Insulation in a Cold Room. Calculate the heat loss per m² of surface area for a temporary insulating wall of a food cold storage room where the outside temperature is 299.9 K and the inside temperature 276.5 K. The wall is composed of 25.4 mm of corkboard having a k of 0.0433 W/m · K.

Determination of Thermal Conductivity. In determining the thermal conductivity of an insulating material, the temperatures were measured on both sides of a flat slab of 25 mm of the material and were 318.4 and 303.2 K. The heat flux was measured as 35.1 W/m². Calculate the thermal conductivity in btu/h · ft · °F and in W/m · K.

Mean Thermal Conductivity in a Cylinder. Prove that if the thermal conductivity varies linearly with temperature as in Eq. (4.1-11), the proper mean value k_m to use in the cylindrical equation is given by Eq. (4.2-3) as in a slab.

Heat Removal of a Cooling Coil. A cooling coil of 1.0 ft of 304 stainless-steel tubing having an inside diameter of 0.25 in. and an outside diameter of 0.40 in. is being used to remove heat from a bath. The temperature at the inside surface of the tube is 40°F and is 80°F on the outside. The thermal conductivity of 304 stainless steel is a function of temperature:

Removal of Heat from a Bath. Repeat Problem 4.2-2 but for a cooling coil made of 308 stainless steel having an average thermal conductivity of 15.23 W/m · K.

Variation of Thermal Conductivity. A flat plane of thickness Δx has one surface maintained at T₁ and the other at T₂. If the thermal conductivity varies according to temperature as

Temperature Distribution in a Hollow Sphere. Derive Eq. (4.2-14) for the steady-state conduction of heat in a hollow sphere. Also, derive an equation which shows that the temperature varies hyperbolically with the radius r.

Insulation Needed for Food Cold Storage Room. A food cold storage room is to be constructed of an inner layer of 19.1 mm of pine wood, a middle layer of cork board, and an outer layer of 50.8 mm of concrete. The inside wall surface temperature is -17.8°C and the outside surface temperature is 29.4°C at the outer concrete surface. The mean conductivities are for pine, 0.151; cork, 0.0433; and concrete, 0.762 W/m · K. The total inside surface area of the room to use in the calculation is approximately 39 m² (neglecting corner and end effects). What thickness of cork board is needed to keep the heat loss to 586 W?

Insulation of a Furnace. A wall of a furnace 0.244 m thick is constructed of material having a thermal conductivity of 1.30 W/m · K. The wall will be insulated on the outside with material having an average k of 0.346 W/m · K, so the heat loss from the furnace will be equal to or less than 1830 W/m². The inner surface temperature is 1588 K and the outer 299 K. Calculate the thickness of insulation required.

Heat Loss Through Thermopane Double Window. A double window called ther-mopane is one in which two layers of glass are separated by a layer of dry, stagnant air. In a given window, each of the glass layers is 6.35 mm thick separated by a 6.35-mm space of stagnant air. The thermal conductivity of the glass is 0.869 W/m · K and that of air is 0.026 over the temperature range used. For a temperature drop of 27.8 K over the system, calculate the heat loss for a window 0.914 m X 1.83 m. (Note: This calculation neglects the effect of the convective coefficient on one outside surface of one side of the window, the convective coefficient on the other outside surface, and convection inside the window.)

Heat Loss from Steam Pipeline. A steel pipeline, 2-in. schedule 40 pipe, contains saturated steam at 121.1°C. The line is covered with 25.4 mm of insulation. Assuming that the inside surface temperature of the metal wall is at 121.1°C and the outer surface of the insulation is at 26.7°C, calculate the heat loss for 30.5 m of pipe. Also, calculate the kg of steam condensed per hour in the pipe due to the heat loss. The average k for steel from Appendix A.3 is 45 W/m · K and the k for the insulation is 0.182.

