Derive the linearization of the nonlinear system

Assignment Help Other Engineering
Reference no: EM131436751

Process Dynamics and Control PROJECT: Dynamics and Control of a Chemical Reactor

Consider a jacketed continuously stirred tank reactor (CSTR) shown below:

1870_Figure.png

The following series of reactions takes place in the reactor:

A →B →C

where A to B is a highly exothermic, first-order reaction and B to C is a first-order reaction that generates a negligible amount of heat. The reactions follow an Arrhenius rate law:

r1 = k10e-E_1/RTCA, r2 = k20e-E_2/RTCB

where k10 and k20 are the pre-exponential factors (given in units of per time), E1 and E2 are activation energies, and R is the gas constant.

The reactor is fed by species A in an inert solvent at temperature T0, concentration CA0 and volumetric flow rate F0. The effluent stream leaves the reactor at concentrations CA, CB, CC, and temperature T. The liquid contents of the reactor are well-mixed (temperature and composition are uniform) and have a constant density (ρ), liquid hold-up volume (V), heat of reactions (?H1 and ?H2 for reaction 1 and 2, respectively) and heat capacity (Cp). The jacket is fed by cooling water at temperature Tj0 and flow rate Fj0. The overall heat transfer coefficient and heat transfer area of the jacket are U and Aht, respectively. The effluent stream leaves the jacket at temperature Tj. The jacket water is assumed to have a uniform temperature.

Process model -

The material and energy balances that describe the dynamic behavior of the process have the following dimensionless form:

 dx1/dθ = 1 - x1 - Da1 exp(E¯1/x3)x1

dx2/dθ = -x2 + Da1 exp(-E¯1/x3)x1 - Da2 exp(-E¯2/x3)x2                                                  (1)

dx3/dθ = x30 - x3 + Da′1exp(-E¯1/x3)x1 + Da′2exp(-E¯2/x3)x2 - U¯(x3 - x4)

dx4/dθ = ∈12(x40 - x4) + U¯∈13(x3 - x4)

where:

x1 = CA/CA0: dimensionless concentration of species A in the reactor

x2 = CB/CA0: dimensionless concentration of species B in the reactor

x3 = T/Tref : dimensionless temperature of the reactor

x4 = Tj/Tref: dimensionless temperature of the jacket

x30 = T0/Tref: dimensionless temperature of reactor inlet stream

x40 = Tj0/Tref: dimensionless temperature of jacket inlet stream

θ = F0t/V: dimensionless time

Tref is the reference temperature and the remaining parameters (Da1, Da2, Da′1, Da′2, E¯1, E¯2, U¯, ∈1, ∈2, and ∈3) are dimensionless process parameters.

Specific Questions -

1. Using mass and energy balances and the assumptions listed in the problem statement above, derive a dynamic model (system of four nonlinear ordinary differential equations for CA, CB, T, and Tj ) for the process (no need to express the model in dimensionless variable form). Clearly explain your derivation.

2. For the model of Eq. 1 and for the following set of parameters: Da1 = 106, Da2 = 107, Da′1 = 1.5 × 106, x30 = 0.025, x40 = 0.025, E¯1 = 1.0066, E¯2 = 1.0532, U¯ = 8, ∈1 = 12.5, ∈2 = 40, ∈3 = 1, and Da′2 = 0 (because B → C generates a negligible amount of heat), compute the three steady-states of the nonlinear algebraic system.

3. Derive the linearization of the nonlinear system of Eq. 1 around each of the steady-states and show proof that the CSTR system has two stable steady-states and one unstable steady-state.

4. Using temperature of the reactor as controlled and measured output (i.e., y = x3 - x3s where x3s is the steady-state value), temperature of the jacket inlet stream as manipulated input (i.e., u = x40 - x40s where x40s is the steady-state value), and temperature of the reactor inlet stream as disturbance (i.e., d = x30 - x30s where x30s is the steady-state value), create a block diagram of the closed-loop system.

5. The operation under the unstable steady-state operating condition is preferable. Using the input, output, and disturbance defined in Problem 4, compute the transfer function of the linearized system.

6. On the basis of this transfer function, design a proportional-integral (PI) controller to stabilize the closed-loop system resulting from the linearized system and the controller.

7. Apply the PI controller to the linearized system and the nonlinear system of Eq. 1, for set-point changes in the reactor temperature, y, of magnitude 0.005 and 0.01 (show the plots of the manipulated input and the controlled output for all the simulation runs). Explain your results.

8. Study the disturbance rejection capabilities of the controller, for step changes in the temperature of the reactor inlet stream, d, of magnitude 0.001 and 0.002. Consider both the linearized system and the nonlinear system (show the plots of the manipulated input and the controlled output for all the simulation runs). Explain your results!

9. Apply the PI controller of Problem 6 to the nonlinear process model with the parameters of Problem 1 and Da′2 = 105. Discuss your results!

