# Principles of Food and Bioprocess Engineering (FS 231) Problems on Heat Transfer

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1 Principles of Food and Bioprocess Engineering (FS 1) Problems on Heat Transfer 1. What is the thermal conductivity of a material 8 cm thick if the temperature at one end of the product is 0 C and the temperature at the other end is 55 C when it is subjected to a constant heat flux of 5 W/m?. A room heater is to be installed to maintain the room temperature at 5 C in winter when the air temperature outside is -0 C. All surfaces of the room, except a wall of area 6 m are well insulated. This wall is made up of a layer of brick (thickness = 10 cm, k = 0.7 W/m K) and a layer of cardboard (thickness =.5 cm, k = 0.07 W/m K). If the convective heat transfer coefficient between the air in the room and the wall is 5 W/m K and that between air outside and the wall is 100 W/m K, what should be the wattage of a heater that will accomplish the task of maintaining the room temperature at 5 C?. A product (c p =,700 J/kg K) is being conveyed in a tube (k = W/m K) of I.D. 8 cm, and O.D. 8.5 cm at the rate of kg/s. Insulation of thickness 1 cm and k = 0.0 W/m K is used to reduce heat loss (per unit length of the pipe) to 0 W/m. The convective heat transfer coefficient between the product and the inside wall is 00 W/m K and that between insulation and air is 5 W/m K. If the temperature of the interface between the insulation and the pipe is 6 C, determine: a. The temperature of the inside wall of the pipe b. The temperature of the outside of the insulation c. The temperature of the product d. The ambient air temperature 4. Milk at 80 C flows through a cylindrical tube (I.D. = 6 cm, O.D. = 1 cm, k = 10 W/m K). An insulation (k = 0.1 W/m K) of thickness 5 cm is used to minimize heat loss to 50 W/m. If the wall temperature is 75 C, Determine a. The outside temperature of the tube b. The outside temperature of the insulation 5. Water is flowing in a 10 m long pipe (I.D. = 5 cm, O.D. 5.5 cm) at the rate of 0.0 kg/s. If the water enters at 0 C and leaves at 60 C, determine the convective heat transfer coefficient between water and the wall of the pipe if the pipe wall is at 95 C. 6. Estimate the percentage increase in the convective heat transfer coefficient for flow in a pipe when the velocity of the fluid is doubled. Consider situations -- laminar and turbulent flow. 7. Water at 100 C enters a 100 m long pipe (k = 10 W/m K) of I.D. 10 cm and O.D. 11 cm at the rate of 0.1 kg/s. What should be the specifications of the insulation (i.e. what should be the thickness and thermal conductivity of the insulation) if the water temperature is not to drop below 70 C at the exit? The convective heat transfer coefficient between air and the insulation is 7 W/m K and the ambient air temperature is 0 C. Neglect the viscosity correction factor.

2 8. Meatballs of diameter 0.8 cm are being transported by means of a perforated conveyor belt moving at the rate of 0 cm/s. In order to cool these meatballs which are initially at 90 C, air at 10 C is passed over them from the opposite direction at a speed of 0 cm/s such that heat transfer to the product takes place from all sides. Determine the convective heat transfer coefficient between air and the meatballs. 9. Water at a bulk temperature of 90 C is being conveyed through a pipe (I.D. = 8 cm, O.D. = 1 cm, k = 15 W/m K). The pipe is covered by insulation (k = 0.04 W/m K) of thickness 4 cm. The convective heat transfer coefficient between the outside of the insulation and ambient air is 10 W/m K and the outside temperature of the insulation is 40 C. If the inside of the pipe is at 75 C, determine the following: a. The convective heat transfer coefficient between the water and the inside pipe b. The temperature at the interface of the pipe and insulation c. The ambient air temperature 10. A hot cylindrical shaped product ( = 800 kg/m, k = 0. W/m K, c p = 4,000 J/kg K) is taken out of the oven when it is at 90 C and placed under ambient conditions (10 C) on its base. The diameter of the cylindrical product is 10 cm and its height is 0 cm. Determine the convective heat transfer coefficient between the product and the ambient air. 11. Cold water (at an average temperature of 15 C) is being conveyed in a pipe (I.D. = 5 cm, O.D. = 6 cm, k = 60 W/m K) at the rate of kg/s. The inside wall of the pipe is at 90 C. The thickness of insulation (k = 0.1 W/m K) outside the pipe is 1 cm and the ambient temperature is 0 C. Determine the convective heat transfer coefficient between the cold water and the inside of the pipe. 1. Water (flowing at 1 kg/s) is being heated from 10 C to 70 C in a pipe of I.D. 1", O.D. 1.", and length 5 m. If the inside of the pipe is at 95 C, determine the convective heat transfer coefficient (inside heat transfer coefficient) between the wall of the pipe and water. 1. Hot water (bulk temperature = 95 C) is being pumped at the rate of 0.1 kg/s in a 90 cm long vertical metal pipe of I.D. cm, and O.D..5 cm. The temperature of the inside of the pipe is 90 C and that of the outside of the pipe is 87 C. If there is no insulation, and the ambient air temperature is 1 C, determine: a. The convective heat transfer coefficient between water and the pipe. b. The convective heat transfer coefficient between the metal pipe and the air. 14. Saturated steam at a pressure of kpa and a quality of 95% is condensing in the annular space of a 5 m long double pipe heat exchanger operating in a counter-current mode. The exiting steam has a quality of 90%. Apple juice (c p =,000 J/kg K) is flowing in the inner tube at the rate of 1 kg/s. It enters at 0 C and exits at 70 C. What is the logarithmic mean temperature difference? 15. A product at 90 C is being conveyed in a pipe (diameter = 5 cm, k = 16 W/m K) of negligible thickness. The convective heat transfer coefficient between ambient air (at 5 C) and the pipe (and also air and the insulation) is 5 W/m K. Will the use of any amount of insulation (k = 0.1 W/m K) result in a decrease in the heat loss from this pipe? Explain.

