Heat Transfer Problem Set Note: 1. Please make the solution according the following headers: Given, To Find, Assumptions, Properties (Give Reference – Book and Table No.), and Schematic if any, Actual solution. 2. Please begin each problem on a new page, so that book keeping is easy. 3. Be neat, your work will not be graded if it is sloppy and unreadable. 4. Final answers must be clearly identified and given to two significant digits. Appropriate units must be used for all intermediate values and the final answer during the calculation.
Conduction-1 Find the heat transfer per unit area through the composite wall shown in figure below. The thermal conductivities are given as: k1 = 50 W/m K, k2 = 30 W/m K, k3 = 6 W/m K, k4 = 50 W/m K, L1=3cm, L2=8cm, L3=5cm, Area of 2 = Area of 3 = 0.5 m2. Also design a composite wall as shown in the figure so as to limit the heat transfer through the wall 80% of its original value by altering thickness of L3.
A cast iron pipe (k = 80 W/m K) having inside diameter 5cm and outside diameter 5.5cm carries steam at 3200c with a convective heat transfer coefficient of 60 W/m2K to reduce the heat loss the pipe is covered with 3cm thick glass wool (k = 0.05 W/m K). If the surroundings are 20 0 C and the combined convection and radiation heat transfer coefficient outside is 18 W/m2 K. calculate the rate heat loss per meter length, intermediate temperatures and overall heat transfer coefficient based on outside surface area. (a)Derive the equation for temperature distribution in a hollow cylinder with internal heat generation. (b) A current of 200 amps is passed through a Ni – chromium wire(k = 17 W/m K) of 3 mm diameter. The resistivity of Ni – cr is 100 cm and the length of the wire is 1m. The wire is submerged in a liquid at 1200C and experiences a convective heat transfer coefficient of 4000 W/m2K. Calculate the surface temp& central Temp of the wire. A refrigerator door has dimensions 120cm x 45cm and is made up of 3 mm thick mild steel of k = 40 W/m K with 5cm of polyurethane foam insulation having k = 0.04 W/m K sandwiched between them. The average values of convective heat transfer coefficient at the interior and exterior of the door are 10 W/m2 K and 13 W/m2K respectively. The temperature inside the refrigerator is 3oC and outside is 20oC. Determine (a) the rate of heat leak through door (b) interface temperatures and (c) overall heat transfer coefficient. Design the refrigerator door by replacing the mild steel by FRP of same thickness having k=8 w/mK. Compare the overall heat transfer coefficient of your design and the original design.
An electric current of 34000 amps flows along a steel plate 1.25 cm thick and 10 cm wide. The temperature of one surface of the plate is at 800C and that of the other is at 950C. Find the
temperature distribution across the plate and the value, position of maximum temperature. Also calculate the total amount of heat generated per meter length of plate and flow of heat from each surface of the plate. Take for steel 12 cm and k = 54 W/m K. In an electronic circuit a resistor of variable cross section is connected between two points maintained at 800C and 200C. Estimate the maximum permissible joulean heating if the resistor configuration is a frustum of cone having 20 and 15mm dia at its extremes and length of the resistor is 2cm. Assume k for resistor 25 W/m K. If the current passing through it 1.5 amps calculate the resistance of the resistor. The insulation boards for air conditioning purposes are made of three layers, middle one being packed grass of 10 cm thick (k = 0.02 W/m-K) and the sides are made of plywood’s of 2 cm thick each (k = 0.12 W/m-K). They are glued with each other. (a) Determine the heat flow per m2 area if one surface is at 350C and other surface is at 200C. Neglect the resistance of glue. (b) Instead of glue, if these three pieces are bolted by four steel bolts of 1 cm diameter at the corners, (k = 40) per m2 area of the board then find the heat flow per m2 area of the board. The pressure vessel of a nuclear reactor is approximated as a large plane wall of thickness L. The inside surface of the wall at x 0 is insulated. The outside surface at x L is maintained at a temperature T2 . The gamma ray heating of the plate can be represented as a heat generation term of the form 𝑞 (𝑥) = 𝑞𝑜 𝑒 −𝛾𝑥 where q o and are positive constants and x is measured from the insulated surface. Develop expressions for the following: a. Temperature distribution in the plate b. Temperature at the insulated surface c. Heat flux at the outer surface One dimensional, steady state conduction with uniform internal energy generation occurs in a plane with a thickness of 50 mm and a constant thermal conductivity of 5 W/m.K. For these conditions, the temperature distribution has the form T(x) = a + bx + cx 2 . The surface at x = 0 has a temperature of T(0) = To = 120C and experiences convection with a fluid for which T = 20C and h = 500 W/m2.K. The surface at x = L is well insulated. a. Applying overall energy balance to the wall, calculate the internal energy generation rate, q b. Determine the Temperature at the center of the wall. A furnace wall is made up of three layers of 20 cm of refractory fire clay brick(K=1.52 W/m-0C), 10 cm Silocel bricks (k 0.318 W/m-0C) and 6 mm steel plate, (k = 45 W/m-0C). The Inside temperature of the furnace on the surface of refractory brick is 11000C and outside temperature on the surface of steel is300C. Determine the heat loss per unit area b) If the heat loss is found to be only 300 W/m2, due to the thin layers of air between brick and steel determine the resistance offered by these air layers c) what is the equivalent thickness of Silocel to offer the same resistance.
