Heat Transfer – Mechanical Engineering Optional PYQs (1995–2026)

Here are all 151 Heat Transfer questions asked in UPSC Mechanical Engineering Optional Paper II from 1995 to 2026, arranged under the 6 topics of the official syllabus. Within each topic the latest questions come first.

Each question ends with a code such as [2024/15m/3b] – the year, the marks, the word limit (where UPSC set one) and the question number in the paper.

2.1] Conduction

39 questions

1. Determine the heat transfer through the composite wall shown in the figure below. Take the thermal conductivities of A, B, C, D and E as 50, 10, 8, 20 and 25 W/m-K, respectively and assume one-dimensional heat transfer:

[Figure: Composite wall of face area $A=1m^2$, drawn in perspective, with heat flow Q from left ($T_1=800{}^{\circ }C$) to right ($T_2=200{}^{\circ }C$). Layers from left to right: A, 5 cm thick (full height); B and C, 10 cm thick, side by side in parallel (B in the upper half, C in the lower half of the height); D, 10 cm thick (full height); E, 5 cm thick (full height).] [2026/10m/1d]

2. Consider a large plane wall of thickness 0.25 m, having thermal conductivity 3 W/m-°C and surface area $10m^2$. The left side of the wall is subjected to a net heat flux of $750W/m^2$. The surface temperature of the left side of the wall is 90 °C. Assuming constant thermal conductivity and no heat generation in the wall, express the differential equation and boundary conditions for steady one-dimensional heat conduction through the wall. Also, obtain a relation for the variation of temperature in the wall by solving the differential equation and find the temperature of the right side of the wall. [2026/10m/1e]

3. At a certain instant of time, the temperature across a large wall of thickness 50 cm is given as $T\left(x\right)=90-80x+16x^2+32x^3-25x^4$ where x is in metres and measured from the left face of the wall as shown in the figure below and T is in °C. If the area of the wall is $10m^2$ and there is no generation of heat in the wall, compute the following: (i) The rate of heat entering and leaving the wall (ii) The rate of heat energy stored in the wall (iii) The rate of temperature change with time at $x=0$ and $x=0.5$ m.

[Figure: Plane wall of thickness 50 cm (hatched) with the x-axis drawn below it, x = 0 at the left face and x = 0.5 m at the right face.] [2025/10m/1d]

4. A long cylindrical rod of radius 10 cm consists of a nuclear reacting material ($k=0.5$ W/m-K) generating $24000W/m^3$ uniformly throughout its volume. The rod is encapsulated within another cylinder whose outer radius is 20 cm and that has a thermal conductivity of 4 W/m-K. The outer surface is surrounded by a fluid at 100 °C, and the convection coefficient between the surface and the fluid is $20W/m^2\mathrm{\text{–}}K$. Find the temperatures of the interface between the two cylinders, at the outer surface and the maximum temperature under steady-state condition. [2025/20m/3a]

5. Deduce an expression for the temperature distribution in an infinite long slab of thickness “L” m under one-dimensional steady state heat conduction. The slab uniformly generates heat of $\dot{q}W/m^3$. One of its surfaces is perfectly insulated and the other surface is maintained at a constant temperature of $T_w$°C. Also plot the temperature profile clearly mentioning the maximum and minimum temperatures and the location. [2024/10m/1c]

6. Heat is generated in a stainless steel plate (thermal conductivity = 22 W/m-K) of thickness 1 cm, at a uniform rate of $600MW/m^3$. The left side of the plate is maintained at 200 °C and the right side is maintained at 100 °C. What will be the (i) temperature distribution across the plate, (ii) location and value of maximum temperature and (iii) heat flux from both sides of the plate and its direction? Assume one-dimensional, steady-state heat conduction. [2023/20m/3a]

7. Hot water is flowing through a pipe made of cast iron having thermal conductivity of 52 W/m-°C, with an average velocity of 1.5 m/s. The inner and outer diameters of the pipe are 3 cm and 3.5 cm, respectively. The pipe passes through a 15 m long section of a basement whose temperature is 15 °C. The temperature of the water drops from 70 °C to 67 °C as it passes through the basement. The heat transfer coefficient on the inner surface of the pipe is $400W/m^2{\mathrm{\text{–}}}^{\circ }C$. Determine the combined convection and radiation heat transfer coefficient at the outer surface of the pipe. [2022/10m/1d]

8. A shaft having diameter of 5 cm rotates in a bearing made of cast iron. The shaft rotates at 4500 r.p.m. The bearing is 15 cm long, 8 cm outer diameter and has thermal conductivity of 70 W/m-K. There is a uniform clearance between the shaft and the bearing of 0.6 mm. The clearance is filled with a lubricating oil having thermal conductivity of 0.14 W/m-K and dynamic viscosity of $0.03N\mathrm{\text{–}}s/m^2$. The bearing is cooled externally by a liquid, and its outer surface is maintained at 40 °C. Disregarding the heat conduction through the shaft and assuming only one-dimensional heat transfer, determine (i) the rate of heat transfer to the coolant, (ii) the surface temperature of the shaft and (iii) the mechanical power wasted by the viscous dissipation in the lubricating oil. [2022/20m/2b]

9. Consider a large plane wall of thickness $L=0.4$ m, thermal conductivity $k=2.3$ W/m°C and surface area $A=20m^2$. The left side of the wall is maintained at a constant temperature of $T_1=80$°C while the right side loses heat by convection to the surrounding air at $T_a=15$°C with a heat transfer coefficient of $h=24W/m^2{}^{\circ }C$. Assuming constant thermal conductivity and no heat generation in the wall, (i) obtain a relation for the variation of temperature in the wall. (ii) evaluate the rate of heat transfer through the wall. [2021/10m/1c]

10. Two walls A and B are maintained at temperatures $T_A$ and $T_B$, respectively. One end of a metal rod of length l is embedded in the wall A, while the other end is fixed to wall B, the rod loses heat by convection to the environment at $T_{\infty }$. Derive an expression to determine (i) the temperature distribution in the rod (ii) the total heat lost by the rod (iii) the heat transferred from the wall A. [2021/20m/4a]

11. Consider the base plate of an 800 W household electric iron with thickness of $L=0.6$ cm, base area = 160 cm${}^2$ and thermal conductivity $k=60$ W/m-K. The inner surface of the base plate is subjected to uniform heat flux generated by the resistance heater inside. When steady operating conditions are reached, the outer surface temperature of the plate is measured to be 112 °C. Disregarding any heat loss through the upper part of the iron– (i) express the differential equation and boundary conditions for steady one-dimensional heat conduction through the plate; (ii) obtain a relation for the variation of temperature in the base plate by solving the differential equation; (iii) evaluate the inner surface temperature. [2020/10m/1b]

12. Consider a 3 m diameter spherical tank that is initially filled with liquid nitrogen at 1 atm and –196 °C. The tank is exposed to ambient air at 15 °C with a combined convective and radiative heat transfer coefficient of 35 W/m${}^2$-K. The temperature of the thin-shelled spherical tank is observed to be almost the same as the temperature of the nitrogen inside. Determine the rate of evaporation of the liquid nitrogen in the tank as a result of the heat transfer from the ambient air if the tank is (i) not insulated, (ii) insulated with 5 cm thick fiberglass wool insulation ($k=0.035$ W/m-K) and (iii) insulated with 2 cm thick superinsulation which has an effective thermal conductivity of 0.00005 W/m-K. Consider the heat of vaporization of liquid nitrogen as 198 kJ/kg and its density as 810 kg/m${}^3$ at 1 atm. [2020/20m/2a]

13. A furnace wall consists of 200 mm inner layer of refractory brick ($k=1.52$ W/m-K), 8 mm thick steel plate ($k=45$ W/m-K) and outer layer of 100 mm thick insulation brick ($k=0.138$ W/m-K). The inner wall of the furnace is maintained at 1150 °C, while the outer wall of the insulation brick is at 40 °C. Calculate the heat transfer through the wall. What is the temperature at the junction of steel plate and outer insulation brick wall? [2019/10m/1d]

14. A 1 mm diameter electric wire is covered with 2 mm thick layer of insulation ($k=0.5$ W/m-K). Air surrounding the wire is at 25 °C and $h=10$ W/m${}^2$-K. The wire temperature is 100 °C. Find the rate of heat dissipation from the insulated wire per unit length. Also, find the maximum value of heat dissipation per unit length for the corresponding critical thickness of insulation. [2019/10m/2c]

15. A plane wall of thickness 0.2 m and $k=1.2W/m\mathrm{\text{–}}K$ has a surface area of 15 m². The left and right sides of the wall are maintained at 120 °C and 50 °C respectively. Using the basic differential equation for heat conduction, determine the variation of temperature within the wall and the temperature at 0.13 m from the left. Also find the rate of conduction through the wall under steady-state conditions. [2018/10m/1b]

16. Heat is generated uniformly in uranium ($k=29.5W/m\mathrm{\text{–}}K$) rods of 5 cm diameter at the rate of $7.5\times {10}^{7}W/m^3$. Cooling water at 120 °C is circulated in the annulus around the rods with heat transfer coefficient of $55kW/m^2\mathrm{\text{–}}K$. Find the maximum temperature of the fuel rods. [2018/10m/3c]

17. A flue gas stream is to be monitored for its temperature using a thermocouple. The thermocouple design needs to be evaluated in terms of its time response to accurately predict the measured temperature. The thermocouple junction can be approximated as a sphere of diameter 0.6 mm, density of the bead material ($\rho$) 8500 kg/m³, thermal conductivity (k) is 30 W/m-K; specific heat (c) is 0.3 kJ/kg-K. The convective heat transfer coefficient (h) between the junction and flue gas is 300 W/m²-K. Determine the time required to read 90% of the initial temperature difference. Neglect radiation effect and change in thermophysical properties with temperature. [2017/20m/2b]

18. A steam pipe (inner diameter = 150 mm and outer diameter = 160 mm) having thermal conductivity 58 W/m K is covered with two layers of insulation of thicknesses 30 mm and 50 mm respectively and thermal conductivities 0.18 W/m K and 0.09 W/m K respectively. The temperature of inner surface of the steam pipe is 320 °C and that of the outer surface of the insulation layers is 40 °C. (i) Determine the quantity of heat lost per metre length of the steam pipe and layer contact temperature. (ii) If the condition of the steam is dry and saturated, find the quality of the steam coming out of one metre pipe assuming that the quantity of steam flowing is 0.32 kg/min. [ Use the data: At 320 °C saturation temperature $h_f=1463kJ/kg$, $h_{fg}=1240kJ/kg$, $h_g=2703kJ/kg$ ]. [2016/20m/3a]

