CHAPTER 2 STEADY STATE HEAT CONDUCTION

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1 CHAPTER 2 STEADY STATE HEAT CONDUCTION 1. A storage chamber of interior dimensions high has its inside maintained at a temperature of whilst the outside is at. The walls and ceiling of the chamber have three layers made of 60 mm thick board ( ) on the inside 90 mm thick insulation ( ) at the mid 240 mm thick concrete ( ) on the outside Neglecting flow of heat through the floor, determine the rate at which heat can flow towards inside of the chamber. (D.S. Kumar, Example 3.13) 2. A 8 mm thick metal plate, having thermal conductivity is exposed to vapor at 100 on one side and cooling water at 30 on another side. The heat transfer coefficients are on vapor side and on water side. Determine the rate of heat transfer and drop in temperature on each side of the plate. Assume area of the plate as unity. (Summer 2014) (Similar to D.S. Kumar, Example 3.40) 3. A composite wall has three layers of material held together by 3 cm diameter aluminium rivet per 0.1m2 of surface. The layer of material consists of 10 cm thick brick with hot surface at, 1cm thick wood with cold surface at. These two layers are interposed by third layer of insulating material 25cm thick. The conductivity of the material are: Assuming one dimensional heat flow, calculate the percentage change in heat transfer rate due to rivets. (Summer 2015) 4. A steel tube of 5 cm inner diameter and 8 cm outer diameter ( ), is covered with an insulation of 3 cm thickness ( ). A hot gas at, flows. Calculate the heat loss from the tube for 20 meter length. Also calculate the temperature at the interface of insulation and steel. Outside air Darshan Institute of Engineering and Technology, Rajkot 1

2 temperature is at,. (May-2012) (Similar to Mahesh Rathod, Example 3.29) 5. A steam pipe 8 cm in diameter is covered with 3 cm thick layer of insulation which has a surface emissivity of 0.9. The surface temperature of the insulation is and the pipe is placed in atmospheric air at. Considering heat loss by both radiation and natural convection calculate: (Dec-2011) I. The heat loss from the 7 m length of pipe. II. The overall heat transfer coefficient and the heat transfer coefficient due to radiation alone. The thermo physical properties of air at mean film temperature of 52 are as following: (where the notations have their usual meaning.) use empirical correlation for horizontal cylinders as ( ). 6. A refrigeration suction line having outer diameter 30 mm is required to be thermally insulated. The outside air convective heat transfer coefficient is. The thermal conductivity of the insulating material is. Determine: I. Whether the insulation will be effective II. Estimate the maximum value of thermal conductivity of insulating material to reduce heat transfer III. The thickness of cork insulation to reduce the heat transfer to 20% (k=0.04 W/m o C) (Summer 2013) Darshan Institute of Engineering and Technology, Rajkot 2

3 CHAPTER 3 FIN HEAT TRANSFER ( ) 1. Two long rods of the same diameter, one made of brass (k = 85 W/m-deg) and the other of copper (k = 375 W/m-deg), having one of their ends inserted into a furnace. At a section 10.5 cm away from the furnace, the temperature of the brass rod is 120. At what distance from the furnace end, the same temperature would be reached in the copper rod. Both rods are exposed to the same environment. (D.S. Kumar, Example 5.1) 2. A rod of 10 mm square section and 160 mm length with thermal conductivity of 50 W/m-deg protrudes from a furnace wall at 200, and is exposed to air at 30 Darshan Institute of Engineering and Technology, Rajkot 1 with convection coefficient 20 W/m 2 -deg. Make calculations for the heat convected upto 80 mm and 158 mm lengths and comment on the result. Adopt a long fin model for the arrangement. (D.S. Kumar, Example 5.4) 3. A rod of 10 mm diameter and 80 mm length with thermal conductivity 16 W/m-deg protrudes from a surface at 160. The rod is exposed to air at 30 with a convection coefficient of 25 W/m 2 -deg. How does the heat flow from this rod get affected if the same material volume is used for two fins of the same length? Assume short fin with end insulated. (D.S. Kumar, Example 5.11) 4. A 5 cm diameter rod, 90 cm long is having its lower face grinded smooth. The remainder of the rod is exposed to the 32 room air and a surface coefficient heat transfer equal to 6.5 W/m 2 -deg exists between the rod surface and the room air. The grinder dissipates mechanical energy at the rate of 35 W. If thermal conductivity of rod material is 41.5 W/m-deg, find the temperature of the rod at the point where the grinding is taking place. (D.S. Kumar, Example 5.21) 5. A thermometric pocket is a hollow brass tube (k = 75 W/m-deg) having outer and inner diameter of 15 mm of 10 mm respectively. The pocket extends to 5 cm depth from the wall of a 15 cm diameter pipe which carries hot air. The heat transfer coefficient between the pocket and air is prescribed by the relation: Nusselt number Nu = (Re) Make calculation for the error in temperature measurement. Presume the following data: (D.S. Kumar, Example 5.31)

