Solution - Fundamentals of Heat and Mass Transfer 6th Edition

PROBLEM 1.1 KNOWN: Thermal conductivity, thickness and temperature difference across a sheet of rigid extruded insulation. FIND: (a) The heat flux through a 2 m × 2 m sheet of the insulation, and (b) The heat rate through the sheet. SCHEMATIC: A = 4 m2

k = 0.029

W m ⋅K

qcond T1 – T2 = 10˚C

T1

T2 L = 20 mm x

ASSUMPTIONS: (1) One-dimensional conduction in the x-direction, (2) Steady-state conditions, (3) Constant properties. ANALYSIS: From Equation 1.2 the heat flux is q′′x = -k

T -T dT =k 1 2 dx L

Solving, q"x = 0.029 q′′x = 14.5

W 10 K × m⋅K 0.02 m

W m2


12 and τ < τ* = 70 deg, h is given by Eq. 9.54. With a Rayleigh number of Ra L =

gβ ( T1 − T2 ) L3

αν

⎡

Nu L = 1 + 1.44 ⎢1 −

⎣

=

(

9.8 m / s 2 0.0031 K −1

) ( 40°C)( 0.03 m )3 = 69, 600

25.9 × 10−6 m 2 / s × 18.2 × 10−6 m 2 / s

⎤ ⎡ 1708 ( 0.923) ⎤ ⎡⎛ 0.5 × 69, 600 ⎞1/ 3 ⎤ ⎟ − 1⎥ ⎥ ⎢1 − ⎥ + ⎢⎜ 0.5 ( 69, 600 ) ⎦ ⎣ 0.5 ( 69, 600 ) ⎦ ⎢⎝ 5830 ⎠ ⎥⎦ ⎣ 1708

Nu L = 1 + 1.44 [ 0.951][ 0.955] + 0.814 = 3.12

h = ( k / L ) Nu L = ( 0.028 W / m ⋅ K / 0.03 m ) 3.12 = 2.91 W / m 2 ⋅ K

(

q conv = 2.91 W / m 2 ⋅ K 4 m 2

) ( 70 − 30) °C = 466 W