Heat Loss with Trial-and-Error Solution. The exhaust duct from a heater has an inside diameter of 114.3 mm with ceramic walls 6.4 mm thick. The average k = 1.52 W/m · K. Outside this wall, an insulation of rock wool 102 mm thick is installed. The thermal conductivity of the rock wool is k = 0.046 + 1.56 × 10^-4 T°C (W/m · K). The inside surface temperature of the ceramic is T₁ = 588.7 K, and the outside surface temperature of the insulation is T₃ = 311 K. Calculate the heat loss for 1.5 m of duct and the interface temperature T₂ between the ceramic and the insulation. [Hint: The correct value of k_m for the insulation is that evaluated at the mean temperature of (T₂ + T₃)/2. Hence, for the first trial assume a mean temperature of, say, 448 K. Then calculate the heat loss and T₂. Using this new T₂, calculate a new mean temperature and proceed as before.]

Heat Loss by Convection and Conduction. A glass window with an area of 0.557 m² is installed in the wooden outside wall of a room. The wall dimensions are 2.44 × 3.05 m. The wood has a k of 0.1505 W/m · K and is 25.4 mm thick. The glass is 3.18 mm thick and has a k of 0.692. The inside room temperature is 299.9 K (26.7°C) and the outside air temperature is 266.5 K. The convection coefficient h_i on the inside wall of the glass and the wood is estimated as 8.5 W/m² · K; the outside h₀ is also estimated as 8.5 for both surfaces. Calculate the heat loss through the wooden wall, through the glass, and the total.

Convection, Conduction, and Overall U. A gas at 450 K is flowing inside a 2-in. steel pipe, schedule 40. The pipe is insulated with 51 mm of lagging having a mean k = 0.0623 W/m · K. The convective heat-transfer coefficient of the gas inside the pipe is 30.7 W/m² · K and the convective coefficient on the outside of the lagging is 10.8. The air is at a temperature of 300 K.

Heat Transfer in Steam Heater. Water at an average of 70°F is flowing in a 2-in. steel pipe, schedule 40. Steam at 220°F is condensing on the outside of the pipe. The convective coefficient for the water inside the pipe is h = 500 btu/h · ft² · °F and the condensing steam coefficient on the outside is h = 1500.

Heat Loss from Temperature Measurements. A steel pipe carrying steam has an outside diameter of 89 mm. It is lagged with 76 mm of insulation having an average k = 0.043 W/m · K. Two thermocouples, one located at the interface between the pipe wall and the insulation and the other at the outer surface of the insulation, give temperatures of 115°C and 32°C, respectively. Calculate the heat loss in W per m of pipe.

Effect of Convective Coefficients on Heat Loss in Double Window. Repeat Problem 4.3-3 for heat loss in the double window. However, include a convective coefficient of h = 11.35 W/m² · K on one outside surface of one side of the window and an h of 11.35 on the other outside surface. Also calculate the overall U.

Uniform Chemical Heat Generation. Heat is being generated uniformly by a chemical reaction in a long cylinder of radius 91.4 mm. The generation rate is constant at 46.6 W/m³. The walls of the cylinder are cooled so that the wall temperature is held at 311.0 K. The thermal conductivity is 0.865 W/m · K. Calculate the center-line temperature at steady state.

Heat of Respiration of a Food Product. A fresh food product is held in cold storage at 278.0 K. It is packed in a container in the shape of a flat slab with all faces insulated except for the top flat surface, which is exposed to the air at 278.0 K. For estimation purposes the surface temperature will be assumed to be 278 K. The slab is 152.4 mm thick and the exposed surface area is 0.186 m². The density of the foodstuff is 641 kg/m³. The heat of respiration is 0.070 kJ/kg · h and the thermal conductivity is 0.346 W/m · K. Calculate the maximum temperature in the food product at steady state and the total heat given off in W. (Note: It is assumed in this problem that there is no air circulation inside the foodstuff. Hence, the results will be conservative, since circulation during respiration will reduce the temperature.)

