(Ecological engineering) Several researchers have examined the feasibility of using wetlands to clean up high volumes of polluted water. Experiemental wetland three (EW3) at the Des Moines Experimental Wetlands site in Illinois has a volume of 15,000,000 dm³ and an inflow of 70,000 dm³ of water per house from the Des Plaines Rier. The outflow eventually returns to the river. During late spring, the river water typically contains of the herbicide atrazine. However, the maximum contaminant level (MCL) under the federal Drinking Water Act is As a first approximation treat EW3 as a perfectly mixed CSTR and assume that atrazine decomposition is first order with k= 0.0025 h ^-1.

(a) First consider that EW3 contains water but no atrazine at the time the flow fromthe river is diverted. Plot C _A as a function of t for the case where the outflow is kept equal to inflow. After what time does C _A reach 99% of its steady-state value? Is it below the MCL?
(b) Next consider the case where EW3 initially contains neither atrazine nor water. Outflow is kept at 50,000 dm ³ /h for 750 h. after which it becomes 70,000 dm ³ /h. Plot the concentration as a function of time and explain how it differs from the plot in part (a). Plot the number of moles of atrazine in EW3 versus time. Why is N _A increasing while C _Aversus time is decreasing?
(c) EW3 is operating initially at the steady-state conditions found in part (a). Suppose that weekly thunderstorms periodically increase the amount of atra zine leached into the Des Plaines River, thereby increasing the concentration in the inflow to EW3. Concentration as a function of time is given by

where the trigonometric argument is in radians. Plot C _A0 and C _A versus t on the same graph. Does the outflow exceed the MCL at any time? Do C _A0 and C _A reach their maximums and minimums at the same times? Does C _Aever exceed C _A0? How can this be?

(d) EW3 is operating initially at the steady-state conditions found in part (a). Suppose that there is a drought lasting 1000 h during which the evaporative flux of water from EW3 is 10,000 dm ³ /h. The overall water balance is such that the wetland volume remains constant, however. No atrazine leaves via evaporation. What does the concentration profile look like? How can this be explained?

The cells in your body need to obtain nutrients, hormones, growth factors, and other molecules present in very low concentrations in the fluid around them. To avoid engulfing a large quantity of this fluid and then intracellularly separating useful from useless molecules, the cells possess what are known as receptors on their surface. These receptors are able to bind interesting molecules or ligands with high affinity, thus capturing molecules for the cell's use (Figure CDP4-C).

(a) You are growing 10 ⁶ cells/mL in a T flask containing 10 mL of media. Each cell has 10 ⁵ receptors on its surface. The association rate constant for the binding of ligands to receptors is 10 ⁶ M ^-1 min^-1 . Calculate the time for 50% of the receptors to bind ligands if you add ligands at a concentration of 10 ^-7 M. Assume irreversible binding and perfect mixing.
(b) Show that the ligand concentration in part (a) is sufficiently in excess so that the binding could be considered pseudo-first order.

Figure CDP4-C

(c) The binding of ligand to receptor is actually a reversible reaction. For the binding of your ligand to receptors, the dissociation rate constant k _r is 0.1 min ^-1 . Using the approximation justified in part (b) and assuming perfect mixing, calculate the percentage of receptors bound 5 min after you add the ligand to the media. (J. Linderman, University of Michigan)

[2nd Ed. P4-34]

(Batch bromination of p-chlorophenyl isopropyl ether) You are in charge of the production of specialty chemicals for your organization and an order comes in for 3 lb of brominated p-chlorophenyl isopropyl ether. You decide to use the technique reported by Bradfield et al. [J. Chem. Soc., 1389 (1949)], who carried out the reaction in 75% acetic acid at 68°F. You have a batch reactor that holds 5 gal (0.670 ft ³ ) of a reacting mixture that can be used. Starting out with a mixture that contains 0.002 lb mol (0.34 lb) of p-chlorophenyl isopropyl ether and 0.0018 lb mol (0.288 lb) of bromine in the 5 gal, you decide to run 10 batches of the mixture to 65% conversion of the p-chlorophenyl isopropyl ether. This procedure will give the desired 3 lb. How long will each batch take?

Additional information:
Kinetics (from Bradfield et al.):

Reaction:

where A is p-chlorophenyl isopropyl ether, B is bromine, and C is monobrominated product .

Rate Law:

Specific reaction rates at 68°F:

k ₁ = 1.98 ft ³ /lb molmin
k₂ = 9.2 x 10 ³ (ft ³ /lb mol) ² min ^-1

[2nd Ed. P4-29]

A liquid organic substance, A, contains 0.1 mol % of an impurity, B, which can be hydrogenated to A:

The material is purified by hydrogenation as a liquid in a continuous well-mixed reactor at 100°C. The feed rate of the liquid is constant at 730 lb/h. The reactor holds 50 gal of liquid, at 500 psig, and the amount of B in the product levels out at 0.001 mol %. What will be the concentration of B in the product if the hydrogen pressure is held at 300 psig? Assume that the reaction behaves as though it were first order with respect to both B and H ₂ , that is, in batch,

Assume perfect gas laws and Henry's law. Also assume the following properties:

[2nd Ed. P4-15]

The gas-phase reaction is to be carried out in an isothermal plug-now reactor at 5.0 atm. The mole fractions of the feed streams are A = 0.20, B = 0.50, and inerts = 0.30.

