# Model 3 A Radiation Cooled Greenhouse

In this model, the greenhouse wall is modeled as a homogenous surface exposed to a concentric outer surface at the uniform temperature of the Martian environment. Radiative heat transfer alone is considered. Convective and conductive heat transfer are neglected.

## The Equations

The greenhouse can be modeled as being composed of three sections of single layer walls separating two thermal reservoirs. In this model, the only relavent thermal resistances are in parallel, so that the total thermal resistance is a mean resistance. Thermal effects at the interfaces are neglected.

Since a half cylinder greenhouse is relatively easy to build, we will assume this is the shape of our greenhouse. The three sections considered are the floor, the overhead arch, and the two end walls (which are treated as a single section).

The radiative heat transfer through each surface is:
$Q_{s} = A_{s} \epsilon_{s} \sigma_{b} ( T_{i}^{4} - T_{o}^{4} )$
$Q_{s} = {\Delta T \over R_{s}}$

$R_{s} = {\Delta T \over A_{s} \epsilon_{s} \sigma_{b} ( T_{i}^{4} - T_{o}^{4} )}$

where:
Qs is the thermal resistance of the individual surface
As is the surface area
$\epsilon_{s}$ is the emissivity of the surface
Ti is the internal temperature of the greenhouse
To is the outside temperature of the MArtian environment
$\sigma_{b}$ is the Stephan-Boltzman constant ($5.67 \cdot 10^{-8} {W \over m^{2} K^{4}}$)
$\Delta T$ is the difference between the internal and external temperatures
Rs is the overall thermal resistance of the surface

There are three thermal resistances in parallel, one for each surface. Their overall thermal resistance is:

$R_{total} = {R_{floor} R_{arch} R_{ends} \over R_{floor} + R_{arch} + R_{ends}}$

The total heat flow rate can be computed from this:

$\Sigma Q = {\Delta T \over R_{total}}$

where:
$\Sigma$Q is the total heat transfer rate
Rtotal is the total thermal resistance
Rfloor is the thermal resistance of the floor
Rarch is the thermal resistance of the overhead arch and windows
Rends is the combined thermal resistance of the end surfaces

For sufficiently thin walls at the temperatures involved for a Martian greenhouse, this heat transfer is primarily from the greenhouse to the Martian air. During the day, heat can be absorbed from sunlight to offset this heat transfer rate out of the greenhouse.

$Q_{solar} = e C_{solar} A_{collector}$

where:
Qsolar is the solar heat absorbed by the greenhouse’s thermal collectors (not necessarily contained within the greenhouse itself)
Csolar is the insolation of the collector
e is the efficiency of the collector

## Daytime Example

Begin by assuming that the greenhouse foundation is lined with reflective foil on both the greenhouse subfloor and ground. Reflective foil has a very low emissivity, and because heat transfer is effectively from surface to surface, using foil on the ground lowers its emissivity as well.

Assume the greenhouse walls are thin, uninsulated polyethylene sheeting.

Let the dimensions of the greenhouse be:
r = 4 m
z = 8 m

Neglect wall thickness.

This implies that:
Afloor = 64 m2
Aends = 50.3 m2
Aarch = 100.5 m2

Let the external and internal conditions be:
Ti = 20 oC
To = -30 oC
$\epsilon_{floor} = 0.05$ (Double-layered Reflective Aluminum Foil)
$\epsilon_{ends} = 0.1$ (Single-layered Reflective Aluminum Foil)
$\epsilon_{arch} = 0.9$ (Single-layered Polyethylene Film)
Csolar = 440 W/m2
e = 0.10

If the area of the greenhouse floor is used as the collector area, the solar heat collected is Qsolar = 2810 W.

The total heat transfer through all three surfaces is $\Sigma Q$ = 21730 W, implying a total thermal resistance of Rtotal = 0.0023 W/K. This is equivalent to only R1.5 insulation in English units, allowing 7.7 times more heat to escape through the greenhouse window than is absorbed from sunlight.

