Inside temperature

The ultimate aim is to determine the steady state inside temperature of the Desert Fridge for a given set of external (outside temperature, humidity etc) and internal (material conductivity, thickness etc) parameters.
Once we have know the steady state mass flux through the Desert Fridge (n.) we can take this as the mass of water vapourising (D. K. Edwards, 1978, p. 145) and therefore determine the heat transfer due to vaporization by:

Where:
q._evap = heat flux by evaporation (W/m2)
n.= mass flux of vapourising water (Kg/m2.s)
L = latent heat of vapourisation which, for water, is given by the following function of temperature (evaluated at the wet bulb temperature in our case I think):

Lwater(T) = 0.0000614342T3 - 0.00158927T2 + 2.36418T - 2500.79 (Yau, 1989)

As the water vapourises at the surface of the outer pot it will draw heat () away from the air inside the inner pot (it may also draw some part of this heat from the outside air by convection and radiation but lets keep it simple for now) which will cause the temperature of the inside the inner pot to decrease thus setting up a temperature gradient from the outside to the inside which will drive a heat flux in the reverse direction.
Eventually the heat fluxes in the opposite directions will reach equilibrium at which point, the surface of the outer pot maintained at the wet bulb temperature of the outside air (Kreith) (which can be determined from a psychometric chart for a given dry bulb and humidity) as shown in the diagram below:

The heat transfer in the reverse direction (q_back) can be determined using the standard method for heat transfer through a composite slab (Wong, 1997, p. 14). This requires the overall heat transfer coefficient which in this case is given by;

Where:
U = overall heat transfer coefficient (W/m2)
h = film (convective) heat transfer coefficient (W/m2)
k = conductive heat transfer coefficient (W/m)
x= thickness (m)

Then the heat flux due to the temperature difference between the outside (T_h) and inside (T_c) air is given by Newton’s law of heating i.e.

Now, although q_evap and q_back will be in steady state equilibrium, this does NOT mean they must be equal (this would imply no temperature difference), it means that the net heat flux will be constant which is given by:

Equation 1

To find a useful expression for q_net we use the same theory for heat transfer through compound walls as used above but here the heat transfer is from the inside air to the vapourising water at the outer surface (at T_wb) opposed to the outside air (at T_h) as before – so we can exclude the h_2 term (which is the convective heat transfer coefficient from outer surface to outside air) so we have:

Where;

Now substituting the expressions derived for q_net, q_evap and q_back, back into equation 1 gives:

Rearranging this equation to make T_c the subject gives:

Bingo!... only problem we still need to know mass flux

Works Cited

D. K. Edwards, V. E. (1978). Transfer Processes: An Introduction to Diffusion, Convection and Radiation. Hemisphere Publishing Corporation.
Yau, R. &. (1989). A Short Course in Cloud Physics. Pergamon press.