ME 353 Heat Transfer 1

M.M. Yovanovich

P1MT97.MWS

Solution to Problem 1 of Midterm Exam, October 1997.

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An aluminum spherical shell having radii r1 and r2 > 1; and

thermal conductivity k. A heater is located inside the aluminum

shell for steady-state testin of insulation materials which are

cast on the outer surface to a thickness t. The system is located

in a large room. The combined outer heat transfer coefficient is

given along with the heater power. The air and room temperatures

are equal. The thermal contact resistance is negligible.

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> restart:

a) Sketch the system which consists of two spherical shells in perfect

contact. The is a thermal source inside the inner shell and the outer

surface of the outer shell has a uniform combined heat transfer coefficient.

b) The equivalent thermal circuit consists of two solid resistances:

Rs1 and Rs2, and a combined film resistance in series arrangement.

The temperature nodes are T1 at r1, T2 at r2 , T3 at r = r2 + t, and

the air Tf and surroundings temperature Tsurr.

The throughput is the heater power Q.

The heat flow rate Q splits into two streams at the outer surface:

a convection stream Qconv and a radiation stream Qrad.

c) Calculate the thermal conductivity of the insulation.

Define the system parameters.

> sys:= (r1 = .15, r2 = .18, t = .12, T1 = 250,
Tf = 20, ho = 30, k1 = 220, Q = 80);

Relation for inner surface temperature.

> T1:= Tf + Q*Rsystem;

> Rsystem:= Rs1 + Rs2 + Rf2;

> Rs1:= 1/(4*Pi*k1)*(1/r1 - 1/r2);

> Rs2:= 1/(4*Pi*k2)*(1/r2 - 1/r3);

> Rf2:= 1/(4*Pi*r3^2*ho);

> r3:= r2 + t;

Substitute the given parameter values into the temperature relation to

get an equation for the unknown conductivity k2.

> evalf(subs(sys, T1 = 250));

Solve for k2.

> k2:= evalf(solve(%, k2), 4)*W/(m*K);

d) Find the temperatures at the inner and outer surfaces of the insulation.

> restart:

Define the system parameters, adding the conductivity of the insulation.

> sys2:=
(r1 = 0.15, r2 = 0.18, t = 0.12, T1 = 250,
Tf = 20, ho = 30, k1 = 220, k2 = 0.06217, Q = 80);

Write the temperature relations for the unknown values T2 and T3.

> T2:= Tf + Q*(Rs2 + Rf2);

> T3:= Tf + Q*Rf2;

> Rs2:= 1/(4*Pi*k2)*(1/r2 - 1/r3);

> Rf2:= 1/(4*Pi*r3^2*ho);

> r3:= r2 + t;

> T2:= evalf(subs(sys2, T2), 5)*C;

> T3:= evalf(subs(sys2, T3), 4)*C;

e) Given the emissivity, calculate the radiative coefficient, hrad.

> restart:

> hrad:= epsilon*sigma*(T3^2 + Tsurr^2)*(T3 + Tsurr);

Define the parameters.

> par:=
(epsilon = 0.87, sigma = 5.67e-8,
T3 = 273 + 22.36, Tsurr = 273 + 20);

> hrad:= evalf(subs(par, hrad), 4)*W/(m^2*K);