CinChE Problem-Solving Strategy Chapter 4 Development of a Mathematical Algorithm. partitioning. procedure

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1 Development o a Mathematical Algorithm Transormation Process Mathematical Model partitioning procedure Mathematical Algorithm The mathematical algorithm is a plan (or blueprint) that identiies the independent variables which satisy the degrees o reedom (DOF) and states the order in which the equations rom the mathematical model are to be solved. The systematic partitioning procedure described below moves equations rom the mathematical model to the mathematical algorithm. Partitioning Procedure. Deine the utional equation or the mathematical algorithm as ollows: [ dependent variables ] [ independent variables ] where the independent variables are those variables in the mathematical model that have known values. The number o independent variables must equal the degrees o reedom (DOF); that is, # o variables minus # o equations. The dependent variables are either all o the unknown variables in the mathematical model or ust those unknown variables o primary interest. We oten ust list the Finds quantities or the dependent variables. The symbol implies a ution. Basically, a utional equation says that knowing the independent variables you can determine the dependent variables. Oten, we replace the symbol with a more descriptive name, like reactor or ilterpress. The steps in the mathematical algorithm illustrate the order in with the equations must be solved to ind the unknown variables. The remainder o this partitioning procedure will help you to develop those steps in the mathematical algorithm. 2. Underline the known variables in the mathematical model using a red peil or pen. These underlining variables are known in the equations. Some o these known variables could be values that you will look up later in a literature source such as the ideal gas constant or the molecular weights o pure chemical substaes. 3. an algebraic equation contains only one unknown variable, then a) Solve algebraically or that unknown variable and simpliy its expression. b) Remove the resulting equation rom the math model, place it as the next step in the math algorithm, and replace its equals sign with a let arrow ( ). c) Circle this unknown variable in the remaining equations o the math model. d) Reapply Step an unknown variable appears in only one equation, then a) Solve algebraically or that unknown variable and simpliy its expression. v , Michael E. Hanyak, Jr., All Rights Reserved Page 4-2

2 Development o a Mathematical Algorithm b) Remove the resulting equation rom the math model, place it beore the next-tothe-last step that was placed in the math algorithm, and replace its equals sign with a let arrow ( ). c) Reapply Step a linear set o algebraic equations exists (i.e., as many unknown variables as equations), then a) Find the smallest set (i.e., 2x2, 3x3, 4x4, etc.) and algebraically rearrange each linear equation in the set into the orm: unknown terms constant(s). b) Remove the rearranged set rom the math model and place it as the next step in the math algorithm using a SOE construct. For example, or a (3x3) set: SOE m FH, m W, m FW in m FH.25 m W m P.45 m W + m FW x FH m FH.35 m FW where the unknown terms are lined up vertically in the set o equations. The three unknown variables (m FH, m W, and m FW ) are to be solved simultaneously using the three coupled linear equations. The symbol "x FH " and "m P " are either a known (i.e., given) variable or a calculated variable that appears in an earlier step o the math algorithm. c) Circle the to-be-solved variables in the remaining equations o the math model. d) Reapply Step a nonlinear set o algebraic equations exists (as many unknown variables as equations), then a) Find the smallest set (i.e., 2x2, 3x3, 4x4, etc.) and algebraically rearrange each equation in the set into the orm: unknown terms plus any constant terms. b) Remove the rearranged set rom the math model and place it as the next step in the math algorithm using a NSOE construct. For example, or a (3x3) set: NSOE A, B, in A 27 X 2 / B X / (m + A / 2 ) ( B) 8 Y The three unknown variables (A, B, and ) are to be solved simultaneously in the coupled nonlinear equations. The symbol "m", "X", and "Y" are either a known (i.e., given) variable or a calculated variable that appears in an earlier step o the math algorithm. c) Circle the to-be-solved variables in the remaining equations o the math model. d) Reapply Step no additional equations remain in the math model, then stop applying the partitioning procedure. You have moved all equations rom the math model to the math algorithm. v , Michael E. Hanyak, Jr., All Rights Reserved Page 4-3