Conduction-2 1. An electric semiconductor device has a rating of 60 mW and for its proper operation the inside temperature should not be more than 800C. It is found that the device can dissipate about 15 mW of heat on its own when placed in an environment at 40 0C. To avoid over heating of the device it is proposed to install aluminum (190 W/mK) square fins of size 0.5 mm 0.5 mm and 10 mm long to provide additional cooling. Find number of fins required if h = 12.5 W/m2K. 2. A steel plate of 1m2 area is provided with 200 fins of diameter 1 cm and length 15 cm made of Cu (k=300 W/mK). The base temperature is at 200oC and environment is at 40oC, with h
= 20 W/m2K. Determine a) fin efficiency b) heat lost from the plate, c) effectiveness of the arrangement. Design the fins in above arrangement by altering the diameter of the fins such that the effectiveness increases to 2. Compare the heat transfer in this case with previous fin arrangement. 3. Aluminium square fins (k = 90 W/m K) of (0.5 mm x 0.5 mm) of 1cm long are provided on the surface of an electronic semiconductor device to carry 40 X10-3Watts of energy generated by the surface. The maximum surface temperature permissible is 80 oC. The temperature of the surrounding medium is 40oC with h = 12.5 W/m2K. Find the number of fins required to dissipate the heat out. Design fin arrangement for required heat dissipation with same configuration and dimensions by altering the fin material as copper (k=300W/mK). 4. An aluminium rod of diameter 2 cm and length 10 cm having k= 210 W/mK is attached to
two surface maintained at 100oC on its either sides.. If the environment is at 30oC with h = 10 W/m2K, Calculate the minimum temperature in the rod and the heat transfer from the rod. 5. A copper fin of diameter 2.5 cm and length 10 cm having k= 300 w/mK is attached to a surface maintained at 200oC. If the environment is at 20oC with h = 12 W/m2K, calculate a) fin efficiency b) heat loss from fin. Modify the fin design by changing its length such that the efficiency increases to 99%. Compare the heat transfer rate of the modified fin with original fin. 6. 15 numbers of fins are attached to a transformer base of 20 x 20 cm2 size which is maintained at 150oC. The thickness of each fin is 2.5 mm and width 20 cm having a height of 2 cm from base. If the ambient is at 30oC with h = 20 W/m2K, calculate the total heat transfer from the surface and the efficiency of the fin and effectiveness of the arrangement. Assume k for fin = 15 W/mK. Design a fin with another material having thermal conductivity k1 having identical dimensions exposed to same environment such that the total heat transfer rate from the fin arrangement should be 500W. 7. (a) Derive the expression for corrected length for rectangular and circular fins. (b)A cylinder of 2.5 cm diameter and 0.25 m length whose surface temperature is maintained at 170 oC is provided with 25 no. of circumferential fins equally spaced along the length of the cylinder. The fins are made up of aluminium (k = 160 W/m K) with 1 mm thickness and 1 cm height from cylinder surface. If the surroundings are maintained at 30oC with a heat transfer coefficient of 200 W/m2K calculate the heat loss to the ambient and its effectiveness.
8. Water and air are separated by a mild-steel plane wall. It is proposed to increase the heat-transfer rate between these fluids by adding straight rectangular fins of 1.27-mm thickness and 2.5-cm length, spaced 1.27cm apart. The air-side and water-side heat-transfer coefficients may be assumed constant with values of 11.4 and 256 W/m2 K respectively. Determine the percent change in total heat transfer when fins are placed on (a) the water side, (b) the air side, and (c) both sides.
