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Heat Transfer

Lecturer : Dr. Rafel Hekmat Hameed University of Babylon

Subject : Heat Transfer College of Engineering

Year : Third B.Sc. Mechanical Engineering Dep.

Steady-State Conduction- One Dimension

HEAT-SOURCE SYSTEMS

We shall confine our discussion to one-dimensional systems, or, more specifically, systems

where the temperature is a function of only one space coordinate.

Plane Wall with Heat Sources

Consider the plane wall with uniformly distributed heat sources. The thickness of the wall in

the x direction is L, and it is assumed that the dimensions in the other directions are sufficiently large

that the heat flow may be considered as one dimensional. The heat generated per unit volume is ˙q,

and we assume that the thermal conductivity does not vary with temperature.

𝑑2𝑇

𝑑𝑥2+

�̇�

𝑘= 0 …..(1)

By integration

𝑑𝑇

𝑑𝑥+

�̇�

𝑘𝑥 = 𝑐1 ; 𝑇 +

�̇�

𝑘

𝑥2

2= 𝑐1𝑥 + 𝑐2 𝑇 = −

�̇�

𝑘

𝑥2

2+ 𝑐1𝑥 + 𝑐2 …..(2)

First: Assume the both surfaces of the plane have the same temperature as shown in figure below

B.C

At x = 0 ; T=Tw

At x = L ; T=Tw

put these B.C. in eq. (2) to find c1 and c2

c2=Tw ; c1=�̇�

2 𝑘𝐿

substitute the values of c1 and c2 in eq. (2)

𝑇 = −�̇�

𝑘

𝑥2

2+

�̇�

2 𝑘𝐿𝑥 + 𝑇𝑤 …….(3) or 𝑇 =

�̇�

𝑘 2(𝐿 − 𝑥)𝑥 + 𝑇𝑤

𝑇 − 𝑇𝑤 =�̇�

𝑘 2(𝐿 − 𝑥)𝑥

2

to find the position of the maximum temperature, differentiating eq. (3), and equating the derivative

to zero

𝑑𝑇

𝑑𝑥=

�̇�

𝑘(

𝐿

2− 𝑥) ;

�̇�

𝑘(

𝐿

2− 𝑥) = 0 x=L/2

So the maximum temperature will occurs at x=L/2, and its value equal

𝑇𝑚𝑎𝑥 =�̇�

𝑘 2(𝐿 −

𝐿

2)

𝐿

2+ 𝑇𝑤 𝑇𝑚𝑎𝑥 =

�̇�

𝑘 2(

𝐿2

4) + 𝑇𝑤 …..(4)

Heat transfer in both surfaces is by conduction

𝑞 = −𝑘𝐴𝑑𝑇

𝑑𝑥 ; =−𝑘𝐴

�̇�

𝑘(

𝐿

2− 𝑥)

𝑥=0 𝑜𝑟 𝑥=𝐿; 𝑞 =

𝐴𝐿

2�̇�

When both surface are considered

𝑞 =𝐴𝐿

2�̇� . 2 = 𝐴𝐿�̇� ……(5)

Heat conducted to each wall surface will dissipated heat to the surrounding at temperature T , so for

one surfaces of plane

𝐴𝐿

2�̇� = ℎ𝐴(𝑇𝑤 − 𝑇∞) ; 𝑇𝑤 =

𝐿

2ℎ𝑞 +̇ 𝑇∞

Second: Assume one surface is insulated and the other at Tw , thickness of the plane 2L as shown in

figure below

B.C

At x =L ; 𝑑𝑇

𝑑𝑥=0

At x = 2L ; T=Tw

The position x=L refers to the insulated face of a given wall

H.W. Derive T, Tmax

Third: Assume one surface is at Tw1 , and the other surface temperature at Tw2 ,thickness of the

plane L as shown in figure below

B.C

At x =0 ; 𝑇=Tw1

At x = L ; T=Tw2

H.W. Derive T, Tmax

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CYLINDER WITH HEAT SOURCES

Consider a cylinder of radius R with uniformly distributed heat sources and constant thermal

conductivity. If the cylinder is sufficiently long that the temperature may be considered a function of

radius only, the appropriate differential equation may be obtained by neglecting the axial, azimuth,

and time-dependent terms

The boundary conditions are

T =Tw at r =R

and heat generated equals heat lost by conduction at the rod surface:

Since the temperature function must be continuous at the center of the cylinder, we could specify that

Then integration yields

From the second boundary condition above,

We could also note thatC1 must be zero because at r =0 the logarithm function becomes infinite.

From the first boundary condition,

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The final solution for the temperature distribution is then

or, in dimensionless form,

where T0 is the temperature at r =0 and is given by

Solved examples

- See solved problems in Holman

EX: 1

Consider a 0.8m-high and 1.5 m-wide double-pane window consisting of two 4-mm-thick layers of

glass (k = 0.78 W/m ·°C) separated by a 10-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 20°C while the temperature of the

outdoors is =10°C. Take the convection heat transfer coefficients on the inner and outer surfaces of

the window to be h1 = 10 W/m2 ·°C and h2 = 40 W/m2 ·°C, which includes the effects of radiation.

