Steady State Heat Transfer
3.4 GRAPHICAL METHOD FOR WALL HEAT TRANSFER AND DESIGNAND DESIGN
In section 3.3, we have seen that the rate of heat flow q, (w/m2) for a plain wall is given by
where k is the overall heat transfer coefficient.
q t t
k t t
f o
h h
f o
= −
= −
+ +
( )
( )
1 1
1 2
d l
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76 Industrial Heating
Therefore,
or overall thermal resistance.
For a multiple-layered wall, the middle term d/l in the above equation will be replaced by for the individ-ual layers having thickness dn and conductivity ln.
The above equation also shows that there is a linear rela-tionship between (tf − to) and 1/k, namely
tf − to= q × k or for any layer
These linear relationships can be used for graphical con-struction to determine the layer temperatures and the rate of heat flow q.
The following data are assumed to be available:
• Furnace source (flame) temperature tf
• Outside temperature to
• Heat transfer coefficients h1 and h2
• Thickness dn and the conductivities of different layers ln First, the thermal resistances for the wall are calcu-lated. For this purpose the flame to wall (inside) and from wall to atmosphere (outside) heat transfer coefficients are also regarded as ficticious resistances.
Let the calculated resistances be
On the y axis, temperature is plotted on any convenient scale.
The temperature point B inside is connected by a straight line to the temperature point C outside. Point A marks the ini-tial temperature of the wall.
k
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These resistances are plotted on the x axis (Figure 3.7).
Steady State Heat Transfer 77
The various wall temperatures tw1…twnare read on the tem-perature axis at the layer boundary and the temtem-perature line BC.
The rate of heat flow q will be
EXAMPLE 3.5
A three-layered wall is made as follows:
• I layer (hot side) fire clay, 240 mm, l = 1.2 w/m°C
• II layer (cera wool), 150 mm, l = 0.12 w/m°C
• III layer (Rock wool), 200 mm, l = 0.03 w/m°C Figure 3.7 Steady state heat flow through a three-layered wall (graphical method).
q k t t t t
AB AC
f o
f o
h n n h
= −
= −
∑
= ×
+ −
( )
( ) 1
1 1
1 2
d l DK340X_C03A.fm Page 77 Tuesday, April 26, 2005 2:32 PM
78 Industrial Heating
The furnace temperature is 1300°C and the outside temperature is 30°C. The heat transfer coefficient on the hot side (h1) is 0.2 kw/m2°C and that on the outside (h2) is 12 w/m2°C.
Determine the wall temperatures graphically.
Solution
First we will calculate the thermal resistances.
The total resistance (k)
Therefore,
The various resistances are plotted against temperature transfer coefficient h1 is very small and cannot be plotted on a convenient scale.
From the graph:
1. Wall inner temperature (tf = tw1) 1300°C 2. Fire clay and cera wool boundary temperature
tw2 = 119°C
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in Figure 3.8. Note that the resistance offered by the heat
Steady State Heat Transfer 79
3. Cerawool and rock wool boundary temperature tw3 = 520°C
4. Outer wall temperature tw3 = 90°C Solution of Example 3.5 by analytical method The overall conductivity of the wall (k) is
Hence, heat flow rate q is
= 0.456 (1300 − 30)
= 580 w/m2
Figure 3.8 Graphical solution (conduction through multilayered wall).
k
h h
=
=
+ + + +
1
0 456
1 1
1 1 1
2 2
3
3 2
d l
d l
d l
.
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80 Industrial Heating
For h1, i.e., the first resistance
For h1 and d 1 resistances
For h1, d1, and d2 resistances
For h1, d 1, d 2, and d3 resistances
3.5 CONVECTION
In convection heat transfer occurs across the boundary between a solid and a fluid (gas or liquid) in contact with it.
The fluid is in motion. Heat transfer will be from the fluid to
t t q
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Steady State Heat Transfer 81
the solid such as hot combustion gases circulating along the object in the furnace or the furnace walls. It may also take place from a hot object or wall to the gases or liquid circulating around it. If the temperature of the hot medium is t1 and that of the cold medium t2 then the heat transferred across the con-tact surface of area A (m2) is (see Figure 3.9(A))
Q = hc(t1− t2)A w (3.23)
where hc is the convective heat transfer coefficient (w/m2°C).
The heat transferred across unit area will be
(3.24) It can be seen that the above equations are similar to that for conduction. However, the only similarity between conduc-tion and convecconduc-tion is the fact that both depend directly on the temperature difference (t1 − t2).
Figure 3.9 Convective heat transfer.
q Q
A h tc t
= = ( 1− 2) (w/m2)
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The heat transfer coefficient is not a single property of either solid or fluid medium but depends on the thermal and other physical properties of both mediums.
Some of the properties related to the heat transfer coef-ficient are density, velocity, viscosity, temperature, conduc-tivity, specific heat, etc. The involvement of a large number of properties makes it impossible to develop a single, univer-sal formulation for calculating the heat transfer coefficient.
It depends on the situation and conditions of the surfaces and fluids involved in transfer. Thus, hc is a variable and not a physical constant like thermal conductivity. For the same pair of fluid and solid, there will be different values of con-vective coefficients, each applicable to a particular situation only. A partial list of factors that influence the coefficient is as follows:
1. Nature of fluid flow — This may be laminar or turbu-lent, or mixed. It is determined from the Reynolds number (Re) which is given by
4
turbulent. For 2320 < Re < 104 the flow is mixed (see 2. Physical properties of the fluid such as density,
con-ductivity, viscosity, specific heat, and pressure.
3. Shape of contact surfaces such as flat plate, tube, and coiled tube, and their dimensions.
4. Roughness of the solid surface which determines the friction.
5. Velocity of the fluid, which may be determined from conditions of flow such as natural and forced. Fluids expand on heating and their density lowers. Hence, a density difference naturally occurs between a cold and a hot fluid taking part in convection. This creates a motion that is natural. Convection under this natural
Re=Ul u
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Chapter 2).
For Re < 2320 the flow is laminar and for Re > 10 it is
Steady State Heat Transfer 83
force is called “natural convection,” which is influ-enced by the coefficient of expansion and gravity.
In many instances the fluid is “forced” to move along the surface. This is done by using a pump, or pan, etc. Convection under such mechanically induced flow is called “forced convection.” Besides the gravity and expansion forces, natural convection also occurs under a concentration difference that is accompanied by mass transfer.
6. Temperature of fluid which affects all physical properties.
7.
over a furnace outer wall, or “internal,” such as fluid The heat transfer coefficients for “external” and
“internal” flow situations are different.
Determination of Convective Heat Transfer Coefficient
Considering the difficulties in calculating the heat transfer coefficient from basic principles, it is determined by experi-mental methods based on the theory of similarities.
Experiments are conducted on small-scale models in which heat transfer takes place in a manner similar to the large-scale phenomena of interest. The theory of similarity requires that the model and the real process must have the same determining criteria and the same mathematical rela-tions between the criteria.
particular situation of interest (e.g., cooling fluid flowing through a hot pipe). These relationships are used to estimate the heat
economical to estimate the transfer coefficients. It also saves The various parameters and properties that are dimensionless numbers. There are many such numbers, some
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flowing through a tube (Figure 3.9(B)).
The flow of fluid may be “external,” such as air flowing
Theory of models and similarity criteria make it easy and Critical relationships are experimentally established for a
considerable time. For details of these methods see [1, 2].
involved in a given heat transfer situation are grouped in transfer coefficients applicable to a real large-scale situation.
84 Industrial Heating
of which are of interest and are given below. These numbers are called “similarity criteria.”
Nusselt number (3.25)
Grashof number (3.26)
Peclet number (3.27)
Reynolds number (3.28)
Prandtl number (3.29)
Stanton number (3.30)
where
h = Heat transfer coefficient (w/m°C) U = Fluid velocity (m/sec)
l = Thermal conductivity of fluid (w/m°C) g = Gravitational attraction (m/sec)
b = Coefficient of volumetric expansion (1/°C)
l = Any suitable dimension of the system, m (e.g., diameter of pipe)
u = Kinematic viscosity (N m2/sec) a = Thermal diffusivity (m2/sec)
r = Density (kg/m3) c = Specific heat (J/kg°C)
ts = Temperature of solid (°C) tfl = Temperature of fluid (°C)
The various dimensionless groups given above are related to various aspects of the fluid flow at the solid-fluid boundary.
Their significance is as follows:
The Reynolds number relates the forces of inertia and vis-cosity in the fluid. It is thus a criteria to distinguish between laminar and turbulent flow.
Nu= hl
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Steady State Heat Transfer 85
The Peclet number shows the ratio between heat transfer by convection and conduction.
The Grashof number considers the ratio between gravi-tation force and viscous force, hence applicable to natural convection.
The Prandtl number is a ratio of two physical proper-ties — viscosity and thermal diffusivity. It can be obtained from handbooks. It relates the viscous and thermal fields in the fluid.
For determining the convective heat transfer coefficient, the Nusselt number (and Stanton number which contains Nu) is the most important group as it contains the unknown coef-ficient h.
Relationships between the dimensionless numbers are constructed to fit the data obtained from model experiments based on the similarity principle. They generally have a form such as
Nu = A(Re)n (Pr)m— Forced Convection or
Nu = K(Gr)x (Pr)y, etc. — Natural Convection
where A and K are constants and m, n, x, y, and so on are the powers.
From such relationships, the Nusselt number is derived and the heat transfer coefficient is calculated. It is again pointed out that the given relationships apply only to a partic-ular situation. We will consider several examples to make this point clear.