19. A plane wall 90 mm thick ($k=0.18W/m{}^{\circ }C$) is insulated on one side while the other side is exposed to environment at 80°C. The rate of heat generation within the wall is $1.3\times {10}^{5}W/m^3$. If the convective heat transfer coefficient between the wall and the environment, (h) is $520W/m^2{}^{\circ }C$, determine the maximum temperature in the wall. Derive the expression used, starting from the steady state one-dimensional heat conduction with heat generation equation. [2015/20m/2b]

20. The outer and inner surfaces of a thick hollow cylinder have areas $1.25m^2$ and $0.25m^2$ respectively. The thickness of the cylinder is 10 cm and the thermal conductivity of the cylinder material is 50 W/m K. Find the radial heat transfer through the cylinder for 100 °C temperature difference at the surfaces. Derive the formula used. [2014/10m/1e]

21. Two long slender rods A and B, made of different materials having same diameter of 12 mm and length 1 m, are attached to a surface maintained at a temperature of 100 °C. The surfaces of the rods are exposed to ambient still air at 20 °C. By traversing along the length of the rods with a temperature sensor, it is found that the surface temperatures of rods A and B are equal at positions 15 cm and 7.5 cm respectively away from the base surface. If material of A is carbon steel with thermal conductivity 60 W/m K, what is the thermal conductivity of rod B? List the assumptions made. Assume that the average convection coefficient for air is $5W/m^{2}K$. Find the ratio of rate of heat transfer for rods A and B. [2014/20m/3b]

22. Inner and outer surfaces of a spherical shell are maintained at temperatures $T_i$ and $T_o$ respectively such that $T_i>T_o$. If inner and outer radii of the shell are $r_i$ and $r_o$ and its conductivity is k, derive an expression for the rate of heat conduction through the shell. Assume steady state and no heat generation within the shell. [2013/10m/5e]

23. A very long AISI 316 stainless steel (K = 14 W/m-K) rod 5 mm in diameter has one end maintained at 100°C. The surface of the rod is exposed to ambient air at 30°C with average convective heat transfer coefficient of 50 $W/m^2\mathrm{\text{–}}K$. Neglecting radiation heat transfer, estimate how long the rod must be to treat it as “infinitely long” to yield a reasonable accurate estimation of heat loss. If the rod is made of copper (K = 350 W/m-K, will the length be different? How much will it be and why? Compare the heat transfer rates for both the rods. The analysis may be based on fin tip heat loss alone. [2012/12m/1b]

24. The volumetric heat generation rate in a cylindrical nuclear reactor fuel rod of 5 cm diameter is $5\times {10}^{7}W/m^3$. Under steady state operating condition, the temperature distribution in the rod is $t=800-4.167\times {10}^5{r}^2$. Density, specific heat and thermal conductivity of rod material are 1100 $kg/m^3$, 800 J/kg-K and 30 W/m-K respectively. (i) What are the rates of heat transfer per unit length of the rod at the centre line (axis) and at the rod surface? (ii) If the reactor power level is suddenly increased to ${10}^{8}W/m^3$, what are the initial time rate of change of temperature at the centre line and at the surface? [2012/20m/2b]

25. A decorative plastic film on a copper sphere having 10 mm diameter is cured in an oven at 75 °C. Upon removal from the oven, the copper sphere is subjected to an air stream at a pressure, temperature and velocity of 1 bar, 23 °C and 10 m/s respectively. How long it will take for the sphere to cool down to 35 °C? State the assumptions made and justify the method of analysis used. For copper, the density, specific heat and thermal conductivity are, respectively, 8933 $kg/m^3$, 388 J/kg-K and 350 W/m-K. The following correlation for forced convection may be used: $N{u}_d=2+[0.4R{e}_d^{1/2}+0.06R{e}_d^{2/3}]P{r}^{0.4}$ For air, kinematic viscosity, thermal conductivity and Prandtl number at the mean film temperature under consideration are $15.53\times {10}^{-6}{m}^2/s$, 0.025 W/m-K and 0.708 respectively. [2010/20m/1c]

26. The thermal conductivity of a hollow sphere of inside radius ($R_i$) and outside radius ($R_o$) is given by $K=K_i+(K_o-K_i)\left(\frac{T-T_i}{T_o-T_i}\right)$ where, $T_i$ = Inner surface temperature, $T_o$ = Outside surface temperature. Prove that (i) The heat flow rate is given by $Q=4\pi R_i{R}_o\left(\frac{K_i+K_o}{2}\right)\left(\frac{T_i-T_o}{R_o-R_i}\right).$ (ii) Also determine the heat loss from a spherical shell whose $D_i=2.5m$ and covered with 30 cm of insulation. The thermal conductivity of insulation is 0.3 W/mK and 0.2 W/mK at inner and outer surface temperatures of 150° C and −15° C respectively. [2009/20m/1d]

27. Consider a large plane wall of thickness L = 0.3 m, thermal conductivity k = 2.5 w/m-K, and surface area $A=12m^2$. The left side of the wall at x = 0 is subjected to a net heat flux of $q=700w/m^2$, while the temperature at that surface is maintained at $T_1=80$ °C. Assuming constant thermal conductivity and no heat generation in the wall: (i) express the differential equation and boundary conditions for steady, one-dimensional heat conduction through the wall (ii) Obtain a relation for the variation of temperature in the wall by solving the differential equation. (iii) Evaluate the temperature of the right surface of the wall. [2008/20m/200w/1b]

28. The diagram shows a truncated conical section fabricated from a material of thermal conductivity K. The circular cross-section of the conical section has the diameter D = ax, where a is a constant and x is the axial distance of the section from the apex of the cone. The temperatures at the two end faces of the conical section (at distances $x_1$ and $x_1$ from the apex) are respectively $T_1$ and $T_2$ while the lateral surface of the truncated cone is thermally insulated. (i) Derive an expression for the temperature distribution $T\left(x\right)$ in symbolic form assuming one-dimensional steady-state conditions. Sketch the temperature distribution. (ii) Calculate the heat rate $q_x$ through the cone in x-direction.

[Figure: Truncated cone lying along a horizontal axis, apex (vertex) on the right, x measured from the apex towards the left (arrow labelled x pointing left). The small end face (right) is at distance $x_1$ from the apex and is at temperature $T_1$; the large end face (left) is at distance $x_2$ from the apex and is at temperature $T_2$. The inclined lateral surface is hatched and labelled “Thermal Insulation”; dimension lines show $x_1$ and $x_2$ measured from the apex.] [2007/40m/2b]

29. Steel ball bearings ($K=50W/m\mathrm{\text{–}}K$, $\alpha =1.3\times {10}^{-5}{m}^2/s$) having a diameter of 40 mm are heated to a temperature of 650 °C and then quenched in a tank of oil at 55 °C. If the heat transfer coefficient between the ball bearings and the oil is $300W/m^2\mathrm{\text{–}}K$, determine: (i) the duration of time the bearings must remain in oil to reach a temperature of 200 °C; (ii) the total amount of heat removed from each bearing during this time; (iii) the instantaneous heat transfer rate from the bearings when they are first immersed in oil and when they reach 200 °C. [2007/20m/5d]

30. The temperature of a gas flowing through a pipe was measured by a mercury-in-glass thermometer, dipped in an oil-filled steel tube welded radially to the pipeline. The thermometer indicates a temperature lower than the gas temperature. How large is the error in the temperature measurement if the thermometer reads 85 °C and the temperature of the pipe wall is 40°C? The steel tube is 125 mm long and has a 1.5 mm thick wall. The thermal conductivity of this tube material is 56 W/m-K and the local heat transfer coefficient between the gas and the tube is $23.5W/m^2\mathrm{\text{–}}K$. In what way the thermometric error can be reduced? [2006/30m/4b]

31. With usual notations, develop an expression for the efficiency of a fin of uniform cross-section when the heat loss from the tip is considered negligible. [2005/30m/6b]

32. A fluid at a temperature $T_{{\infty }_1}$, with a heat transfer coefficient $h_1$ flows on one side of a slab at x = 0 and another fluid at a temperature $T_{{\infty }_2}$ with a heat transfer coefficient $h_2$ flows on the other side of the slab at x = L. Both the fluids flow in opposite directions. Develop an expression for the heat flow Q through an area ‘A’ of the slab. Calculate the heat transfer rate through an area of one square metre of the slab for the following data: $T_{{\infty }_1}=150{}^{\circ }C$, $h_1=300W/m^{2}K$, $T_{{\infty }_2}=50{}^{\circ }C$, $h_2=600W/m^{2}K$, thickness of slab = 5 cm and K = 20 W/m K. [2004/30m/6a]

33. A small hemispherical oven is built of two insulating materials. The inner layer is of fire brick, 125 mm thick and outer layer of 85% magnesia, 40 mm thick. The inner surface of the oven is at 800°C and the heat transfer coefficient for the outer surface is 10 $W/m^2\mathrm{\text{–}}K$. The room temperature is 20° C. Calculate the heat loss through hemisphere is the inside radius is 0.6 m. The thermal conductivities of fire brick and 85% magnesia are 0.31 and 0.05 W/m-K respectively. Also, calculate the temperature at contact between insulating materials and at the outer surface. [2003/20m/6b]

34. Heat is transferred along the axis of a truncated conical cylinder of length $l$, radius $r_1$ at the shorter end and radius $r_2$ at the bigger end. The circumference of the cylinder is completely insulated. Develop an expression to calculate heat transfer along the axis of the cylinder. Assume no variation of conductivity with temperature. [2002/20m/200w/5b]

35. A spherical thermocouple of 2.5 mm diameter is used to measure the temperature of air flowing in a pipe. Initially both the thermocouple and the air are at a temperature of 30°C. The air is heated to a temperature of 235°C and maintained at this temperature. Find the time required for the thermocouple to reach 200°C. Also find out the time constant of the thermocouple and comment on the suitability of this thermocouple to measure unsteady state temperature. For thermocouple material take density = 9000 $kg/m^3$, specific heat 0.4 kJ / kg K and thermal conductivity = 30 W/m K. Convective heat transfer coefficient between thermocouple surface and the air is 120 $W/m^{2}K$. [2002/30m/6a]

36. A thin rod of length $l$ has its two ends connected to two reservoirs at temperatures $t_1$ and $t_2$ respectively. The rod has the perimeter P and its cross-sectional area A. The rod loses heat to the environment at temperature, $t_{\infty }$. The convection coefficient of heat transfer between the rod and the environment is h. Assuming the thermal conductivity of the material of the rod to be constant k, derive an expression for (i) the temperature distribution in the rod and (ii) the heat transfer from the rod to the environment. [2000/15m/7b]

37. Hot air at a temperature of 80 °C is flowing through a steel pipe of 10 cm diameter, The pipe is covered with two layers of different insulating materials of thickness 6 cm and 3 cm, their corresponding thermal conductivities are 0.23 and 0.37 W/m K respectively. The inside and outside heat transfer coefficients are 60 and $10W/m^{2}K$. The atmospheric temperature is 27 °C. Find the rate of heat loss from a 60 m length of pipe. Neglect resistance of steel pipe. [1999/20m/200w/5c]

