Transient-heat-conduction temperature

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.

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. 

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. 

SOLUTION A hot metal ball is allowed to cool in ambient air. The differential I equation for the variation of temperature within the ball is to be obtained. Analysis The ball is initially at a uniform temperature and is coaled uniformly from the entire outer surface. Also, the temperature at any point in the ball changes with time during cooling. Therefore, this is a one-dimensional transient heat conduction problem since the temperature within the ball changes with the radial distance rand the time t. That is, T = T r, t). 

We start this chapter with the analysis of lumped systems in which the temperature of a body varies with time but remains uniform throughout at any time. Then we consider the variation of temperature with time as well as position for one-dimensional heat conduction problems such as those associated with a large plane wall, a long cylinder, a sphere, and a semi infinite medium using transient temperature charts and analytical solutions. Finally, we consider transient heat conduction in multidimensional systems by utilizing the product solution. 

Noi we demonstrate the use of the method of separation of variables by applying it to the onc-dimcnsional transient heat conduction problem given in Hqs. 4—12, First, we express the dimensionless temperature function 6 X, t) as a product of a funcliou of X only and a function of t only as... 

The temperature of the body changes from the initial temperature 7) to the temperature of tlte surroundings at the end of the transient heat conduction proce.ss. I hus, the maximum amount of heat that a body can gain (or lose if Tj > TJ) is simply the change in the energy content of the body. Thai is. 

S-73 Consider transient heat conduction in a plane wall whose left surface (node 0) i.s maintained at. >0°C while the tiglil surface (node 6) is subjeeted to a solar heal flux of 600 W/m. The wall is initially at a uniform temperature of 50°C. Express the explicit finite difference fomiulalion of the boundary nodes 0 and 6 for the case of no heal generation. Also, obtain die finite difference formulaiioti for the total amount of heat transfer at the left boundary during the first three lime steps. 

Consider transient heat conduction in a plane wall with variable heal generation and constant thermal conductivity. The nodal network of (he medium consists of nodes 0, 1, 2, 3, and 4 with a uniform nodal spacing of A.r. The wall is initially at a specified temperaWre. The temperature at the right bound ary (node 4) is specified. Using the energy balance approach, obtain the explicit finite difference formulation of the boundary... 

Transient mass difliision in a stationary medium is analogous to transient heat transfer provided that the solution is dilute and thus the density of the medium p is constant. In Chapter 4 we presented analytical and graphical solutions for one-dimensional transient heat conduction problems in solids with constant properties, no heat generation, and uniform initial temperature. The analogous one-dimensional transient mass diffusion problems satisfy these requirements ... 

Analysis This problem is analogous to the one-dimenslonal transient heat conduction problem in a semi-infinite medium with specified surface temperature, and thus can be solved accordingly. Using mass fraction for concentration since the data are given in that form, the solution can be expressed as... 

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. 

The numerical solution of a transient heat conduction problem is of particular importance when temperature dependent material properties or bodies with... 

This transient heat conduction problem can be used as a model for the following real process. A fire resistant wall is rapidly heated on its outer side (x = <5) as a result of a fire. We are interested in the temperature rise over time of the other side of the wall at x = 0. The assumption of an adiabatic surface at x = 0 results in a faster temperature rise than would be expected in reality. This assumption therefore leaves us on the safe side. 

The three bodies — plate, very long cylinder and sphere — shall have a constant initial temperature d0 at time t = 0. For t > 0 the surface of the body is brought into contact with a fluid whose temperature ds d0 is constant with time. Heat is then transferred between the body and the fluid. If s < i90, the body is cooled and if i9s > -i90 it is heated. This transient heat conduction process runs until the body assumes the temperature i9s of the fluid. This is the steady end-state. The heat transfer coefficient a is assumed to be equal on both sides of the plate, and for the cylinder or sphere it is constant over the whole of the surface in contact with the fluid. It is independent of time for all three cases. If only half of the plate is considered, the heat conduction problem corresponds to the unidirectional heating or cooling of a plate whose other surface is insulated (adiabatic). 

The equations, (2.171) for the temperature distribution in the plate as well as (2.173) and (2.174) for the released heat have been repeatedly evaluated and illustrated in diagrams, cf. [2.34] and [2.35]. In view of the computing technology available today the direct evaluation of the relationships given above is advantageous, particularly in simulation programs in which these transient heat conduction processes appear. The applications of the relationships developed here is made easier, as for large values of t+ only the first term of the infinite series is required, cf. section 2.3.4.5. Special equations for very small t+ will be derived in section 2.3.4.6. In addition to these there are also approximation equations which are numerically more simple than the relationships derived here, see [2.74]. 

