a rectangular cavity; (a) velocity vectors; (b) streamlines; (c) isotherms; (d) liquid isocomps [18].

Experiments revealed physical conditions much more complex than those predicted by the model during early stages of solidification [123]. Shadowgraphs (Fig. 26) revealed two counterrotating thermal convection cells that developed immediately after the mold walls were chilled. Although dendritic crystals began forming on the sidewalls within 15 s of

284

PATRICK J. PRESCOm AND FRANK P. INCROPERA

FIG.26. Shadowgraphs taken during multidirectional solidification of aqueous ammonium chloride in a rectangular cavity: (a) r = 1 min; (b) I = 2.5 min; (c) t = 5 min; (d) t = 7.5 min [181.

cooling, the vigor of the thermal convection cells caused most of the crystals to be detached and to be advected with the liquid. Since solid NH,Cl is relatively dense, many of the equiaxed crystals collected in two piles on the bottom of the cavity near the sidewalls (Fig. 26b). Solidification along the bottom wall generated cold but solutally buoyant interdendritic liquid, and a fine-scale salt finger convection pattern was observed (Figs. 26a, b). As solidification continued, there was a transition from many salt fingers to fewer, well-defined plumes (Figs. 26a-d) and regions along the bottom, where equiaxed crystals collected, were especially conducive to pluming and channel formation. Eventually, the plumes emanating from these regions were enveloped by the mushy zone advancing from the sidewalls, which essentially transformed the plumes into A-segregate channels in the sidewall mushy regions [18, 1231. Within 2 min, coherent mushy regions formed along the sidewalls 118, 1231, and interdendritic liquid ejected from the mushy zone was entrained in thermally driven downflows along the vertical liquidus interfaces. How-


285

ever, because of solutal buoyancy forces associated with water-enriched liquid emerging from the sidewall mushy region, flows along the vertical liquidus interfaces turned abruptly inward and then upward, exhibiting the fishhook structures shown in Fig. 26b. In Fig. 26d, water-enriched liquid is seen to be accumulating from the top left and right regions of the cavity. Eventually these pockets joined along the top and formed the first of several double-diffusive convection cells, which significantly affected solidification conditions in the adjacent mushy regions [18, 1231. These double-diffusive convection cells also interacted with plumes ascending from the middle of the bottom mushy layer. Within 10 min, small detached dendritic crystals were observed in the upper region of the melted zone, near the liquidus interface. This slurry region and the A-segregate channels, from which the dendrite fragments were believed to have originated [18, 1231, are shown in Fig. 13. Equiaxed crystals were also discharged from the bottom mushy region and advected by ascending plumes, The advection of equiaxed dendritic crystals has been observed by others [124-1271, and it is not yet clear whether fragments are fractured by flow-induced drag, remelted due to coarsening, or created by a combination of fracture and remelting effects. The numerical model [18] did not account for the advection of crystallites and hence could not predict the formation of mounds of equiaxed crystals on the floor of the mold. At a vertical position of approximately 75 mm, the model also predicted extended mushy zone growth from each sidewall and an associated merging of liquidus interfaces to form a “bridge” of dendrites across the melted zone. The predicted macrosegregation pattern at t = 30 rnin (Fig. 27) shows severely segregated freckles near the bottom, where vertical channels formed during early stages of solidification, and an A-segregate pattern along the sidewalls near the top. Although the model is able to predict several key trends, it is limited by its inability to account for slurry behavior and nonequilibrium conditions that existed in the experiments [18, 1231. Furthermore, the numerical simulations were unable to resolve many fine-scale features of the doublediffusive convection that occurred during solidification. Models that account for two-phase, nonequilibrium behavior [49,52,53] would have to be implemented with a much finer numerical discretization to capture such details. The computational effort required for such improvements is presently prohibitive.

4. Other Configurations The continuum model [43] has been used to simulate solidification of semitransparent analog alloys for conditions involving mixed convection

286

PATRICK .I. PRESCOTT AND FRANK P. INCROPERA

A segregate

NHqCI-Rich

Water-Rich

19.9-22.4 22.4-24.9 24.9-27.3 FIG.27. Macrosegregation predicted at chloride [18].

f =

=

27.3-29.7 29.7-32.1 32.1-34.6

30 min for solidification of aqueous ammonium

[ 1281, combined buoyancy- and surface-tension-driven convection [ 129, 1301, and buoyancy-driven convection in a horizontal annular region [ 1311. Solidification of an aqueous NH,C1 solution flowing downward in a vertical rectangular channel was studied numerically [ 1281, and predictions have been compared with experimental results [1321. Mixed convection conditions resulted from the interaction of thermosolutal buoyancyinduced flows with the forced through flow. For a flow Reynolds number of Re = 200, solutally buoyant interdendritic liquid was able to penetrate the liquidus interface, thereby discharging water-enriched liquid into the main flow near the top of the channel. These conditions favored the development of distinct channels in the mushy zone, facilitating the exchange of liquid between mushy and melted zones. With Re = 800, interdendritic flow was confined to the mushy region, and channel develop-


