ME 4310 Heat Transfer Summer II, 2013 Example Problems

Transcription

1 ME 4310 Heat Transfer Summer II, 2013 Example Problems Dr. Bade Shrestha G-218, Department of Mechanical and Aerospace Engineering

2 Example 1 (Conduction) One face of a copper plate 3 cm thick is maintained at 400 o C, and the other face is maintained at 100 o C. How much heat is transferred through the plate?

3 Example 2 (Convection) Air at 20 o C blows over a hot plate 50 by 75 cm maintained at 250 o C. The convection heat transfer coefficient is 25 W/(m 2. o C). Calculate the heat transfer.

4 Example 3 (Multimode) Assuming that the plate in Example 2 is made of carbon steel 2 cm thick and that 300 W is lost from the plate surface by radiation, calculate the inside plate temperature.

5 Example 4 (Plane Wall) Consider a large plane wall of thickness L=0.2 m, thermal conductivity k= 1.2W/m K and surface area A = 15 m2. The two side of wall are maintained at constant temperature of T1=120 o C and T2 = 50 o C, respectively. Determine a) the variations of temperature within the wall and the value of temperature at x=0.1m and b) the rate of heat conduction through the wall under steady conditions.

6 Example 5 (k(t)) Consider a 2 m high and 0.7 m wide bronze plate whose thickness is 0.1 m. One side of the plate is maintained at a constant temperature of 600 K while the other side is maintained at 400 K. The thermal conductively of the bronze plate can be assumed to vary linearly in that temperature range as k(t)=ko(1+βt) where ko=38 W/mK and β = 9.21 X 10-4 K -1. Determine the rate of heat conduction through the plate assuming steady state conditions.

7 Example 6 (heat generation) A plane wall of thickness 0.1m and thermal conductivity 25 W/mK having uniform volumetric heat generation of 0.3MW/m3 is insulated on one side, while the other side is exposed to a fluid at 92 o C. The convection heat transfer coefficient between the wall and the fluid is 500 W/m2K. Determine the maximum temperature in the wall.

8 Example 7 (Multilayer) A thermo pane window consists of two pieces of glass 7 mm thick that enclose an air space 7 mm thick. The window separates room air at 20 o C from outside ambient air at -10 C. The convection coefficient associated with the inner (room-side) surface is 10 W/m2K. If the convection coefficient associated with the outer (ambient) air is 80 W/M2k, what is the heat loss through a window that is 0.8m long by 0.5 m wide? Neglect radiation, and assume the air enclosed between the glasses to be stagnant.

9 Example 8 (Over all heat transfer) Two by four wood studs have actual dimensions of 4.13X9.21 cm and a thermal conductivity of 0.1 W/m2K. A typical wall of a house is constructed as shown in the figure. Calculate the over all heat transfer coefficient and R value of the wall.

10 Example 9 (Fin) Compare the temperature distributions in a straight cylindrical rod having a diameter of 2 cm and a length of 10 cm and exposed to a convection environment with h = 25 W/m2K for three fin materials: copper (k=385w/m2k), stainless steel (k=17w/mk) and glass (k=0.8 W/mK). Also compare the relative heat flows and fin efficiencies.

11 Example 10 (Fin) An aluminum fin (K=200 W/mK) 3.0 mm thick and 7.5 cm long protrudes from a wall. The base is maintained at 300 o C, and the ambient temperature is 50 o C with h=10w/m2k. Calculate the heat loss from the fin per unit depth of material.

12 Example 12 (Unsteady) A steel ball (c= 0.46 kj/kg K, k=35w/mk) 5 cm in diameter and initially at a uniform temperature of 450 C is suddenly placed in a control environment in which the temperature is maintained at 100 C. The convective heat coefficient is 10 W/m2K. Calculate the time required for the ball to attain a temperature of 150 C.

13 Example 13 A large block of steel (k=45w/mk, α = m 2 /s) is initially at a uniform temperature of 35 o C. The surface is exposed to heat flux (a) by suddenly raising the surface temperature to 250 o C and (b) through a constant surface heat flux of 3.2x10 5 W/m 2. Calculate the temperature at a depth of 2.5 cm after a time of 0.5 min for the both these cases.