Temperature Rise in Heating Wire. A current of 250 A is passing through a stainless-steel wire having a diameter of 5.08 mm. The wire is 2.44 m long and has a resistance of 0.0843 Ω. The outer surface is held constant at 427.6 K. The thermal conductivity is k = 22.5 W/m · K. Calculate the center-line temperature at steady state.

Critical Radius for Insulation. A metal steam pipe having an outside diameter of 30 mm has a surface temperature of 400 K and is to be insulated with an insulation having a thickness of 20 mm and a k of 0.08 W/m · K. The pipe is exposed to air at 300 K and a convection coefficient of 30 W/m² · K.

Curvilinear-Squares Graphical Method. Repeat Example 4.4-1 but with the following changes:

Heat Loss from a Furnace. A rectangular furnace with inside dimensions of 1.0 × 1.0 × 2.0 m has a wall thickness of 0.20 m. The k of the walls is 0.95 W/m · K. The inside of the furnace is held at 800 K and the outside at 350 K. Calculate the total heat loss from the furnace.

Heat Loss from a Buried Pipe. A water pipe whose wall temperature is 300 K has a diameter of 150 mm and a length of 10 m. It is buried horizontally in the ground at a depth of 0.40 m measured to the center line of the pipe. The ground surface temperature is 280 K and k = 0.85 W/m · K. Calculate the loss of heat from the pipe.

Heating Air by Condensing Steam. Air is flowing through a tube having an inside diameter of 38.1 mm at a velocity of 6.71 m/s, average temperature of 449.9 K, and pressure of 138 kPa. The inside wall temperature is held constant at 204.4°C (477.6 K) by steam condensing outside the tube wall. Calculate the heat-transfer coefficient for a long tube and the heat-transfer flux.

Trial-and-Error Solution for Heating Water. Water is flowing inside a horizontal 1-in. schedule 40 steel pipe at 37.8°C and a velocity of 1.52 m/s. Steam at 108.3°C is condensing on the outside of the pipe wall and the steam coefficient is assumed constant at 9100 W/m² · K.

Heat-Transfer Area and Use of Log Mean Temperature Difference. A reaction mixture having a c_pm = 2.85 kJ/kg · K is flowing at a rate of 7260 kg/h and is to be cooled from 377.6 K to 344.3 K. Cooling water at 288.8 K is available and the flow rate is 4536 kg/h. The overall U_o is 653 W/m² · K.

Heating Water with Hot Gases and Heat-Transfer Area. Water flowing at the rate of 13.85 kg/s is to be heated from 54.5 to 87.8°C in a heat exchanger by 54 430 kg/h of hot gas flowing counterflow and entering at 427°C (c_pm = 1.005 kJ/kg · K). The overall U_o = 69.1 W/m² · K. Calculate the exit-gas temperature and the heat-transfer area.

Cooling Oil and Overall U. Oil flowing at the rate of 7258 kg/h with a c_pm = 2.01 kJ/kg · K is cooled from 394.3 K to 338.9 K in a counterflow heat exchanger by water entering at 294.3 K and leaving at 305.4 K. Calculate the flow rate of the water and the overall U_i if the A_i is 5.11 m².

Laminar Flow and Heating of Oil. A hydrocarbon oil having the same physical properties as the oil in Example 4.5-5 enters at 175°F inside a pipe having an inside diameter of 0.0303 ft and a length of 15 ft. The inside pipe surface temperature is constant at 325°F. The oil is to be heated to 250°F in the pipe. How many lb_m/h oil can be heated? (Hint: This solution is trial and error. One method is to assume a flow rate of, say, m = 75 lb mass/h. Calculate the N_Re and the value of h_a. Then make a heat balance to solve for q in terms of m. Equate this q to the q from the equation q = h_aA ΔT_a. Solve for m. This is the new m to use for the second trial.)

