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physics
process dynamics control

Questions and Answers of Process Dynamics Control

Consider the PCM furnace module of Appendix E. Assume that hydrocarbon temperature THCis the output variable and that air flow rate FAis the input variable.(a) Develop an FOPTD model from response
Data for a person with type 1 diabetes are available as both MATLAB and Excel data files on the book web site. 1 Glucose measurements (y) were recorded every five minutes using a wearable sensor that
Unit step response data are given in Table E7.15 for a process with gain K = 2. Fit the data to a first order model with no time delay. Next use linear regression to fit a first-order discrete-time
The following data were collected from a cell concentration sensor measuring absorbance in a biochemical stream. The input u is the flow rate deviation (in dimensionless units) and the sensor output
Fig. E7.12 presents the response of a system to a unit step in the input.(a) Use these data to derive an FOPTD model of this system.(b) Plot the response of the model and compare with the data.(c)
The response of a system to a unit step change in the input (occurring at time 0) is shown in Fig. E7.11.(a) Derive a second-order plus time delay model approximation for the system. Provide values
Noisy data for the step response of a boiler temperature T to a decrease in air flow rate q from 1000 to 950 cfm are shown below. Develop a FOPTD model using a method from Chapter 7. Be sure to use
The output response data y shown in Table E7.9 were generated from a step change in input u from 1 to 5 at time t = 0. Develop a transfer function model of the form Y(s) Ke-0s (T,8 +1)(t,s + 1) U(s)
The level in a tank responds as a first-order system to changes in its inlet flow. The data shown below were gathered after the inlet flow was increased quickly from 1.5 to 4.8 gal/min.(a) Determine
Assume that step response data obtained from an FOPTD system with K = τ = 1 are available. Determine the accuracy of the FOPTD approximate model derived from these data using the Sundaresan and
A single-tank process has been operating for a long period of time with the inlet flow rate qi equal to 30.1 ft3/min. After the operator increases the flow rate suddenly at t = 0 by 10%, the
Consider the following cascade connection of isothermal reactors. A first-order reaction A †’ B occurs in both reactors, with reaction rate constant k. The volumes of liquid in the
Example 5.1 derives the gain and time constant for a first-order model of a stirred tank heating process. (a) Simulate the response of the tank temperature to a step change in heat input from
A two-input / two-output process involving simultaneous heating and liquid-level changes is illustrated in Fig. E6.20. Find the transfer function models and expressions for the gains and the time
The system equations for two liquid surge tanks in series areUsing state-space notation, determine the matrices A, B, C, and E, assuming that the level deviations is h€²1 and h€²2
An operator introduces a step change in the flow rate qi to a particular process at 3:05 A.M., changing the flow from 500 to 520 gal/min. The first change in the process temperature T (initially at
Show that the liquid-level system consisting of two interacting tanks (Fig. 6.11) exhibits over damped dynamics; that is, show that the damping coefficient in Eq. 6-57 is larger than one. di h2 hi R2
A process has the block diagramDerive an approximate first order plus time delay transfer function model. 2e-de Y(s) |U(s)· (2s + 1)(0.4s + 1) (0.4s + 1)(s + 1)
composition analyzer is used to measure the concentration of a pollutant in a wastewater stream. The relationship between the measured composition Cm and the actual composition C is given by the
Consider the transfer function(a) What are the gain, time delay, time constants, poles, and zeros of G(s)?(b) Will the step response of this transfer function exhibit (i) inverse response or (ii)
The transfer function relating the blood pressure of a patient to the infusion rate of a blood pressure drug is given bywhere Î¸1 = 30 s, Î¸2 = 45 s, and
Consider a process model:For a step input, show that:(a) y(t) can exhibit an extremum (maximum or minimum value) in the step response only if(b) Overshoot occurs only for
For a lead€“lag unit,show that for a step input of magnitude M:(a) The value of y at t = 0+ is given by y(0+) = KM Ï„a/Ï„1.(b) Overshoot occurs only for
The following transfer function is not written in a standard form:(a) Put it in standard gain/time constant form.(b) Determine the gain, poles and zeros.(c) If the time-delay term is replaced by a
Consider the transfer functionWhat is y(t †’ ˆž) for the following inputs:(a) Step input of height M(b) Unit impulse input Î´(t)(c) Sin t(d) Unit rectangular pulse
A heater for a semiconductor wafer has first-order dynamics, that is, the transfer function relating changes in temperature T to changes in the heater input power level P iswhere K has units
Consider the fourth-order plus time delay system represented below:Assume that a step change is applied in the system input U(s). Will the time required for the output Y(s) to reach steady state
A tank used to dampen liquid flow rate surges is known to exhibit second-order dynamics. The input flow rate changes suddenly from 180 to 210 gal/min. An operator notes that the tank level changes as
A process has the transfer function(a) For a step change in the input U(s) = 2/s, sketch the response y(t) (you do not need to solve the differential equation). Show as much detail as possible,
Using the step responses of (1) an integrating element and (2) a first-order process to an input change of magnitude M. (a) Show that the step response for an input change M of a first-order
For a stirred-tank heater, assume the transfer function between the heater input change u(t) (cal/sec) and the tank temperature change y(t)(∘C) can be modeled as G(s) = 5 / 3s + 1 (a) Using
An additive process model is depicted in the figure below. For (unit impulse) (a) Derive the response Y(s) and describe y(t) quantitatively.(b) Simulate the response and identify its major
