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Examples and Problems Problem 1.31 Chips of width L = 15 mm on a side are mounted to a substrate that is installed in an enclosure whose walls and air are maintained at a temperature of Tsur = T = 25oC. The chips have an emissivity of = 0.60 and a maximum allowable temperature of Ts = 85oC. a) If heat is rejected from the chips by radiation and natural convection, what is the maximum operating power of each chip? The convection coefficient depends on the chip-to-air temperature difference and may be approximated as h = C(Ts – T)1/4, where C = 4.2 W/m2.K5/4. Problem 1.31 (Contd.) b) If a fan is used to maintain air flow through the enclosure and heat transfer is by forced convection, with h = 250 W/m2.K, what is the maximum operating power? Problem 1.44 Radioactive wastes are packed in a long, thinwalled cylindrical container. The wastes generate thermal energy non-uniformly according to the relation q qo 1 r ro 2 , where q is the local rate of energy generation per unit volume, q o is a constant, and ro is the radius of the container. Steady-state conditions are maintained by submerging the container in a liquid that is at T and provides a uniform convection coefficient h. Obtain an expression for the total rate at which energy is generated in a unit length of the container. Use this result to obtain an expression for the temperature Ts of the container wall. Problem 1.63 A rectangular forced air heating duct is suspended from the ceiling of a basement whose air and walls are at a temperature of T = Tsur = 5oC. The duct is 15 m long, and its cross-section is 350 mm × 200 mm. a) For an uninsulated duct whose average surface temperature is 50oC, estimate the rate of heat loss from the duct. The surface emissivity and convection coefficient are approximately 0.5 and 4 W/m2.K, respectively. b) If heated air enters the duct at 58oC and a velocity of 4 m/s and the heat loss corresponds to the result of part (a), what is the outlet temperature? The density and specific heat of the air may be assumed to be r = 1.10 kg/m3 and cp = 1008 J/kg.K, respectively. Problem 2.22 Uniform internal heat generation q 5107 at W/m3 is occurring in a cylindrical nuclear reactor fuel rod of 50-mm diameter, and under steady-state conditions the temperature distribution is of the form T(r) = a + br2, where T is in degrees Celsius and r is in meters, while a = 800oC and b = -4.167 × 105 oC/m2. The fuel rod properties are k = 30 W/m.k, r = 1100 kg/m3, and cp = 800 J/kg.K. Problem 2.22 (contd.) a) What is the rate of heat transfer per unit length of the rod at r = 0 (the centerline) and r = 25 mm (the surface)? b) If the reactor power level is suddenly increased to q 2 108 W/m3, what is the initial time rate of temperature change at r = 0 and r = 25 mm? Problem 2.26 (a) & (b) One-dimensional, steady-state conduction with uniform internal energy generation occurs in a plane wall with a thickness of 50 mm and a constant thermal conductivity of 5 W/m.K. For these conditions, the temperature distribution has the form, T(x) = a + bx + cx2. The surface at x = 0 has a temperature of T(0) = To = 120oC and experiences convection with a fluid for which T = 20oC and h = 500 W/m2.K. The surface at x = L is well insulated. Problem 2.26 (contd.) a) Applying an overall energy balance to the wall, calculate the internal energy generation rate q . b) Determine the coefficients a, b, and c by applying the boundary conditions to the prescribed temperature distribution. Use the results to calculate and plot the temperature distribution. Problem 3.2 (a) The rear window of an automobile is defogged by passing warm air over its inner surface. a) If the warm air is at T,i = 40oC and the corresponding convection coefficient is hi = 30 W/m2.K, what are the inner and outer surface temperatures of 4-mm-thick window glass, if the outside ambient air temperature is T,o = -10oC and the associated convection coefficient is ho = 65 W/m2.K? Problem 3.29 The diagram shows a conical section fabricated from pure aluminum. It is of circular cross section having diameter D = ax1/2, where a = 0.5 m1/2. The small end is