Heat Transfer

Abstract

Heat transfer is the movement of energy from one point to another by virtue of a difference in temperature. Heating and cooling are manifestations of this phenomenon, which is used in industrial operations and in domestic activities. Increasing energy costs and in some cases inadequate availability of energy will require peak efficiency in heating and cooling operations. An understanding of the mechanisms of heat transport is needed in order to recognize limitations of heating and cooling systems, which can then lead to adoption of practices that circumvent these limitations. In industrial and domestic heating and cooling, energy use audits can be used to determine total energy use and energy use distribution within the process, to identify areas of high energy use, and to target these areas for energy conservation measures.

Problems

• 6.1. How many inches of insulation would be required to insulate a ceiling such that the surface temperature of the ceiling facing the living area is within 2 °C of the room air temperature? Assume a heat transfer coefficient on both sides of the ceiling of 2.84 W/(m²· K) and a thermal conductivity of 0.0346 W/(m · K) for the insulation. The ceiling is 1.27-cm-thick plasterboard with a thermal conductivity of 0.433 W/(m · K). Room temperature is 20 °C and attic temperature is 49 °C.

• 6.2. If a heat transfer coefficient of 2.84 W/(m² · K) exists on each of the two inside faces of 6.35-mm-thick glass separated by an air gap, calculate the gap that could be used such that the rate of heat transfer by conduction through the air gap would equal the rate of heat transfer by convection. What would be the rate of heat transfer if this gap is exceeded?

  • The thermal conductivity of air is 0.0242 W/(m · K). Solve for temperatures of 20 °C and −12 °C on the outside surfaces of the glass. Would the calculated gap change in value for different values of the surface temperatures? Would there be any advantage to increasing the gap beyond this calculated value?

• 6.3. A walk-in freezer 4 × 6 m and 3 m high is to be built. The walls and ceiling consist of 1.7-mm-thick stainless steel (k = 14.2 W/(m · K)), 10 cm thick of foam insulation (k = 0.34 W/(m · K)), a thickness of corkboard (k = 0.043 W/(m · K)), and 1.27 cm thick of wood siding (k = 0.43 W/(m · K)). The inside of the freezer is maintained at −40 °C. Ambient air outside the freezer is at 32 °C. The heat transfer coefficient is 5 W/(m² · K) on the wood side of the wall and 2 W/(m² · K) on the stainless steel side.

  • (a) If the outside air has a dew point of 29 °C, calculate the thickness of the corkboard insulation that would prevent condensation of moisture on the outside wall of this freezer.

  • (b) Calculate the rate of heat transfer through the walls and ceiling of this freezer.

• 6.4. (a) Calculate the rate of heat loss to the surroundings and the quantity of steam that would condense per hour per meter of a l.5-in. (nominal) schedule 40 steel pipe containing steam at 130 °C. The heat transfer coefficient on the steam side is 11,400 W/(m² · K) and on the outside of the pipe to air is 5.7 W/m² · K. Ambient air averages 15 °C in temperature for the year. The thermal conductivity of the steel pipe wall is 45 W/(m · K).

  (b) How must energy would be saved in 1 year (365 days, 24 h/day) if the pipe is insulated with 5-cm-thick insulation having a thermal conductivity of 0.07 W/(m · K). The heat transfer coefficients on the steam and air sides are the same as in (a).

• 6.5. The rate of insolation (solar energy impinging on a surface) on a solar collector is 475 W/m². Assuming that 80% of the impinging radiation is absorbed (the rest is reflected), calculate the rate at which water can be heated from 15 °C to 50 °C in a solar hot water heater consisting of a spiral-wound, horizontal, 1.9-cm inside-diameter polyethylene pipe 30 m long. The pipe has a wall thickness of 1.59 mm. The projected area of a horizontal cylinder receiving radiation is the diameter multiplied by the length. Assume a heat transfer coefficient of 570 W/(m²· K) on the water side and a heat transfer coefficient of 5 W/(m²· K) on the air side, when calculating heat loss to the surroundings after the radiant energy from the sun is absorbed. The thermal conductivity of the pipe wall is 0.3 W/m · K. Ambient temperature is 7 °C.