287

ment was reduced significantly. The effects of chill wall temperature and nominal alloy composition were also studied, and the system was found to be most sensitive to composition. Although experimental trends were predicted by the simulations, good quantitative agreement was precluded by (1) the inability to experimentally effect a sudden change in chilled wall temperature, (2) uncertainty in the nominal alloy composition, and (3) uncertainties associated with model assumptions, such as equilibrium freezing, constant thermophysical properties, and mushy zone permeability [132]. The effects of thermo/diffusocapillary convection were delineated in numerical simulations of solidification for aqueous NH,CI in a 20 X 20-mm cavity with a top-free surface [1291. Without buoyancy effects (a zerogravity environment), conditions associated with thermocapillary convection were compared with those for thermo/diffusocapillary convection and found to be virtually identical, indicating that diffusocapillary effects were negligible. In the absence of gravity, surface tension significantly affected solidification by retarding solid growth near the top of the cavity, where surface tension driven flow is strongest. In a 1-g environment, surface tension driven flow along the top free surface inhibits the discharge of interdendritic liquid from the mushy zone into the melt during early stages of solidification. Soon thereafter, however, solutal buoyancy forces discharge interdendritic liquid from the mushy zone into a region just below the top-free surface, thereby subdividing the melt into a top cell driven by capillary forces and a bottom cell driven by thermal buoyancy. Eventually, conditions approached those associated exclusively with buoyancy-driven convection [1291. Differences between predicted and experimental results were associated with the times at which significant events occurred and with the thicknesses of the solid and mushy layers [130]. The continuum model [43] was also used to simulate the solidification of an aqueous sodium carbonate solution (Na,CO,-H,O) from the inner surface of a horizontal annulus [131]. To assess the accuracy of predicted results, the conditions of the simulation were chosen to match those of a related experimental study [133]. Although solutal upflow from the top portion of the mushy zone was predicted and observed, there was a significant difference in the flow structures. In the simulation, a single thin plume of cold but solutally buoyant fluid ascended from the uppermost region of the mushy zone. In contrast, shadowgraphic and dye injection flow visualization revealed a wide band (across the top quadrant of the mushy zone) of buoyant plumes with a very complex, perhaps turbulent, flow structure. Also, whereas the numerical model predicted the development, downward movement and eventual erosion of a single doublediffusive interface, multiple interfaces were observed in the experiments.

288


Moreover, the actual evolution of double-diffusive convection patterns occurred much more slowly than did conditions predicted by the model. Finally, while good agreement was obtained between measured and predicted results for the thickness of the fully solid layer, the thickness of the mushy zone was overpredicted. Discrepancies were attributed to numerically induced false diffusion, the lack of a turbulence model to account for the complex mixing that occurred at the top of the cavity, and uncertainties associated with the mushy zone permeability, which was thought to be highly anisotropic in the experiments [131]. The disagreement with mushy zone thickness may also be attributed to uncertainties in the actual alloy composition and, perhaps, to constitutional undercooling at dendrite tips 1681. B. METAL ALLOYS

Although the solidification models discussed in Section I11 are able to simulate solid-liquid phase change in many physical systems (e.g., freezing and melting in Arctic waters, food processing, and magma chambers), they were developed specifically for simulating the casting of metal alloys. Metals are unique by virtue of their low Prandtl numbers. The Prandtl number, which is the ratio of diffusivities for momentum and heat, is an important parameter in buoyancy-induced convection, since it affects the relative thicknesses of velocity and thermal boundary layers, which, in turn, determine the regions over which thermal buoyancy and viscous forces are significant. The Prandtl number of most molten metals is on the order of lo-' to lo-*, which is two to three orders of magnitude smaller than values typical of aqueous solutions. Amberg [S7] simulated the solidification of an Fe-1 wt% C alloy (Pr = 0.174) in a rectangular cavity (100 mm tall by 200 mm wide), cooled through its vertical sidewalls with a constant and uniform heat flux (60 kW/m2). Approximately one hour was required to completely solidify the casting. The simulation was based on model equations from Hills et al. [41], which were integrated numerically using a finite-volume approach with an explicit time-marching scheme for most terms in the governing equations and an implicit scheme for heat diffusion terms. Thermal convection within the melt affected the shape of the liquidus interface during early stages of solidification, whereas solidification within the mushy zone wa6 primarily conduction-dominated. During intermediate stages of solidification, the entire mold cavity was occupied by a mushy zone, and a weak solutally driven convection cell provided solute transport that enriched the top and depleted the bottom region of carbon. However, the


289

final carbon concentration was between 0.97 and 1.03 wt% throughout most of the casting. Amberg’s modeling procedures [57] were also employed to simulate experiments in which Sn-10 wt% Pb and Pb--15 wt% Sn alloys were solidified in a square (100 X 100-mm) mold [70]. The experiments were performed with a constant temperature difference maintained between the vertical walls, while the temperatures of both walls were gradually reduced from values exceeding the liquidus temperature to values below the solidus temperature. The measured wall temperatures were used as boundary conditions in the simulations, Thermal and solutal buoyancy forces augmented each other during the solidification of the Sn-10 wt% Pb alloy (Pr = 0.0151, and predicted fluid velocities, along with lines of constant solid fraction, are shown in Figs. 28a-c at times of 3,6, and 9 min, respectively [70]. At t = 3 min (Fig. 28a), a strong, counterclockwise thermally driven convection cell in the melt is responsible for retarding the growth of the mushy zone near the top of the mold. Since solutal buoyancy forces augment thermal buoyancy forces, interdendritic fluid descends through the mushy zone and is discharged into the melt along the bottom of the cavity, and at later times, dendrites are remelted along the bottom as a result of the accumulation of Pb-rich fluid (Figs. 28b, c). Figures 29a, b show predicted macrosegregation patterns at 9 and 45 min, respectively, and reveal significant Pb enrichment along the bottom of the casting. Good agreement was obtained between predicted and measured macrosegregation results (Figs. 30a-c). Because of opposing thermal and solutal buoyancy forces during solidification of the Pb-15 wt% Sn alloy [70], convection and macrosegregation patterns are more complex than those of the Sn-10 wt% Pb alloy. Fluid velocities are shown in Figs. 31a-c for 12, 24, and 36 min, respectively. At t = 12 min, the mushy zone thickness is highly nonuniform as a result of strong, counterclockwise thermal convection in the melt. However, with increasing time, the thermally driven cell vanishes, as solutally driven Convection becomes dominant (Figs. 31b, c). The clockwise solutal convection cell in the mushy zone is also responsible for the highly irregular distributions of solid fraction in the mushy zone (Fig. 31) and the irregular macrosegregation pattern (Fig. 32). Although fully melted channels in the mushy zone were not predicted, Fig. 32 displays an A-segregate pattern of slanted adjacent regions of positive (Sn-rich) and negative (Pb-rich) segregation. Also, a wedge (analogous to a cone) of Sn-rich material was predicted to form in the top right region of the casting (Fig. 32b) and is attributed to solutal convection during the intermediate and later stages of solidification. Although macrosegregation measurements were made at

