Theoretical Concept of Equivalent Wall

Background to the Equivalent Wall Model

Relationship Between Response Factors and Z-transfer Function Coefficients
Determining the z-transfer function coefficients from the response factors

A simplified model involves creating a fictitious multi-layer wall with properties selected so that its dynamic response to the transient conditions is the same as a given assembly of equivalent thickness. Thus, for example, a homogenous wall material could be defined with a specific conductivity, density, specific heat, etc., to give the same dynamic (and steady-state) response as a steel-framed wall with gypsum wallboard sheathing. Formal relationships describing the effect of structure on the dynamic thermal behavior of walls have been developed from the integral formulae for heat flow across wall surfaces in a finite time interval [Kossecka 1992]. These relationships include quantities called thermal structure factors. Thermal structure factors appear in expressions for the asymptotic heat flow across the surfaces of a wall, for boundary conditions independent of time. Correlations between thermal structure factors, response factors and z-transfer function coefficients have been derived and analyzed [Kossecka 1996, Kossecka 1998, Kossecka and Kosny 1997].

A general approach is the equivalent wall method [Kossecka and Kosny 1997, Kosny et al. 1998]. This method has been used by ORNL for dynamic evaluations of several complex wall technologies. During these projects a detailed three-dimensional computer model was developed for each wall. Dynamic measurements of these wall systems were carried out by hot-box testing. A full-scale representative (8 x 8 ft) cross-section of the clear wall area of the wall system was used to determine its dynamic thermal performance. A dynamic test typically consists of three basic stages:
•    steady-state stage (constant temperatures on both sides of the wall);
•    thermal ramp (rapid change of the temperature on the one side of the wall); and
•    stabilizing stage (the wall is maintained at the second set of constant boundary temperatures until steady-state heat transfer occurs).

These computer models were then used to generate equivalent walls. An equivalent wall has a simple one-dimensional multi-layer structure and the same thermal properties as the actual wall (total resistance and thermal capacitance). Its dynamic thermal behavior is identical to the actual wall. The equivalent-wall concept for complex thermal structures expresses the role of storage effects in heat flow through an element. It leads to the definition of structure factors (dimensionless quantities representing the fractions of heat stored in the wall volume, in transition between two different states of steady heat flow, which are transferred across each wall surface). These quantities, together with total transmittance and capacity, provide the basic thermal characteristics of the structure. Even for a complex thermal bridge configuration, response factors, steady-state R-values and thermal structure factors have the same values for both walls (the “complex” wall and the equivalent wall) [Kossecka and Kosny 1996, Kossecka and Kosny 1997, Kosny et al. 1998]. A validation of the equivalent wall theory was performed using dynamic test and finite- difference computer modeling results [Kosny et al. 1998]. A dynamic thermal performance of the EPS Form Wall System was analyzed based on dynamic hot-box testing. The Oak Ridge National Laboratory’s Buildings Technology Center guarded hot-box facility served for this test that took about 240 hours to complete. The same wall configuration was modeled for dynamically changing boundary conditions. The finite-difference computer code HEATING 7.2 [Childs 1993] was used for dynamic modeling (HEATING 7.2 was previously calibrated during steady-state tests). Thermal mass validation of the model was made using the dynamic boundary conditions monitored during the hot-box test. Computer-generated results were compared with the results of the dynamic test. Also, a series of response factors, heat capacity, and R-value were computed using finite-difference computer modeling. They were compared with response factors generated for the equivalent wall. Very good agreement was found between test and computer modeling results. The response factors for the equivalent wall were almost identical to those for the nominal wall.

The equivalent wall technique is a relatively simple way to use whole-building energy simulations (using DOE-2.1 or BLAST) for buildings containing complex assemblies. The traditional method used by DOE-2.1 modelers is to generate a series of response factors or transfer functions for the complex wall and modify DOE-2.1 or BLAST source codes to permit this type of wall data input. The equivalent wall method represents all the thermal information about the wall using only five numbers (R, C, and three thermal structure factors). This is much simpler than the alternate use, ascribed to a three-dimensional model, where a long series of response factors (for massive walls, 60 to as many as 150 numbers, multiplied by 3) must be accompanied by the troublesome modification of the program source code to enable this type of wall data input.