9. An experimental device that produces excess heat is passively cooled. The addition of pin fins to the casing of this device is being considered to augment the rate of cooling. Consider a copper pin fin 0.25 cm in diameter that protrudes from a wall at 95°C into ambient air at 25°C. The heat transfer is mainly by natural convection with a coefficient equal to 10 W/m2 K. Calculate the heat loss, assuming that (a) the fin is “infinitely long” and (b) the fin is 2.5 cm long and the coefficient at the end is the same as around the circumference. Finally, (c) how long would the fin have to be for the infinitely long solution to be correct within 5%? 10. A brass rod in the form of a fin 100 mm long and 5 mm in diameter extends horizontally from a casting which is at 200C. The air temperature is 20C and provides a heat transfer coefficient of 30 W/m2-K. What is the heat transfer from the rod? Evaluate the temperature of the rod at 50 mm from the base and at the free tip. Now, if this fin is replaced by two identical fins of 50 mm length. All other parameters and dimensions remain the same. What is the heat transfer from this combination? Evaluate the temperature of the fin at the tip. Comment on your results.
Conduction-3 1. (a) Derive temperature distribution equation for a lumped system in terms of Fourier and Biot numbers. (b)A spherical steel ball of 2cm diameter is cooled in a controlled ambience from 6000C to 500C in 10minutes. If the body is a cube of 2cm side of same material, what would be the time required for cooling to achieve the same temperature at the end of the cooling process. 2. (a)Differentiate between Biot and Fourier numbers. Signify these numbers in the analysis of lumped heat capacity system. (b) An egg with mean diameter of 4 cm and initially at 20oC is placed in a boiling water pan for 4 minutes and found to be boiled to the consumer’s taste. For how long should a similar egg for same consumer be boiled when taken from a refrigerator at 5oC? Take the following properties for egg: k = 10 W/m-K, ρ = 1200 kg/m3, Cp = 2 kJ/kg K and heat transfer coefficient = 100 W/m2 K.
3. (a) Signify the application of Heisler charts in solving the problem of transient conduction. (b) A long cylindrical iron bar [ρ = 7800 kg/m3, C = 460 J/kg. oC, and k = 60 W/(m oC)] of diameter D = 5cm, initially at temperature T0 = 700oC, is exposed to a cool air stream at Tα = 100 oC. The heat transfer coefficient between the air stream and the surface of the iron bar is h = 80 W/m 2.oC. Determine the time taken for the centre temperature to reach 200 0 C. Also find the temperature at a radius of 1 cm at that instant of time 4. (a) What do you understand by the term time constant ?State its significance in the design of Thermocouple junctions. (b) A water carrying pipe of 15 cm dia is buried under earth to prevent freezing of water in winter. Assuming the surrounding temperature falls to -15 0 C and sustain for a period of 16 hrs in a day determine the minimum depth at which the pipe need to be laid down. 5. (a) Signify the importance of Heisler charts in the analysis of transient conduction. (b) A 2 cm diameter stainless steel ball ρ = 7800 kg/m3, C = 0.46 kJ/(kgoC) and k = 60 W/(moC) is uniformly heated to T0 = 800oC. It is to be hardened by suddenly dropping it into an oil bath at T α = 50oC. If the
quenching occurs when the ball reaches 100oC and the heat transfer coefficient between the oil and the sphere is 300 W/m2.oC, how long should the ball be kept in the oil bath? If 100 balls are to be quenched, determine the rate of heat removal from the oil bath per minute needed to maintain the oil temperature at 50oC.
6. (a) Illustrate that in lumped heat capacity system the temperature variation of the body is a function of Biot and Fourier numbers. (b) A long steel rod of 16 cm diameter initially at 20oC is placed in a furnace at 400oC, having h = 200 W/m2K. Calculate the time required for the centre of the rod to reach 250 oC. Also calculate the temperature at this instant at a radius of 4 cm from surface. Use k = 20 W/mK, α = 6 x 10-4 m2/s. 7. (a)Analyse the application of Fourier and Biot numbers in the analysis of lumped heat capacity system. (b) A slab of 15 cm thick is originally at a temperature of 500oC. It is suddenly immersed in a liquid at 100oC resulting in a heat transfer coefficient of 1000 W/m2K. Determine the temperature at the centreline and on the surface 30 minutes after immersion. Also calculate the total thermal energy removed per unit area during this period. Take α = 6.1 x 10-6 m2/s, k = 40 W/m K. ρ = 7800 kg/m3 and C = 840 J/kg K.
