Model 2 The Greenhouse as Flow Heat Exchanger
The greenhouse can be modeled as a heat exchanger using two gas flows (the air inside the greenhouse and the air outside it) with some ratio of capacity rates, Rc:
<math>R_{c} = \frac{{C_{min}}}{{C_{max}}}</math>
where Cmin and Cmax are the minimum and maximum capacity rates, respectively. The heat capacity rate is the maximum amount of heat a flowing fluid can absorb, equal to its specific heat at constant pressure, cp, times its mass flow rate, m’. The minimum capacity rate is whichever can absorb the least heat, and the maximum capacity rate is that of the other. Further, one gas stream would be the cold stream (the air inside) and the other the hot stream (the air outside), so the two capacity rates could also be written as Chot and Ccold.
This heat exchanger would have some effectiveness, e:
<math>e = \frac{{C_{hot} \cdot \Delta T_{hot}}}{{T_{hot in} - T_{cold in}}}</math>
where <math>\Delta</math>Thot is the change in temperature for the air between entering the greenhouse from its own heater and returning to the heater, Thot in is the temperature to which the greenhouse air must be heated to keep the average temperature acceptable, and Tcold in is the temperature of the martian air before it gets heated by contact with the greenhouse.
The heat transfer area number, or number of thermal units, of the heat exchanger is:
<math>NTU = \frac{{h_{overall} \cdot A}} {{C_{min}}}</math>
where hoverall is the overall heat transfer coefficient of the heat exchanger.
The density of the Martian atmosphere is sufficiently low that if the pressure in the greenhouse is near 1000mb, then:
<math>C_{max} = C_{hot}</math>
and
<math>\frac{{C_{min}}}{{C_{max}}} = 0</math>
In that case,
e = 1 – eNTU
which can be used to derive the overall heat transfer coefficient of the greenhouse.
See also:
