Question: A cylindrical shell has inner radius r 1 , outer radius r 2 , and length L . a . The 1 - D ,

A cylindrical shell has inner radius

r_{1},

outer radius

r_{2},

and length

L .

.

The

1 -

,

steady

-

state, constant properties heat diffusion energy equation in a cylinder with no energy generation is

\frac{1}{r} \frac{d}{d r} (r \frac{d T}{d r}) = 0

Solve this differential equation with boundary conditions

T (r_{1}) = T_{1}

and

T (r_{2}) = T_{2}

to show that the solution is

T (r) = T_{2} + \frac{l n (\frac{r}{r_{2}})}{l n (\frac{r_{1}}{r_{2}})} (T_{1} - T_{2})

.

Apply the Fourier law to determine an expression for the radial heat flow rate

q (r) .

.

Use your

q (r)

solution from b

.

to show that the thermal resistance of a cylindrical shell is

l n \frac{\frac{r_{2}}{r_{1}}}{2 k L} .

.

Plot

T (r)

for

T_{1} = 50 C, T_{2} = 22 C, r_{1} = 2 c m,

and

r_{2} = 12 c m .

A cylindrical shell has inner radius

r_{1},

outer radius

r_{2},

and length

L .

.

The

1 -

,

steady

-

state, constant properties heat diffusion energy equation in a cylinder with no energy generation is

\frac{1}{r} \frac{d}{d r} (r \frac{d T}{d r}) = 0

Solve this differential equation with boundary conditions

T (r_{1}) = T_{1}

and

T (r_{2}) = T_{2}

to show that the solution is

T (r) = T_{2} + \frac{l n (\frac{r}{r_{2}})}{l n (\frac{r_{1}}{r_{2}})} (T_{1} - T_{2})

.

Apply the Fourier law to determine an expression for the radial heat flow rate

q (r) .

.

Use your

q (r)

solution from b

.

to show that the thermal resistance of a cylindrical shell is

l n \frac{\frac{r_{2}}{r_{1}}}{2 k L} .

.

Plot

T (r)

for

T_{1} = 50 C, T_{2} = 22 C, r_{1} = 2 c m,

and

r_{2} = 12 c m .

Plot using Matlab.

A cylindrical shell has inner radius r 1 , outer

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