CinChE Problem-Solving Strategy Chapter 4 Development of a Mathematical Model. formulation. procedure

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1 nhe roblem-solvng Strategy hapter 4 Transformaton rocess onceptual Model formulaton procedure Mathematcal Model The mathematcal model s an abstracton that represents the engneerng phenomena occurrng n the conceptual model. It contans four thngs a set of equatons, a lst of assumptons, a degrees-of-freedom analyss, and a lst of varable defntons. If standard nomenclature has been defned for all problems, then the varable descrptons are optonal. Mathematcal models are developed from frst prncples; that s, any aoms, laws, or abstractons assumed and regarded as representng the hghest possble degree of generalzaton. The frst prncples n ths ntroductory course are Newton's second law of moton, materal balances, phase equlbrum, and energy balance as appled to chemcal process unts. The systematc formulaton procedure s descrbed below. ormulaton rocedure 1. Decde what frst prncple(s) apply for the problem by analyzng the nds quanttes: When asked to fnd flow rates or amounts ( m,, V or m, n, V ) and/or compostons ( w,, y, z, or cˆ ), you must wrte materal balances. When no chemcal reactons occur or when the chemcal reactons are known, use ths general form to wrte mass or mole balances: d( amount) sys flow n flow out ± reacton = dt When chemcal reactons do occur, but the reactons are not known, use ths form to wrte atom balances, one for each chemcal element: d( atom) sys flow n flow out = dt When asked to fnd heat (Q) and/or work (W), you must wrte the energy balance. That s, apply the general form: d( energy) sys flow n flow out ± Q ± W s = dt. Determne the type of materal balances to wrte mass?, mole?, or atom? Use mass balances when most nformaton s gven n terms of mass. Use mole balances when most nformaton s gven n terms of moles. Use atom balances when no chemcal reactons are gven for a reactor; otherwse, use mole balances when the chemcal reactons are known. Go to Step 3 to wrte mass or mole balances; otherwse, go to Step 4 to wrte atom balances. 3. Wrte the total balance and all component balances, one for each chemcal substance, when no chemcal reactons occur or when the chemcal reactons are known. Wrte total flow rates (or amounts) as symbols, even when ther values are known. Wrte component flow rates (or amounts) as symbols, even when ther values are known. v , Mchael E. Hanyak, Jr., All Rghts Reserved age 4-5

2 nhe roblem-solvng Strategy hapter 4 Wrte total and component balances n lnear form (each term s a number*varable 1 ). Wrte known compostons as numbers n the balances (e.g., 0.35 or 0.45 n f ). Wrte unknown component quanttes as a varable (e.g., N or n f, N) Lne up common terms n ths set of materal balance equatons. Go to Step 5. onsult hapter 5 n ths nhe manual to learn how to wrte the mass or mole balances. 4. Wrte the total mass balance and the atom balances when chemcal reactons occur, but the chemcal reactons are not known. Wrte total flow rates (or amounts) as symbols, even when ther values are known. Wrte component flow rates (or amounts) as symbols, even when ther values are known. Wrte total mass and atom balances n lnear form (each term s a number*varable 1 ). Wrte known compostons as numbers n the balances (e.g., 0.35 or 0.45 n f ). Wrte unknown component quanttes as a varable (e.g., N or n f, N) By the conservaton of mass, total mass s conserved when chemcal reactons occur; however, component masses, component moles, and total moles are not necessarly conserved. onsult hapter 5 n ths nhe manual to learn how to wrte the total mass and atom balances. 5. Update the Lst of Assumptons based on what terms have been dropped from the materal balances. or eample, add no reactons when t s applcable. Note that the absence of a flow n or flow out term s already mpled n the assumpton for the process type; that s, batch, sem-batch, and sem-contnuous process. Also, the assumpton for a process operaton of steady state already mples that the term to the rght of the equal sgn n any balance s zero. 6. Wrte mture equatons, one for each mture that has a least one unknown composton. or a flowng mture (stream), wrte total flow equals the sum of the component flows. or a statonary mture, wrte total amount equals the sum of the component amounts. Wrte known compostons as numbers n mture equatons (e.g., 0.35 or 0.45 n f ). If all mtures n the conceptual model have known compostons, then no mture equatons are wrtten. See hapter 5 n the nhe manual to wrte the mture and composton equatons for a contnuous, batch, sem-batch, or sem-contnuous process. 7. Select one wrtten lnear equaton as the check equaton, because t s not lnearly ndependent. A lnear equaton s dependent f: 1) t s a multple of one of the other equatons, or ) t can be form by algebracally combnng one or more the other equatons together. or mass or mole balances, proceed as follows: All wrtten component balances and mture equatons can be algebracally combned to form the total balance equaton. Thus, one of the lnear equatons s not ndependent. The last wrtten equaton (usually a mture equaton) s to be chosen as the dependent or check equaton. or atom balances, proceed as follows: The total mass balance s chosen as the dependent or check equaton. The chosen dependent equaton s not part of the math model, but t s labeled wth the symbol or the word check whch s crcled. Ths equaton wll be used later as a comprehensve check for any human errors that mght occur when manually solvng the materal balances n the Numercal Soluton. v , Mchael E. Hanyak, Jr., All Rghts Reserved age 4-6

3 nhe roblem-solvng Strategy hapter 4 8. Label the ndependent equatons, wrtten thus far n the mathematcal model, wth numbers begnnng wth one. Also crcle those numbers, n order to dentfy those equatons as beng part of the mathematcal model. or eample,,,,,, etc. 9. Mentally, do a degrees-of-freedom analyss on ths lnear set of ndependent equatons; that s, subtract the number of equatons from the number of varables (DO = # vars - # eqns). If the DO <= zero, then these equatons were not properly wrtten. If the DO >= one, then you have properly wrtten these equatons. The DO must equal the number of flow rates (or amounts) known n the conceptual model, before you can solve for all of the unknown varables n the lnear set. If no flow rates (or amounts) are known n the conceptual model, then you must assume one. That s, you pck a bass for a flow rate (or amount) on whch to do the calculatons later n the Numercal Soluton. Add ths bass to the Lst of Assumptons; for eample, = 100 mol / h or m 100 kg. = Symbols that represent constants lke R for the gas constant and M for molecular weght are not counted as varables n the mathematcal model. 10. If the DO s greater than the number of flow rates (or amounts) known n the conceptual model (mass, molar, and/or volumetrc quanttes), you must add more equatons to the mathematcal model untl ts DO becomes equal to number of flow rates (or amounts) known n the conceptual model. heck the conceptual model to see f any nformaton has not been used yet. Wrte an equaton for each pece of nformaton that has not be used. or eample, molar reacton converson, reacton yeld, ecess reactants, ecess ar, etc. These equatons should contan varables that appear n the materal balances. Thus, they usually do not ncrease the varable count but ncrease the equaton count n the DO analyss. Re-calculate the DO usng all equatons that now appear n the mathematcal model. If t does not equal the number of flow rates (or amounts) known n the conceptual model, then add more equatons, as descrbed below, untl t does. 11. Based on what s known (mass, molar, or volumetrc quanttes) and what varables are used n the materal balances (mass or molar quanttes), you can proceed one of two ways: If the known and balance quanttes are the same (.e., mass or moles), then suffcent equatons est n the lnear set to solve for all unknown flow rates (or amounts). Thus, proceed to Step 1 below. If the known and balance quanttes are not the same, then more equatons must be added to the mathematcal model to relate known quanttes to the unknown quanttes used n the materal balance. Wrte equatons to go from V m n or V m n, and then re-calculate the DO for all equatons that are now n the mathematcal model. onsult Table gotche n hapter 3 of the nhe manual for the nterrelatonshps between mass, molar, and volumetrc quanttes. 1. Eamne the nds quanttes and add more equatons to the mathematcal model, f necessary. Wrte equatons to relate the mass or molar quanttes n the materal balances to any mass, molar, or volumetrc flow rates (or amounts) n the nds lst. That s, wrte equatons to go from m V or n m V. Wrte composton equatons when mass fractons, mole fractons, volume fractons and/or concentratons are to be found. or eample, wrte for mass or mole fractons: v , Mchael E. Hanyak, Jr., All Rghts Reserved age 4-7

4 nhe roblem-solvng Strategy hapter 4 m j= m w j or, j= j wrtten for each component j or m j= m w j or n j= n j wrtten for each component j onsult Table gotche n hapter 3 of the nhe manual for the nterrelatonshps between mass, molar, and volumetrc quanttes. 13. Show a degrees-of-freedom (DO) analyss for all equatons wrtten thus far n the mathematcal model. 14. heck f the DO agrees wth the number of known varables that appear n the conceptual model. Remember that symbols for constants lke R and M are not counted as varables. Numerous constants lke molecular weghts can be found n lterature tables. or eample, Table B.1 n Elementary rncples of hemcal rocesses by elder and Rousseau [005]. If t does not, then you may need to wrte more equatons. If the problem statement drectly or ndrectly ndcates that multple phases (gas, lqud, and/or sold) co-est n the process unt or set of process unts, you wll need to apply approprate frst prncples for phase equlbrum, such as functonal equatons lke Tdp = vlet[, V f = 1, Z ] for vapor-lqud equlbrum and dstrbuton coeffcent relatonshps for water solublty and envronmental parttonng coeffcents. onsult hapter 6 n the nhe manual to fnd the approprate relatonshps for phase equlbrum. After addng the approprate equatons, re-check your DO. 15. If applcable because of what s left n the nds lst (.e., a temperature, pressure, heat, and/or work) that has not been accounted for n the mathematcal model so far, wrte the energy balance wth ts mture enthalpy relatonshps. or eample, the energy to heat process Stream so that ts temperature s hgher n Stream s gven by: ˆ ˆ H H + Q Hˆ = hm[ T, X ] Hˆ = hm[ T,, X ] where Ĥ s the specfc enthalpy of the mture n the -th stream ( or ) n unts of energy/mass or energy/mole. Usng the above procedure, you should be able to develop a workable mathematcal model. The net three pages present three eamples for mathematcal models that were developed usng the above procedure. or these three eamples, hydrogen and oygen react to form water, and 50 mol% of the oygen gets converted (.e., the amount reacted over the amount fed equals 0.50). The frst eample s for a contnuous process where the chemcal reacton s known and mole balances are wrtten to count the molecules. In these mole balances, an etent-of-reacton term s wrtten to account for those molecules beng consumed or produced by chemcal reacton. or eample, term R I accounts for two hydrogen molecules beng consumed per reacton event. The second eample s also for a contnuous process, but the chemcal reacton s presumed not to be known and atom balances are wrtten to count the atoms. The thrd eample s a batch process where the chemcal reacton s known and mole balances are wrtten to count the molecules between the ntal and fnal tmes. Note that no materal flows nto or out of the system boundary for a batch process. v , Mchael E. Hanyak, Jr., All Rghts Reserved age 4-8

5 nhe roblem-solvng Strategy hapter 4 ontnuous Reactor Eample { when the chemcal reactons are gven } Dagram Assumptons: T = 5 = 3 atm h = 100 mol / s, H,, N reactor T 0 = 3 atm h = gas H, O, N, 1. contnuous process. steady state 3. M s deal gas Gvens: nds: Rn I: H + O H O V n L / h 50 mol% converson of O Mathematcal Model Total: 1 R I H : 0.50 H R I O : R I N : 0.40 N H O: n, + R I # vars = 7 M : n = H + + N + # eqns = onverson: deal gas: DO = # vars = 10 V = R T # eqns = 7 DO = 3 Snce three varables [, T, ] and a constant [R] are known, the math model can therefore be solved! The symbol R wthout a dot and subscrpt wll always represent the deal gas constant. The symbol R I wth a dot and/or a subscrpt wll always represent the etent of reacton. Eample net unts for the R g mol of H g rns I term n the H balance are: R I s. What does R I mean? g rns s If no reactons est n a contnuous problem, then the materal balances would not contan R terms. v , Mchael E. Hanyak, Jr., All Rghts Reserved age 4-9

6 nhe roblem-solvng Strategy hapter 4 ontnuous Reactor Eample { when the chemcal reactons are not gven } Dagram Assumptons: T = 5 = 3 atm h = 100 mol / s, H,, N reactor T 0 = 3 atm h = gas H, O, N, 1. contnuous process. steady state 3. M s deal gas Gvens: nds: 50 mol% converson of O V n L / h Mathematcal Model Total: m m H atom: (0.50 ) H n, O atom: (0.10 ) 1 n, N atom: (0.40 ) N # vars = 6 M : n = H + + N + # eqns = onverson: deal gas: DO = # vars = 9 V = R T # eqns = 6 DO = 3 Snce three varables [, T, ] and a constant [R] are known, the math model can therefore be solved! In any atom balance, each term has net unts of number of atoms per tme. or eample, the number of H atoms n the feed stream s gven by the epresson: {note a mol and g-mol are synonymous} g atoms of H g mol of H g mol of (0.50 ) s g mol of H g mol of m s m When the flow rate s kg-mol/h or lb-mol/mn, the net unts are kg-atoms/h or lb-atoms/mn, respectvely. v , Mchael E. Hanyak, Jr., All Rghts Reserved age 4-10

7 nhe roblem-solvng Strategy hapter 4 Batch Reactor Eample Dagram ntal tme, t closed contaner Gvens: T = 5 = 1 atm h n, H,, N Rn I: H + O H O 50 mol% converson of O fnal tme, t f closed contaner nds: Tf 0 f = 1.5 atm hf = gas n f f, H f, =? f, N f, V f n L Assumptons: 1. batch process. unsteady state 3. deal gas behavor 4. n = 100 g-mol Mathematcal Model Total: 1 RI = n f n H : RI = n f,h 0.50 n O : 1 RI = n f, 0.10 n N : 0 = n f, N 0.40 n H O: RI = n f, # vars = 7 M : n f = n f, H + n f, + n f, N + n f, # eqns = n n f, onverson: n V deal gas: f f f f Snce three varables [ n, T, f f DO = v , Mchael E. Hanyak, Jr., All Rghts Reserved age 4-11 # vars = 10 = n RT # eqns = 7 DO = 3 ] and a constant [R] are known, the math model can therefore be solved! The symbol R wthout a dot and subscrpt wll always represent the deal gas constant. The symbol R I wth a subscrpt wll always represent the etent of reacton. Eample net unts for the RI term n g mol of H the H balance are: R I s ( g rns). What does RI mean, physcally? g rns If no reactons est n a batch problem, then the materal balances would not contan RI terms.