Frequently Asked Questions about the ASTEC Systems

8. How do you compare mathematical equations with established ASHRAE engineering formulations?

DEMONSTRATING ASHRAE AND ICC/ASTEC SURFACE TEMPERATURE EQUATION EQUIVALENCE

    ASHRAE 1989 Fundamentals Handbook
    Chapter 26. Air-Conditioning Cooling Load.

Page 26.3: Heat Gain Through Exterior Walls and Roofs
Heat Gain Through Exterior Surfaces. The heat balance at a sunlit surface gives the heat flux into the surface q/A in BTU/h ft², as:

q/A = αI_t + h₀(t₀-t_s)-εδR

Where
    σ         =     absorptance of the surface for solar radiation
    I_t        =     total solar radiation incident on the surface, BTU/h*ft²
    h₀       =     coefficient of heat transfer by long wave radiation and convection at the outer surface, BTU/h*ft²*°F
    t_s       =     roof or wall surface temperature, ˚F
    t₀        =     outdoor air temperature, ˚F
    ε         =     hemispherical emittance of the surface
    δR     =     difference between the long wave radiation incident on the surface from the sky and surroundings and the radiation emitted by a blackbody at outdoor air temperature, BTU/h*ft²"

Analysis of ASHRAE terms /definitions relative to the terms /definitions used in the ICC/ASTEC report.

ASHRAE		ICC/ASTEC
q/A	->	conduction heat flux, q" conduction = K/L (T_surface –T_inside )
It	->	solar irradiation, G
α	->	solar absorptivity, α = 1 - p
t0	->	outside air temperature, T_air
ts	->	surface temperature, T_surface
ε	->	thermal emissivity, ε_thermal
ho	->	convective coefficient, h_film, plus a "radiation coefficient' not used in report, but assume its name "h_r", h_o = h_film + h_r
σR	->	radiation from surrounding surface (σT⁴_surr ) minus air temperature blackbody radiation (σT⁴_air), σR=σT⁴_surr - T⁴_air

Substituting the ICC/ASTEC values into the ASHRAE equation gives:

q"conduction = (l-_P)G + (h_film+h_r)(T_air-T_surf) - εthermal (σT⁴_surr - T⁴_air)

Substituting with: q"conduction = K / L (T_surf – T_inside) ; and expanding:

(h_film+h_r)(T_air-T_surf) = h_film(T_air-T_surf) + h_r(T_air-T_surf) where the radiation heat transfer coefficient, hr, is defined1 as:

h_r=εσ(T⁴_air-T⁴_surf)/(T_air-T_surf)

and substituting this gives:

(h_film+h_r)(T_air-T_surf) = h_film(T_air-T_surf) + ε_thermalσ(T⁴_air - T⁴_surf)

Substituting the above into the top equation:

K/L(T_surf - T_inside) = (l-p)G + h_film(T_air - T_surf) + εσT⁴_air - εσT⁴_surf - ε_thermalσ T⁴_surr + ε_thermalσ T⁴_air

If it is assumed that T_surr ≈ T_air then ε_thermalσ T⁴_surr ≈ ε_thermalσ T⁴_air and the equation reduces to the form:

(1-p)G = ε_thermalσ (T4_surf- T⁴_air) + h_film (T_surf- T_air) + K/L (T_surf - T_inside)

As shown, by starting with the ASHRAE equation in the Fundamentals Handbook, the equivalence between the two equations can be demonstrated.

1 P. 281; Principles of Heat Transfer, Frank Kreith, 3rd Ed., 1973, Intext Press; ISBN 0-7002-2411-X

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