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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.

09.03 - Shortcut Estimation of Saturation Properties Click here. 20 1

Shortcut estimation of thermodynamic properties (sample calculation) can be very quick and sometimes reasonably accurate.(6min, uakron.edu) As a follow-up exercise, it is suggested to adapt the shortcut vapor pressure equation in combination with Eqn. 2.45 and the pathway of Fig. 2.6c to rapidly estimate stream properties. Briefly, all you need is an "IF" statement that checks whether the T is less than Tsat at the given P. If so, then H=Href+CpΔT+Hvap. If not, then H=Href+CpΔT. This can be a quick and convenient method to estimate stream attributes of a process flow diagram. One equation per cell and you're done. This sample calculation illustrates the process for the heat duty of a butane vaporizer and compares the PREOS to the methods of Chapter 2 (ie. Eq. 2.45 etc.)

Comprehension Questions: Suppose you want to tabulate the entropy (S) of your stream attributes by this approach.
1. How would you compute the Sig(T,P)-Sig(Tref,Pref) contribution?
2. How would you compute ΔSvap?
3. Compute "S" for propane at 355K and 3MPa relative to the liquid at 230K and 0.1MPa by this approach.
4. Compute "H" for propane at 355K and 3MPa relative to the liquid at 230K and 0.1MPa by this approach.
5. Compute H and S for the same conditions/reference using the PREOS.
6. Explain the discrepancies between the two approaches. e.g. compare the Hvap values and the (HV-Hig) values, where HV represents the enthalpy of the vapor phase, not the heat of vaporization (Hvap).

08.07 - Implementation of Departure Functions Click here. 20 3

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?

17.12 Energy Balances for Reactions Click here. 20 1

Equilibrium constants and adiabatic reactor calculations with Excel (uakron.edu, 6 min) We previously discussed adiabatic reactor calculations in Section 3.6 with application to the dimethyl ether process. At that time, we accepted the expression for equilibrium constant as given. In Chapter 17, we must recognize how to compute the equilibrium constant for ourselves. This presentation illustrates the calculations for Example 17.9. These kinds of calculations often occur in the context of an overall process, rather than in isolation. Therefore, the presentation shows how to apply Eqn 3.5b with pathway 2.6c to characterize the enthalpies of process streams and solve for the extent of reaction and adiabatic outlet temperature simultaneously.

Comprehension Questions:

1. Suppose the reactor inlet feed was: kmol/hr of 110 N2, 300 H2, 15NH3 and 16 CH4. Solve for the adiabatic reactor temperature and extent of reaction in that case.
2. Suppose the actual conversion was only 80% of the equilibrium conversion and the inlet feed was the same as given in part 1. Solve for the adiabatic reactor temperature and extent of reaction in that case.
3. Compute the stream attributes for this entire process assuming 85% of the equilibrium conversion and a feed (kmol/h) of 105 N2, 300 H2, 20 CH4 at 10bars and 200C. The distillation column operates at 10 bars with a partial condenser and splits of 99.99% on N2 and 2% on NH3. The recycle ratio is 19:1. Assume the compressors are 100% efficient and the reactor operates adiabatically with an inlet temperature of 400K and a pressure of 100bars. Report the molar flow rates of all outlet stream components.

08.07 - Implementation of Departure Functions Click here. 20 5

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 - Scott+TPT EOS. (uakron.edu) A sample derivation (8min) for the compressibility factor given that (A-Aig)TV/RT = -2ln(1-2ηP) - 18.7ηPβε/[1+0.36βεexp(-5ηP)]. This equation of state is a little complicated, but the derivation is no problem if you just go slow and steady. The remainder of this screencast shows a sample calculation (21min) to solve the resulting equation of state at a given value of pressure and temperature following the methodology of "visualizing the vdW EOS." This problem was adapted from an actual test problem. This screencast is live so the audio is inferior, but it gives insight into questions that real students have. 
Comprehension Questions:
1. Derive an expression for the internal energy departure function of this EOS.
2. Is there a sign error on one of terms in this video? Check the derivation independently.
3. Derive the Z factor given (A-Aig)TV/RT = -2ln(1-2bρ) - (9.5aρ/RT)/{1-a/bRT[1-4bρ+4(bρ)2]}.

04.03 The Macroscopic View of Entropy Click here. 20 1

Once we establish equations relating macroscopic properties to entropy changes, it becomes straightforward to compute entropy changes for all sorts of situations. To begin, we can compute entropy changes of ideal gases (learncheme, 3 min). Entropy change calculations may also take a more subtle form in evaluating reversibility (learncheme, 3min). 

Comprehension Questions: 

1. Nitrogen at 298K and 2 bars is adiabatically compressed to 375K and 5 bars in a continuous process. (a) Compute the entropy change. (b) Is this process reversible, irreversible, or impossible?
2. Nitrogen at 350K and 2 bars is adiabatically compressed to 575K and 15 bars in a piston/cylinder. (a) Compute the entropy change. (b) Is this process reversible, irreversible, or impossible?
3. Steam at 450K and 2 bars is adiabatically compressed to 575K and 15 bars in a continuous process. (a) Compute the entropy change. (b) Is this process reversible, irreversible, or impossible?
4. Steam at 450K and 2 bars is isothermally compressed to 8 bars in a continuous process. (a) Compute the entropy change. (b) Is this process reversible, irreversible, or impossible?

09.06 - Fugacity Criteria for Phase Equilibria Click here. 20 1

When liquid is added to an evacuated tank of fixed volume, equilibrium is established between the vapor and liquid. (3min,learncheme.com) The fugacity criterion characterizes this equilibrium as occurring when the escaping tendency from each phase is equal.

08.07 - Implementation of Departure Functions Click here. 20 1

Internal Energy Departure - PR EOS starting from Helmholtz Departure (uakron.edu,9min) This sample derivation supplements what is in the textbook by starting from the Helmholtz departure function. It also includes a few intermediate steps to help clarify how the formal equations in the textbook were developed. Hopefully, seeing this content from slightly different perspectives will make it a little easier to comprehend. See also the derivation for (U-Uig).

Comprehension Questions: Starting from the Helmholtz Departure function and referring to the above results...

1. Derive the internal energy departure function for the "modified vdW" EOS.
2. Derive the entropy departure function for the "modified vdW" EOS. (Hint: A=U-TS)
3. Derive the internal energy departure function for the "Scott+TPT" EOS.

11.05 - Modified Raoult's Law and Excess Gibbs Energy Click here. 20 2

Extending the M1 derivation of the activity coefficient to multicomponent mixtures  (uakron.edu, 14min) is straightforward but requires careful attention to the meaning of the subscripts and notation. It is a good warmup for derivations of more sophisticated activity models. This presentation begins with a brief review of the M1 model and its relation to the Gibbs excess function, then systematically explains the notation as it extends from the binary case to multiple components.

Comprehension Questions
1. Derive the activity coefficient for the multicomponent M2 model.
2. Derive the activity coefficient for the multicomponent Redlich-Kister model.

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