¡Descarga 10-mathcad-problems-in-chem-eng y más Ejercicios en PDF de Métodos Numéricos solo en Docsity!
Page 1
ABSTRACT
Current personal computers provide exceptional computing capabilities to engineer- ing students that can greatly improve speed and accuracy during sophisticated prob- lem solving. The need to actually create programs for mathematical problem solving has been reduced if not eliminated by available mathematical software packages. This paper summarizes a collection of ten typical problems from throughout the chemical engineering curriculum that requires numerical solutions. These problems involve most of the standard numerical methods familiar to undergraduate engineer- ing students. Complete problem solution sets have been generated by experienced users in six of the leading mathematical software packages. These detailed solutions including a write up and the electronic files for each package are available through the INTERNET at www.che.utexas.edu/cache and via FTP from ftp.engr.uconn.edu/ pub/ASEE/. The written materials illustrate the differences in these mathematical software packages. The electronic files allow hands-on experience with the packages during execution of the actual software packages. This paper and the provided resources should be of considerable value during mathematical problem solving and/ or the selection of a package for classroom or personal use.
iNTRODUCTION
Session 12 of the Chemical Engineering Summer School*^ at Snowbird, Utah on
- (^) The Ch. E. Summer School was sponsored by the Chemical Engineering Division of the American Society for Engineering Education.
Michael B. Cutlip, Department of Chemical Engineering, Box U-222, University of Connecticut, Storrs, CT 06269-3222 ([email protected]) John J. Hwalek, Department of Chemical Engineering, University of Maine, Orono, ME 04469 ([email protected]) H. Eric Nuttall, Department of Chemical and Nuclear Engineering, University of New Mexico, Albuquerque, NM 87134-1341 ([email protected]) Mordechai Shacham, Department of Chemical Engineering, Ben-Gurion Uni- versity of the Negev, Beer Sheva, Israel 84105 ([email protected]) Joseph Brule, John Widmann, Tae Han, and Bruce Finlayson, Department of Chemical Engineering, University of Washington, Seattle, WA 98195- ([email protected]) Edward M. Rosen, EMR Technology Group, 13022 Musket Ct., St. Louis, MO 63146 ([email protected]) Ross Taylor, Department of Chemical Engineering, Clarkson University, Pots- dam, NY 13699-5705 ([email protected])
A COLLECTION OF TEN NUMERICAL PROBLEMS IN
CHEMICAL ENGINEERING SOLVED BY VARIOUS
MATHEMATICAL SOFTWARE PACKAGES
Page 2 A COLLECTION OF TEN NUMERICAL PROBLEMS
August 13, 1997 was concerned with “The Use of Mathematical Software in Chemical Engineering.” This session provided a major overview of three major mathematical software packages (MathCAD, Mathematica, and POLYMATH), and a set of ten problems was distributed that utilizes the basic numerical methods in problems that are appropriate to a variety of chemical engineering subject areas. The problems are titled according to the chemical engineering principles that are used, and the numerical methods required by the mathematical modeling effort are identified. This problem set is summarized in Table 1.
- Problem originally suggested by H. S. Fogler of the University of Michigan
** Problem preparation assistance by N. Brauner of Tel-Aviv University
Table 1 Problem Set for Use with Mathematical Software Packages
SUBJECT AREA PROBLEM TITLE
MATHEMATICAL MODEL PROBLEM
Introduction to Ch. E.
Molar Volume and Compressibility Factor from Van Der Waals Equation
Single Nonlinear Equation
1
Introduction to Ch. E.
Steady State Material Balances on a Sep- aration Train*
Simultaneous Lin- ear Equations
2
Mathematical Methods
Vapor Pressure Data Representation by Polynomials and Equations
Polynomial Fit- ting, Linear and Nonlinear Regres- sion
3
Thermodynamics Reaction Equilibrium for Multiple Gas Phase Reactions*
Simultaneous Nonlinear Equa- tions
4
Fluid Dynamics Terminal Velocity of Falling Particles Single Nonlinear Equation
5
Heat Transfer Unsteady State Heat Exchange in a Series of Agitated Tanks*
Simultaneous ODE’s with known initial conditions.
