COVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: MECHANICAL ENGINEERING COURSE: MCE 524
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MCE 524: Heat Transfer Contributor: Dr.S.O Oyedepo Q1 Calculate the rate of heat loss from a furnace wall per unit area. The wall is constructed from an inner layer of 0.5-cm-thick steel (k = 40 W/m K) and an outer layer of 10-cm zirconium brick (k = 2.5 W/m K) as shown in Fig. Q1. The inner- surface temperature is 900 K and the outside surface temperature is 460 K. What is the temperature at the interface? Fig. Q1: Schematic diagram of furnace wall Q2 The outside surface of a cylindrical cryogenic container is at 10 C. The outside radius is 8 cm. There is a heat flow of 65.5 W/m, which is dissipated to the surroundings both by radiation and convection. The convection coefficient has a value of 4.35 W/m 2 K. The radiation factor F = 1. Determine the surrounding temperature. Q3 In a solar flat plate heater some of the heat is absorbed by a fluid while the remaining heat is lost over the surface by convection the bottom being well insulated. The fraction absorbed is known as the efficiency of the collector. If the flux incident has a value of 800 W/m 2 and if the collection temperature is 60 C while the outside air is at 32 C with a convection coefficient of 15 W/m 2 K, determine the collection efficiency. Also find the collection efficiency if collection temperature is 45 C. Q4 Choose the correct statement in each question. 3
(i) A pipe carrying steam at about 300 C traverses a room, the air being still at 30 C. The major fraction of the heat loss will be by (a) conduction to the still air (b) convection to the air (c) radiation to the surroundings (d) conduction and convection put together. (ii)a satellite in space exchanges heat with its surroundings by (a) conduction (b) convection (c) radiation (d) conduction as well as convection. (iii) (iv) For the same temperature drop in the temperature ranges of 300 400 C the heat flow rate will be highest by (a) conduction process (b) convection process (c) radiation process (d) other factors should be known before any conclusion. In the cold season a person would prefer to be near a fire because (a) the conduction from the fire will be better (b) the convection will be better if he is near the fire (c) direct unimpeded radiation will provide quick warmth (d) combined conduction and convection will be better. (a) A finned tube hot water radiator with a fan blowing air over it is kept in rooms during winter. The major portion of the heat transfer from the radiator to air is due to: (a) radiation (b) convection (c) conduction (d) combined conduction and radiation. (v) For a specified heat input and a given volume which material will have the smallest temperature rise (Use data book if necessary) (a) steel (b) aluminium (c) water (d) copper. Q5 A thin metal sheet receives heat on one side from a fluid at 80 C with a convection coefficient of 100 W/m2K while on the other side it radiates to another metal sheet parallel to it. The second sheet loses heat on its other side by convection to a fluid at 20 C with a convection coefficient of 15 W/m2K. Determine the steady state temperature of the sheets. The two sheets exchange heat only by radiation and may be considered to be black and fairly large in size. Q6 A steel tube having k = 46 W/m C has an inside diameter of 3.0 cm and a tube wall thickness of 2 mm. A fluid flows on the inside of the tube producing a convection coefficient of 1500 W/m 2 C on the inside surface, while a second fluid flows across the outside of the tube producing a convection coefficient of 197 W/m 2 C on the outside tube surface. The inside fluid temperature is 223 C while the outside fluid temperature is 57 C. Calculate the heat lost by the tube per meter of length. Q7 A furnace wall is of three layers, first layer of insulation brick of 12 cm thickness of conductivity 0.6 W/mK. The face is exposed to gases at 870 C with a convection coefficient of 110 W/m 2 K. This layer is backed by a 10 cm layer of firebrick of conductivity 0.8 W/mK. There is a contact resistance between the layers of 2.6 10 4 m 2 C/W. The third layer is the plate backing of 10 mm thickness of conductivity 49 W/mK. The contact resistance between the second and third layers is 1.5 10 4 m2 C/W. The plate is exposed to air at 30 C with a convection coefficient of 15 W/m 2 K. Determine the heat flow, the surface temperatures and the overall heat transfer coefficient. Q8 Steam having a quality of 98% at a pressure of 1.37 X 105 N/m 2 is flowing at a velocity of 1 m/s through a steel pipe of 2.7-cm OD and 2.1-cm ID. The heat transfer coefficient at the inner 4
