Theoretical Concept of Equivalent Wall

Equivalent Wall Model

This discussion presents the idea of the “thermally equivalent wall”; a plane multi-layer structure with dynamic characteristics similar to those of a complex structure in which three-dimensional heat flow occurs. The effect of internal thermal structure on the dynamic characteristics of walls is analyzed below. It introduces the idea of structure factors and shows the conditions they impose on response factors.

Relationships between response factors and thermal structure factors

Relationships between the structural and dynamic thermal characteristics of building walls are analyzed below to answer the question how should the response factors (or transfer function coefficients) be modified to account for the effects of thermal bridges in computerized whole building energy calculations

The simplest method is to calculate the overall resistance of a wall with imperfections - solving the steady state heat transfer problem - and multiply response factors by the resulting correction factor. This approach is suitable for light walls for which storage effects are insignificant.

However, imperfections in plane walls do not only change the resistance of the walls but also modify their dynamic properties, which may be represented by response factors in computer whole building energy simulations. To account for this effect, the general conditions between structural and dynamic characteristics for walls must be noted. Such conditions follow from the asymptotic formulae for the heat flow across the surfaces of the separated wall element due to temperature difference on its two sides.

Consider the heat flow through an element of a building envelope of complex material and geometrical structure, embedded in a plane wall, homogeneous in every cross section, parallel to the wall surfaces. It is assumed that the thermophysical properties of the structure-thermal conductivity k, density r and specific heat c-are constant in time. The element, together with its nearest neighborhood, is represented by the region D. Region D is bounded by the inner surface facing room temperature T; the outer surface facing environmental temperature T; and the adiabatic surface of the cut which separates it from those parts of the wall where the heat flow can be considered as one-dimensional.

For sufficiently large t asymptotic expressions for the total heat flow in time interval [0, t], across the internal and external surface of a wall element, for ambient temperatures held constant at T_i and T_e, and initial conditions zero, the following simple form [Kossecka 1992, 1996, 1998, Kossecka and Kosny 1996, 1997]:

	(19)
	(20)

where R and C denote the total thermal resistance and capacity of the wall element of volume V, adiabatically cut from the surroundings:

(21)

whereas j_ii, j_ie and j_ee are given by:

	(22)
	(23)
	(24)

wherej is the dimensionless temperature for the problem of steady-state heat transfer through a wall, for ambient temperatures T_i = 0 and T_e = 1.

Dimensionless quantities j_ii and j_ie are called the “thermal structure factors” for a wall or a wall detail. For a plane wall they are determined directly by its structure, represented by the thermal capacity and resistance distribution across its thickness. In a three-dimensional case, to calculate them effectively, one has to solve the steady state heat transfer problem.

The following identity is satisfied:

(25)

Structure factors j_ii and j_ie for a wall composed of n plane homogeneous layers, numbered from 1 to n with layer 1 at the interior surface, are given as follows:

	(26)
	(27)
	(28)

where: R_m and C_, denote the thermal resistance and capacity of the m-th layer respectively, whereas R_i-m and R_m-e denote the resistance for heat transfer from surfaces of the m-th layer to inner and outer surroundings, respectively. Structure factors j_ii and j_ie take values from the range (0, 1), whereas j_ie from the range (0, 1/4). For a homogeneous wall, contact resistances being neglected, j_ii = j_ee =1/3, j_ie = 1/6.

Products Cj_ii, Cj_ie, Cj_ee, are equivalent to the thermal mass factors, introduced by Anderson [1989]; see also ISO/DIS 9869.2. 1991.

Correspondence with the dynamic thermal characteristics of walls one deals with using Laplace transform method is as follows [Kossecka 1998]:

(29)

where A(s), B(s) and D(s) are elements of the s-transfer matrix for a multi-layer wall.

The s-transfer function 1/B may be represented as [Brown and Stephenson 1993]:

(30)

where t _n are the time constants of the wall, i.e. the poles of 1/B are at s = -1/t_n. Differentiating (A30) gives:

(31)

Compatibility of Equations (A29) and (A31) gives the condition:

(32)

which means that walls of the same value of the product RCj_ie, have the same sum of time constants. For multi-layer walls, in general, sum of the first five time constants is about 0.9 of the product R×C×j_ie, whereas first time constant t ₁ is between 0.5 to 0.9 of this quantity.

Compatibility of the asymptotic formula (19) with the expressions for the heat flow, given in terms of the response factors, yields the following constraints conditions [see Kossecka 1992, 1996, 1998, Kossecka and Kosny 1996, 1997]:

	(33)
	(34)

Conditions analogous to (33), (34), for the dimensionless z-transfer function coefficients b_n , c_n, d_n, have the form [Kossecka 1998]:

	(35)
	(36)

Response factors Y_n , with n ³ 1, describe the storage effects - heat fluxes after the time of duration of the triangular temperature impulse. They are all positive. High value of the structure factor j_ie, indicates that response factors with number n ³ 1, are comparatively large; on the contrary, small value of this structure factor indicates that they are comparatively small. However equations (33), (34) must be satisfied simultaneously with (4), which states that the sum of all response factors must be equal to 1. Therefore the larger are the values of Y_n for n ³ 1, the smaller is the value Y_o and vice versa. Taking into account that Y_n should be relatively smooth functions of the number n, one can expect that, for given C, response factors corresponding to small values of structure factors decay relatively quickly whereas those corresponding to large values of structure factors decay relatively slowly. Constraint condition (32), for the time constants t _n, indicates the same.

One can say thus, that thermal structure factors, together with total thermal resistance R and capacity C, determine the dynamic thermal properties of a wall element - through the conditions they impose on response factors. Those conditions however do not determine the response factors in a unique way, but rather play the role of constraints. Nevertheless one may expect that walls with the same total thermal resistance, capacity and structure factors, have also similar dynamic characteristics - response factors - even if they are quite different in details.

Relationships between structure factors and response factors, and also z-transfer function coefficients, have the same form for plane and composite walls. This analogy constitutes the basis of the notion of the “thermally equivalent wall”: the plane wall of the same dynamic properties as a complex structure, which may be used as its substitute in the whole building thermal modeling.