4 Air temperature 160 and pipe wall temperature 40 HEAT TRANSFER ( ) Reynolds number and thermal conductivity of air W/m-deg Darshan Institute of Engineering and Technology, Rajkot 2

5 CHAPTER 4 TRNASIENT HEAT CONDUCTION 1. A potato with mean diameter of 4 cm is initially at. It is placed in boiling water for 5 minute and 30 seconds and found to be boiled perfectly. For how long should be a similar potato for the same consumer be boiled when taken from cold storage at. (Summer-2015) (Similar to D.S. Kumar, Example 6.7) Use lumped system analysis and take thermophysical properties of potato as 2. A solid sphere of 1 cm made up of steel is at initially at temperature. Properties of steel:, Density =, Sp. Heat = Calculate the time required for cooling it up to I. cooling medium is air at with in the following two cases II. cooling medium is water at with (Winter-2013) (Similar to D.S. Kumar, Example 6.7) Darshan Institute of Engineering and Technology, Rajkot 1

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7 CHAPTER 5 & 6 RADIATION HEAT TRANSFER ( ) 1. Determine the view factor from any one side to any other side of the infinitely long triangular duct whose cross section is given in figure 1. (Cengel; Example 13.4) Figure 1 Infinitely long triangular duct 2. A furnace is shaped like a long equilateral triangular duct, as shown in Figure 2. The width of each side is 1 m. The base surface has an emissivity of 0.7 and is maintained at a uniform temperature of 600 K. The heated left-side surface closely approximates a blackbody at 1000 K. The right-side surface is well insulated. Determine the rate at which heat must be supplied to the heated side externally per unit length of the duct in order to maintain these operating conditions. (Cengel; Example 13.9) Figure 2 The triangular furnace 3. Determine the rate of heat loss by radiation from a steel tube of outside diameter 7 cm and length 3 m at a temperature of 227 brick conduit of 0.3 m side and at 27 if the tube is located within a square. Take emissivity of steel and brick as 0.79 and 0.93 respectively. (Summer 2014) )(Similar to D.S. Kumar; Example 8.30) 4. Two large parallel plates with emissivity (ε) = 0.5 each, are maintained at different temperatures and are exchanging heat only by radiation. Two equally large radiation Darshan Institute of Engineering and Technology, Rajkot 1

8 shields with surface emissivity 0.05 are introduced in parallel to the plates. Find percentage reduction in net radiative heat transfer. (May 2011) (Similar to D.S. Kumar; Example 8.38) 5. Calculate the net radiation heat transfer per m 2 area of two large plates placed parallel to each other at temperatures of and respectively. ( ) and ( ). If a polished aluminum shield is placed between them, find the % reduction in heat transfer, ( ). (Summer 2015) (Similar to D.S. Kumar; Example 8.38) Darshan Institute of Engineering and Technology, Rajkot 2

