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Homework assignments posted here are subject to correction in class or through other means.  Problems as assigned here are for your convenience but are not a substitute for obtaining assignments in class.  

Assignments as issued in class supersede these assignments unless otherwise noted.

Homework Assignment: 1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Homework #1 (Due January 19, 2010)

Modeling and Computing

Problems for Class:

1. You will likely use three primary computer based tools during this course. One will be a programming language, such as FORTRAN or Visual Basic. The second will be a CAS, such as Mathcad or Maple. The last will be a spreadsheet. For the program (or language) you expect to use in each of these three categories, you should determine the number of significant digits in the mantissa for a floating point variable and an integer variable. You may express this as a range of base 10 values. Note that in some cases this will involve consulting a manual or help file, but in others may require another approach.

2. Explain why the machine epsilon is often used as a convergence criterion in an iterative calculation, instead of continuing to iterate until the difference in values between successive iterations is zero.

3. Give one example of an engineering problem where each of the following classes of numerical methods can come in handy. If possible, draw from your experience in class and in readings or from any professional experience you have had to date.

bulletRoots of equations
bulletLinear algebraic equations
bulletCurve fitting; regression and interpolation
bulletOptimization
bulletIntegration
bulletOrdinary differential equations
bulletPartial differential equations

Problems for Submission:

4. Examine your engineering textbooks and find four examples where mathematical models are used to describe the behavior of physical systems. List the independent and dependent variables as the parameters and forcing functions.

5. Problem 3.8 in your text

6P (based on 3.3 in your text) Determine the machine epsilon for the computers in the computer lab. You may program in any language that requires you to manually write the code to determine the machine epsilon (you may not use the MATLAB system variable eps, for example).

You may use one of the templates available on the course web site for programming in VB or FORTRAN. Download the ZIP file and expand it to a directory called epsilon on your Z: drive.

FORTRAN: In your Z:\epsilon\ folder there are several files. Start Compaq Visual Fortran and open the workspace epsilon.dsw in the aforementioned folder. The only source file you need to edit is epsilon.f90. All of the other files, however, are required to generate the working windows program. The comments in epsilon.f90 contain information on the variables already defined for you that you must use for the program to function as intended. You may need to declare other variables in your code.

VB: In your Z:\epsilon\ folder is the MachineEpsilon project file. You will edit the Epsilon module. The form and its associated code, including the interface between the form and the global variables is already created. Note the comments in the module file.

In either language, you can compile the code without editing anything. The results will be meaningless, but you will be able to see how the finished program will operate. There is very little error checking, so take care to use appropriate input, most importantly keep the stepping factor between 0 and 1.

In both cases, submit a printout of your code and a screenshot of the result you obtain. Does the result change if your inputs change? Note that the "proper" way is to start with a value of one with a stepping factor of 0.5.

Assignment Learning Objectives:

bulletEstablish the role of modeling in chemical engineering (3, 4)
bulletAnalyze models in terms of applicable numerical methods (3)
bulletShow the role of the machine epsilon and related truncation errors in computing (1, 2, 5, 6)

Reading Assignments:

bullet Thursday (1/14): Chapters 1-3: Computers and Numerical Methods
bullet Tuesday (1/19): Chapter 4: Taylor Series; Error Propogation and Uncertainty
   
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Homework #2 (Due January 26, 2010)

Taylor Series, Root-Finding

Problems for Class:

1. Examine the Taylor series about x =0 for ex when x=4. The fourth term in the series, x3/3! is larger than the third term, x2/2! Does this mean the series is divergent? Explain the apparent anomaly.

2. 4.12 a, b from your text

3. 5.6 from your text

Problems for Submission:

4. 4.8 from your text

5. In a coffeepot, water flows by gravity from a reservoir into a heated tube. The water is heated from room temperature (20 °C) to 100 °C. Some of the water is vaporized, and the expansion of steam from the volume occupied by the equivalent mass of water to the volume occupied by the steam at atmospheric pressure provides the work to move the water from the heated tube at the base of the coffeepot to the top of the coffeepot. The water then flows through the basket of coffee grounds via gravity. Assuming no energy is lost to the surroundings, what is the minimum amount of energy that must be supplied by the electric heater (which heats the water in the tube) to make 750 mL of coffee? The base of the reservoir is 5 cm above the heated tube. The top of the basket into which the coffee flows is 20 cm above the heated tube. CLEARLY STATE THE MODEL EQUATIONS REQUIRED TO SOLVE THIS PROBLEM.

