# Mole Balances

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## Questions and Problems

—Yogi Berra, New York Yankees Sports Illustrated, June 11, 1984

The subscript to each of the problem numbers indicates the level of difficulty, i.e., A, least difficult; B, moderate difficulty; C, fairly difficult; D, (double black diamond), most difficult. For example, P1-5B means “1” is the Chapter number, “5” is the problem number, “B” is the problem difficulty, in this case B means moderate difficulty.

Before solving the problems, state or sketch qualitatively the expected results or trends.

### Questions

• Q1-1A i>clicker. Go to the Web site (http://www.umich.edu/~elements/5 e/01chapliclicker_chl_ql.html) and view at least five i>clicker questions. Choose one that could be used as is, or a variation thereof, to be included on the next exam. You also could consider the opposite case: explaining why the question should not be on the next exam. In either case, explain your reasoning.

• Q1-2A Rework Example 1-1 using Equation (3-1), and then compare reaction rates. How do these values compare with those calculated in the example?

• Q1-3A In Example 1-2, if the PFR were replaced by a CSTR what would be its volume? Q1-4A Rework Example 1-2 for a constant volume batch reactor to show the time to reduce the number of moles of A to 1% if its initial value is 20 minutes, suggest two ways to work this problem incorrectly.

• Q1-5A Read through the Preface. Write a paragraph describing both the content goals and the intellectual goals of the course and text. Also describe what’s on the Web site and how the Web site can be used with the text and course.

• Q1-6A View the photos and schematics Chapter 1 Professional Reference Shelf (http://www.umich.edu/~elements/5e/01chap/prof.html). Write a paragraph describing two or more of the reactors. What similarities and differences do you observe between the reactors on the Web (e.g., www.loebequipment.com), on the Web site, and in the text? How do the used reactor prices compare with those in Table 1-1? Q1-7A Critique one of the Learn ChemE Videos for Chapter 1 (http://www.umich.edu/~elements/5e/01chap/learncheme.html) for such things as (a) value, (b) clarity, (c) visuals and (d) how well it held your interest. (Score 1-7; 7 = outstanding, 1= poor)

• Q1-8A What assumptions were made in the derivation of the design equation for: (a) The batch reactor (BR)? (b) The CSTR? (c) The plug-flow reactor (PFR)? (d) The packed-bed reactor (PBR)? (e) State in words the meanings of −rA and rA.

• Q1-9A Use the mole balance to derive an equation analogous to Equation (1-7) for a fluidized CSTR containing catalyst particles in terms of the catalyst weight, W, and other appropriate terms.

Figure Q1-6 Fluidized Bed CSTR.

Computer Simulations and Experiments

• P1-1A (a) Revisit Example 1-3.

Wolfram

(i) Describe how CA and CB change when you experiment with varying the volumetric flow rate, υ0, and the specific reaction rate, k, and then write a conclusion about your experiments.

(ii) Click on the description of reversible reaction A ⇆ B to understand how the rate law becomes rA=k[CACBKe]. Set Ke at its minimum value and vary k and υ0. Next, set Ke at its maximum value and vary k and υ0. Write a couple sentences describing how varying k, v0, and Ke affect the concentration profiles.

(iii) After reviewing Generating Ideas and Solutions on the Web site (http://www.umich.edu/~ele-ments/5e/toc/SCPS,3rdEdBook(Ch07).pdf), choose one of the brainstorming techniques (e.g., lateral thinking) to suggest two questions that should be included in this problem.

Polymath

(iv) Modify the Polymath program to consider the case where the reaction is reversible as discussed in part (ii) above with Ke = 3.

• P1-2B Schematic diagrams of the Los Angeles basin are shown in Figure P1-2B. The basin floor covers approximately 700 square miles (2 × 1010 ft2) and is almost completely surrounded by mountain ranges. If one assumes an inversion height in the basin of 2,000 ft, the corresponding volume of air in the basin is 4 × 1013ft3. We shall use this system volume to model the accumulation and depletion of air pollutants. As a very rough first approximation, we shall treat the Los Angeles basin as a well-mixed container (analogous to a CSTR) in which there are no spatial variations in pollutant concentrations.