Reference no: EM131436751

Questions Cloud

Identify and summarize the two articles chosen : Health care Providers may volunteer to be accredited by an external body. The Centers of Medicare and Medicaid provide an updated list of approved accrediting organizations. For this discussion, we are focusing on the acute care hospital. Below ar..
Calculate the total variable and fixed costs of the hotel : A hotel group prepares financial statements on a quarterly basis. The senior management is reviewing the performance of one hotel and making plans for next year. The managers have in front of them the results for this year (based on some actual re..
Identify a sourcing plan for your new hires : Identify a Sourcing Plan for your new hires (the representative exempt and non-exempt position only). In this section, you are answering the question, "How are we going to find suitable candidates?"
Stating that deviance is critical to society : If all deviance is constructed by society, can it be deconstructed in the same manner? In other words, is deviance a static or dynamic concept? Is Durkheim right in stating that deviance is critical to society? Explain and provide examples to supp..
Derive the linearization of the nonlinear system : CBE 107 - Process Dynamics and Control Winter 2017 PROJECT: Dynamics and Control of a Chemical Reactor. Derive the linearization of the nonlinear system of Eq. 1 around each of the steady-states and show proof that the CSTR system has two stable st..
North-south economic divide audits effect on trade : What is the significance of the "North/South Economic Divide" audits effect on trade, Economic growth and wealth, as well as the citizens in the countries if the North and South.
Contrast the collective-group oriented cultures : Compare and Contrast the collective/ group oriented cultures of Asia and the Individual orientation of the Western cultures in terms of how they create competitive advantages for themselves.
Relative nature of deviance : Considering the relative nature of deviance, what leads to someone being labeled "deviant"? Provide examples to support your statements.
Explain the concept of globalization : Define and explain the concept of Globalization and the forces that combined to expand globalization to many countries after WWII. Explain the different modern day reasons for firms to go global.

Reviews

len1436751

3/22/2017 6:36:01 AM

Results and Discussion. Address the questions posed in the previous section (Problems 2-9). Point values are indicated for each item. Make sure to include all required plots, with a brief explanation of what they represent. Interpret your results adequately. Appendices. Include listings of all programs and source code you used to generate your results. You have to submit a hard copy of your report.

len1436751

3/22/2017 6:35:54 AM

You are expected to work independently on the project. In order to do the simulation part of the project, you can write your own code or use a software package like MATLAB. The project should be clearly written (deductions, up to 10 points, will be made for messiness, poor organization, and poor writing), according to the following format: Title page. Please include the title of the project and your name. Abstract. Write a paragraph that summarizes your most significant results. Include the controller parameters you determined for Problem 6. Table of contents. Indicate page numbers for all report sections and figures. Introduction. Briefly describe the problem, the dimensionless mathematical model, and the strategy and methods you used to obtain your results. Include your derivation of the four nonlinear ordinary differential equations of Eq. 1 in the introduction (Problem 1).

Write a Review

 

Other Engineering Questions & Answers

  Adopt a condensing power plant

Adopt a condensing power plant with a feedwater tank (MV) with the wiring diagram and information - High-pressure turbines are isentropic efficiency of 0.88.

  Determine the resistance between terminals a and b

What resistance in parallel with 120Ω results in an equivalent resistance of 48Ω. Determine the resistance between terminals a and b for the network shown in Figure.

  Calculate the single pass conversion of a in the reactor

The reaction A giving 2B+C takes pace in a catalytic reactor. The reactor effluent is sent to a separator. Calculate the single pass conversion of A in the reactor and molar ratio of recycle to feed.

  Draw a circuit diagram to provide rotary actuation

Modify the design of the circuit shown in FIGURE 3 and draw a circuit diagram to provide rotary actuation in both directions using a single direction pump (to replace the bi-directional pump) and providing a means of preventing over-speeding in th..

  Estimate effect of misalignment on the compressive strength

Estimate the effect of misalignment on the compressive strength of a carbon- epoxy (AS4-3501) composite fabricated by hand lay-up of prepreg tape, vacuum bagged and oven cured, resulting in a fiber volume fraction Vf = 0.6.

  Instrumentation measurement & lab thermal sensors lab

For the comparator below, complete the design so that the threshold voltage is 1.25V which means that if the input is below 1.25V, the output will be HIGH and if its greater, it will be low.

  Design the digital filter using the bilinear transformation

Design the digital filter using the bilinear transformation and show that the transfer function of the resulting filter may be approximately expressed as:

  Generate a sequence of exponentially

Consider an exponentially distributed RV with a=0, use the inverse transform method to generate a sequence of exponentially distributed random numbers for different values of the parameter b. Plot the histograms (probability density function (pdf))..

  Support vector machine this is a homework based on kernel

this is a homework based on kernel methods and support vector machines. ltbrgtsolution should legible well-organized

  Provide a description of what functionality the screen

Within the Discussion Board area, write 400 words that respond to the following questions with your thoughts, ideas, and comments. This will be the foundation for future discussions by your classmates. Be substantive and clear, and use examples to..

  Decision analysis problem by constructing the payoff table

Formulate a decision analysis problem by constructing the payoff table with the information given above. That is, describe your alternatives, states of natures, and calculate the payoff for each alternative and state of nature pair

  Discuss the contribution of transistors to the development

Discuss the contribution of transistors to the development of technology. Also, describe the impact of transistors on the environment and at least one current effort to address this issue.

Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd