Heat transfer – lecture 2
Fourier’s Law
The basic equation for the analysis of heat conduction is Fourier’s law, which
is based on experimental observations and is:
where the heat flux q”n (W/m2) is the heat transfer rate in the n direction per
unit area perpendicular to the direction of heat flow, kn (W/mK) is the thermal
conductivity in the direction n, and ∂T/∂n (K/m) is the temperature gradient in the
direction n. The thermal conductivity is a thermophysical property of the material,
which is, in general, a function of both temperature and location; that is, k = k(T, n).
For isotropic materials, k is the same in all directions, but for anisotropic
materials such as wood and laminated materials, k is significantly higher along the
grain or lamination than perpendicular to it. Thus for anisotropic materials, k can
have a strong directional dependence.
Because the thermal conductivity depends on the atomic and molecular
structure of the material, its value can vary from one material to another by several
orders of magnitude. The highest values are associated with metals and the lowest
values with gases and thermal insulators.
For three-dimensional conduction in a Cartesian coordinate system, the
Fourier law of can be extended to
where
and i, j, and k are unit vectors in the x, y, and z coordinate directions, respectively.
General Heat Conduction Equations
The general equations of heat conduction in the rectangular and cylindrical
coordinate systems shown in Fig. 2.1 can be derived by performing an energy
balance.
Cartesian coordinate system:
Cylindrical coordinate system:
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where q is the volumetric energy addition (W/m3), ρ the density of the
material (kg/m3), and c the specific heat (J/kgK) of the material.
Figure 2.1. Differential control volumes in Cartesian and cylindrical coordinates.
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Boundary and Initial Conditions
Each of the general heat conduction equations is second order in the spatial
coordinates and first order in time. Hence, the solutions require a total of six
boundary conditions (two for each spatial coordinate) and one initial condition. The
initial condition prescribes the temperature in the body at time t = 0. The three types
of boundary conditions commonly encountered are that of constant surface
temperature (the boundary condition of the first kind), constant surface heat flux
(the boundary condition of the second kind), and a prescribed relationship between
the surface heat flux and the surface temperature (the convective or boundary
condition of the third kind). The precise mathematical form of the boundary
conditions depends on the specific problem.
For example, consider one-dimensional transient condition in a semi-infinite
solid that is subject to heating at x = 0. Depending on the characterization of the
heating, the boundary condition at x = 0 may take one of three forms.
For constant surface temperature
For constant surface heat flux,
and for convection at x = 0,
where h (W/m2K) is the convective heat transfer coefficient and T∞ is the
temperature of the hot fluid in contact with the surface at x = 0.
Besides the foregoing boundary conditions as above, other types of boundary
conditions may arise in heat conduction analysis. These include boundary conditions
at the interface of two different materials in perfect thermal contact, boundary
conditions at the interface between solid and liquid phases in a freezing or melting
process, and boundary conditions at a surface losing (or gaining) heat simultaneously
by convection and radiation.
STEADY ONE-DIMENSIONAL CONDUCTION
In this section we consider one-dimensional steady conduction in a plane wall,
and a hollow cylinder. The objective is to develop expressions for the temperature
distribution and the rate of heat transfer. The concept of thermal resistance is
utilized to extend the analysis to composite systems with convection occurring at the
boundaries.
Plane Wall
Consider a plane wall of thickness L made of material with a thermal
conductivity k, as illustrated in Fig. 2.2. The temperatures at the two faces of the
wall are fixed at Ts,1 and Ts,2 with Ts,1 > Ts,2.
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Figure 2.2. One-dimensional conduction through a plane wall.
For steady conditions with no internal heat generation and constant thermal
conductivity, the appropriate form of the general heat conduction equation is
with the boundary conditions expressed as
Integration of first equation with subsequent application of the boundary
conditions of second equation gives the linear temperature distribution
and application of Fourier’s law gives
where A is the wall area normal to the direction of heat transfer.
Hollow Cylinder
Figure 2.3 shows a hollow cylinder of inside radius r1, outside radius r2, length
L, and thermal conductivity k. The inside and outside surfaces are maintained at
constant temperatures Ts,1 and Ts,2, respectively with Ts,1 > Ts,2.
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Figure 2.3.Radial conduction through a hollow cylinder.
For steady-state conduction in the radial direction with no internal heat
generation and constant thermal conductivity, the appropriate form of the general
heat conduction equation
is
with the boundary conditions expressed as
Following the same procedure as that used for the plane wall will give the
temperature distribution
and the heat flow
Thermal Resistance
Thermal resistance is defined as the ratio of the temperature difference to the
associated rate of heat transfer. This is completely analogous to electrical resistance,
which, according to Ohm’s law, is defined as the ratio of the voltage difference to the
current flow. With this definition, the thermal resistance of the plane wall and the
hollow cylinder are, respectively
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When convection occurs at the boundaries of a solid, it is convenient to define
the convection resistance from Newton’s law:
where h is the convection heat transfer coefficient and T∞ is the convecting fluid
temperature. It follows from equation above that
Composite Systems
The idea of thermal resistance is a useful tool for analyzing conduction
through composite members.
Composite Plane Wall. For the series composite plane wall and the
associated thermal network shown in Fig. 2.4, the rate of heat transfer q is given by
Figure 2.4. Series composite wall and its thermal network.
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Once q has been determined, the surface and interface temperatures can be
found
Composite Hollow Cylinder. A typical composite hollow cylinder with both
inside and outside experiencing convection is shown in Fig. 2.5. The figure includes
the thermal network that represents the system.
