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08.07 - Implementation of Departure Functions Click here. 20 2

Helmholtz Example - Modified vdW EOS (uakron.edu, 13min) A sample derivation of the Helmholtz departure implicit in the Gibbs departure given Z = 1 + abρ/(1+)^3 - (9.5aρ/RT)/(1+aρ/RT). Note that the limits of integration matter for this EOS. The audio is inferior for this live video, but it responds to typical questions and confusion from students in the audience. Some students might find it helpful to hear the kinds of questions that students ask. The responses slow the derivation down so that no steps are skipped and key steps are reiterated multiple times. Just turn the volume up!
Comprehension questions:
1. Which part of this EOS is non-zero at the zero density limit of integration?
2. Is there a sign error on one of the terms in this video? Check the derivation independently.
3. Derive the Helmholtz departure given Z = 1 + 4bρ/(1-)2 - (9.5aρ/RT)/{1-a/bRT[1-4bρ+4(bρ)2]}.
4. Derive the Helmholtz departure given Z = 1 + 4bρ/(1-2) - (9.5aρ/RT){1+4/bRT[1-2(bρ)2]}/{1-a/bRT[1-4bρ+4(bρ)2]})/{1-a/bRT[1-4bρ+4(bρ)2]}

08.07 - Implementation of Departure Functions Click here. 20 1

Helmholtz Example - vdW EOS (uakron.edu, 18min) This video begins with a brief review of the connection of the Helmholtz departure with all other departures then shows four sample derivations assuming that Z is given by the vdW EOS: (1) the Helmholtz departure , (2) the internal energy departure from the Helmholtz departure. (3) the Helmholtz energy from the internal energy (4) the Z factor from the Helmholtz departure. The procedures illustrated here can be applied to any EOS starting with any part (U, A, or Z) as given to derive any other departure: ZUHAGS.
Comprehension Questions: The virial EOS for SW fluids can be written as: Z = 1 + Bρ/RT where B = 4b+[4b(λ^3-1)] [exp(βε)-1], b = πNAσ^3/6.
1. Derive an expression for the Helmholtz departure.
2. Use the result of #1 to derive the internal energy departure.
3. Use the result of #2 to derive the Helmholtz departure. What is the integration constant in this case?

08.07 - Implementation of Departure Functions Click here. 20 2

Helmholtz Energy - Mother of All Departure Functions. (uakron.edu, 10min) This screencast begins with a brief perspective on energy and free energy as they relate to concepts from Chapter 1 and through to the end of the course. Then it focuses on how the Helmholtz departure function is one of the most powerful due to the relations that can be developed from it. The Helmholtz departure is relatively easy to develop from a density integral of the compressibility factor. Then the internal energy departure can be derived from a temperature derivative. Alternatively, if the internal energy departure is given, the Helmholtz energy can be inferred by integration, and the compressibility factor can be derived from a density derivative. 
Comprehension Questions: (Hint: some of the following may be answered in later videos below.)
1. Write an equation that takes you from the Helmholtz energy departure function to Z.
2. Write an equation that takes you from the Helmholtz energy departure function to (U-Uig)/RT.
3. Derive the internal energy departure function for the vdW EOS using Eqn. 8.22.
4. Derive the Helmholtz energy departure function for the vdW EOS using Eqn. 8.25.
5. Use the result of #4 to derive the internal energy departure function for the vdW EOS.

07.08 Matching The Critical Point Click here. 20 1

Visualizing the vdW EOS (uakron.edu, 16min) Building on solving for density, describes plotting dimensionless isotherms of the vdW EOS for methane at 5 temperatures, two subcritical, two supercritical, and one at the critical condition. From these isotherms in dimensionless form, it is possible to identify the critical point as the location of the inflection point where the temperature first exits the 3-root region. This method can be adapted to any equation of state, whether it is cubic or not. The illustration was adapted from a sample test problem. This screencast also addresses the meaning of the region where the pressure goes negative, with a (possibly disturbing) story about a blood-sucking octopus.