6. Consider two functions g(x) and h(x), related in such a way that g’(x)=h(x) and h’(x)=g(x) and that g(0)=0 and h(0)=1. Find the Taylor series expansions for g(x) and h(x) about x=0 using only this information

Assignment Learning Objectives:

bulletUse the Taylor series to approximate functions (1, 6)
bulletEstimate the error associated with a series of calculations (4)
bulletDetermine the conditioning of functions (2)
bulletModel systems using chemical engineering principles (5)
bulletApply numerical methods for finding roots of algebraic equations (3)

Reading Assignments:

bulletThursday (1/21): pp. 99-130 Bracketing Methods for Root Finding
bulletTuesday (1/26): Chapter 6: Open Methods for Root Finding
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Homework #3 (Due February 2, 2010)

Open and Closed Case for Root-Finding

Problems for Class:

1.         A cylindrical tank in a horizontal configuration (lying on its side)  has a radius of 2 m and a length of 5 m.  If the volume of liquid in the tank is 8 m3 , find the maximum depth of the liquid.
2.         6.2 (by hand) (no 4th edition equivalent)
3.         Cutlip 2.2  Note that solutions to 2.1 are available on the book website.

Problems for Submission:

4.         6.14 (by spreadsheet) (no 4th edition equivalent)
5.         8.4
6.         8.10
7.         Cutlip 9.3

Assignment Learning Objectives:

bulletDevelop model equations for geometric systems (1)
bulletApply root-finding techniques to chemical engineering models (3, 5, 6, 7)
bulletExamine the importance of functional form in fixed-point iteration (6)
bulletApply numerical methods for finding roots of algebraic equations (1-6)

Reading Assignments:

bulletThursday (1/28):    pp. 176-184; 212-215        Computer applications of root-finding; epilogue
                            pp. 217-230; Chapter 9   Introduction to Linear Systems; Gauss Elimination
bulletTuesday (2/02):     Chapter 11                            Gauss-Siedel (HW3 Due)
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Homework #4 (Due February 9, 2010)

Beat the System

Problem for Class:

1.            Chapra 9.11

2.            Cutlip 8.8a, b

3.            The following reactions are proposed to be descriptive of a chemical process:

     1)  CH4  + CO2   --> 2CO   + 2H2  

     2)  CO   + H2O   --> CO2  + H2  

      3)  CH4  + H2O   --> CO   + 3H2  

      4)  CH4  + 2H2O  --> CO2  + 4H2

Determine how many of these reactions are independent, and give the equations for two of the possible independent reactions.

Problems for Submission:

4.            Chapra 12.9 (no 4th edition equivalent)

5.            Cutlip 7.14

6.            Cutlip 2.3b,c

7.            Consider heat conduction through the walls of a pipe when the inside wall temperature is 200oC and the outside wall temperature is 80oC and the thickness of the wall is 0.05 m.  The inside radius (r0) is 0.05m and the outside radius is 0.1m.  The differential equation that describes the temperature distribution is

 

Applying a finite difference representation of the derivatives of T with respect to r gives a set of equations of the following form:

 

where

 

and                                                                         T1=200 °C and T11=80 °C

Solve for the temperature distribution using a Gauss Siedel program you develop using the Excel/VBA template on the course website. Check your solution by using the matrix inversion approach in Excel.  Remember that r is not constant.

Assignment Learning Objectives:

bulletSolve systems of linear equations using Gaussian Elimination and other methods  (1)
bulletDevelop systems of equations describing chemical engineering systems (2-5)
bulletApply numerical methods for finding roots of systems of algebraic equations (2-6)
bulletSolve systems of linear equations using Gauss-Siedel Iteration (7)
bulletApply concepts of rank and independence to chemical systems (3)

Reading Assignments:

  • Thursday (2/04):                  Chapter 12: Applications, Epilogue: Part III
  • Tuesday (2/09):                    Chapter 17 Curve-Fitting. HW4 Due, HW5 Assigned
  • Thursday (2/11):                  Exam 1 [Modeling, Taylor Series, Root-finding, Linear Systems]
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    Homework #5 (Due February 28, 2010)

    An assignment you’ll always regress

    Problems for Class:
    1. 18.1
    2. 20.2
    3. 20.8
    4. Estimate the enthalpy of superheated steam at 741 oF and 400 psia using both linear and quadratic interpolation.

