Sample Heat Transfer Problems With Solutions

Sample Heat Transfer Problems with Solutions References are from: from: Incropera, F.P. and De Witt, Witt, D.P., Introduction D.P., Introduction to Heat Transfer, John nd Wiley and Sons, Sons, 2 Edition, New York, 1996.

S.1 The heat flux through a wood slab 50 mm thick, whose inner and outer surface temperatures 2 are 40 and 20°C, respectively, has been determined to be 40 W/m . What is is the thermal conductivity of the wood?

2 S.2 The inner and outer o uter surface temperatures of a glass window 5 mm thick are 15 and 5°C, respectively. What is the heat heat loss through a window window that is 1 m by 3 m on a side? The thermal conductivity of glass is 1.4 W/mK.

3 S.3 The convection heat transfer coefficient between a surface at 40°C and ambient air at 20°C is 2 20 W/m K. Calculate the heat flux leaving the surface by convection.

4 S.4 Air at 300°C flows over a flat plate of dimensions 0.50 m by 0.25 m. If the convection heat 2 transfer coefficient is 250 W/m K, determine the heat transfer rate from the air to one side of the  plate when the plate is maintained at 40°C.

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S.5 A surface of area 0.5 m , emissivity 0.8, and t emperature 150°C is placed in a large, evacuated chamber whose walls wa lls are maintained at 25°C. What is the rate at which radiation is emitted by the surface? What is the net rate at which radiation is exchanged between the surface and the chamber walls?

6 S.6 A hot water pipe with outside radius r1 has a temperature t emperature T 1. A thick insulation applied to reduce the heat loss has an outer radius r 2 and temperature T 2. On T±r coordinates, T±r coordinates, sketch the temperature distribution in the insulation for one-dimensional, steady-state heat transfer with constant properties. Give a brief explanation, justifying justifying the shape o f the curve shown.

7 S.7 Sections of the Alaskan pipeline run above the ground and are supported by vertical steel 2 shafts (k = 25 W/mK) which are 1 m long and have a cross-sectional area of 0.005 m . Under  normal operating conditions, the temperature variation along the length of a shaft is known to be 2 governed by an expression e xpression of the form: T = T = 100 ± 150 x 150 x + 10 x 10 x where T and T and x  x have units of °C and meters, respectively. Temperature variations are negligible negligible over the shaft cross section. Evaluate the t he temperature and conduction heat rate at the shaft/pipeline joint ( x = 0) and at the shaft/ground interface ( x  x = 1 m). Explain the difference d ifference in heat rates.

8 S.8 The rear window of o f an automobile is defogged by passing war m air at 40°C over its inner  2 surface, and the associated co nvection coefficient is 30 W/m K. Under conditions for which the outside ambient air temperature is -10°C and the a ssociated convection coefficient is 65 W/m2K, what are the inner and o uter surface temperatures of the window? The window glass is 4 mm thick.

9 S.9 The rear window of o f an automobile is defogged by attaching att aching a thin, transparent, film-type heating element to its inner inner surface. By electrically heating this this element, a uniform heat flux flux may be established at the inner surface. What is the electrical power that must be pro vided per  unit window area to maintain maintain an a n inner surface temperature of 15°C when the interior air  2 temperature and convection coefficient are 25°C and 10 W/m K and the exterior (ambient) air  2 temperature and convection coefficient are -10°C and 65 W/m K? The window glass is 4 mm thick.

10 S.10 For laminar flow over a flat plate, the local heat transfer coefficient, h x, is known to vary as -1/2  x , where x where x is the distance from the leading edge ( x=0)  x=0) of the plate. What is the ratio of the average coefficient between the leading edge and some location x location x on the plate to the local coefficient at x at x??

11 S.11 A fan that can provide air speeds up to 50 m/s is to be used in a low speed wind tunnel with atmospheric air at 25°C. If one wishes to use the wind tunnel to study stud y flat plate boundary layer  8  behavior up to Reynolds numbers of 10 , what is the minimum plate length? At what distance 5 from the leading edge wou ld transition occur, if the critical Reynolds number is 5×10 ?

12 S.12 Experimental results for heat transfer over a flat plate with an extremely rough surface were found to be correlated by an expression of the form  Nu x 0.04 Re x0.9 Pr 1 / 3 !

where Nu x is the local value of the Nusselt number at a position x position x measured from the leading edge of the plate. Obtain an expression for the ratio of the average heat transfer coefficient  between the leading edge and a location x location x to the local heat transfer tr ansfer coefficient at x at x..

