Introduction to the Problem-Solving Methodology (PSM)

Transcription

1 Problem Name: Problem Description: Date: P01 Simple Reactor Introduction to the Problem-Solving Methodology (PSM) Your Name: Problem Session Objectives To learn the five stages of the problem solving methodology. To learn about the material balances for a continuous reactor. To complete a conceptual model using the problem statement. To write a mathematical model using the conceptual model. To construct a mathematical algorithm using the mathematical model. To create the numerical solution using the mathematical algorithm. To generate heuristic observations about the numerical solution, mathematical algorithm, mathematical model, and conceptual model. Reference Readings Felder and Rousseau, 3 rd Edition, Section 4.1, Process Classification. Felder and Rousseau, 3 rd Edition, Section 4.2, Balances. Felder and Rousseau, 3 rd Edition, Section 4.3, Material Balance Calculations. Felder and Rousseau, 3 rd Edition, Section 4.6, Chemical Reaction Stoichiometry. Felder and Rousseau, 3 rd Edition, Section 4.7, Balances on Reactive Processes. Felder and Rousseau, 3 rd Edition, Chapter 3, Processes and Process Variables. Review Materials Hanyak s Development of a Conceptual Model, CinChE Manual, Chapter 4. Hanyak s Development of a Mathematical Model, CinChE Manual, Chapter 4. Hanyak s Development of a Mathematical Algorithm, CinChE Manual, Chapter 4. Hanyak s Development of a Numerical Solution, CinChE Manual, Chapter 4. Hanyak s Development of the Heuristic Observations, CinChE Manual, Chapter 4. Hanyak s Anatomy of a Math Model and Algorithm, CinChE Manual, Chapter 4. Interaction 1: Topic: Simple Continuous Reactor Background: Welcome to the eleaps problem session about a chemical reactor. Save this script document to the desktop. Click here to open and save the solution template also to the desktop. Close all internet browser windows. Open the two saved documents with Adobe Reader. In the solution template document, right click and select Print, choose Document and Markups under Comments and Forms, and print it to get a PAPER COPY. Print to a color printer for the best effect. You will fill-in this paper copy as you do the problem session. Close the template document and then delete it, since it is no longer needed. v , Michael E. Hanyak, Jr., All Rights Reserved Page 1 of 22

2 P01 eleaps Problem Solution Template Coaching Script and Solution Template This coaching script contains two kinds of pages script and template. They are arranged similar to the left and right pages in a book. The left page is an interaction in the coaching script. The right page is the current focus in the solution template that is associated with the left coaching script page. How you navigate through the coaching script depends up the type of computer that you are using a personal computer with a mouse or an Apple ipad with a stylus pen. In either case, you have opened this coaching script using the Acrobat Reader program that is installed on your computer and not the Acrobat Reader plug-in found in a web browser. Please complete the first interaction in the first coaching script page. Then, proceed to navigate through the coaching script based upon your computer type, as describe below. Personal Computer with a Mouse The Acrobat Reader program should have displayed this coaching script in its two-page view mode. If not, then select the View/Page Display/Two Page Scrolling option from the menu bar. In the two-page view mode, the left column of pages will be the coaching script, while the right column of pages will be the current focus in the solution template. You can magnify the view (i.e., zoom in) so that the coaching script page is readable. Then, you can use the horizontal scroll bar to move between the left page (the coaching script) and its right page (the template solution). After you manually complete a portion of your PAPER COPY of the problem solution template (as directed by its associated coaching script interaction), you can then delete the boxes in the right page to view the correct answers. You can also view the popup notes found in the right page. You proceed to the next script Interaction by scrolling down to the next set of two pages in the Acrobat Reader program. Apple ipad with a Rubber-Domed Stylus Pen The Acrobat Reader app for the ipad (downloaded from the App Store) does not support the two-page view mode. To simulate this viewing mode, select the Single Page option under Document Modes in the menu bar. In the Single Page mode, you will be able to horizontally swipe between the left page (the coaching script) and its right page (the template solution). After you manually complete a portion of your PAPER COPY of the problem solution template (as directed by its associated coaching script interaction), you can then delete the boxes in the right page to view the correct answers. You can also view the popup notes found in the right page. You proceed to the next script Interaction by swiping pass the current right page in the Acrobat Reader app. If you quickly tap the Home button on the ipad twice, you can conveniently switch between the Adobe Reader and any other apps.