Figure 2.5. Series composite hollow cylinder and its thermal network.
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The rate of heat transfer q is given by
Once q has been determined, the inside surface Ts,1, the interface temperature
T2, and the outside surface temperature Ts,2 can be found
Overall heat transfer
In many applications of heat transfer two fluids at different temperatures are
separated by a solid wall. Heat is transferred from the fluid at the higher
temperature to the wall, conducted through the wall, and then finally transferred
from the cold side of the wall into the fluid at the lower temperature. This series of
convective and conductive heat transfer processes is known as overall heat transfer.
Overall heat transfer takes place, above all in heat exchangers. Here, for
example, a hot fluid flowing in a tube gives heat up, via the wall, to the colder fluid
flowing around the outside of the tube. House walls are also an example for overall
heat transfer. They separate the warm air inside from the colder air outside. The
resistance to heat transfer should be as large as possible, so that despite the
temperature difference between inside and outside, only a small amount of heat will
be lost through the walls. In contrast to this case, the heat transfer resistances
present in a heat exchanger should be kept as small as possible; here a great
amount of heat shall be transferred with a small temperature difference between the
two fluids in order to keep thermodynamic (exergy) losses as small as possible.
As these examples show, the calculation of the overall heat transfer is of
significant technical importance. This problem is dealt with in the next sections.
The overall heat transfer coefficient
The following analysis is based on the situation shown in Fig. 2.6. A flat or
curved wall separates a fluid at temperature ϑ1 from another with a temperature ϑ2
< ϑ1. At steady state heat Q, flows from fluid 1 through the wall to fluid 2, as a
result of the temperature difference ϑ1 − ϑ2. The heat flow Q is transferred from
fluid 1 to the wall which has an area A1 and is at temperature ϑW1. With α1 as the
heat transfer coefficient, it follows that
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For the conduction through the wall
Here λm is the mean thermal conductivity of the wall, δ its thickness and Am the
average area. Finally an analogous relationship exists for the heat transfer from the
wall to fluid 2
Fig. 2.6. Temperature profile for heat transfer through a tube wall bounded by two fluids.
The unknown wall temperatures ϑW1 and ϑW2, can be eliminated from the
three equations for Q. This means that Q can be calculated by knowing only the fluid
temperatures ϑ1 and ϑ2. This results in
where
is valid.
The overall heat transfer coefficient k, for the area A is defined above, where
A is the size of any reference area. Next equation shows that kA can be calculated
using the quantities already introduced for convective heat transfer and conduction.
In practice values for k are often given and used. This can be seen for
example in Polish building regulations where a minimum value for k is set for house
walls. This is to guarantee a sufficient degree of insulation in each house that is built.
This sort of statement of k is tacitly related to a certain area. For flat walls this is the
area of the wall A1 = A2 = Am; for tubes mostly the outer surface A2, which does not
normally differ greatly from A1 or Am.
Value 1/kA represents the resistance to overall heat transfer. It is made up of
the single resistances of each transfer process in the series; the resistance to
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convective transfer between fluid 1 and the wall, (1/α1A1), the conduction resistance
in the wall, (δ/λmAm) and the resistance to convective transfer between the wall and
fluid 2, (1/α2A2). This series approach for overall heat transfer resistance is
analogous to that in electrical circuits, where the total resistance to the current is
found by the addition of all the single resistances in series. Therefore, the three
resistances which the heat flow Q must pass through, are added together. These
three are the resistance due to the boundary layer in fluid 1, the conduction
resistance in the wall and the resistance to transfer associated with the boundary
layer in fluid 2.
The temperature drop due to these thermal resistances behaves in exactly the
same manner as the voltage drop in an electrical resistor, it increases as the
resistance goes up and as the current becomes stronger. From above equations we
get that
From this the temperature drop in the wall and the boundary layers on both
sides can be calculated. To find the wall temperatures the equations
are used.
For the overall heat transfer through a pipe, when it is taken into account that
a pipe of diameter d and length L has a surface area of A = πdL. Then it follows that
where d1 is the inner and d2 the outer diameter of the pipe.
Multi-layer walls
The analogy to electrical circuits is also used to extend the relationships for
overall heat transfer to walls with several layers. Walls with two or more layers are
often used in technical practice. A good example of these multi-layer walls is the
addition of an insulating layer made from a material with low thermal conductivity λis.
Fig. 2.7 shows a temperature profile for a wall that consists of a number of layers.
The resistance to heat transfer for each layer in series is added together and this
gives the overall heat transfer resistance for the wall as
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In curved walls the average area of a layer Ami is calculated using the inner
and outer areas. Each layer touches its neighbour so closely that there is no
noticeable temperature difference between the layers. If this was not true a thermal
contact resistance, similar to the contact resistance that appears in electric circuits,
would have to be considered.
Fig. 2.7. Temperature profile for overall heat transfer through a flat wall of three layers of different
materials
The temperature drop ϑi − ϑi+1 in the i-th layer is proportional to the heat
flow and the resistance to conduction. This is analogous to a voltage drop across a
resistor in an electric circuit. It follows that
In tubes which consist of several layers e.g. the actual tube plus its insulation,
is extended to
The i-th layer is bounded by the diameters di and di+1. The first and last
layers, which are in contact with the fluid, can also be layers of dirt or scale which
develop during lengthy operation and represent an additional conductive resistance
to the transfer of heat.
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