CHME 7320 (CRN 36904) Spring 2019 PDF

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGIEERING MATHEMATICS CHME 7320-01 (CRN 36904)

CONFERENCE HOURS:

OFFICE HOURS:

MONDAYS WEDNESDAYS

6:20 – 8:00 p.m. 6:20 – 8:00 p.m.

.

CLASSROOM

007 Behrakis Health Sciences Center

By Appointments Only By Appointments Only

PROFESSOR: Dr. BEHROOZ (BARRY) SATVAT, Sc. D., P. E. 465 Snell Engineering Center [email protected] (617) 373-3461 TEACHING ASSISTANTS: NONE

REQUIRED TEXTBOOK : Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

REFERENCES : Jeffrey, Allen, “Advanced Engineering Mathematics”, Harcourt/Academic Press, Massachusetts (2002).

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING

SPRING 2019

CHEMICAL ENGIEERING MATHEMATICS CHME 7320-01 (CRN 36904)

Course Description

Chemical Engineering Mathematics Course includes: Formulation and solutions of problems involving advanced calculus as they arise in chemical engineering systems. Methods covered will include ordinary differential equations, series solutions, and complex variables. Applications involving Laplace transforms, partial differential equations, and matrix operations. Vectors and tensors. Optimizations methods. Emphasis will be on methods for formulating the problems.

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGIEERING MATHEMATICS CHME 7320-01 (CRN 36904) REQUIREMENTS AND POLICIES o There will be homework assignments which will be collected regularly. o No late homework assignments will be accepted. o Regular and punctual class attendance is required. o Students should come to class prepared to discuss homework and reading assignments. o There will be one mid-term exam and one final. o Total of homework assignments makes up 10% of the course grade. o The mid-term exam makes up 40% of the course grade. o The final exam makes up 50% of the course grade. o All Exams will be closed-books and closed-notes. o No programming of notes on your calculators are allowed. o The use of computers is not allowed during the Exams. o With the exception of extraordinary medical situations, no make-ups will be given. o A commitment to the principles of academic integrity is essential to the mission of Northeastern University. The promotion of independent and original scholarship ensures that students derive the most from their educational experience and their pursuit of knowledge. Academic dishonesty violates the most fundamental values of an intellectual community and undermines the achievements of the entire University. Additional details about the policy are located here: http://www.northeastern.edu/osccr/academicintegrity/index.html

MID-TERM EXAM FINAL EXAM

WEDNESDAY WEDNESDAY

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2/27/2019 4/17/2019

NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGIEERING MATHEMATICS CHME 7320-01 (CRN 36904) REQUIRED TEXTBOOK: Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995). MONDAY

1/07/2019

Introduction and Chapter 1

WEDNESDAY

1/09/2019

Chapter 1 and Chapter 2

MONDAY

1/14/2019

Chapter 2

WEDNESDAY

1/16/2019

Chapter 2

MONDAY

1/21/2019

NO CLASS (Martin Luther King Jr.’s Birthday)

WEDNESDAY

1/23/2019

Class Problem Set # 1

MONDAY

1/28/2019

Chapter 3

WEDNESDAY

1/30/2019

Chapter 3 SOLUTION TO HW ASSIGNMENT # 1

MONDAY

2/04/2019

Class Problem Set # 2

WEDNESDAY

2/06/2019

Chapter 4 SOLUTION TO HW ASSIGNMENT # 2

MONDAY

2/11/2019

Chapter 5

WEDNESDAY

2/13/2019

Class Problem Set # 2 SOLUTION TO HW ASSIGNMENT # 3

MONDAY

2/18/2019

NO CLASS (Presidents’ Day)

WEDNESDAY

2/20/2019

Class Problem Set # 3 HW ASSIGNMENT # 5 DUE SOLUTION TO HW ASSIGNMENT # 4

MONDAY

2/25/2019

Class Problem Set # 3 REVIEW FOR THE MID-TERM EXAM

HW ASSIGNMENT # 1 DUE

HW ASSIGNMENT # 2 DUE

HW ASSIGNMENT # 3 DUE

HW ASSIGNMENT # 4 DUE

SOLUTION TO HW ASSIGNMENT # 5 WEDNESDAY

2/27/2019

MID-TERM EXAM (Chapters 1-5)

