3.40 A truncated solid cone is of circular cross section, and its diameter is related to the axial coordinate by an expression of the form D=a x3 /2, where

a=1.0m1/2

The sides are well insulated, while the top surface of the cone at x1 is maintained at T 1 and the bottom surface at x2 is maintained at T 2.

a) Obtain an expression for the temperature distribution T(x).b) What is the rate of heat transfer across the cone if it is constructed of

pure aluminum with x1= 0,075 m, T 1=100oC, x2=0,225 m, and T 2=20oC.

3.57 A composite cylindrical wall is composed of two materials of thermal conductivity k A and k B, which are separated by a very thin, electric resistance heater for which interfacial contact resistances are negligible.

Liquid pumped through the tube is at a temperature T ∞,i and provides a convection coefficient hi at the inner surface of the composite. The outer surface is exposed to ambient air, which is at T ∞,o and provides a convection

coefficient of ho. Under steadystate conditions, a uniform heat flux of qh' ' is

dissipated by the heater.

a) Sketch the equivalent thermal circuit of the system and express all resistances in terms of relevant variables.

b) Obtain an expression that may be used to determine the heater temperature, T h.

c) Obtain an expression for the ratio of heat flows to the outer and inner

fluids qo' , q i

'. How might the variables of the problem be adjusted to

minimize this ratio?

3.73 A The energy transferred from the anterior chamber of the eye through the cornea varies considerably depending on whether a contact lens is worn. Treat the eye as a spherical system and assume the system to be at steady state. The convection coefficient h0 is unchanged with and without the contact lens in place. The cornea and the lens cover one-third of the spherical surface area.

Values of the parameters representing this situation are as follows:

r1= 10,2 mm k 1= 0,35 W/m∙ K r2= 12,7 mm k 2= 0,80 W/m∙ Kr3= 16,5 mm ho= 6 W/m2∙ KT ∞,o= 21oC h2= 12 W/m2∙ KT ∞,i= 37oC

a) Construct the thermal circuits, labeling all potentials and flows for the systems excluding the contact lens and including the contact lens. Write resistance elements in terms of appropriate parameters.

b) Determine the heat loss from the anterior chamber with and without the contact lens in place.

c) Discuss the implication of your results.

3.79 The air inside a chamber at T ∞,i=50oC is heated convectively with hi= 20 W/m2 ∙K by a 200-mm-thick wall having a thermal conductivity of 4 W/m ∙K and a uniform heat generation of 1000 W/m3. To prevent any heat generated within the wall from being lost to the outside of the chamber at T ∞,o=25oC with ho= 5 W/m2∙K, a very thin electrical strip heater is placed on the outer

wall to provide a uniform heat flux, qo' ' .

a) Sketch the temperature distribution in the wall on T- X coordinates for the condition where no heat generated within the wall is lost to the outside of the chamber.

b) What are the temperatures at the wall boundaries, T(0) and T(L), for the conditions of part (a)?

c) Determine the value of qo' ' that must be supplied by the strip heater so that

all heat generated within the wall is transferred to the inside of the chamber.

d) If the heat generation in the wall were switched off while the heat flux to the strip heater remained constant, what would be the steady-state temperature, T(0), of the outer wall surface?

3.106 Consider the plane wall, long cylinder, and sphere shown schematically, each with the same characteristic length a, thermal conductivity k , and uniform volumetric energy generation rate q̇

a) On the same graph, plot the steady-state dimensionless temperature, [T(x or r) - T(a)]/[(q̇. a2)/2k], versus the dimensionless characteristic length, x /a or r / a, for each shape.

b) Which shape has the smallest temperature difference between the center and the surface? Explain this behavior by comparing the ratio of the volume-to-surface area.

c) Which shape would be preferred for use as a nuclear fuel element? Explain why.

Answer :

3.40 (3.30)

(a)4 qxdx

π a2 x3=−k dT

4 qxπ a2

∫x1

xdxx3

=−k∫T 1

T

dT

4 qxπ a2 [ 12x2 ]=−k (T−T 1 )