## Chapter 4: Transient Conduction | |

## 4.1 Transient StateTransient 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
where L, L = _{c}L; for a cylinder, L = _{c}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
where T is the initial temperature of the solid, and _{i}T is the temperature of the fluid. This equation assumes that _{∞}T remains constant through the process. The newly introduced variable _{∞}θ/ θ is the "normalized temperature," which measures the temperature in relative terms. That is, the normalized temperature measures the _{i}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 is the initial wall temperature. This is illustrated in _{i}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
where 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 conditionAnother possible case is to have convection off the surface of the wall (Fig. 4.1b). In this case,
where ## 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 Figure 4.2 One Dimensional GeometriesThe 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:
where T is the initial temperature, and _{i}T is the ambient temperature;
_{∞}
where
where
## 4.4.2 Reading the GraphsThere 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 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}, 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.
See graphs for Plane wall, Cylinder, or Sphere. (Download printable version) ## 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, x*,t), T(_{2}y*,t), and T(_{3}z*,t).
## Chapter 4 Appendices- Chart for Gaussian Error Functions
- Graphical Solution for Plane wall
- Graphical Solution for Cylinder
- Graphical Solution for Sphere
- Download printable version of all charts
Click here for References 4 | 5 | 6 | 7 | 8 | >>
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