9 CHAPTER 7 CONVECTION HEAT TRANSFER ( ) 1. A hot square plate 40cm x 40cm at 100 C is exposed to atmospheric air at 20 C. Make calculations for the heat loss from both surfaces of the plate, if (a) plate is kept vertical (b) plate is kept horizontal. The following empirical correlations have been suggested: Nu = (Gr Pr) 0.33 for vertical position of plate, and Nu = 0.72 (Gr Pr) 0.25 for upper surface Nu = 0.35 (Gr Pr) 0.25 for lower surface [Ans: W] [D.S. Kumar 11.9, GTU - JAN 2013] 2. Calculate the rate of heat loss from a human body which may be considered as vertical cylinder 30 cm in diameter and 175 cm high in still air at 15 C.The skin temperature is 35 C and emissivity at the skin surface is 0.4. Neglect sweating and effect of clothing. Use Nu = 0.13 (Gr Pr) [Ans: W] [D.S. Kumar 11.15] 3. Estimate the heat transfer from a 40W incandescent bulb at 120 C to 20 C quiescent air. Approximate the bulb as a 50 mm dia. Sphere. What percentage of power is lost by free convection? The approximate co-relation is, ( ). [Ans: W; 19.62%] 4. A steam pipe 8 cm in diameter is covered with 3 cm thick layer of insulation which has a surface emissivity of 0.9.The surface temperature of the insulation is 80 C and the pipe is placed in atmospheric air at 24 C. Considering heat loss by both radiation and natural convection calculate: (1) The heat loss from the 7 m length of pipe. (2) The overall heat transfer coefficient & heat transfer co-efficient due to radiation alone. Use empirical correlation for horizontal cylinders as, ( ) [Ans: W; W/m 2 - C; W/m 2 - C][GTU DEC 2011] 5. Water at 10 C, flows over a flat plate (at 90 C) measuring 1 m X 1 m, with a velocity of 2 m/s. Determine, (a) The length of plate over which the flow is laminar (b) The rate of heat transfer up to the above length (c) The rate of heat transfer from the entire plate. Useful correlation: ( ) ( ) [ ( ) ]( ) [Ans: 0.139m; KW; 471KW][GTU MAY 2013] 6. Air at 20 C is flowing over a flat plate which is 200mm wide and 500mm long. The plate is maintained at 100 C. Find the heat loss from the plate if the air is flowing Darshan Institute of Engineering and Technology, Rajkot 1

10 parallel to 500mm side with 2m/s velocity. What will be the effect on heat transfer if the flow is parallel to 200mm side? For laminar flow over flat plate, use following correlation: ( ) ( ) [Ans: W, 85.6 W] [R. K. Rajput; 7.15] 7. Air at 20 C and at a pressure of 1 bar is flowing over a flat plate at a velocity of 3m/sec. If the plate is 280 mm wide and at 56 C, calculate the following quantities at x = 280 mm. a. Hydrodynamic boundary layer thickness b. Local and average friction coefficient c. Shearing stress due to friction d. Thickness of thermal boundary layer e. Local and average convective heat transfer coefficient f. Rate of heat transfer by convection g. Total drag force on the plate and h. Total mass flow rate through the boundary [Ans: Using Blasius Solution:-6.26mm; ; ; N/m 2 ; 6.88mm; 6.43W/m 2 K; 12.86W/m 2 K; 36.29W; N; kg/s] [4.6; P. K. NAG] 8. A plate of length 750mm has been placed longitudinally in a stream of crude oil which flows with a velocity of 5 m/sec. If the oil has a specific gravity of 0.8 and kinematic viscosity of 1 x 10-4 m 2 /sec, calculate, a. Boundary layer thickness at the middle of plate b. Shear stress at the middle of plate and c. Friction drag on one side of the plate. [Ans: m; N/m 2 ; N] 9. Air at 20 C and at atmospheric pressure flows at a velocity 4.5 m/s past a flat plate with a sharp leading edge. The entire plate surface is maintained at a temperature of 60 C. Assuming that the transition occurs at a critical Reynolds number of , find the distance from the leading edge at which the boundary layer changes from laminar to turbulent. At the location calculate: (1) thickness of hydrodynamic and thermal boundary layer, (2) Local and average heat transfer coefficients, (3) Heat transfer rate from both sides per unit width of plate. Use ( ) ( ) Assume cubic velocity profile and approximate method. [Ans: 12.34mm; 13.55mm; 3.05W/m 2 K; 6.1W/m 2 K; 917.4W] [GTU MAY 2012] [4.7; P. K. NAG] 10. The air at atmospheric pressure and temperature of 30 C flows over one side of plate of a velocity of 90 m/min. This plate is heated and maintained at 100 C over its entire length. Find out the following at 0.3 and 0.6 m from its leading edge. (1) Thickness of velocity boundary layer and thermal boundary layer. (2) Mass flow rate Darshan Institute of Engineering and Technology, Rajkot 2