14 Example 14 (boundary layer) Air at 27 C and 1 atm flows over a flat plate at a speed of 2 m/s. Calculate the boundary layer thicknesses at distances of 20 and 40 cm from the leading edge of the plate. Calculate the mass flow that enters the boundary layer between x=20 and x=40 cm.

15 Example 14a (Laminar) Air at 27 C and 1 atm flows over a flat plate at a speed of 2 m/s. Calculate the boundary layer thicknesses at distances of 20 and 40 cm from the leading edge of the plate. Calculate the mass flow that enters the boundary layer between x=20 and x=40 cm. And assuming that the plate is heated over its entire length to a temperature of 60 C, calculate the heat transferred in the first 20 cm of the plate and the first 40 cm of the plate.

16 Example 14b contd. For the flow system in example 14, calculate the drag force exerted on the first 40 cm of the plate using the analogy between fluid friction and heat transfer.

17 Example 14c The leading edge of a wing is to be heated to a constant temperature of 3 o C to prevent ice formation. How much heat must be supplied to the heating system per meter of wing span? (length of the heating edge is 10 cm, stream velocity is 200 Km per hour and ambient temperature is -15 o C)

18 Example 15 Assuming a transition Reynolds number of 5X 10 5, determine the distance from the leading edge of a flat plate at which the transition will occur for the following fluids when u = 1 m/s and temperature = 27 o C; atmospheric air, engine oil, water and mercury.

19 Example 16 (Tub. Heat) Air at 20 o C and 1 atm flows over a flat plate at 35 m/s. The plate is 75 cm long and is maintained at 60 C. Assuming unit depth: a) calculate the heat transfer from the plate b) critical distance from the leading edge when the flow becomes turbulent. c) and thickness of the boundary layers at the critical distance and the end of the plate.

20 Example 17 (turb. H. T.) A flat plate of width 1m is maintained at a uniform surface temperature of Tw = 150 o C by using independently controlled, heat generating rectangular modules of thickness a = 10 mm and length b = 50 mm. Each module is insulated from its neighbors, as well as on its back side. Atmospheric air flows at 25 o C over the plate at a velocity of 30 m/s. Find the required power generation (W/m3), in a module positioned at a distance 700 mm from the leading edge. Find the maximum temperature in the heat-generating module. (Take k =5.2 W/mK; cp = 320 j/kg K and ρ = 2300 kg/m3 for the module).

21 Example 18 (cylinder) Air at 1 atm and 35 o C flows across a 5 cm diameter cylinder at a velocity of 50 m/s. The cylinder surface is maintained at a temperature of 150 o C. Calculate the heat loss per unit length of the cylinder.

22 Example 19 (sphere) Air at 1 atm and 27 o C blows across a 12 mm diameter sphere at a free stream velocity of 4 m/s. A small heater inside the sphere maintains the surface temperature at 77 o C. Calculate the heat lost by the sphere.

23 Example 20 (Lam. Pipe) Water at 60 C enters a tube of 2.54 cm diameter at a mean flow velocity of 2 cm/s. Calculate the exit water temperature if the tube is 3 m long and the wall temperature is constant at 80 C. (neglect the entrance effect).

24 Example 21 (Entrance) Water at 60 C enters a tube of 2.54 cm diameter at a mean flow velocity of 2 cm/s. Calculate the exit water temperature if the tube is 3 m long and the wall temperature is constant at 80 C. (including the entrance effect).

25 Example 22 (Free Conv.) A large vertical plate 4 m high is maintained at 60 C and exposed to atmosphere air at 10 C. Calculate the heat transfer if the plate is 10 m wide. Find the location where boundary layer becomes turbulent. And maximum velocity in the boundary layer at this location and position of maximum. Find the boundary layer thickness at this position.

26 Example 23 (LMTD) Water at the rate of 68 Kg/min is heated from 35 to 75 o C by an oil having a specific hate of 1.9 kj/kg o C. The fluids are used in a counter flow double pipe heat exchanger, and the oil enter the exchanger at 110 o C and leaves at 75 o C. Calculate the overall heat transfer coefficient if inner diameter of the pipe is 30 mm and the outer annulus diameter is 50 mm, and the length of the heat

27 Example contd. If the overall heat-transfer coefficient is 320 W/m 2 o C, and instead of the double pipe heat exchanger of the previous example, it is desired to use a shell and tube exchanger with water making one shell pass and the oil making two tube pass,calculate the new area.

