Heat and Mass

Transfer (ME 209)

"Solved Problems"

Part 1References

Heat Transfer "A Practical Approach" by Yunus Cengel

Fundamentals of Heat and Mass Transfer by Incropera

Heat Transfer by J.P. Holman

1 )Consider a 1.2-m-high and 2-m-wide double-pane window consisting of two 3-mm-thick layers of glass )k =0.78 W/m°C) separated by a 12-mm-wide stagnant air

space )k = 0.026 W/m °C). Determine the steady rate of heat transfer through this double-pane window and the temperature of its inner surface for a day during which the room is maintained at 24°C while the temperature of the outdoors is -5°C. Take the convection heat transfer coefficients on the inner and outer surfaces

of the window to be 10 W/m2 °C and 25 W/m2 °C respectively.2 )A 2-m1.5-m section of wall of an industrial furnace burning natural gas is not

insulated, and the temperature at the outer surface of this section is measured to be 80°C. The temperature of the furnace room is 30°C, and the combined convection and radiation heat transfer coefficient at the surface of the outer furnace is 10 W/m2°C. It is proposed to insulate this section of the furnace wall with glass wool insulation )k = 0.038 W/m°C) in order to reduce the heat loss by 90 percent. Assuming the outer surface temperature of the metal section still remains at about

80°C; determine the thickness of the insulation that needs to be used.3 )Water is boiling in a 25-cm-diameter aluminum pan )k = 237 W/m °C) at 95°C.

Heat is transferred steadily to the boiling water in the pan through its 0.5-cm-thick flat bottom at a rate of 800 W. If the inner surface temperature of the bottom of the pan is 108°C, determine )a) the boiling heat transfer coefficient on the inner surface

of the pan, and )b) the outer surface temperature of the bottom of the pan.4) Two 5-cm-diameter, 15–cm-long aluminum bars )k = 176 W/m°C) with ground

surfaces are pressed against each other with a pressure of 20 atm )h = 11,400 W/m2C). The bars are enclosed in an insulation sleeve and, thus, heat transfer

from the lateral surfaces is negligible. If the top and bottom surfaces of the two-bar system are maintained at temperatures of 150°C and 20°C, respectively, determine )a) the rate of heat transfer along the cylinders under steady conditions and

)b) the temperature drop at the interface.5 )An electric current is passed through a wire 1 mm in diameter and 10 cm long. The

wire is submerged in liquid water at atmospheric pressure, and the current is increased until the water boils. For this situation h = 5000 W/m2°C, and the water temperature will be 100 °C. How much electric power must be supplied to the wire

to maintain the wire surface at 114 °C?6) Steam at 320°C flows in a cast iron pipe )k = 80 W/m · °C)

whose inner and outer diameters are 5 cm and 5.5 cm, respectively. The pipe is covered with 3-cm-thick glass wool insulation with k = 0.05 W/m ·°C. Heat is lost to the surroundings at 5°C by natural convection and radiation, with a combined heat transfer coefficient of h2= 18 W/m · °C. Taking the heat transfer coefficient inside the pipe to be h1= 60 W/m2 °C, determine the rate of heat loss from

the steam per unit length of the pipe. Also determine the temperature drops across the pipe shell and the insulation.

7) Consider a 2-m-high electric hot water heater that has a diameter of 40 cm and maintains the hot water at 55°C. The tank is located in a small room whose average temperature is 27°C, and the heat transfer coefficients on the inner and outer surfaces of the heater are 50 and 12 W/m2°C, respectively. The tank is placed in another 46-cm-diameter sheet metal tank of negligible thickness, and the space between the two tanks is filled with foam insulation )k = 0.03 W/m °C). The thermal resistances of the water tank and the outer thin sheet metal shell are very small and can be neglected. Determine the heat loss from the tank. If 3 cm thick fiber glass insulation is used to wrap the entire tank with K = 0.035°C what will be

the heat loss 8)Consider a large 3-cm-thick stainless steel plate )k = 15.1 W/m °C) in which heat is

generated uniformly at a rate of 5 105 W/m3. Both sides of the plate are exposed to an environment at 30°C with a heat transfer coefficient of 60 W/m °C. Explain where in the plate the highest and the lowest temperatures will occur, and

determine their values.9 )In a nuclear reactor, 1-cm-diameter cylindrical uranium rods cooled by water from

outside serve as the fuel. Heat is generated uniformly in the rods )k= 29.5 W/m · °C) at a rate of 7107 W/m3. If the outer surface temperature of rods is 175°C,

