The process design division of your company has asked you to mathematically model to describe vapor-liquid equilibria to enable them to design large scale separation columns for binary systems. In particular, your company wants to separate ethanol from water. They need a guideline for relating x_i, y_i, T and p necessary for process simulation. Alternatively, they wish to be able to set up T-x-y, p-x-y and x-y diagrams.

Procedure

1. We will first find experimental data for vapor liquid equilibria for ethanol-water system. Do the internet search and locate some experimental data T-x-y at a pressure of 1 atm. Since most distillation processes are performed at atmospheric pressure, experimental T-x-y data at 1 atm pressure will be ideal for our work.

2. We will use these data to calculate experimental activity coefficients 1 and 2 at different xs and ys. Remember ₁ (T, P, x₁, x₂) = ₁ P/x₁P₁^Sat(T), ₂ (T, P, x₁ ,x₂) =₂ P/x₂P₂^Sat(T)

3. In the above equations, we have assumed that the vapor phase behaves as an ideal gas.

4. Use just determined s to calculate experimental molal excess Gibbs Free Energy G ^E (T, P, x₁, x₂) = RT [x₁ ln ₁ (T, P, x₁, x₂)] + x₂ ln ₂ (T, P, x₁, x₂)].

5. In addition to tabulating the values of s, x, y and T, you should also tabulate the values of G^E/RTx₁x₂ and RTx₁x₂/G^E .

6. Now you have experimental values of G^E . You wish to fit G^E to three mathematical models. These are 1) one parameter Margules model ( G^E/RTx₁x₂ = A), 2) two-parameter Margules model [G^E/RTx₁x₂ = A + B(x₁ -x₂)] and 3) two-parameter van Laar model [RTx₁x₂/G^E = A + B(x₁ x₂)

7. Now perform the least square fits and determine the best values of A for one parameter Margules model, A and B for two parameter Margules model and A and B for van Laar model.

8. The above models for G^E do not contain pressure and temperature explicitly. In general, the effect of pressure is negligible. However, temperature does have a significant impact. The parameters in the above models are indeed temperature dependent. Fortunately, the temperature effect on vapor liquid equilibria primarily comes through the temperature dependence of the pure component vapor pressures in the above equations. Typically, a temperature increase of 10C raises the vapor pressure by a factor of 2. However, the activity coefficients over the same temperature change by only a few percentage points which is within the experimental uncertainty. As a result, the temperature dependence of G^E is neglected too. Please keep in mind that this will not apply to liquid-liquid equilibria as no vapor pressures would be involved there.

9. Now you are ready to predict vapor liquid equilibria for a wide range of conditions. Please remember, you cannot expect very good results for conditions that are far removed from the conditions of underlying experimental data. Also please remember, these models do not explicitly contain T and P.

10. The procedure is outline below:

a. Choose a mathematical model for G^E (T, P, x₁, x₂).

b. Determine the parameters of that model from the limited experimental data as follows.

c. Use modified Raoults law to determine the numerical values of ₁ and ₂.

d. Use Equation (1) to calculate numerical values of G^E at different experimental values of liquid compositions.

e. Try and fit these values to three models: 1) Margules one constant model, 2) Margules two-constant model and 3) two-constant van Laar model.

f. Determine the associated parameters from these linear fits or least squares analysis.

g. Construct p-x-y and T-x-y diagrams. The procedures will be discussed in class.

11. Mathematical models for G^E :

a. One parameter Margules Model The model contains only one parameter, A.

^E/RTX₁x₂= , ₁ = ²₂ , 2= ²₁

b. Two parameter Margules Model This model has two parameters A and B.

G^E/RTx₁x₂ = + (x₁ x₂)..(1),

G^E/RTx₁x₂ = (x₁ + x₂) + (x₁ + x₂)..(2),

G^E/RTx₁x₂ = ( + )x₁ + ( )x₂ ..(3),

G^E/RTx₁x₂ = (₂₁)x₁ + (₁₂)x₂ (4),

₁ = x^2₂ (₁₂ + 2(_{21 12})x₁ (5),

₂ = x^2₁ (₂₁ + 2(_{12 21})x₂ (6)

c. Two parameter van Laar Model The model has two parameters A and B.

