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ǣǤ Q1. Heat loss from a rectangular fin: Calculate the heat loss from a rectangular fin for the following conditions:
Solution
Q2. Maximum temperature in a lubricant. An oil is acting as a lubricant for a pair of cylindrical surfaces such as those shown in Fig. The angular velocity of the outer cylinder is 7908 rpm. The outer cylinder has a radius of 5.06 cm, and the clearance between the cylinders is 0.027 cm. What is the maximum temperature in the oil if both wall temperatures are known to be 158°F? The physical properties of the oil are assumed constant at the following values:
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Q3. Current-carrying capacity of wire: A copper wire of 0.040 in. diameter is insulated uniformly with plastic to an outer diameter of 0.12 in. and is exposed to surroundings at 100°F. The heat transfer coefficient from the outer surface of the plastic to the surroundings is 1.5 Btu/hr ft2 . F. What is the maximum steady current, in amperes, that this wire can carry without heating any part of the plastic above its operating limit of 200°F? The thermal and electrical conductivities may be assumed constant at the values given here:
Solution
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Q4. Temperature rise in an electrical wire. (a) A copper wire, 5 mm in diameter and 15 ft long, has a voltage drop of 0.6 volts. Find the maximum temperature in the wire if the ambient air temperature is 25°C and the heat transfer coefficient h is 5.7 Btu/hr. ft2. oF. (b) Compare the temperature drops across the wire and the surrounding air.
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Q5. Heat conduction from a sphere to a stagnant fluid: A heated sphere of radius R is suspended in a large, motionless body of fluid. It is desired to study the heat conduction in the fluid surrounding the sphere in the absence of convection. (a) Set up the differential equation describing the temperature Tin the surrounding fluid as a function of r, the distance from the center of the sphere. The thermal conductivity k of the fluid is considered constant. (b) Integrate the differential equation and use these boundary conditions to determine the integration constants: (c) From the temperature profile, obtain an expression for the heat flux at the surface. Equate this result to the heat flux given by "Newton's law of cooling" and show that a dimensionless heat transfer coefficient (known as the Nusselt number) is given by
in which D is the sphere diameter. This well-known result provides the limiting value of Nu for heat transfer from spheres at low Reynolds and Grashof numbers. (d) In what respect are the Biot number and the Nusselt number different? Solution
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Q6. Viscous heating in slit flow. Find the temperature profile for the viscous heating problem shown in Fig., when given the following boundary conditions: at x = 0, T = T0; at x = b, qx = 0.
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Q7. Heat conduction in an annulus: (a) Heat is flowing through an annular wall of inside radius r0 and outside radius r1 The thermal conductivity varies linearly with temperature from ko at To to k1 at T1. Develop an expression for the heat flow through the wall. (b) Show how the expression in (a) can be simplified when (r0 - r1)/ r0 is very small. Interpret the result physically.
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Q8. Forced-convection heat transfer in flow between parallel plates: A viscous fluid with temperatureindependent physical properties is in fully developed laminar flow between two flat surfaces placed a distance 2B apart. For z < 0 the fluid temperature is uniform at T = T1,. For z > 0 heat is added at a constant, uniform flux qo at both walls. Find the temperature distribution T(x, z) for large z. (a) Make a shell energy balance to obtain the differential equation for T(x, z). Then discard the viscous dissipation term and the axial heat conduction term. (b) Recast the problem in terms of the dimensionless quantities
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Q9. Heat loss from a circular fin: (a) Obtain the temperature profile T(r) for a circular fin of thickness 2B on a pipe with outside wall temperature To. Make the same assumptions that were made in the study of the rectangular fin. (b) Derive an expression for the total heat loss from the fin.
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Q10. Radial temperature gradients in an annular chemical reactor. A catalytic reaction is being carried out at constant pressure in a packed bed between coaxial cylindrical walls with inner radius ro and outer radius r1. Such a configuration occurs when temperatures are measured with a centered thermowell, and is in addition useful for controlling temperature gradients if a thin annulus is used. The entire inner wall is at uniform temperature To, and it can be assumed that there is no heat transfer through this surface. The reaction releases heat at a uniform volumetric rate Sc, throughout the reactor. The effective thermal conductivity of the reactor contents is to be treated as a constant throughout. (a) By a shell energy balance, derive a second-order differential equation that describes the temperature profiles, assuming that the temperature gradients in the axial direction can be neglected. What boundary conditions must be used? (b) Rewrite the differential equation and boundary conditions in terms of the dimensionless radial coordinate and dimensionless temperature defined as WĂŐĞϭϬŽĨϭϮ
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Explain why these are logical choices. (c) Integrate the dimensionless differential equation to get the radial temperature profile. To what viscous flow problem is this conduction problem analogous? (d) Develop expressions for the temperature at the outer wall and for the volumetric average temperature of the catalyst bed. (e) Calculate the outer wall temperature when ro = 0.45 in., r1 = 0.50 in., keff = 0.3 Btu/hr. ft.° F, To = 900°F, and Sc = 4800 cal/hr cm3. (f) How would the results of part (e) be affected if the inner and outer radii were doubled? Answer: (e) 888°F
Solution:
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