3 Interaction 2: Topic: Simple Reactor Background: A simple reactor problem is used to introduce you to the Problem- Solving Methodology that we will use in this course. Conceptual Model Question: The first step in the problem-solving methodology (PSM) is to analyze the problem statement and create a conceptual model composed of a labeled Diagram, Other Givens, Finds, and initial Assumptions. Chapter 4 in the white CinChE manual presents the guidelines for the development of a conceptual model. This development contains five sub-parts. First, you identify the process operation and represent it as a diagram. View the problem statement and possible diagrams to the right of this script page. Which process diagram applies to this problem? Option 1: Diagram (a). Option 2: Diagram (b). Option 3: Diagram (c). Option 4: Diagram (d). Feedback 1: Incorrect! Diagram (a) is for a batch process, where no material flows in or out. Feedback 2: Incorrect! Diagram (b) is for a continuous process, with one income stream and two out going streams. Feedback 3: Correct! Diagram (c) is for a continuous process with a feed stream entering and a product stream exiting. Feedback 4: Incorrect! Diagram (d) is for a semi-batch process, where some material flows out. Leave the development of a conceptual model browser window open until further notice. v , Michael E. Hanyak, Jr., All Rights Reserved Page 2 of 22

4 Problem Statement Chemists in our Research and Development Department have discovered a catalyst, which will produce styrene monomer from toluene and methanol in one step by the reaction pathway: C 7 H 8 + CH 3 OH C 8 H 8 + H 2 O + H 2 toluene methanol styrene water hydrogen Based on their experimental reactor studies, our chemists report that 80 mole % of the toluene is converted to styrene monomer when an equimolar mixture of methanol and toluene are fed to a small-scale reactor at 520ºC and 300 kpa. For a feed rate of L/s, our company needs to know what the molar flow rate and composition would be for the product stream, which exits the adiabatic reactor as a gas at 337ºC and 270 kpa. The mass flow of this gas stream is also needed in kilograms per hour. As a sophomore chemical engineer in our company, please provide a welldocument solution to this problem using our standard problem-solving methodology. Diagram (a) (b) (c) (d)

5 Interaction 3: Topic: Simple Reactor Background: Second, you label the process state of each material stream in the sketched diagram by analyzing the problem statement. The process state is a stream's temperature, pressure, flow rate, and composition. You also label the phase state of the stream; that is, whether it is a gas, a liquid, a solid, vapor-liquid, etc. When a property is not given, you label it with a question mark. Conceptual Model Question: View the problem statement with its conceptual diagram to the right of this script page. As indicated by the yellow and green highlights, complete labeling the process state of each material stream on Page 1 in your PAPER COPY of the template document. Since the phase of the feed stream is not given, you must determine it or assume it. Currently, you do not have the knowledge to determine it; but, you can use your engineering judgment to assume it. What is the phase state of Stream F? Option 1: liquid. Option 2: gas. Option 3: solid. Option 4: I don t know! Feedback 1: Incorrect! The temperature of the feed stream is well above the normal boiling temperatures for toluene and methanol ( C and 64.7 C at 1 atm, see Table B.1 in F&R, 3rd Edition). Feedback 2: Correct! Since the product stream is a gas, the feed stream must also be a gas because its temperature is higher. Also, the feed stream is an ideal gas, because its pressure is less then or equal to 3 atm (303 kpa or 3.03 bar). Feedback 3: Incorrect! See the explanation under Feedback 1. Feedback 4: Incorrect! The phase state is one of the other three. Try again! v , Michael E. Hanyak, Jr., All Rights Reserved Page 3 of 22

6 Problem Statement Chemists in our Research and Development Department have discovered a catalyst, which will produce styrene monomer from toluene and methanol in one step by the reaction pathway: C7Hlo + CH30H --+ CsHs + Hz0 + H2 toluene methanol Styrene water hydrogen Based on their experimental reactor studies, our chemists report that 80 mole % of the toluene is converted to styrene monomer when an equimolar mixture of methanol and toluene are fed to a small-scale reactor at 520 C and 300 kpa. For a feed rate of Lls, our company needs to know what the molar flow rate and composition would be for the product stream, which exits the adiabatic reactor as a gas at 337OC and 270 kpa. The mass flow of this gas stream is also needed in kilograms per hour. As a sophomore chemical engineer in our company, please provide a welldocument solution to this problem using our standard problem-solving methodology. other Givens:

7 Interaction 4: Topic: Simple Reactor Background: View the problem statement with its labeled diagram to the right of this script page. As shown in the diagram, molar variables were chosen for the flow rates and compositions. The flow rate variables associated with streams in the diagram could be mass or molar quantities, but NOT volumetric flow rates. Also, the composition variables could be mass or mole fractions, but NOT volume fractions or concentrations. After reading the problem statement, you must decide which to use mass or molar quantities. Conceptual Model Which reasons below suggest that molar quantities should be chosen Question: for the diagram? Option 1: Most given and find quantities are in mass units. Option 2: Reaction material balances based on moles are simpler to write. Option 3: Most given and find quantities are in molar units. Option 4: Reaction material balances based on mass are easier to write. Feedback 1: Incorrect! Only one mass quantity is to be found. Feedback 2: Correct! Mole balances with chemical reactions are easier to solve. Feedback 3: Correct! One molar quantity is given, and two others are to be found. Feedback 4: Incorrect! Mass balances with chemical reactions are harder to write and solve. v , Michael E. Hanyak, Jr., All Rights Reserved Page 4 of 22

8 Problem Statement Chemists in our Research and Development Department have discovered a catalyst, which will produce styrene monomer from toluene and methanol in one step by the reaction pathway: C7Hlo + CH30H --+ CsHs + Hz0 + H2 toluene methanol Styrene water hydrogen Based on their experimental reactor studies, our chemists report that 80 mole % of the toluene is converted to styrene monomer when an equimolar mixture of methanol and toluene are fed to a small-scale reactor at 520 C and 300 kpa. For a feed rate of Lls, our company needs to know what the molar flow rate and composition would be for the product stream, which exits the adiabatic reactor as a gas at 337OC and 270 kpa. The mass flow of this gas stream is also needed in kilograms per hour. As a sophomore chemical engineer in our company, please provide a welldocument solution to this problem using our standard problem-solving methodology. other Givens:

9 Interaction 5: Topic: Simple Reactor Background: As shown in the diagram to the right of this script page and also in your PAPER COPY of the template document, the product stream was labeled to contain toluene (TL), methanol (ME), styrene monomer (SM), water (WA), and hydrogen (H2). Why? Conceptual Model Question: Since only partial conversion of the toluene occurs, left over toluene and methanol must appear in the product. Furthermore, the toluene that reacts with methanol, based on the chemical reaction, produces styrene monomer, water, and hydrogen. These chemical components (or compounds) must appear in the product stream. What percent conversion of toluene would lead to NO toluene and methanol appearing in the product stream? Option 1: 0%. Option 2: 35%. Option 3: 70%. Option 4: 100%. Feedback 1: Incorrect! A molar conversion of 0% for toluene would mean that no reactions take place. Thus, all of the toluene and methanol that enters the reactor would leave in the product stream. Feedback 2: Incorrect! A molar conversion of 35% for toluene would mean that some of the toluene reacts with the methanol (on a 1:1 molar basis). The unreacted toluene and methanol that entered the reactor would leave in the product stream. Feedback 3: Incorrect! A molar conversion of 70% for toluene would mean that some of the toluene reacts with the methanol (on a 1:1 molar basis). The unreacted toluene and methanol that entered the reactor would leave in the product stream. Feedback 4: Correct! A molar conversion of 100% for toluene would mean that all of the toluene reacts entirely with methanol (on a 1:1 molar basis). Thus, no toluene and methanol would leave in the product stream. As another example, let's say the feed to the reactor was NOT equimolar but 40% toluene and 60% methanol on a molar basis. For 100% conversion, no toluene would appear in the product stream, but some methanol would, because toluene is the limiting reactant. v , Michael E. Hanyak, Jr., All Rights Reserved Page 5 of 22

10 Problem Statement Chemists in our Research and Development Department have discovered a catalyst, which will produce styrene monomer from toluene and methanol in one step by the reaction pathway: C7Hlo + CH30H --+ CsHs + Hz0 + H2 toluene methanol Styrene water hydrogen Based on their experimental reactor studies, our chemists report that 80 mole % of the toluene is converted to styrene monomer when an equimolar mixture of methanol and toluene are fed to a small-scale reactor at 520 C and 300 kpa. For a feed rate of Lls, our company needs to know what the molar flow rate and composition would be for the product stream, which exits the adiabatic reactor as a gas at 337OC and 270 kpa. The mass flow of this gas stream is also needed in kilograms per hour. As a sophomore chemical engineer in our company, please provide a welldocument solution to this problem using our standard problem-solving methodology. other Givens:

11 Interaction 6: Topic: Simple Reactor Background: Third, you list other given information that appears directly and/or is implied in the problem statement. For example, the chemical reaction was written and then balanced. Conceptual Model Question: Fourth, you list the quantities with units that are to be found. For example, the total molar flow rate of Stream P was written. View the problem statement with its conceptual diagram to the right of this script page. As indicated by the yellow and green highlights, complete labeling the rest of the Givens and Finds on Page 1 in your PAPER COPY of the template document. Note that for the "Finds," you choose the units for the important variables to be calculated. These units are either given in the problem or selected by you. Why was kgmol/h picked for the molar flow rate of Stream P? Option 1: It does not matter what you pick! Option 2: It does matter what you pick! Feedback 1: Incorrect! See the explanation under Feedback 2. Feedback 2: Correct! Because you are asked to find the mass flow rate of the product in kg/h, anything other than kgmol/h, such as g-mol/h or lbmol/h, will probably require more unit conversions when doing the calculations. v , Michael E. Hanyak, Jr., All Rights Reserved Page 6 of 22

12 Problem Statement Chemists in our Research and Development Department have discovered a catalyst, which will produce styrene monomer from toluene and methanol in one step by the reaction pathway: C7Hlo + CH30H --+ CsHs + Hz0 + H2 toluene methanol Styrene water hydrogen Based on their experimental reactor studies, our chemists report that 80 mole % of the toluene is converted to styrene monomer when an equimolar mixture of methanol and toluene are fed to a small-scale reactor at 520 C and 300 kpa. For a feed rate of Lls, our company needs to know what the molar flow rate and composition would be for the product stream, which exits the adiabatic reactor as a gas at 337OC and 270 kpa. The mass flow of this gas stream is also needed in kilograms per hour. As a sophomore chemical engineer in our company, please provide a welldocument solution to this problem using our standard problem-solving methodology. other Givens:

13 Interaction 7: Topic: Simple Reactor Background: Fifth and final sub-part in the development of a conceptual model, you begin listing the initial assumptions, in order to complete the conceptual model. For Assumption 1, identify the process type, either continuous, batch, semi-batch, or semi-continuous. For Assumption 2, identify the operation state, either steady state or unsteady state for the material within the imaginary system boundary. As indicated by the yellow highlight to the right of this script page, complete labeling the first two assumptions on Page 1 in your PAPER COPY of the template document. Conceptual Model What would be the third assumption? Question: Option 1: Stream P is a non-ideal gas. Option 2: Stream F is an ideal gas. Feedback 1: Incorrect! Because the pressure of Stream P is less then or equal to 3 atm (303 kpa or 3.03 bar), it can be treated as an ideal gas. Feedback 2: Correct! Because the pressure of Stream F is less then or equal to 3 atm (303 kpa or 3.03 bar), it can be treated as an ideal gas. Before continuing, close the development of a conceptual model browser window. v , Michael E. Hanyak, Jr., All Rights Reserved Page 7 of 22

14 Continuous Reactor Problem Statement Chemists in our Research and Development Department have discovered a catalyst, which will produce styrene monomer from toluene and methanol in one step by the reaction pathway: C7Hlo + CH30H --+ CsHs + Hz0 + H2 toluene methanol Styrene water hydrogen Based on their experimental reactor studies, our chemists report that 80 mole % of the toluene is converted to styrene monomer when an equimolar mixture of methanol and toluene are fed to a small-scale reactor at 520 C and 300 kpa. For a feed rate of Lls, our company needs to know what the molar flow rate and composition would be for the product stream, which exits the adiabatic reactor as a gas at 337OC and 270 kpa. The mass flow of this gas stream is also needed in kilograms per hour. As a sophomore chemical engineer in our company, please provide a welldocument solution to this problem using our standard problem-solving methodology. other Givens: v , Michael E. Hanyak, Jr., All Rights Reserved 1 of 6

15 Interaction 8: Topic: Simple Reactor Background: The second step in the problem-solving methodology (PSM) is to review the conceptual model and create a mathematical model composed of first principle equations, additional equations, and a degrees-of-freedom analysis. Mathematical Model Question: Chapter 4 in the white CinChE manual presents the guidelines for the development of a mathematical model. Using Guidelines 1 to 5, you are to finish writing the first-principle equations in your PAPER COPY of the template document in the first six equations, as indicated by the yellow highlights to the right of this script page. In the total balance, the reaction term is a number times the extent of reaction. The number is found by summing the product coefficients in the BALANCED chemical reaction and then subtracting the sum of the reactant coefficients in that same reaction. In a component balance, the number in a reaction term is the component's coefficient from the chemical reaction with a negative sign for a reactant or a positive sign for a product. If the component does not participate in the chemical reaction, then its number is zero; that is, the reaction term is not written. Writing material balances is an accounting technique on the chemical substances that are associated with the system boundary. You can count atoms, molecules, or mass. Why was the accounting done the way it is shown in the first six equations of the mathematical model? Option 1: because most information in the conceptual model is given in terms of atoms. Option 2: because most information in the conceptual model is given in terms of mass. Option 3: because most information in the conceptual model is given in terms of moles. Feedback 1: Incorrect! The conceptual model contains mostly molar quantities. You use atom balances only when you know that chemical reactions are occurring, but you have NOT be given what those reactions are. If you do not know the reactions, you can not write the extent-of-reaction terms [ R s or ξ s ( Xi) ] that appear in the mole balances. Feedback 2: Incorrect! The conceptual model contains mostly molar quantities. You rarely use mass balances when chemical reactions are occurring, because the resulting equations tend to be more complex than the mole balances. Feedback 3: Correct! Since the conceptual model contains mostly molar quantities, you use mole balances when chemical reactions are occurring, and you know the chemical reactions. The mole balance equations tend not to be as complex as the mass balance equations v , Michael E. Hanyak, Jr., All Rights Reserved Page 8 of 22

