Chapter 11
Exercises
Supporting Files and Up-to-Date Exercise Material Is Available on the Book's Companion Website
11.1. Problems
1. If the energy balance at the lower flammability limit (LFL) of a mixture of flammable and inert gas can be expressed as:
where:
is LFL value of the fuel expressed in volume fraction.
is the energy absorbed per unit volume of fuel.
is the energy absorbed per unit volume of inert.
is the energy generated or released per unit volume of flammable gas mixture.a. Write the corresponding energy balance equation at the UFL limit
b. From oxygen calorimetry, using 
, where 
is heat generated per unit volume of oxygen, derive the expression for and .
, where is heat generated per unit volume of oxygen, derive the expression for and .2. A flammable vapor is composed of 63.5% ethyl acetate, 20.8% ethyl alcohol, and rest toluene by volume.
a. Based on the two sets of LFL values as provided in Table 11.1, estimate the LFL of the mixture using Le Chatelier's formula.
b. The sensitivity to input data illustrates the importance of having accurate data at hand. The LFL value of the given mixture is most sensitive to the LFL value of ______________.
3. Using Antoine's equation for vapor pressure and assuming ideal vapor liquid behavior, calculate flash point of a water–ethanol liquid mixture containing 40 mol% ethanol.
4. Consider the following set of reactions as observed in the nitric acid oxidation of 2-octanol to 2-octanone (Eizenberg et al., 2006).
A (2-octanol) + B (nitrosonium ion) → P (2-octanone) + 2B (nitrosonium ion) …(11.1)
P (2-octanone) + B (nitrosonium ion) → E (carboxylic acid) …(11.2)
Table 11.1
LFL Values for Chemicals in Example 1
| Chemical | LFL Values (Vol.% in Air) | |
| Lewis and Elbe (2013) | National Fire Protection Association (1986) | |
| Ethyl acetate | 2.5 | 2.0 |
| Ethyl alcohol | 4.3 | 3.3 |
| Toluene | 1.4 | 1.2 |
The overall set of reactions is exothermic in nature, and hence there is a requirement of heat removal to maintain the reactor temperature at around 0 °C. Under normal process conditions, only partial conversion (7.5%) of 2-octanone to carboxylic acid is achieved. However, if the operating temperature exceeds 5 °C, owing to increased reaction rate leading to insufficient heat removal, runaway conditions will develop. Please use the information as provided next to address the following:
a. Develop a steady state mathematical model of the reactor system involving these reactions to represent the system behavior under normal process conditions.
b. Extend the developed model to represent the dynamic system behavior with deviations from normal process conditions.
c. Develop a table for Hazard and Operability Study (HAZOP) keywords applicable for studying the given runaway reaction.
d. Implement the developed dynamic model to perform HAZOP analysis of the system.
5. Consider the following study (Švandová et al., 2005):
Hydrolysis reaction of propylene oxide: Propylene oxide hydrolyses to monopropylene glycol and on further consecutive reactions to higher glycols as illustrated in Figure 11.1.
For a three continuously stirred tank reactor (CSTR) system in series, consider Vk as the volume of the kth reactor, Ci,k as the molar concentration of ith component, Rj,k as the reaction rate of the jth reaction in the kth CSTR, Ej as the activation energy of the jth reaction, Kj as the reaction rate constant of the jth reaction, and Tk as the temperature of the kth CSTR. Also assume that all the reactions follow the Arrhenius equation and are of first order. The reaction parameters are provided in Table 11.2. Initial values are provided in Tables 11.3 and 11.4. Cooling medium as water (not shown in Figure 11.2) is flowing from the jacket of the first reactor to the second reactor and finally to the third reactor. The process moves to unsafe conditions once reactor temperature exceeds 503.15 K.
Find the following:
a. Mass and energy balance equation as a function of time, using subscripts for each reactor.
b. HAZOP study looks into potential process deviation scenarios using various guidewords. Based on that, examine the temperature in reactors if the CPropylene oxide,0 is varied from 5 to 30 mol/s, as six different scenarios.
c. In a similar vein, examine the effect of a sudden drop of cooling water flow rate from 6.8 to 5.8 mol/s.