MONDAY

3/04/2019

NO CLASS (SPRING BREAK)

WEDNESDAY

3/06/2019

NO CLASS (SPRING BREAK)

MONDAY

3/11/2019

SOLUTIONS TO MID-TERM EXAM

WEDNESDAY

3/13/2019

Vector Analysis

MONDAY

3/18/2019

Matrices Application of Matrices in Solving System of Equations

WEDNESDAY

3/20/2019

Chapter 9

MONDAY

3/25/2019

Class Problem Set # 4

WEDNESDAY

3/27/2019

Chapter 9

4

HW ASSIGNMENT # 6 DUE

SOLUTION TO HW ASSIGNMENT # 6

MONDAY

4/01/2019

Chapter 10

HW ASSIGNMENT # 7 DUE

WEDNESDAY

4/03/2019

Chapter 10 SOLUTION TO HW ASSIGNMENT # 7

MONDAY

4/08/2019

Class Problem Set # 5

WEDNESDAY

4/10/2019

REVIEW FOR THE FINAL EXAM SOLUTION TO HW ASSIGNMENT # 8

MONDAY

4/15/2019

NO CLASS (Patriots’ Day)

WEDNESDAY

4/17/2019

FINAL EXAM

HW ASSIGNMENT # 8 DUE

CLASS PROBLEMS & HOMEWORK PROBLEMS

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS CHME 7320-01 (CRN 36904) REQUIRED TEXTBOOK:

Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

CLASS PROBLEM SET # 1 1. Solve: (dy/dx) – y = x y5

2. Consider a steam pipe 20cm in diameter which is insulated with a material 6 cm thick. Thermal conductivity of the insulation is k = 0.0003 cal/sec.0C.cm. Find the heat loss per unit length of the pipe per hour if the surface of the pipe is 2000C and that of the outer surface of the insulation is 300C. 3. Solve: (dy/dx) + y = y2 (Cos x – Sin x) 4. Solve: (2x Sin 3y) dx + (3x2 Cos 3y + 2y) dy = 0 5. As illustrated in chapter 17 of B.S.L. the general equation of diffusion with homogeneous chemical reaction is can be obtained as: -DAB (d2CA/dZ2) + k1’’’ CA = 0

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With the boundary conditions: CA = CA0

at

Z=0

(dCA/dZ) = 0

at

Z=L

Show that:

Cosh { b1[1- (Z/L)]} (CA/CA0) =  Cosh b1

Where: b1 = [ k1’’’L2 / DAB]1/2

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS

CHME 7320-01 (CRN 36904) REQUIRED TEXTBOOK: Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

CLASS PROBLEM SET # 2 6. A rod of length L has one end maintained at temperature T0 and is exposed to an environment of temperature T∞. An electrical heating element is placed in the rod so that heat is generated uniformly along the length at a rate q. Derive an expression (a) for the temperature distribution in the rod and (b) for the total heat transferred to the environment. Obtain an expression for the value of q which will make the heat transfer zero at the end which is maintained at T0. 7. A gas containing an entrained mist is nonvolatile tar is located inside the cylinder of a reciprocating compressor. It is desired to determine the work required to compress the gas adiabatically and reversibly from its present pressure of 0.33 atm to a pressure of 1.0 atm. The following information is available: a) the gas i) ii) iii) iv) v)

vi) b) The tar i) ii) iii) iv)

Molecular weight is 24 Specific heat at constant volume is constant at 6.2 Btu/(lb mole)( 0F) Obeys the perfect-gas equation of state Initial temperature is 1700F or 6300R Initial pressure is 0.33 atm Final pressure will be 1.0 atm The air is always present as a mist in the ratio 0.2 lb of tar per pound of tarfree gas The volume of the tar may be neglected in comparison with the volume of the associated tar-free gas The temperature of the tar is always the same as the temperature of the associated tar-free gas The specific heat of the tar is 0.5 Btu/(lb)( 0F)

c) The compression cylinder i) The initial cylinder volume is 0.4 ft3 ii) The initial cylinder pressure is 0.33 atm iii) The final cylinder pressure will be 1.0 atm iv) The compression process is reversible and adiabatic in the sense that friction and heat transfer to or from the cylinder, piston, and associated machinery may be neglected

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS CHME 7320-01 (CRN 36904) REQUIRED TEXTBOOK: Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

CLASS PROBLEM SET # 2 (Continued) 8.