8. When a thermocouple is moved from one medium to another medium at a different temperature, the thermocouple must be given sufficient time to come to thermal equilibrium with the new conditions before a reading is taken. Consider a 0.10cm diameter copper thermocouple wire (spherical shaped) originally at 150°C. Determine the temperature response when this wire is suddenly immersed in (a) water at 40°C ( h= 80 W/m2 K) and (b) air at 40°C (h =10W/m2 K). 9. A long 20 cm diameter cylindrical shaft made of stainless steel 304 comes out of an oven at a uniform temperature of 600C. The shaft is then allowed to cool in an environment chamber at 200C with an average heat transfer coefficient of 80 W/m2.C. Determine the temperature at the center of the shaft 45 min after the start of the cooling process. Also, determine the heat transfer per unit length of the shaft during this time period. (k = 14.9 W/m-K, Cp = 477 J/kgK, = 7900 kg/m3). 10. In a fabrication process, steel components are formed hot and then quenched in water. Consider a 2.0-m-long, 0.2-m-diameter steel cylinder, initially at 400°C, that is suddenly quenched in water at 50°C. If the heat transfer coefficient is 200 W/m2 K, calculate the following 20 min after immersion: a. The center temperature b. The surface temperature c. The heat transferred to the water during the initial 20 min
Forced Convection (a) Discuss your understanding of Prandtl mixing length theory. What is its significance? (b)Determine the heat transfer and drag force exerted on the flat plate of 4 m long at 1000C when air is flowing over it with a velocity of 10 m/s and at 27oC. Width of the plate is 1 m and assumes the flow is fully turbulent from leading edge. Also find the velocity boundary layer thickness at the edge. (a) Elucidate the concept of eddy diffusivity of momentum. Comment on the role of eddy diffusivity of heat in turbulent heat transfer. o o (b) Atmospheric air at 30 flows over a flat plate of 3mx1m maintained at 70 C with a velocity of 10 m/s. Calculate the distance from the leading edge at which transition occurs. Find the
thickness of the hydrodynamic boundary layer and thermal boundary layer at 0.5 m from the edge. (a) Differentiate between Reynolds’s analogy and Colburn analogy. (b) Air enters at a temperature of 60oC and flows through a 2.5 cm diameter tube with a velocity of 0.8 m/s. It can be heated either by (i) condensing steam on its outer surface or (ii) by electrical resistance heating. Calculate the value of heat transfer coefficient in both cases. Assume fully developed flow. (a) Analyse the effect of Prandtl number on thermal and hydrodynamic boundary layers. (b) Air enters at a temperature of 60oC and flows through a 2.5 cm diameter tube with a velocity of 0.8 m/s. It can be heated either by (i) condensing steam on its outer surface or (ii) by electrical resistance heating. Calculate the value of heat transfer coefficient in both cases. Assume fully developed flow. (a) Illustrate the development of hydrodynamic boundary layer inside a pipe. (b) Nitrogen at 30oC flows over a plate maintained at 70oC with a free stream velocity of 10 m/s. (a) Determine the local and average values of convective heat transfer coefficient (b) calculate the values of the boundary layer thickness (velocity, and thermal) and (c) also determine the location at which the flow turns turbulent considering critical Reynolds number = 5 x 105. (a)Apply the concept of entrance length to hydrodynamic and thermal boundary layers. (b) Air at a temperature of 40oC flows over a flat plate of 2m long maintained at 120oC with a velocity of 5 m/s. Determine the average heat transfer coefficient and rate of heat transfer between the plate and air per metre width. (a) Obtain an expression for the velocity profile in laminar flow through a pipe and show that the average velocity is half of maximum velocity. (b) Water at 60oC enters a 2.5 cm diameter tube with a velocity of 2 cm/s and is heated keeping wall temperature constant at 100oC by condensing steam on outside. Take the length of the pipe as 6 m and find the exit temperature of water. Assume the flow is fully developed.
