Question

The steady-state temperature distribution inside a solid object is described by the following expression, where x is the spatial co-ordinate: T(x) =2x³- 3x² + x +10

If the thermal conductivity and thickness of the solid are 10 W.m^-1.K^-1 and 0.8 m respectively, what will be the form of heat flux expression?

At what point within the solid does the heat flux reach a maximum (or minimum)?

Answer 1

Part a

From the Fouriers law of heat conduction at steady state,

Heat flux

Q = - kdT/dx

Q = - k d(2x³ - 3x² + x +10) /dx

Do the differentiation and we get

Q = - k (6x² - 6x + 1)

The above equation is the form of heat flux expression

Put the value of thermal conductivity k = 10 W/m-K

And thickness x = 0.8 m

Q = - 10*(6*0.8*0.8 - 6*0.8 + 1)

Q = - 0.4 W/m2

Part b

From the expression of heat flux

Q = - k (6x² - 6x + 1)

Do the differentiation

dQ/dx = - kd(6x² - 6x + 1)/dx

To find the maximum or minimum, Slope = 0

dQ/dx = - k(12x - 6) = 0

12x = 6

x = 0.5

Now do the second derivative

d²Q/dx² = - 12k

d²Q/dx² < 0

At x = 0.5 the heat flux would be maximum.

Maximum heat flux

Qmax = - k (6x² - 6x + 1)

= - 10*(6*0.5*0.5 - 6*0.5 + 1)

= - 5 W/m2

Negative sign shows that temperature decreases with x.