Relationship Between Response Factors and Z-transfer Function Coefficients

Three-dimensional z-transfer function coefficients may be used effectively for the dynamic simulations of the heating and cooling loads if the whole building simulation program enables this type of wall data input. Method of derivation of the conduction z-transfer function coefficients from the response factors for three-dimensional wall assemblies is presented. It is assumed that the response factors, which represent surface heat flow due to triangular temperature excitations, at discrete time instants, may be calculated with the help of a computer code to simulate three-dimensional heat conduction. They are used as the “input data”, to determine z-transfer function coefficients from the set of linear equations, which includes relationships with the response factors and compatibility conditions. This infinite set of equations is to be solved applying cut-off and using minimization procedures. In terms of the response factors, heat flux across the interior surface of a wall element at time instant n d , Q_i,n _d , can be represented as follows [Kusuda 1969, Clarke 1985]:

(1)

where R is the resistance of the wall, {T_i,n _d} and {T_e,n _d} are sequences of the ambient temperatures values, and {X_n} and {Y_n } are sequences of the response factors. As far as three-dimensional problems are considered, all quantities are to be understood as average values over the surface of a wall element adiabatically cut from the surroundings.

The z-transform of the interior heat flux, Z{Q_i } is related to the z-transforms of the interior and exterior temperature, Z{T_i } and Z{T_e }, by the following equation [Jury 1964]:

(2)

where Z{X} and Z{Y} are the z-transforms of the sequences of the response factors X_n and Y_n.:

(3)

The condition, response factors X_n and Y_n should satisfy:

(4)

is equivalent to the following condition for the z-transforms Z{X} and Z{Y}:

(5)

Let now Z{X } and Z{Y} be given as the quotients:

(6)

where

(7)

b_n and c_n are the normalized conduction z-transfer function coefficients, which correspond to the coefficients b_n , c_n from the ASHRAE Handbook Fundamentals [1989, 1997] multiplied by R. M_b, M_c and M_d represent numbers of coefficients significantly different from zero.

Equation (2) can be rewritten in the form:

(8)

Equation (1) for Q _i,n _d , for n exceeding M_b, M_c and M_d, assuming d_o = 1, is now replaced by [Stephenson and Mitalas, 1971]:

(9)

Relationships (6) for the z-transforms, rewritten in the form:

(10)

are equivalent to the convolution type relationships between the response factors Y_n , X_n and conduction transfer function coefficients b_n , c_n , d_n :

(11)

The conditions (5), for Z{Y} and Z{X} now have the form:

(12)

which gives the following conditions for the z-transfer function coefficients:

(13)

Determining the z-transfer function coefficients from the response factors

On the basis of Equations (11) and (13) one may try to determine z-transfer function coefficients from the response factors Y_n , X_n . This is the most straightforward method; z-transfer function obtained in this way are expected to “exactly” reproduce the output for any input function composed of straight-line segments, joining the coordinates that correspond to its z-transform coefficients.

Assuming that values of the coefficients with indices above some n are negligibly small and d_o = 1, we obtain the following set of linear equations

	(14.1)
	(14.2)
	(14.3)
	(14.4)
	(14.5)

	(14.n)
	(14.n+1)
	(14.n+2)

	(15.1)
	(15.2)
	(15.3)
	(15.4)
	(15.5)

	(15.n)
	(15.n+1)
	(15.n+2)

When the structure factors are calculated together with the resistance and the response factors, one may use conditions imposed by the structure factors on the z-transfer function coefficients, see Equations (31) and (32) below, as subsidiary equations.

One may use more equations than the number of unknowns and apply minimizing procedures to get the solution. Maximum indices M_b, M_c, M_d, of the coefficients b_n , c_n, d_n, depend on the specific dynamic thermal properties of given wall assembly. In general, total number of the z-transfer function coefficients increases with the resistance and mass of the wall; however it is not the rule. Trying different kinds of the cut-off one should control the following quantity:

(16)

E_r is the relative error of the heat flux in steady state conditions, simulated using the z-transfer function method.

Solving back equations (11) for Y_n, X_n , with d_o = 1, gives the recurrence formulae, which may be used to additionally verify the solution obtained for the z-transfer function coefficients:

(17)

(18)