determine the temperature at their center.10) Consider a long resistance wire of radius r1 = 0.2 cm and thermal conductivity kwire = 15 W/m · °C in which heat is generated uniformly as a result of resistance heating at a constant rate of qv = 50 W/m3. The wire is embedded in a 0.5-cm-thick layer of ceramic whose thermal conductivity is kceramic = 1.2 W/m · °C. The outer surface temperature of the ceramic layer is measured to be 45°C, .and is surrounded by air at 30 °C with heat transfer coefficient is of 10 W/m2C. Determine the temperatures at the center of the resistance wire and the interface of the wire and the ceramic layer under steady conditions. 11) Steam in a heating system flows through tubes whose outer diameter is 3 cm and whose walls are maintained at a temperature of 120°C. Circular aluminum fins )k =180 W/m · °C) of outer diameter 6 cm and constant thickness of 2 mm are attached to the tube. The space between the fins is 3 mm, and thus there are 200 fins per meter length of the tube. Heat is transferred to the surrounding air at 25°C, with a combined heat transfer coefficient of h = 60 W/m2· °C. Determine the increase in heat transfer from the tube per meter of its length as a result of adding fins. )Fin efficiency = 95%).12) A hot surface at 100°C is to be cooled by attaching 3-cm-long, 0.25-cm-diameter aluminum pin fins )k =237 W/m · °C) to it, with a center-to-center distance of 0.6 cm. The temperature of the surrounding medium is 30°C, and the heat transfer coefficient on the surfaces is 35 W/m2°C. Determine the rate of heat transfer from the surface for a 1-m 1-m section of the plate. Also determine the overall effectiveness of the fins. .)Fin efficiency = 95.9%).

13) Consider steady two-dimensional heat transfer in a long solid body whose cross section is given in the figure. The temperatures at the selected nodes and the thermal conditions at the boundaries are as shown. The thermal conductivity of the body is k = 45 W/m · °C, and heat is generated in the body uniformly at a rate of qv = 6 × 106 W/m3. Using the finite difference method with a mesh size of Δx = Δy= 5.0 cm, determine the temperatures at nodes:

14) Consider steady two-dimensional heat transfer in a long solid body whose cross section is given in the figure. The measured temperatures at selected points of the outer surfaces are as shown. The thermal conductivity of the body is k = 45 W/m · °C, and there is no heat generation. Using the finite difference method with a mesh size of Δx = Δy= 2.0 cm, determine the temperatures at the indicated points in the medium.

15) Consider steady two-dimensional heat transfer in a long solid bar whose cross section is given in the Figure 1 )a) and )b). The measured temperatures at selected points of the outer surfaces are as shown. The thermal conductivity of the body is k = 20 W/m · °C, and there is no heat generation. Using the finite difference method with a mesh size of Δx = Δy= 1.0 cm. Determine the temperatures at the indicated points in the medium

FIGURE (1)

16) Consider steady two-dimensional heat transfer in a long solid body whose cross section is given in the figure. The temperatures at the selected nodes and the thermal conditions on the boundaries are as shown. The thermal conductivity of the body is k = 180 W/m°C, and heat is generated in the body uniformly at a rate of qv = 107 W/m3. Using the finite difference

method with a mesh size of Δx = Δy= 10 cm, determine the temperatures at nodes 1, 2, 3, and 4.

17) Consider steady two-dimensional heat transfer in an L-shaped solid body whose cross section is given in the figure. The thermal conductivity of the body is k = 45 W/m · °C, and heat is generated in the body at a rate of qv = 5 × 106 W/m3. The right surface of the body is insulated, and the bottom surface is maintained at a uniform temperature of 120°C. The entire top surface is subjected to convection with ambient air at T∞ = 30°C with a heat transfer coefficient of h = 55 W/m2 · °C, and the left surface is subjected to heat flux at a uniform rate of 8000 W/m2. The nodal network of the problem consists of 13 equally spaced nodes with Δx = Δy= 1.5 cm. Five of the nodes are at the bottom surface and thus their temperatures are known. Obtain the finite difference equations at the remaining eight nodes.

Solutions1) A double-pane window consists of two 3-mm thick layers of glass separated by a 12-mm wide stagnant air space. For specified indoors and outdoors temperatures, the rate of heat loss through the window and the inner surface temperature of the window are to be determined.

Assumptions 1 Heat transfer through the window is steady since the indoor and outdoor temperatures remain constant at the specified values. 2 Heat transfer is one-dimensional since any significant temperature gradients will exist in the direction from the indoors to the outdoors. 3 Thermal conductivities of the glass and air are constant. 4 Heat transfer by radiation is negligible.