3 16. Cold water flowing in a 10 m long pipe (diameter 5 cm, k = 0 W/m K) of negligible thickness is being heated by hot water flowing in the outer pipe in the same direction. If the inlet temperatures of the hot and cold water are 90 C and 10 C respectively, and the exit temperature of the cold water is 40 C, determine the overall heat transfer coefficient between the hot and the cold water. The mass flow rate of the hot water is 1.5 kg/s and that of the cold water is 1 kg/s. Assume that the energy lost by the hot water to the surroundings is,000 Watts and that the properties of the hot and cold water are the same. 17. A certain product at 90 C is being conveyed in a metal pipe (L = 10 m, I.D. 4 cm, O.D. 6 cm) at the rate of 50 kg/hr. The temperature difference between the inside and outside of the metal pipe is 5 C. Insulation (k = 0.4 W/m K) of thickness 0.5 cm is used to minimize heat loss from the pipe to 50 W/m. If the convective heat transfer coefficient between the ambient air (at 5 C) and the insulation is 0 W/m K, determine the convective heat transfer coefficient between the hot water and the inside of the pipe. 18. A solid cylinder (diameter = 0.1 m, height = 0. m, k = 50 W/m K, c p = 4,190 J/kg K, = 814 kg/m ) is taken out of the refrigerator when its temperature is 5 C and placed under ambient conditions (air temperature = 5 C). Determine the time taken for the center of the cylinder to reach 0 C. Assume that the convective heat transfer coefficient between air and the cylinder is 40 W/m K. Neglect heat transfer from the flat surfaces. 19. A dough mix (at 0 C) is placed in a tray (100 cm x 10 cm x 4 cm) inside an oven at 150 C. Determine the center temperature of the product after 1 hr. The density, specific heat and thermal conductivity of the dough mix are 950 kg/m, 4,000 J/kg K, and 0.5 W/m K respectively. Assume that the convective heat transfer coefficient between the air in the oven and the product is 100 W/m K and that the product is exposed to air from all sides. 0. a. Determine the time taken for the center of a spherical apple (8 cm dia.) to reach 5 C when immersed in water at C. The initial temperature of the apple was 0 C. The density, specific heat, and thermal conductivity of apple are 800 kg/m,,700 J/kg K, and 0.4 W/m K respectively. The convective heat transfer coefficient between water and the apple is 75 W/m K. b. Determine the time taken for the center of the apple to reach 5 C using the lumped parameter analysis. c. Why is there a large difference in the times obtained from parts (a) and (b)? 1. A product in a cylindrical can of radius cm, and height 8 cm is taken out of a retort when the center temperature is 85 C. The ambient air is at 5 C and the convective heat transfer coefficient between the product and the air is 0 W/m K. Neglect the resistance to heat conduction offered by the can. The density, specific heat and thermal conductivity of the product are 900 kg/m,,800 J/kg K, and 0.4 W/m K respectively. a. Determine the center temperature of the product after 1 hour b. Outline the procedure for determining the time taken for the center to reach 0 C

