Explanation Given information: The thickness of the steel plate is 20 mm , the emissivity of the outer surface is 0.30 , the heat flux is 5 kW / m 2 , the thermal conductivity is 25 W / m ⋅ K , the convective heat transfer coefficient is 10 W / m 2 ⋅ K and the surrounding air temperature is 20 ° C . The schematic diagram of the system is shown below: Figure-(1) Write the expression for the net heat flux. q ˙ 0 = h ( T L − T surr ) + ε σ ( T L 4 − T surr 4 ) ...... (I) Here, the net heat flux is q ˙ 0 , the convective heat transfer coefficient is h , the emissivity of the plate is ε , the Stefan’s Boltzmann constant is σ , the outer surface temperature is T L , the temperature of surrounding is T surr . Write the expression for the Fourier law of heat conduction. q ˙ 0 = − k d T d x d T d x = − q ˙ 0 k ...... (II) Here, the temperature is T . Calculation: Integrate Equation (II) on the both side. ∫ ( d T d x ) = ∫ ( − q ˙ 0 k ) T x = − q ˙ 0 k x + C 1 ...... (III) Here, the arbitrary constant is C 1 . Apply boundary conditions in Equation (III) to find the arbitrary constant C 1 . Substitute L for x and T L for T x in Equation (III). T L = − q ˙ 0 k L + C 1 C 1 = T L + q ˙ 0 k L Substitute ( T L + q ˙ 0 k L ) for C 1 in Equation (III). T x = − q ˙ 0 k x + ( T L + q ˙ 0 k L ) = q ˙ 0 k ( L − x ) + T L ...... (IV) Substitute 0 for x and T 0 for T x in Equation (IV). T 0 = q ˙ 0 k ( L − 0 ) + T L = q ˙ 0 k L + T L ...... (V) Substitute 5 kW / m 2 for q ˙ 0 , 10 W / m 2 ⋅ K for h , 20 ° C for T surr , 0.30 for ε and 5.67 × 10 − 8 W / m 2 ⋅ K 4 for σ in Equation (I). ( 5 kW / m 2 ) = [ ( 10 W / m 2 ⋅ K ) { T L − ( 20 ° C ) } + ( 0.30 ) ( 5.67 × 10 − 8 W / m 2 ⋅ K 4 ) { T L 4 − ( 20 ° C ) 4 } ] ( 5 × 10 3 W / m 2 ) = [ ( 10 W / m 2 ⋅ K ) { T L − ( 20 + 273 ) K } + ( 0.30 ) ( 5.67 × 10 − 8 W / m 2 ⋅ K 4 ) { T L 4 − { ( 20 + 273 ) K } 4 } ] ( 5 × 10 3 W / m 2 ) = [ ( 10 W / m 2 ⋅ K ) { T L − ( 293 K ) } + ( 1.701 × 10 − 8 W / m 2 ⋅ K 4 ) { T L 4 − ( 293 K ) 4 } ] ( 5 × 10 3 W / m 2 ) = [ ( 10 W / m 2 ⋅ K ) T L − ( 2930 W / m 2 ) + ( 1

6th Edition

ISBN: 9780073398198

Author: CENGEL

Publisher: RENT MCG

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Chapter 2, Problem 67P

To determine

The inside surface temperature of the furnace front.

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Chapter 2 Solutions

HEAT+MASS TRANSFER:FUND.+APPL.

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