located at x1 = 25 mm and the large end at x2 = 125 mm. The end temperatures are T1 = 600 K and T2 = 400 K, while the lateral surface is well insulated. a) Derive an expression for the temperature distribution T(x) in symbolic form, assuming 1-D conditions. Sketch the temperature distribution. b) Calculate the heat rate qx. Example 3.5 The possible existence of an optimum insulation thickness for radial systems is suggested by the presence of competing effects associated with an increase in this thickness. In particular, although the conduction resistance increases with the addition of insulation, the convection resistance decreases due to increasing outer surface area. Hence there may exist an insulation thickness that minimizes heat loss by maximizing the total resistance to heat transfer. Resolve this issue by considering the following system. Example 3.5 (Contd.) 1. A thin-walled copper tube of radius ri is used to transport a low-temperature refrigerant and is at a temperature Ti that is less than that of the ambient air at T∞ around the tube. Is there an optimum thickness associated with application of insulation to the tube? Problem 3.73 Consider 1-D conduction in a plane composite wall. The outer surfaces are exposed to a fluid at 25oC and a convection heat transfer coefficient of 1000 W/m2.K. The middle wall B experiences uniform heat generation q B , while there is no generation in walls A and C. The temperatures at the interfaces are T1 = 261oC and T2 = 211oC. a) Assuming a negligible contact resistance at the interfaces, determine the volumetric heat generation q B and the thermal conductivity kB. Problem 3.73 (Contd.) b) Plot the temperature distribution, showing its important features. c) Consider conditions corresponding to a loss of coolant at the exposed surface of material A ( h = 0). Determine T1 and T2 and plot the temperature distribution throughout the system. Problem 3.101 A thin flat plate of length L, thickness t, and width W »L is thermally joined to two large heat sinks that are maintained at a temperature To. The bottom of the plate is well insulated, while the net heat flux to the top surface of the plate is known to have a uniform value of qo'' . a) Derive the differential equation that determines the steady-state temperature distribution T(x) in the plate. b) Solve the foregoing equation for the temperature distribution, and obtain an expression for the rate of heat transfer from the plate to the heat sinks. Problem 3.102 Consider the flat plate of problem 3.101, but with the heat sinks at different temperatures, T(0) = To and T(L) = TL, and with the bottom surface no longer insulated. Convection heat transfer is now allowed to occur between this surface and a fluid at T∞, with a convection coefficient h. a) Derive the differential equation that determines the steady-state temperature distribution T(x) in the plate. Problem 3.134 As more and more components are placed on a single integrated circuit (chip), the amount of heat that is dissipated continues to increase. However, this increase is limited by the maximum allowable chip operating temperature, which is approximately 75oC. To maximize heat dissipation, it is proposed that a 4 × 4 array of copper pin fins be metallurgically joined to the outer surface of a square chip that is 12.7 mm on a side. Problem 3.134 (contd.) Problem 3.134 (contd.) a) Sketch the equivalent thermal circuit for the pin-chip-board assembly, assuming onedimensional, steady-state conditions and negligible contact resistance between the pins and the chip. In variable form, label appropriate resistances, temperatures, and heat rates. b) For the conditions prescribed in Problem 3.27, what is the maximum rate at which heat can be dissipated in the chip when the pins are in place? That is, what is the value of qc for Tc = 75oC? The pin diameter and length are Dp = 1.5 mm and Lp = 15 mm. Problem 4.10 A pipeline, used for transport of crude oil, is buried in the earth such that its centerline is a distance of 1.5 m below the surface. The pipe has an outer diameter of 0.5 m and is insulated with a layer of cellular glass 100 mm thick. What is the heat loss per