290


x=o

x=lOcm

x= 0

x= 10 cm

x=O

x= i o c m

FIG.28. Velocity vectors and liquid volume fraction distributions predicted for solidification of a Sn-10 wt% Pb alloy: (a) t = 3 min; (b) t = 6 min; (c) 9 min. Velocity arrow lengths equal to the grid spacing correspond to 5 cm/s and 0.1 mm/s in the fully melted and mushy zones, respectively. Isopleths are drawn for liquid volume fractions of 1.0, 0.8, and 0.6. (Reprinted with permission from Shahani et al. [70], 1992, ASM International.)

only a few locations, reasonably good agreement was obtained between predicted and measured results (Figs. 33a-d). Detailed predictions of the evolution of convection and macrosegregation during solidification of a Pb-19 wt% Sn alloy in an experimental test cell are provided by Prescott and Incropera [1341. Solidification occurred in axisymmetric, annular mold of stainless steel, cooled along its outer vertical wall. The heat flux was expressed as

E z

a

9%

II

%

13% 0 II

1 5%

x

x =o

x = lorn

b

x =o

x=lOn

FIG.29. Macrosegregation patterns predicted for a Sn-10 wt% Pb alloy at (a) t = 9 min and (b) t = 45 min (reprinted with permission from Shahani et al. [70], 1992, ASM International).

29 1

292


a Wt. % Pb

20 -

vertical section VII, at xm9.47 cm

18 -

0

experiment

-elmulation 16-

~

0

87 0

'

I

I

I

1

2

4

6

8

10

Y (cm) FIG.30. Measured and predicted macrosegregation patterns for a Sn-10 wt% Pb alloy at (a) x = 9.47 cm, (b) x = 8 cm, and (c) x = 3.5 cm. The cooled surface corresponds to x = 0. (Reprinted with permission from Shahani el al. [70], 1992, ASM International.)

where T'Jr) is the local wall temperature, T, is the coolant temperature, and U is an overall heat transfer coefficient. Values of T, = 13°C and U = 35 W/m2 K were prescribed to match experimental conditions [135, 1361. According to predictions, a thermal convection cell is established in the melt once cooling is initiated [134]. The cell is driven by a radial temperature gradient confined within 10 mm of the mold wall, and the central portion of the melt becomes thermally stratified during the initial 90 s of cooling. Moreover, convective mixing reduces the temperature gradient throughout the melt, thereby delaying the onset of solidification relative to conduction-dominated conditions. At t = 120 s, solid dendrites begin precipitating at the bottom of the cooled mold wall, thereby forming a two-phase (mushy) zone. As cooling continues, the mushy zone grows, with the liquidus interface moving vertically upward and radially inward. h il result of phase equilibrium requirements, the precipitation of solid is accompanied by Sn enrichment of interdendritic liquid, which ifiduces solutal buoyancy forces acting upward on the interdendritic liquid. f i e

-

293


b Wt. Yo Pb 20

-

18

-

, 16

-

14

-

12

-

vertical section VI, at x=8 cm

0

0 experiment -simulation

o

10 -

0

8

I

I

0

0

0

0

-

1

1

20 vertical section 111, at xz3.5 cm 0 experiment -simulation

-o r

8

0

1

I

1

2

4

6

1

8

10 Y (cm)

FIG.30. Continued.

294

PATRICK J. PRESCOlT AND FRANK P. INCROPERA

z

1 -

x =o

x=lOm

h

1 -

,

x =o

x=lOm

C

x

,

x =o

I

*

x=lOm

FIG.31. Velocity vectors and liquid volume fraction distributions predicted for solidification of a Pb-15 wt% Sn alloy: (a) t = 12 min; (b) t = 24 min; (c) t = 36 min. Velocity arrow lengths equal to the grid spacing correspond to 5 cm/s and 0.1 mm/s in the fully melted and mushy zones, respectively. Isopleths are drawn for liquid volume fractions of 1.0, 0.8,0.6, 0.4, and 0.2. (Reprinted with permission from Shahani et al. [70], 1992, ASM International.)

solutal buoyancy forces oppose thermal buoyancy forces caused by the radial temperature gradient, and because the density of Sn is significantly less than that of Pb, solutal forces dominate within the mushy zone ( N = -14). Hence, solidification within the mushy zone can be regarded as providing an upward momentum source for the interdendritic liquid. As the mushy zone continues to grow, the influence of solutal buoyancy increases, and Figs. 34 and 35 show predicted convection conditions at t = 155 s and t = 195 s, respectively. Field plots of velocity vectors, streamlines, isotherms, and liquid isocomposition lines are drawn on r-z planes, with the outer and inner radii represented by the left and right

a

18% 16%

12%

I

14%

x=o

1

x=l Ocm

b 24%

22% 20% 1 8%

16%

1 2%

1 4%

I

I

x=o

x=l Ocm

FIG.32. Macrosegregation patterns predicted for a Pb-15 wt% Sn alloy at (a) t = 30 min and (b) f = 120 min. (Reprinted with permission from Shahani el al. [70], 1992, ASM International.)