38. Explain the terms fin efficiency and fin effectiveness. Derive the expressions for these for the fin which is insulated at the end. Discuss the conditions when installation of fins on a heat transfer surface will not necessarily increase heat transfer rate. [1996/25m/7a]

39. The cylinder of an engine is 1 m [?] long and has an outside diameter of 6 cm. The outside surface temperature of the cylinder is 200°C when the ambient temperature is 30°C. The film coefficient of heat transfer is $25W/m^{2}K$. The cylinder is provided with 12 longitudinal straight fins of 0.1 cm thickness and 3 cm length. The thermal conductivity of cylinder and fin material is 75W/mk. Assuming that the fins have insulated tips, determine. (i) percentage increase in heat dissipation due to addition of fins. (ii) the temperature at the centre of the fin and (iii) find efficiency and fin effectiveness. [1996/35m/7b]

2.2] Convection over a flat plate

14 questions

1. Wind is blowing at 50 km/hr along a 12 m long and 5 m high wall of a house. The atmospheric air is at 4 °C and the wall surface temperature is 10 °C. Determine the rate of heat loss from the wall by convection. The properties of air at 1 atmosphere and at mean film temperature may be taken as: $k=0.0248W/m\mathrm{\text{–}}{}^{\circ }C$ $\nu =1.412\times {10}^{-5}{m}^2/s$ $Pr=0.72$ Use the Nusselt number correlation for combined flow: $Nu=(0.037{Re}^{0.8}-871){Pr}^{0.33}$ [2026/10m/2c]

2. Incompressible fluid having free stream velocity of “u” m/s and temperature of T°C flows over a flat plate maintained at a constant temperature of $T_w$°C ($T\ne T_w$). Flow is within the laminar region. Draw the relative thicknesses of thermal and hydrodynamic boundary layers developed on the flat plate for three fluids having (i) Pr < 1, (ii) Pr = 1 and (iii) Pr > 1. Justify your answer appropriately. (Draw three diagrams for three fluids for better clarity) [2024/10m/1e]

3. A stainless steel plate of 1 m length, 1 m width and 10 mm thickness is kept horizontal. The thermal conductivity of the plate is 10 W/mK. Bottom surface of the plate is exposed to hot gases at 700°C with heat transfer coefficient at the bottom surface of $50W/m^{2}K$. Top surface of the plate is cooled by air at 69°C and flowing parallel to the top surface. Any part of the plate should not exceed a maximum permissible temperature of 400°C to avoid failure. Find the minimum permissible velocity of the air required to ensure the plate does not get over-heated beyond the permissible limit. Neglect the heat loss from the side surfaces of the plate and assume one-dimensional heat transfer. Use the following correlation to solve the problem: $\overline{N{u}_L}=P{r}^{0.333}[0.037R{e}_L^{0.8}-871]$ Take the appropriate properties of the air from the table attached at the end. [2024/20m/4b]

4. Calculate the displacement thickness and momentum thickness of a laminar boundary layer, in terms of the nominal boundary layer thickness $\delta$, for the following velocity distribution: $\frac{u}{U_0}=sin\left(\frac{\pi }{2}\frac{y}{\delta }\right)$ [2021/20m/2b]

5. Air at 20 °C and at atmospheric pressure flows at a velocity of 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 $5\times {10}^5$, find the distance from the leading edge at which the boundary layer changes from laminar to turbulent. At this location, also calculate (i) thicknesses of hydrodynamic and thermal boundary layers and (ii) local and average convective heat transfer coefficients. Assume cubic velocity profile and approximate method. At mean temperature of 40 °C, the properties of air are the following: Density $=1.128kg/m^3$ Kinematic viscosity $=16.96\times {10}^{-6}{m}^2/s$ $k=0.02755W/m\mathrm{\text{–}}K$ $Pr=0.7$ [2018/10m/1e]

6. Air flows over a surface, 2 m in length, oriented in the direction of flow and of sufficient breadth, maintained at 150 °C. The pressure is 1 atm and the bulk air temperature is 30 °C. If the air velocity is 12 m/s, find the average heat transfer coefficient. [ Use the following physical properties of air at film temperature, that is $(150+30)/2=90{}^{\circ }C$: $\rho =0.962kg/m^3$, $\mu =2.131\times {10}^{-5}kg/ms$ $k=0.031W/mK$, $C_p=1.01kJ/kgK$ $Nu=0.332{Re}^{1/2}{Pr}^{1/3}$ for laminar flow $Nu=0.0296{Re}^{0.8}{Pr}^{1/3}$ for turbulent flow ]. [2016/20m/2b]

7. A 2-stroke motor cycle petrol engine cylinder consists of 15 fins on its outer surface. If the outside and inside diameters of each fin are 200 mm and 100 mm respectively, the average fin surface temperature is 475°C and the atmospheric air temperature is 25°C, calculate the heat transfer rate from the fins for the following cases: (i) when the motor cycle is stationary; (ii) when the motor cycle is running at a speed of 60 kmph. The fin may be idealized as a single horizontal plate of the same area, and the significant length may be taken as $L=0.9d_o$, where $d_o$ is the outer diameter of the fin. Assume $d_o$ as 200 mm. The properties of air may be taken as follows: $k=4.266\times {10}^{-2}W/m{}^{\circ }C$; $\nu =40.61\times {10}^{-6}{m}^2/s$; $Pr=0.677$ For turbulent flow (forced convection): $Nu=0.036{\left(Re\right)}^{0.8}{\left(Pr\right)}^{0.33}$ For natural convection: $Nu=0.54{\left(Gr.Pr\right)}^{1/4}$ if $\left(Gr.Pr\right)<{10}^9$ $Nu=0.10{\left(Gr.Pr\right)}^{0.33}$ if $\left(Gr.Pr\right)>{10}^9$. [2015/20m/4a]

8. Do you think that velocity boundary layer and thermal boundary layer depend on Prandtl number? If yes, explain properly. Also explain, with the help of neat sketches, the significance of relative thickness of velocity boundary layer and thermal boundary layer for the following: (i) Liquid metals (ii) Oils. [2013/10m/4c]

9. A steel strip emerges from hot roll section of a steel mill at a constant speed of 0.1 m/s and a temperature of 500°C. The length, width and thickness of the steel strip are 10 m, 1 m and 0.003 m respectively. Air at a mean velocity and free stream temperature of 20 m/s and 20°C respectively flows transversely over the strip. Density, specific heat and emissivity of steel strip, respectively, are 7850 $kg/m^3$, 620 J/kg and 0.7. The thermal conductivity, kinematic viscosity and Prandtl number for air at 20°C are respectively 0.044 W/m-K, $4.5\times {10}^{-5}{m}^2/s$ and 0.68. Neglect the variation in strip temperature across its width and thickness. (i) Write the governing equation for the temperature distribution along the length of the strip. (ii) Neglecting radiation from the steel strip, derive an expression for the temperature of steel strip. (iii) Neglecting radiation from the steel strip, calculate the steel strip temperature at the trailing edge. The following correlation for convection heat transfer may be used: $\overline{Nu}=0.037R{e}_{x\mathrm{\text{–}}l}^{0.8}P{r}^{1/3}$ Symbols have the usual meaning. [2012/20m/4a]

10. Define the term Nusselt number and explain its significance in the case of forced convection heat transfer. Atmospheric air at $T_{\infty }=275K$ and a free-stream velocity $u_{\infty }=20m/s$ flows over a flat plate L = 1.5 m long that is maintained at a uniform temperature $T_w=325K$. (i) Calculate the average heat transfer coefficient $h_m$ over the region where the boundary layer is laminar. (ii) Find the average heat transfer coefficient over the entire length L = 1.5 m of the plate. (iii) Calculate the total heat transfer rate Q from the plate to the air over the length L = 1.5 m and width W = 1 m. Assume transition occurs at $R_e=2\times {10}^5$. [2004/30m/6b]

11. Define Reynolds number, Prandtl number, Stanton number and Nusselt number, and give their physical significance. What is meant by fully developed flow in a pipe? Illustrate it with the help of a figure. State the parameters upon which the entrance length depends. [2003/20m/200w/5b]

12. Air at 30°C flows over a horizontal plate heated to 70°C at a speed of 3 m/s. Calculate the convective heat transfer coefficient and the rate of heat transfer between the plate and the air. The length of the plate is 2 m and width 1 m. Nusselt number is given by: $Nu=0.664{Re}^{1/2}{Pr}^{1/3}$. The properties of air at 50°C are $\nu =17.95\times {10}^{-6}{m}^2/s$, $K=0.0283W/mK$, $\rho =1.093kg/m^3$, $C_p=1.005kJ/kgK$. [2002/30m/6b]

13. Using dimensional analysis, derive an expression for the Nusselt number in terms of Prandtl and Grashof numbers for free convection heat transfer system. [1999/20m/200w/5b]

14. Show by dimensional analysis that in forced convection $Nu=\varphi [Re,Pr,U^2/C_{p}T]$ [?] when friction heating in the fluid cannot be neglected. [1995/10m/200w/5c(ii)]

2.3] Analogy between heat and momentum transfer

6 questions

1. A thin flat plate, that is 0.2 m × 0.2 m on a side, is oriented parallel to an atmospheric air stream having a velocity of 40 m/s. The air is at a temperature of $T_{\infty }=20{}^{\circ }C$, while the plate is maintained at $T_s=120{}^{\circ }C$. The air flows over the top and bottom surfaces of the plate, and measurement of the drag force reveals a value of 0.075 N. What is the rate of heat transfer from both sides of the plate to the air? (Assume ${\rho }_{air}=0.995kg/m^3$, ${\nu }_{air}=20.92\times {10}^{-6}{m}^2/s$, ${Pr}_{air}=0.7$, $k_{air}=30\times {10}^{-3}W/m\mathrm{\text{–}}K$) [2025/10m/2c]

2. A windshield of a car, having dimensions as 0.6 m high and 1.8 m long, is electrically heated and is subjected to parallel winds at 1 atm, 0 °C and 80 km/hr. The electrical power consumption is observed to be 50 W, when the exposed surface temperature of the windshield is 4 °C. Disregarding the radiation and heat transfer from the inner surface and using the momentum heat transfer analogy, determine the drag force the wind exerts on the windshield. The properties of air at 0 °C and 1 atm may be taken as: $\rho =1.292kg/m^3,C_p=1.006kJ/kg\mathrm{\text{–}}K,Pr=0.7362$ [2022/10m/3c]