An uninsulated steam pipe of diameter 0.5 m is used to transport high temperature steam from one building to another. The pipe outside surface temperature is 150C and it is exposed to ambient air at 4C. Air moves over the pipe in cross flow with a velocity of 5 m/s. What is the heat loss per unit length of the pipe? Water is heated from 15C to 65C as it flows through a 3 cm ID, 3 m long tube. The tube is equipped with a resistance heater that provides uniform heating throughout the surface of the tube. The outside surface of the heater is well insulated so that all heat generated goes into the fluid in steady state operation. If the system is to provide hot water at the rate of 10 litre per minute, determine the power rating of the heater. Also, calculate the inside surface temperature of the pipe at the exit location. Consider flow of water over a flat plate with a velocity of 2 m/s. The length of the plate is 5m. The properties of water are = 1000kg/m3, = 0.0007975 Pa.s. Determine the following a. Hydrodynamic boundary layer thickness at x = 0.1 m and 1 m b. Shear stress at x = 0.1 m and 1 m c. Thermal boundary layer thickness at x = 0.1 m and 1 m (Pr = 5) Comment on these values.
Free Convection: (a)Analyse the correlations applied to turbulant free convection on a vertical plate. (b)The glass door of a furnace is having dimensions of height 0.75 m and width 1.5 m at a temperature of 230oC. If the outside air temperature is 25oC estimate the heat loss from the door to atmosphere.
(a) Illustrate the development of hydrodynamic and thermal boundary layers on a vertical plate in
natural convection. (b) A thin horizontal circular plate of 20 cm dia maintained at 100oC is kept horizontally in a large pool of water at 20oC in convecting heat from both of its faces. Determine the amount of heat input required to maintain the surface temperature of the plate at100oC. Assume β for water = 0.75 x 10-3K.
(a) Compare and contrast between Prandtl’s and Rayleigh’s numbers. Signify Rayleigh’s number with respect to natural convection. (b) A vertical pipe of 15 cm dia and 3 m long whose surface is maintained at 100oC is kept in a room air 20oC. Determine the rate of heat lost from pipe (i) when it is 3 m long (ii) if the pipe is of 1 m length (iii) if the pipe is 50 cm long. (a) Identify which configuration, vertical or horizontal position of flat plate will result more heat transfer. Justify. (b) A vertical pipe of 15 cm dia and 3 m long whose surface is maintained at 100oC is kept in a room air 20oC. Determine the rate of heat lost from pipe (a) when it is 3 m long (b) if the pipe is of 1 m length (c) if the pipe is 50 cm long. (a) In laminar natural convection over a vertical plate which one, constant wall temperature or constant heat flux, will exibit high heat transfer coefficient? Illustrate with relavent theory. (b) An air-conditioned duct carries air at an average temperature of 10oC. The duct size is 300 mm x 200 mm and is exposed to the surrounding air at 30oC. Find the heat gain by the air in the duct per metre length. Assume 200 mm side is vertical. (a) Illustrate the phenomenon of natural convection in enclosed spaces with sketches. (b) A hot plate of 15 cm2 area is maintained at a temperature of 200oC and exposed to still air at 30oC temperature. When the smaller side of the plate is held vertical, convective heat transfer rate is 14% higher than when bigger side of the plate is held vertical. Determine dimensions of the plate. Also calculate heat transfer rates in both cases. Use the following relation: Nu= 0.59 (Gr Pr)0.25 (a) Signify the application of a non-dimensional quantity to identify the mode of convection i.e natural or combined. Quote examples. (b) A vertical 90 cm high and 1.2m wide doubled pane window consists of two sheets of glass separated by 2 cm air gap. If the glass surface temperatures across the air gap are measured as 12 oC and 2oC, respectively, determine the rate of heat transfer through the window.
Consider a 0.6 m X 0.6 m thin square plate in a room at 30°C. One side of the plate is maintained at a temperature of 90°C, while the other side is insulated, as shown in Fig. determine the rate of heat transfer from the plate by natural convection if the plate is a) vertical, b) horizontal with hot surface facing up, and c) horizontal surface facing down. A copper fin of diameter 4 mm and length 25 mm is used to transfer heat from a base surface at a temperature of 350 K to ambient air at 300 K by natural convection. Calculate the heat transfer rate, effectiveness and efficiency of the fin. Assume the fin tip to be adiabatic.
