Transient heat conduction problem

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 ... 

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... 

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

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. 

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). 

Therefore, the proper form of the dimensionless time is t atlL, which is called the Fonrier number Fo, and we recognize Bi = kJliL as the Biot number defined in Section 4—1. Then the formulation of the one-diniensional transient heat conduction problem in a plane wall can be expressed in nondimensional form as... 

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... 

This completes the analysts for the solution of one-dimensional transient heat conduction problem in a plane wall. Solutions in other geometries such as a long cylinder and a sphere can be determined using the same approach. The results for all three geometries arc summarized in Table 4—1. The solution for the plane wall is also applicable for a plane wall of thickness L whose left surface at, r = 0 is insulated and the right surface at.t = T. is subjected to convection since this is precisely the mathematical problem we solved. 

Using the one-term approximation, the solutions of onedimensional transient heat conduction problems are expressed analytically as... 

C How does the finite difference formulation of a transient heat conduction problem differ from that of a steady heat conduction problem Wlial does Ihe term pAil.xCp(T — Tlij/Ar represent in the transient finite difference formulation ... 

C Is there any liiniialion on Ihe size of the time step Ar in Ihe solution of transient heat conduction problems using (o) the explicit method and (fc) the implicit method ... 

S-68C Express the general stability criterion for the explicit method of solution of transient heat conduction problems. 

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 solution of a transient heat conduction problem can be found in three different ways ... 

The solution of transient heat conduction problems using the Laplace transformation consists of three steps ... 

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. 

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. 

The latter of the two methods offers a practical, simple applicable solution to transient heat conduction problems and should always be applied for sufficiently small Biot numbers. 

The simple, explicit difference method for transient heat conduction problems... 

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... 

Consider the following transient heat conduction problem in a slab.[1-3] The governing equation is ... 

Heat conduction with a constant boundary condition at x =0 was considered in example 4.1. The same technique can be applied for time dependent boundary conditions. Consider the transient heat conduction problem in a slab.[4] The governing equation is ... 

M. S. Bhatti, Limiting Laminar Heat Transfer in Circular and Flat Ducts by Analogy with Transient Heat Conduction Problems, unpublished paper, Owens-Corning Fiberglass Corporation, Granville, Ohio, 1985. 

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 Laplace-transform based boundary integral equation was proposed by Rizzo and Shippy (1970) to solve transient heat conduction problems. 

Brian PLT (1961) A finite-difference method of high-order accuracy forthe solution of three-dimensional transient heat conduction problems. AIChE J 7 367-370... 

Thomas BG, Samarasekera IV, Brimacombe JK (1984) Comparison of numerical modeling techniques for complex, two-dimensional, transient heat conduction problems. Meteill Tians 15B 307-318... 

If we follow an element of the film, then we can recast Eq. 5.193 as a transient heat conduction problem. With f = z/b, where b is one-half the film thickness and t = z/Vo, we are now solving the following DEQ (differential equation) ... 

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