Solution to Homework Problem 6

The heat flux in the rod and the outward heat flow.

Now that we have obtained the temperature distribution in the rod Equation (3.16), the heat flux at any location is determined using Fourier's law. The heat flux (W/m2) is

(3.17)

The outward directed component of the vector at z = 0 and z = L and respectively
(3.18)
(3.19)
where is the outward drawn normal to the surface. the net outflow from the rod is then
(3.20)
where Q  = net outflow of heat from the rod, watts
  Ac = cross sectional area of rod, m2

Start with Equation (3.20).

To determine dT/dz we have to differentiate the temperature profile with respect to z. The temperature distribution is given by equation (3.16)

where
Differentiating equation (3.16)

(3.16a)
At z = 0 and z = L.

(3.16b)
(3.16c)

Inserting Equations (3.16b) and (3.16c) into equation (3.20) and simplifying we obtain
(3.16d)

The coefficients are
 
 
 
 
 
 
 
 
 
 
 

The positive sign says that the net heat flow is out of the rod.

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