Properties The thermal conductivity of the glass and air are given to be kglass = 0.78 W/m°C and kair = 0.026 W/m°C.

Analysis The area of the window and the individual resistances are

The steady rate of heat transfer through window glass then becomes

The inner surface temperature of the window glass can be determined from

2) An exposed hot surface of an industrial natural gas furnace is to be insulated to reduce the heat loss through that section of the wall by 90 percent. The thickness of the insulation that needs to be used is to be determined. Also, the length of time it will take for the insulation to pay for itself from the energy it saves will be determined.

Assumptions 1 Heat transfer through the wall is steady and one-dimensional. 2 Thermal conductivities are constant. 3 The furnace operates continuously. 4 The given heat transfer coefficient accounts for the radiation effects.

Properties The thermal conductivity of the glass wool insulation is given to be k = 0.038 W/m°C.

Analysis The rate of heat transfer without insulation is

In order to reduce heat loss by 90%, the new heat transfer rate and thermal resistance must be

Air

R1 R2 R3 RoRi

T1 T2

Insulation

RoT

Rinsulation

Ts

L

and in order to have this thermal resistance, the thickness of insulation must be

3) Heat is transferred steadily to the boiling water in an aluminum pan. The inner surface temperature of the bottom of the pan is given. The boiling heat transfer coefficient and the outer surface temperature of the bottom of the pan are to be determined.

Assumptions 1 Steady operating conditions exist. 2 Heat transfer is one-dimensional since the thickness of the bottom of the pan is small relative to its diameter. 3 The thermal conductivity of the pan is constant.

Properties The thermal conductivity of the aluminum pan is given to be k = 237 W/m°C.

Analysis (a) The boiling heat transfer coefficient is

(b) The outer surface temperature of the bottom of the pan is

4) Two cylindrical aluminum bars with ground surfaces are pressed against each other in an insulation sleeve. For specified top and bottom surface temperatures, the rate of heat transfer along the cylinders and the temperature drop at the interface are to be determined.

Assumptions 1 Steady operating conditions exist. 2 Heat transfer is one-dimensional in the axial direction since the lateral surfaces of both cylinders are well-insulated. 3 Thermal conductivities are constant.

Properties The thermal conductivity of aluminum bars is given to be k = 176 W/m°C. The contact conductance at the interface of aluminum-aluminum plates for the case of ground surfaces and of 20 atm 2 MPa pressure is hc = 11,400 W/m2C (Table 3-2).

Analysis (a) The thermal resistance network in this case consists of two conduction resistance and the contact resistance, and are determined to be

95C

108C

600 W 0.5 cm

Ri Rglass Ro

T1 T2

Bar Bar

Interface

Then the rate of heat transfer is determined to be

Therefore, the rate of heat transfer through the bars is 142.4 W.

(b) The temperature drop at the interface is determined to be

6)

7)An electric hot water tank is made of two concentric cylindrical metal sheets with foam insulation in between. The fraction of the hot water cost that is due to the heat loss from the tank and the payback period of the do-it-yourself insulation kit are to be determined.

Assumptions 1 Heat transfer is steady since there is no indication of any change with time. 2 Heat transfer is one-dimensional since there is thermal symmetry about the center line and no variation in the axial direction. 3 Thermal conductivities are constant. 4 The thermal resistances of the water tank and the outer thin sheet metal shell are negligible. 5 Heat loss from the top and bottom surfaces is negligible.

Properties The thermal conductivities are given to be k = 0.03 W/m°C for foam insulation and k = 0.035 W/m°C for fiber glass insulation

Analysis We consider only the side surfaces of the water heater for simplicity, and disregard the top and bottom surfaces (it will make difference of about 10 percent). The individual thermal resistances are

The rate of heat loss from the hot water tank is

The amount and cost of heat loss per year are

If 3 cm thick fiber glass insulation is used to wrap the entire tank, the individual resistances becomes

The rate of heat loss from the hot water heater in this case is

Tw

Ro

T2

Rfoam

Tw

Rfiberglass Ro

T2

Rfoam

8) Both sides of a large stainless steel plate in which heat is generated uniformly are exposed to convection with the environment. The location and values of the highest and the lowest temperatures in the plate are to be determined.

Assumptions 1 Heat transfer is steady since there is no indication of any change with time. 2 Heat transfer is one-dimensional since the plate is large relative to its thickness, and there is thermal symmetry about the center plane 3 Thermal conductivity is constant. 4 Heat generation is uniform.

Properties The thermal conductivity is given to be k =15.1 W/m°C.

