A rationale for nodal placement for heat flow calculations in walls

Abstract

A building wall consists of one or more layers of homogeneous material. To examine its thermal behaviour in unsteady conditions using finite difference methods, a thin layer of resistance r L and capacity c L is replaced by a T section of lumped values r L /2, c L , r L /2 – 3 elements—and a thicker layer might be divided into say three equal slices so as to be represented by the sequence ( r L /6, c L /3, r L /3, c L /3, r L /3, c L /3, r L /6), 7 elements in all. A wall may consist of several such layers. The wall is bounded by exterior and interior nodes which, according to choice, are respectively, the ambient or the wall exterior surface, and the wall interior surface or the room index node. An index node implies a further resistance. The simple finite difference model consists of the sum of such elements. A method of design is presented here which is based on the integral properties of the wall. With isothermal outside and inside nodes, the values of the series of wall decay times z j are found. The wall is supposed driven by temperatures whose values are known at intervals of δ, typically 1 h . From the sequence of z j values, together with the value of δ, the minimum number of capacities N needed to represent the wall can be determined. The response of the wall to a ramp excitation (of 1 K/δ) can be expressed in terms of amplitudes q j ∗ where the eigennumber j has values 0– N. A model wall having assumed values for the elements r 1, c 2, r 3, c 4,…, c 2 N , r 2 N+1 is set up and the corresponding amplitudes q j ̂ found. From them a measure SS is found which is based on the sums of squares of the differences between functions of the q j ∗ and q j ̂ values; SS provides a measure of the difference in response between the real wall and its model. By systematic variation of the values of r j and c j , SS can be reduced to some acceptable value. The 2 N+1 values of the model elements, so found, may be expected to be significantly fewer than the number required by conventional wall division.