3 Development o a Mathematical Algorithm Dimensional Consistey Analysis Ater completing the above Partitioning Procedure, some equations in the mathematical model must be checked or consistey o units beore doing the numerical solution. the net units o terms in any equation o the math model do not match, then unit conversions must be applied to that equation to make it dimensionally consistent. Consider the ollowing example math model and its algorithm: Example Mathematical Model and ts Algorithm or a Continuous Reactor Givens: Feed at lb m /h; Product at 2ºC and 2 psig; P b atm This math model and its algorithm is based on a Finds: Product in mol/s, mole ractions, and / min modiied version o the math model on Page 4-9. Mathematical Model: Mathematical Algorithm: [ n, x, ] reactor[ m, T, P ] () total: n F n P R (2) (3) (4) H 2 :.5 n F n PH, 2 2 R O 2 :. n F n PO, 2 R N 2 :.4 n F n PN, 2 (5) H O: n + 2 R 2 P, WA ( ) (6) conv :. n n.5. n F PO, 2 F (7) MF m F / n F (8) M.5M +.M +.4M (9to) xp, n P, / n P or H 2, O2, N 2 (2) n RT / P F H2 O2 N2 P P P P P P P F P P (8). M.5M +.M +.4M (7) 2. n F m F / MF (4) (6) n PN, 2.4n F n PO, 2 (.5)(. n F ) (3) 5. R. n n / F H2 O2 N2 (, 2) ( ) F PO (2) (5) n PH, 2.5 n F n PH, 2 n P, WA 2 R / 2 () 8. n P n F R (9to) 9. xp n P / n P (2). n RT / P,, or H 2, O2, N 2 P P P P The units consistey analysis or this example is shown below. First, you list the base units or amount, time, and energy. Sie these are the units o variables appearing in the material balaes (Eqs. to 5), mixture equations (none), composition equations (Eqs. 9 to ), reaction conversion (Eq. 6), reaction yields or selectivities (none), and energy balae (none), no units analysis is required on these types o equations in the math model. Second, all other remaining equations, except or mixture molecular weights, must be checked. As exempliied below, only those remaining equations that require units conversion are listed in the units consistey analysis using the appropriate steps rom the math algorithm and the variable units rom the Givens and Finds ound in the coeptual model. Note that you are to list both the model equation and algorithm step numbers in this analysis. Example o the Units Consistey Analysis Base Units: mole, second, no energy, liter {indicate length or volume only when needed} (7) 2. (3) 5. (2). mol lbm lb mol g mol h s h lb lb mol 36 s R in m g rxns g mol g rxns s s g mol ( C + ) mol K.826 atm 6 s min s atm mol K min psig + atm psi Finally, you are to provide the units or one extent-o-reaction variable. The conversion actors rom the units consistey analysis must be added to the math model or an E-Z Solve numerical solution, or these same actors must be added to the math algorithm or a manual or Excel numerical solution. v , Michael E. Hanyak, Jr., All Rights Reserved Page 4-4

4 Development o a Mathematical Algorithm SOE and NSOE Transormations Ater completing the above Partitioning Procedure, the mathematical algorithm may contain one or more linear SOE constructs and/or one or more nonlinear NSOE constructs. a sotware system like E-Z Solve, Mathcad, or Mathematica will be used to solve simultaneously all equations in the mathematical model, then urther processing o the SOE and NSOE constructs is not necessary. When the numerical solution is going to be done manually, each SOE and NSOE constructs must be urther transormed within the mathematical algorithm as described below. SOE Construct { transormed only when the numerical solution is going to be done manually } When the size o a linear SOE is greater than (3x3), urther processing o this type o construct in the mathematical algorithm is not necessary, because it will be solved by automated means in the numerical solution using a sotware system like your pocket calculator, Excel, Matlab, or E-Z Solve. For a SOE with two or three linear algebraic equations, it must to be manually transormed within the mathematical algorithm using the algebraic operations described in Appendix Sections C.4 and C.5. An example o this transormation is shown next. Symbolic Solution Strategy or a inear Set Multiple irst two equations, each by a constant: X 2Y 2Z Y + 2Z 6 2 X + 4Z mp multiple by +2 multiple by -4 to get the ollowing: 2X 4Y 4Z + 4Y 8 Z 24 2 X + 4Z mp then add the three equations together to eliminate variables X and Y: 8 Z m P 24 Z 24 8 Back substitute into the 3 rd and 2 nd equations to ind X and Y: 24 mp 2 X + 4 mp 8 24 mp Y Check the symbolic solution using the irst equation: X Y 6 m 4 m P 3 m mp 6 mp 2 mp 2 OK P P The above transormation process, called Gaussian Elimination, is to be worked out manually on a separate piece o scrap paper. Then, the yellow-highlighted symbolic equations in the above example v , Michael E. Hanyak, Jr., All Rights Reserved Page 4-5

