Example 2.1 plots responses for changes in input flows for

A perfectly stirred, constant-volume tank has two input streams, both consisting of the same liquid. The temperature and flow rate of each of the streams can vary with time. Tl Stream 1 ------,1 wl '+' T2 Stream 2 w2 T3 Stream 3 w3 Figure E2.1 (a) Derive a dynamic model that will describe transient operation. Make a degrees of freedom analysis assuming that both Streams 1 and 2 come from upstream units (i.e., their flow rates and temperatures are known functions of time). (b) Simplify your model, if possible, to one or more differential equations by eliminating any algebraic equations. Also, simplify any derivatives of products of variables. Notes: w; denotes mass flow rate for stream i. Liquid properties are constant (not functions of temperature).

A completely enclosed stirred-tank heating process is used to heat an incoming stream whose flow rate varies. T; w--~ Heating coil Figure E2.2 Q The heating rate from this coil and the volume are both constant. (a) Develop a mathematical model (differential and algebraic equations) that describes the exit temperature if heat losses to the ambient occur and if the ambient temperature (Ta) and the incoming stream's temperature (T;) both can vary. (b) Discuss qualitatively what you expect to happen as T; and w increase (or decrease). Justify by reference to your model. Notes: p and CP are constants. U, the overall heat transfer coefficient, is constant. As is the surface area for heat losses to ambient. T; > Ta (inlet temperature is higher than ambient temperature).

Two tanks are connected together in the following unusual way in Fig. E2.3. (a) Develop a model for this system that can be used to find \(h_{1}\), \(h_{2}\), \(w_{2}\), and \(w_{3}\) as functions of time for any given variations in inputs. (b) Perform a degrees of freedom analysis. Identify all input and output variables. Notes: The density of the incoming liquid, \(\rho\), is constant. The cross-sectional areas of the two tanks are \(A_{1}\) and \(A_{2}\). \(w_{2}\) is positive for flow from Tank 1 to Tank 2. The two valves are linear with resistances \(R_{2}\) and \(R_{3}\). Equation Transcription: Text Transcription: h_1 h_2 w_2 w_3 rho A_1 A_2 R_2 R_3

Consider a liquid flow system consisting of a sealed tank with noncondensible gas above the liquid as shown in Fig. E2.4. Derive an unsteady-state model relating the liquid level h to the input flow rate q;. Is operation of this system independent of the ambient pressure Pa? What about for a system open to the atmosphere? You may make the following assumptions: (i) The gas obeys the ideal gas law. A constant amount of mgiM moles of gas are present in the tank. (ii) The operation is isothermal. (iii) A square root relation holds for flow through the valve. Figure E2.4

Two surge tanks are used to dampen pressure fluctuations caused by erratic operations of a large air compressor. (See Fig. E2.5.) (a) If the discharge pressure of the compressor is Pd(t) and the operating pressure of the furnace is P1 (constant), develop a dynamic model for the pressures in the two surge tanks as well as for the air mass flows at points a, b, and c. You may assume that the valve resistances are constant, that the valve flow characteristics are linear, e.g., wb = (P1 - P2)/Rb, that the surge processes operate isothermally, and that the ideal gas law holds. (b) How would you modify your model if the surge tanks operated adiabatically? What if the ideal gas law were not a good approximation?

A closed stirred-tank reactor with two compartments is shown in Fig. E2.6. The basic idea is to feed the reactants continuously into the first compartment, where they will be preheated by energy liberated in the exothermic reaction, which is anticipated to occur primarily in the second compartment. The wall separating the two compartments is quite thin, thus allowing heat transfer; the outside of the reactor is well insulated; and a cooling coil is built into the second compartment to remove excess energy liberated in the reaction. Tests are to be conducted initially with a single-component feed (i.e., no reaction) to evaluate the reactor's thermal characteristics. (a) Develop a dynamic model for this process under the conditions of no reaction. Assume that q0, T;, and Tc all may vary. (b) Make a degrees of freedom analysis for your modelidentifying all parameters, outputs, and inputs that must be known functions of time in order to obtain a solution. (c) In order to estimate the heat transfer coefficients, thereactor will be tested with T; much hotter than the exit temperature. Explain how your model would have to be modified to account for the presence of the exothermic reaction. (For purposes of this answer, assume the reaction is A~ Band be as specific as possible.) Notes: Overall heat transfer coefficient and surface area between compartments. Uc, Ac: Overall heat transfer coefficient and surface area of cooling tube. V1: Volume of Compartment 1. Vz: Volume of Compartment 2.

Using the blending process described in Example 2.1, calculate the response of x to a change in x1 (the disturbance from 0.4 to 0.5 and a change in w2 from 200 to 100 kg/min. Plot the response using appropriate software for 0 ,; t ,; 25 minutes. Explain physically why the composition increases or decreases, compared to case (d) in Fig. 2.2.