    Problems for Submission:
    5. Using the following finite difference expression for the second derivative:

    It was found that the error of the approximation was 10-1 using a step size Dx of 10-1. How small should Dx be so that the error of the approximation equals 10-4?

    6. A heterogeneous reaction is known to occur at a rate described by the following Langmuir-Hinshelwood expression:

    The following data was obtained at 400K:

    PA (atm)

    1

    0.9

    0.8

    0.7

    0.6

    0.5

    0.4

    PR (atm)

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    r

    3.4 x 10-5

    3.6 x 10-5

    3.7 x 10-5

    3.9 x 10-5

    4.0 x 10-5

    4.1 x 10-5

    4.2 x 10-5

    A) Using multiple linear regression, try to estimate the values of k1, KA and KR
    B) Try part A) again, this time using nonlinear regression.
    C) The rate constant k1 is later determined from initial rate measurements as 0.015 mol/(s·g -catalyst·atm). Re-estimate KA and KR using this information and linear regression.

    Make sure you report appropriate units with all calculated values!

    7. Consider the following data on the index of refraction of aqueous solutions of sucrose at 20 oC:

    Percent Sucrose

    Index of Refraction

    10

    1.3479

    15

    1.3557

    20

    1.3639

    25

    1.3723

    30

    1.3811

    35

    1.3902

    Estimate the composition of a water/sucrose solution which has an index of refraction of 1.3606 by using interpolation and by using regression.

     

    Assignment Learning Objectives:

    bulletInterpolate amongst known data points using first and second order methods (1, 3, 4, 7)
    bulletApply least-squares regression methods to linear, multi-variable linear, and nonlinear systems (2, 6, 7)
    bulletAnalyze the error associated with numerical differentiation (5)

    Reading Assignments:

    bullet

    Tuesday (2/16):                     Exam 1 [Modeling, Taylor Series, Root-finding, Linear Systems]

    bullet

    Thursday (2/18):                  Chapter 20: Epilogue: Applications of Curve Fitting

    bullet

    Tuesday (2/23):                    Chapter 23: Numerical Differentiation

    bullet

    Thursday (2/25):                  Chapter 21: Numerical Integration

    bullet Tuesday (3/1):                      Part 4 Prologue, Chapter 13: Optimization
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    Homework #6 (Due March 2, 2010)

    The integral of the derivative results in this assignment

    Problems for Class:

    1. Using steam table data, estimate the value of the following integral at 700oF for superheated steam using both the trapezoidal method and Simpson’s rule:

    2. 23.1
    3. 23.14 (using Mathcad or Matlab)
    4. Consider the integration of a known function for a fixed interval. If the step size is cut in half, what would happen to the total truncation error for:
    A. the trapezoidal rule?
    B. Simpson’s rule?

    Problems for Submission:

    5. 24.6
    6. 24.8   You should review the section on Gauss Quadrature for guidance
    7. 24.9
    8. 24.11

    Assignment Learning Objectives:

    bulletApply methods of numerical differentiation to models (2, 3, 5, 7)
    bulletApply methods of numerical integration to model equations (1, 6, 8)
    bulletAnalyze the error associated with numerical integration (4)

    Reading Assignments:

    bulletThursday (2/25): Chapter 21: Numerical Integration
    bulletTuesday (3/2): Part 4 Prologue, Chapter 13: 1-D Optimization (HW 6 Due, HW 7 Assigned)
    bulletThursday (3/4): Chapter 14: Gradient Methods of Optimization.
    bulletTuesday (3/9): Chapter 15: Linear Programming (HW 7 Due, HW 8 Assigned)
    bulletThursday (3/11): Exam 2
    bulletTuesday (3/16): Spring Break
    bulletThursday (3/18): Spring Break
    bulletTuesday (3/23): Felder Rousseau Chapter 11
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    Homework #7 (Due March 9, 2010)

    Optimize your time as you prepare for the exam

    Problems for Class:

    1.            15.11

    2.            16.7

    3.            16.12

    4.             Find a journal article referring to the use of a heuristic (random search) optimization technique in a field related to chemical engineering.  Some algorithms to choose from include: simulated annealing, tabu search, genetic algorithms, and neural networks.  Each member of the class should choose a different method.  I suggest you limit yourself to journals obtainable online.

