Question

The heat generation rate in a plane wall of 0.24 m thickness is 0.4 MW/m^3. The wall is exposed on both sides to convection at 30°C.
a) Determine and compare the maximum temperatures for k=25, k=50, k=200 and k= 410 W/mK assuming h=250W/m^2 K.
b) Determine and compare the maximum temperatures for h=50, 250, 500 and 1000 W/m^2K with k =W/mK

Answer 1

t=0-24m Is • Igen agen = 0.4 W/m² Too = 30°c Too h As both sides conditions are same, then we can assume that both sides and temperature would be same. Too h ik + Igen x= - t to x = t - - For ID Plane wall with heat generation d²T dx² Boundary conditions At T=Ts On entegration, we get T(x) = agen x² + 4x + HE- ---- At X= T=Ts 4 2 2K At X3 At x= x= 2 ts = Lamm = 54:)+5 -0 Ts = agen (45) +6-2 ak 2 + We U - & from O get c = 0 get = Ts - qgen (t² wo + 2K 4 :. Tbx)= Igen x² + Ts-ggen (t² here (n=&J + T 2k 2k 4 T(x)= qgen. X +² +Ts 2k surrounding through convection As heat is dissipating to the h (Ts - Too) agent & Too I gan Xt Ta from we get A $ T(4) 2 ť) + Igen t + Too Ik h

Date: As we know maximum temperature should be surrounding because of Therefore, at X=0 or centre of wall same On both sides. ca) when h= 250 W/m² K & K = 25 w/mk Τζο) = #Julient + Too 2K h TO) = - 0.4x10°x (0-24) 0.4x1080.24 + + 30 2x 25X4 250 TO) 20 + + 30 250 T (0) = -115.2+ 384 +30 298.8°C When h= 250 W/m²K & k=50 wink TO) = 0:4X106 x (0.24) 0.4X108 80.24 2x 50X4 250 T(o) = -57.6 + 384 +30 [To) = 356.4°C] When h= 250 W/m²k & k = 200 wlonk T(o)= -0.4X106 x (0.24)? 0.4X106x0.24 + 2X200 X 4 - 14.4 + 384 +30 399.6°C When h= 250 W/m²k & k = 410 wlink TO) 4060975 °c When h= 50 wink fk = 25 Wink T(0) = -0.4 X106x(0.24% 0-4 X10 X 0.24 + 2 x 25x4 To) 115.2 + 1920+ 30 1884.8°C When h= 250 wlikk &k= 25W/mk t)298.8°c) When h = 500 wink & k = 25 W/mk Ito) (6) + 30 so TO) = 106.8°c when h= looo W/m²K TO) = 10.8°c tk = 25 w/mk
As we can see from the calculatios that at constant convective heat transfer coeffiicient and increasing conductive heat transfer coefficient the temperature is increasing slowly while at constant conductive heat transfer coefficient and increasing convective heat transfer coefficient the temperature is decreasing at a very fast rate because of faster convection heat transfer.