3. A 2 m × 3 m flat plate is suspended in a room and subjected to air flow parallel to its surfaces along its 3 m long side. The free stream temperature and velocity of air are 20 °C and 7 m/s respectively. The total drag force acting on the plate is measured to be 0.86 N. Determine the average convective heat transfer coefficient for the plate. The properties of air at 20 °C and 1 atm are $\rho =1.204kg/m^3$, $C_p=1.007kJ/kg\mathrm{\text{–}}K$, Pr = 0.7309. [2020/10m/3c]

4. What is Reynolds-Colburn analogy? What do you mean by Colburn’s j-factor? [2018/20m/2b]

5. The average friction coefficient, when an incompressible fluid flows over a stationary flat surface with free stream velocity of 10 m/s, is 0.008. The average temperature of the plate and free stream fluid temperature are respectively at 120°C and 20°C respectively. Fluid properties known: density = 0.88 $kg/m^3$ and specific heat = 1001 J/kg-K, Pr = 0.65. Estimate the average rate of heat transfer per unit area of the plate. Flow over the plate is laminar. [2012/12m/5b]

6. Show that: $Nu{\left(Pr\right)}^{-1/3}=\{St{\left(Pr\right)}^{2/3}\}Re$. [2001/20m/200w/5d]

2.4] Convection in tubes and free convection

26 questions

1. Thin square plates of size 2.5 m × 2.5 m are coming out of an oven at 280 °C in a material processing facility. The plates are cooled by blowing ambient air at 20 °C horizontally parallel to their surfaces. Determine the air velocity above which the natural convection effects on heat transfer are less than 12% and thus are negligible. The kinematic viscosity of air at 1 atmosphere may be taken as $\nu =2.859\times {10}^{-5}{m}^2/s$. [2026/10m/3c]

2. The velocity and temperature profiles for laminar flow in a tube of radius $r_0=10$ mm have the form at a particular axial location $u\left(r\right)=0.1(1-\frac{r^2}{r_0^2})$ $T\left(r\right)=344.8+75\frac{r^2}{r_0^2}-18.8\frac{r^4}{r_0^4}$ with units of m/s and K, respectively. Determine the corresponding value of the mean (or bulk) temperature $T_m$ at this axial position. [2025/10m/3c]

3. The heat transfer rate due to free convection from a vertical surface, 1 m high and 0.6 m wide, to quiescent air that is 20 K colder than the surface is known. What is the ratio of the heat transfer rate for that situation to the rate corresponding to a vertical surface, 0.6 m high and 1 m wide, when the quiescent air is 20 K warmer than the surface? Neglect heat transfer by radiation and any influence of temperature on the relevant thermophysical properties of air. The correlation between Nusselt number and Rayleigh number is given as ${\overline{Nu}}_L=0.10{Ra}_L^{0.25}$. [2025/10m/4c]

4. Saturated liquid refrigerant at – 7°C flows through a horizontal copper (k = 330 W/mK) tube of inside diameter 25 mm, thickness 2.5 mm and length 10 m. The tube is exposed to surrounding air at 20°C. Find the exit dryness fraction of the refrigerant from the tube if the flow rate is 0.0012 kg/s and latent heat of evaporation is 400 kJ/kg. Take the property values of air at 280 K as given below: $\rho =1.271kg/m^3$, k = 0.0246 W/mK, $\gamma =1.4\times {10}^{-5}{m}^2/s$, Pr = 0.717 For natural convection from a horizontal tube, the following correlation be used: $\overline{N{u}_f}=\left(0.48\right){[Gr\cdot Pr]}^{0.25}$ Neglect the temperature difference between the copper tube and the refrigerant. Also neglect the thermal resistance of copper tube. [2024/20m/2b]

5. A vertical flat plate is maintained at a temperature of “$T_w$°C” and exposed to a stagnant atmospheric air “$T_a$°C”. If $T_w<T_a$, show the shape of thermal and velocity boundary layers developed on the surface of the plate. Also show the variation “$h_x$” along the vertical surface of the plate. Assume the flow is within the laminar region and consider only free convection. [2024/5m/4c(i)]

6. A fluid flows through a tube exposed to constant heat flux condition. The flow is in the turbulent flow regime for which the Dittus-Boelter correlation is applicable for determining the Nu number as given below: $N{u}_d=\left(0.023\right)R{e}^{0.8}P{r}^{0.4}$ In order to reduce the surface temperature of the tube, it is suggested to double the velocity of the flow. Find the percentage increase in the heat transfer coefficient due to increased velocity. [2024/5m/4c(ii)]

7. Water at 30 °C enters a 1.5 cm diameter horizontal tube with a velocity of 1 m/s. The tube wall is maintained at a constant temperature of 90 °C. Calculate the length of the tube if the exit water temperature is 65 °C. One may assume that the flow is turbulent, fully developed and the internal surface of the tube is smooth. The properties of water are given: Thermal conductivity ($k$) = 0.656 W/m-K Density ($\rho$) = $984.4kg/m^3$ Kinematic viscosity ($\nu$) = $0.497\times {10}^{-6}{m}^2/s$ Specific heat ($C_p$) = 4178 J/kg-K Prandtl number (Pr) = 3.12 Friction factor ($F$) = 0.079 (Reynolds number)${}^{-0.25}$ Reynolds number $\equiv R{e}_D$ Average Nusselt number $\left(N{u}_D\right)=\frac{\left(\frac{F}{2}\right)(R{e}_D-1000)Pr}{1+12.7{\left(\frac{F}{2}\right)}^{1/2}(P{r}^{2/3}-1)}$ Also, calculate the water temperature at the middle of the tube and pressure drop across the tube. [2023/20m/4b]

8. A solar collector, as shown in Fig. 3(b) below, having dimensions as 1 m wide and 5 m long, has constant spacing of 3 cm between the glass cover and the collector plate. Air enters the collector at 30 °C and at a rate of $0.15m^3/s$ through the 1 m wide edge and flows along the 5 m long passageway. If the average temperatures of the glass cover and the collector plate are 20 °C and 60 °C, respectively, determine (i) the net rate of heat transfer to the air in the collector and (ii) the temperature rise of air as it flows through the collector.

[Figure: Fig. 3(b) – Flat-plate solar collector shown in perspective: a flat rectangular passage of uniform height 3 cm between the top glass cover (Glass cover at 20 °C) and the collector plate (Collector plate at 60 °C), with insulation beneath the plate (labelled Insulation); air enters along one 1 m wide edge (arrows labelled ‘Air at 30 °C, $0.15m^3/s$’), flows along the 5 m long passageway (dashed streamlines) and leaves upward through an outlet at the far end (vertical arrows)]

The properties of air at 1 atm and an estimated average temperature of 35 °C may be taken as: $\rho =1.145kg/m^3,k=0.02625W/m{\mathrm{\text{–}}}^{\circ }C,\nu =1.655\times {10}^{-5}{m}^2/s,C_p=1007J/kg{\mathrm{\text{–}}}^{\circ }C,Pr=0.7268$ [2022/20m/3b]

9. The following equation has been proposed for the heat transfer coefficient in natural convection from long vertical cylinders to air at atmospheric pressure: ${\bar{h}}_c=\frac{536.5{(T_s-T_{\infty })}^{0.33}}{T}$ where T = the film temperature = $\frac{(T_s+T_{\infty })}{2}$ and T is in the range 0 to 200°C. The corresponding equation in dimensionless form is $\frac{{\bar{h}}_{c}L}{K}=C{\left(GrPr\right)}^m$. Compare the two equations to determine the values of C and m such that the second equation will give the same results as the first. Use properties of dry air at 100°C and one atmosphere: $K=0.0307$ W/(mk), $g=9.8m/se{c}^2$ $\mu =21.673\times {10}^{-6}NS/m^2$ $C_p=1022$ J/(kg K) The absolute pressure of one atmosphere = $101,000N/m^2$ The gas constant R (for air) = 287 J/kg K Symbols have their usual meaning. [2021/10m/1d]

10. Assume that the velocity distribution in the turbulent core for tube flow may be represented by $\frac{u}{u_c}={(1-\frac{r}{r_o})}^{\frac{1}{7}}$ where $u_c$ is the velocity at the centre of the tube and $r_o$ is the tube radius. Using the Blasius relation for friction factor, derive an equation for the thickness of the laminar sublayer. For this problem the average flow velocity may be calculated using the turbulent velocity distribution. Assume linear profile in sublayer. [2021/20m/3b]

11. Liquid mercury flows at a rate of 1.25 kg/s through a copper tube of diameter 20 mm. Mercury enters the tube at 20 °C and leaves the tube at 30 °C. Calculate the tube length for constant heat flux at the wall which is maintained at an average temperature of 40 °C. The empirical correlation for heat transfer for liquid metal flowing through a tube is given as $\overline{Nu}=7+0.025{\left(Pe\right)}^{0.8}$, where Pe is Peclet number, $Pe=Pr\cdot Re$. The properties of mercury at 25 °C are as follows: $\rho =13582kg/m^3$ $C_P=140J/kg\mathrm{\text{–}}K$ $k=8.69W/m\mathrm{\text{–}}K$ $\nu =1.5\times {10}^{-7}{m}^2/s$ $Pr=0.0248$ [2019/10m/1e]

12. Explain clearly what is “thermally developed zone” in case laminar flow through a tube both for (i) constant wall temperature case (ii) constant heat flux case. [2017/10m/4c]

13. A 2-stroke motor cycle petrol engine cylinder consists of 15 fins on its outer surface. If the outside and inside diameters of each fin are 200 mm and 100 mm respectively, the average fin surface temperature is 475°C and the atmospheric air temperature is 25°C, calculate the heat transfer rate from the fins for the following cases: (i) when the motor cycle is stationary; (ii) when the motor cycle is running at a speed of 60 kmph. The fin may be idealized as a single horizontal plate of the same area, and the significant length may be taken as $L=0.9d_o$, where $d_o$ is the outer diameter of the fin. Assume $d_o$ as 200 mm. The properties of air may be taken as follows: $k=4.266\times {10}^{-2}W/m{}^{\circ }C$; $\nu =40.61\times {10}^{-6}{m}^2/s$; $Pr=0.677$ For turbulent flow (forced convection): $Nu=0.036{\left(Re\right)}^{0.8}{\left(Pr\right)}^{0.33}$ For natural convection: $Nu=0.54{\left(Gr.Pr\right)}^{1/4}$ if $\left(Gr.Pr\right)<{10}^9$ $Nu=0.10{\left(Gr.Pr\right)}^{0.33}$ if $\left(Gr.Pr\right)>{10}^9$. [2015/20m/4a]