Can a tank with the outflow rate fixed by a constant speed pump reach a steady state if the inlet flow rate undergoes a step change? Why, or why not? If the transfer function is G(s) = K/s, is it
A thermometer with first-order time constant = 0.1 min and gain = 1.0 is placed in a temperature bath (25∘C). After the thermometer comes to equilibrium with the bath, the temperature of the bath
A thermometer has first-order dynamics with a time constant of 1 sec and is placed in a temperature bath at 120∘F. After the thermometer reaches steady state, it is suddenly placed in a bath at
Show that when K = 1, the ramp response of the third-order system G(s)= K / (τs+1)3 lags behind the input signal (u = at) by three time constants, once the output is changing linearly in
A vertical, cylindrical tank is filled with water at 20∘C. The tank is insulated at the top and bottom, with diameter of 0.5 m and height of 1.0 m. The overall heat transfer coefficient is U = 120
An operator tests the dynamic behavior of a furnace in order to identify the transfer function relating the furnace temperature (output) to the heat input. The operator performs a series of step
onsider the model of the electrically heated stirred-tank system in Section 2.4.3. Subscript e refers to the heating element:(a) Derive transfer functions relating changes in outlet temperature T to
A stirred-tank blending system can be described by a first-order transfer function between the exit composition x and the inlet composition x1 (both are mass fractions of solute):X′(s) /
Consider the transfer function model in Exercise 4.2. For an initial condition of y(0) = 4 and a step change in u of magnitude 2 (at t = 0), calculate the response, y(t).
The dynamic behavior of a pressure sensor / transmitter can be expressed as a first-order transfer function (in deviation variables) that relates the measured value Pm to the actual pressure,
Consider the following transfer function:G(s)= Y(s) / U(s) = 3e−s / 10s + 1(a) What is the steady-state gain?(b) What is the time constant?(c) If U(s) = 4/s, what is the value of the output y(t)
Consider a transfer function:Y(s)/U(s) = d/bs + c(a) What is the steady-state gain?(b) For a step change of magnitude Min the input, will the output response be bounded for all values of constants
A horizontal cylindrical tank shown in Fig. E4.13a is used to slow the propagation of liquid flow surges in a processing line. Figure E4.13b illustrates an end view of the tank and wtis the width of
Solve this ODE using a Symbolic software program:All initial conditions for y and its derivatives are zero. d'y dềy + 16y 176 + 105y = 1 + 86y dt? dr dt di3 dt
For the three stirred-tank system of Exercise 3.20 (Part (b)1), use symbolic system software to find the exit con-centration of tank 3, c3(t), after a rectangular pulse in ci(t) occurs at t = 0. The
Three stirred-tanks in series are used in a reactor train (see Fig. E3.20). The flow rate into the system of some inert species is maintained constant while tracer tests are conducted. Assuming that
A liquid storage facility can be modeled bywhere y is the liquid level (m) and u is an inlet flow rate (m3/s). Both are defined as deviations from the nominal steady-state values. Thus, y = u = 0 at
A continuous, stirred-tank reactor is initially full of water with the inlet and exit volumetric flow rates of water having the same numerical value. At a particular time, an operator shuts off the
Use Laplace transforms to find the solution to the following set of equationsfor x = eˆ’t and zero initial conditions. dy + У2 %3D х dt dy2 + Зу, 3 2у, dt 2y1
For the equation,Use the partial fraction method, to find y(t). ÿ + 5ỷ + 6y(t) = 7, ý(0) = 0, y(0) = 1
The dynamic model between an output variable y and an input variable u can be expressed by(a) Does this system exhibit an oscillatory response after an arbitrary change in u?(b) What is the
The dynamic model for a process is given by where u(t) is the input function and y(0) and dy/dt (0) are both zero.What are the functions of the time (e.g., eˆ’t/Ï„) in the
Figure E3.3 shows a pulse function,u(t).(a) From the information shown in Fig. 3.3, calculate the pulse width, tw.(b) Express u(t) as the sum of simpler functions (some perhaps translated in time),
Derive Laplace transforms of the input signals shown in Figs. E3.2 and E3.3 by summing component functions found in Table 3.1. 10 f(t) 3 t (min) Figure E3.2 Slope = -a u(t) tw Figure E3.3 Triangular
Find the Laplace transforms of the following functions, using the information in Table 3.1. (However, some of the individual terms in these functions may not have Laplace transforms.)(a) f(t) = 5 +
Consider the PCM distillation column module of Appendix E, in which a 50%–50% mixture of methanol (MeOH) and ethanol is separated.Do the following sequence of simulations:(a) Change the Vapor Flow
Consider the PCM Furnace module of Appendix E, which is used to preheat a high-molecular-weight hydro-carbon feed (C16 – C26) to a cracking unit at a petroleum refinery.Do the following
Plot the level response for a tank with constant cross-sectional area of 4 ft2 as a function of time for the following sequence of events; assume an initial level of 1.0 ft with the drain open, and
Consider the unusual piping diagram for the four tanks in Fig. E2.22 in which both the flow rates F1and F2are split between two streams entering the upper and lower tanks (denoted by the fractions in
Perform a degrees of freedom analysis for the model in Eqs. 2-64 through 2-68. Identify parameters, output variables, and inputs (manipulated and disturbance variables).
Example 2.1 plots responses for changes in input flows for the stirred tank blending system. Repeat part (b) and plot it. Next, relax the assumption that V is constant, and plot the response of x(t)
Sketch the level response for a bathtub with cross-sectional area of 8ft2 as a function of time for the following sequence of events; assume an initial level of 0.5 ft with the drain open. The inflow
Recall the stirred-tank heating process with variable holdup as described in Section 2.4.2. Recalculate the degrees of freedom for this example under the following separate circumstances (be sure to

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