16 Continuous Reactor v , Michael E. Hanyak, Jr., All Rights Reserved 2 of 6

17 Interaction 9: Topic: Simple Reactor Background: Guidelines 1 to 4 produced the six material balance equations shown in your PAPER COPY of the template document for the mathematical model. Guideline 5 adds any additional assumptions to the conceptual model. For this example, no additional assumptions where added. Mathematical Model Question: At this point in the development of the mathematical model, you should inspect the material balances for correctness. For example, the reaction terms in the component balances should sum to the same reaction term in the total balance. Component flow rate terms with mole fractions should sum to a term in the total balance. Component flow rate terms without mole fractions should have a proper sign, positive for an incoming flow and negative for an outgoing flow. Using Guideline 6, you are to finish writing the mixture equation for Stream P in the check equation on your PAPER COPY of the template document, as indicated by the yellow highlight to the right of this script page. Guideline 7 was used to select the CHECK equation, and the application of Guideline 8 resulted in numbering the linear independent equations in the mathematical model. Applying Guideline 9, what are the degrees of freedom for the first six linear independent equations? Option 1: zero. Option 2: one. Option 3: two. Feedback 1: Incorrect! A DOF = 0 means that you have as many unknown variables as equations. Since a flow rate is known in the problem and we have not used it yet in the mathematical model, the DOF cannot be zero. Try counting again. Feedback 2: Incorrect! A DOF = 1 means that you have one more unknown variable as you do equations. However, the degrees of freedom are greater than one for the first six equations. Try counting again. Feedback 3: Correct! The DOF for the first six equations is two. Since a flow rate is known in the problem, we would like to have the DOF be one. We need to add one or more equations to the math model, in order to get the DOF to become one. v , Michael E. Hanyak, Jr., All Rights Reserved Page 9 of 22

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19 Interaction 10: Topic: Simple Reactor Background: Guidelines 6 and 7 produced the mixture equation and identified it as the CHECK. Guideline 9 lead to the degrees of freedom (DOF) equal to two. Mathematical Model Question: Using Guideline 10, you should inspect the conceptual model to see if any information has not been used yet in the mathematical model. Based on that inspection, you are to finish writing the conversion equation (Eq. 7) and re-calculating the DOF on your PAPER COPY of the template document, as indicated by the yellow highlights to the right of this script page. The conversion equation increased the equation count by one, but it did not increase the variable count. Since the DOF now equals 1, is a molar flow rate given in the conceptual model? Option 1: yes. Option 2: no. Feedback 1: Incorrect! A molar flow rate is NOT given in the conceptual model; however, a volumetric flow rate for Stream F is known. Feedback 2: Correct! A volumetric flow rate for Stream F is given. You must find a way to relate the volumetric flow rate to the molar flow rate that appears in the material balances. v , Michael E. Hanyak, Jr., All Rights Reserved Page 10 of 22

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21 Interaction 11: Topic: Simple Reactor Background: Using Guide 11, you are to write an equation in the mathematical model that relates the volumetric and molar flow rates of Stream F in Eq. 8 on your PAPER COPY of the template document, as indicated by the yellow highlight to the right of this script page. The third assumption in the conceptual model provides a clue for you. Mathematical Model Question: Also, you are to re-calculate the degrees of freedom. What variables in the conceptual model do you know that would satisfy the DOF in the mathematical model? Option 1: the T and P of Stream F and the T and P of Stream P. Option 2: the T and P of Stream F, as well as its two mole fractions. Option 3: the T and P of Stream F, its volumetric flow rate, and the gas constant. Feedback 1: Incorrect! See the explanation under Feedback 3. Feedback 2: Incorrect! See the explanation under Feedback 3. Feedback 3: Correct! The re-calculate DOF is four. Since the ideal gas law was applied to Stream F, its temperature, pressure, and volumetric flow rate are known. You can find values for the gas constant in the back cover of the Felder and Rousseau textbook. v , Michael E. Hanyak, Jr., All Rights Reserved Page 11 of 22