A 6-m-long section of an 8-cm-diameter horizontal hot water pipe passes through a large room whose temperature is 20°C. If the outer surface temperature of the pipe is 70°C, determine the rate of heat loss from the pipe by natural convection.
Heat Exchangers A liquid chemical (Ethylene Glycol) flows through a thin walled copper tube of 12 mm diameter at the rate of 0.5 kg/sec, water flows in opposite direction at the rate 0.37 kg/sec through the annular space formed by this and a tube diameter of 20mm. The liquid chemical enters and leaves at 100oC and 60oC, while water enters at 10oC. Find the length of the tube required. Also
find the length of the tube required if the water flows in the same direction as liquid chemical. (a) Signify NTU and LMTD. (b) In a double pipe parallel flow heat exchanger water is heated from 15oC to 50oC and hot oil is cooled from 130oC to 60oC (a) Determine the exit temperatures if the flow is made counter flow, without changing flow rates (c) If the length of the tubes is increased keeping parallel flow configuration determine the minimum temperature to which the oil can be cooled and its effectiveness.
In a parallel flow double pipe heat exchanger hot water enters at the rate of 10 kg/min and at a temperature of 70oC and leaves at 50oC. The cold water enters at 25oC with a flow rate of 25 kg/min. Calculate the area of heat exchanger required (i) if heat transfer coefficients inside and outside the pipes are 60 W/m2K, (ii) If the hot water flow rate is doubled without changing inlet temperature what will be the exit temperature of both fluids? In a parallel flow double pipe heat exchanger hot water enters at the rate of 10 kg/min and at a temperature of 70oC and leaves at 50oC. The cold water enters at 25oC with a flow rate of 25 kg/min. Calculate the area of heat exchanger required (i) if heat transfer coefficients inside and outside the pipes are 60 W/m2K, (ii) If the hot water flow rate is doubled without changing inlet temperature what will be the exit temperature of both fluids? (a) Derive the expression for LMTD in counter flow double pipe heat exchanger. (b) In a double pipe heat exchanger oil flows through the annulus with a convective heat transfer coefficient of 1500 W/m2K which is heated by using hot water flowing through the Cu tube (k = 300 W/m K) of inside diameter 2 cm and outer diameter 2.54 cm having convective heat transfer coefficient of 2500 W/m2K. The fouling factor on waterside is 0.0004 m2 oC/W and the fouling factor on oil side is 0.0009 m2 oC/W. (a) Determine the overall heat transfer coefficient based on outside area per metre length. (b) Also find the overall heat transfer coefficient neglecting fouling resistances on both sides. In petrol engine arranged for evaporative cooling the steam formed at 100oC is condensed in the radiator where it surrounds the tube through which cooling air at 18oC is made to flow with a mean air velocity of 5.5 m/sec. The radiator consists of 12.5 cm long parallel tubes and 8 mm diameter tubes. This is connected to an engine, which develops 8.75 kW of power with brake thermal efficiency of 20 per cent. If 35 per cent of heat supplied to the engine is dissipated at the radiator, calculate no. of tubes and temperature of air at exit of the radiator. For air, Cp = 1005 J/kg K, ρ = 1.4 kg/m3 and overall heat transfer coefficient through the tube = 19 W/m2K. A double pipe heat exchanger is used to cool the lubricating oil from 90 oC to 40oC using water available at 10oC. The mass flow rate of the oil is 0.15 kg/s with specific heat 2.13 kJ/kg K and that of water is 0.3 kg/s. Determine the heat transfer area required if U = 50 W/m2K for (a) parallel flow arrangement (b) counterflow arrangement. Design a heat exchanger under same conditions such that heat transfer rate should increase to 20kW. In indirect heat exchanger, feed water is heated from 250C to 950C by condensing the steam at 1500C. Water flows through the tube of inner diameter 3cm and outer diameter 3.5cm having k = 60W/mK. If the flow rate of water is 1 kg/s. find the length of pipe required. The condensing heat transfer coefficient on outside the pipe is 12000 W/m2K.Take the feed water properties 4.62 10 4 kg/m s, k 0.653 W/m K, C p 4200 J/kg K
Steam in the condenser of a power plant is to be condensed at a temperature of 30 °C with cooling water from a nearby lake, which enters the tube of the condenser at 14 °C and leaves at 22 °C. The surface area of the tube is 45 m2, and the overall heat transfer coefficient is 2100 w/m2. °C . Determine the mass flow rate of the cooling water needed and the rate of condensation of the steam in the condenser.