Single layer polyethylene window material simply allows too much heat to escape. 8 times the solar thermal collector area would be required to operate a greenhouse in this configuration, which is impractical.

This can be alleviated by changing the window/arch material to one with a lower emissivity. Transparent aramid plastics such as vectran are available with emissivities as low as 0.5. Using one of those in this system could reduce the heat loss rate by almost half, but would still allow too great a heat loss rate for solar heating to be effective.

Decreasing the window area could also be effective. Reducing the arch area by half would also reduce the heat loss by almost half, with only a slight reduction of the illuminated area inside the greenhouse.

A third approach could be to reduce the relative temperature. If the greenhouse bulk internal temperature were maintained at only 4 oC rather than 20oC, the reduction in radiated heat would also be nearly 40%.

Taken together, these three improvements – use aramid windows, decrease window area, and use a bulk temperature less than the optimum – can decrease the greenhouse heat loss to 4540 W. This is still 1.6 times what is taken in by the solar collector, but a larger collector can be used to compensate.

The less than optimum bulk temperature can be compensated for by employing cold frames within the greenhouse, effectively creating a greenhouse within a greenhouse.

If the transmission of light through the aramid arch is only 70%, the insolation of the cold frames is only 340 W/m2. However, because they operate in a warmer environment, collect heat with better overall efficiency, and can be made with lower emissivity materials such as metal and fiberglass sheeting, they can have emissivities as low as 0.3, heat loss densities as low as 34 W/m2, and solar heat gains of 68 W/m2 or more. These cold frames won’t require pressure walls, and can be built with much greater relative collector area – so much so that they may be able to absorb enough solar heat during the Martian day to not only prevent their freezing over but remain comfortably at 20 oC. They can even accumulate excess heat for the night.

## Nighttime Example

Temperatures drop precipitously on Mars at night due to the thin atmosphere.

For the greenhouse described above, let:
To = -50 oC
Ti = 0 oC
Qsolar = 0 W

In this case, the heat flow rate out of the main greenhouse, given the adaptations listed above, would accelerate to 5830 W. Note that the bulk internal temperature is maintained at roughly the freezing point of water. If water is used as the thermal mass of this greenhouse, it will eventually begin to freeze. This phase transition would release latent heat at a rate of 220000 J/kg, stabilizing the temperature near the thermal mass at the freezing point. At this heat loss rate, 1 kg of thermal mass will freeze every 40 seconds. The addition of 10T water would provide sufficient latent heat to continue at this rate for nine days before the phase transition was complete and the bulk temperature finally fell below freezing.

The total energy required to remelt the ice generated would be enormous. However, this internal environment is not substantially different from that in the greenhouse during the daytime – there is only 4 oC difference. Well insulated cold boxes can be adapted to function in this environment, without allowing their internal temperatures to drop below 10 oC. Plants can survive this degree of temperature drop if it occurs slowly and in darkness.

## Conclusions of this Model

The rate of heat transfer by radiation is quite large – easily the largest component of heat loss from the greenhouse. The radiative heat loss is so large that solar heating alone is insufficient to maintain the entire greenhouse at optimal growing temperatures without solar thermal collectors extending beyond the greenhouse.

It is advantageous to use windows with as small an area as possible, made of low emissivity materials. Also, if internal cold boxes are used to maintain optimum temperatures near the plants instead of trying to keep the entire greenhouse structure at plant grown temperatures, the rest of the greenhouse can be maintained at temperatures which are near freezing but still livable.

A larger greenhouse should also be considered, as it would have a larger floor area for collecting solar heat and could contain more thermal mass.

## Limitations of this Model

This model neglects conductive and convective heat transfer, which are substantial both inside and outside the greenhouse. The proper use of cold boxes will depend on the convective conditions within the greenhouse, which is neglected in this model.

Discussion of other models for greenhouse thermodynamics can be found under Thermodynamics of the greenhouse.