295

35

:.

30-. 25

vertical section VII, at x-9.47cm 0 experiment -simulation

'

0

-

20 -

15-+

10

= I

1

35 -.

:.

30-.

25

vertical section VI, at x=8 cm

(


-

20 -

15

-4

10 1

I

I

I

I

C Wt. X Sn 35

'

vertical section 111, atx=3.5cm

.


30-. 25

-

0

d Wt. % Sn 35

.

30-. 25

vertical section II. atx-2cm 0 experiment -simulation

-

FIG.33. Continued.

297

298

PATRICK .I.PRESCOTT AND FRANK P. INCROPERA

edges of each plot, respectively. Also, the bottom and vertical mold walls are included in these plots. At t = 155 s (Fig. 34), the mushy zone covers approximately 75% of the inside surface of the outer mold wall. Fluid is exchanged between the mushy and melt zones in a relatively confined region near the top of the mushy zone (Fig. 34b), where a strong, solutally driven upflow, emerging from the mushy zone, interacts with thermally driven downflow in the bulk melt. The interaction turns both flows radially inward, thereby constricting the thermal cell and returning discharged interdendritic fluid to the mushy zone. Since the Pb-Sn system is characterized by a large Lewis number (Le = Sc/Pb = 8600), fluid within the solutal convection cell readily exchanges energy with the bulk liquid but largely retains its composition. Hence, warm, Sn-enriched fluid from the melt is advected toward the mushy zone, establishing conditions conducive to the development of channels within the mushy zone. A channel, although not fully melted, is aligned vertically and is located along the mold wall for z* 2 0.5. It is delineated by a thick dashed line in Fig. 34a. As solidification continues, the channel along the outer mold wall grows and continues to ingest interdendritic fluid, which is advected to the top of

a -+

b 18.7 mmls

d

C Wrniw4.04254 Wrnax= 0.03687

Trni,-j=279.2T Tmax=289.0°C

f%li"=O.1900

FIG.34. Convection conditions in a solidifying Pb-19 wt% Sn alloy after 155 s of cooling: (a) velocity vectors; (b) streamlines; (c) isotherms; (d) liquid isocomps (reprinted with permission from Prescott and Incropera [134], 1994, ASME).


299

the mold cavity. In addition, thermosolutal interactions along the liquidus interface create small recirculation cells in which interdendritic and bulk liquid are exchanged. This flow pattern was identified as the genesis of an A-segregate pattern in the final casting [134]. The momentum associated with the thermal convection cell gradually decreases, as radial temperature gradients in the melt diminish and opposing solutal buoyancy forces increase. At t = 195 s, the thermal cell is confined to the bottom half of the fully melted zone (Fig. 35b), while the solutal cell encompasses the mushy zone and the top half of the bulk melt, which has become solutally stratified (Fig. 35d). The thermal cell is completely extinct by 240 s, beyond which a large solutally driven cell occupies the entire mold cavity and provides for the recirculation of interdendritic liquid. Figure 36 shows the predicted evolution of macrosegregation between 240 and 600 s. At t = 240 s, macrosegregation is characterized by a pattern of A segregates [7, 1371 (Fig. 36a), which formed in the outer periphery of the ingot as a result of thermosolutal convection during the early stages of solidification. The overall mass fraction solid is 10.4% at 240 s, and

b

a -

A

-

C

d ffLn=O. 1900

9.43 mmls

FIG.35. Convection conditions in a solidifying Pb-19 wt% Sn alloy after 195 s of cooling: (a) velocity vectors; (b) streamlines; (c) isotherms; (d) liquid isocomps (reprinted with permission from Prescott and Incropera [134], 1994, ASME).

a

b

C

d

L

FIG. 36. Predicted macrosegregation patterns in a Pb-19 wt% Sn alloy after (a) 240 s, (b) 300 (reprinted with permission from Prescott and Incropera [134],ASME).

I

(c) 360 s, (d) 480 s, and (e) 600 s


301

macrosegregation intensifies with time, as solidification and solutal convection in the mushy zone enrich the lower and outer regions of the ingot with Pb, while transporting Sn to the top and downward along the inner radius. During the intermediate stages of solidification, a large cone of Sn-rich material forms on the top of the mold and extends deep into the ingot. The development of this cone segregate is illustrated in Figs. 36b-e. By 600 s, the overall mass fraction solid is 45.4%, and although the solid fraction in the top part of the cone segregate is less than lo%, solid fractions exceed 35% throughout the rest of the ingot, with Pb-rich zones near the outer radius being 60% solidified. At this time the permeability is sufficiently small throughout the cavity to inhibit fluid recirculation and hence further development of macrosegregation. Since discrete channels form in actual ingots and interdendritic flow patterns are, therefore, three-dimensional [136], local composition variations may exceed those shown in Fig. 36, which are predicted for axisymmetric conditions. In a related study, the effect of cooling rate on convection conditions and macrosegregation was studied by comparing simulations with five different values of the overall heat transfer coefficient between 35 and 3000 W/m2 K [601. Because the extent of the mushy zone and time over which the permeability is large are reduced with increased cooling rate, macrosegregation is less severe. The macrosegregation predicted for U = 3000 W/m2 * K is shown in Fig. 37, and except for small regions near the top and bottom of the casting, the Sn concentration is within 0.5% of the nominal (19%) concentration. The predicting thermosolutal convection patterns and the macrosegregation trends were verified through comparisons with measurements and metallographic examinations [ 1361. However, predicted and measured results were characterized by several significant differences. Figures 38a-c show predicted and measured cooling curves at three different vertical locations within the mold cavity during the first 600 s of cooling [136]. Overall, the predicted cooling rate exceeds the measured cooling rate, and differences may be attributed to uncertainties in prescribed thermodynamic properties, which, for example, affect the thermal capacitance of the system, and to uncertainties in the overall heat transfer coefficient, which governs the rate of heat transfer from the system. However, the most conspicuous disagreement relates to the absence of the undercooling and recalescence effects in the predicted cooling curves between 150 and 300 s. The measured cooling curves in Figs. 38a-c reveal a period of time, immediately after the commencement of solidification, during which the temperature increases for r* 5 0.8. This nonequilibrium phenomenon, termed recalescence [68], occurs following nucleation in an undercooled melt. Because the model assumed local thermodynamic equilibrium, such