Analysis The lowest temperature will occur at surfaces of plate while the highest temperature will occur at the midplane. Their values are determined directly from

9) A nuclear fuel rod with a specified surface temperature is used as the fuel in a nuclear reactor. The center temperature of the rod is to be determined.

Assumptions 1 Heat transfer is steady since there is no indication of any change with time. 2 Heat transfer is one-dimensional since there is thermal symmetry about the center line and no change in the axial direction. 3 Thermal conductivity is constant. 4 Heat generation in the rod is uniform.

Properties The thermal conductivity is given to be k = 29.5 W/m°C.

Analysis The center temperature of the rod is determined from

10)

T =30°C

h=60 W/m2.°C2L=3 cm

k

gT =30°C

h=60 W/m2.°C

g

175°C

Uranium rod

11)

12) The number of fins, finned and unfinned surface areas, and heat transfer rates from those areas are

/ n 1

0 006 0 00627777

m

m) m)

2

) . ) .

Then the total heat transfer from the finned plate becomes

The rate of heat transfer if there were no fin attached to the plate would be

A

Q hA T Tb

no fin2

no fin no fin2 2

m m m

W / m C m C W

) )) )

) ) ) . )) )) )

1 1 1

35 1 100 30 2450

Then the fin effectiveness becomes

13) T T T T Tg l

kleft top right bottom nodenode 4 0

2

where

The finite difference equations for boundary nodes are obtained by applying an energy balance on the

volume elements and taking the direction of all heat transfers to be towards the node under consideration:

where

Substituting, T1 = 280.9°C, T2 = 397.1°C, T3 = 330.8°C,

(b) The rate of heat loss from the bottom surface through a 1-m long section is

14)

There is symmetry about the horizontal, vertical, and diagonal lines passing through the midpoint, and thus we need to consider only 1/8th of the region. Then,

Therefore, there are there are only 3 unknown nodal temperatures,

, and thus we need only 3 equations to determine

them uniquely. Also, we can replace the symmetry lines by insulation and utilize the mirror-image concept when writing the finite difference equations for the interior nodes.

Solving the equations above simultaneously gives

15)

(a) There is symmetry about the insulated surfaces as well as about the diagonal line. Therefore , and

are the only 3 unknown nodal temperatures. Thus we need only 3 equations to determine them

uniquely. Also, we can replace the symmetry lines by insulation and utilize the mirror-image concept when writing the finite difference equations for the interior nodes.

Also,

Solving the equations above simultaneously gives

(b) There is symmetry about the insulated surface as well as the diagonal line. Replacing the symmetry lines by insulation, and utilizing the mirror-image concept, the finite difference equations for the interior nodes can be written as

Solving the equations above simultaneously gives

180

200

180

150 180 200 180 150

150 180 200 180 150

180

200

180

1 2 3

4 5 6

7 8 9

Insulated

180

200

150 180 200

3

1 2

4

Insulated

100

120

140

120 120

3

1 2

4

Insulated

100

120

140

16)

There is symmetry about a vertical line passing through the middle of the region, and thus we need to consider only half of the region. Then,

Therefore, there are there are only 2 unknown nodal temperatures, T1 and T3, and thus we need only 2 equations to determine them uniquely. Also, we can replace the symmetry lines by insulation and utilize the mirror-image concept when writing the finite difference equations for the interior nodes.

Noting that and substituting,

The solution of the above system is

17)

We observe that all nodes are boundary nodes except node 5 that is an interior node. Therefore, we will have to rely on energy balances to obtain the finite difference equations. Using energy balances, the finite difference equations for each of the 8 nodes are obtained as follows:

100 100 100 100

120

150

3

1 2

0.1 m

g

200 200 200 200

120

150

4

Node 1:

Node 2:

Node 3:

Node 4:

Node 5:

Node 6:

Node 7:

Node 8:

where l = 0.015 m, k = 45 W/mC, h = 55 W/m2C, and T =30C.

This system of 8 equations with 8 unknowns is the finite difference formulation of the problem.

(b) The 8 nodal temperatures under steady conditions are determined by solving the 8 equations above simultaneously with an equation solver to be

T1 =163.6C, T2 =160.5C, T3 =156.4C, T4 =154.0C, T5 =151.0C, T6 =144.4C,

T7 =134.5C, T8 =132.6C

100 100 100 100

120

150

3

1 2

0.1 m

g

200 200 200 200

120

150

4

100 100 100 100

120

150

3

1 2

0.1 m

g

200 200 200 200

120

150

4

100 100 100 100

120

150

3

1 2

0.1 m

g

200 200 200 200

120

150

4