5 Development o a Mathematical Algorithm would be written to the right o the SOE construct in the mathematical algorithm. NSOE Construct { transormed only when the numerical solution is going to be done manually } Each NSOE construct is to be transormed into an TERATE construct. An TERATE construct indicates that the set o non-linear equations in the NSOE construct will be solved by a trial-anderror technique in the numerical solution. As an example, consider the mathematical model below or the vapor-liquid equilibrium o a pure compound. The ugacity coeicient utions (phi and phi) can be determined rom an equation o state like the Peng-Robinson or Soave-Redlich-Kwong equation. The mathematical algorithm below shows how to ind the saturation temperature (T) or a given pressure (P). t was generated using the Partitioning Procedure described above. Note that utional equations are nonlinear equations. Mathematical Model phi[ T, P] phi[ T, P] # vars 4 #eqns 3 do where subscript means pure component. Mathematical Algorithm [ T ] tsat[ P ] NSOE T,, in phi[ T, P] phi[ T, P] The TERATE construct below implies that the saturation temperature T is to be calculated using a trial-and-error technique. When doing the numerical solution, you would guess a value or T, solve the three equations or (T), and then see how close (T) is to zero. (T) is not close enough, then you would guess a new value or T and then recalculate (T). You continue this iteration process until (T) is close enough to zero, say within two to our decimal digits o zero (e.g.,. to.). TERATE T in phi[ T, P] phi[ T, P] ( T) UNT ( T ) How do you transorm a NSOE construct into an TERATE construct or a scalar unknown variable? all o the unknown variables in the NSOE are all vector quantities like mole ractions and/or distribution coeicients, then the transormation at the end o this section is to be ollowed. However, i the NSOE construct contains at least one scalar unknown variable like temperature or low rate, the transormation process below is to be ollowed: a. Pick a scalar unknown variable appearing in the equations o the NSOE that you want as your iteration variable. n general, you should select this iteration variable such that () you can produce a reasonable irst guess or it, and (2) it appears in the most number o NSOE equations. To the right o the NSOE line in the mathematical algorithm, write the TERATE line using the iteration variable, as illustrated in the above example. At this point in the mathematical algorithm, the iteration variable will have a guessed value. Now, circle this iteration variable wherever it appears in the NSOE equations. v , Michael E. Hanyak, Jr., All Rights Reserved Page 4-6

6 Development o a Mathematical Algorithm n the above example, the three unknowns each appear in two equations, but the saturation temperature (T) was selected as the iteration variable because a reasonable irst guess would be one-hal o the critical temperature (T c ) or the pure chemical compound. Note that the saturation temperature must lie within in the range o to T c in Kelvin. The iteration variable T would be circled in the last two equations o the NSOE because it now has a value. b. Apply Steps 3 to 6 rom the above Partitioning Procedure to move equations rom the NSOE construct to the TERATE construct. Continue this procedure until only one equation is let in the NSOE construct. At this point, this equation will have all o its unknown variables circled. This last equation will become the iteration ution or the TERATE construct. c. Rearrange the last equation in the NSOE so that everything is on the let o the equals sign and zero appears on the right o the equals sign. Move the let-hand expression to the TERATE construct and write as its last equation the ollowing iteration ution: ( iteration_variable ) let-hand expression as illustrated in the above example or the iteration ution (T). d. To the right o the line in the NSOE construct o the mathematical algorithm, complete the TERATE construct by writing the line o UNT (iteration_variable) as illustrated in the above example or (T). e. Check the ust-completed TERATE construct to see i it contains any SOE and/or nested NSOE constructs. For each SOE construct, use the Symbolic Solution Strategy or a inear Set, described above, to transorm it into solvable symbolic equations. For each nested NSOE construct, transorm it into an TERATE construct as described above. What does an TERATE construct or a scalar unknown variable mean mathematically? Basically, you are plotting (x) versus the unknown variable x in order to see where the curve crosses the x-axis, as illustrated below: (x) (x) curve root x When the curve crosses the x-axis, you have a root that satisies the (x) ution. The curve could possibly cross the x-axis several times, thus giving multiple roots that satisy the iteration ution. For example, an iteration ution that is a cubic polynomial could have three real roots. No guarantee exists that the trial-and-error iteration process, done during the numerical solution, will converge to the correct root when multiple roots exist. You are responsible to insure that the iteration process does provide you with the correct root. By using the problem context and a reasonable irst guess or the iteration variable, you hopeully can ind the correct root. An alternative to the trail-and-error technique is to use a numerical technique like the Regula-Falsi method or Newton s Rule, as described in Appendix A.2a to A.2e o the Felder and Rousseau textbook [25], 3 rd Edition. Also, see Appendix A.2h about convergee criteria and a possible alse v , Michael E. Hanyak, Jr., All Rights Reserved Page 4-7

7 Development o a Mathematical Algorithm solution. A alse solution can occur when the (x) curve is relatively lat with the x-axis within some range o x values and crosses over the x-axis within that range o x values. To guard against a alse solution, you should not only see how close (x) is to zero, but you should also see i the last two guesses or the unknown variable x are close in value too. How do you transorm a NSOE construct into an TERATE construct or a vector unknown variable? This type o TERATE construct is where you would be simultaneously iterating on a vector o similar quantities like mole ractions or distribution coeicients. You apply the below transormation process only i all unknown variables in the NSOE are vector quantities. As an example, consider the rigorous mathematical model or the vapor-liquid equilibrium o a multicomponent mixture o chemical compounds. The ugacity coeicient utions (phi and phi) can be determined rom an equation o state like the Peng-Robinson or the Soave-Redlich-Kwong. The mathematical algorithm below shows how to ind the equilibrium temperature (T), liquid composition ( x ), and vapor composition ( y ) or a given pressure (P), vapor raction ( ), and total composition ( z ), ectors x, y, and z each represent mole ractions, with chemical components in the mixture. Mathematical Model. ˆ / ˆ ˆ phi[ T, P, x] ˆ phi[ T, P, y] + z y + x y K x K x y or, 2,..., or, 2,..., or, 2,..., or, 2,..., or, 2,..., x is vector elements x, x2,, x y is vector elements y, y2,, y z is vector elements z, z2,, z K is vector elements K, K,, K 2 Degrees-o-Freedom Analysis: # vars # eqns do + 2 Mathematical Algorithm A [ T, x, y ] vlet[ P,, z ].. ˆ 2. NSOE,,,, ', ˆ T x y K s ' s in ˆ / ˆ ˆ phi[ T, P, x] ˆ phi[ T, P, y] z y x y K x K x y or, 2,..., or, 2,..., or, 2,..., or, 2,..., or, 2,..., When Steps to 7 o the above Partitioning Algorithm are applied to the mathematical model, the mathematical algorithm to the let is the result o that transormation. t produces one direct calculation and a nonlinear solve with (5 + ) unknown variables. The scalar unknown is T, while the other ive unknowns are vector quantities with elements each. Sie the mathematical algorithm to the let contains a NSOE construct, it must be urther transormed as shown in Mathematical Algorithm B below. v , Michael E. Hanyak, Jr., All Rights Reserved Page 4-8

