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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