22 Continuous Reactor v , Michael E. Hanyak, Jr., All Rights Reserved 2 of 6

23 Interaction 12: Topic: Simple Reactor Background: Using Guide 12, you are to write the additional equations need to complete the mathematical model (i.e., Eqs. 9 to 15) on your PAPER COPY of the template document, as indicated by the yellow highlights to the right of this script page. Mathematical Model Question: Using Guide 13, you are to show the degrees of freedom (DOF) for the completed math model. After reviewing Guide 14, what are the five additional variables needed to satisfy the DOF of nine? Option 1: the T and P of Stream P, the two mole fractions of Stream F, and the molar conversion appearing in the conceptual model. Option 2: the molecular weights of the five chemical components (i.e., compounds) appearing in the mathematical model. Feedback 1: Incorrect! The T and P of Stream P are not required to solve the problem. The two mole fractions and molar conversion already appear in the mathematical model as numbers. Feedback 2: Correct! The five molecular weights are considered constants in the mathematical model, since you can look them up in Table B.1 of the Felder and Rousseau textbook. To simplify the counting process in future problems that we will solve, we will establish the convention NOT to count constants, like the gas constant and molecular weight, as variables. If we had followed this convention for this problem, the DOF would be three and not nine. Before continuing, close the development of a mathematical model browser window. v , Michael E. Hanyak, Jr., All Rights Reserved Page 12 of 22

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25 Interaction 13: Topic: Simple Reactor Background: The third step in the problem-solving methodology (PSM) is to transform the mathematical model into a mathematical algorithm. A mathematical algorithm does not tell you how to solve, but it identifies the order in which the equations are to be solved. Mathematical Algorithm Question: Click here to see the guidelines for the development of a mathematical algorithm. Using Guideline 1, you are to finish writing the functional form for the "reactor" mathematical algorithm on Page 3 in your PAPER COPY of the template document, as indicated by the yellow highlights to the right of this script page. For this problem only, we are counting constants as being variables. Thus, we must list those constants as independent variables in the functional form for the mathematical algorithm. In future problems, we will not list constants because we will not count them as variables. Could a mathematical model have more than one mathematical algorithm? Option 1: yes. Option 2: no. Feedback 1: Correct! A problem could have two parts that use the same mathematical model. One part might ask to solve the mathematical model for a specific set of known variables. The other part could be for another set of known variables. The two mathematical algorithms would be different, but they both would have the same mathematical model. Feedback 2: Incorrect! See the explanation under Feedback 1. v , Michael E. Hanyak, Jr., All Rights Reserved Page 13 of 22

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27 Interaction 14: Topic: Simple Reactor Background: Using Guidelines 2 to 3, identify the first equation and write it as the first step in the mathematical algorithm on Page 3 in your PAPER COPY of the template document, as indicated by the yellow highlights to the right of this script page. Using Steps 3 to 6 of the Partitioning Procedure, you are to complete the development of the mathematical algorithm on your PAPER COPY of the template document, as indicated by the yellow highlights to the right of this script page. v , Michael E. Hanyak, Jr., All Rights Reserved Page 14 of 22

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29 Interaction 15: Topic: Simple Reactor Background: Inspect the complete mathematical algorithm to the right of this script page for correctness. Start with the first assignment; look at each variable in the expression on the right side of the arrow. Ask yourself if this variable is specified or a constant; that is, appears in the right side of the functional form for the mathematical algorithm. If yes, move on to the next variable in the expression. If it is not a specified variable or constant, then it must be a calculated variable. Ask yourself, has it been calculated by a previous step in the mathematical algorithm. If yes, then move on to the next variable in the expression checking it for correctness. If all variables in the expression have been checked, then move on to the next assignment and check it for correctness. If no, then the assignment is not in the correct position of the mathematical algorithm. Fix it by repositioning it to a correct place. Mathematical Algorithm The mathematical algorithm to the right of this script page has eleven Question: steps, all of which are assignment statements. Why is that (select one or more that apply)? Option 1: because Step 6 in the Partitioning Procedure did not apply. Option 2: because Step 4 in the Partitioning Procedure did apply. Option 3: because Step 3 in the Partitioning Procedure did apply. Option 4: because Step 5 in the Partitioning Procedure did apply. Feedback 1: Correct! Step 6 of the Partitioning Procedure was never used. If it was used, the mathematical algorithm would have fewer steps, because a nonlinear set of equations would need to be solved simultaneously. Thus, this NSOLVE construct would be counted as one step, but it would contain more than one equation. Feedback 2: Incorrect! Step 4 of the Partitioning Procedure was never used. If it was, it would have created an assignment for an equation. Feedback 3: Correct! Step 3 of the Partitioning Procedure was the only one needed to transform the mathematical model into the mathematical algorithm. Steps 4 to 6 were not used, because all of the equations could be solved one by one. Feedback 4: Incorrect! Step 5 of the Partitioning Procedure was never used. If it was used, the mathematical algorithm would have fewer steps, because a linear set of equations would need to be solved simultaneously. Thus, this SOLVE construct would be counted as one step, but it would contain more than one equation. v , Michael E. Hanyak, Jr., All Rights Reserved Page 15 of 22

