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**1. Homework Statement**

A cylinder has lenght [itex] L [/itex], inner diameter [itex] R_1 [/itex] and outer diameter [itex] R_2 [/itex]. The temperature on the inner cylinder surface is [itex] T_1 [/itex] and on the outer cylinder surface [itex] T_2 [/itex]. There is no temperature variation along the cylinders lenght-axis. Assume that the heat conductivity [itex] k [/itex] is temperature dependent and given by

[tex] k = aT^{\nu} [/tex]

where [itex] a [/itex] is a constant. Find [itex] T(r), r > 0 [/itex].

**2. Homework Equations**

Fourier's law

[tex] \boldsymbol{j} = -k \nabla T [/tex]

Temperature gradient

[tex] \nabla T = \frac{dT}{dr} \hat{e_r} [/tex]

where [itex] \hat{e_r} [/itex] is a unit vector in radial direction.

**3. The Attempt at a Solution**

The stationary heat flow outwards is

[tex] \dot{Q} = -k\frac{dT}{dr}2\pi rL [/tex]

rearranges to

[tex] dT = -\frac{\dot{Q}}{2\pi kL}\frac{dr}{r} [/tex]

integration from [itex] r_1 [/itex] to [itex] r [/itex] gives

[tex] T - T_1 = ??? [/tex]

Not sure what to do here when [itex] k [/itex] is not constant.