Heating Air by Condensing Steam. Air at a pressure of 101.3 kPa and 288.8 K enters inside a tube having an inside diameter of 12.7 mm and a length of 1.52 m with a velocity of 24.4 m/s. Condensing steam on the outside of the tube maintains the inside wall temperature at 372.1 K. Calculate the convection coefficient of the air. (Note: This solution is trial and error. First assume an outlet temperature of the air.)

Heat Transfer with a Liquid Metal. The liquid metal bismuth at a flow rate of 2.00 kg/s enters a tube having an inside diameter of 35 mm at 425°C and is heated to 430°C in the tube. The tube wall is maintained at a temperature of 25°C above the liquid bulk temperature. Calculate the tube length required. The physical properties are as follows (H1): k = 15.6 W/m · K, c_p = 149 J/kg · K, μ = 1.34 ×‐ 10^–3 Pa · s.

Heat Transfer from a Flat Plate. Air at a pressure of 101.3 kPa and a temperature of 288.8 K is flowing over a thin, smooth, flat plate at 3.05 m/s. The plate length in the direction of flow is 0.305 m and its temperature is 333.2 K. Calculate the heat-transfer coefficient assuming laminar flow.

Chilling Frozen Meat. Cold air at -28.9°C and 1 atm is recirculated at a velocity of 0.61 m/s over the exposed top flat surface of a piece of frozen meat. The sides and bottom of this rectangular slab of meat are insulated and the top surface is 254 mm by 254 mm square. If the surface of the meat is at -6.7°C, predict the average heat-transfer coefficient to the surface. As an approximation, assume that either Eq. (4.6-2) or (4.6-3) can be used, depending on the N_Re,L.

Heat Transfer to an Apple. It is desired to predict the heat-transfer coefficient for air being blown past an apple lying on a screen with large openings. The air velocity is 0.61 m/s at 101.32 kPa pressure and 316.5 K. The surface of the apple is at 277.6 K and its average diameter is 114 mm. Assume that it is a sphere.

Heating Air by a Steam Heater. A total of 13 610 kg/h of air at 1 atm abs pressure and 15.6°C is to be heated by passing over a bank of tubes in which steam at 100°C is condensing. The tubes are 12.7 mm OD, 0.61 m long, and arranged in-line in a square pattern with S_p = S_n = 19.05 mm. The bank of tubes contains six transverse rows in the direction of flow and 19 rows normal to the flow. Assume that the tube surface temperature is constant at 93.33°C. Calculate the outlet air temperature.

Natural Convection from an Oven Wall. The oven wall in Example 4.7-1 is insulated so that the surface temperature is 366.5 K instead of 505.4 K. Calculate the natural convection heat-transfer coefficient and the heat-transfer rate per m of width. Use both Eq. (4.7-4) and the simplified equation. (Note: Radiation is being neglected in this calculation.) Use both SI and English units.

Losses by Natural Convection from a Cylinder. A vertical cylinder 76.2 mm in diameter and 121.9 mm high is maintained at 397.1 K at its surface. It loses heat by natural convection to air at 294.3 K. Heat is lost from the cylindrical side and the flat circular end at the top. Calculate the heat loss neglecting radiation losses. Use the simplified equations of Table 4.7-2 and those equations for the lowest range of N_Gr N_Pr. The equivalent L to use for the top flat surface is 0.9 times the diameter.

Heat Loss from a Horizontal Tube. A horizontal tube carrying hot water has a surface temperature of 355.4 K and an outside diameter of 25.4 mm. The tube is exposed to room air at 294.3 K. What is the natural convection heat loss for a 1-m length of pipe?

Natural Convection Cooling of an Orange. An orange 102 mm in diameter having a surface temperature of 21.1°C is placed on an open shelf in a refrigerator held at 4.4°C. Calculate the heat loss by natural convection, neglecting radiation. As an approximation, the simplified equation for vertical planes can be used with L replaced by the radius of the sphere (M1). For a more accurate correlation, see (S2).

