Numerische Integration — Bindeglied zwischen Theorie und Praxis

Abstract

To do examinations into the steady and unsteady-state behavior as well as into the feedback-controlled behavior of a heating plant system, it is necessary to find mathematical models of any subsystem which are able to describe the performance of such a heating plant. The consideration of dynamic processes becomes more and more important on investigations of building heating plants, because of their influence on the economics, but also on the determination of an energy-optimal operation of such a system.An often used working method in developing such mathematical models for HVAC&R investigations is to think of a network which means numerically to look at the considered technical systems as an interconnection of a lot of subsystems. This can be done quite well in modelling all subsystems with the method of lumped parameters and in solving simultaneously the set of ordinary differential equations (ODE). According to the equations of motion, mass and energy conservation one has to put through changes to get these ODEs because of the fact that one has to consider systems with distributed parameters which consist in partial differential equation (PDE).The change from PDE (one-dimensional case) to ODE will done by the aid of finite-difference approximations for the derivative terms (time and space) in respect to the numerical integration methods which have been the Euler-Cauchy, the Runge-Kutta-Merson and the Crank-Nicolson method in this contribution. The so far received results will be investigated according to any considered method especially the values of the steady-state-behavior of an analytical approach compared with those ones of the numerical methods taking into account the number of nodes which is required for a well solution in any case. Special attention is given to the fact that in some circumstances the heat balance will not equal to zero. This balance has to be regarded for a definite distance of time as the variation of stored heat energy (as a sum over every segment with the computed mean temperature) equalized to the cumulative net heat flow formed by the differences between every input and output heat flow of any segment.For practical considerations of counter-clockwise operated heat exchanger model with known system parameters has been extracted for the theoretical investigations. The first part of this contribution deals with the functionalism and the action of the proposed numerical methods (the kind of finite differencing-scheme, grid spacing of time and space). The second part belongs to the results of the system performance of the technical component. The resulting differences which are produced by the used numerical method will be considered by special investigations of the asymptotic behavior according to the grid spacing (number of nodes).