This is an attempt to map out a math model for the system of liquid wine in a pot open to the atmosphere at 1 bar, with a horizontal liquid surface of area A. The entire wine is assumed to have been brought to the operating temperature T, and the air above in the pot is also at T. So the time to preheat to the operating temperature is not included.
All the wine is assumed to be at constant water and ethanol concentrations spatially, except, possibly, in close proximity to the interface with the air (see below). The overall mass balances for the wine, and the mass balance for the ethanol component are written as follows:
$$\frac{dm}{dt}=-\left(k_W\frac{p_W}{RT}+k_E\frac{p_E}{RT}\right)A\tag{1}$$
$$\frac{d(mx)}{dt}=-k_E\frac{p_E}{RT}A\tag{2}$$where m is the total moles of wine in the tank, x is the mole fraction ethanol, the k's are the mass transfer coefficients of ethanol and water (cm/sec) to the air, the p's are the partial pressures of ethanol and water at the interface. Implicit in these equations is that the partial pressures of ethanol and water in the far-field air are negligible. The mass transfer coefficients are the largest uncertainties in this analysis, since I have not yet found correlations for these as a function of the operating conditions.
If we subtract Eqn. 1 from Eqn. 2, we obtain $$m\frac{dx}{dt}=-k_E\frac{p_E(1-x)}{RT}A+k_W\frac{p_Wx}{RT}A\tag{3}$$Eqns. 1 and 3 are our starting equations.
Let's next next turn to the Vapor-Liquid Equilibrium behavior which applies at the interface between the liquid and vapor. The first thing I wan to call attention to is the relationship between the equilibrium vapor pressure of pure ethanol and pure water. Over the temperature range of interest (60 C to 100 C), the equilibrium vapor pressure is almost exactly equal to 2.2 times the equilibrium vapor pressure of water: $$P^*_{E}(T)=2.2P^*_W(T)\tag{4}$$
@Dullard has pointed out that the VLE behavior of the ethanol-water system does not satisfy Raoult's Law. However, in the present region of system operation, at law mole fractions of ethanol (x < 0.1), the VLE behavior of this non-ideal system will approach: $$p_w=P^*_W(1-x)\tag{5}$$$$p_E=\gamma P^*_Ex\tag{6}$$where #\gamma# is the infinite dilution activity coefficient of ethanol in water. Experimental values of ##\gamma## reported in the literature (
http://www.ddbst.com/en/EED/ACT/ACT Ethanol;Water.php) are approximately 3.2. Therefore, combining Eqns. 5 and 6 gives: $$p_E=7.0P^*_Wx\tag{7}$$The figure below shows observed VLE behavior for the binary system ethanol-water at combined pressures ranging from 1/8 bar to 1/2 bar.
Eqns. 4 and 7 describe this behavior very accurately at ethanol mole fractions < 0.1. Therefore, we can confidently use these equations in our model calculation.
If we now substitute Eqns. 4 and 7 into Eqns. 1 and 3, we obtain:
$$\frac{dm}{dt}=-(k_W(1-x)+7k_Ex)A\frac{P^*_W}{RT}\tag{8}
$$$$m\frac{dx}{dt}=-(7k_E-k_W)Ax(1-x)\frac{P^*_W}{RT}\tag{9}$$
In Eqn. 9, if we neglect the value of x compared to 1, and neglect the change in m (as described In Eqn. 8) relative to its initial value ##m_0##, the equation reduces to $$\frac{d\ln{x}}{dt}=-\frac{1}{\tau}\tag{10}$$where the characteristic decay time ##\tau## is given by$$\tau=\frac{m_0}{(7k_E-k_W)A\frac{P^*_W}{RT}}\tag{11}$$
Based on this final approximate equation, the only thing left to do now is to estimate values of the mass transfer coefficients and provide an initial value for the initial number of moles ##m = m_0##. I will continueue unless there are substantial objections to the development I have presented.
@ArtZ, what is the initial volume of wine you intend to put in the 4.5 quart stock pot? Are you sure that stock pot is 12" in diameter?
