Transient heat conduction

Time dependent or transient temperature fields appear when the thermal conditions at the boundaries of the body change. If, for example, a body that initially has a constant temperature is placed in an environment at a different temperature then heat will flow over the surface of the body and its temperature will change over time. At the end of this time dependent process a new steady-state temperature distribution will develop. 

In the following sections we will discuss simple solutions, which are also important for practical applications, of the transient heat conduction equation. The problems in the foreground of our considerations will be those where the temperature field depends on time and only one geometrical coordinate. We will discuss the most important mathematical methods for the solution of the equation. The solution of transient heat conduction problems using numerical methods will be dealt with in section 2.4. 

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Heatshield thickness and weight requirements are determined using a thermal prediction model based on measured thermophysical properties. The models typically include transient heat conduction, surface ablation, and charring in a heatshield having multiple sublayers such as bond, insulation, and substmcture. These models can then be employed for any specific heating environment to determine material thickness requirements and to identify the lightest heatshield materials. 

Transient Heat Conduction. Our next simulation might be used to model the transient temperature history in a slab of material placed suddenly in a heated press, as is frequently done in lamination processing. This is a classical problem with a well known closed solution it is governed by the much-studied differential equation (3T/3x) - q(3 T/3x ), where here a - (k/pc) is the thermal diffuslvity. This analysis is also identical to transient species diffusion or flow near a suddenly accelerated flat plate, if q is suitably interpreted (6). 

Figure 4. Temperature profiles in transient heat conduction.

Mills and Gilchrist (270) analysed the heat transfer that occurs when closed cell foams are subjected to impact, to predict the effect on the uniaxial compression stress-strain curve. Transient heat conduction from the hot compressed gas to the cell walls occurs on the 10 ms... 

Transient Heat Conduction Problem Using Constant Strain Triangle... 

Three-Dimensional Transient Heat Conduction Problem With Convection... 

What would the constant strain finite element equations look like for the transient heat conduction problem with internal heat generation if you were to use a Crank-Nicholson time stepping scheme ... 

Figure 12.4. Transient heat conduction in the single-particle model (from Botterill, 1975) ...

In the emulsion phase/packet model, it is perceived that the resistance to heat transfer lies in a relatively thick emulsion layer adjacent to the heating surface. This approach employs an analogy between a fluidized bed and a liquid medium, which considers the emulsion phase/packets to be the continuous phase. Differences in the various emulsion phase models primarily depend on the way the packet is defined. The presence of the maxima in the h-U curve is attributed to the simultaneous effect of an increase in the frequency of packet replacement and an increase in the fraction of time for which the heat transfer surface is covered by bubbles/voids. This unsteady-state model reaches its limit when the particle thermal time constant is smaller than the particle contact time determined by the replacement rate for small particles. In this case, the heat transfer process can be approximated by a steady-state process. Mickley and Fairbanks (1955) treated the packet as a continuum phase and first recognized the significant role of particle heat transfer since the volumetric heat capacity of the particle is 1,000-fold that of the gas at atmospheric conditions. The transient heat conduction equations are solved for a packet of emulsion swept up to the wall by bubble-induced circulation. The model of Mickley and Fairbanks (1955) is introduced in the following discussion. 

Example 5.2 Semi-infinite Solid with Constant Thermophysical Properties and a Step Change in Surface Temperature Exact Solution The semi-infinite solid in Fig. E5.2 is initially at constant temperature Tq. At time t — 0 the surface temperature is raised to T. This is a one-dimensional transient heat-conduction problem. The governing parabolic differential equation... 

Temperature profiles can be determined from the transient heat conduction equation or, in integral models, by assuming some functional form of the temperature profile a priori. With the former, numerical solution of partial differential equations is required. With the latter, the problem is reduced to a set of coupled ordinary differential equations, but numerical solution is still required. The following equations embody a simple heat transfer limited pyrolysis model for a noncharring polymer that is opaque to thermal radiation and has a density that does not depend on temperature. For simplicity, surface regression (which gives rise to convective terms) is not explicitly included. 

Problem Solve the one-dimensional, transient heat conduction problem with the following boundary conditions ... 

For non-stationary heat conduction in a semi-infinite stationary medium the onedimensional transient heat conduction without heat production, we have next parabolic differential equation... 

We continue our discussion of transient heat conduction by analyzing systems which may be considered uniform in temperature. This type of analysis is called the lumped-heat-capacity method. Such systems are obviously idealized because a temperature gradient must exist in a material if heat is to be conducted into or out of the material. In general, the smaller the physical size of the body, the more realistic the assumption of a uniform temperature throughout in the limit a differential volume could be employed as in the derivation of the general heat-conduction equation. 

In most practical situations the transient heat-conduction problem is connected... 

For a discussion of many applications of numerical analysis to transient heat-conduction problems, the reader is referred to Refs. 4, 8, 13, 14, and 15. 

Richardson. P. D.. and Y. M. Shum Use of Finite-Element Methods in Solution of Transient Heat Conduction Problems, ASME Pap. 69-WA./H1-3fi. 

Transient Heat Conduction in Semi-Infinite Solids 240 Contact of Two Semi-Infinite Solids 245... 

Transient Heat Conduction in a Plane Wall 313 Two-Dimensional Transient Heat Conduction 324... 

The coverage of Chapter 4, Transient Heat Conduction, is now expanded to include (1) the derivation of the dimensionless Biot and Fourier numbers by nondimensionalizing the heat conduction equation and the boundary and initial... 

Chapter 4 "Transient Heat Conduction is revised greatly, as explained previously, by including the theoretical background and the mathematical details of the analytical solutions. 

Another material property that appears in the transient heat conduction analysis is the thermal diffusivity, which represents how fast heat diffuses through a material and is defined as... 

Now consider a sphere, with density p, specific heat c, and outer radius R. The area of the sphere normal to the direclion of heat transfer at any location is A — 4vrr where r is the value of the radius at that location. Note that the heat transfer area A depends on r in this case also, and thus it varies with location. By considering a thin spherical shell element of thickness Ar and repeating tile approach described above for the cylinder by using A = 4 rrr instead of A = InrrL, the one-dimensional transient heat conduction equation for a sphere is determined to be (Fig. 2-17)... 

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