RTx₁x₂/G^E = A +B(x₁ - x₂ ). (7)

RTx₁x₂/G^E = A(x₁ + x₂) + B(x₁ - x₂ ). (8)

RTx₁x₂/G^E = (A +B)x₁ + (A- B))x₂ (9)

RTx₁x₂/G^E = (1/A₂₁)x₁ + (1/A₁₂)x₂ (10)

₁ = A₁₂ {1 + (A₁₂ x₁/ A₂₁ x₂)} ²(11)

₂ = A₂₁ {1 + (A₂₁ x₂/ A₁₂ x₁)} ². (12)

12. Fit the G^E/RTx₁x₂ data to all three models and determine the best values of the parameters for each case. Then you can use the corresponding equations to evaluate s over the entire composition range. This will enable you to calculate T-x-y and P-xy diagrams over the full composition range.

13. Adopt the usual convention that component 1 is the more volatile component than 2. It implies that it has a higher vapor pressure at a given temperature than the component 2. You can calculate the vapor pressure of any chemical specie from Antoine equation ln P_i^Sat = _{i i} /C_i+ T Here the values of A, B, C can be looked up from any reference book. Just be certain to use appropriate units as specified by the source.

14. In cases where the binary mixture forms an azeotrope, one may be able to use the azeotrope information (T, P, azeotropic composition) to ones advantage to calculate the model parameters. At azeotropic composition, x₁ = y ₁and x₂ = y₂. Hence

₁ ^az(, , x₁ x₂ ) = /P₁^Sat

₂ ^az((, , x₁ , x₂ ) = /P₂^Sat

Hence if you know the azeotropic composition and T and P of the azeotrope, one can easily calculate the activity coefficients of both components at the azeotropic conditions. One can then use these s to calculate the parameters.

a. One parameter Margules Model = ln₁^az/x₂² or 6 = ln₂^az/x₁² If the As turn out to be different, which one would you choose? Since the formation of an azeotrope is a highly non-ideal effect, and the activity coefficient is a measure of non-ideality, one may assume that the larger value of |_i | carries more significant information. Hence choose A coming from that component.

b. Two parameter Margules Model Here the model parameters can be easily computed from the azeotropic point information

₁₂ = [2 (1 /x₂)] ₁^az /x₂ + 2₂^az /x₁,

₂₁ = [2 (1 /x₁)] ₂^az /x₁ + 2₁^az /x₂

Once A₁₂ and A₂₁ are known, you can use Equations 5 and 6 to calculate s at any composition.

c. Two parameter van Laar Model Here the model parameters can be easily computed from the azeotropic point information

A₁₂ = ₁^az{1 + (x_{2 . 2}^az /x_{1 . 1}^az)}² ,

A₂₁ = ₂^az{1 + (x_{1 . 1}^az /x_{2 . 2}^az)}

Once these parameters are known, you can use Equations (11) and (12) to compute s at any value of x.

15. Construction of T-x-y diagram at a fixed P and P-x-y diagram at a fixed T These will be discussed in class.

16. Check for thermodynamic consistency of the experimental The measured experimental data must conform to data. The Gibbs-Duhem Equation has to be satisfied. x_i[d ln _i (T, P, x₁, x₂)] = 0, Of course, for the binary system, the above equation reduces to x₁ [d ln ₁ (T, P, x₁, x₂)] + x₂ [d ln ₂ (T, P, x₁, x₂)] =0 The activity coefficients ₁ and ₂ in their common mixture are not completely independent of one another. You can integrate the above equation numerically to check if the sum of the two terms is equal to zero or close to zero.

for given value of T-x-y foe ethanol and water system at 1 atm:

T C: 95.2, 91.8, 87.3, 84.7, 83.2, 82,81, 80.1, 79.1, 78.3, 78.2, 78.1, 78.2, 78.3

X: 0.05, 0.1, 0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,0.94, 0.96,0.98,1

y 0.377, 0.527, 0.656, 0.713, 0.746, 0.771, 0.794, 0.822, 0.858, 0.912, 0.942, 0.959, 0.78, 1

please show the calculation and graph