A jacketed vessel is used to cool a process stream as shown in Fig. E2.8. The following information is available: (i) The volume of liquid in the tank V and the volume of coolant in the jacket V1 remain constant. Volumetric flow rate qp is constant, but q1 varies with time. (ii) Heat losses from the jacketed vessel are negligible. (iii) Both the tank contents and the jacket contents are well mixed and have significant thermal capacitances. (iv) The thermal capacitances of the tank wall and the jacket wall are negligible. (v) The overall heat transfer coefficient for transfer between the tank liquid and the coolant varies with coolant flow rate: U= KqJ8 where U [ =] Btu/h ft2 oF q] [ = l ft3/h K =constant

Solve the nonlinear differential equation (2-61) for \(q_{i}=0\), either analytically or numerically, to obtain h(t). Assume A = 2, \(C_{v}{ }^{*}=0.5\), \(\rho=60\), g/gc = 1, and h(0) = 10, and that units of these parameters are consistent. Equation Transcription: Text Transcription: q_i = 0 C_v ^* = 0.5 rho = 60

Irreversible consecutive reactions A --> B --> C occur in a jacketed, stirred-tank reactor as shown in Fig. E2.10. Derive a dynamic model based on the following assumptions: (i) The contents of the tank and cooling jacket are well mixed. The volumes of material in the jacket and in the tank do not vary with time. (ii) The reaction rates are given by r1 = k1e-E1/RTcA [=]mol A/h L r2 = k2e-E2/RTcs [=]mol B/h L (iii) The thermal capacitances of the tank contents and the jacket contents are significant relative to the thermal capacitances of the jacket and tank walls, which can be neglected. (iv) Constant physical properties and heat transfer coefficients can be assumed. Note: All flow rates are volumetric flow rates in Llh. The concentrations have units of moUL. The heats of reaction are !J.H1 and

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 Vis constant, and plot the response of x(t) and V(t) for the change in w1 for t = 0 to 15 minutes. Assume that w2 and w remain constant.

A process tank has two input streams-Stream 1 at mass flow rate w1 and Stream 2 at mass flow rate w2. The tank's effluent stream, at flow rate w, discharges through a fixed valve to atmospheric pressure. Pressure drop across the valve is proportional to the flow rate squared. The cross-sectional area of the tank, A, is 5 m2, and the mass density of all streams is 940 kg/m3. (a) Draw a schematic diagram of the process and write an appropriate dynamic model for the tank level. What is the corresponding steady-state model? (b) At initial steady-state conditions, with w1 = 2.0 kg/sand w2 = 1.2 kg/s, the tank level is 2.25 m. What is the value of the valve constant (give units)? (c) A process control engineer decides to use a feed forward controller to hold the level approximately constant at the set-point value (hsp = 2.25 m) by measuring w1 and manipulating w2. What is the mathematical relation that will be used in the controller? If the w1 measurement is not very accurate and always supplies a value that is 1.1 times the actual flow rate, what can you conclude about the resulting level control? (Hint: Consider the process initially at the desired steady-state level and with the feedforward controller turned on. Because the controller output is slightly in error, w2 of. 1.2, so the process will come to a new steady state. What is it?) What conclusions can you draw concerning the need for accuracy in a steady-state model? for the accuracy of the measurement device? for the accuracy of the control valve? Consider all of these with respect to their use in a feedforward control system.

The liquid storage tank shown in Fig. E2.13 has two inlet streams with mass flow rates w1 and w2 and an exit stream with flow rate w3. The cylindrical tank is 2.5 m tall and 2 m in diameter. The liquid has a density of 800 kg/m3. Normal operating procedure is to fill the tank until the liquid level reaches a nominal value of 1.75 m using constant flow rates: w1 = 120 kg/min, w2 = 100 kg/min, and w3 = 200 kg/min. At that point, inlet flow rate w1 is adjusted so that the level remains constant. However, on this particular day, corrosion of the tank has opened up a hole in the wall at a height of 1 m, producing a leak whose volumetric flow rate q4 (m3/min) can be approximated by where h is height in meters. (a) If the tank was initially empty, how long did it take for the liquid level to reach the corrosion point? (b) If mass flow rates w1, w2, and w3 are kept constant indefinitely, will the tank eventually overflow? Justify your answer.

Consider a blending tank that has the same dimen- sions and nominal flow rates as the storage tank in Exercise 2.13 but that incorporates a valve on the outflow line that is used to establish flow rate w3. (For this exercise, there is no leak in the tank as in Exercise 2.13.) In addition, the nominal inlet stream mass fractions of component A are x1 = Xz = 0.5. The process has been operating for a long time with constant flow rates and inlet concentrations. Under these conditions, it has come to steady state with exit mass fraction x = 0.5 and level h = 1.75 m. Using the information below, answer the following questions: (a) What is the value of w3? the constant, Cv? (b) If x1 is suddenly changed from 0.5 to 0.6 without changing the inlet flow rates (of course, x2 must change as well), what is the final value of x3? How long does it take to come within 1% of this final value? (c) If w1 is changed from 120 kg/min to 100 kg/min without changing the inlet concentrations, what will be the final value of the tank level? How long will it take to come within 1% of this final value? (d) Would it have made any difference in part (c) if the concentrations had changed at the same time the flow rate was changed? Useful information: The tank is perfectly stirred. W3 = Cv Vh