Table 11.2
| Parameters | Reaction (R1) | Reaction (R2) | Reaction (R3) |
| K (m3 mol−1 s−1) | 96,000 | 9600 | 960 |
| E (J mol−1) | 75,362 | 82,899 | 91,189 |
 (J mol−1) |
−91,360 | −111,000 | −244,600 |
Table 11.3
Initial Conditions of the Reactor
| CPropylene oxide,0 (mol/s) | 13 |
| CWater,0 (mol/s) | 6.8 |
| T0 (C) (feed temperature) | 26 |
| T0,1 (K) | 470 |
| T0,2 (K) | 425.7 |
| T0,3 (K) | 400 |
| Vi=1,2,3 (m3) | 1 |
Table 11.4
Initial Conditions around Heat Exchanger
| UA (kW/K) | 10 |
| Cooling medium | Water |
| Fwater (mol/s) | 105 |
Table 11.5
Quantitative Representation of Qualitative HAZOP Keywords
| Parts/Components | Qualitative Failure Mode | Quantitative Representation |
| Valve | Open failure/closed failure | |
| Heat exchanger | Fouling, plugging | |
| Separator | Open failure/closed failure |
6. For performing a dynamic simulation to assist an HAZOP study, explain how you will represent quantitative guidewords for the qualitative keywords in Table 11.5.
7. Correct the order of steps below for performing a multivariate statistical analysis and explain reasons behind it:
a. Develop analysis plan.
b. Define the multivariate technique.
c. Estimate multivariate model.
d. Validate the multivariate model.
e. Evaluate the assumptions underlying the multivariate technique.
f. Estimate multivariate model.
g. Assess overall model fit.
h. Interpret variate(s).
i. Define research problem and objective.
8. Lead dioxide, a compound found in waste batteries, can be reduced to lead acetate with the addition of a reducing agent and acetic acid. A certain batch process adds both acetic acid and sulfuric acid in a stoichiometric amount to convert the initial lead dioxide slurry to lead sulfate slurry. Even though acetic acid is regenerated in the process, it is necessary to validate the same. Accordingly, between each batch, the laboratory tests for excess sulfuric acid in the solution by reacting it with a BaCl2 solution. In the test, any presence of sulfuric acid is indicated by the resulting precipitation in the form of BaSO4. It is also given that, if any lead acetate remains unreacted (which can only happen in absence of sulfuric acid), it will react with BaCl2 and PbCl2 will precipitate out of the solution. Now, if a process proceeds along the following steps in each batch and, in just one batch, less than required sulfuric acid is added, illustrate how and what problems might come up and keep growing. Also explain how would you fix these problems.
a. Measured amount of lead dioxide, acetic acid, and sulfuric acid is mixed in a CSTR.
b. After completion of the reaction, a sample is sent to the laboratory to test acid concentrations.
c. Initially, the total acid equivalent is estimated through titration.
d. Sulfuric acid concentration is estimated by the BaCl2 testing.
e. Acetic acid concentration is estimated by subtracting the sulfuric acid concentration from the initial total acid equivalent.
9. Explain how you will develop a molecular dynamics–based model to understand evaporation characteristics of a droplet. Consider the initial system to be an FCC crystal of a Xe/N2 system with liquid Xe (with initial diameter D) at the center of the cube surrounded by N2.
10. Assume that the heat of formation for ethane, 
, is not known and consider the following reaction, which includes ethane as a product.
, is not known and consider the following reaction, which includes ethane as a product.
a. Summarize how can be determined using the above reaction, indicate any needed information, and write an expression using the usual notations.
can be determined using the above reaction, indicate any needed information, and write an expression using the usual notations.b. Compare the reaction below with the reaction above and state why this new reaction is more useful or less useful for determining .
.
11. Applying the linear combination of atomic orbitals (LCAO) approach:
a. List the 6-31G∗∗ basis functions to be used for calculations involving molecule hydrogen fluoride (HF), a toxic gas, from H(1s2) and F(1s22s22p5) and indicate the number of Gaussian primitive functions for each in parentheses needed for this LCAO application.
b. State the total number of 6-31G∗∗ basis functions and the total number of Gaussian primitive functions to be used for HF.
12. Consider the following set of models: AM1, Hartree-Fock, MP2, B3LYP, and QCISD(T). Indicate for the scenarios below which model(s) you would choose and why:
a. Equilibrium structure of a large molecule at low or moderate cost.
b. Enthalpy of an isodesmic reaction at low or moderate cost.
c. Accurate enthalpy of reaction for medium-sized molecule at moderate cost.
13. A room has a relatively small opening either at the ceiling or the floor level. A fire in the room develops energy at the rate of 
kW. Derive an expression for dP/dt and the mass flow rate out of the room as a function of and a number of other variables. Name the main assumptions. The expression you should arrive at is: 
.
kW. Derive an expression for dP/dt and the mass flow rate out of the room as a function of and a number of other variables. Name the main assumptions. The expression you should arrive at is: .14. Discuss how the above equation can lead to
a. An analytical expression for the pressure build-up in a hermetically closed room.
b. An analytical expression for the mass flow through an opening, when the opening is either at floor or ceiling level.