Natural Convection in Enclosed Horizontal Space. Repeat Example 4.7-3 but for the case where the two plates are horizontal and the bottom plate is hotter than the upper plate. Compare the results.

Natural Convection Heat Loss in Double Window. A vertical double plate-glass window has an enclosed air-gap space of 10 mm. The window is 2.0 m high by 1.2 m wide. One window surface is at 25°C and the other at 10°C. Calculate the free convection heat-transfer rate through the air gap.

Natural Convection Heat Loss for Water in Vertical Plates. Two vertical square metal plates having dimensions of 0.40 X 0.40 m are separated by a gap of 12 mm and this enclosed space is filled with water. The average surface temperature of one plate is 65.6°C and the other plate is at 37.8°C. Calculate the heat-transfer rate through this gap.

Heat Loss from a Furnace. Two horizontal metal plates having dimensions of 0.8 X 1.0 m comprise the top of a furnace and are separated by a distance of 15 mm. The lower plate is at 400°C and the upper at 100°C, and air at 1 atm abs is enclosed in the gap. Calculate the heat-transfer rate between the plates.

Boiling Coefficient in a Jacketed Kettle. Predict the boiling heat-transfer coefficient for the vertical jacketed sides of the kettle given in Example 4.8-1. Then, using this coefficient for the sides and the coefficient from Example 4.8-1 for the bottom, predict the total heat transfer.

Boiling Coefficient on a Horizontal Tube. Predict the boiling heat-transfer coefficient for water under pressure boiling at 250°F for a horizontal surface of -in.-thick stainless steel having a k of 9.4 btu/h · ft · °F. The heating medium on the other side of this surface is a hot fluid at 290°F having an h of 275 btu/h · ft² · °F. Use the simplified equations. Be sure to correct this h value for the effect of pressure.

Condensation on a Vertical Tube. Repeat Example 4.8-2 but for a vertical tube 1.22 m (4.0 ft) high instead of 0.305 m (1.0 ft) high. Use SI and English units.

Condensation of Steam on Vertical Tubes. Steam at 1 atm abs pressure and 100°C is condensing on a bank of five vertical tubes each 0.305 m high and having an OD of 25.4 mm. The tubes are arranged in a bundle spaced far enough apart that they do not interfere with each other. The surface temperature of the tubes is 97.78°C. Calculate the average heat-transfer coefficient and the total kg condensate per hour.

Condensation on a Bank of Horizontal Tubes. Steam at 1 atm abs pressure and 100°C is condensing on a horizontal tube bank with five layers of tubes (N = 5) placed one below the other. Each layer has four tubes (total tubes = 4 × 5 = 20) and the OD of each tube is 19.1 mm. The tubes are each 0.61 m long and the tube surface temperature is 97.78°C. Calculate the average heat-transfer coefficient and the kg condensate per second for the whole condenser. Make a sketch of the tube bank.

Mean Temperature Difference in an Exchanger. A 1-2 exchanger with one shell pass and two tube passes is used to heat a cold fluid from 37.8°C to 121.1°C by using a hot fluid entering at 315.6°C and leaving at 148.9°C. Calculate the ΔT_lm and the mean temperature difference ΔT_m in K.

Cooling Oil by Water in an Exchanger. Oil flowing at the rate of 5.04 kg/s (c_pm = 2.09 kJ/kg · K) is cooled in a 1-2 heat exchanger from 366.5 K to 344.3 K by 2.02 kg/s of water entering at 283.2 K. The overall heat-transfer coefficient U_o is 340 W/m² · K. Calculate the area required. (Hint: A heat balance must first be made to determine the outlet water temperature.)

Heat Exchange Between Oil and Water. Water is flowing at the rate of 1.13 kg/s in a 1-2 shell-and-tube heat exchanger and is heated from 45°C to 85°C by an oil having a heat capacity of 1.95 kJ/kg · K. The oil enters at 120°C and leaves at 85°C. Calculate the area of the exchanger if the overall heat-transfer coefficient is 300 W/m² · K.