Comprehension Questions:

1. What are the dimensions of the quantity (bP/RT)?
2. Starting with the expression for Z(ρ,T), rewrite the vdW EOS to solve for the quantity (bP/RT) in terms of () and (a/bRT).
3. Consider the following EOS: Z = 1 + 2/(1-2) - (a/bRT) /(1-)2. Estimate the value of bPc/(RTc) for this EOS.
4. Consider the following EOS: Z = 1 + 2/(1-2) - (a/bRT) /(1-)2. Estimate the value of (a/bRTc) for this EOS.
5. Compute the values of a(J-cm3/mol2) and b(cm3/mol) for methane according to this new EOS.

09.10 - Saturation Conditions from an Equation of State Click here. 20 1

Solving for the saturation pressure using PREOS.xls simply involves setting the temperature and guessing pressure until the fugacities in vapor and liquid are equal. (5min, learncheme.com) It is not shown, but it would also be easy to set the pressure and guess temperature until the fugacities were equal in order to solve for saturation temperature. One added suggestion would be to type in the shortcut vapor pressure (SCVP) equation to give an initial estimate of the pressure. Rearranging the SCVP can also give an initial guess for Tsat when given P. This presentation illustrates a sample calculation for toluene to explore when the vapor is the stable, when the liquid is the stable phase, and when the phases are roughly in equilibrium.

Comprehension Questions:

1. Estimate the vapor pressure (MPa) of n-pentane at 450K according to the PREOS. Compare your result to the value from Eq. 2.47 (SCVP) and to the Antoine equation using the coefficients given in Appendix E. What do you think explains the observations that you make?
2. Estimate the saturation temperature (K) of n-pentane at 3.3 MPa according to the PREOS. Compare your result to the value from Eq. 2.47 (SCVP) and to the Antoine equation using the coefficients given in Appendix E. What do you think explains the observations that you make?
3. Estimate the vapor pressure (MPa) of n-pentane at 223K according to the PREOS. Compare your result to the value from Eq. 2.47 (SCVP) and to the Antoine equation using the coefficients given in Appendix E. What do you think explains the observations that you make?
4. Estimate the saturation temperature (K) of n-pentane at 3.3 kPa according to the PREOS. Compare your result to the value from Eq. 2.47 (SCVP) and to the Antoine equation using the coefficients given in Appendix E. What do you think explains the observations that you make?

05.2 - The Rankine cycle Click here. 20 1

Rankine Example Using Steam.xls (uakron.edu, 15min) High pressure steam (254C,4.2MPa, Saturated vapor) is being considered for application in a Rankine cycle dropping the pressure to 0.1MPa; compute the Rankine efficiency. This demonstration applies the Steam.xls spreadsheet to get as many properties as possible.

Comprehension Questions:

1. Why does the proposed process turn out to be impractical?

2. What would you need to change in the process to make it work? Assume the high and low temperature limits are the same. Be quantitative.

3. What would be the thermal efficiency of your modified process?

06.2 Derivative Relations Click here. 20 1

Assembling your derivative toolbox including the triple product rule, (uakron.edu, 13min) Beginning with the fundamental property relation, substitutions lead to Eqns. 6.4-6.7. Differentiating these and equating through exact differentials leads to Eqns. 6.29-6.32 (aka. Maxwell's Relations). Combining Maxwell's Relations with Eqns. 6.4-6.7 leads to Eqns. 6.37-6.41. With these tools in hand, and Eqn. 6.15 (aka. Triple Product Rule), you have all the tools you need to quickly transform any derivative into "expressions involving Cp, Cv, P, V, T, and their derivatives." This capability is fundamental to obtaining expressions for U, H, and S from any given equation of state for any chemical of interest. Four sample derivations are illustrated: (∂U/∂V)T, (∂T/∂S)V, (∂T/∂V)S, (∂S/∂V)A,

Comprehension Questions:

1. Transform the following into "expressions involving Cp, Cv, P, V, T, and their derivatives:" (∂T/∂V)S.

2. Transform the following into "expressions involving Cp, Cv, P, V, T, and their derivatives." Your expression may involve absolute values of S as long as they are not associated with any derivative. (∂T/∂U)P.

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