Refined Model 54

A refined version of the model is needed in some situations, in particular when the balloon expansion is large or the device is operated in water. The balloon model is refined by generalizing the treatment of some phenomena and by relaxing some assumptions. The improved version of the model includes the following:

- expansion of the balloon volume with internal pressure,

- more general modeling of water permeation to include osmosis, - gas solubility into water.

In this section, we analyze each of these refinements.

The balloon volume is considered fixed in the first version of the model. This assumption is justified if the balloon expansion is negligible compared to its volume at rest. When parylene is used for the balloon wall, this assumption is usually acceptable. However, when silicone is used, the low elastic modulus allows the balloon to expand significantly, and the fixed volume assumption is not accurate. The variation of volume V

with pressure is approximated by a linear relationship





0 atm

( ) 1 ( )

V t V _  p t p _ (2-31)

where p is the balloon internal pressure, patm is the external atmospheric pressure, V0 is

and  is a proportionality constant. To be more accurate, a full finite-element simulation of the balloon expansion as a function of pressure should be done. However, the fabrication process produces large variations in the geometric parameters of the balloons, and these variations would strongly affect the results of the simulation. Therefore, the accuracy of the simulation would not be guaranteed to be better than a simple model like the linear approximation above. The dependence of the volume on pressure causes the equations to be much harder to solve. To simplify the solution, the volume-pressure relation above is not inserted directly into the mass balance equations. In each time step of the numerical solution of the mass balance equations, the balloon volume is considered fixed. The solution of the mass balance equations is then used to calculate the current balloon volume, which is updated for the following time step.

The modeling of water permeation is generalized. The objective is to extend the model to the case of a balloon containing an electrolyte solution and operating in water (or in an aqueous solution), where water can permeate in either direction due to osmosis. The permeation of water is assumed proportional to the chemical potential difference between the inside (H2Oi) and the outside (H2Oo) of the balloon





H2O( ) H2O H2O( ) H2O H2Oi( ) H2Oo( ) .

A A

f t c t c t t

h







    (2-32)

If an electrolyte (i.e., a solution of some solute in water) is present inside the balloon, the chemical potential of water inside the balloon is calculated using the equations for ideal and non-ideal solutions illustrated in Section B.1.3. The chemical potential of pure water can be taken as a reference, because we are only interested in the chemical potential gradients, and that term would cancel anyway when taking the difference

H2O H2Oi H2O H2O vh sol ( ) ( ) ln ( ) ln . ( ) ( ) n t t RT x t RT n t i n t        (2-33)

The term xH2O is the molar fraction of water in the liquid phase, and  is the activity

coefficient, which accounts for the non-ideality of the solution. The term nsol represents

the moles of solute in solution. The total amount of solute in the balloon is fixed, because no solute is assumed to permeate into or out of the balloon. However, the amount of solute actually in solution may be smaller, due to the solubility limit. The amount nsol

cannot be higher than the solubility of the solute in water, and it is, therefore, capped at that value, while the excess solute is assumed to precipitate in solid form (with negligible volume). The term ivh is the Van’t Hoff factor (indicated by the traditional i, with a

subscript to avoid confusion with current), which accounts for the dissociation of the solute molecules in water. The chemical potential of water is affected not only by the composition, but also by the pressure of the other gases in contact with water. However, as explained by Equation (2-29), the effect of pressure is negligible at the typical balloon operational pressure.

The chemical potential of water outside the balloon is calculated in different ways depending on whether the environment is a liquid or air. If the environment is a water solution, Equation (2-33) holds there too. If the environment is air, we consider the relative humidity instead of the molar fraction.

The permeability coefficient of water is calculated from the water vapor permeability given in literature. In literature, the permeation between a phase at 90% relative humidity and one at 0% relative humidity is usually measured. We calculate the difference in chemical potential between these two phases, and we divide the

permeability coefficient by this amount, to find the permeability coefficient per unit of chemical potential gradient. This is used as the coefficient cH2O in Equation (2-33).

In the first version of the model, O2, H2, and N2 were considered to be present

only in the gas phase. However, these gases are soluble in water, and they are present also in the liquid phase. The amount of each substance x is then split between the two phases

( ) ( ) ( ).

x xl xg

n t n t n t (2-34)

The subscript l indicates the liquid phase, whereas the subscript g indicates the gas phase. We assume that Henry’s law of absorption holds (Section B.1.3), so that the concentration of each gas in solution is proportional to its partial pressure. We can then write

H2O

( ) ( ) ( )

xl x x

n t S p t n t (2-35)

where Sx is the solubility coefficient of component x, expressed in moles of gas per moles

of water per unit of partial pressure. Assuming that Henry’s law holds at every instant is equivalent to assuming that the gases are always at equilibrium in the two phases. In other words, the equilibrium condition is assumed to be reached very quickly relative to the time scale of the other phenomena in the system. The volume of the liquid phase is considered to be unaffected by the presence of the gases in solution. The main effect of including gas solubility is that more gas is required to generate the same pressure increase, because some of the gas is stored in the liquid phase, where it does not contribute to the pressure.

The presence of water vapor in the gas phase should be included in the model as well. However, the vapor pressure of water at room temperature (or at human body temperature) is much lower than the partial pressure of the gases, and it can be neglected to simplify the equations.