# Sample problems _ Midterm Exam

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Sample Problems _ Midterm Exam3-16. Consider the mixing process shown in Figure P3-8.

The purpose of this process is to blend a stream, weak in component A, with another stream, pure A. The density of stream 1, V1, can be considered constant because the amount of A in this stream is small. The density of the outlet stream is, of course, a function of the concentration and is given by V 3 (t )!a b3 c % 3 t The flow through valve 1 is given by f1 (t ) ! CV 1 .vp1 (t ) The flow through valve 2 is given by f 2 (t ) ! CV 2 .vp 2 (t ) Finally, the flow through valve 3 is given by f 3 (t ) ! CV 3(P1 G1 (P2 G2

(P3 G3 The relationship between the valve position and the signal is given by vp1(t) = a1 + b1.[m1(t) d1] and vp2(t) = a2 + b2.[m2(t) d2] where a1, b1, d1, a2, b2, d2, a3, and b3 = known constants CV1, CV2, CV3 = valve coefficients of valve 1, 2, and 3, m3/(s. psi 0.5) vp1(t), vp2(t) = valve position of valves 1 and 2, respectively, a dimensionless fraction P1, P2 = pressure drop across valves 1 and 2, respectively (constants), psi P3(t) = pressure drop across valve 3, psi G1, G2 = specific gravity of streams 1 and 2, respectively (constants), dimensionless G3(t) = specific gravity of stream 3, dimensionless

Develop the mathematical model that describes how the forcing functions m1(t), m2(t), and cA1 (t) affect h(t) and cA3 (t). Be sure to show the units of all the gains and time constants.

3-17. Consider the tank shown in Figure P3-9.

Figure P3-9 Sketch for Problem 3-17. A 10% ( 0.2%) by weight NaOH solution is being used for a caustic washing process. In order to smooth variations in flow rate and concentration, an 8000-gal tank is being used as surge tank. The steady-state conditions are as follows: V(0) = 4000 gal, fi(0) = fo(0) = 2500 gph (gal per hour), ci(0) = co(0) = 10 wt% The tank contents are well mixed, and the density of all streams is 8.8 lb/gal. (a) An alarm will sound when the outlet concentration drops to 9.8 wt% (or rises to 10.2 wt %). Assume that the flows are constant. (i) Derive the dynamic model relating the outlet concentration to the inlet concentration. Obtain the numerical values of all gains and time constants. (ii) Because of an upset, the inlet concentration, ci(t), drops to 8% NaOH instantaneously. Determine how long it will take before the alarm sounds. (b) Consider now that the inlet flow, fi(t), can vary, whereas the outlet flow is maintained constant at 2500 gph. Therefore, the volume in the tank can also vary. (iii) Develop the differential equation that relates the volume in the tank to the flows in and out. (iv) Develop the differential equations that relate the outlet concentration of NaOH to the inlet flow and inlet concentration. (v) Obtain the numerical values of all gains and time constants. (vi) Suppose now that the inlet flow to the tank drops to 1000 gph. Determine how long it takes to empty the tank.

20 Minutes Quiz 17/3/2009Name: ID:

The blending tank shown in the Figure below may be assumed to be perfectly mixed.

Blending Vessel The input variables are the solute concentrations and flows of the inlet streams, c1 (t), c2(t) [kg/m3 ], f1(t), and f2(t) [m3/min] . The volume of liquid in the tank, V [m3], can be assumed constant, and variation of stream densities with composition may be neglected. (a) Obtain the dynamic model relating the outlet composition c(t), kg/m3, and outlet flow f(t), m3/min, to the four input variables (b) Write the expressions for the time constant and gains of the blender in terms of the parameters of the system. (c) Calculate the numerical values of the time constants and gains for a blender that is initially mixing a stream containing 80 kg/m3 of solute with a second stream containing 30 kg/m3 of the solute to produce 4.0 m3/min of a solution containing 50 kg/m3 of the solute. The volume of the blending vessel is 40 m3.

(2) A semi t react r means t at one or more reagents, which do not react, are entered completel into a batch container and the final reagent is then added under controlled conditions. No product is withdrawn until the entire final reagent has been added and the reaction has proceeded to the required extent. See Figure 3.

Figure 3: Semi-batch reactor Consider the isothermal reaction A + B C being carried out by the controlled addition o f reagent B to the semi batch reactor shown below. The entire reagent A is dumped once into the reactor. Assume that the reaction is fi t wit tt t A B (i.e. second order overall). Assume constant liquid density for the system. a- Write the mathematical model that describes the system. b- Solve the model for the concentrations of A, B and C, and for the solution volume, V, after 0.1 hour (Take t = 0.1 hour) and the given data below: Reaction rate constant, k = 10 liter/h.mol Initial concentrations: CA(0) = 1.0 mol/L, CB(0) = CC(0) = 0.0 mol/L Charge rate, F = 100 liter/h Charge concentration, CBo = 1.0 mol/liter Initial volume of solution in the reactor, V(0) = 100 liter. TMB: d ! F 0 dt CMB on A: VC A ! kVC A C B t CMB on B: VC B ! FC Bo kVC C B t V ! M i Ci

But we do not have information about on C:

and Mi to get CC, then CC can be found from CMB

d( C C ) ! 00 dt

C CB