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 5

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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