A substance γ is being formed by the reaction of two substances α and β in which a grams of α and b grams of β form (a+b) grams of γ. If initially there are x0 grams of α, y0 grams of β, and none of γ present and if the rate of formation of γ is proportional to the product of the quantities of α and β uncombined, express the amount (z grams ) of γ formed as a function of time t.

9. Consider the operation of an annealing furnace for a continuous rod material as shown below. The rod moves into the furnace at a constant velocity of magnitude V ft/sec. For steady operation, the temperature of the rod as it passes the entrance to the furnace may be taken as a constant, TE. The heat transfer coefficient for convection from the rod to the air is h Btu/hr.ft2.0F, and the rod material has a conductivity of k Btu/hr.ft.0F. The radius of the rod is r ft. Variation of temperature across the cross section of the rod may be neglected because r is small. TE and T0 are constant. The rod may be assumed to be very long. Set up and solve the boundary-value problem required to determine the temperature of the rod outside the furnace as a function of the distance from the furnace entrance, z. Define any additional symbols which you use.

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS CHME 7320-01 (CRN 36904) REQUIRED TEXTBOOK: Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

CLASS PROBLEM SET # 2 (Continued) 10. Consider steam at temperature T0 is flowing through a finned tube (following figure). The outer radius of the tube and that of the fins are R0 and R, respectively. The thickness of the fins is δ, the heat transfer coefficient between the fins and the ambient is h, and the ambient temperature is T∞. Find the temperature distribution in the fins.

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS CHME 7320-01 (CRN 36904) REQUIRED TEXTBOOK:

Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

CLASS PROBLEM SET # 3 11.

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS CHME 7320-01 (CRN 36904) REQUIRED TEXTBOOK:

Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

CLASS PROBLEM SET # 3 (Continued) 12. Consider the steady state heat transfer in the infinitely long fin shown below. Determine the temperature profile as a function of x.

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS

CHME 7320-01 (CRN 36904)

REQUIRED TEXTBOOK:

Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS

CHME 7320-01 (CRN 36904)

REQUIRED TEXTBOOK:

Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

CLASS PROBLEM SET # 4

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS CHME 7320-01 (CRN 36904) REQUIRED TEXTBOOK:

Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

CLASS PROBLEM SET # 5 16. Find the inverse transform of the following: a)

b)

c)

S +1 F(S) = ─────── S2 + S -6 S F(S) = ─────── S2 + 4 1 F(S) = ─────── S(S2 + 4)

17. Solve the following differential equations: a)

y’ – y = eat

B.C.

@t=0

y = -1

b)

y’’ + 2 y’ + y = t e-t

B.C.

@t=0

y = 1 and y’ = -2

c)

yiv + 2 y’’+ y =0

B.C.

@t=0

y=y’=y’’’=0 and y’’=1

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS CHME 7320-01 (CRN 36904) REQUIRED TEXTBOOK:

Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

CLASS PROBLEM SET # 5 (Continued)

18. A vertical cylindrical tank, 2 ft in inside diameter and 10 ft high is open to the atmosphere through a small vent in the top. Two sharp-edged orifices, each 0.50 inch in diameter, are located in the side of the tank, one being two feet vertically above the other. When the two orifices are discharged water from the full tank, it is noted that the streams cross each other at a horizontal distance of 2 ft from the outer wall. How long will be required for the liquid level in the tank to fall to the center of the upper orifice? It is agreed to neglect air friction on the orifice jets and to assume that the discharge rate from a sharp-edged orifice is given by : –––––––

V= 0.6 √ 2 gh Where V is the velocity through the orifice in ft/sec, g is 32.2 ft/sec2, and h is the level of fluid in feet above the orifice.