6
Mass Transfer Diffusion with Chemical Reaction in a One Dimensional Slab
Simultaneous ODE’s with split boundary condi- tions.
7
Separation Processes
Binary Batch Distillation** Simultaneous Dif- ferential and Non- linear Algebraic Equations
8
Reaction Engineering
Reversible, Exothermic, Gas Phase Reac- tion in a Catalytic Reactor*
Simultaneous ODE’s and Alge- braic Equations
9
Process Dynamics and Control
Dynamics of a Heated Tank with PI Tem- perature Control**
Simultaneous Stiff ODE’s
10
Page 4 A COLLECTION OF TEN NUMERICAL PROBLEMS
use in conjunction with their courses.
THE TEN PROBLEM SET The complete problem set is given in the Appendix to this paper. Each problem statement carefully identifies the numerical methods used, the concepts utilized, and the general problem content.
APPENDIX ( Note to Reviewers - The Appendix which follows can either be printed with the article or provided by the authors as a Acrobat PDF file for the disk which normally accompanies the CAEE Journal. File size for the PDF document is about 135 Kb.)
A COLLECTION OF TEN NUMERICAL PROBLEMS Page 5
1. M OLAR VOLUME AND C OMPRESSIBILITY FACTOR FROM VAN D ER WAALS E QUATION
1.1 Numerical Methods
Solution of a single nonlinear algebraic equation.
1.2 Concepts Utilized
Use of the van der Waals equation of state to calculate molar volume and compressibility factor for a gas.
1.3 Course Useage
Introduction to Chemical Engineering, Thermodynamics.
1.4 Problem Statement
The ideal gas law can represent the pressure-volume-temperature (PVT) relationship of gases only at low (near atmospheric) pressures. For higher pressures more complex equations of state should be used. The calculation of the molar volume and the compressibility factor using complex equations of state typically requires a numerical solution when the pressure and temperature are specified. The van der Waals equation of state is given by
where
and
The variables are defined by
P = pressure in atm V = molar volume in liters/g-mol T = temperature in K R = gas constant ( R = 0.08206 atm.liter/g-mol.K) Tc = critical temperature (405.5 K for ammonia) P (^) c = critical pressure (111.3 atm for ammonia)
P a V
+^ ------ 2 -
(^) ( V – b ) = RT
a
R^2 T (^) c^2 P (^) c
b
RT (^) c 8 P (^) c
A COLLECTION OF TEN NUMERICAL PROBLEMS Page 7
2. S TEADY S TATE M ATERIAL BALANCES ON A S EPARATION T RAIN
2.1 Numerical Methods
Solution of simultaneous linear equations.
2.2 Concepts Utilized
Material balances on a steady state process with no recycle.
2.3 Course Useage
Introduction to Chemical Engineering.
2.4 Problem Statement
Xylene, styrene, toluene and benzene are to be separated with the array of distillation columns that is shown below where F, D, B, D1, B1, D2 and B2 are the molar flow rates in mol/min.
