Question

A one-dimensional plane wall of thickness 2L=80 mm experiences uniform thermal generation of q= 1000 W/m^3 and is convectively cooled at x=±40 mm by an ambient fluid characterized byT infinity=20degreesC. If the steady-state temperature distribution within the wall is T(x) = a(L2-x2)+b where a=15oC/m2and b=40oC, what is the thermal conductivityof the wall? What is the value of the convection heat transfer coefficient?

Answer 1

`h=1.99998\ W/m^2.C`

`k=33.333\ W/m.C`

Step-by-step explanation:

Considering the one dimensional and steady state:

From Heat Conduction equation considering the above assumption:

`(\partial^2T)/(\partial x^2)+(\dot e_(gen))/(k)=0` Eq (1)

Where:

k is thermal Conductivity

`\dot e_(gen)` is uniform thermal generation

`T(x) = a(L^2-x^2)+b`

`(\partial\ T(x))/(\partial x)=(\partial\ a(L^2-x^2)+b)/(\partial x)=-2ax\\(\partial^2\ T(x))/(\partial x^2)=(\partial^2\ -2ax)/(\partial x^2)=-2a`

Putt in Eq (1):

`-2a+(\dot e_(gen))/(k)=0\\ k=(\dot e_(gen))/(2a)\\ k=(1000)/(2*15)\\ k=33.333\ W/m.C`

Energy balance is given by:

`Q_(convection)=Q_(conduction)`

`h(T_L-T_(inf))=-k((dT)/(dx)) _L` Eq (2)

`T(x) = a(L^2-x^2)+b`

Putting x=L

`T(L) = a(L^2-L^2)+b\\T(L)=b\\T(L)=40^oC`

`(dT)/(dx)=(d(a(L^2-x^2)+b)/(dx)=-2ax\\Put\ x\ =\ L\\(dT)/(dx)=-2aL\\((dT)/(dx))_L=-2*15*0.04=-1.2`

From Eq (2)

`h=(-k*-1.2)/((40-20)) \\h=(-33.333*-1.2)/((40-20))\\h=1.99998\ W/m^2.C`