302


20-21.5%

19.5-20°/o

U19-19.5%

m

18.5-19%

FIG. 37. Final macrosegregation pattern predicted for a Pb-19 wt% Sn alloy with = 3000 W/m2 . K (reprinted with permission from Prescott and Incropera [60],1991, ASM International).

U

phenomena could not be predicted. Furthermore, inferences drawn from the measured recalescence patterns indicated the existence of a convecting slurry [136], whereas the model assumed the mushy zone to be stationary. Although attempts have been made to account for nonequilibrium phenomena and solid-particle transport in the melt using the continuum model [61], there is a clear need to incorporate more sophisticated models for such phenomena [52, 531. Finally, temperature fluctuations were observed in measured cooling curves prior to and just after the onset of solidification. Such fluctuations are indicative of vigorous convection occurring within the molten metal and disappear following recalescence (Fig. 38), since the mushy zone dampens fluid motion. Predicted cooling curves are smooth, even before solidification, indicating failure to resolve pertinent features of the flow in the simulation. Measured macrosegregation patterns from six experimental ingots [1361 are compared with model predictions [134] in Figs. 39a-c. Zero A%Sn


303

corresponds to no macrosegregation, positive values represent Sn enrichment, and negative values represent Sn depletion. The agreement between predicted and measured macrosegregation patterns is reasonable, and although the data are scattered, they confirm general trends predicted by the model. That is, the concentration of Sn increases with increasing height and, at z * = 0.5 and 0.83, with decreasing radius. However, since the uncertainty in the measured results is only +0.40%Sn [135, 1361, it does not account for the scatter, which is as large as 4%. The circumferential variation of macrosegregation in an experimental ingot [1361 is plotted in Fig. 40, which shows %Sn as a function of 8 at ( r * , z*) = (0.3,0.83). The plot indicates clearly the three-dimensional nature of the macrosegregation field. If conditions were axisymmetric, the variation of %Sn with 8 would be within the measurement uncertainty interval. Since the data in Fig. 39 were taken at different circumferential positions, the scatter is largely attributable to existence of a threedimensional macrosegregation pattern. Locations in Fig. 39 that exhibit pronounced scatter are believed to be within segregated regions associated with channeling during solidification. Metallographic examination of ingot sections revealed pockets of eutectic material, providing further evidence that macrosegregation is three-dimensional, due to the development of discrete channels in the mushy zone during solidification 11361. The volume-averaged two-phase model of solidification [49] was used to simulate equiaxed solidification of an Al-4 wt% Cu alloy in a 50 X 50-mm cavity cooled from one vertical sidewall [81]. Since the interdendritic liquid is enriched with copper, thermal and solutal buoyancy forces augment each other. The model accounted for nucleation, growth, and advective transport of grains. However, the floating or settling of dendrites was not modeled, because the solid density was prescribed to be equal to the liquid density. Results demonstrated the model’s ability to predict recalescence, macrosegregation, and grain size distribution. Solidification of a metal alloy, chilled from below, has been simulated by Heinrich et af. [138, 1391 using a finite-element formulation [721. The Pb-10 wt% Sn alloy was characterized by unstable solutal gradients when cooled from below, and the resultant solutal upwelling produced channels in the mushy zone and freckles in the final casting [15, 108, 1141. Calculations were performed for small physical domains (e.g., 5 X 10 mm and 2.5 X 4.5 mm) [138, 1391 so that solidification and transport behavior could be resolved within the channel region using a grid of 20 X 30 elements. The effect of lateral cooling or heating through sidewalls was investigated, and it was concluded that channels tended to form along the sidewalls, especially with lateral cooling. Sidewall heating had the effect of moving the channels inward slightly [139].

304


a310 305 300 295 T

290

(“C:) 285 280 275 270 0

60

120

180

240

300

360

420

480

540

600

Time ( 5 )

FIG.38. Measured and predicted cooling curves during the solidification of a Pb-19 wt% Sn alloy in a cylindrical annular mold: (a) z* = 0.083 (near bottom); (b) z* = 0.50 (midheight); (c) z* = 0.83 (near top) (reprinted with permission from Prescott et al. [136], 1994, ASME).