14. A hot plate of 100 cm height and 25 cm wide is exposed to atmospheric air at 25 °C. The surface temperature of the plate is 95 °C. Find the heat loss from both the surfaces of the plate. Also find the change in heat loss if the height of the plate is reduced to 50 cm and the width is increased to 40 cm. Use the following relations: $Nu=0.59{(Gr\cdot Pr)}^{0.25}\mathrm{\text{if}}Gr\cdot Pr<{10}^9$ $=0.10{(Gr\cdot Pr)}^{0.33}\mathrm{\text{if}}Gr\cdot Pr>{10}^9$ The properties of air are: $\rho =1.06kg/m^3$, $C_p=1004J/kg\mathrm{\text{–}}K$, $k=0.029W/m\mathrm{\text{–}}K$, $\nu =18.97\times {10}^{-6}{m}^2/sec$ [2013/15m/4b]

15. Water enters a tube of dia D and length L. Inlet and outlet temperatures are $T_1$ and $T_2$ respectively and wall temperature is $T_0$. Convective heat transfer is h. (i) Derive the following expression for heat transfer: $Q=\frac{h\pi DL(T_1-T_2)}{\ell n\left(\frac{T_1-T_0}{T_2-T_0}\right)}$ (ii) What is the significance of the logarithmic term in part (i) above? (iii) State assumptions used in part (i). [2011/20m/3a]

16. A heat transfer equipment utilises 5 mm diameter, 10 cm long smooth circular cross-sectioned conduits drilled horizontally in a plate longitudinally on which a constant heat flux is imposed uniformly. Air enters each conduit at 27 °C with a mean velocity of 3.0 m/s and leaves at 77 °C. If 20 conduits are arranged in the plate, calculate the following: (i) total rate of heat removed from the plate, (ii) exit conduit wall temperature, and (iii) local heat transfer coefficient at the exit of each conduit. Sketch the variation in local heat transfer coefficient along the length of conduit. The following correlations are known for convection heat transfer: (1) $N{u}_d=4.364$, for fully developed laminar flow, (2) $N{u}_d=0.023R{e}_d^{0.8}P{r}^{0.4}$, for turbulent flow, (3) $N{u}_z=1.3{\left[\frac{R{e}_{d}Pr}{\left(z/d\right)}\right]}^{1/3}$, developing flow with $\left[\frac{R{e}_{d}Pr}{\left(z/d\right)}\right]>10$ ‘z’ is the axial location from the inlet along the length of conduit. Properties of air at 52 °C: $C=1006J/kg\mathrm{\text{–}}K$, $\rho =1.1774kg/m^3$, $\nu =18.22\times {10}^{-6}{m}^2/s$, $K=0.028W/m\mathrm{\text{–}}K$, $Pr=0.703$. [2010/20m/3b]

17. For a circular tube, explain with the help of neat sketches (i) Hydrodynamic entry region and hydrodynamically developed flows. (ii) Thermal entry region and thermally developed flows. [2009/15m/3c]

18. A hot plate of 15 $c{m}^2$ area maintained at a temperature of 200° C is exposed to still air at 30° C temperature. When the smaller side of the plate is held vertical, convective heat transfer rate is 14% higher than when the bigger side of the plate is held vertical. Determine the dimensions of the plate. Neglect internal temperature gradient of the plate thickness. Also determine the heat transfer rates in both the cases. Use the following relation: $Nu=0.59{(Gr\cdot Pr)}^{0.25}$ Take the following properties of air:

Temperature (°C)	$\rho$ ($kg/m^3$)	$C_p$ (kJ/kg-K)	$\mu$ ($N\mathrm{\text{–}}s/m^2$)	K (W/m-K)
30	1.165	1.005	$18.6\times {10}^{-6}$	0.0267
115	0.910	1.009	$22.65\times {10}^{-6}$	0.0331
200	0.746	1.026	$26\times {10}^{-6}$	0.0393

[2009/30m/4b]

19. A 30 cm × 30 cm horizontal duct made of sheet metal carries warm air. The duct is maintained at 65°C and is 10 m long. Heat loss from duct to the ambient air takes place due to free convection only. If the ambient air is at 25°C, calculate the heat loss from the duct. Use the following free convection correlations. Heat transfer from the upper surface of a horizontal plate: $Nu=0.54{\left(R{a}_L\right)}^{1/4}$; Heat transfer from the lower surface of a horizontal plate: $Nu=0.27{\left(R{a}_L\right)}^{1/4}$; Heat transfer from vertical plate: $Nu=0.59{\left(R{a}_L\right)}^{1/4}$. The properties of air at 45° are: $Pr=0.695$, $\nu =1.85\times {10}^{-5}{m}^2/sec$, k = 0.028 w/mK. [2008/20m/2b]

20. A rectangular cooper plate 10 cm × 50 cm, having a mass of 1 kg and at a temperature of 100°C, is suspended vertically in still air at 20°C so that 50 cm side is vertical. Neglecting radiation effect, find heat transfer coefficient due to natural convection and initial rate of cooling of the plate in °C/minute. Take $C_p$ for copper = 383 J/kg-K. The properties of air at mean temperature 60°C are: $\rho =1.06kg/m^3$, $\nu =18.97\times {10}^{-6}{m}^2/s$, $Pr=0.696$, $C_p=1.005kJ/kg\mathrm{\text{–}}K$, $k=28.96\times {10}^{-3}W/m\mathrm{\text{–}}K$, $\mu =20.1\times {10}^{-6}N\mathrm{\text{–}}s/m^2$. You may use the following correlation: $Nu=0.1{\left(GrPr\right)}^{1/3}$. Will the result change if 10 cm side is vertical? Why? [2006/20m/1d]

21. Define Reynolds number, Prandtl number, Stanton number and Nusselt number, and give their physical significance. What is meant by fully developed flow in a pipe? Illustrate it with the help of a figure. State the parameters upon which the entrance length depends. [2003/20m/200w/5b]

22. Calculate the number of tubes required for a single pass steam condenser of 2.5 m length to handle 20000 hg/hr of dry saturated steam at 60°C. The cooling water enters the tubes at 20°C and leaves at 30°C. The tubes are of 25 mm outside diameter and 22.5 mm inside diameter. The thermal conductivity of tube material is 100 W/m-K. The water velocity is 1.5 m/s. Assume that the steam side film coefficient is 4500 $W/m^2\mathrm{\text{–}}K$. For water use, $Nu=0.023R{e}^{0.8}P{r}^{0.4}$ The properties of water at 25°C are $\rho =996.95kg/m^3$, $C_p=4178J/kg\mathrm{\text{–}}K$, $k=60.78\times {10}^{-2}W/m\mathrm{\text{–}}K$, $\nu =0.9055\times {10}^{-6}{m}^2/s$, Pr = 6.22, $h_{fg}$ (at 60°C) = 2358.5 kJ/kg. [2003/40m/6a]

23. A 3 m long 25 mm dia tube is held at 100°C by steam jacketing. Water flows through the tube @ 180 kg/hr at 20°C. Calculate the rate of heat transfer from the tube to water. For water (60°C): $C_p=4.178kJ/kgK$; $k=0.66\left(W/mK\right)$; $\rho =983kg/m^3$ and $\mu =0.47\times {10}^{-3}kg/ms$. [2001/30m/7a]

24. Explain dropwise condensation and film condensation. Why heat transfer rate is higher in dropwise condensation than in film condensation? [1996/20m/200w/5b]

25. Distinguish between nucleate boiling and film boiling and discuss a typical pool boiling curve. [1995/10m/200w/5a(i)]

26. A drop of oil put on a hot plate dances over the plate before finally evaporating, why? [1995/10m/200w/5a(ii)]

2.5] Radiation

33 questions

1. Consider two black surfaces $A_1$ and $A_2$ as shown in the figure below. Define the term ‘radiation shape factors’. Also, find a general expression for the energy exchange between these surfaces when they are maintained at different temperatures $T_1$ and $T_2$, respectively. If both the surfaces are at the same temperature, find out the reciprocity relationship between them:

[Figure: Two small plane surfaces drawn as tilted squares, $A_1$ (lower left) and $A_2$ (upper right), arbitrarily oriented, each with a pair of perpendicular lines crossing at its centre; a straight line of length r joins the two centres. $T_1$ is printed beside the right edge of $A_1$ and $T_2$ beside the upper-right edge of $A_2$.] [2026/10m/4c]

2. A 120 W electric bulb has a filament temperature of 3005 °C. Assuming the filament to be black, calculate (i) the diameter of the filament wire if the length is 250 mm and (ii) the efficiency of the bulb based on visible radiation if the visible radiation lies in the wavelength range from 0.4 μm to 0.75 μm. Assume Stefan-Boltzmann constant ($\sigma$) as $5.67\times {10}^{-8}W/m^2\mathrm{\text{–}}{K}^4$. The black body radiation functions are given in the table:

$\lambda T$ (μm-K)	$f_{0\to \lambda }$
1300	0.004317
1400	0.007791
2400	0.140266
2500	0.161366

[2025/10m/1e]

3. Explain the procedure to arrive at Stefan-Boltzmann law from the Planck’s law. Also find the total emissive power of a black sphere of 5 cm diameter maintained at 500 K. Take $\sigma =5.67\times {10}^{-8}W/m^2{K}^4$. [2024/5m/2c(i)]

4. Explain the procedure of arriving at Wien’s displacement law from Planck’s law. Also find the temperature of the sun if the wavelength at which maximum monochromatic emissive power is received is 0.55 μm. Take Wien’s constant = 2.9 mm K. [2024/5m/2c(ii)]

5. An industrial furnace (blackbody) is emitting radiation at 2700 °C. Calculate the following: (i) Spectral emissive power at $\lambda =1.2\mu m$ (ii) Wavelength at which the emissive power is maximum (iii) Maximum spectral emissive power (iv) Total emissive power Use Planck’s distribution law equation given below: $E_{b\lambda }=\frac{C_1}{{\lambda }^5[exp\left(C_2/\lambda T\right)-1]}$ where $C_1=3.742\times {10}^{8}W\mathrm{\text{–}}\mu m^4/m^2$, $C_2=1.438\times {10}^4\mu m\mathrm{\text{–}}K$. Take $\sigma =5.67\times {10}^{-8}W/m^2\mathrm{\text{–}}{K}^4$. [2023/10m/1d]

6. Define the total and spectral black body emissive powers. How are they related to each other? [2022/5m/1e(i)]

7. Consider two identical bodies, one at 1000 K and the other at 1500 K. Which body emits more radiation in the shorter wavelength region? Which body emits more radiation at a wavelength of 20 μm? [2022/5m/1e(ii)]

8. (i) Show that the effective conductance, $\left(A_1{\bar{F}}_{12}\right)$ for two black, parallel plates of equal area connected by re-radiating walls at constant temperature is $A_1{\bar{F}}_{1-2}=A_1\left(\frac{1+F_{1-2}}{2}\right).$ (ii) Determine the steady-state temperatures of two radiation shields placed in the evacuated space between two infinite planes at temperatures of 555 K and 278 K. The emissivity of all surfaces is 0.8. [$\sigma$ = Stefan-Boltzmann constant = $5.670\times {10}^{-8}W/m^2{K}^4$].