Figure P1-2B Schematic diagrams of the Los Angeles basin.

(http://www.umich.edu/~ekments/5e/web_mod/la_basin/index.htm)

We shall perform an unsteady-state mole balance (Equation (1-4)) on CO as it is depleted from the basin area by a Santa Ana wind. Santa Ana winds are high-velocity winds that originate in the Mojave Desert just to the northeast of Los Angeles. Load the Smog in Los Angeles Basin Web Module. Use the data in the module to work parts 1-12 (a) through (h) given in the module. Load the Living Example Polymath code and explore the problem. For part (i), vary the parameters υ0, a, and b, and write a paragraph describing what you find.

There is heavier traffic in the L.A. basin in the mornings and in the evenings as workers go to and from work in downtown L.A. Consequently, the flow of CO into the L.A. basin might be better represented by the sine function over a 24-hour period.

• P1-3B This problem focuses on using Polymath, an ordinary differential equation (ODE) solver, and also a nonlinear equation (NLE) solver. These equation solvers will be used extensively in later chapters. Information on how to obtain and load the Polymath Software is given in Appendix D and on the CRE Web site.

(a) There are initially 500 rabbits (x) and 200 foxes (y) on Professor Sven KöttloVs son-in-law, Stépân Dolez’s, farm near Rica, Jofostan. Use Polymath or MATLAB to plot the concentration of foxes and rabbits as a function of time for a period of up to 500 days. The predator-prey relationships are given by the following set of coupled ordinary differential equations:

dxdt=k1xk2xy

dxdt=k3xyk2y

Constant for growth of rabbits k1 = 0.02 day-1

Constant for death of rabbits k2 = 0.00004/(day × no. of foxes)

Constant for growth of foxes after eating rabbits k3 = 0.0004/(day x no. of rabbits)

Constant for death of foxes k4 = 0.04 day-1

What do your results look like for the case of k3 = 0.00004/(day × no. of rabbits) and tfinal = 800

days? Also, plot the number of foxes versus the number of rabbits. Explain why the curves look the way they do. Polymath Tutorial (https://www.youtube.comfwatch?v=nyjmt6cTiL4)

1. Using Wolfram in the Chapter 1 LEP on the Web site, what parameters would you change to convert the foxes versus rabbits plot from an oval to a circle? Suggest reasons that could cause this shape change to occur.

2. We will now consider the situation in which the rabbits contracted a deadly virus. The death rate is rDeath = kDx with feD = 0.005 day-1. Now plot the fox and rabbit concentrations as a function of time and also plot the foxes versus rabbits. Describe, if possible, the minimum growth rate at which the death rate does not contribute to the net decrease in the total rabbit population.

3. Use Polymath or MATLAB to solve the following set of nonlinear algebraic equations

x3y4y2+3x=1

6y29xy=5

with inital guesses of x = 2, y = 2. Try to become familiar with the edit keys in Polymath and MATLAB. See the CRE Web site for instructions

Screen shots on how to run Polymath are shown at the end of Summary Notes for Chapter 1 or on the CRE Web site, ivwiv.umich.e<iu/~eIemertts/5e/so/tiv«re/poh’m«tfi-tutorial.html.

Interactive Computer Games

• P1-4A Find the Interactive Computer Games (ICG) on the CRE Web site, (http://www.umich.edu/~elements/5e/icg/index.html). Read the description of the Kinetic Challenge module (http://www.umich.edu/~elements/5e/icm/kinchall.html) and then go to the installation instructions (http://www.umich.edu/~elements/5e/icm/install.htmi) to install the module on your computer. Play this game and then record your performance number, which indicates your mastery of the material.