11 which enters the boundary layer between 0.3 m and 0.6 m per metre depth of plate. Assume unit width of plate. [Ans: 8.30mm; 9.119mm; 11.73mm; 12.88mm; kg/s] [GTU MAY 2012] Use following properties of fluid at required temperature, Example Temp ρ Cp ν x 10 Fluid K µ Pr No. C kg/m 3 KJ/kg-deg m 2 /sec W/m-deg kg/m-hr 1 Air Air Air Air Water Air Air Air Air Darshan Institute of Engineering and Technology, Rajkot 3

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13 CHAPTER 9 HEAT EXCHANGERS HEAT TRANSFER ( ) 1. Exhaust gases (Cp = 1.12 kj/kg-deg) flowing through a tubular heat exchanger at the rate of 1200 kg/hr are cooled from 400 C to 120 C. The cooling is affected by water (Cp = 4.18 kj/kg-deg) that enters the system at 10 C at the rate of 1500 kg/hr. If the overall heat transfer co-efficient is 500 kj/m 2 -hr-deg, what heat exchanger area is required to handle the load for (a) Parallel flow and (b) Counter flow arrangement? [Ans: m 2, m 2 ] [14.9, D. S. Kumar] 2. A counter flow concentric tube heat exchanger is used to cool the lubricating oil of a large industrial gas turbine engine. The oil flows through the tube at 0.19 kg/s (Cp = 2.18 kj/kg-k), and the coolant water flows in the annulus in the opposite direction at a rate of 0.15 kg/sec (Cp = 4.18 kj/kg-k). the oil enters the coolant at 425 K and leaves at 345 K while the coolant enters at 285 K. how long must the tube be made to perform this duty if the heat transfer co-efficient from oil to tube surface is 2250 W/m 2 K and from tube surface to water is 5650 W/m 2 K? The tube has a mean diameter of 12.5 mm and its wall presents negligible resistance to heat transfer. [Ans: 7.21 m] [14.17, D. S. Kumar] 3. A one-shell two-tube pass heat exchanger having 3000 thin wall brass tubes of 20 mm diameter has been installed in a steam power plant with a heat load of 2.3X10 8 W. the steam condenses at 50 C and the cooling water enters the tubes at 20 C at the rate of 3000 kg/s. Calculate the overall heat transfer co-efficient, the tube length per pass, and the rate of condensation of steam. Take the heat transfer co-efficient for condensation on the outer surfaces of the tubes as W/m 2 K and the latent heat of steam as 2380 kj/kg. further presume the following fluid properties: c = 4180 J/kg-K, μ = 855 X 10-6 Ns/m 2, k = W/m-k and Pr = 5.83 [Ans: 6524 W/m 2 K, 4.82 m, kg/s] [14.18, D. S. Kumar] 4. A heat exchanger is to be designed to condense 8 kg/s of an organic liquid (tsat = 80 C; hfg = 600 kj/kg) with cooling water available at 15 C and at a flow rate of 60 kg/s. The overall heat transfer co-efficient is 480 W/m 2 -deg. Calculate: Darshan Institute of Engineering and Technology, Rajkot 1