A counter flow double pipe heat exchanger is to heat water from 20C to 80C at a rate of 1.2 kg/s. The heating is to be accomplished by geothermal water available at 160C at a mass flow rate of 2 kg/s. The inner tube is thin walled and has a diameter of 1.5 cm. If the overall heat transfer coefficient of the heat exchanger is 640 W/m2, determine the length of the heat exchanger required to achieve the desired heating.
Radiation A thin aluminium sheet with an emissivity of 0.1 on both sides is placed between two very large parallel plates that are maintained at uniform temperatures T1 = 800 K and T2 = 500 K and have emissivities 1= 0.2 and 2= 0.2, respectively. Determine the net rate of radiation heat transfer between the two plates per unit surface area of the plates and compare the result to that without the shield. Consider a cylindrical furnace with radius = Height = 1 m. The top surface (surface 1) and the base surface (surface 2) of the furnace have emissivities 1 = 0.8 and 1 = 0.4, respectively, and are maintained at uniform temperatures T1 = 700 K and T2= 500 K. The side surface (surface 3) closely approximates a blackbody and is maintained at a temperature of T3 = 400K. Determine the net rate of radiation heat transfer at each surface during steady operation and explain how these surfaces can be maintained at specified temperatures. A furnace is shaped like a long equilateral triangular duct. The width of each side is 1 m. The base surface has an emissivity of 0.7 and is maintained at uniform temperature of 600 K. The heated left side surface closely approximates a blackbody at 1000K. 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.
(a)Derive the expression for surface resistance and shape resistance using electrical analogy. (b)Two areas 1 and 2 are (having emissivity 0.3 and 0.6 respectively) in the form of circular rings, coaxial and are in two parallel planes at a distance of 10cm. For area 1 the inner radius is 5 cm, the outer radius is 10 cm and the corresponding values for area 2 are 8 cm and 20 cm respectively. Make calculation for the heat transfer between these two areas. The following relation may be used for calculating the shape factor between two circular areas located coaxially in two parallel planes.
2 X X 2 4B 2C 2 2 F1-2 = 2 B R R Where, B = 1 ; C 2 ; X 1 B 2 C 2 and H is the distance between the two areas. H H (a) Distinguish between Lambert’s cosine law and Wein’s displacement law. (b) Two circular discs of diameter 50 cm are placed opposite to each other at a distance of 1 m apart. The discs are maintained at 500oC and 200orespectively. Calculate the heat transfer between them if (a) both are perfectly black (b) they are having emissivity 0.8.
(a) Signify the importance of superposition rule in finding shape factor. (b)The filament of a bulb is approximately at a temperature of 2855 K. If the filament is assumed to e gray what fraction of its energy is visible in the range of 0.35 μm? Take the filament surface area 10 x 10-6 m2 and consumes 60 W. Also find its efficiency.
(a) Derive Stefan Boltzmann’s law from Plank’s law. (b) Two large parallel planes having emissivities 0.3 and 0.4 are maintained at a temperature 800oC and500oC respectively. A radiation shield of emissivity 0.05 on both sides is placed in
between. Determine the (i) rate of heat transfer per unit area if shield is not there (ii) heat transfer with the shield (iii) the temperature of the shield. (a) Derive Stefan Boltzmann’s law from Plank’s law. (b) Two large parallel planes having emissivities 0.3 and 0.4 are maintained at a temperature 800oC and500oC respectively. A radiation shield of emissivity 0.05 on both sides is placed in between. Determine the (i) rate of heat transfer per unit area if shield is not there (ii) heat transfer with the shield (iii) the temperature of the shield. (a) Compare and contrast the following: Absorptivity and emissivity, reflectivity and transitivity, grey body and black body. (b) A 2.5 cm diameter pipe whose surface is maintained at 1000 K having emissivity 0.1 is enclosed inside a large pipe of diameter7.5cm maintained at 350 K. Determine the heat loss from the inner pipe to outer pipe per metre length if the emissivity of the outer pipe is 0.3.
(a)Analyse the correlations applied to turbulant free convection on a vertical plate. (b)The glass door of a furnace is having dimensions of height 0.75 m and width 1.5 m at a temperature of 230oC. If the outside air temperature is 25oC estimate the heat loss from the door to atmosphere.