Solidification of Al-Cu alloys from below was simulated by Diao and Tsai [140, 1411, and their predictions agreed very well with experimental data [4, 1421. However, the density gradient was both thermally and solutally stable, and thus, no buoyancy-induced convection occurred. Fluid flow was driven solely by shrinkage, rendering the problem one-dimensional. In a subsequent study, Diao and Tsai considered under-riser macrosegregation [143]. Overall, stable buoyancy conditions were imposed, and fluid flow was driven mainly by shrinkage associated with phase change. However, because of sudden expansion experienced by downward flow from the riser into the main mold cavity, recirculation cells were induced in the top comer regions of the cavity and were responsible for macrosegregation in the under-riser region. The effects of shrinkage-driven convection on solidification were considered by Chiang and Tsai [%I, and an attempt was made to compare the effects of shrinkage to those of buoyancy [144] in a 1% Cr steel, chilled from the bottom and sidewalls of a rectangular cavity. However, their calculations were based on an unrealistically large contraction ratio, [( ps - p , ) / p , ] X loo%, of 20% and a thermal expansion coefficient more than four orders of magnitude smaller than an appropriate value. Further-

305


120

180

240

300

360

420

480

540

600

360

420

480

540

600

Time (s)

C

0

60

120

180

240

300 Time (s)

FIG.38. Continued.

more, solutal buoyancy was completely ignored. Combined shrinkage- and buoyancy-driven convection during the solidification of an AI-Cu alloy was also simulated by Xu and Li [19] using a continuum model [59]. However, the effects of shrinkage and buoyancy were not delineated in a systematic manner.

306

PATRICK J . PRESCOTT AND FRANK P. INCROPERA

I

I

I

I

I

I

3

1

I

1

I

I

I

I

ij:i I

2'

4 -

1

1

I

_

= 0.083

j

-

E-3

E-5 E-6 __ Model 0

2 c v

-

)

.

f : 0 -

! m

8 -2 -

-4

i

>

l

l

/

l

/

,

l

l

/

l

l

l

l

l

l

I

The effects of shrinkage and buoyancy on macrosegregation were compared recently for the solidification of a Pb--19.2 wt% Sn alloy from the side of a rectangular cavity [20]. Nine numerical simulations were performed, covering a matrix of conditions involving three cooling rates and (1) thermosolutal convection without shrinkage, (2) thermosolutal convection with shrinkage, and (3) shrinkage-driven flow without buoyancy effects. Results demonstrated that buoyancy exerts the dominant influence on macrosegregation and that shrinkage effects are important only near the solidus interface under extreme cooling conditions. A scaling analysis reinforced conclusions drawn from the numerical simulations [20] and suggested similar behavior for an Al-Cu alloy. Scaling also suggested that, for a Cu-Sn system, shrinkage effects may be important relative to buoyancy, since thermal and solutal buoyancy forces nearly offset each other in the mushy zone. Although continuous or direct-chill (DC) casting is of great practical importance and the subject of many investigations, simulations of convection heat and mass transfer in the melt and mushy zones are sparse. Existing models are somewhat restrictive, treating, for example, only conduction [145], only transport within the mushy zone [146- 1481, or transport only within the melted region [149]. Convection and solidification

307


b

6

~

I

I

,

I

I

I

I

I

T

i

-

1

I

Z'

F

-

= 0.50

4 -

E-4 E-5 E-6

0

-Model

2 c

-

-

[ I ) -

8 a

: . 0

0 -

-2

A

!

P

0

0

w A 0

8

Q

-

0

1

-4

1

I

1

1

1

1

,

1

,

1

,

1

/

1

~

I

I

E-5 E-6 I

A

0

m

-

-2 -

-4

I

,

l

,

l

,

l

,

l

,

l

l

l

/

l

l

l

~

FIG.39. Continued.

in a twin-roll continuous casting process were simulated by Ha et al. [150] using a continuum model [431, although solute transport was not considered and buoyancy was neglected. These effects were also ignored by Farouk et al. [151,152], who simulated solidification in a twin-belt continuous caster, with turbulence treated using a low-Reynolds-number model

308


24

23

22

s

8 21

20

19

I

0

I

10

I

20

,

I

30

I

I

L

40

I

50

,

l

I

60

I

I

70

80

e (deg.1 FIG.40. Circumferential variation of measured Sn concentration at r* (reprinted with permission from Prescott et al. [136], 1994, ASME).

5

0.30 and Z*

90

-

0.83

[1531. Shyy et al. [154] modeled steady-state continuous ingot casting, with electron-beam melting (EBM), and included the effects of strand motion, thermal buoyancy, surface tension, and turbulence on convection in the melt and mushy zones. The simulation incorporated an adaptive grid over which continuum equations were integrated. However, the effects of solute transport on macrosegregation and buoyancy were not considered. Finally, a single-domain model, which accounted for thermal and solutal buoyancy in both liquid and mushy zones, was used to simulate DC casting of an A1-4.5 wt% Cu alloy [155]. The effects of thermal and solutal buoyancy forces and casting speed were studied, and although the results of the investigation were considered to be preliminary, they indicated that solutal buoyancy contributed significantly to increased copper concentration along the centerline of the ingot.

V. Strategies for Intelligent Process Control

Buoyancy-induced fluid flow, which is primarily responsible for most forms of macrosegregation and affects grain structure in castings, is not directly controllable. One means of eliminating buoyancy effects is to