Outlet Temperature and Effectiveness of an Exchanger. Hot oil at a flow rate of 3.00 kg/s (c_p = 1.92 kJ/kg · K) enters an existing counterflow exchanger at 400 K and is cooled by water entering at 325 K (under pressure) and flowing at a rate of 0.70 kg/s. The overall U = 350 W/m² · K and A = 12.9 m². Calculate the heat-transfer rate and the exit oil temperature.

Radiation to a Tube from a Large Enclosure. Repeat Example 4.10-1 but use the slightly more accurate Eq. (4.10-5) with two different emissivities.

Baking a Loaf of Bread in an Oven. A loaf of bread having a surface temperature of 373 K is being baked in an oven whose walls and the air are at 477.4 K. The bread moves continuously through the large oven on an open chain belt conveyor. The emissivity of the bread is estimated as 0.85, and the loaf can be assumed to be a rectangular solid 114.3 mm high × 114.3 mm wide × 330 mm long. Calculate the radiation heat-transfer rate to the bread, assuming that it is small compared to the oven and neglecting natural convection heat transfer.

Radiation and Convection from a Steam Pipe. A horizontal oxidized steel pipe carrying steam and having an OD of 0.1683 m has a surface temperature of 374.9 K and is exposed to air at 297.1 K in a large enclosure. Calculate the heat loss for 0.305 m of pipe from natural convection plus radiation. For the steel pipe, use an ε of 0.79.

Radiation and Convection to a Loaf of Bread. Calculate the total heat-transfer rate to the loaf of bread in Problem 4.10-2, including the radiation plus natural convection heat transfer. For radiation first calculate a value of h_r. For natural convection, use the simplified equations for the lower N_GrN_Pr range. For the four vertical sides, the equation for vertical planes can be used with an L of 114.3 mm. For the top surface, use the equation for a cooled plate facing upward, and for the bottom, a cooled plate facing downward. The characteristic L for a horizontal rectangular plate is the linear mean of the two dimensions.

Heat Loss from a Pipe. A bare stainless-steel tube having an outside diameter of 76.2 mm and an ε of 0.55 is placed horizontally in air at 294.2 K. The pipe surface temperature is 366.4 K. Calculate the value of h_c + h_r for convection plus radiation and the heat loss for 3 m of pipe.

Radiation Shielding. Two very large and parallel planes each have an emissivity of 0.7. Surface 1 is at 866.5 K and surface 2 is at 588.8 K. Use SI and English units.

Radiation from a Craft in Space. A space satellite in the shape of a sphere is traveling in outer space, where its surface temperature is held at 283.2 K. The sphere "sees" only outer space, which can be considered as a black body with a temperature of 0 K. The polished surface of the sphere has an emissivity of 0.1. Calculate the heat loss per m² by radiation.

Radiation and Complex View Factor. Find the view factor F₁₂ for the configuration shown in Fig. P4.11-3. The area A₄ and A₃ are fictitious areas (C3). The area A₂ + A₄ is called A₍₂₄₎ and A₁ + A₃ is called A₍₁₃₎. Areas A₍₂₄₎ and A₍₁₃₎ are perpendicular to each other. [Hint: Follow the methods in Example 4.11-5. First, write an equation similar to Eq. (4.11-48) which relates the interchange between A₃ and A₍₂₄₎. Then relate the interchange between A₍₁₃₎ and A₍₂₄₎. Finally, relate A₍₁₃₎ and A₄.]

Radiation Between Parallel Surfaces. Two parallel surfaces each 1.83 × 1.83 m square are spaced 0.91 m apart. The surface temperature of A₁ is 811 K and that of A₂ is 533 K. Both are black surfaces.

