# Show that the steady state temperature distribution in a cylindrical spine is the solution to: where

Show that the steady state temperature distribution in a

cylindrical spine is the solution to:

where mc = (Phc/kA)0.5,

and mr = (Phr/kA)0.5. Use

the given data to find the temperature distribution at steady state in the

above fin. Data: L = 15 cm, D = 1 cm, TB = 350

C, Tf = 27 C, k = 50 W/m·C, hc = 15

W/m2 ·C, _ = 0.75. The tip of the fin is losing energy

by both radiation and convection.

[Hint: You may solve this problem analytically or

numerically. The analytical solution follows the method described in Section 8

of Chapter IVa. Since we have linearized the differential equation

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Show that the steady state temperature distribution in a

cylindrical spine is the solution to:

where mc = (Phc/kA)0.5,

and mr = (Phr/kA)0.5. Use

the given data to find the temperature distribution at steady state in the

above fin. Data: L = 15 cm, D = 1 cm, TB = 350

C, Tf = 27 C, k = 50 W/m·C, hc = 15

W/m2 ·C, _ = 0.75. The tip of the fin is losing energy

by both radiation and convection.

[Hint: You may solve this problem analytically or

numerically. The analytical solution follows the method described in Section 8

of Chapter IVa. Since we have linearized the differential equation by choosing

the heat transfer coefficient, as shown in Problem 38, an iterative solution is

required regardless of the solution method we choose (hr,

therefore, mr are unknowns). The initial guess to estimate hr

for thermal radiation is obtained by ignoring thermal radiation (i.e., by

setting mr = 0). Upon obtaining the initial guess for

temperature, we then include thermal radiation but evaluate hr at the

temperature calculated in the previous iteration. At the end of each trial, we

find:

where k is the iteration index. The iteration is

terminated when _ _ _s where _s

is the specified convergence criterion].

Problem 38

A cylindrical spine used as a fin to enhance the rate of

heat transfer from a surface. Our goal is to find the one-dimensional

temperature distribution in the spine. Write the conservation equation of

energy for the control volume shown in the figure. Show that temperature at

every location on this fin and at every time is found from the solution to:

where = T(,

t) Tf, a = _c/k, mc

= (Phc/_Ac)0.5, and mr

= (Phr/_Ac)0.5. In these relations, hc

is the heat transfer coefficient for heat transfer by convection, and hr

is the heat transfer coefficient for heat transfer by radiation. Note that hr

is defined as:

Specify the required initial and boundary conditions.

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