15% Xylene 25% Styrene 40% Toluene 20% Benzene
F=70 mol/min
D
B
D 1
B 1
D 2
B 2
7% Xylene 4% Styrene 54% Toluene 35% Benzene
18% Xylene 24% Styrene 42% Toluene 16% Benzene
15% Xylene 10% Styrene 54% Toluene 21% Benzene
24% Xylene 65% Styrene 10% Toluene 1% Benzene
Figure 1 Separation Train
Page 8 A COLLECTION OF TEN NUMERICAL PROBLEMS
Material balances on individual components on the overall separation train yield the equation set
Overall balances and individual component balances on column #2 can be used to determine the molar flow rate and mole fractions from the equation of stream D from
where XDx = mole fraction of Xylene, XDs = mole fraction of Styrene, XDt = mole fraction of Toluene, and XDb = mole fraction of Benzene. Similarly, overall balances and individual component balances on column #3 can be used to determine the molar flow rate and mole fractions of stream B from the equation set
Xylene: 0.07D 1 + 0.18B 1 + 0.15D 2 +0.24B 2 =0.15 × 70
Styrene: 0.04D 1 + 0.24B 1 + 0.10D 2 +0.65B 2 =0.25 × 70
Toluene: 0.54D 1 + 0.42B 1 + 0.54D 2 +0.10B 2 =0.40 × 70
Benzene: 0.35D 1 + 0.16B 1 + 0.21D 2 +0.01B 2 =0.20 × 70
Molar Flow Rates: D = D 1 + B 1
Xylene: XDxD = 0.07D 1 + 0.18B 1 Styrene: XDsD = 0.04D 1 + 0.24B 1 Toluene: XDtD = 0.54D 1 + 0.42B 1 Benzene: XDbD = 0.35D 1 + 0.16B 1
Molar Flow Rates: B = D 2 + B 2
Xylene: XBx B = 0.15D 2 + 0.24B 2 Styrene: XBsB = 0.10D 2 + 0.65B 2 Toluene: XBt B = 0.54D 2 + 0.10B 2 Benzene: XBb B = 0.21D 2 + 0.01B 2
(a) Calculate the molar flow rates of streams D 1 , D 2 , B 1 and B 2. (b) Determine the molar flow rates and compositions of streams B and D.
Page 10 A COLLECTION OF TEN NUMERICAL PROBLEMS
tation when using a least-squares objective function. The Clausius-Clapeyron equation which is useful for the correlation of vapor pressure data is given by
where P is the vapor pressure in mmHg and T is the temperature in °C. Note that the denominator is just the absolute temperature in K. Both A and B are the parameters of the equation which are typi- cally determined by regression. The Antoine equation which is widely used for the representation of vapor pressure data is given by
where typically P is the vapor pressure in mmHg and T is the temperature in °C. Note that this equa- tion has parameters A , B , and C which must be determined by nonlinear regression as it is not possi- ble to linearize this equation. The Antoine equation is equivalent to the Clausius-Clapeyron equation when C = 273.15.
log ( P ) A B T +273.
log ( P ) A B T + C
(a) Regress the data with polynomials having the form of Equation (9). Determine the degree of polynomial which best represents the data. (b) Regress the data using linear regression on Equation (10), the Clausius-Clapeyron equation. (c) Regress the data using nonlinear regression on Equation (11), the Antoine equation.
A COLLECTION OF TEN NUMERICAL PROBLEMS Page 11
4. R EACTION E QUILIBRIUM FOR M ULTIPLE G AS P HASE R EACTIONS
4.1 Numerical Methods
Solution of systems of nonlinear algebraic equations.
4.2 Concepts Utilized
Complex chemical equilibrium calculations involving multiple reactions.
4.3 Course Useage
Thermodynamics or Reaction Engineering.
4.4 Problem Statement
The following reactions are taking place in a constant volume, gas-phase batch reactor.
A system of algebraic equations describes the equilibrium of the above reactions. The nonlinear equilibrium relationships utilize the thermodynamic equilibrium expressions, and the linear relation- ships have been obtained from the stoichiometry of the reactions.
In this equation set and are concentrations of the various species at equilibrium resulting from initial concentrations of only CA0 and CB0. The equilibrium constants KC1 , KC2 and KC3 have known values.
A + B ↔ C + D
B + C ↔ X + Y
A + X ↔ Z
K C 1
C C C D
C A C B
= ---------------- K C 2
C X C Y
C B C C
= ----------------- K C 3
C Z
C A C X
C A = C A 0 – C D – C Z C B = C B 0 – C D – C Y
C C = C D – C Y C Y = C X + C Z
C A , C B , C C , C D , C X , C Y C Z
Solve this system of equations when CA 0 = CB 0 = 1.5, , and starting from four sets of initial estimates. (a) (b) (c)
K C 1 = 1.06 K C 2 = 2.63 K C 3 = 5
C D = C X = C Z = 0
C D = C X = C Z = 1
C D = C X = C Z = 10
A COLLECTION OF TEN NUMERICAL PROBLEMS Page 13
6. H EAT E XCHANGE IN A S ERIES OF TANKS
6.1 Numerical Methods
Solution of simultaneous first order ordinary differential equations.
6.2 Concepts Utilized
Unsteady state energy balances, dynamic response of well mixed heated tanks in series.
6.3 Course Useage
Heat Transfer.
6.4 Problem Statement
Three tanks in series are used to preheat a multicomponent oil solution before it is fed to a distillation column for separation as shown in Figure (2). Each tank is initially filled with 1000 kg of oil at 20 °C. Saturated steam at a temperature of 250 °C condenses within coils immersed in each tank. The oil is fed into the first tank at the rate of 100 kg/min and overflows into the second and the third tanks at the same flow rate. The temperature of the oil fed to the first tank is 20 °C. The tanks are well mixed so that the temperature inside the tanks is uniform, and the outlet stream temperature is the temper- ature within the tank. The heat capacity, Cp , of the oil is 2.0 KJ/kg. For a particular tank, the rate at which heat is transferred to the oil from the steam coil is given by the expression
(18)
where UA = 10 kJ/min· °C is the product of the heat transfer coefficient and the area of the coil for each tank, T = temperature of the oil in the tank in , and Q = rate of heat transferred in kJ/min.
Energy balances can be made on each of the individual tanks. In these balances, the mass flow rate to each tank will remain at the same fixed value. Thus W = W 1 = W 2 = W 3. The mass in each tank
will be assumed constant as the tank volume and oil density are assumed to be constant. Thus M = M 1 = M 2 = M 3. For the first tank, the energy balance can be expressed by
Accumulation = Input - Output
Note that the unsteady state mass balance is not needed for tank 1 or any other tanks since the mass
Q = UA T ( (^) steam – T )
° C
T0 =20oC
W 1 =100 kg/min
Steam
T
Steam
T 2
Steam
T
T1 T2 T 3
Figure 2 Series of Tanks for Oil Heating
MC (^) p
dT 1 dt
----------- = W C (^) p T 0 + UA T ( (^) steam – T 1 ) – W C (^) p T 1
Page 14 A COLLECTION OF TEN NUMERICAL PROBLEMS
in each tank does not change with time. The above differential equation can be rearranged and explic- itly solved for the derivative which is the usual format for numerical solution.
Similarly for the second tank
For the third tank
dT 1 dt
----------- =[ W C (^) p ( T 0 – T 1 ) + UA T ( (^) steam – T 1 )] ⁄( MC (^) p )
dT 2 dt
----------- =[ W C (^) p ( T 1 – T 2 ) + UA T ( (^) steam – T 2 )] ⁄( MC (^) p )
dT 3 dt ----------- =[ W C^^ p ( T^2 – T 3 )^ + UA T (^^ steam – T 3 )]^ ⁄(^ MC^ p )
Determine the steady state temperatures in all three tanks. What time interval will be required for T 3 to reach 99% of this steady state value during startup?
Page 16 A COLLECTION OF TEN NUMERICAL PROBLEMS
(a) Numerically solve Equation (23) with the boundary conditions of (24) and (25) for the case where CA0 = 0.2 kg mol/m^3 , k = 10 -3^ s-1^ , DAB = 1.2 10-9^ m^2 /s, and L = 10 -3^ m. This solution should utilized an ODE solver with a shooting technique and employ Newton’s method or some other technique for converging on the boundary condition given by Equation (25). (b) Compare the concentration profiles over the thickness as predicted by the numerical solu- tion of (a) with the analytical solution of Equation (26).
A COLLECTION OF TEN NUMERICAL PROBLEMS Page 17
8. B INARY BATCH D ISTILLATION
8.1 Numerical Methods Solution of a system of equations comprised of ordinary differential equations and nonlinear algebraic equations.
8.2 Concepts Utilized
Batch distillation of an ideal binary mixture.
8.3 Course Useage
Separation Processes.
8.4 Problem Statement
For a binary batch distillation process involving two components designated 1 and 2, the moles of liq- uid remaining, L , as a function of the mole fraction of the component 2, x 2 , can be expressed by the fol- lowing equation
where k 2 is the vapor liquid equilibrium ratio for component 2. If the system may be considered ideal, the vapor liquid equilibrium ratio can be calculated from where Pi is the vapor pressure of component i and P is the total pressure. A common vapor pressure model is the Antoine equation which utilizes three parameters A , B , and C for component i as given below where T is the temperature in °C.
The temperature in the batch still follow the bubble point curve. The bubble point temperature is defined by the implicit algebraic equation which can be written using the vapor liquid equilibrium ratios as
(29)
Consider a binary mixture of benzene (component 1) and toluene (component 2) which is to be considered as ideal. The Antoine equation constants for benzene are A 1 = 6.90565, B 1 = 1211.033 and C 1 = 220.79. For toluene A 2 = 6.95464, B 2 = 1344.8 and C 2 = 219.482 (Dean^1 ). P is the pressure in mm Hg and T the temperature in °C.
dL dx 2
--------- L
x 2 ( k 2 – 1 )
k (^) i = P (^) i ⁄ P
P (^) i 10
A B T + C –^ --------------- =
k 1 x 1 + k 2 x 2 = 1
The batch distillation of benzene (component 1) and toluene (component 2) mixture is being car- ried out at a pressure of 1.2 atm. Initially, there are 100 moles of liquid in the still, comprised of 60% benzene and 40% toluene (mole fraction basis). Calculate the amount of liquid remaining in the still when concentration of toluene reaches 80%.
A COLLECTION OF TEN NUMERICAL PROBLEMS Page 19
Addition Information The notation used here and the following equations and relationships for this particular problem are adapted from the textbook by Fogler.^2 The problem is to be worked assuming plug flow with no radial gradients of concentrations and temperature at any location within the catalyst bed. The reactor design will use the conversion of A designated by X and the temperature T which are both functions of location within the catalyst bed specified by the catalyst weight W. The general reactor design expression for a catalytic reaction in terms of conversion is a mole balance on reactant A given by
The simple catalytic reaction rate expression for this reversible reaction is
where the rate constant is based on reactant A and follows the Arrhenius expression
and the equilibrium constant variation with temperature can be determined from van’t Hoff’s equa- tion with
The stoichiometry for and the stoichiometric table for a gas allow the concentrations to be expressed as a function of conversion and temperature while allowing for volumetric changes due to decrease in moles during the reaction. Therefore
and
(a) Plot the conversion (X), reduced pressure (y) and temperature (T × 10 -3^ ) along the reactor from W = 0 kg up to W = 20 kg. (b) Around 16 kg of catalyst you will observe a “knee” in the conversion profile. Explain why this knee occurs and what parameters affect the knee. (c) Plot the concentration profiles for reactant A and product C from W = 0 kg up to W = 20 kg.
F A 0
dX dW
--------- (^) =– r ' (^) A
C C
K C
k k (@ T =450° K )
E A
R
T
= exp – ----
∆ C ˜^ P = 0
K C K C ( @ T =450° K )
∆ H R
R
T
= exp – ----
2 A C
C A C A 0
1 – X
1 +ε X
-----------------^
P
P 0
T 0
T
------ C A 0
1 – X
1 – 0.5 X
----------------------^ y
T 0
T
y
P
P 0
C C
0.5 C A 0 X
1 – 0.5 X
------------------------^ y
T 0
T
Page 20 A COLLECTION OF TEN NUMERICAL PROBLEMS
The pressure drop can be expressed as a differential equation (see Fogler^2 for details)
or
The general energy balance may be written at
which for only reactant A in the reactor feed simplifies to
d P P 0
dW
P 0
P
T
T 0
dy dW
T
T 0
dT dW
U (^) a ( T (^) a – T ) + r ' (^) A ( ∆ H (^) R ) F (^) A 0 ( (^) ∑θ i C (^) Pi + X ∆ C ˜^ P )
dT dW
U (^) a ( T (^) a – T ) + r ' (^) A ( ∆ H (^) R ) F (^) A 0 ( C (^) PA )