309

perform solidification processes in a zero-gravity environment, and although studies have indicated that such measures would prevent the formation of freckles in directionally solidified castings [109,156,157], they are not practical for large-scale industrial operations. Macrosegregation can also be reduced by increasing the cooling rate, but the cooling rate is often limited by uncontrollable factors such as contact resistance between the mold and ingot. Moreover, because of the thermal resistance imposed by the outer solid shell as it forms, the effective cooling rates in interior regions of an ingot are necessarily smaller than those near the mold wall. Hence, macrosegregation is more difficult to control in large ingots. In addition to gaining a better understanding of the mechanisms by which macrosegregation occurs, some researchers have explored process control options for which macrosegregation is inhibited. Possible options include the use of inertial, centrifugal, or electromagnetically induced forces to activity control macrosegregation. Kou et al. [36] performed experiments in which a cylindrical mold was rotated about its axis during the solidification of a Sn-Pb alloy. A n axially moving cooling jacket surrounded the mold, and immersion heaters maintained a melt above the mushy zone. The entire system was mounted on a turntable, and thermocouples connected through a slip ring were used to monitor temperatures during the solidification process. Without rotation, Pb-rich segregates formed in the center of the ingot (positive centerline segregation) as a result of the upward concavity of the liquidus and solidus interfaces. The shape of the interfaces allowed Pb-rich interdendritic liquid to flow downward and radially inward, along a path that was nearly parallel to the liquidus interface. Centrifugal forces due to rotation therefore inhibited denser Pb-rich fluid from migrating toward the center of the ingot, and a more uniform radial composition profile was achieved. However, it was reported that if rotational speeds became too large, segregates formed along a ring between the axis and outer edge of the ingot. Petrakis et al. [158] also investigated rotation as a means of controlling macrosegregation. The apparatus was a modification of that used by Kou et al. [36], and again, the phase-change material was tin alloyed with lead. In these experiments the ingot remained stationary, but swirl was introduced into the melt (above the mushy zone) by rotating immersion heaters. They found that with sufficient swirling of the melt, the shape of the mushy zone could be changed from concave to convex upward. Depending on rotational speed and solidification rate, rotation was found to have a beneficial effect on macrosegregation. The aforementioned studies involved Sn-Pb alloys for which thermal and solutal buoyancy forces augmented each other. The objectives of introducing swirl were to (1) oppose the effect of the buoyancy forces with a

310

PATRICK J. PRESCOTI AND FRANK P. INCROPERA

centrifugal force and/or (2) stabilize the liquid density gradient by enhancing heat transfer in the melt, rendering the isotherms, liquid isocomps, and liquidus and solidus interfaces nearly planar (horizontal). However, when solutal and thermal buoyancy forces oppose each other during solidification, the liquid density gradient cannot be completely stabilized, and, when the system is cooled from below, double-diffusive finger and plume convection may occur. Hence, in contrast to the aforementioned studies, Sample and Hellawell [lo71 found that steady mold rotation had little or no effect on freckle formation in unidirectionally solidified ingots of NH,Cl-H,O and Pb-Sn that produce solutally unstable density gradients when solidified from below. However, it was discovered that, by slowly rotating an ingot about an axis making a small angle with the direction of gravity (i.e., a precessional movement), freckle formation could be suppressed in directionally solidified ingots [107, 1081. Although the original motivation for this work was to continuously change the orientation of the gravitational field relative to the ingot, it was surmised that suppression of channel formation was due primarily to the shearing action associated with movement of the bulk melt along the liquidus interface [107]. The shearing action was induced by the precessional motion and minimized perturbations in the liquid density gradient [108]. The findings of Sample and Hellawell [107, 1081 are supported by the numerical simulations and experiments of Neilson and Incropera [112, 159, 1601. Both numerical simulations [ 1591 and experiments [ 1601 indicate that relatively slow, steady rotation of the mold during unidirectional solidification of a binary alloy has a negligible effect on the formation of channels and freckles, although steady rotation does have the effect of “organizing” the flow in the melt overlying the mushy zone [159]. Furthermore, the hypothesis of Sample and Hellawell [ 1081 regarding the physical basis of channel formation is consistent with additional calculations [1591, for which intermittent rotation was applied during unidirectional solidification. When a stationary mold holding a quiescent liquid is suddenly rotated, an initial “spinup” period follows, during which the liquid gradually approaches steady, solid-body rotation. Angular momentum is transferred from the mold walls to the liquid by shear forces in boundary layers along the mold wall known as Ekman layers. Shear forces in Ekman layers are responsible for minimizing perturbations in field variables along the liquidus interface, thereby eliminating a prominent mechanism for channel nucleation [112]. Although the Ekmann layers would eventually vanish as the liquid achieves solid-body rotation, mold rotation can be terminated, creating a “spindown” period during which the liquid gradually loses its angular momentum.


311

Simulations were performed [159] using the continuum equations for binary solid-liquid phase change 1431 in axisymmetric coordinates. In addition to equations for conservation of total mass, axial and radial momentum, energy, and species, an equation was introduced for conservation of swirl ( a = o r 2 ) in a system containing liquid, mushy, and solid zones [159]. The equation is of the form d 2CL CL -(pn)fV~(pVR)=V~(~vn)-----(n-n,) dt r ar K

where the mixture swirl is a mass fraction-weighted average of solid- and liquid-phase swirls, = f,n, + fin,.The third term on the right side of Eq. (42) is a Darcy damping term. The fourth term on the right side of Eq. (42) is an advection-like source term, which arises in a manner similar to advection-like source terms in Eqs. (4) and (5). The last term on the right side of Eq. (42) accounts for solid phase acceleration [45], which must be prescribed. Figures 41-43 show predicted convection conditions at various times during solidification of an aqueous NH,C1 solution from below with intermittent rotation [159]. The end of the first spinup period occurs at t = 90 s, for which conditions are shown in Fig. 41. Swirl (Fig. 41c) is responsible for centrifugal forces that drive counterrotating convection cells in the melt (Fig. 41b). The Ekman layer along the liquidus interface (Figs. 41a, b) diminishes perturbations in temperature (and salt concentration) along the interface (Fig. 41d). However, a solutally driven plume emerges from the most elevated region of the mushy zone, which corresponds to the outer mold wall (Figs. 41a, b). The melt achieved a near state of solid-body rotation at t = 90 s, at which point mold rotation was halted to reduce the potential for eventual channel formation. Figure 42 shows predicted conditions during the subsequent spindown period ( t = 130 s). The centrifugally induced convection cell above the liquidus interface has reversed direction, restoring nearly horizontal isotherms. The simulation continued with alternating spinup and spindown periods, and conditions at t = 19 min (Fig. 43) reveal the existence of channels at the centerline and outer radius, but not at intermediate radii. The aforementioned numerical results suggest that intermittent rotation, with a half-period approximately equal to the theoretical spinup time,

