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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 qio,0 “now” through a wall into an inside space held at zero temperature depends upon the outside temperature To,0 “now” and earlier values To,–δ, To,–2δ, To,–3δ,… (δ is usually taken as 1 h) and qio,0 can be closely estimated through response factors as ∑k = 0∞δφkTo,–kδ,. Alternatively, qio,0 can be estimated from transfer coefficients as ∑k = 0N bkTo,−kδ −∑k = 1N dkqio−kδ. The φk values can be found directly, either from Solutions (1) and (3), or from Solution (2) alone. dks follow directly from the decay times zj provided by Solution (3), together with choice of δ. The bks then follow from the φks and the dks. 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.