.BondLib.Examples.Biosphere.Cmoist

Information

Parameters:

  pao:     Atmospheric pressure (default value = 87.848 kPa)

  vai:     Air volume of Biosphere (default value = 35000 m3)

Comments:

This model presents maybe one of the nicest examples emonstrating the usefulness of bond graph modeling.

If you check the literature, you will notice that evaporation and condensation are not an area of physics that is very well ploughed. For example, our model uses Teten's law to compute the saturation vapor pressure. However, different authors either use different formulae or at least slightly different numerical parameter values. The simulation results obtained in this fashion differ by a few percent one from another.

Hence the student, who created the model, Francisco Luttmann, had a hard time finding good equations to mathematically capture the phenomena of evaporation and condensation. His model was pretty good, but he still made mistakes that he didn't find. In particular, his latent heat capacitance model was totally wrong. He had forgotten to multiply the capacitance value by lambda, i.e., his capacitance value was off by a factor of 2500.

When using bond graphs to describe evaporation and condensation, we first need to select the two adjugate variables. In fact, we only have one choice to make.

In accordance with Luttmann's model, I chose the humidity ratio as the effort variable, thus:

[e] = kg_water/kg_air

The units of effort are kg_water per kg_air.

Since the product of effort and flow must be power, and since we know that power is being expressed in this model as kJ/h, it follows that:

[f] = [P]/[e] = kJ.kg_air/(h.kg_water)

Since the bondgraphic resistance is defined by the equation:

e = R*f

it follows that:

[R] = [e]/[f] = h.kg_water²/(kJ.kg_air²)

Comparing these units with those used by Luttmann, we see that Luttmann's units were off by kg_water/kg_air. Hence the bondgraphic latent heat resistance must be multiplied by the humidity ratio, i.e., the effort variable, which is not very surprising, as the same happens in the thermal domain.

Since the bondgraphic capacitance is defined by the equation:

f = C*der(e)

it follows that:

[C] = [f].h/[e] = kJ.kg_air²/kg_water²

Since the bondgraphic resistor is being multiplied by e, the bondgraphic capacitance must be divided by e, such that the product of resistance times capacitance remains a time constant. Thus the units of the physical capacitance are:

[gamma] = [C]*[e] = kJ.kg_air/kg_water

However, Luttmann let his capacitance be described as air mass, i.e., the product of air density, da, and air volume, vai, thus:

[gamma] = [da]*[vai] = kg_air

Consequently, his units were off by kJ/kg_water, which are exactly the units of lambda, the latent heat of vaporization. Hence the equation was corrected to be:

gamma = lambda*da*vai

References:

1. Luttmann, F. (1990), A Dynamic Thermal Model of a Self-sustaining Closed Environment Life Support System, Ph.D. dissertation, Department of Nuclear and Energy Engineering, University of Arizona.