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31 Interaction 16: Topic: Simple Reactor Background: After completing the Partitioning Procedure, some equations in the mathematical algorithm must be analyzed for consistency of units. Click here to view the guidelines for the Dimensional Consistency Analysis, and then click here to view the results of its application to the mathematical algorithm on the right. Mathematical Algorithm Question: Two forms of the Check equation exist -- a standard form and its normalized form. The standard form says the total flow rate is equal to the sum of its component flow rates. The normalized form says the sum of the composition quantities must equal one. The normalized form is gotten by dividing the total flow rate through both sides of the standard form equation. These two forms are shown after the mathematical algorithm on Page 3 in your PAPER COPY of the template document and also as Page 3 to the right of this script page, only to make a point. We will not write the check equation as being part of the mathematical algorithm. You must choose which form to use when doing your check on the Numerical Solution under the Heuristic Observation part of the Problem-Solving Methodology. Which CHECK equation should be chosen, because it will check more steps in the mathematical algorithm? Option 1: standard form; that is, total = sum of the components. Option 2: normalized form; that is, component composition must sum to one. Feedback 1: Incorrect! The standard form would only check the first eight steps in the mathematical algorithm. Feedback 2: Correct! The normalized form would check the first thirteen steps in the mathematical algorithm. The last two steps would be checked using the total mass balance. Before continuing, close the development of a mathematical algorithm browser window. v , Michael E. Hanyak, Jr., All Rights Reserved Page 16 of 22

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33 Interaction 17: Topic: Simple Reactor Background: The fourth step in the problem-solving methodology (PSM) is to generate the Numerical Solution using the mathematical algorithm as a blueprint or guide. Click here to see the guidelines for the development of a numerical solution. Using Guidelines 1 and 2, you are to finish writing the basis and the givens on Page 4 in your PAPER COPY of the template document, as indicated by the yellow highlights to the right of this script page. Also, you are to finish the calculations on Page 4, using Guideline 3. Numerical Solution Because a chemical reaction is occurring, the extent of reaction Question: [ R or ξ ( Xi) ] appears in the total and component mole balances. What are the proper units for the extent of reaction? Option 1: kg-mol/h. Option 2: g-rxns/h. Option 3: g-mol/h. Option 4: kg-rxns/h. Feedback 1: Incorrect! The units of kg-mol/h represent the number of molecules flowing per unit of time. Feedback 2: Incorrect! The units of g-rxns/h represent the number of chemical reactions occurring per unit of time, when the molar flow rate has units of g-mol/h. Feedback 3: Incorrect! The units of g-mol/h represent the number of molecules flowing per unit of time. Feedback 4: Correct! The units of kg-rxns/h represent the number of chemical reactions occurring per unit of time [ R or ξ ( Xi) ], when the molar flow rate has units of kg-mol/h. v , Michael E. Hanyak, Jr., All Rights Reserved Page 17 of 22

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35 Interaction 18: Topic: Simple Reactor Background: Using Guidelines 5 to 6, you are to account for precision and box-in the answers on Page 4 in your PAPER COPY of the template document, as indicated by the yellow highlights to the right of this script page. Numerical Solution What is the precision for this problem? Question: Option 1: two digits. Option 2: three digits. Option 3: four digits. Feedback 1: Correct! The "Finds" quantities are to be reported to 2-digits of precision. The least significant digits in the problem statement are two digits. Feedback 2: Incorrect! See the explanation under Feedback 1. Feedback 3: Incorrect! See the explanation under Feedback 1. Before continuing, close the development of a numerical solution browser window.. v , Michael E. Hanyak, Jr., All Rights Reserved Page 18 of 22