• 6.6. A swept surface heat exchanger cools 3700 kg of tomato paste per hour from 93 °C to 32 °C. If the overall heat transfer coefficient based on the inside surface area is 855 W/m² · K, calculate the heating surface area required for concurrent flow and countercurrent flow. Cooling water enters at 21 °C and leaves at 27 °C. The specific heat of tomato paste is 3560 J/(kg · K).

• 6.7. Design the heating and cooling section of an aseptic canning system that processes 190 L per minute of an ice cream mix. The material has a density of 1040 kg/m³ and a specific heat of 3684 J/(kg · K).

• 6.8. Calculate the number of units of swept surface heat exchangers required to heat the material from 39 °C to 132 °C. Each unit has an inside heat transfer surface area of 0.97 m². The heating medium is steam at 143 °C. Previous experience with a similar unit on this material was that an overall heat transfer coefficient of 1700 W/(m² · K) based on the inside surface area may be expected.

• 6.9. Calculate the number of units (0.9 m² inside surface area per unit) required for cooling the sterilized ice cream mix from 132 °C to 32 °C. The cooling jacket of the swept surface heat exchangers is cooled by Freon refrigerant from a refrigeration system at −7 °C. Under these conditions, a heat transfer coefficient of 855 W/(m² · K) based on the inside surface area may be expected.

• 6.10. A small swept surface heat exchanger having an inside heat transfer surface area of 0.11 m² is used to test the feasibility of cooking a slurry in a continuous system. When the slurry was passed through the heat exchanger at a rate of 168 kg/h, it was heated from 25 °C to 72°C. Steam at 110 °C was used. The slurry has a specific heat of 3700 J/(kg · K).

  • What is the overall heat transfer coefficient in this system?

  • If this same heat transfer coefficient is expected in a larger system, calculate the rate at which the slurry can be passed through a similar swept surface heat exchanger having a heat transfer surface area inside of 0.75 m², if the inlet and exit temperatures are 25 °C and 72 °C, respectively, and steam at 120 °C is used for heating.

• 6.11. A steam-jacketed kettle has an inside heat transfer surface area of 0.43 m² that is all completely covered by the product. The product needs to be heated from 10 °C to 99 °C. The product contains 80% water and 20% nonfat solids. Previous experience has established that an overall heat transfer coefficient based on the inside surface area of 900 W/(m² · K) may be expected. The kettle holds 50 kg of product. Condensing steam at 120 °C is in the heating jacket. The contents are well stirred continuously during the process.

  • Calculate the time required for the heating process to be completed.

  • Determine the nearest nominal size steel pipe that can be used to supply steam to this kettle if the rate of steam flow through the pipe is to average a velocity of 12 m/s.

• 6.12. A processing line for a food product is being designed. It is necessary to estimate the number of kettles that is required to provide a production capacity of 500 kg/h. The cooking process involving the kettles requires heating the batch from 27 °C to 99 °C, simmering at 99 °C for 30 minutes and filling the hot product into cans. Filling the kettles and emptying requires approximately 15 minutes. The specific heat of the product is 3350 J/(kg · K). The density is 992 kg/m³.

  • Available for heating are cylindrical vessels with hemispherical bottoms with the hemisphere completely jacketed. The height of the cylindrical section is 25 cm. The diameter of the vessel is 0.656 m. Assume the vessels are filled to 85% of capacity each time. The overall heat transfer coefficient based on the inside surface area averages 600 W/(m² · K). How many kettles are required to provide the desired production capacity? Steam condensing at 120 °C is used in the jacket for heating.

• 6.13. A process for producing frozen egg granules is proposed where a refrigerated rotating drum that has a surface temperature maintained at −40 °C contacts a pool of liquid eggs at 5 °C. The eggs freeze on the drum surface and the frozen material is scraped off the drum surface at a point before the surface reenters the liquid eggs. The frozen material, if thin enough, will be collected as frozen flakes. In this process, the thickness of frozen egg that forms on the drum surface is determined by the dwell time of the drum within the pool of liquid. An analogy of the process, which may be solved using the principles discussed in the section on freezing water, 7.5 h is the freezing of a slab directly in contact with a cold surface (i.e., h is infinite). It is desired that the frozen material be 2 mm thick on the drum surface.

