## Estimate Thermodynamic Parameters from Data

#### Thermodynamic Parameter Estimation

Using data from an ebulliometer, determine parameters for the Wilson activity coefficient model using the measured data for an ethanol-cyclohexane mixture at ambient pressure. Use the results to determine whether there is:

- An azeotrope in the system and, if so, at what composition
- The values of the activity coefficients at the infinitely dilute compositions
- gamma
_{1}at x_{1}=0 - gamma
_{2}at x_{1}=1

- gamma

The liquid and vapor compositions of this binary mixture are related by the following thermodynamic relationships

where *y _{1}*is the vapor mole fraction,

*P*is the pressure,

*x*is the liquid mole fraction,

_{1}*gamma*is the activity coefficient that is different than 1.0 for non-ideal mixtures, and

_{1}*P*is the pure component vapor pressure. The same equation also applies to component 2 in the mixture with the corresponding equation with subscript 2.

_{1}^{sat}The Wilson equation is used to predict the activity coefficients *gamma _{2}* and

*gamma*over the range of liquid compositions.

_{2}There are correlations for *P ^{sat}_{1}* and density (

*rho*) for many common pure components from BYU's DIPPR database. In this case

*P*is a function of temperature according to

_{1}^{sat}and density *rho* or molar volume *v* is also a function of temperature according to

The number of degrees of freedom in a multi-component and multi-phase system is given by DOF = 2 + #Components - #Phases. In this case, there are two phases (liquid and vapor) and two components (ethanol and cyclohexane). This leads to two degrees of freedom that must be specified. In this case, we can chose to fix two of the four measured values for this system with either *x _{1}*,

*y*,

_{1}*P*, or

*T*. It is recommended to fix the values of

*x*and

_{1}*P*as shown in the tutorial below.

#### Background on Parameter Estimation

A common application of optimization is to estimate parameters from experimental data. One of the most common forms of parameter estimation is the least squares objective with (model-measurement)^2 summed over all of the data points. The optimization problem is subject to the model equations that relate the model parameters or exogenous inputs to the predicted measurements. The model predictions are connected by common parameters that are adjusted to minimize the sum of squared errors.

#### Tutorial on Parameter Estimation

#### Nonlinear Confidence Interval

Nonlinear confidence intervals can be visualized as a function of 2 parameters. In this case, both parameters are simultaneously varied to find the confidence region. The confidence interval is determined with an F-test that specifies an upper limit to the deviation from the optimal solution

with p=2 (number of parameters), n=number of measurements, theta=[parameter 1, parameter 2] (parameters), theta^{*} as the optimal parameters, SSE as the sum of squared errors, and the F statistic that has 3 arguments (alpha=confidence level, degrees of freedom 1, and degrees of freedom 2). For many problems, this creates a multi-dimensional nonlinear confidence region. In the case of 2 parameters, the nonlinear confidence region is a 2-dimensional space. Below is an example that shows the confidence region for the dye fading experiment confidence region for forward and reverse activation energies.

The optimal parameter values are in the 95% confidence region. This plot demonstrates that the 2D confidence region is not necessarily symmetric.

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