unit length of pipe under conditions for which heated oil at 120oC flows through the pipe and the surface of the earth is at temperature of 0oC? Problem 4.23 A hole of diameter D = 0.25 m is drilled through the center of a solid block of square cross section with w = 1 m on a side. The hole is drilled along the length, l = 2 m, of the block, which has a thermal conductivity of k = 150 W/m.K. The outer surfaces are exposed to ambient air, with T,2 = 25oC and h2 = 4 W/m2.K, while hot oil flowing through the hole is characterized by T,1 = 300oC and h1 = 50 W/m2.K. Determine the corresponding heat rate and surface temperatures. 2nd Major Exam (062) An aluminum heated plate is being cooled by air flowing over both sides and parallel to the plate as shown in the figure below with T=25oC and h=30 W/m2K. At time t = 0, the plate is 200C. i. Find the Biot number and check the validity of lumped analysis ii. Find the plate temperature at t=10sec. iii. Find the time rate of change of the plate temperature at t=0. (Hint: Neglect radiation) Properties of Aluminum: k=200W/m.K, c=900 J/kg.K, r=2700kg/m3. 2nd Major Exam (062) L=1m L=1m =5mm Aluminum Plate U=12m/s Air T=25C Problem 5.111 (modified) A plane wall of thickness 20 mm is insulated on the left face and subjected to convection condition on the right face, as shown below. Problem 5.111 (modified, contd.) a) Consider the 5-node network shown schematically. Write the implicit form of the finite-difference equations for the network and determine temperature distributions for t = 50, 100, and 500 s using a time increment of Δt = 1 s. b) Use the one-term approximation given in section 5.5 to obtain the temperature at the same location and times as in (a). Compare the two results. Solution of Problem 5.111 x L / 4 0.020 / 4 0.005m h x 500 0.005 Bi 16 k 15 Explicit method p 1 p p 2 FoT 1 2 FoT 1 2 1 Node 1: T Node 2: T p 1 p p p Fo T T 1 2 FoT 2 3 2 1 Node 3: T p 1 p p p Fo T T 1 2 FoT 3 4 3 2 Node 4: T p 1 p p p Fo T T 5 1 2 FoT 4 4 3 Node 5: p 1 p p T5 2 FoT 1 2 Fo 2 Bi FoT 5 2 Bi FoT 4 Stability condition: 1 2 Fo 0 & 1 2 Fo 2 Bi Fo 0 1 Fo 2 1 Bi t 2.45 Implicit method Node 1: 1 2 FoT 1p 1 2 FoT Node 2: 1 2 FoT 2p 1 p 1 p p 1 Fo T T T 1 3 2 Node 3: 1 2 FoT 3p 1 p 1 p p 1 Fo T T T 2 4 3 p 1 1 2 Fo T Node 4: 4 p 1 p p 1 Fo T T 5 T 3 4 p 1 1 2 Fo 2 Bi Fo T Node 5: 5 p 1 p T 2 1 2 FoT p 1 p T 5 2 Bi FoT 4 Problem 6.26 Forced air at T = 25oC and V = 10 m/s is used to cool electronic elements on a circuit board. One such element is a chip, 4 mm by 4 mm, located 120 mm from the leading edge of the board. Experiments have revealed that flow over the board is disturbed by the elements and that convection heat transfer is correlated by an expression of the form 0.85 1 3 Nu x 0.04 Re x Pr Estimate the surface temperature of the chip if it is dissipating 30 mW. Problem 8.13 Consider a cylindrical nuclear fuel rod of length L and diameter D that is encased in a concentric tube. Pressurized water flows through the annular region between the rod and the tube at a rate m , and the outer surface of the tube is well insulated. Heat generation occurs within the fuel rod, and the volumetric generation rate is known to vary sinusoidally with distance along the rod. That is q,x q o sin x L, where q o (W/m3) is a constant. A uniform convection coefficient h may be assumed to exist between the surface of the rod and the water. Problem 8.13 (Contd.) a) Obtain expressions for the local heat flux q’’(x) and the total heat transfer q from the fuel rod to the water. b) Obtain an expression for the variation of the mean temperature Tm(x) of the water with distance x along the tube. c) Obtain an expression for the variation of the rod surface temperature Ts(x) with distance x along the tube. Develop an expression for the x location at which this temperature is maximized. Problem 8.31 To cool a summer home without using a vapor compression refrigeration cycle, air is routed through a plastic pipe (k = 0.15 W/m.K, Di = 0.15 m, Do = 0.17 m) that is submerged in an adjoining body of water. The water temperature is nominally at T = 17oC, and a convection coefficient of ho = 1500 W/m2.K is maintained at