4 . Determine the time taken for a cube-shaped food product of volume 7 cm to reach 90 C when dipped in oil at 110 C. The initial temperature of the product is 0 C. The convective heat transfer coefficient between the oil and the food product is 1000 W/m K. The density, specific heat, and thermal conductivity of the food product are 950 kg/m,,600 J/kg K, and 0.4 W/m K respectively.. A cube shaped product (each side of the cube = 5 cm) at 5 C is taken out of the refrigerator and dipped in hot water at 85 C. Use = 950 kg/m, k = 0.5 W/m K, c p =,700 J/kg K in this problem. The h value required for solving this problem is 650 W/m K. a. Are the properties (, k, and c p) required for solving the problem, the properties of the product or that of the water? b. What does h (convective heat transfer coefficient) represent in THIS problem? Be specific. c. How long will it take for the center of the cube to reach 50 C? 4. A cylindrical product (height = 10 cm, diameter = 8 cm) is taken out of the oven when it is at 90 C and placed under ambient conditions (5 C). If the convective heat transfer coefficient between the product and the ambient air is 5 W/m K, determine the temperature at the center of the product after 0 minutes. The density, specific heat, and thermal conductivity of the product are 8,169 kg/m, 460 J/kg K, and 10 W/m K respectively. 5. Determine the temperature at the center of a brick-shaped object (6 cm x 7 cm x 8 cm) after 1 hour, if it is taken out of the freezer when it is at -5 C and placed near a fan where the ambient air temperature is 5 C. The density, specific heat, and thermal conductivity of the product are 800 kg/m, 500 J/kg K, and 0.5 W/m K respectively. Neglect the convective resistance between the product and the air. 6. A cube-shaped object (length of each side = 4 cm) is placed in a blast freezer with air at -10 C being blown at 5 cm/s around all sides of the object. If the initial temperature of the object is 40 C, determine the time taken for the center of the object to reach 5 C. The density, specific heat, and thermal conductivity of the product are 900 kg/m,,000 J/kg K, and 5 W/m K respectively. The convective heat transfer coefficient between the object and the air is 0 W/m K. 7. A product (k = 0.5 W/m K, = 950 kg/m, c p = 4,000 J/kg K) in a cylindrical container of height 10 cm and radius 4 cm is taken out of the oven when it is at 90 C and placed under ambient conditions (0 C). The convective heat transfer coefficient between the container and ambient air is 5 W/m K. a. Determine the center temperature of the product after 1 hour and 45 minutes. b. Outline the procedure for determining the time taken for the center temperature of the product to reach 40 C. 8. A hot product (c p = 4,000 J/kg K, = Pa s, = 900 kg/m, k = 0.5 W/m K) is being conveyed in a 0 m long metal pipe of diameter 6 cm and negligible thickness at the rate of 0.5 kg/s. The inlet and exit temperatures of the hot product are 90 C and 80 C respectively. In order to reduce the heat loss to half its current value, insulation of 5 mm thickness is

5 added to the outside of the pipe. If the convective heat transfer coefficient between the ambient air (which is at 10 C) and either the pipe or the insulation is 100 W/m K, determine the thermal conductivity of the insulation to be used. MAKE NECESSARY ASSUMPTIONS. 9. A cube shaped product (each side of the cube = 5 cm) at 5 C is taken out of the refrigerator and placed in a conventional oven at 150 C. The properties of the product are: = 950 kg/m, k =.5 W/m K, c p =,700 J/kg K. The convective heat transfer coefficient between the product and the air in the oven is 10 W/m K. How long will it take for the center of the cube to reach 80 C? 0. A product (c p = 4,100 J/kg K, = Pa s, = 980 kg/m, k = 0.5 W/m K) is being pumped through a 10 m long un-insulated horizontal pipe (I.D. = 4 cm, O.D. = 5 cm, k = 8 W/m K) at the rate of 0. kg/s. If the bulk temperature of the product is 85 C, the outside wall temperature is 80 C, and the ambient air temperature is 10 C, a. Determine the convective heat transfer coefficient between the ambient air and the outside surface of the pipe. b. How much insulation (k = 0.1 W/m K) should be added to the outside of this pipe to ensure that heat loss from the pipe will decrease as a result of the addition of insulation? 1. a. A 10 m long double tube heat exchanger (inside pipe: ID = 4.5 cm, OD = 4.8 cm; outside pipe: ID = 9.5 cm, OD = 10 cm) is used in a counter-current configuration to heat a product (c p = 4,00 J/kg K, = 0.00 Pa s, = 950 kg/m, k = 0.4 W/m K) from 5 C to 70 C using 90 C hot water (c p = 4,00 J/kg K, = Pa s, = 1000 kg/m, k = 0.6 W/mK), with the product flowing in the inside pipe. The flow rate of the product and hot water are 0. kg/s and 0.4 kg/s respectively. Neglecting heat losses to the surroundings, determine the overall heat transfer coefficient. b. When rapid initial rate of cooling is desired, is a counter-current or a co-current configuration preferable? EXPLAIN.. When the lumped parameter analysis is applicable, and under similar initial and ambient conditions, will it take a cube (length of each side = D) or a sphere (diameter = D), both made of the same material, a longer time to heat up (or cool down)? EXPLAIN.. A room (6 m long, m wide, and.5 m high) is being heated by a 1.5 kw heater. All of the walls, except one wall are interior walls and hence have negligible heat loss through them. The wall that is in contact with the outside is m in length and has two layers of insulation. The inner layer has a thickness of.5 cm and a thermal conductivity of 0.18 W/m K while the outer layer has a thickness of 4 cm and a thermal conductivity of 0. W/m K. Determine the temperature at the interface between the two insulating layers. Assume that the convective heat transfer coefficient between air in the room and the wall is 100 W/m K and that between the outside air (which is at -5 C) and the wall is 500 W/m K. 4. A product in a cylindrical vessel (height = 0 cm and diameter = 1 cm) is taken out of a refrigerator when it is at 4 C and placed on its base in a room where the ambient temperature is 40 C. Determine the convective heat transfer coefficient between the product ( = 800 kg/m, k = 0. W/m K, c = 4,000 J/kg K) and the ambient air. p