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37 Interaction 19: Topic: Simple Reactor Background: The fifth step in the problem-solving methodology (PSM) is to generate the Heuristic Observations about the numerical solution, the mathematical algorithm, the mathematical model, and the conceptual model. Click here to see the guidelines for the development of the heuristic observation. Using Guideline 1, you are to finish writing the heuristic observations for the numerical solution on Page 5 in your PAPER COPY of the template document, as indicated by the yellow highlights to the right of this script page. Heuristic Observations What equation checks the last two steps in the numerical solution? Question: Option 1: the sum of the mole fractions equals one. Option 2: total mass flow in equals total mass flow out. Feedback 1: Incorrect! The normalized mixture equation for Stream P checks only the first thirteen equations in the numerical solution. Feedback 2: Correct! The total mass balance checks the last two steps in the numerical solution. Since total mass is always conserved during chemical reactions, using the total mass balance is a very good check equation, whenever the mathematical model contains molar flow rate variables. v , Michael E. Hanyak, Jr., All Rights Reserved Page 19 of 22

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39 Interaction 20: Topic: Simple Reactor Background: Guideline 2 was applied to generate a heuristic observation. Specifically, how would the mathematical algorithm change if the total molar flow rate of Stream P where known and you wanted to calculate the volumetric flow rate of Stream F? Heuristic Observations Question: The answer to this what if question is shown as the mathematical algorithm on Page 5 in your PAPER COPY of the template document and also as Page 5 to the right of this script page. Compared to the original mathematical algorithm, how complex is the new algorithm? Option 1: less complex. Option 2: about the same. Option 3: more complex. Feedback 1: Incorrect! See the explanation under Feedback 3. Feedback 2: Incorrect! See the explanation under Feedback 3. Feedback 3: Correct! The new algorithm is more complex, because it contains a SOLVE construct. The original algorithm has no such construct. v , Michael E. Hanyak, Jr., All Rights Reserved Page 20 of 22

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41 Interaction 21: Topic: Simple Reactor Background: Guidelines 3 and 4 were applied to give the heuristic observations that answer the following "what if" questions: How would the mathematical model change if the total mole balance was the CHECK equation or if the IDEAL GAS assumption was dropped? How would the conceptual model and the mathematical model change if a second reaction were occurring? The answers to these questions are shown on Page 6 in your PAPER COPY of the template document and also as Page 6 to the right of this script page. Heuristic Observations The function eos[... ] is for an equation of state. Which ONES of the Question: following ways could be used to represent the function "eos"? Option 1: table. Option 2: graph. Option 3: equations. Option 4: computer program. Feedback 1: Correct! Any function can be represented by a table, a graph, a set of equations, or a computer program like HYSYS or Excel Solver. Feedback 2: Correct! See the explanation under Feedback 1. Feedback 3: Correct! See the explanation under Feedback 1. Feedback 4: Correct! See the explanation under Feedback 1. Before continuing, close the development of heuristic observations browser window.. v , Michael E. Hanyak, Jr., All Rights Reserved Page 21 of 22

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43 Interaction 22: Topic: Simple Reactor Background: Thank you for completing this problem session. Please place your filled-in PAPER COPY of the template document in your technical journal. Click here for the correct solution to the template document. If you so desire, you could print this eleaps script (two pages per sheet and on both side of a sheet) and place it also in your technical journal. If you have any questions or concerns about the problem session, please contact your instructor. Read the Important Observations below and consult the Reference Readings and Review Materials. Problem Session Observations The problem solving methodology is a multi-step procedure to solve the problem. The differential material balances are used for a continuous process. The conceptual model concisely presents the problem in terms of variables. The mathematical model presents the first principles to model the problem. The mathematical algorithm presents the order in which to solve the equations. The numerical solution identifies the basis and presents the final answers. The heuristic observations provide important reflections about the solution. Reference Readings Felder and Rousseau, 3 rd Edition, Section 4.1, Process Classification. Felder and Rousseau, 3 rd Edition, Section 4.2, Balances. Felder and Rousseau, 3 rd Edition, Section 4.3, Material Balance Calculations. Felder and Rousseau, 3 rd Edition, Section 4.6, Chemical Reaction Stoichiometry. Felder and Rousseau, 3 rd Edition, Section 4.7, Balances on Reactive Processes. Felder and Rousseau, 3 rd Edition, Chapter 3, Processes and Process Variables. Review Materials Hanyak s Development of a Conceptual Model, CinChE Manual, Chapter 4. Hanyak s Development of a Mathematical Model, CinChE Manual, Chapter 4. Hanyak s Development of a Mathematical Algorithm, CinChE Manual, Chapter 4. Hanyak s Development of a Numerical Solution, CinChE Manual, Chapter 4. Hanyak s Development of the Heuristic Observations, CinChE Manual, Chapter 4. Hanyak s Anatomy of a Math Model and Algorithm, CinChE Manual, Chapter 4. v , Michael E. Hanyak, Jr., All Rights Reserved Page 22 of 22