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS CHME 7320-01 (CRN 36904) REQUIRED TEXTBOOK : Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

HOMEWORK ASSIGNMENT # 1

DUE DATE

:

JANUARY 14, 2019

CHAPTER ONE

:

Problems 1.3, 1.5, 1.8

CHAPTER TWO

:

Problems 2.1, 2.6, 2.7

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS CHME 7320-01 (CRN 36904) REQUIRED TEXTBOOK : Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

HOMEWORK ASSIGNMENT # 2 DUE DATE

:

JANUARY 28, 2019

SPECIAL PROBLEMS: A)

(2x + y-1) dx + (y-1 – xy-2) dy = 0

B)

(dy/dx) – (2/x)y = y4

C)

(d3y/dx3) + 2 (d2y/dx2) – 3 (dy/dx) = 0

D)

(d5y/dx5) – 2 (d4y/dx4) + (d3y/dx3) = 0

E)

(1+x2) (d2y/dx2) + x (dy/dx) + ax = 0

F)

ρ (dθ/dρ) – 2/ρ (dρ/dθ) = 0

G)

(dy/dx)2 -4x (dy/dx) +6y = 0

H)

y (d2y/dx2) + (dy/dx)2 = (dy/dx)

I)

Solve simultaneously: (dx/dt) + (dy/dt) + y –x = e2t (d2x/dt2) + (dy/dt) = 3 e2t

J)

Using method of variation of parameter, solve: y'' – 8 y' +16 y = 6x e4x

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS CHME 7320-01 (CRN 36904) REQUIRED TEXTBOOK : Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

HOMEWORK ASSIGNMENT # 3 DUE DATE

:

CHAPTER THREE :

FEBRUARY 04, 2019 Problems 3.2, 3.6, 3.9, 3.10

SPECIAL PROBLEM: A) A container is maintained at a constant temperature of 8000F and is fed with a pure gas A at a steady state rate of 1 lb mole/min; the product gas stream is withdrawn from the container at the rate necessary to keep the total pressure constant at a value of 3 atm. The container contents are vigorously agitated, and the gas mixture is always well mixed. The following irreversible second-order gas-phase reaction occurs in the container: 2A ———→ B At a temperature of 8000F, the reaction-rate constant for the reaction has the numerical value of 1,000 ft3/lb mole/min. Both A and B are perfect gases. Because of their low temperature, no reaction occurs in the lines leading to and from the vessel. If under steady-state conditions, the product stream is to contain 33 1/3 mole % B, how large (in cubic feet) should be the volume of the reaction container? B) After the steady-state of (a) has been attained, the valve on the exit pipe of the isothermal vessel is abruptly closed. The feed rate is controlled so that the total tank pressure is maintained at 3 atm. If the mixing is still perfect, how many minutes will it take (after the instant of closing the valve) for the tank contents to be 90 mole % B?

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERING SPRING 2019 CHEMICAL ENGINEERING MATHEMATICS CHME 7320-01 (CRN 36904) REQUIRED TEXTBOOK : Richard G. Rice and Duong D. Do, “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, Inc., New York (1995).

HOMEWORK ASSIGNMENT # 4 DUE DATE

:

FEBRUARY 11, 2019

SPECIAL PROBLEMS: 1) Use the method of Frobenius to obtain the general solution of each of the following differential equations, valid near x=0. a)

2x (d2y/dx2) + (1-2x) (dy/dx) – y = 0

b)

x(1-x) (d2y/dx2) -2 (dy/dx) + 2y = 0

a)

Obtain a solution in power series for:

2) (dy/dx) = x + y2 + 1 When y = 0 at x = 0. Evaluate the coefficients of the terms up to x6. b)

Compare the results of (a) with that obtained by solution in Taylor Series. Calculate the value of y at x = -1, and note that the series converges rapidly only when x is a fraction.

3) Solve the following differential equation: y'' + xm y = 0

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NORTHEASTERN UNIVERSITY DEPARTMENT OF CHEMICAL ENGINEERIN...