• 6.14. Calculate the dwell time of the drum surface within the liquid egg pool. Assume that on emerging from the liquid egg pool, the frozen material temperature will be 2 °C below the freezing point.

  • If the drum has a diameter of 50 cm and it travels 120E (¹/₃ of a full rotation) after emerging from the liquid egg pool before the frozen material is scraped off, calculate the rotational speed of the drum needed to satisfy the criterion stipulated in (a), and calculate the average temperature of the frozen material at the time it is scraped off the drum. The density of liquid eggs is 1012 kg/m³ and frozen eggs 1009 kg/m³. Calculate the thermophysical properties based on the following compositional data: 75% water, 12% protein, 12% fat, and 1% carbohydrates.

• 6.15. In an experiment for pasteurization of orange juice, a 0.25-in. outside-diameter tube with 1/32-in.-thick wall was made into a coil and immersed in a water bath maintained at 95 °C. The coil was 2 m long, and when the juice was pumped at the rate of 0.2 L/min, the juice temperature changed from 25 °C to 85 °C. The juice contained 12% total solids. Calculate:

The overall heat transfer coefficient.

• The inside local heat transfer coefficient if the ratio ho/hi is 0.8.

• The inside local heat transfer coefficient, hi is directly proportional to the 0.8 power of the average velocity. If the rate at which the juice is pumped through the system is increased to 0.6 L/min, calculate the tube length needed to raise the temperature from 25 °C to 90 °C.

• 6.16. Calculate the surface area of a heat exchanger needed to pasteurize 100 kg/h of catsup by heating in a one-pass shell-and-tube heat exchanger from 40 °C to 95 °C. The catsup density is 1090 kg/m³. The flow behavior index is 0.5, and the consistency index is 0.5 and 0.35 Pa ·sⁿ at 25 °C and 50 °C, respectively. Estimate the thermal conductivity and specific heat using correlations discussed in Chaps. 5 and 7. The catsup contains 0.4% fiber, 33.8% carbohydrate, and 2.8% ash, and the balance is water. The catsup is to travel within the heat exchanger at a velocity of 0.3 m/s. Steam condensing at 135 °C is used for heating, and a heat transfer coefficient of 15,000 W/(m² · K) may be assumed for the steam side. The heat exchanger tubes are type 304 stainless steel with an inside diameter of 0.02291 m and an outside diameter of 0.0254 m.

• 6.17. A box of beef that was tightly packed was originally at 0 °C. It was inadvertently left on a loading dock during the summer when ambient conditions were 30 °C. Assuming that the heat transfer coefficient around the box averages 20 W/(m²· K), calculate the surface temperature and the temperature at a point 2 cm deep from the surface after 2 h. The box consisted of 0.5-cm-thick fiberboard with a thermal conductivity of 0.2 W/(m · K) and had dimensions of 47 × 60 × 30 cm. Because the box material has very little heat capacity, it may be assumed to act as a surface resistance, and an equivalent heat transfer coefficient may be calculated such that the resistance to heat transfer will be the same as the combined conductive resistance of the cardboard and the convective resistance of the surface heat transfer coefficient. The meat has a density of 1042 kg/m³, a thermal conductivity of 0.44 W/(m · K), and a specific heat of 3558 J/(kg · K).

• 6.18. In the operation of a microwave oven, reduced power application to the food is achieved by alternatively cutting on and off the power applied. To minimize excessive heating in some parts of the food being heated, the power application must be cycled such that the average temperature rise with each application of power does not exceed 5 °C followed by a 10-second pause between power applications. If 0.5 kg of food is being heated in a microwave oven having a power output of 600 W, calculate fraction of full power output that must be set on the oven controls such that the above temperature rise and pause cycles are satisfied. The food has a specific heat of 3500 J/(kg · K).