6 5. A product ( = 975 kg/m, k = 0.55 W/m K, c p = 4,100 J/kg K, = Pa s) at a temperature of 80 C is being conveyed at the rate of 0.5 kg/s in a 1. m long horizontal uninsulated stainless steel pipe (I.D. =.5 cm, O.D..4 cm, k = 16 W/m K). The inside surface temperature of the pipe is 75 C. Determine the convective heat transfer coefficient between product and the pipe. 6. A cylindrical-shaped product (k = 0.54 W/m K, = 105 kg/m, c p =,45 J/kg K) 8 cm in diameter and 0.1 m in height is taken out of the oven when it is at 90 C and placed under ambient conditions (0 C). Determine the center temperature of the product after 90 minutes if the convective heat transfer coefficient between the product and ambient air is 5 W/m K. 7. A well-insulated tubular heat exchanger (I.D. and O.D. of inner pipe are.6 cm and.8 cm resp.; I.D. and O.D. of outer pipe are 4. cm and 4.4 cm resp.; Length = 0 m) is operating in a counter-current mode. Hot water (c p = 400 J/kg K, = Pa s, k = 0.6 W/m K) enters the outer tube at a flow rate of 0. kg/s and a temperature of 90 C and exits at 70 C. The product (c p = 4000 J/kg K, = Pa s, k = 0.4 W/m K) enters the inner tube at 10 C and exits at 60 C. a. What is the overall heat transfer coefficient for transfer of energy from the hot water to the product? b. What is the convective heat transfer coefficient between the product and the inner wall of the inner pipe? 8. A solid sphere ( = 800 kg/m, c p = 4000 J/kg K, k = 0.4 W/m K) of diameter 1 cm is taken out of the refrigerator when it is at 15 C and placed in a water bath that is at 45 C. How long would it take for its center temperature to reach 0 C?

7 Answers to Problems on Heat Transfer W/m K W. a. 6.1 C b. 8. C c. 6.5 C d. 16. C 4. a C b. 6. C W/m K 6. a. 5.7 % b. 74 % 7. Any combination of R and k that satisfies the following equation: o ins Example: R o = 0.06 m (i.e. thickness of insulation = 5 mm), k ins = 0.1 W/m K 8. 4 W/m K 9. a. 5 W/m K b C c. 7. C W/m K 11.,405 W/m K 1. 8,49 W/m K 1. a.,805 W/m K b. 17 W/m K C 15. Yes 16. 1,57 W/m K W/m K ,10 s (= 4.5 hrs) C 0. a. 74 min b min c. Because the lumped parameter analysis is not valid 1. a C b. Iterate between N Fo and TR by using an assumed time, t. 85 s. a. Product b. Between the surface of the product and water c min C 5..6 C 6. 1 min 7. a. T = 45. C b. Iterate between N Fo and TR by using an assumed time, t 8. k = 0.4 W/m K 9. T =,14 s 0. a. h o = 7. W/m K b. Adding any amount of insulation will decrease the heat loss 1. a. U = 1,110 W/m K b. co-current configuration. Both will take the same time. 8.7 C W/m K

8 W/m K C 7. a. 7 W/m K b. 106 W/m K 8. 0 s

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