                    In class, you will be expected to briefly explain the method and its application in a presentation to last no longer than 5 minutes.  Any images you wish to use should be submitted to me electronically at least 1 hour before class.

    Problems for Submission:

    5.            13.11

    6.            16.6

    7.            16.10

    8.            You are required to produce an alloy that has at least 30% Pb and at least 30% Zn by mixing a number of available Pb-Zn-Sn alloys.  Find the cheapest blend using the method of Lagrange Multipliers.

     

    Available Alloys

    Analysis (%)

    Cost ($/lb)

     

    Pb

    Zn

    Sn

     

    1

    20

    20

    60

    6.0
    2

    10

    40

    50

    6.3

    3

    40

    50

    10

    7.5
    4

    50

    30

    20

    8.0

     

    9.            Submit the complete text of the journal article (from problem 4), along with a summary (<1 page) indicating how the technique was used.  You should also include comments on why this method was chosen by the author and provide a brief description of the method.

    Assignment Learning Objectives:

    bullet Apply methods of unidimensional optimization (5)
    bullet Find optimal input values for models using computer software (1, 2, 3, 6, 7)
    bulletFormulate optimization problems (1, 2, 3, 6, 7, 8)
    bulletDevelop optimization models and find optimal parameters for those models using Lagrange multipliers (8)
    bulletDevelop skills required for life-long learning (4, 9)
    bulletDescribe heuristic methods of optimization (4, 9)

    Reading Assignments:

    bulletThursday (3/11): Exam 2
    bulletTuesday (3/16): Spring Break
    bulletThursday (3/18): Spring Break
    bulletTuesday (3/23): Felder Rousseau Chapter 11
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    Homework #8 (Due March 25, 2010)

    Returning to Where the Fun Began

    Problems for Class:

    1. Problem 11.4 from Felder/Rousseau (FR)
    2. Problem 11.25 (FR)

    Problems for Submission:

    3. Problem 11.16 (FR)
    4. Problem 11.26 (FR)

    Assignment Learning Objectives:

    bulletWrite and solve separable differential equations representing material and energy balances

    Reading Assignments:

    bulletTuesday (3/23):                    Felder Rousseau Chapter 11. HW 7 Due, HW 8 Assigned
    bullet Thursday (3/25):                  Chapter 25: Numerical Approaches to solving ODEs, HW8 Due, HW9 Assigned
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    Homework #9 (Due April 1, 2010)

    Don’t be overanalytical

    In addition to the instructions for each problem, you should also:

    bulletCompletely classify the equation you are solving (which determines the analytical method you use to solve)
    bulletPlot your results.

    Problems for Class:

    1. Desalinization plants are used to purify sea water so it is suitable for drinking. Sea water containing 8 g salt/kg solution is pumped into a well-mixed tank at a rate of 10 kg/min. Assume that the balance of the solution is pure water. Because of faulty design work, water is evaporating from the tank at a rate of 0.5 kg/min. The salt solution leaves the tank at a rate of 10 kg/min.

    A) If the tank is filled initially with 1000 kg of the inlet solution, how long after the outlet pump is turned on will the tank run dry?

    B) Determine the salt concentration in the tank as a function of time.

    Problems for Submission:

    2. A spherical ice cube (an "ice sphere") that is 5 cm in diameter is removed from a 0 °C freezer and placed on a mesh screen at room temperature Ta=20 °C. What will be the diameter of the ice cube as a function of time out of the freezer (assuming that all the water that has melted immediately drips through the screen)?

    3. Compound A diffuses through a 4-cm-long tube and reacts as it diffuses. The equation governing diffusion with reaction is

    At one end of the tube, there is a large source of A at a concentration of 0.1 M. At the other end of the tube there is an adsorbent material that quickly adsorbs any A, making the concentration 0M. If D = 1 × 10-6 cm2/s and k = 4 × 10-6 s-1, what is the concentration of A as a function of distance in the tube?

     

    Assignment Learning Objectives:

    bulletWrite and solve separable differential equations representing material and energy balances
    bulletSolve first and second order differential equations using analytical methods

    Reading Assignments:

    bulletThursday (3/25):                      Chapter 25: Numerical Approaches to solving ODEs. HW 8 Due, HW 9 Assigned
    bulletTuesday (3/30):                    Runge-Kutta Methods
    bulletThursday (4/1):                      Chapter 27: Boundary-Value Problems. HW 9 Due, HW 10 Assigned
    bulletTuesday (4/6):                  Review shell balances
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    Homework #10 (Due April 13, 2010)

    Everything Changes With Time

    Problems for Class:

    1. 25.1
    2. 25.8 (Analytically and numerically with Runge-Kutta)
    3. 28.3 (your choice of method)
    4. 28.12

    Problems for Submission:

    5. 10.2 from Cutlip

    6. Consider the semibatch reactor discussed in class, now with the following reaction scheme:

    where concentrations are given in mol/L and the reaction rates in mol/L-sec. Determine the reaction time necessary to attain the maximum concentration of the intermediate B.

    Assignment Learning Objectives:

    bulletWrite and solve separable differential equations representing material and energy balances
    bulletSolve first and second order differential equations using analytical and numerical methods

    Reading Assignments:

    bulletTuesday (4/8):                      Chapter 25: Numerical Approaches to solving ODEs. HW 9 Due, HW 10 Assigned
    bulletThursday (4/10):                  Runge-Kutta Methods, Review shell balances
    bulletTuesday (4/15):                    Chapter 27: Boundary-Value Problems. HW 10 Due, HW 11 Assigned
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    Homework #11 (Due April 22, 2010)

    Oodles of ODEs

    Problems for Submission:

    Problem for Class:
     1. Chapra Problem 28.10 solved using a computer program of your choice.

    Problems for Submission:

    2. Consider the cooling fin pictured below.

    A shell balance results in the following energy balance expression

    where T =Tw at z = 0 and at z = L. When h and k are constant, an analytical solution of this BVP can be obtained. But in general, the heat transfer coefficient h will be a function of z and the thermal conductivity of the fin material k will very with temperature. Assume the following forms for h and k:

    A) Consider the temperature dependent of k and show that the governing equation for this system is

    B) Solve this problem to three significant figures in T using the SOR method and the following values:

    h0 = 40 Btu/hr-ft2-oF k0 = 60 Btu/hr-ft-oF a = 0.02 oF-1
    Tw = 450 oF T0 = 77 oF Ta = 90 oF
    L = 1.5 in B = 0.02 in

    3.            Cutlip Problem 8.3

    Assignment Learning Objectives:

    bulletFind solutions to BVP numerically (2,3)
    bulletUse engineering software to solve systems of ordinary differential equations (1-3)

    Reading Assignments:

    bulletThursday (4/15):                  Modeling and Simulation
    bulletTuesday (4/20):                    HW 11 Due, HW 12 Assigned
    bulletThursday  (4/22):                 Exam 3
    bulletTuesday (4/27):                    HW 12 Due
    bulletThursday (4/29):                  Review
    bulletTuesday (5/4):                      Comprehensive Final Exam (10:45-1:15)
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    Homework #12 (Due April 29, 2010)

    You Can Simulate This If Needed

    Problems for Submission:

    1. Problem 10.13 from Felder-Rousseau (FR).

    2. Develop a simulation in Aspen for the same scenario in 10.13a. Compare results with your spreadsheet and hand calculations. Why the differences? Correct for those differences in your Aspen simulation wherever possible.

    Assignment Learning Objectives:

    bulletUse engineering software to simulate model processes

    Reading Assignments:

    bulletThursday  (4/22):                 Exam 3
    bulletTuesday (4/27):                    HW 12 Due
    bulletThursday (4/29):                  Review
    bulletTuesday (5/4):                      Comprehensive Final Exam (10:45-1:15)
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    Assignment Learning Objectives:

    Reading Assignments:

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    Assignment Learning Objectives:

    Reading Assignments:

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