if the thickness of the plate is L and its thermal conductivity k is constant. T

Question

if the thickness of the plate is L and its thermal conductivity k is constant. The plane wall shown in Fig. 1.12 has one surface maintained at T1 and the other at T2. The temperature at the center plane is measured to be T3. And the rate of heat flow through the wall is q. Assuming that the thermal conductivity of the wall varies linearly with temperature, find an expression for the thermal conductivity as a function of temperature and the rate of heat flow through the wall. Show that the gradient of the temperature distribution. T , at point P shown in Fig. 1.5 as a vector normal to the isothermal surface passing through P and pointing in the direction of

Explanation / Answer

consider an element of thickness dx at a distance x from the T1 end of the wall

temperature difference across it to be dT

so by fourier's law

q/A=-k(T)*dT/dX

where k=mT+c (since it varies linearly with temp)

(q/A)*dX=-(mT+c)dT

on integrating for(x from 0 to L & T from T1 to T3)

q*L/A=m(T12-T32)/2 + c(T1-T3)

q*L/(A*(T1-T3)) = m(T1+T3)/2 + c

similarly integrating for(X from L to 2L & T from T3 to T2 ) we get

q*L/(A(T3-T2)) = m(T3+T2)/2 +c

solving above two equations we get

m= -2qL*(T1+T2)/A*((T1-T3)(T3-T2)(T1-T2))

c=qL*(T12+T22)/A*(T1-T3)(T3-T2)(T1-T2)

thus k=mT+c

rate of heat floe through the wall is given to be q

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