surface, where condensation occurs, is 567 W/m2 K. A dirt film at the inner surface adds a unit thermal resistance of 0.18 m 2 K/W. Estimate the rate of heat loss per meter length of pipe if (a) the pipe is bare, (b) the pipe is covered with a 5-cm layer of 85% magnesia insulation. For both cases assume that the convection heat transfer coefficient at the outer surface is 11 W/m 2 K and that the environmental temperature is 21 C. Also estimate the quality of the steam after a 3-m length of pipe in both cases. Q9 A fin in the form of a ring of 0.25 mm thickness and 15 mm OD and 15 mm long is used on an electric device to dissipate heat. Consider the outer surface alone to be effective and exposed to air at 25 C with a convection coefficient of 40 W/m 2 K. The conductivity of the material is 340 W/mK. If the heat output is 0.25 W and if the device is also of the same OD, determine the device temperature with and without the fin. Q10 A solar collector plate is exposed to a flux of 900 W/m2. Heat is collected by water pipes fixed at 12 cm pitch with a water temperature of 48 C. The plate is 2 mm thick and has a conductivity of 204 W/mK. If the losses over the plate is accounted by a convection coefficient of 15 W/m2K to air at 30 C, determine the maximum temperature in the plate and also the rate of heat collection by the water per pitch width and 1 m length. Q11 For the boundary conditions for the plate shown in Fig. Q11 determine using analytical method the temperature at the midpoint p, under steady two dimensional conduction. (use up to 5 terms in the series summation). Fig. Q11. Problem model Q12 A rectangle 0.5 m 1 m has both the 1 m sides and one 0.5 m side at 200 C. The other side is having a temperature distribution given by T = 200 + 400 sin (π x/0.5) where x is in m and T in C. Locate the y values at x = 0.5 m at which the temperatures will be 300, 400, 500 C. Also locate the values of x for y = 1 m at which these temperatures occur. Q13 The temperature distribution and boundary condition in part of a solid is shown in Fig. Q13. Determine the Temperatures at nodes marked A, B and C. Determine the heat convected over surface exposed to convection. k = 1.5 W/mK. 5
Fig. Q13. Q14 A part of a solid with temperatures at the nodes and the boundaries are shown in Fig. Q14. Determine the temperature at node A and also the heat flow over the convecting surface. The top surface is exposed to convection at 300 C with h = 10 W/m 2 K. Fig. Q14 Q15 Nitrogen at a pressure of 0.1 atm flows over a flat plate with a free stream velocity of 8 m/s. The temperature of the gas is 20 C. The plate temperature is 20 C. Determine the length for the flow to turn turbulent. Assume 5 10 5 as critical Reynolds number. Also determine the thickness of thermal and velocity boundary layers and the average convection coefficient for a plate length of 0.3 m. Properties are to be found at film temperature. Q16 A thin conducting plate separates two parallel air streams. The hot stream is at 200 C and 1 atm pressure. The free stream velocity is 15 m/s. The cold stream is at 20 C and 2 atm pressure and the free stream velocity is 5 m/s. Determine the heat flux at the mid - point of the plate of 1 m length. Q17 Air at 1 atm with a temperature of 500 C flows over a plate 0.2 m long and 0.1 m wide. The Reynolds number is 40,000. (Flow is along the 0.2 m side). Determine the rate of heat transfer from the plate at 100 C to air 50 C. If the velocity of flow is doubled and the pressure is increased to 5 atm, determine the percentage change. The properties of air are read from tables and interpolated for film temperature of 75 C. Q18 6
A radioactive sample is to be stored in a protective box with 4-cm-thick walls and interior dimensions of 4 cm X 4 cm X 12 cm. The radiation emitted by the sample is completely absorbed at the inner surface of the box, which is made of concrete. If the outside temperature of the box is 25 C but the inside temperature is not to exceed 50 C, determine the maximum permissible radia- tion rate from the sample, in watts. Q19 A surface with A = 2 cm 2 emits radiation as a blackbody at T= 1000 K. (a) Calculate the radiation emitted into a solid angle subtended by 0 2π and 0 θ π 6 (b) What fraction is the energy emitted into the above solid angle of that emitted into the entire hemispherical space? Q20 Greenhouse effect is nothing but trapping of radiation by letting in radiation of short wavelength and shutting out radiation of long wavelength. A green house has a roof area of 100 m 2 perpendicular to the solar inclination. The material has a transmissivity of 0.9 up to a wavelength of 4 µm and zero beyond. The solar flux has a value of 800 W/m 2. The total wall area is 600 m 2. It the inside is to be maintained at 22 C while the outside is at 5 C, determine the maxi- mum value of overall heat transfer coefficient for heat flow through the walls. The temperature of solar radiation may be taken as 5000 K. MODEL ANSWERS A1 Assumptions Assume that steady state exists, neglect effects at the corners and edges of the wall, and assume that the surface temperatures are uniform. The rate of heat loss per unit area can be calculated from equation given below: 7
N.B: The temperature drop across the steel interior wall is only 1.4 K because the thermal resistance of the wall is small compared to the resistance of the brick, across which the temperature drop is many times larger. A3 Solution: The heat lost by convection = Q = ha(t 1 T 2 ) Fig. A3 8
A5 The energy balance provides (Fig. A5) heat received convection by Sheet 1 = heat radiation exchange between sheet 1 and 2. = heat convected by sheet 2. 9
A7 The data and equivalent circuit are shown in Fig A7. Using equation: Fig. A7. Composite wall. 10
Note: The contact drops and drop in the metal plate are very small. The insulation resistances and outside convection are the controlling resistances. A9 The heat is lost from the surface of the device by convection without fin: 11
Fig. A9 A11 Using equation: 12
A13 Considering A 13
A15 Film temperature = ( 20 + 20)/2 = 0 C As density and kinematic viscosities will vary with pressure, dynamic viscosity is read from tables. 14
A17 15
A19 (a) The radiation energy emitted by an area A streaming through a differential solid angle dω = sinθdθd in any direction is given by 16