Name:

Student #:

Qi: Q2: Q3: Q4: Q5: Total:

CHE222S - Process Modeling and Dynamics

Final Test

April 29, 2019

Closed Book

All work to be marked must appear on front of page. Use back of page for rough work only.

The last page of this exam includes an aid sheet which you may detach and keep aside for reference.

Do not write more than what is necessary. When asked to provide a justification, please keep your answers to less than 2 sentences.

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1). 10 Marks Consider the following process model of a home assuming well mixed interiors.

dT mCpd- = Q UA(TTamb)

Here T is the temperature of the home, mCp is the thermal mass, Q is the heat input (solar and furnace), UA is the overall heat transfer coefficient, Tm is the ambient temperature.

Will the time taken to reach a steady state following a perturbation in Tm increase, decrease, or stay the same when the following parameters are increased. Justify your answer ?

a) mCp

b)UA

Will the steady state temperature of the home following a perturbation in Tm increase, decrease, or stay the same when the following parameters are increased. Indicate why?

C) MCP

d)UA

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2) 20 marks

The following initial value problem needs to be solved numerically. Y' + 4v = 2 - e_ 4t

a) A step size of h = 1 is chosen to solve this equation using Explicit Euler's method. Justify whether this choice of step size is good for use with the chosen method. If not, suggest a step size that will be stable for use with Explicit Euler's.

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b) Show two iterations of Explicit Euler's integration using a step size of 0.25 and y(0) =

2 to solve the above ordinary differential equation.

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Use a step size of h = .25 and show two iterations of the second order Runge Kutta (RK23/midpoint method) for numerical integration of the initial value problem with the same initial conditions.

Comment on the difference between the solutions in part b) and c), specifically regarding their accuracy.

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3)15 Marks

Using Laplace transforms (see Table at the end), solve the differential equation given below.

dy = 2— e _4t

dt y(t = 0) = 2

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4) 20 Marks Consider the following unconstrained optimization problem in two variables ? The goal is to minimize this objective function.

f(x,y)=x2 +y2 -12x-8y+0.5xy

a) Using Newton's method calculate the next value of the solution after one iteration starting from an initial guess of [0, 0]

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b) Using a gradient descent method with an alpha of 0. 1, calculate the next value of the solution after one iteration starting from the same initial guess as a).

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c) Which of the values of the solutions in parts a) and b) is likely to be closer to the final optimal solution and why?

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5) (35 Marks) Consider the figure below that shows a system for heating a continuous flow mixing tank using a hot plate. Assume that the hot plate temperature can be changed instantaneously by adjusting the rate of heat input into the hot plate, and assume uniform temperature within the tank. Here, p and Cp are the density and specific heat capacity of the liquid in the two streams, UA is the heat transfer coefficient between the surface of the plate and the water, F is the volumetric flowrate out, V is the volume of water in the tank, TA,in and TBJn are the inlet temperature and FA and FB are the volumetric flow rate of the two streams A and B respectively, T is the outlet temperature (and hence the temperature within the tank), and Tp is the temperature of the surface of the hotplate.

FA, TA, in

Stream A

Electricity

Continuous flow mixing tank

a) Write the unsteady state mass balance equations for the tank, and simplify them to obtain a model for the rate of change of Volume of the tank. State all assumptions made in answering all the following questions.

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b) Write down the unsteady state energy balance equations for the tank to obtain a model for the rate of change of Temperature in the tank? State all assumptions made.

c) Perform a degree of freedom analysis and identify the inputs and outputs in the model.

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d) Now assume that the Volume and the flow rates are constant and the flow rate out of the tank F is equal to the sum of the flow rates in. Perform a degree of freedom analysis and identify the inputs and outputs in the model.

e) Assume that the flow rates are constant and define the deviation variables for each input and output in the model, and re-write the model in terms of the deviation variables.

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f) Take the Laplace Transform of the model equations expressed in deviation form and express the system of ordinary differential equations as a system of algebraic equations.

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g) Derive all the transfer functions between the inlet temperatures (TA,in and TB,in and T all in deviation form) and the outlet temperature (in deviation form T).

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h) What are the gains and time constants of (i) the transfer functions between T and TA,I, and (ii) the transfer function between T and TBj,, and (iii) the transfer function between T and T (all in deviation form)?

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i) Sketch the time domain response of T (in deviation form) when the change in the inlet temperature of the stream A, TA,I (in deviation form) is a unit step (assume TB,I and Tp do not change) using the appropriate transfer function you developed in part h). Note you do not need to solve the ODE.

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s-a

1.35"(2n-1).j

S11-21

S

S2

/

U- I

s(s2 +3a2)

(s2 +

s cos (b)-a sin (b)

'. ., -

(s-a)'

e

S

e"F(s)

F(s-c)

J F(u)du

F(s)G(s)

sF(s)- f(0)

f F(s)-:

24. f(cr) !F1J

26. 5(1-c)

e Dirac Delta Function

28. tic (t)g(r) e{g(t+c)}

30, t'f(t), n = 1,2 (-1)" F(s)

32. f f(v)dv

34. f(t+T)=f(t) f T ef(t)dt

I -e

36. f(t) s2F(s)-sf(0)-f'(0)

f(0) _2f (0).- f(2) (0)- (0)

Table of Laplace Transforms f = {F (s)} F(s) = {f(t)} f(t) = {F(s)} F(s) =

1. 1

3. f'. n=1.2,3....

7. sin (at)

9. ,sin(at)

ii. sin (at)-atcos(at)

13. Cos (at)-at sin (at)

15. sin (at +h)

17. sinh(ar)

19. e sin (bt)

21. e"sinh(bt)

23. t"e'. n=1.2,3,

25. u.(t)=u(r-c)

Heaviside Function

27. it, (t)f(t -c)

29. ef(t)

31.

33. f'f(t-r)g(r)dr

35. 1(r)

37. f(") (t)

2. e

4. tp>-1

6. n=1,2,3..

8. Cos (at)

10. tcos(at)

12. siu(ar)+afcos(at)

14. cos (at)+al sin (c;t)

16. cos (at +b)

18. cosh(at)

20, e cos(br)

22. e" cosh Or)

S

1?

s,,+l

2s4 a

S 2 + CI 2

2as

(s -a)

2a

(s2 1a2)

s(s2 -a2)

(t2 +a2)?

S sin(b) + a cos(b)

s2 +a 2 a

2 a2

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