A 4m times 6m plane slab (thermal conductivity k, thickness L, surface emmissivi
ID: 2324582 • Letter: A
Question
A 4m times 6m plane slab (thermal conductivity k, thickness L, surface emmissivity ep and surface temperature Ts) is exposed to a large wall enclosure at Tw and air at Ta. The slab surface interacts with the enclosed wall by radiation and with the air by convection. Temperature at the bottom side of the slab is T1. The slab does not generate heat. A. Let the slab k=1.17 W/m.k, L=0.03 m, ep=0.75, Tw=25C, ew=0.7, Ta=217 C, h=30 W/m^2.K. Find the steady state temperature of the slab Ts. Let the slab k=12 W/m.K, L=0.1 m, ep=0.75, Ts = 217C and Tw=25C, Ta=25C, h=6 W/m^2.K. The slab base is kept at T1 by heating coils. Find qr (radiation from the surface to the wall), qc (convection from the surface of the air) and the base temperature T1 under steady state condition. We let 0 C = 273 K.Explanation / Answer
Energy Balance
Energy gain from radiation= Energy gain Slab
Energy gain by radiation= Energy from sun radiation (Qs)+Energy from wall Radiation(Qw) -Energy from slab emissivity (Qslab)
As in problem size wall is mention large
Therefore Qs=0
Qw=ewAsTw4
s= Stefan Boltzman constant, 5.67x10-8 W/m2×K4
Qslab=epAsTs4
Energy gain Slab=Heat transfer Convection+ Heat transfer Conduction
Heat transfer Convection=hA(Ts-Ta)
Heat transfer Conduction=kA(Ts-Tl)/L
A=6x4 m2
Part A
Heat transfer Conduction=0
hA(Ts-Ta)= ewAsTw4- epAsTs4
Therefore adjusting eqn
Ta=273+217=490 K, Tw=25+273=298 K
30(Ts-490)=(0.7x(298)4-0.75x(Ts)4)x5.67x10-8
5291005291Ts-2592592592592.59+0.75(Ts)4-5520305291=0
5291005291Ts +0.75(Ts)4-2598112897883.79=0
Using numerical technique
Trapezoidal rule
Guess Ts value 480 & 490 after 4 iteration
Final Ts= 483.3 K =210.3° C
Part B
Radiation from surface to wall= Qslab=epAsTs4
=0.75x24x5.67x10-8 (217+273)4
= 5883.556 W
Convection from surface to air= hA(Ts-Ta)
= 6x24x(490-298)
= 27648 w
Heat generated by coil= -kA(Ts-Tl)/L
kA(Ts-Tl)/L+ hA(Ts-Ta)+ epAsTs4=0
Therefore
Tl= (hA(Ts-Ta)+ epAsTs4)/kA+Ts
=606.43 K
=333.43 °C
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