8 Development o a Mathematical Algorithm Mathematical Algorithm B [ T, x, y ] vlet[ P,, z ].. 2. TERATE T in NSOE,,, ˆ ', ˆ x y K s ' s in z y x y K x ˆ / ˆ K ˆ phi[ T, P, x] or, 2,..., or, 2,..., or, 2,..., or, 2,..., ˆ phi[ T, P, y] or, 2,..., ( T) x y UNT ( T ) Sie the NSOE in Mathematical Algorithm A contains one scalar unknown variable, the transormation process described above or a scalar unknown variable is applied. t produces an TERATE construct on the equilibrium temperature T with the iteration ution (T) being the dieree between the sum o the liquid mole ractions minus the sum o the vapor mole ractions. The generation o the TERATE construct produces another NSOE with ive vector quantities, each containing elements; that is, the NSOE has 5 equations with 5 unknowns. Sie the NSOE to the let contains all vector unknown variables, it must be urther transormed as shown in Mathematical Algorithm C below. Transormation to an TERATE Construct or a ector Unknown ariable Mathematical Algorithm C [ T, x, y ] vlet[ P,, z ].. 2. TERATE T in TERATE K in SOE x, y in ˆ phi[ T, P, x] ˆ phi[ T, P, y] ˆ / ˆ K x + y z K x y UNT K K ( T) x y UNT ( T ) or, 2,..., or, 2,..., or, 2,..., or, 2,..., or, 2,..., a. Pick a vector unknown variable in the NSOE to be the iteration vector variable. Select it such that () you can produce a reasonable irst guess or it, and (2) it appears in the most number o equations. b. Place an TERATE line to the right o the existing NSOE line using the iteration variable. At this point, the iteration vector variable will have guessed values. Now, circle these variables wherever they appear in the NSOE equations. c. Apply Steps 3 to 6 rom the above Partitioning Algorithm to move equations rom the NSOE to the TERATE. Continue this procedure until only one vector equation is let in the NSOE, and it must contain the iteration vector variable. d. Solve or the unknown vector variable in last remaining equation o the NSOE. Move it to the TERATE construct and write it as vector_variable right-hand expression with a prime mark ( ) on the vector_variable. e. Complete the TERATE construct by writing the line o UNT vector_variable vector_variable with a bar over each variable to the right o the line in the NSOE.. Process urther any SOE and/or nested NSOE constructs in the newly-generated vector TERATE construct. v , Michael E. Hanyak, Jr., All Rights Reserved Page 4-9

9 Development o a Mathematical Algorithm Sie the inner-most TERATE construct or the vector iteration variable o K in the above Mathematical Algorithm C contains a SOE construct, that SOE must be urther processed using the Symbolic Solution Strategy or a inear Set, described above, in order to transorm it into two solvable symbolic vector equations. The inal mathematical algorithm is shown next or the vapor-liquid equilibrium o a mixture o multiple chemical compounds. Final Mathematical Algorithm or Multicomponent apor-iquid Equilibrium Mathematical Algorithm ET [ T, x, y ] vlet[ P,, z ].. 2. TERATE T in TERATE in ( ) or, 2,..., x z / K + ˆ phi[ T, P, x] ˆ phi[ T, P, y] K y K x ˆ / ˆ K UNT K K ( T) x y UNT ( T ) or, 2,..., or, 2,..., or, 2,..., or, 2,..., Math Algorithm ET contains an outer iteration loop on the equilibrium temperature and an inner iteration loop on the distribution coeicients or each -th component; that is, vector elements K, K 2,, K. When doing the numerical solution, you would guess a value or T, solve all ive vector equations and the one scalar equation in the outer iteration loop or (T), and then see how close (T) is to zero. (T) is not close enough, then you would guess a new value or T and then recalculate (T). You continue this iteration process until (T) is close enough to zero. For each guessed value or T, one must converged simultaneously the vector K in the inner loop using the method o successive substitution. You would guess values or K, K 2,, K, solve the ive vector equations in the inner iteration loop, and then see how close the calculated K, K 2,, K are to the guessed K, K 2,, K. they are not close enough, then you ' would use the calculated K as your new guess K. You continue this inner iteration ' process until each element o K and K are close enough in value. For the outer iteration loop, temperature was selected as the iteration variable because it was the only scalar variable, and one can reasonably generate an initial guess or it; that is, one can use sat T zt where the saturation temperature o each pure component can be ound rom its Antoine equation at the given pressure. For the inner iteration loop, the distribution coeicients were selected as the iteration vector values because one can reasonably generate initial guesses using sat Raoult s aw; that is, K P / P or each -th component where the saturation pressure o each pure component can be ound rom its Antoine equation at the guessed temperature. ariable x or y was not selected as the iteration vector variable, because an initial guess or K is easier to generate. An alternative to the successive substitution technique or convergee in the inner loop is to use the Wegstein method, as described in Appendix A.2 to A.2g o the Felder and Rousseau textbook [25], 3 rd Edition. Also, see Appendix A.2h about convergee criteria and a possible alse solution. v , Michael E. Hanyak, Jr., All Rights Reserved Page 4-2