14 a. The number of tubes required. The tubes are to be of 25 mm outer diameter, 2 mm thickness and 4.85 m length. b. The number of tube passes. The velocity of the cooling water is not to exceed 2 m/s. [Ans: 478 tubes, 6 passes] [14.19, D. S. Kumar] 5. A surface condenser used in a steam power plant deals with kg of steam per hour at a pressure of 4.15 kn/m 2 and 0.9 dryness fraction. The cooling medium will be water that enters the condenser at 15 C and leaves at 25 C. From previous experience, a water velocity of 1.5 m/s is maintained through the tubes and the overall co-efficient of heat transfer is estimated at 3500 W/m 2 -K. Calculate : a. Mass flow rate of water, b. Surface area required for the given duty and c. Passes and number of tubes. The tubes used in condenser are 20 mm outside diameter, 1.5 mm thick and the space limitation restricts the condenser length to 4 meters. At the condensing pressure, steam has saturation temperature ts = 29.5 C and latent heat of vaporization hfg = 2435 kj/kg. Presume that the condensate coming out of the condenser is saturated water at the condenser pressure, i.e., there is no under cooling and the steam losses only latent part of its heat. [Ans: X 10 6 kg/hr, m 2, 1158 tubes, 2 passes] [14.22, D. S. Kumar] 6. Calculate the surface area required for a heat exchanger which is required to cool 3600 kg/hr of benzene (Cp = 1.74 kj/kg-k) from 75 C to 45 C. The cooling water (Cp = 4.18 kj/kg-deg) at 15 C has a flow rate of 2500 kg/hr. consider the following arrangements: a. Single pass counter flow b. 1-4 exchanger (one shell pass and four tube passes) c. Cross flow single pass with water mixed and benzene unmixed. The overall heat transfer co-efficient for each configuration is approximated to be 0.3kW/m 2 -K. [Ans: 4.87 m 2, 5.29 m 2, 5.18 m 2 ] [14.35, D. S. Kumar] Darshan Institute of Engineering and Technology, Rajkot 2

15 7. A counter flow heat exchanger is used to cool 2000 kg/hr of oil (cp = 2.5 kj/kg-k) from 105 C to 30 C by the use of water entering at 15 C. If the overall heat transfer co-efficient is expected to be 1.5 kw/m 2 K, make calculations for the water flow rate, the surface area required and the effectiveness of heat exchanger. Presume that the exit temperature of the water is not to be exceed 80 C. Use NTU-effectiveness approach. [Ans: kg/hr, 3.55 m 2, 0.833] [14.41, D. S. Kumar] 8. In a surface condenser, the water flowing through a series of tubes at the rate of 200 kg/hr is heated from 15 C to 75 C. The steam condenses on the outside surface of tubes at atmospheric pressure and the overall co-efficient of heat transfer is estimated at 860 kj/m 2 -hr-deg. Use NTU method to work out the length of tube and the steam condensation rate. Presume that the tube is 25 mm in diameter. At the condensing pressure, steam has saturation temperature ts = 100 C and the latent heat of vaporization hfg = 2160 kj/kg. Further, the steam is initially just saturated and the condensate leaves the exchanger without sub-cooling, i.e., only the latent heat of condensing steam is transferred to water. Take specific heat of water as 4 kj/kg-k. [Ans: 14.5 m, kg/hr] [14.45, D. S. Kumar] 9. A tube type heat exchanger is used to cool hot water from 80 C to 60 C. The task is accomplished by transferring heat to cold water that enters the heat exchanger at 20 C and leaves at 40 C. Should this heat exchanger operate under counter flow or parallel flow conditions? Also determine the exit temperatures if the flow rates of fluids are doubled. [Ans: C, C] [14.47, D. S. Kumar] Darshan Institute of Engineering and Technology, Rajkot 3