The number of b and d wall transfer coefficients

Abstract

There are three solutions to the Fourier continuity equation from which the parameters describing heat transmission through a building wall can be found directly: Solution (1)—a “slope” solution, Solution (2)—a sinusoidal solution, and Solution (3)—a transient solution. The heat flow q_io,0 “now” through a wall into an inside space held at zero temperature depends upon the outside temperature T_o,0 “now” and earlier values T_o,–δ, T_o,–2δ, T_o,–3δ,… (δ is usually taken as 1 h) and q_io,0 can be closely estimated through response factors as ∑_{k = 0}^∞δφ_kT_o,–kδ,. Alternatively, q_io,0 can be estimated from transfer coefficients as ∑_{k = 0}^N b_kT_o,−kδ −∑_{k = 1}^N d_kq_io−kδ. The φ_k values can be found directly, either from Solutions (1) and (3), or from Solution (2) alone. d_ks follow directly from the decay times z_j provided by Solution (3), together with choice of δ. The b_ks then follow from the φ_ks and the d_ks. The article sketches this sequence and with examples provides an estimate of the number N^′ of wall decay times needed to determine the set of d wall transfer coefficients and of the smaller number N + 1 of d coefficients and N + 2 of b coefficients involved in the summations; they depend upon the massiveness of the wall and the accuracy sought. This sequence differs from the classical approach to finding these quantities which is based on Solution (2), but in conjunction with Laplace and z-transforms; it is outlined in Appendix C.

Nomenclature

bk	=	transmission transfer coefficient (W/m2K)
dk	=	time-based transfer coefficient (—)
cp	=	specific heat (J/kg K)
c	=	thermal capacity per unit area of a wall element or layer (J/m2K)
C	=	thermal capacity per unit area of the entire wall (J/m2K)
C1, C2j	=	constants (W/m2)
e	=	elements in the wall transmission matrix (various)
i, j, k, n	=	integers
j	=	√(–1)
J	=	∑dk = ∏(1 – exp(δ/zj)) (—)
N	=	number that determines dN+1 < ϵ.J
N′	=	number that determines βN’ + 1 < ϵ.J
q	=	heat flux (W/m2)
r	=	thermal resistance of unit area of a wall layer (m2K/W)
R	=	thermal resistance of unit area of entire wall (m2K/W)
t	=	time (s, h)
T	=	temperature (K)
u	=	√(cr/z) (—)
U	=	thermal transmittance or U value (W/m2K) (= 1/R)
x	=	distance (m)
X	=	layer thickness (m)
zj	=	jth wall decay time (s, h)

Greek symbols

β	=	exp(–δ/z) (—)
δ	=	sampling time interval (s, h)
ϵ	=	accuracy (—)
λ	=	thermal conductivity (W/mK)
ρ	=	density (kg/m3)
φ	=	response factors (W/m2δK)

Subscripts

i	=	inside
i, j, k, n	=	integer variables
o	=	outside

Additional information

Notes on contributors

Morris Grenfell Davies

Morris Grenfell Davies, PhD, DSc, was a Reader.

Miren Karmele Urbikain

Miren Karmele Urbikain, PhD, is an Assistant Professor.