9.56. For a reactor operating in the steady state, the spatial distribution of temperature in any component, e. g., a fuel rod, is given by equation (9.16), assuming the medium to be isotropic and the thermal conductivity independent of temperature. In a transient situation, however, such as when the reactor is being started up or shut down, the steady-state condition is not applicable. In deriving the steady-state equation (9.15) for one­dimensional heat flow, of which equation (9.16) is the general form, a heat (or energy) balance was obtained by equating the difference between the rates of heat flow into and out of a volume element to the rate of internal heat generation.

9.57.

For the transient situation, the energy balance (or conservation) must include the time rate of change of internal energy in the volume element. This is given by cpp{A dx){dt! d%), where cp is the specific heat of the material and p is its density ; A dx is the volume of the element under consideration (§9.42) and dtid% is the rate of increase of temperature t with time 0. Hence, equation (9.15) becomes

or

d2t _ Q(x) 1 dt

dx2 к a dO ’

where a, called the thermal diffusivity of the material, is defined by

The general form of equation (9.26) for the spatial distribution of tem­perature in a conducting medium with internal heat generation under tran­sient conditions is

which may be compared with the steady-state equation (9.16).

9.58. In the analysis of transient behavior, the appropriate form of equation (9.27) is generally solved by numerical methods using a computer; for this purpose, the differential equation must be expressed in finite — difference form. To illustrate the general principles used in these (and many other) calculations, the simple case will be considered of conduction in one dimension in a component (or components) which may be regarded as being made up of several adjacent volume elements (or cells).

9.59. The system can be treated as a series of grid points or nodes, with the characteristic properties, e. g., the temperature, of each cell being rep­resented by the value at its nodal point. Three such cells, designated і — 1, і, and i + l, respectively, with the corresponding nodal points, assumed to be at the cell centers, are shown in Fig. 9.9; the nodal temperatures are

th and ti+1. The dimension of each cell in the x direction is Ax, and the volume is A Ax, where A is the cross-sectional area of the cell.

9.60.

Suppose, for the present, that there is no internal heat generation. Conservation of energy then requires that the difference between the rates of heat flow into and out of the cell і be equal to the rate of increase of internal energy cpp(A Ax^dt^dQ), If qt-u is the rate of heat flow in the x direction from cell і — 1 into cell /, and qii+г is the rate of heat flow out

qu+1 = кA

assuming к to be independent of temperature. If these values are inserted into equation (9.28), it is found that

ti_l + ti+l — 2tj _ 1

(Ax)2 add

It can be readily shown that the left side is the linear-difference form of + cPt/dx2 so that equation (9.29) is the equivalent of equation (9.26) with Q(x) equal to zero.

9.61.

In order to express dtt/dQ in linear-difference form, the nodal approach is extended into a time coordinate 0 as in Fig. 9.10, where n — 1, n, and n + 1 indicate successive times. For a time step A0 from n

to n + 1, the temperature change Att in the cell і is tin+1 — ti n, so that

dti^ M = tj, n+i ~ Kn d% A0 A0

Upon insertion of this result into equation (9.29) for the time 0„ and rearranging, it is found that

It is thus seen that if the temperatures tn at the various nodal points are known at the time 0„, the temperature tn + 1 at the time 0„ + 1, i. e., after a specified time increment Д0, can be calculated directly from equation (9.31). When the values of t at all the nodal points at the time 0„ + 1 are known, the calculation can proceed to the next time step, i. e., to time 0n+2, and so on. This procedure is referred to as the explicit method for solving the differential equation (9.26) with Q(x) equal to zero. Internal heat gener­ation may be included, however, without affecting the general principles.

9.62. A simplification of equation (9.31) is possible if the time and distance increments Д0 and Ax are chosen so that the dimensionless quantity

aA0

(АхГ

It is apparent from equation (6.31) that, in these circumstances, the tem­perature at a node і after a time step is equal to the mean of the temper­atures at the two adjacent nodes before the time step, i. e., fl> + 1 =

9.63. In any event, it has been shown that for a numerical solution of equation (9.31) to be possible, аД0/(Д*)2 must be positive and less than

0. 5. This limitation on аД0/(Д*)2 places a restriction on the magnitude A0 of the time step, relative to (Дх)2/а, that can be used in the computations. Consequently, when the calculations must be made for an extended time period, a large number of time intervals may have to be used. Furthermore, it is found that if the time intervals are small, instabilities arise in applying the numerical techniques.

9.64. When, for one reason or another, longer time intervals are re­quired than are permitted by the explicit method, the implicit form of the difference equation is used. The spatial finite-difference temperature terms are expressed at the advanced time point n + 1 instead of at the point n as in the explicit procedure. That is to say, equation (9.30) is inserted into equation (9.29) in which the temperatures are now tt_l n + 1, ti+l n+1, and

ti n +1- It is then found that

9.65. In order to solve this equation for the temperatures at the time 0n+1 from the known value ti n at the preceding time 0n, it is necessary to write a set of simultaneous equations for all the space points at each time step. These equations can be represented by a tridiagonal matrix, i. e., a matrix in which only the main diagonal and the two adjacent diagonals are nonzero. Such a matrix is readily solved on a computer by the Gauss elimination procedure [4]. The implicit method requires a greater calcu — lational effort at each time step than does the explicit method, but larger time steps can be taken and stability problems are eased; as a result, the overall computer time is often less than for the explicit method.

9.66. In the foregoing discussion a simple situation has been treated to provide some insight into the general principles used in solving transient problems in heat transmission. In practice, boundary conditions, e. g., a convection boundary at the fuel cladding-coolant interface, must be in­cluded in the calculations. Allowance may also be required for internal heat generation, which is usually time dependent. In addition, in some cases it may be necessary to determine the spatial temperature distributions in more than one dimension. It should be noted, too, that a, which is a characteristic property of the material, may change from one nodal point to another. The calculations are thus often quite complex and are per­formed by means of appropriate computer codes.