13 S.13 Consider steady, parallel flow of o f atmospheric air over a flat plate. The air has a temperature and free stream velocity of 300 K and 25 m/s. a) Evaluate the boundary layer thickness at distances of  x  x = 1, 10, and 100 mm from the leading edge. If a second plate were installed parallel to and at a distance of 3 mm from the first plate, what would be the distance from the leading edge at which the boundary layer merger would occur?  b) Evaluate the surface shear stress and the t he y-velocity component at the outer edge edg e of the  boundary layer for the single plate at x at x = 1, 10, and 100 mm. c) Comment on the validity of o f the boundary layer appro ximations. ximations.

14 S.14 Consider water at 27°C in parallel flow over an isothermal, 1 m long flat plate with a velocity of 2 m/s. Plot the variation variat ion of the local heat transfer coefficient with distance along t he  plate. What is the value of the average coefficient?

15 S.15 A thin, flat plate o f length L = 1 m separates two airstreams that are in parallel flow over  opposite surfaces of the plate. One a irstream has a temperature of 200°C and a velocity o f 60 m/s, while the other has a temperature t emperature of 25°C and a velocity of 10 m/s. What is the heat flux  between the two streams at the t he midpoint of the plate?

16 S.16 Consider a rectangular fin that is used to cool a motorcycle engine. The T he fan is 0.15 m long and at a temperature of 250°C, while the motorcycle is moving at 80 km/hr in air at 27°C. The air is in parallel flow over both surfaces of o f the fin, and turbulent flow conditions may be assumed to exist throughout. What is the rate rat e of heat removal per unit width of fin?

17 S.17 One-hundred electrical components, each dissipating 25 W, are attached to one surface of a 0.2 m square copper plate, and all of the dissipated d issipated energy is transferred to water in parallel flow over the opposite surface. A protuberance protu berance at the leading edge of o f the plate acts to trip the  boundary layer, and the p late itself may be assumed to be isothermal isot hermal.. The water wat er velocity and temperature are 2 m/s and 17°C. 17 °C. a) What is the temperature of the copper plate? 2  b) If each component has a plate contact surface area of 1 cm and the corresponding area-4 2 specific contact resistance is 2×10 m K/W, what is the component temperature? t emperature?

18 S.18 A square aluminum plate 5 mm thick and 200 mm on a side is cooled while vertically suspended in quiescent air a ir at 40°C. Determine the average heat transfer coefficient for the plate when its temperature is 15°C.

19 S.19 An aluminum plate, heated to a uniform temperature of 227°C, is allowed allowed to cool while vertically suspended in a room roo m where the ambient air and surroundings are at 27°C. The plate is 0.3 m square with a thickness t hickness of 15 mm and an e missivity missivity of o f 0.25. a) Develop an expression for the t he time rate of change of the p late temperature, assuming the temperature to be uniform at any t ime.  b) Determine the initial rate of cooling (K/s) when t he plate temperature is 227°C. c) Justify the uniform plate temperature assumption.

20 S.20 The plate described descr ibed in the previous problem has been used in an experiment to determine the free convection heat transfer t ransfer coefficient. At an instant of time when the p late temperature was 127°C, the time rate of o f change of this temperature was observed to be -0.0465 K/s. What is the corresponding free convection convect ion heat transfer coefficient? Compare this result with an estimate  based on a standard empirical correlation.

21 S.21 A household oven door of 0.5 m height and 0.7 m width reaches an average surface temperature of 32°C during operation. Estimate the heat loss to the room with ambient air at 22°C. If the door has an emissivi e missivity ty of 1.0 and the surroundings are a lso at 22°C, comment on the heat loss by free convection convect ion relative to that by radiation.

22 S.22 An enclosure has an a n inside area of 100 m2, a nd its inside surface is black and is maintained at a constant temperature. A small opening in the enclosure has an area of 0.02 m2. The radiant  power emitted from this opening is 70 W. What is the temperature of the interior enclosure wall? If the interior surface is maintained at this temperature, but is now polished, what will be the value of the radiant power emitted from the opening?

23 S.23 Under steady state operation, a 50 W incandescent light bulb has a surface temperature of  135°C when the room air is at a temperature of 25°C. If the bulb may be approximated as a 60 mm diameter sphere with a diffuse, gray surface o f emissivity emissivity 0.8, what is the radiant heat transfer from the bulb surface to its surroundings?