[Figure: A rectangular box (enclosure) drawn in perspective: the top face is plate $A_1$ and the bottom face is plate $A_2$ (equal, parallel plates); the four side faces are labelled ‘Re-radiating Walls’] [2021/20m/3a]

9. Daylight and incandescent light may be approximated as blackbody emissions at the effective surface temperatures of 5800 K and 2800 K, respectively. From the principles of radiation, show that incandescent lightbulbs are inefficient as light sources in the visible spectrum wavelength range of 0.40 μm to 0.76 μm. The blackbody radiation function, $f_{\lambda }$ as a function of $\lambda T$ (μm-K) given in the table may be used. The blackbody radiation function, $f_{\lambda }$ represents the fraction of radiation emitted from a blackbody at temperature $T$ in the wavelength band from $\lambda =0$ to $\lambda$, to the total radiation energy emitted by the blackbody in the wavelength range 0 to ∞.

Blackbody radiation function, $f_{\lambda }$

$\lambda T$ (μm-K)	$f_{\lambda }$	$\lambda T$ (μm-K)	$f_{\lambda }$
200	0.000000	6200	0.754140
400	0.000000	6400	0.769234
600	0.000000	6600	0.783199
800	0.000016	6800	0.796129
1000	0.000321	7000	0.808109
1200	0.002134	7200	0.819217
1400	0.007790	7400	0.829527
1600	0.019718	7600	0.839102
1800	0.039341	7800	0.848005
2000	0.066728	8000	0.856288
2200	0.100888	8500	0.874608
2400	0.140256	9000	0.890029
2600	0.183120	9500	0.903085
2800	0.227897	10000	0.914199
3000	0.273232	10500	0.923710
3200	0.318102	11000	0.931890
3400	0.361735	11500	0.939959
3600	0.403607	12000	0.945098
3800	0.443382	13000	0.955139
4000	0.480877	14000	0.962898
4200	0.516014	15000	0.969981
4400	0.548796	16000	0.973814
4600	0.579280	18000	0.980860
4800	0.607559	20000	0.985602
5000	0.633747	25000	0.992215
5200	0.658970	30000	0.995340
5400	0.680360	40000	0.997967
5600	0.701046	50000	0.998953
5800	0.720158	75000	0.999713
6000	0.737818	100000	0.999905

[2019/10m/4c]

10. Liquid air boiling at −153 °C is stored in a spherical condenser of diameter 320 mm. The container is surrounded by a concentric spherical shell of diameter 360 mm in a room at 27 °C. The space between the two spheres is evacuated. The surfaces of the spheres are flushed with aluminium of emissivity 0.03. Taking the latent heat of vaporization of liquid air as 210 kJ/kg, find the rate of evaporation of liquid air. [2018/10m/4c]

11. What is the shape factor in case of radiative heat exchange? Discuss the four (4) basic shape factor laws. [2017/10m/1c]

12. A small disc-shaped earth satellite, 1 m in diameter circles the earth (radius 6250 km) at a distance of 300 km from the surface. The flat surface of the disc is oriented tangential to the earth’s surface. The satellite surface has an emissivity of 0.3 and is at −18°C. Calculate the net rate at which energy is leaving the satellite. Assume that: (i) The average earth surface temperature is 27°C and the earth is black body. (ii) The satellite is in shadow of the earth and (iii) The part of the satellite surrounding not occupied by the earth is black and at 0 K. (iv) Stefan-Boltzmann constant $\left(\sigma \right)=5.67\times {10}^{-8}W/m^2\cdot K^4$.

[Figure: Satellite 1 (a small disc) located 300 km above the surface of Earth 2 (a circle of radius 6250 km); dashed lines from the satellite run tangent to the earth’s circle and to its centre; the region around (Surrounding 3) is shown by a curved boundary above the satellite; Sun Rays arrive from below the earth, so the satellite is in the earth’s shadow.] [2017/20m/7b]

13. Derive Wien’s displacement law from Planck’s law of radiation. [2016/10m/1d]

14. A small sphere (of outside diameter = 60 mm) with a surface temperature of 300°C is located at the geometric centre of a large sphere (of inside diameter = 360 mm) with an inner surface temperature of 15°C. Calculate how much of emission from the inner surface of the large sphere is incident upon the outer surface of the small sphere. Assume that both bodies approach black body behaviour. What is the net interchange of heat between the two spheres? [2015/10m/1c]

15. A boiler furnace of 3 m height is made in the shape of a frustum of a cone with the bottom diameter 5 m ($d_1$) and top diameter 6 m ($d_2$). The emissivity of both the surfaces is 0.9. The bottom surface is at 1000°C and the top surface is at 500°C. Considering the inclined surface is refractory surface, find (i) the radiation heat transfer from the bottom to top surface and (ii) the inclined surface temperature. Radiation Shape Factor $F_{1-2}=0.4$. [2015/10m/3c]

16. A thin radiation shield having equal emissivities on both sides is introduced parallel to and in between two large planes with emissivities 0.8 and 0.5 respectively. Determine the emissivity of the radiation shield to reduce the heat transfer rate by 92% of the original. [2014/10m/1d]

17. Liquid $N_2$ enters a thin-walled 20 mm diameter tube at 77 K and flows steadily. The outer surface of the tube has an emissivity of 0.02. This tube is placed concentrically in another tube of 50 mm inner diameter, whose inner surface emissivity is 0.05. The inner surface of the outer tube is maintained at 300 K and the space in between the tubes is evacuated. Determine the heat gained by the liquid $N_2$ per unit length of the tube. If a thin-walled radiation shield with emissivity 0.02 on both sides is inserted midway concentrically between inner and outer tubes, calculate the % change in heat gained by liquid $N_2$ per unit length of the tube. [2014/20m/4a]

18. Explain Wien’s displacement law. Assuming sun to be a blackbody with a constant surface temperature of 5780 K, calculate the wavelength at which it will have the maximum spectral emissive power. [2013/10m/1d]

19. An electric furnace consisting of two flat surface heaters, top and bottom, is used to heat treat a coating that is applied to both surfaces of a thin metal plate inserted midway between the heaters. The heaters and the plate are 2 m × 2 m on a side, and each heater is separated from the plate by a distance of 0.5 m. Each heater is well insulated on its back side and has an emissivity of 0.9 at its exposed surface. The plate and side walls have emissivities of 0.6 and 0.3 respectively. Under steady operating conditions, both heaters are at 800 K while the side walls are at 400 K. View factor between the heaters and the plate is 0.62. (i) Sketch the system and its equivalent thermal network and label all pertinent resistances and potentials. (ii) Calculate the associated resistances and driving potentials. (iii) Calculate the required electrical power. [2012/20m/8a]

20. Sunlight of 1 $kW/m^2$ falls directly on a window which is a vertical sheet of plain glass. Sun is in southwest and 30° above horizon. Window points towards the south. Estimate the amount of solar energy that, (i) enters the building through the window, (ii) is reflected by the window. Given black body emissivity power table:

$\lambda T$	$f\left(\lambda T\right)$
1900	0.05210
2000	0.06672
2100	0.08305
15000	0.96893
16000	0.97377
17000	0.97765

[2011/15m/1c]

21. A 0.5 m diameter disc heater is horizontally placed and enclosed concentrically in a hemispherical shaped surface. The surface of the enclosure having an emissivity of 0.7 is maintained at 500 K. The disc heater, having emissivity of 0.8 is maintained at 1200 K. The diameter of the hemisphere is 2 m and the remaining base area enclosed is open to surroundings at 300 K and may be considered as black with reference to radiation exchange. Sketch the schematic and thermal radiation network. Using thermal network method, calculate the heat exchange between heater and hemispherical enclosure and that between heater and surroundings. Neglect convection heat transfer. Assume heater and hemispherical surface are opaque, diffuse and gray. [2010/20m/3c]

22. Reflectivity measurements are often used to obtain surface radiation properties. Diffuse spectral reflectivity of a fire brick wall maintained at 500 K is shown below:

[Figure: Graph of spectral reflectivity ${\rho }_{\lambda }$ (vertical axis, marks 0.2, 0.5, 0.9, 1.0) against wavelength $\lambda$ (μm) (horizontal axis, marks 0, 2.4, 12, → ∞). Step function: ${\rho }_{\lambda }=0.9$ for 0 < λ < 2.4 μm; ${\rho }_{\lambda }=0.5$ for 2.4 < λ < 12 μm; ${\rho }_{\lambda }=0.2$ for λ > 12 μm (continuing to ∞).]

The wall is exposed to a bed of coal at 2500 K. Determine the total hemispherical emissivity and emissive power of the fire brick wall. What is the total absorptivity of the wall to the irradiation resulting from emission by coal? Use the following Plank’s radiation functions:

λT (μmK)	F(0-λ)
1200,	0.0025
6,000	0.7393
30,000	0.9912

Assume that the brick wall is opaque and the spectral distribution of irradiation at the brick wall approximates that due to emission from a black body at 2500 K. [2008/40m/2a]

23. Consider a diffuse circular disc of diameter D and area $A_j$ and a plane diffuse surface of area $A_i\ll A_j$. The surface are parallel and $A_i$ is located at a distance L from the centre of $A_j$. Obtain an expression for view factor. [2007/20m/200w/1d]

24. The configuration of a furnace can be approximated as an equilateral triangular duct which is sufficiently long that the end effects are negligible. The hot wall is maintained at $T_1=900K$ and has an emissivity ${\epsilon }_1=0.8$. The cold wall is at $T_2=400K$ and has an emissivity ${\epsilon }_2=0.8$. The third wall is reradiating zone for which $Q_3=0$. The accompanying sketch illustrates the configuration. Calculate the net radiation heat flux leaving the hot wall.