312

PATRICK J. PRESCOTI' AND FRANK P. INCROPERA

-

2 . 3 8 mn/s

,x=l .000

b

An* = 0.100

C

r

-

r-++ FIG.41. Predicted convection conditions at t = 90 s (end of spinup) during unidirectional solidification of an aqueous ammonium chloride solution with intermittent rotation (10 rpm): (a) velocity vectors; (b) streamlines; (c) isoswirls; (d) isotherms. (Reprinted from Neilson and Incropera [159]. Copyright 1993, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 IGB, U.K.)

suppresses the formation of channels, and therefore freckles, throughout most of a unidirectionally solidified casting. This conclusion was validated experimentally [160]. Figures 44a-c are photographs taken from above a cylindrical mold, in which aqueous ammonium chloride was solidified from below. Figure 44a shows that channels formed only along a ring near the outer radius of the mold during intermittent rotation with a half-period of 90 s. Without rotation (i.e., a static casting), channels formed at random


313

b

&,=0.413

An*= 0.041

r+-I

FIG. 42. Predicted convection conditions at t = 130 s (spindown) during unidirectional solidification of an aqueous ammonium chloride solution with intermittent rotation (10 rpm): (a) velocity vectors; (b) streamlines; (c) isoswirls; (d) isotherms. (Reprinted from Neilson and Incropera [159]. Copyright 1993, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 IGB, U.K.)

locations on the horizontal plane shown in Fig. 44b. Figure 44c shows that when the mold is rotated intermittently at a frequency twice that of Fig. 44a, channels form along two concentric rings. One ring, containing nine channels, formed along the outer mold wall, and the other ring, containing five channels, formed within a radial interval from 60-65% of the outer mold radius.

314


b

r

Advances in Heat Transfer, Volume 28: Transport Phenomena in Materials Processing

Advances in Heat Transfer Transport Phenomena in Materials Processing

Advances in Heat Transfer, Volume 39 (Advances in Heat Transfer)

Advances in Heat Transfer, Volume 35 (Advances in Heat Transfer)

Advances in Heat Transfer, Volume 34 (Advances in Heat Transfer)

Advances in Heat Transfer, Volume 37 (Advances in Heat Transfer)

Transport Phenomena in Fires (Developments in Heat Transfer)

Transport Phenomena In Fuel Cells (Developments in Heat Transfer)

Advances in Heat Transfer, Volume 30: Transport Phenomena in Crystal Growth

Advances in Transport Phenomena: 2010

Advances in Heat Transfer, Volume 25

Advances in Heat Transfer, Volume 36

Advances in Heat Transfer, Volume 36

Advances in Heat Transfer, Volume 7

Advances in Heat Transfer, Volume 9

Advances in Heat Transfer, Volume 17

Advances in Heat Transfer, Volume 36

Advances in Heat Transfer, Volume 16

Advances in Heat Transfer, Volume 8

Advances in Heat Transfer, Volume 33

Advances in Heat Transfer, Volume 13

Advances in Heat Transfer, Volume 4

Advances in Heat Transfer, Volume 1

Advances in Heat Transfer, Volume 19

Advances in Heat Transfer, Volume 23

Advances in Heat Transfer, Volume 18

Advances in Heat Transfer, Volume 31

Advances in Heat Transfer, Volume 21

Advances in Heat Transfer, Volume 10

Advances in Heat Transfer, Volume 26

Advances in Numerical Heat Transfer

Advances in Heat Transfer, Volume 28: Transport Phenomena in Materials Processing

Advances in Heat Transfer Transport Phenomena in Materials Processing

Advances in Heat Transfer, Volume 39 (Advances in Heat Transfer)

Advances in Heat Transfer, Volume 35 (Advances in Heat Transfer)

Advances in Heat Transfer, Volume 34 (Advances in Heat Transfer)

Advances in Heat Transfer, Volume 37 (Advances in Heat Transfer)

Transport Phenomena in Fires (Developments in Heat Transfer)

Transport Phenomena In Fuel Cells (Developments in Heat Transfer)

Advances in Heat Transfer, Volume 30: Transport Phenomena in Crystal Growth

Advances in Transport Phenomena: 2010

Advances in Heat Transfer, Volume 25

Advances in Heat Transfer, Volume 36

Advances in Heat Transfer, Volume 36

Advances in Heat Transfer, Volume 7

Advances in Heat Transfer, Volume 9

Advances in Heat Transfer, Volume 17

Advances in Heat Transfer, Volume 36

Advances in Heat Transfer, Volume 16

Advances in Heat Transfer, Volume 8

Advances in Heat Transfer, Volume 33

Advances in Heat Transfer, Volume 13

Advances in Heat Transfer, Volume 4

Advances in Heat Transfer, Volume 1

Advances in Heat Transfer, Volume 19

Advances in Heat Transfer, Volume 23

Advances in Heat Transfer, Volume 18

Advances in Heat Transfer, Volume 31

Advances in Heat Transfer, Volume 21

Advances in Heat Transfer, Volume 10

Advances in Heat Transfer, Volume 26

Advances in Numerical Heat Transfer

Recommend Documents