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Departure Function Overview (11:22) (msu.edu)
The philosophy and overall approach for using departure functions.

10.04 - Multicomponent VLE & Raoult's Law Calculations Click here. 66.6667 3

This example shows how to use VLookup with the xls Solver to facilitate  multicomponent  VLE calculations for ideal solutions: bubble, dew, and isothermal flash. (15min, uakron.edu) The product xls file serves as a starting point for multicomponent VLE calculations with activity models and for adiabatic flash and stream enthalpy calculations. This video shows sample calculations for the bubble, dew, and flash of propane, isobutane, and n-butane, like Example 10.1.

Note: This is a companion file in a series. You may wish to choose your own order for viewing them. For example, you should implement the first three videos before implementing this one. Also, you might like to see how to quickly visualize the Txy analog of the Pxy phase diagram. If you see a phase diagram like the ones in section 11.8, you might want to learn about LLE phase diagrams. The links on the software tutorial present a summary of the techniques to be implemented throughout Unit3 in a quick access format that is more compact than what is presented elsewhere. Some students may find it helpful to refer to this compact list when they find themselves "not being able to find the forest because of all the trees."

Comprehension Questions - Assume the reboiler composition for the column in Example 10.1 was zi={0.2,0.3,0.5} for n-butane, isopentane, and n-pentane, respectively.

a)  Calculate the temperature at which the boiler must operate in order to boil the bottoms product completely at 8 bars.
b)  Assuming the bottoms product liquid is in equilibrium with the liquid in the boiler, calculate the temperature of boiler and composition of the vapor in the boiler.
c)  Suppose this stream is to be boiled again and the vapor returned to the column with a ratio of 2 parts vapor to 1 product. (FYI: this is known as "boilup ratio.") Find the relevant temperature and compositions.

Wilson's model concepts (2:44) (msu.edu)

The background on the assumptions and development of Wilson's activity coefficient model.

Comprehension Questions:

1. What value is assumed by Wilson's model for the coordination number (z)?
2. What are the values of Λ21 and Λ12 at infinite temperature, according to Wilson's equation?
3. Solve for x1+x2Λ12 in terms of volume fraction (Φ1) and mole fraction (x1) at infinite temperature.
4. What type of phase behavior is impossible to represent by Wilson's equation?

This example shows how to incorporate activity calculations into Excel for solutions that follow the Margules 1-parameter (M1) model.(9min, uakron.edu)

You should be able to adapt this procedure along with the procedure for the multicomponent ideal solutions to create a multicomponent M1 model. If you are having trouble, the video for the multicomponent SSCED model illustrates a very similar procedure. You can check your answers by putting in the same component twice. For example, instead of an equimolar binary mixture, input a quaternary mixture with 0.25 moles of methanol, 0.25 methanol (ie. type it as if it was another component), 0.25 of benzene and 0.25 of benzene. If you don't get the same results as for the binary equimolar system, check your calculations.Note: This is a companion file in a series. You may wish to choose your own order for viewing them. For example, you should implement the first three videos before implementing this one. Also, you might like to see how to quickly visualize the Txy analog of the Pxy phase diagram. If you see a phase diagram like the ones in section 11.8, you might want to learn about LLE phase diagrams. The links on the software tutorial present a summary of the techniques to be implemented throughout Unit3 in a quick access format that is more compact than what is presented elsewhere. Some students may find it helpful to refer to this compact list when they find themselves "not being able to find the forest because of all the trees."

Comprehension Questions: Assume the SCVP model (Eq. 2.47).
1. Develop a Pxy diagram for the IPA+water system like Figure 10.8c, guessing values of A12 until you match the maximum pressure (azeotrope). What value of A12 did you find? (Hint: A12 is not the same as A12*RT.)
2. Develop a Pxy diagram for the acetone+chloroform system like Figure 10.9c, guessing values of A12 until you match the minimum pressure (azeotrope). What value of A12 did you find? (Hint: A12 is not the same as A12*RT.)
3. Develop a Pxy diagram for the acetone+acetic acid system like Figure 10.9a, guessing values of A12 until you match the pressure at x1=0.5 (305mmHg). What value of A12 did you find? (Hint: A12 is not the same as A12*RT.)

Principles of Probability I, General Concepts, Correlated and Conditional Events. (msu.edu, 17min) (Flash)
Comprehension Questions:
1. Estimate the probability of pulling an king from a randomly shuffled deck of 52 cards.
2. A coin is flipped 5 times. Estimate the probability that heads is observed three of the 5 times.
3. A die (singular of dice) is a cube with the numbers 1-6 inscribed on its 6 faces. If you roll the die 7 times, what is the probability that 5 will be observed on all 7 rolls?

