General Heat Conduction Equation

The general heat-conduction equation, along with the familiar diffusion equation, are both consequences of energy conservation and, like we have just seen for the Navier-Stokes equation, require a first-order approximation to the solution of Boltz-man s equation. 

Substituting this expression for Q and the first-order expression we found earlier for the pressure-tensor (equation 9.63) into equation 9.66 yields the general heat conduction equation ... 

Beginning with the three-dimensional heat-conduction equation in cartesian coordinates [Eq. (l-3a)], obtain the general heat-conduction equation in cylindrical coordinates [Eq. (1-36)]. 

To analyze a transient heat-transfer problem, we could proceed by solving the general heat-conduction equation by the separation-of-variables method, similar to the analytical treatment used for the two-dimensional steady-state problem discussed in Sec. 3-2. We give one illustration of this method of solution for a case of simple geometry and then refer the reader to the references for analysis of more complicated cases. Consider the infinite plate of thickness 2L shown in Fig. 4-1. Initially the plate is at a uniform temperature T, and at time zero the surfaces are suddenly lowered to T = T,. The differential equation is... 

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. 

Eq. 2-38 is the general heat conduction equation in rectangular coordinates. In the case of constant thermal conductivity, it reduces to... 

The general heat conduction equation in cylindrical coordinates can be obtained from an energy balance on a volume element in cylindrical coordinates, shown in Fig. 2-23, by following the steps just outlined. It can also be obtained directly from Eq. 2-38 by coordinate transformation u ing the... 

Substituting this to the first law of thermodynamics of a given system gives the general heat conduction equation as... 

The scheme of second-order accuracy (unconditionally stable in the asymptotic sense). Before taking up the general case, our starting point is the existing scheme of order 2 for the heat conduction equation possessing the unconditional asymptotic stability and having the form... 

The explicit difference scheme. The schemes considered in Section 1 may be generalized to the case of the heat conduction equation with several spatial variables. 

One way of covering this for the heat conduction equation is to construct a homogeneous conservative scheme by means of the integro-interpo-lation method. To make our exposition more transparent, we may assume that the coefficient of heat conductivity k = k(x) is independent of t. The general case k = k(x,t) will appear on this basis in Section 8 without any difficulties. 

Because of this, there is a real need for designing the general method, by means of which economical schemes can be created for equations with variable and even discontinuous coefhcients as well as for quasilinear non-stationary equations in complex domains of arbitrary shape and dimension. As a matter of experience, the universal tool in such obstacles is the method of summarized approximation, the framework of which will be explained a little later on the basis of the heat conduction equation in an arbitrary domain G of the dimension p with the boundary F... 

The general methodology provides proper guidelines for the selection rules in studying one-dimensional heat conduction equations... 

The physical argument presented above is consistent with the mathematical nature of the problem since tlie heat conduction equation is second order (i.e., involves second derivative.s with respect to the space variables) in all directions along which heat conduction is significant, and the general solution of a second-order linear differential equation involves two surbitrary constants for each direction. That is, the number of boundary conditions that needs to be specified in a direction is equal to the order of the differential equation in that direction. 

In the application of the heat conduction equation in its general form (2.8) a series of simplifying assumptions are made, through which a number of special differential equations, tailor made for certain problems, are obtained. A significant simplification is the assumption of constant material properties A and c. The linear partial differential equations which emerge in this case are discussed in the next section. Further simple cases are... 

If A and eg change with temperature, cf. section 2.1.4, a closed solution to the heat conduction equation cannot generally be found, which only leaves the possibility of using a numerical solution method. We will show how temperature dependent properties are accounted for by using the example of the plate, m = 0 in (2.274). The transfer of the solution to a cylinder or sphere (m = 1 or 2 respectively) is... 

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