Here the Laplace operator V2t has the form given in 2.1.2 for cartesian and cylindrical coordinate systems. We will once again consider the transient heat conduction problem solved for the plate, the infinitely long cylinder and the sphere in section 2.3.4 A body with a constant initial temperature is brought into contact with a fluid of constant temperature tfy so that heat transfer takes place between the fluid and the body, whereby the constant heat transfer coefficient a is decisive. 

This time is reached after 40 steps with M = 1, 20 steps for M = 2, 8 steps for M = 5 and finally 4 steps for M = 10. The temperatures for M = 1 and M = 2 agree very well with each other and with the analytical solution. The values for M = 5 yield somewhat larger deviations, while the result for M = 10 is useless. This large step produces temperature oscillations which are physically impossible. In [2.57], p. 122, a condition for the restriction of the step size, so that oscillations can be avoided, is given for a transient heat conduction problem with boundary conditions different from our example. The transfer of this condition to the present task delivers the limit... 

In the second chapter we consider steady-state and transient heat conduction and mass diffusion in quiescent media. The fundamental differential equations for the calculation of temperature fields are derived here. We show how analytical and numerical methods are used in the solution of practical cases. Alongside the Laplace transformation and the classical method of separating the variables, we have also presented an extensive discussion of finite difference methods which are very important in practice. Many of the results found for heat conduction can be transferred to the analogous process of mass diffusion. The mathematical solution formulations are the same for both fields. 

In the previous two examples, the temperature (dependent variable) at x = 0 was specified. The same technique can be applied for the case where the derivative of the dependent variable is known at the boundary x = 0 (flux boundary conditions). Consider the transient heat conduction problem in a slab. [4] The governing equation in dimensionless form is... 

Y Tanigawa, T. Akai, R. Kawamura, and N. Oka, Transient heat conduction and thermal stress problems of a nonhomogeneous plate with temperature-dependent material properties, J. Thermal... 

Unsteady-state or transient heat conduction commonly occurs during heating or cooling of grains. It involves the accumulation or depletion of heat within a body, which results in temperature changes in the kernel with time. The rate at which heat is diffused out of or into a kernel or layer of kernels is dependent on the thermal diffusivity coefficient, a, of the grain ... 

The thermal diffusivity can also be measured directly by employing transient heat conduction. The basic differential equation (Fourier heat conduction equation) governing heat conduction in isotropic bodies is used in this method. A rectangular copper box filled with grain is placed in an ice bath (0°C), and the temperature at its center is recorded [44]. The solution of the Fourier equation for the temperature at the center of a slab is used ... 

On the other hand, an estimate of the response time of the slabs assembly can be easily taken as the time needed for the mid plane to undergo 99% of a sudden drop of the wall temperature. The solution of such transient heat conduction problem gives the characteristic time as (Carslaw et al., 1986, Bird et al., 1%0) ... 

The temperature distribution in the solid polymer sample can well be apvproximated by the Fourier equation for transient heat conduction within a medium of constant thermal diffusivity, i.e. 

The resistive heating can conduct to an increase of the temperature until values at which the yield strength admissible for steel is very low. The thermal effect of lightning strikes on objects can be simulated by transient heat conduction in soHds. 

Chiu and co-workers [44] measured the cylindrical orthotropic thermal conductivity of spiral woven fabric composites using a mathematical model that they had devised previously. A parameter estimation technique was used to evaluate the thermal properties of spiral woven fabric composites to verify the predictability of the mathematical model. Good agreement was found between the temperatures measured in a transient heat conduction experiment and those calculated using the prediction equations formulated by the estimated parameters. 

We begin our treatment of transient heat conduction by analyzing a simplified case. In this situation we consider a solid which has a very high thermal conductivity or very low internal conductive resistance compared to the external surface resistance, where convection occurs from the external fluid to the surface of the solid. Since the internal resistance is very small, the temperature within the solid is essentially uniform at any given time. 

The temperature field in the mold is governed by the three-dimensional transient heat conduction equation with constant properties. The equation is... 

The temperature field in the filter is described by the equation of transient heat conduction with heat sources in axisymmetric coordinates ... 

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