11.2. Application of KiSThelP Software for Explosive Decomposition Reaction
Kinetic and Statistical Thermodynamical Package (KiSThelP) is a cross-platform free open-source program developed to estimate molecular and reaction properties from electronic structure data. Some key features are gas-phase molecular thermodynamic properties, thermal equilibrium constants and kinetic calculations. Theories like transition state theory (TST) and RRKM are employed. KiSThelP provides graphical front-end capabilities designed to facilitate calculations and interpreting results (data can be read directly from quantum chemistry results). These features make this program well-suited to support and enhance students learning and can serve as a very attractive courseware as well, taking the teaching content directly from results in molecular and kinetic modeling (S. Canneaux et al., 2014).
H + HN3 → NH2 + N2
11.2.1. Context
For many years, a large number of experimental studies in the area of gaseous detonation are carried out in order to test detonation propagation models. The experimental study of the thermal decomposition of energetic molecules is dangerous. Thus, explosive decomposition modeling meets some problems owing to the lack of kinetic data on elementary reactions. Also, kinetic data extrapolation at large temperatures is difficult from an experimental point of view.
Therefore, in the absence of experimental results, investigation by means of theoretical computations is useful in order to provide kinetic parameters and, then, significant insight into reaction mechanisms.
11.2.2. Aim
This example acts as a basic introduction to KiSThelP to compute a reaction heat and a reaction rate constant.
11.2.3. General Methodology
To determine thermodynamic and kinetic properties from quantum results, the use of statistical mechanics is required for relating the microscopic properties of individual atoms and molecules to the macroscopic bulk properties of materials that are under study in your laboratory (Figure 11.3).
In order to estimate macroscopic properties, a number of files are required. KiSThelP typically processes file containing quantum chemistry results. Once the data are imported in KiSThelP for a single species, a lot of properties are delivered at different temperatures and pressures. Gaussian, GAMESS, and NWChem output files are supported. Here, data are taken from Gaussian calculations.
The studied reaction mechanism (involving the limiting step forming the intermediate H2N3) is (see Henon, E., Bohr, F., 2000. Journal of Molecular Structucture (Theochem) 531, 283–299 for more details):
H + HN3 → H2N3 → NH2 + N2
1. For the above-mentioned reaction, the following quantum mechanics calculations are required before using KiSThelP:
a. Geometry optimization and vibrational frequency calculations for H, HN3, H2N3, NH2, N2.
b. Transition state (TS) determination for the rate determining step.
These calculations produced the following files1: H.out, HN3.out, H2N3.out, NH2.out, N2.out, TS.out. In this example, the level of theory was: DFT(M06)/6-311G∗∗.
2. Reaction heat: using the menu option “Calculation/Keq”, and providing the file names for the two reactants (H.out, HN3.out) and for the two final products (N2.out, NH2.out) leads to the following result:
Thermodynamics properties are predicted, specially the heat of reaction ΔH°. Here, this quantity has been plotted in the temperature range 298 to 2000 K.
3. Kinetic parameters prediction: Using the menu option “Calculation/TST/Eck” and providing the file names for the two reactants (H.out, HN3.out) and the transition state (TS.out) leads to the following result:
As can be seen on Figure 11.4, the rate constant is predicted as a function of the temperature (log_10 K = f(1/T)). Here, tunneling effect has been taken into account, slightly increasing the reaction rate (red curve). Numerous possibilities are available in KiSThelP. On the bottom left panel of Figure 11.4, thermodynamics and kinetics parameters are reported for a given temperature. Also, the temperature dependency of the rate constant can be presented using the two- and three-parameter conventional Arrhenius equation.
Available in KiSThelP, the variational transition state theory (VTST) emphasizes the variational effect on the location of the transition state at a given temperature. Additional information is needed on the potential energy surface compared to the conventional TST. The reaction path must be obtained from a quantum chemistry program by performing a so-called IRC calculation. Such a reaction path is illustrated in Figure 11.5 at 700 K.
11.3. Application to a Reaction of Atmospheric Interest
CH3OCH2OCH2O → H + CH3OCH2OCHO
11.3.1. Context
Because of the increasing use as industrial solvents and fuel additives of oxygenated compounds such as ethers, their degradation in the atmosphere has received significant attention in the past few years.
11.3.2. Aim
In polluted atmosphere the title reaction can exhibit pressure dependence. The purpose of this example is to provide an initial introduction to predict pressure-dependent rate constants.
11.3.3. General Methodology
The same methodology as previously illustrated in example one can be applied here. But, in addition to the required quantum chemistry data (for CH3OCH2OCH2O, transition state), the Lennard-Jones collision parameters must be supplied. It can be noticed that a database of these collision parameters is made available in KiSThelP. The level of theory used to obtain the presented results can be found in Henon et al. (2003).
Using the KiSThelP menu “k/RRKM” and interactively providing the collision parameters leads to the following results (here the fall-off region is well characterized):
Several modes of representation of the thermodynamic and kinetic results are available: temperature range, pressure range, or a three-dimensional representation varying simultaneously temperature and pressure (Figure 11.6).
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