Chapter 4: Transient Conduction
4.1 Transient State
Transient state is a state of non-equilibrium, when the temperatures are still changing with time. Some simple examples are the pre-heating time of ovens, when you are waiting for the temperature inside the oven to reach a specified temperature. However, even in a heated oven, cooking occurs by transient conduction because the heat is conducted through the food as it cooks. The internal temperature of the food, therefore, is usually increasing during cooking.
4.2 Lumped Body Model
The simplest case of transient conduction is the lumped body model. This model considers the heat transfer between a solid and a surrounding fluid. The main assumption in the model is that the solid can be approximated to have a uniform temperature; that is, there are no temperature gradients within the solid. Furthermore, the surrounding fluid must be large enough so that its temperature remains a constant throughout the process. This approximation can be made when the following condition is satisfied:
where h is the convection coefficient, k is the conductivity of the solid, and Lc is the characteristic length of the solid, defined as the volume/area ratio. For a wall of thickness 2L, Lc = L; for a cylinder, Lc = r/2; for a sphere, it is r/3. The number Bi is a dimensionless parameter called the Biot number.
For a lumped body, the temperature of the solid which is introduced into the fluid at time t = 0 is given by:
where A is the surface area of the solid, m is the mass of the solid, cp is the specific heat of the solid, Ti is the initial temperature of the solid, and T∞ is the temperature of the fluid. This equation assumes that T∞ remains constant through the process. The newly introduced variable θ/ θi is the "normalized temperature," which measures the temperature in relative terms. That is, the normalized temperature measures the percent change in temperature rather than exact change in temperature in °C. Once the normalized temperature is known, the actual temperature of the solid can be calculated based on initial temperature and the surrounding temperature.
4.3 Semi-Infinite Solid
The second approximation often applied to analyze transient conduction is the semi-infinite approximation. Semi-infinite solids can be visualized as very thick walls with one side exposed to some fluid. The other side, since the wall is very thick, remains unaffected by the fluid temperature. This condition is expressed as T(∞, t) = Ti, where Ti is the initial wall temperature. This is illustrated in Figure 4.1. The condition at the exposed side of the wall is called the boundary condition.
Figure 4.1 Semi-infinite wall
4.3.1 Constant temperature boundary condition
One possible condition for the wall surface is a constant temperature (Fig. 4.1a). In this case, the temperature inside the wall at time t, at distance x from the surface, is given by:
where Tsurface is the constant wall temperature and α is the thermal diffusivity of the wall. Thermal diffusivity, described in Chapter 2, is related to the thermal conductivity, and is the measure of how quickly heat is dissipated in a material. The function erf is called the Gaussian Error Function, and values for erf(x) are often tabulated or available in graphical form for convenience. Both a table and a graph for erf(x) are attached (Click Here).
4.3.2 Surface convection boundary condition
Another possible case is to have convection off the surface of the wall (Fig. 4.1b). In this case, h is assumed to be constant at all times. The temperature in the wall at time t and distance x is:
where erfc is called the Complementary Error Function, equal to 1 - erf(x). The values for erfc(x) are also attached (Click Here).
4.4 One-Dimensional Conduction
One-dimensional transient conduction refers to a case where the temperature varies temporally and in one spatial direction. For example, temperature varies with x and time. Three cases of 1-D conduction are commonly studied: conduction through a plate, in a cylinder, and in a sphere. The three geometries are shown in Figure 4.2. In all three cases, the surface of the solid is exposed to convection.
Figure 4.2 One Dimensional Geometries
The exact analytical solutions to the three cases are very complicated. However, an approximate solution can be obtained by using graphical tools. The graphs allow you to find the centerline temperature at any given time, and the temperature at any location based on the centerline temperature. The graphs for the three geometries are attached.
4.4.1 ParametersThree parameters are needed to use the charts: Normalized centerline temperature, the Fourier Number, and the Biot Number. The definition for each parameter are listed below:
i. Normalized centerline temperature
where T0 is the centerline temperature, Ti is the initial temperature, and T∞ is the ambient temperature;
ii. Fourier Number
where α is the thermal diffusivity, t is time, and L is the half-thickness for a plate or the radius for cylinder or sphere;
iii. Biot Number
where h is the convection coefficient, k is the thermal conductivity, and L is as defined above.
Note, for cylindrical and spherical cases, that the L used in the calculation of Bi in Equation 4.7 is not the same as Lc used in the calculation for Bi defined in Equation 4.1.
4.4.2 Reading the Graphs
There are two types of charts for each geometry: the first for finding the centerline temperature, and the second for finding the temperature at any location.
To find the centerline temperature, first calculate Fo and Bi-1. The first chart has Fo on the horizontal axis, θ0* on the vertical, and lines representing different values of Bi-1. Locate the line corresponding to the Bi-1 you have calculated. Next, locate the value of Fo on the x-axis, and draw a vertical line from the axis so that it intersects the Bi-1 line. From the intersection point, draw a horizontal line. The value at which the horizontal line crosses the y-axis is the normalized centerline temperature.
To find the temperature at any location within the solid, we now will use the second chart. The second chart has Bi-1 on the horizontal axis, normalized temperature on the vertical axis, and lines corresponding to normalized distance, x/L or r/r0. Calculate the normalized distance for the point which you are interested in, and locate the corresponding line on the graph. Using the calculated value of Bi-1, draw a vertical line through the graph. Draw a horizontal line from the point of intersection between normalized distance curve and the vertical line. The point at which the horizontal line intersects the y-axis is the normalized temperature at that location.
4.5 Multi-dimensional Conduction
The analysis of multidimensional conduction is simplified by approximating the shapes as a combination of two or more semi-infinite or 1-D geometries. For example, a short cylinder can be constructed by intersecting a 1-D plate with a 1-D cylinder. Similarly, a rectangular box can be constructed by intersecting three 1-D plates, perpendicular to each other. In such cases, the temperature at any location and time within the solid is simply the product of the solutions corresponding to the geometries used to construct the shape. For example, in a rectangular box, T(x*,y*,z*,t) - the temperature at time t and location x*, y*, z* - is equal to the product of three 1-D solutions: T1(x*,t), T2(y*,t), and T3(z*,t).
Chapter 4 Appendices
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