Scatchard-Hildebrand Theory (6:53) (msu.edu)

Have you ever heard 'Like dissolves like'? Here we see that numerically. The Scatchard-Hildebrand model builds on the van Laar equation by using pure component information. Scatchard and Hildebrand replaced the energy departure with the experimental energy of vaporization. Because this is related to the 'a' parameter in the van Laar theory, they developed a parameter called the 'solubility parameter', but based it on the energy of vaporization. Interestingly, the model reduces to the one parameter Margules equation when the molar volumes are the same.

Comprehension Questions:

1. Based on the Scatchard-Hildebrand  model, arrange the following mixtures from  most compatible to least compatible.  (a) Pentane+hexane,   (b) decane+decalin,  (c) 1-hexene+dodecanol,   (d) pyridine+methanol,
Most compatible                                                                     Least compatible

_____                          ______                             ______                          ______

Principles of Probability II, Counting Events, Permutations and Combinations. This part discusses the binomial and multinomial coefficients for putting particles in boxes. The binomial and multinomial coefficient are used in section 4.2 to quantify configurational entropy. (msu.edu, 16min) (Flash) You might like to check out the sample calculations below before attempting the comprehension questions.
Comprehension Questions:
1. Write the formulas for the binomial coefficient, the multinomial coefficient, and the multinomial with repetition.
2. Ten particles are distributed between two boxes. Compute the number of possible ways of achieving 7 particles in Box A and 3 particles in Box B.
3. Note that the binomial distribution is a special case of the multinomial distribution where the number of categories is 2. Also note that the total number of events for a multinomial distribution is given by M^N where M is the number of categories (aka. outcomes, e.g. boxes) and N is the number of objects (aka. trials, e.g. particles). The probability of a particular observation is given by the number of combinations divided by the total number of events. Compute the probability of observing 7 particles in Box A and 3 Particles in Box B.
4. Ten particles are distributed between three boxes. Compute the probability of observing 7 particles in Box A, 3 particles in Box B, and zero particles in Box C.
5. Ten particles are distributed between three boxes. Compute the probability of observing 3 particles in Box A, 3 particles in Box B, and 4 particles in Box C.

The van Laar Equation (5:54) (msu.edu)

The van Laar equation uses the random mixing rules discussed in Section 12.1 with the internal energy to approximate the excess Gibbs Energy. What we learn is that it is possible to develop models using fundamental principles. Though this model is not used widely in process simulators, it provides a stepping stone to more advanced models.

Txy Phase Diagram Showing LLE and VLE Simultaneously (9min,uakron.edu)

The binary Txy phase diagram of methanol+benzene is visualized with sample calculations of the SSCED model with several values of the nonideality (kij) parameter. The calculations show the liquid-liquid equilibrium (LLE) phase boundary as well as the vapor-liquid equilibrium (VLE) boundary. As the estimated nonideality (kij) increases, the LLE boundary crashes into the VLE. It is so exciting that it makes a thermo nerd wax poetic about the "valley of Gibbs."

Comprehension Questions:

1. The LLE phase boundary moves up as the nonideality increases. Which way does the VLE contribution move? Explain how this relates to the molecules' escaping tendencies.
2. How would this phase diagram change if the pressure was increased to, say, 10 bars?
3. What value of kij is required to make the LLE binodal barely touch the VLE at 1 bar?
4. What value of kij is required to make the LLE binodal barely touch the VLE at 10 bars?

09.10 - Saturation Conditions from an Equation of State Click here. 60 1

We can combine the definition of fugacity in terms of the Gibbs Energy Departure Function with the procedure of visualizing an equation of state to visualize the fugacity as characterized by the PR EOS. (21min, uakron.edu) This amounts to plotting Z vs. density, similar to visualizing the vdW EOS. Then we simply type in the departure function formula. Since the PR EOS describes both vapors and liquids, we can calculate fugacity for both gases and liquids. Taking the reciprocal of the dimensionless density ( V/b=1/(bρ) ) gives a dimensionless volume. When the dimensionless pressure (bP/RT) is plotted vs. the dimensionless volume, the equal area rule indicates the pressure where equilibrium occurs and this can be checked by comparing the ln(f/P) values for the liquid and vapor roots. When the pressure is not exactly saturated, we may still be in the 3-root region. Then you need to check the fugacity to determine which phase is stable.

Concept Questions:

1. What equation can we use to estimate the fugacity of a compressed liquid relative to its saturation value?
2. How accurate is that equation relative to the change in pressure when we are close to saturation?
3. The video shows a graph of ln(f/P) vs. P. Which phase gives the lower value of fugacity when you are to the right of the intersection point? (ie. vapor or liquid?)