A = 0.8 m × 1.5 m = 1.2 m2

the steady rate of heat transfer through the window becomes

The inner surface temperature of the window

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EX:2

A 3-m-high and 5-m-wide wall consists of long 16-cm × 22-cm cross section horizontal bricks (k=

0.72 W/m ·°C) separated by 3-cm-thick plaster layers (k 0.22 W/m · °C). There are also 2-cm-thick

plaster layers on each side of the brick and a 3-cm-thick rigid foam (k = 0.026 W/m · °C) on the inner

side of the wall, as shown in figure below. The indoor and the outdoor temperatures are 20°C and

10°C, and the convection heat transfer coefficients on the inner and the outer sides are h1 = 10 W/m2

· °C and h2 = 25 W/m2 · °C, respectively. Assuming one-dimensional heat transfer and disregarding

radiation, determine the rate of heat transfer through the wall.

There is a pattern in the construction of this wall that repeats itself every 25-cm distance in the vertical direction.

There is no variation in the horizontal direction. Therefore, we consider a 1-m-deep and 0.25-m-high portion

of the wall, since it is representative of the entire wall.

The three resistances R3, R4, and R5 in the middle are parallel, and

their equivalent resistance is determined from

or 4.38/0.25= 17.5 W per m2 area. The total area of the wall is A = 3 m × 5 m = 15 m2. Then the rate of heat

transfer through the entire wall becomes

Q ·total = (17.5 W/m2)(15 m2) = 263 W

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EX:3

A spherical tank, 1 m in diameter, is maintained at a temperature of 120◦C and exposed to a convection

environment. With h=25W/m2 ◦C and T∞ =15◦C, what thickness of urethane foam should be added to

ensure that the outer temperature of the insulation does not exceed 40◦C? What percentage reduction

in heat loss results from installing this insulation?

EX:4 A steel pipe with 5-cm OD is covered with a 6.4-mm asbestos insulation [k =0.166 W/m.oC] followed

by a 2.5-cm layer of fiberglass insulation [k =0.0485 W/m.oC]. The pipe-wall temperature is 315◦C,

and the outside insulation temperature is 38◦C. Calculate the interface temperature between the

asbestos and fiberglass.

EX:5

A 3-mm-diameter and 5-m-long electric wire is tightly wrapped with a 2-mmthick plastic cover whose

thermal conductivity is k =0.15 W/m · °C. Electrical measurements indicate that a current of 10 A

passes through the wire and there is a voltage drop of 8 V along the wire. If the insulated wire is

exposed to a medium at T= 30°C with a heat transfer coefficient of h= 12 W/m2 · °C, determine the

temperature at the interface of the wire and the plastic cover in steady operation. Also determine

whether doubling the thickness of the plastic cover will increase or decrease this interface temperature

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In steady operation, the rate of heat transfer becomes equal to the heat

generated within the wire, which is determined to be

To answer the second part of the question, we need to know the critical radius

of insulation of the plastic cover.

which is larger than the radius of the plastic cover. Therefore, increasing the

thickness of the plastic cover will enhance heat transfer until the outer radius

of the cover reaches 12.5 mm.

EX: 6

Consider a large plane wall of thickness L = 0.4 m, thermal conductivity k = 2.3 W/m · °C, and surface

area A =20 m2. The left side of the wall is maintained at a constant temperature of T1 = 80°C while

the right side loses heat by convection to the surrounding air at T = 15°C with a heat transfer coefficient

of h = 24 W/m2 · °C. Assuming constant thermal conductivity and no heat generation in the wall, (a)

express the differential equation and the boundary conditions for steady one-dimensional heat

conduction through the wall, (b) obtain a relation for the variation of temperature in the wall by solving

the differential equation, and (c) evaluate the rate of heat transfer through the wall.

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H.W

Consider a steam pipe of length L= 20 m, inner radius r1 =6 cm, outer radius r2 = 8 cm, and thermal

conductivity k =20 W/m · °C. The inner and outer surfaces of the pipe are maintained at average

temperatures of T1 =150°C and T2 = 60°C, respectively. Obtain a general relation for the temperature

distribution inside the pipe under steady conditions, and determine the rate of heat loss from the steam

through the pipe.

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H.W

A long homogeneous resistance wire of radius r0 = 0.2 cm. and thermal conductivity K= 7.8 W/m.C

is being used to boil water at atmospheric pressure by the passage of electric current. Heat is generated

in the wire uniformly as a result of resistance heating at a rate of g· = 2400 W/cm3 If the outer surface

temperature of the wire is measured to be Ts =226°C, obtain a relation for the temperature distribution,

and determine the temperature at the centerline of the wire when steady operating conditions are

reached.