Suppose that the fed-batch bioreactor in Fig. 2.11 is converted to a continuous, stirred-tank bioreactor (also called a chemostat) by adding an exit stream. Assume that the inlet and exit streams have the same mass flow rate F and thus the volume of liquid V in the chemostat is constant. (a) Derive a dynamic model for this chemostat by modifying the fed-batch reactor model in Section 2.4.9. (b) Derive the steady-state relationship between growth rate f.L in Eq. 2-93 and dilution rateD where by definition, D = FIV. Suggest a simple control strategy for controlling the growth rate based on this result. (c) An undesirable situation called washout occurs when all of the cells are washed out of the bioreactor and thus cell mass X becomes zero. Determine the values of D that result in washout. (Hint: Washout occurs if dX/dt is negative for an extended period of time, until X= 0.) (d) For the numerical values given below, plot the steadystate cell production rate DX as a function of dilution rate D. Discuss the relationship between the values of D that result in washout and the value that provides the maximum production rate. The parameter values are: f.Lm = 0.20 h - 1; Ks = 1.0 g/1,

In medical applications the chief objectives for drug delivery are: (i) to deliver the drug to the correct location in the patient's body, and (ii) to obtain a specified drug concentration profile in the body through a controlled release of the drug over time. Drugs are often administered as pills. In order to derive a simple dynamic model of pill dissolution, assume that the rate of dissolution r d of the pill in a patient is proportional to the product of the pill surface area and the concentration driving force: rd = kA(cs - Caq) where Caq is the concentration of the dissolved drug in the aqueous medium, Cs is the saturation value, A is the surface area of the pill, and k is the mass transfer coefficient. Because Cs .. Caq, even if the pill dissolves completely, the rate of dissolution reduces to r d = kAcs. (a) Derive a dynamic model that can be used to calculate pill mass M as a function of time. You can make the following simplifying assumptions: (i) The rate of dissolution of the pill is given by r d = kAcs. (ii) The pill can be approximated as a cylinder with radius r and height h. It can be assumed that hlr .. 1. Thus the pill surface area can be approximated as A= 2-rrrh. (b) For the conditions given below, how much time is required for the pill radius r to be reduced by 90% from its initial value of r0? p = 1.2 g/ml ro = 0.4 em h = 1.8 em Cs = 500 g/L k = 0.016 em/min

Bioreactions are often carried out in batch reactors. The fed-batch bioreactor model in Section 2.4.9 is also applicable to batch reactors if the feed flow rate F is set equal to zero. Using the available information shown below, determine how much time is required to achieve a 90% conversion of the substrate. Assume that the volume V of the reactor contents is constant. Available information: (i) Initial conditions: X(O) = 0.05 giL, S(O) = 10 giL, P(O) = 0 g/L. (ii) Parameter values: V = 1 L, f.Lm = 0.20 hr-1, Ks = 1.0 g/L, Y XIS = 0.5 g/g, YPIX = 0.2 g/g.

Sketch the level response for a bathtub with crosssectional area of 8 ft2 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 and outflow are initially equal to 2 ft3/min. (a) The drain is suddenly closed, and the inflow remains constant for 3 min (0.:::;, t.:::;, 3). (b) The drain is opened for 15 min; assume a time constant in a linear transfer function of 3 min, so a steady state is essentially reached (3.:::;, t.:::;, 18).

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).

Surge and storage tanks are important dynamic processes in a chemical plant. We can investigate their behavior by using simple experiments at home. Obtain a translucent paper cup (available at fine fast-food restaurants) approximately 6 to 8 in high (or more). Puncture the cup on the side near the bottom with a small hole (- 118 in). (a) Fill the cup to the top and record how long it takes for the cup to empty. Try other heights (h) and record the time to empty Cte) (and repeat some of the trials due to experimental error). Plot the results (h vs. te). Is the relationship between h and te linear or nonlinear? Note this data is related to the case if you measure the height vs. time in a single experiment. (b) Is the outflow rate constant with respect to time? Explain why, or why not. (c) Develop a nonlinear dynamic model (ODE) for the process that describes the height vs. time:

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 that level and outflow rate are linearly related. The steady-state inflow and outflow are initially equal to 2 ft3/min. The graph should show numerical values of level vs. time. (a) The drain is suddenly closed, and the inflow remains constant for 3 min (0 _:<:;_ t _:<:;_ 3). (b) The drain is opened for 15 min, keeping the inflow at 2 ff /min, where a steady state is essentially reached (3 _:<:;_ t _:<:;_ 18). (c) The inflow rate is doubled to 4 ft3/min for 15 min (18 _:<:;_ t _:<:;_ 33). (d) The inflow rate is returned to its original value of 2 ft3/min for 17 min (33 _:<:;_ t _:<:;_50).