Mathematical Model

The heat transfer governing equations for steady state laminar natural convection in cavities are the mass, momentum and energy conservation equations in x, y and z axis for an incompressible fluid [Versteeg, 1995]. These equations can be expressed in conservative form:

Conservation of mass:

where T0 is the reference temperature, and is calculated by the mean temperature distribution of the exterior glass surface and its result is averaged with the temperature of the isothermal wall, so:

Hx, Hy y Hz are the lengths of the edge surfaces of the cubic cavity, Hgx is the thickness of the glass and Tci it is the temperature of the wall 2.

The boundary conditions for the momentum equation are:

u(0,y, z)= v(0,y, z)= w(0,y, z)= 0 u(Hx, y,z)= v(Hx, y,z)= w(Hx, y,z)= 0

u(x,0,z)= v(x,0,z)= w(x,0,z)= 0 (6)

u(x, Hy, z)= v(x, Hy, z)= w(x, Hy, z)= 0 u(x, y,0)= v(x, y,0)= w(x, y,0)= 0

u(x, y,Hz)= v(x, y,Hz)= w(x, y,Hz)= 0

The boundary conditions for the energy equation are:

Wall 1

— k“ (X’0’Z ) = ЧГ3 (X’0’ Z ) (7)


Wall 2

T(0,y, z)= Tci (8)

Wall 3

— ka T(x, Hy, Z ) = ЧГ3 ( Hy, z)


Wall 4

d — dT

— ka (Ях ’У’z)=~kg (Hx, y,z)h qr4(Hx, y, z)h Sa, f

Wall 5

— ka (x’ y ’ Hz ) = qr5 (x y ’ Hz )


Wall 6


— k a —(x’ y,0) = qr6(x ’ y,0)


where qr1(x,0,z), qr2(x, y,0), qr3(x, Hy, z), qr4(Hx, y,z), qr5(x, y,Hz) and qr6(x, y,0) are the energy flux from the radiative exchange between the wall surfaces, Saf is the absorbed energy by the solar control coating and Tg(x, y,z) is the glass temperature for Hx<x< Hx+Hx2, where


Hx is the thickness glass. The temperature gradients (Hx, y,z) in the glazing were


evaluated by using the heat conduction model.

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