Modeling and Design of Net ZEBs as Integrated Energy Systems - Modeling, Design, and Optimization of Net-Zero Energy Buildings - page 23

The boundary conditions will include convective heat transfer, absorbed
solar radiation (a heat source), and longwave radiation exchange with other
surfaces.
After taking the Laplace transform of Eq. (2.1) and some algebra, the
equations for the conditions at the two surfaces may be expressed in the
so-called cascade equation matrix form (Beccali et al. , 2005b) as follows
(assuming heat flux q is positive into the wall on both sides):
(2.1a)
The parameter k is the thermal conductivity, L is thickness, γ is equal to
( s / α ) 1/2 and s is the Laplace transform variable. For frequency domain
analysis, including admittance calculations, s is set equal to jω ( s = jω )
where j = √−1 and ω = 2 π / P . For diurnal analysis, the period P = 86,400
s. For a multilayered wall we can multiply the cascade matrices for each
successive layer to get an equivalent wall cascade matrix that relates
conditions at one surface of the wall to those at the other surface, thus
eliminating allintermediatenodeswithnoapproximationrequiredandno
discretization:
(2.1b)
The effective cascade matrix of the wall is expressed as
(2.1c)
The cascade matrix for a simple conductance (per unit area), u , can be
shown to be given by
Usually, the interior surface temperatures of the room are of primary
interest. Consider for example a wall made up of an inner (room side)
storagemasslayerandinsulationontheexteriorasshownin Figure2.3 .The
effective cascade matrix can be represented by