the outer surface of the pipe. Problem 8.31 (contd.) If air from the home enters the pipe at a temperature of Tm,i = 29oC and volumetric flow rate of Vi = 0.025 m3/s, what pipe length L is needed to provide a discharge temperature of Tm,o = 21oC? What is the fan power required to move the air through this length of pipe if its inner surface is smooth? Problem 7.88 A tube bank uses an aligned arrangement of 30mm-diameter tubes with ST = SL = 60 mm and a tube length of 1 m. There are 10 tube rows in the flow direction (NL = 10) and 7 tubes per row (NT = 7). Air with upstream conditions of T = 27oC and V = 15 m/s is in cross flow over the tubes, while a tube wall temperature of 100oC is maintained by steam condensation inside the tubes. Determine the temperature of air leaving the tube bank, the pressure drop across the bank, and the fan power requirement. Problem 11.7 The condenser of a steam power plant contains N = 1000 brass tubes (kt = 110 W/m.K), each of inner and outer diameters, Di = 25 mm and Do = 28 mm, respectively. Steam condensation on the outer surfaces of the tubes is characterized by a convection coefficient of ho = 10,000 W/m2.K. a) If cooling water from a large lake is pumped through the condenser tubes at m c 400kg / s , what is the overall heat transfer coefficient Uo based on the outer surface area of a tube? Properties of the water may be approximated as μ = 9.60 × 10-4 N.s/m2, k = 0.60 W/m.K, and Pr = 6.6. Problem 11.7(contd.) b) If, after extended operation, fouling provides a resistance of R''f,i = 10-4 m2.K/W, at the inner surface, what is the value of Uo? c) If water is extracted from the lake at 15oC and 10 kg/s of steam at 0.0622 bars are to be condensed, what is the corresponding temperature of the water leaving the condenser? The specific heat of the water is 4180 J/kg.K. Problem 11.44 A shell-and-tube heat exchanger is to heat 10,000 kg/h of water from 16 to 84oC by hot engine oil flowing through the shell. The oil makes a single shell pass, entering at 160oC and leaving at 94oC, with an average heat transfer coefficient of 400 W/m2.K. The water flows through 11 brass tubes of 22.9-mm inside diameter and 25.5-mm outside diameter, with each tube making four passes through the shell. (a) Assuming fully developed flow for the water, determine the required tube length per pass. Problem 11.25 In a diary operation, milk at a flow rate of 250 liter/hour and a cow-body temperature of 38.6oC must be chilled to a safe-to-store temperature of 13oC or less. Ground water at 10oC is available at a flow rate of 0.72 m3/h. The density and specific heat of milk are 10300 kg/m3 and 3860 J/kg.K, respectively. a) Determine the UA product of a counterflow heat exchanger required for the chilling process. Determine the length of the exchanger if the inner pipe has a 50-mm diameter and the overall heat transfer coefficient is U = 1000 W/m2.K. Problem 11.25 (contd.) b) Determine the outlet temperature of the water. c) Using the value of UA found in part (a), determine the milk outlet temperature if the water flow rate is doubled. What is the outlet temperature if the flow rate is halved? Problem 12.29 The spectral, hemispherical emissivity of tungsten may be approximated by the distribution depicted below. Consider a cylindrical tungsten filament that is of diameter D = 0.8 mm and length L = 20 mm. The filament is enclosed in an evacuated bulb and is heated by an electrical current to a steadystate temperature of 2900 K. a) What is the total hemispherical emissivity when the filament temperature is 2900 K? Problem 12.29 (contd.) b) Assuming the surroundings are at 300 K, what is the initial rate of cooling of the filament when the current is switched off? Problem 13.50 A very long electrical conductor 10 mm in diameter is concentric with a cooled cylindrical tube 50 mm in diameter whose surface is diffuse and gray with an emissivity of 0.9 and temperature of 27oC. The electrical conductor has a diffuse, gray surface with an emissivity of 0.6 and is dissipating 6.0 W per meter of length. Assuming that the space between the two surfaces is evacuated, calculate the surface temperature of the conductor.