[Figure: Cross-section of the duct: an equilateral triangle with both base angles marked 60°. Left inclined wall labelled ① with $T_1=900K$, ${\epsilon }_1=0.8$; right inclined wall labelled ② with $T_2=400K$, ${\epsilon }_2=0.8$; base wall labelled ③ (reradiating, $Q_3=0$).] [2007/20m/2a]

25. Two large parallel plates at $T_1=1000K$ and $T_2=750K$ have emissivities ${\epsilon }_1=0.5$ and ${\epsilon }_2=0.8$ respectively. A radiation shield having an emissivity of ${\epsilon }_{3,1}=0.12$ on one side and an emissivity of ${\epsilon }_{3,2}=0.08$ on the other side is placed between the plates. Calculate the heat transfer rate by radiation per square meter with and without radiation shield. Take $\sigma =5.67\times {10}^{-8}W/m^2{K}^4$. [2005/30m/6a]

26. The surface of two circular discs parallel to each other, each with 1 $m^2$ area, are exchanging thermal radiation with each other and with the surrounding walls. The temperatures of disc 1, disc 2 and the walls are 800K, 500 K and 300 K respectively. The shape factor between discs is 0.5. Compute with help of network analysis the net radiative heat transfer between the two discs. [2001/30m/7b]

27. The total emissivity of clean chromium as a function of temperature T K is approximately given as $\epsilon =0.38(1-\frac{263}{T})$ Obtain an expression for the absorptivity of solar radiation as a function of surface temperature. The intensity of solar radiation at earth’s surface may be calculated from the following data: Diameter of sun $1.39\times {10}^6$ km Surface temperature of sun = 600 K. Distance between sun and earth = $149\times {10}^6$ km Assume sun to behave as a black body. [2000/30m/7c]

28. Show that the shape factor $F_{1R2}$ for two parallel black surfaces of equal area connected by re-radiating walls at constant temperature is given by $F_{1R2}=(1+F_{1-2})/2$ where $F_{1R2}$ is the shape factor of surface 1 with respect to surface 2, when connected by re-radiating surface R. [1998/20m/6a]

29. A furnace is in the shape of a cylinder of 2 m diameter and 2 m height. The top is at 1000 K while the base is at 700 K. The emissivity of the surface is 0.6. The enclosing curved surface acts as non-absorbing re-radiating surface. Determine the radiation exchange between the top and bottom surfaces. The shape factor between the top and bottom surfaces when sides are open is 0.5. [1998/40m/6b]

30. Explain the process of heat exchange by radiation between two bodies. Define the various laws and terms used in estimation. [1997/20m/200w/1c]

31. Distinguish between Black Body Radiation and Grey Body Radiation. Can the displacement law of radiation be applied to Grey Body Radiation? [1995/15m/7a]

32. Define Plant’s distribution Law and derive Stefan Boltzmann law from Plant’s distribution equation. [1995/15m/7b]

33. Show that where the heat transfer between two black surfaces is influenced by a third adjoining re-radiating surface in steady state the geometric factor will be given by ${\bar{F}}_{12}=F_{12}+\left[\frac{1}{\frac{A_1}{A_2}\frac{1}{F_2}\frac{1}{F_{13}}}\right]$ [?] Where F, A have usual meaning and suffix 1, 2, 3 are for the three surfaces respectively. [1995/30m/7c]

2.6] Heat exchangers

33 questions

1. Steam at 40 °C is condensed in a power plant using cooling water available at 12 °C. The flow rate of steam is 0.15 kg/s. The cooling water is circulated through a bank of 6 m long and 1.25 cm internal diameter thin copper tubes at a mean velocity of 3.5 m/s, and leaves at a temperature of 20 °C. The tubes are nearly isothermal at 40 °C. Calculate the average heat transfer coefficient between the water and the tubes, and find the number of tubes needed to achieve the indicated heat transfer rate in the condenser. The properties of water at an average temperature may be taken as: $\rho =998.8kg/m^3$ $C_p=4184.5J/kg\mathrm{\text{–}}{}^{\circ }C$ $h_{fg}\left(water\right)=2406kJ/kg$ [2026/20m/3a]

2. What does the NTU of a heat exchanger represent? Is a heat exchanger with a very large NTU necessarily a good one? [2026/10m/4a(i)]

3. Consider an oil-to-oil double-pipe heat exchanger, whose flow arrangement is not known. The temperature measurements indicate that the cold oil enters at 18 °C and leaves at 50 °C, while the hot oil enters at 75 °C and leaves at 40 °C. Assuming the mass flow rates of both the oils to be identical, determine the effectiveness and flow configuration of this heat exchanger. [2026/10m/4a(ii)]

4. A shell and tube type condenser is employed in a steam power plant to handle 35000 kg/h of dry and saturated steam at 50 °C. The cooling water enters the condenser at 15 °C and leaves at 25 °C. The tubes are of 22.5 mm inside diameter and 25 mm outside diameter. The water flows through the tubes at an average velocity of 2 m/s. The heat transfer coefficient on steam side is $5001W/m^2\mathrm{\text{–}}K$. Calculate the following: (i) The mass flow rate of water (in kg/s) (ii) The heat transfer surface area (based on $U_0$) (iii) The number of tubes required for water flow (iv) The number of tube passes in condenser if the length of each tube pass should not be more than 2.5 m Assume that condensate coming out from the condenser is saturated water and resistance of tube wall is negligible. For waterside heat transfer coefficient, use the correlation $Nu=0.023{Re}^{0.8}{Pr}^{0.4}$. Take latent heat of steam as 2374 kJ/kg. The properties of water at mean bulk temperature of 20 °C are as follows: $Pr=6.98$ $\rho =998.9kg/m^3$ $C_p=4.180kJ/kg\mathrm{\text{–}}K$ $\nu =1.0006\times {10}^{-6}{m}^2/s$ $k_f=0.59859W/m\mathrm{\text{–}}K$ Assume fully developed flow through tubes and a single shell is used. [2025/20m/4a]

5. A shell and tube heat exchanger used in a thermal power plant is designed to condense 500 kg/s of saturated steam entering the condenser at 20 kPa to saturated water. Cooling water enters the heat exchanger at 35°C and leaves at 45°C while flowing through copper tubes of 50 mm diameter with negligible thickness. Overall heat transfer coefficient is estimated to be $1500W/m^{2}K$. Find the following for the heat exchanger: (i) Total water flow rate required. (ii) Number of tubes required if water velocity = 1.0 m/s in the tube. (iii) Length of each tube. (iv) Total length of the tubes. Take the following property values: $C_p$ of water = 4.2 kJ/kg.K Density of water = $1000kg/m^3$ For Steam: $T_{saturation}=60$°C $h_{fg}=2600$ kJ/kg. [2024/20m/3a]

6. Write down the basic assumptions for LMTD method in case of heat exchanger analysis. [2023/5m/1e(i)]

7. Write down in which case LMTD method and in which case NTU method will be applicable in basic heat exchanger analysis. [2023/5m/1e(ii)]

8. A shell and tube heat exchanger operates with two shell passes and four tube passes. The shell side fluid is ethylene glycol, which enters at 140 °C and leaves at 80 °C with a flow rate of 4500 kg/hr. Water flows in the tubes, entering at 35 °C and leaving at 85 °C. The overall heat transfer coefficient for this arrangement is $850W/m^2{\mathrm{\text{–}}}^{\circ }C$. Calculate the flow rate of water required and the area of the heat exchanger. The specific heat of ethylene glycol may be taken as 2.742 J/g-°C and the specific heat of water may be taken as 4.175 J/g-°C. For NTU relations, the following figure may be used.

[Figure: Effectiveness–NTU chart for ‘Two shell passes, 4, 8, 12, etc., tube passes’, with a sketch at the top showing 2 shells, shell fluid ${\left(\dot{m}c\right)}_s=C_s$ and tube fluid ${\left(\dot{m}c\right)}_t=C_t$. Vertical axis: Effectiveness ε, % (0 to 100, gridlines every 10); horizontal axis: Number of transfer units, $NT{U}_{max}=AU/C_{min}$ (0 to 5, gridlines every 0.5). Curves for $C_{min}/C_{max}$ = 0, 0.25, 0.50, 0.75 and 1.00, all starting from the origin and rising steeply, the curve for 0 being highest and that for 1.00 lowest (about 72% at NTU = 5)] [2022/10m/4c]

9. A counter-flow double-pipe heat exchanger is to heat water from 20 °C to 80 °C at a rate of 1.2 kg/s. The heating is to be accomplished by geothermal water available at 160 °C 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/m${}^2$-K, determine the length of the heat exchanger required to achieve the desired heating. Take specific heat of water and geothermal fluid as 4.18 kJ/kg-K and 4.31 kJ/kg-K respectively. [2020/20m/4b]

10. A counterflow double-pipe heat exchanger is used to heat glycerin from 20 °C to 50 °C by hot water, which enters the thin-walled 2 cm diameter tubes at 80 °C and leaves at 40 °C. The total length of the heat exchanger is 60 m. The convection heat transfer coefficient is 25 W/m${}^2$-K on the glycerin side (in the annulus) and 160 W/m${}^2$-K on the water side (in the inner tube). Determine the rate of heat transfer in the heat exchanger (i) before any fouling and (ii) after fouling with a fouling factor of 0.0006 m${}^2$-K/W occurred on the glycerin side of the inner tube. The thermal resistance of the inner tube is negligible. [2019/20m/3b]

11. Water ($C_p=4.187kJ/kg\mathrm{\text{–}}K$) is heated at the rate of 1.4 kg/s from 40 °C to 70 °C by an oil ($C_p=1.9kJ/kg\mathrm{\text{–}}K$) entering at 110 °C and leaving at 60 °C in a counterflow heat exchanger. If the overall heat transfer coefficient is $350W/m^2\mathrm{\text{–}}K$, calculate the surface area required. Using the same entering fluid temperature and the same oil flow rate, calculate the exit temperature of oil. Also calculate the exit temperatures of oil and water and the rate of heat transfer, when the water flow rate is halved. [2018/20m/4a]

12. In an open heart surgery, under hypothermic conditions, the patient’s blood is cooled before surgery and rewarmed afterwards. It is proposed that a concentric tube counterflow heat exchanger of length 0.5 m is to be used for this purpose, with a thin walled inner tube having diameter of 55 mm. If water at 60°C and 0.1 kg/s is used to heat the blood entering the heat exchanger at 18°C at a flow rate of 0.01 kg/s, what is the temperature of the blood leaving the heat exchanger? One may assume, overall heat transfer coefficient (U) = 500 W/m²-K, specific heat of blood and water are respectively $C_{P_{blood}}=3.5kJ/kg\mathrm{\text{–}}K$, $C_{P_{water}}=4.187kJ/kg\mathrm{\text{–}}K$. [2017/20m/3a]

13. Write down the assumptions to analyze a counterflow heat exchanger using LMTD (Log mean temperature difference) method and also write down the expression for LMTD in a counterflow heat exchanger with the help of terminal temperatures. [2017/10m/8c]

14. Define effectiveness, NTU and heat capacity ratio in case of heat exchanger analysis, and also establish a relationship among them for counterflow heat exchanger. [2016/10m/1c]