ICG Kinetics Challenge 1 Performance #___________________

### Problems

• P1-5A The reaction

A + B → 2C

takes place in an unsteady CSTR. The feed is only A and B in equimolar proportions. Which of the following sets of equations gives the correct set of mole balances on A, B, and C? Species A and B are disappearing and species C is being formed. Circle the correct answer where Analysis the mole balances are correct.

5. None of the above.

• P1-6R The reaction

A → B

is to be carried out isothermally in a continuous-flow reactor. The entering volumetric flow rate υ0 is 10 dm3/h. (Note: FA = CAv. For a constant volumetric flow rate υ = υ0, then CAO = FAO/UO = ([5mol/h]/[10dm3/h])0.5mol/dm3.)

Calculate both the CSTR and PFR reactor volumes necessary to consume 99% of A (i.e., when the entering molar flow rate is 5 mol/h, assuming the reaction rate −rA is

1. rA = k with k=0.05molhdm3 [Ans.: VCSTR = 99 dm3]

2. rA = kCA with 0.0001 s−1

3. rA = kC2A with k=300dm3molh [Ans.: VCSTR = 660 dm3]

4. Repeat (a), (b), and/or (c) to calculate the time necessary to consume 99.9% of species A in a 1000 dm3 constant-volume batch reactor with CA0 = 0.5 mol/dm3.

• P1-7A Enrico Fermi (1901-1954) Problems (EFP). Enrico Fermi was an Italian physicist who received the Nobel Prize for his work on nuclear processes. Fermi was famous for his “Back of the Envelope Order of Magnitude Calculation” to obtain an estimate of the answer through logic and then to make reasonable assumptions. He used a process to set bounds on the answer by saying it is probably larger than one number and smaller than another, and arrived at an answer that was within a factor of 10. See http://methforam.orj/worfeshops/sum96/tnteriitsc/sheiIa2.htm!.

Enrico Fermi Problem

(a) EFP #1. How many piano tuners are there in the city of Chicago? Show the steps in your reasoning.

1. Population of Chicago___________

2. Number of people per household___________

3. Etc.___________

An answer is given on the CRE Web site under Summary Notes for Chapter 1.

(b) EFP #2. How many square meters of pizza were eaten by an undergraduate student body population of 20,000 during the Fall term 2016?

(c) EFP #3. How many bathtubs of water will the average person drink in a lifetime?

• P1-8A What is wrong with this solution? The irreversible liquid phase second order reaction (r=kCA2)

2Ak1B  k=0.03dm3/mols

is carried out in a CSTR. The entering concentration of A, CA0, is 2 molar, and the exit concentration of A, CA is 0.1 molar. The volumetric flow rate, υ0, is constant at 3 dm3/s. What is the corresponding reactor volume?

Analysis

1. Mole Balance

V=FA0FArA

2. Rate Law (2nd order)

rA=kCA2

3. Combine

V=FA0FAkCA2

6. V=(60.3)mo1s(0.03dm3mo1s)(2mo1dm3)2=47.5dm3

For more puzzles on what’s “wrong with this solution,” see additional material for each chapter on the CRE Web site home page, under “Expanded Material.”

1. For further elaboration of the development of the general balance equation, see not only the Web site www.umich.edu/~elements/5e/index.html but also

FELDER, R. M., and R. W ROUSSEAU, Elementary Principles of Chemical Processes, 3rd ed. New York: Wiley, 2000, Chapter 4.

SANDERS, R.J., The Anatomy of Skiing. Denver, CO: Golden Bell Press, 1976.

2. A detailed explanation of a number of topics in this chapter can be found in the tutorials.

CRYNES, B. L., and H. S. FOGLER, eds., AlChE Modular Instrudion Series E: Kinetics, Vols. 1 and 2. New York: AIChE, 1981.

3. A discussion of some of the most important industrial processes is presented by

AUSTIN, G. T., Shreve’s Chemical Process Industries, 5th ed. New York: McGraw-Hill, 1984.

4. Short instructional videos (6-9 minutes) that correspond to the topics in this book can be found at http://www.karncfieme.com/.