Radiation Between Adjacent Perpendicular Plates. Two adjacent rectangles are perpendicular to each other. The first rectangle is 1.52 × 2.44 m and the second 1.83 × 2.44 m, with the 2.44-m side common to both. The temperature of the first surface is 699 K and that of the second is 478 K. Both surfaces are black. Calculate the radiant heat transfer between the two surfaces.

Radiation and Heating of Planes. Two plane disks each 1.25 m in diameter are parallel and directly opposite each other. They are separated by a distance of 0.5 m. Disk 1 is heated by electrical resistance to 833.3 K. Both disks are insulated on all faces except the two faces directly opposite each other. Assume that the surroundings emit no radiation and that the disks are in space. Calculate the temperature of disk 2 at steady state and also the electrical energy input to disk 1. (Hint: The fraction of heat lost from area number 1 to space is 1 – F₁₂.)

Radiation by Disks to Each Other and to Surroundings. Two disks each 2.0 m in diameter are parallel and directly opposite each other and are separated by a distance of 2.0 m. Disk 1 is held at 1000 K by electric heating and disk 2 at 400 K by cooling water in a jacket at the rear of the disk. The disks radiate only to each other and to the surrounding space at 300 K. Calculate the electric heat input and also the heat removed by the cooling water.

View Factor by Integration. A small black disk is vertical, with an area of 0.002 m², and radiates to a vertical black plane surface that is 0.03 m wide and 2.0 m high and is opposite and parallel to the small disk. The disk source is 2.0 m away from the vertical plane and placed opposite the bottom of the plane. Determine F₁₂ by integration of the view-factor equation.

Gas Radiation to Gray Enclosure. Repeat Example 4.11-7 but with the following changes:

Gas Radiation and Convection to a Stack. A furnace discharges hot flue gas at 1000 K and 1 atm abs pressure containing 5% CO₂ into a stack having an inside diameter of 0.50 m. The inside walls of the refractory lining are at 900 K and the emissivity of the lining is 0.75. The convective heat-transfer coefficient of the gas has been estimated as 10 W/m² · K. Calculate the rate of heat transfer q/A from the gas by radiation plus convection.

Laminar Heat Transfer of a Power-Law Fluid. A non-Newtonian power-law fluid banana purée flowing at a rate of 300 lb_m/h inside a 1.0-in.-ID tube is being heated by a hot fluid flowing outside the tube. The banana purée enters the heating section of the tube, which is 5 ft long, at a temperature of 60°F. The inside wall temperature is constant at 180°F. The fluid properties as given by Charm (C1) are ρ = 69.9 lb_m/ft³, c_p = 0.875 btu/lb_m · °F, and k = 0.320 btu/h · ft · °F. The fluid has the following rheological constants: n = n' = 0.458, which can be assumed constant, and K = 0.146 lb_f · sⁿ · ft^-2 at 70°F and 0.0417 at 190°F A plot of log K versus T°F can be assumed to be a straight line. Calculate the outlet bulk temperature of the fluid in laminar flow.

Heating a Power-Law Fluid in Laminar Flow. A non-Newtonian power-law fluid having the same physical properties and rheological constants as the fluid in Example 4.12-1 is flowing in laminar flow at a rate of 6.30 × 10^-2 kg/s inside a 25.4-mm-ID tube. It is being heated by a hot fluid outside the tube. The fluid enters the heating section of the tube at 26.7°C and leaves the heating section at an outlet bulk temperature of 46.1°C. The inside wall temperature is constant at 82.2°C. Calculate the length of tube needed in m. (Note: In this case the unknown tube length L appears in the equation for h_a and in the heat-balance equation.)