28 Example 24 (Overall HT coefficient) Hot oil be cooled in a double tube counter flow heat exchanger. The copper inner tubes have a diameter of 2 cm and negligible thickness. The inner diameter of the outer tube (the shell) is 3 cm. Water flows through the tube at a rate of 0.5 kg/s, and the oil through the shell at a rate of 0.8 kg/s. Taking the average temperatures of water and oil to be 45 o C and 80 o C, respectively, determine the overall heat coefficient of this heat exchanger.

29 Example 25 (Fouling) A double pipe (shell-and-tube) heat exchanger is constructed of a stainless steel (k=15.1 W/mK) inner diameter D i =1.5 cm and outer diameter D o =1.9 cm and an outer shell of inner diameter 3.2 cm. The convective heat coefficient is given to be h i = 800 W/m 2 k on the inner surface of the tube and h o = 1200 W/m 2 K on the outer surface. For a fouling factor of R fi = m 2.K/W on the inside tube and R fo = m 2.K/W on the shell side, determine: a) the thermal resistance of the heat exchanger per unit length. b) the overall heat transfer coefficient U i and U o base on the inner and outer surface areas of the tube respectively.

30 Example 26 (Radiator) A test is conducted to determine the overall heat transfer coefficient in an automotive radiator that is a compact cross-flow water-to-air heat exchanger with both fluids unmixed. The radiator has 40 tubes of internal diameter 0.5 cm and length 65 cm in a closely spaced plate-fin matrix. Hot water enters the tubes 90 o C at a rate of 0.6 Kg/s and leaves at 65 o C. Air flows across the radiator through the inter-fin spaces and is heated from 20 o C and 40 o C. Determine the overall heat transfer coefficient, U i of this radiator based on the inner surfaces area of the tubes.

31 Example 27 (NTU) A couter flow double tube heat exchanger is to used to heat water from 20 o C to 80 o C at a rate of 1.2 Kg/s. The heating is to be accomplished by geothermal water available at 160 o C at a mass flow rate of 2 kg/s. The inner tube is thin-walled and has a diameter of 1.5 cm. The overall heat transfer coefficient of the heat exchanger is 640 W/m 2 K. Using the NTU method determine the length of the exchanger.

32 Example 28 (NTU) Hot oil is to be cooled by water in a 1-shell-pass and 8- tube-passes heat exchanger. The tubes are thin-walled and are made of copper with an internal diameter of 1.4 cm. The length of each tube pass in the heat exchanger is 5 m, and the overall heat transfer coefficient is 310 W/m 2 K. Water flows through the tubes at a rate of 0.2 kg/s, and the oil through the shell at a rate of 0.3 kg/s. The water and oil enter at temperatures of 20 o C and 150 o C, respectively. Determine the rate of heat transfer in the heat exchanger and the outlet temperatures of water and the oil.

33 Example 29 (NTU) Hot oil at 100 o C is used to heat in a shelland-tube heat exchanger. The oil makes six tube passes and the air makes one shell pass; 2.0 kg/s of air are to be heated from 20 to 80 o C. The specific heat of the oil is 2100 J/kg o C an its flow rate is 3.0 kg/s. Calculate the area required for the heat exchanger for U = 200 W/m 2 o C.

34 Example 30 (NTU) A counter flow double pipe heat exchanger is used to heat 1.25 kg/s of water from 35 o to 80 o C by cooling and oil ( cp=2 kj/kg o C) from 150 o to 85 o C. The overall heat transfer coefficient is 850 W/m 2 C. A similar arrangement is to be built at another plant location, but it is desired to compare the performance of the single counter flow heat exchanger with two smaller counter flow heat exchangers connected with series on the water side and in parallel on the oil side. The oil flow is split equally between two exchangers, and it may be assumed that the overall heat transfer coefficient for the smaller exchangers is the same as for the large exchanger. If the smaller exchangers cost 20 % more per unit of surface area, which would be the most economical arrangement- the one large exchanger or two equal sized small exchangers?