15. In a large steam power plant, a shell and tube type condenser is used which has the following data: Heat exchange data = 2100 MW Number of shell passes = 1 Number of tubes = 31500 Number of tube passes = 2 Diameter of each tube = 25 mm Condensation temperature = 50 °C Mass flow rate of cooling water = $3.4\times {10}^{4}kg/s$ Heat transfer coefficient on the steam side = $11400W/m^{2}K$ Inlet water temperature = 20 °C Heat transfer coefficient on the water side = $8018W/m^{2}K$ Using only ε-NTU method, calculate– (i) the outlet temperature of cooling water; (ii) the length of tube pass. [ Properties of water at 27 °C are: $C_p=4.18kJ/kgK$, $\mu =855\times {10}^{-6}Ns/m^2$, $k=0.613W/mK$ and $Pr=5.83$ ] Neglect the thermal resistance due to tube wall. [2016/20m/4c]

16. Steam at atmospheric pressure enters the shell of a surface condenser, in which the water flows through a tube bundle, at the rate of 0.05 kg/s. The inner diameter of the tube is 25 mm. The overall heat transfer coefficient, ($U_i$) based on the inner diameter is $230W/m^2{}^{\circ }C$. The inlet and outlet temperatures of water are 15°C and 70°C, respectively. The condensation of steam takes place on the outside surface of the tubes. Calculate the following: (i) The effectiveness of the heat exchanger and NTU. (ii) Length of each tube. (iii) The rate of steam condensation. Assume $C_p$ of water $=4.18\frac{kJ}{kg{}^{\circ }C}$; $h_{fg}=2257\frac{kJ}{kg}$ (latent heat of condensation). [2015/10m/1d]

17. The vapour, at the saturation temperature of an oil flowing at the rate of 500 kg/min, enters a heat exchanger tube, at 355 K and condenses while it is cooled by water flowing at the rate of 3600 kg/min entering the concentric tube of a parallel-flow heat exchanger at 286 K. Assuming overall heat transfer coefficient of $475W/m^{2}K$, latent heat of oil as 600 kJ/kg K, calculate the number of tubes required of 25 mm outer diameter and 2 mm thick with a length of 4.87 m. What will be the number of tube passes, if cooling water velocity should not exceed 2 m/s? Take $C_p$ for water as 4.18 kJ/kg K and density of water as $1000kg/m^3$. [2014/10m/2c]

18. In a balanced counterflow heat exchanger, where ${\dot{m}}_c{C}_{p,c}={\dot{m}}_h{C}_{p,h}$, show that– (i) $\Delta T_1=\Delta T_2=\Delta T$ at any section; (ii) the temperature profiles of two fluids are parallel and linear. [2013/20m/3c]

19. In a double-pipe heat exchanger, ${\dot{m}}_h{C}_{p,h}=0.5{\dot{m}}_c{C}_{p,c}$. The inlet temperatures of hot and cold fluids are $T_{h,i}$ and $T_{c,i}$. Determine an expression, in terms of $T_{h,i}$, $T_{c,i}$ and $T_{h,o}$, for the ratio of area of counterflow heat exchanger to that of parallel-flow heat exchanger, which will give same hot fluid outlet temperature $T_{h,o}$. Also find out the ratio, if $T_{h,i}=150{}^{\circ }C$, $T_{c,i}=30{}^{\circ }C$ and $T_{h,o}=90{}^{\circ }C$. [2013/25m/4a]

20. A tubular gas heater heats air flowing at the rate of 5.5 kg/s from 20°C to 75°C using saturated steam condensing at 1.3 bar (saturation temperature, $t_{sat}$ = 107°C). It is proposed to double the flow rate of air to heat the same for the same rise in temperature in the same gas heater. One way of doing this is to increase the condensing pressure of saturated steam. What should be the pressure needed if the overall heat transfer coefficient remains the same for both the operating conditions? Specific heat of air = 1.005 kJ/kg-K. For steam, the following pressure and the corresponding saturation temperature are known:

p (bar)	2	3	4	5	6	7	8	9
$t_{sat}$(°C)	120.2	133.5	143.6	151.8	158.8	165.0	170.4	175.4

[2012/20m/6b]

21. A cross flow heat exchanger consists of a bundle of 32 straight 0.6 m long tubes in a rectangular duct of cross sectional area of 0.6 $m^2$. Hot water at 150 °C and a mean velocity of 0.5 m/s enters each tube having inner and outer diameters of 10.2 mm and 12.5 mm respectively. Atmospheric air at 10 °C enters the heat exchanger with a volumetric flow rate of 1 $m^3/s$. The mean convective heat transfer coefficient on the outside air flow is 400 $W/m^2\mathrm{\text{–}}K$. Assume tube side flow is fully developed and negligible thermal resistance due to tube wall. Heat transfer is only between air and water. Calculate exit temperatures of water and air and the total heat transfer rate. The following properties are known: For air: at 10 °C, $\rho =1.2407kg/m^3$, at 40 °C, $\rho =1.1181kg/m^3$, $C_p=1007J/kg\mathrm{\text{–}}K$. For water: $\rho =922kg/m^3$, $C_p=4297J/kg\mathrm{\text{–}}K$, $K=0.688W/m\mathrm{\text{–}}K$, $\mu =188\times {10}^{-6}N\mathrm{\text{–}}s/m^2$, $Pr=1.18$. Take cross flow correction factor as 0.8. Heat transfer correlations: (1) $N{u}_d=4.364$, for fully developed laminar tube flow. (2) $N{u}_d=0.023R{e}_d^{0.8}P{r}^{0.3}$, for turbulent flow in tube. (3) $\epsilon =\frac{1-e^{\{-C_R[1-e^{-N}]\}}}{C_R}$, minimum fluid mixed and the other unmixed in cross flow exchanger $\epsilon =1-e^{-\{[1-e^{-\left(C_{R}N\right)}]\cdot \frac{1}{C_R}\}}$, minimum fluid unmixed and the other mixed in cross flow exchanger, where $\epsilon$: effectiveness, $C_R$: capacity ratio, N: Number of Transfer Units. Symbols have the usual meaning. [2010/20m/4c]

22. An oil is cooled to 100° C in a parallel flow heat exchanger by transferring its heat to cooling water, that leaves the exchanger at 30° C. However, it is now required that the oil must be cooled down to 75° C by increasing the length of heat exchanger, while oil and water flow rates, their inlet temperatures and other dimensions of the exchanger keeping constant. The inlet temperatures of water and oil are 15° C and 150° C respectively. If the original cooler was 1 metre long, determine (i) outlet temperature of water in the new cooler and (ii) length of the new cooler. [2009/30m/4a]

23. A counterflow, concentric tubes heat exchanger is designed to heat water from 20 °C to 80°C using hot oil flowing through the annulus. The oil temperature gets reduced from 160 °C to 140 °C. The nominal diameter of the inner tube is 20 mm and the corresponding overall heat transfer coefficient is $500W/m^2\mathrm{\text{–}}K$. The heat transfer rate from the oil is 3000 watts. Determine the length of the exchanger. Because of fouling after some days the outlet temperature of water reduced to 65°C for the same flow rates and same inlet conditions. Determine the outlet temperature of oil, the fouling factor and the new heat transfer rate. Sketch the heat exchanger arrangement and the temperature profiles. [2006/30m/4a]

24. Illustrate, with neat sketches, the temperature profile for hot and cold fluids as a function of the distance along the path for (i) parallel flow heat exchangers, (ii) counter flow heat exchangers, (iii) condenser and (iv) gas-heated boiler. [2005/20m/5a]

25. In a double pipe counterflow heat exchanger, 10,000 kg/h of an oil, having a specific heat of 2095 J/kg K, is cooled from 80° C to 50° C by 8000 kg/h of water entering at 25° C. Determine the heat exchanger area for an overall heat transfer coefficient of 300 $W/m^{2}K$. Take $C_p$ of water as 4180 J/kg K. [2004/20m/5a]

26. Show that the effectiveness of a parallel flow heat exchanger is given by $\epsilon =\frac{1-exp\{-NTU(1+C_{min}/C_{max})\}}{1+C_{min}/C_{max}}$ [2003/20m/200w/5a]

27. Calculate the number of tubes required for a single pass steam condenser of 2.5 m length to handle 20000 hg/hr of dry saturated steam at 60°C. The cooling water enters the tubes at 20°C and leaves at 30°C. The tubes are of 25 mm outside diameter and 22.5 mm inside diameter. The thermal conductivity of tube material is 100 W/m-K. The water velocity is 1.5 m/s. Assume that the steam side film coefficient is 4500 $W/m^2\mathrm{\text{–}}K$. For water use, $Nu=0.023R{e}^{0.8}P{r}^{0.4}$ The properties of water at 25°C are $\rho =996.95kg/m^3$, $C_p=4178J/kg\mathrm{\text{–}}K$, $k=60.78\times {10}^{-2}W/m\mathrm{\text{–}}K$, $\nu =0.9055\times {10}^{-6}{m}^2/s$, Pr = 6.22, $h_{fg}$ (at 60°C) = 2358.5 kJ/kg. [2003/40m/6a]

28. Define the terms: capacity ratio, effectiveness and number of transfer units. Derive an expression for effectiveness for a parallel flow concentric heat exchanger in terms of capacity ratio and number of transfer units. [2000/20m/200w/5a]

29. In a concentric tube heat exchanger, cold water is heated by the heat released by the steam condensation. The water enters the tube at 10°C and leaves at 50°C. The steam pressure is maintained at 1.01325 bar (atmospheric). If the length of the tube is increased to 3 times, what would be the outlet temperature of water? Assume overall heat transfer coefficient remains constant for both the cases. Specific heat for water may be assumed to be 4187 kJ/kg-K,. [2000/15m/7a]

30. A counter flow heat exchanger of heat transfer area of $16.7m^2$ is to cool oil ($C_{ph}=2000$ J/kg s) with water ($C_{pc}=4170$ J/kg s). The oil enters at $t_{h,in}=100{}^{\circ }C$ and $m_h=2.0$ kg/sec while the water enters at $t_{c,in}=20{}^{\circ }C$ and $m_c=0.42$ kg/s. The overall heat transfer coefficient is $U=300W/m^2{}^{\circ }C$. Calculate the exit temperature of water $t_{c,out}$ and the total heat transfer rate Q. [1999/40m/6b]

31. Define the terms (i) over all heat transfer coefficient (ii) log mean temperature difference (iii) effectiveness of heat exchangers. [1997/15m/4a]

32. A counter flow double pipe heat exchanger is used to heat water using oil as the heating fluid. The heat exchanger area is 16 sq.m and it is observed that water inlet and outlet temperatures are 35°C and 75°C while the corresponding oil temperatures are 110°C and 75°C. If water flows at the rate of 68 kg/min calculate the overall heat transfer coefficient for the heat exchanger. Take specific heat of oil as 1.9 kJ/kg°C. [1997/20m/4b]

33. Estimate the exit water temperature for the above heat exchanger if the water flow rate is reduced to 40 kg/min with oil continuing to flow at the earlier rate. Assume the effectiveness of the heat exchanger under these conditions to be 0.74. [1997/25m/4c]

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