Heat Transfer in a Jacketed Vessel with a Paddle Agitator. A vessel with a paddle agitator and no baffles is used to heat a liquid at 37.8°C. A steam-heated jacket furnishes the heat. The vessel's inside diameter is 1.22 m; the agitator diameter is 0.406 m and it is rotating at 150 rpm. The wall surface temperature is 93.3°C. The physical properties of the liquid are ρ = 977 kg/m³, c_p = 2.72 kJ/kg · K, k = 0.346 W/m · K, and μ = 0.100 kg/m · s at 37.8°C and 7.5 × 10^–3 at 93.3°C. Calculate the heat-transfer coefficient to the wall of the jacket.

Heat Loss from Circular Fins. Use the data and conditions from Example 4.13-2 and calculate the fin efficiency and rate of heat loss from the following different fin materials:

Heat Loss from Longitudinal Fin. A longitudinal aluminum fin as shown in Fig. 4.13-3a (k = 230 W/m · K) is attached to a copper tube having an outside radius of 0.04 m. The length of the fin is 0.080 m and the thickness is 3 mm. The tube base is held at 450 K and the external surrounding air at 300 K has a convective coefficient of 25 W/m² · K. Calculate the fin efficiency and the heat loss from the fin per 1.0 m of length.

Heat Transfer in Finned Tube Exchanger. Air at an average temperature of 50°C is being heated by flowing outside a steel tube (k = 45.1 W/m · K) having an inside diameter of 35 mm and a wall thickness of 3 mm. The outside of the tube is covered with 16 longitudinal steel fins with a length L = 13 mm and a thickness of t = 1.0 mm. Condensing steam inside at 120°C has a coefficient of 7000 W/m² · K. The outside coefficient of the air has been estimated as 30 W/m² · K. Neglecting fouling factors and using a tube 1.0 m long, calculate the overall heat-transfer coefficient U_i based on the inside area A_i.

Dimensional Analysis for Natural Convection. Repeat the dimensional analysis for natural convection heat transfer to a vertical plate as given in Section 4.14. However, do as follows:

Dimensional Analysis for Unsteady-State Conduction. For unsteady-state conduction in a solid, the following variables are involved: ρ, c_p, L (dimension of solid), t, k, and z (location in solid). Determine the dimensionless groups relating the variables.

Temperatures in a Semi-Infinite Plate. A semi-infinite plate is similar to that in Fig. 4.15-2. At the surfaces x = 0 and x = L, the temperature is held constant at 200 K. At the surface y = 0, the temperature is held at 400 K. If L = 1.0 m, calculate the temperature at the point y = 0.5 m and x = 0.5 m at steady state.

Heat Conduction in a Two-Dimensional Solid. For two-dimensional heat conduction as given in Example 4.15-1, derive the equation to calculate the total heat loss from the chamber per unit length using the nodes at the outside. There should be eight paths for one-fourth of the chamber. Substitute the actual temperatures into the equation and obtain the heat loss.

Steady-State Heat Loss from a Rectangular Duct. A chamber that is in the shape of a long, hollow, rectangular duct has outside dimensions of 3 × 4 m and inside dimensions of 1 × 2 m. The walls are 1 m thick. The inside surface temperature is constant at 800 K and the outside constant at 200 K. The k = 1.4 W/m · K. Calculate the steady-state heat loss per unit m of length of duct. Use a grid size of Δx = Δy = 0.5 m. Also, use the outside nodes to calculate the total heat conduction. Use a spreadsheet for the calculation.

Two-Dimensional Heat Conduction and Different Boundary Conditions. A very long, solid piece of material 1 by 1 m square has its top face maintained at a constant temperature of 1000 K and its left face at 200 K. The bottom face and right face are exposed to an environment at 200 K and have a convection coefficient of h = 10 W/m² · K. The k = 10 W/m · K. Use a grid size of Δx = Δy = m and calculate the steady-state temperatures of the various nodes.

Nodal Point at Exterior Corner Between Insulated Surfaces. Derive the finite-difference equation for the case for the nodal point T_n,m at an exterior corner between insulated surfaces. The diagram is similar to Fig. 4.15-6c except that the two boundaries are insulated.