(a) What is the steady-state volumetric flow rate at any point in the reactor if the pressure drop due to fluid friction can be ignored? [Ans.: (1 - 0.2X).]
(b) What are the expressions for the concentrations of A, B, and D as a function of conversion at any point along the reactor?
(c) What is the feed concentration (units: mol/ dm ³ ) of A if the feed temperature is 55°C?
(d) Determine how large the plug-flow reactor must be to achieve a conversion (based on A) of 0.70 if the temperature in the reactor is uniform (55°C), the volumetric feed rate is 50 dm ³ /min, and the rate law at 55°C is

-r = 2.5 C _A^(1/2) C _B kmol/m ³min

(Ans.: V = 50.21 dm ³.)

(e) Plot the concentrations, volumetric flow rate, and conversion as a function of reactor length. The reactor diameter is 7.6 cm.
(f) How large would a CSTR have to be to take the effluent from the PF reactor in part (d) and achieve a conversion of 0.85 (based on the feed of A to the plug-flow reactor) if the temperature of the CSTR is 55°C?
(g) How many 1-in.-diameter pipe tubes, 20 ft in length, packed with a catalyst, are necessary to achieve 95% conversion of A starting with the original stream? Plot the pressure and conversion as a function of reactor length. The particles are 0.5 mm in diameter and the bed porosity is 45%.
(h) Calculate the PFR size to achieve 70% of the equilibrium conversion and the CSTR size necessary to raise the conversion of the PFR effluent to 85% of the equilibrium conversion if their temperatures were uniform at 100°C. The activation energy for the reaction is 30 kJ/mol, and the reaction is reversible with an equilibrium constant at 100°C of 10 (m ³ /kmol) ^1/2 . (Ans: V _PFR = 8.56 dm ³ , V _CSTR = 6.45 dm ³ )

[2nd Ed. P4-8]

You are designing a reactor system for carrying out the constant-density liquid-phase reaction

which has the rate law

(a) What system (i.e., type and arrangement) of flow reactors, either one alone or two in series, would you recommend for continuous processing of a feed of pure A in order to minimize the total reactor volume? (90% conversion of A is desired.)
(b) What reactor size(s) should be used?
(c) Plot the conversions and concentrations of A and B as a function of plug-flow reactor volume.

Additional information:

k₁ = 10.0 (lb-mol/ft ³) ^0.5 h ^-1
k ₂ = 6.0 ft ³/lb mol
feed = 100 lb mol/h of pure A
C _AO = 0.25 lb mol/ft ³
total pressure = 2000 kPa

[1st Ed. P4-14]

The liquid-phase reaction is carried out in a semibatch reactor. The reactor volume is 1.2 m ³ . The reactor initially contains 5 mol of B at a concentration of 0.015 kmol/m ³ . A at an aqueous concentration of 0.03 kmol/m ³ is fed to the reactor at a rate of 4 dm ³ /min. The reaction is first order in A and half order in B with a specific reaction rate of k = 6 (m ³ /kmol) ^1/2 / min. The activation energy is 35 kJ /mol. The feed rate to the reactor is discontinued when the reactor contains 0.53 m ³ of fluid.

(a) Plot the conversion, volume, and concentration as a function of time. Calculate the time necessary to achieve:
(b) 97% conversion of A.
(c) 59% conversion of B.
(d) The reaction temperature is to be increased from 25°C to 70°C and the reaction is to be carried out isothermally. At this temperature the reaction is reversible with an equilibrium constant of 10 (m ³ /kmol) ^1/2 . Plot the conversion of A and B and the equilibrium conversion of A as a function of time.
(e) Repeat part (d) for the case when reactive distillation is occurring. Study the effect of the evaporation rate on conversion.

[2nd Ed. P4-27]

The irreversible liquid-phase acid-catalyzed isomerization reaction

is carried out isothermally in a semibatch reactor (Figure CDP4-J). A 2 M solution of H ₂ SO ₄ is fed at a constant rate of 5 dm³ / min to a reactor that ini- tially contains no sulfuric acid. The initial volume of pure A solution in the reactor is 100 dm³ . The concentration of pure A is 10 mol / dm³ . The reaction is first order in A and first order in catalyst concentration, and the specific reaction rate is 0.05 dm³ /molmin ^-1 . The catalyst, of course, is not consumed during the reaction.

Figure CDP4-J

(a) Determine both the number of moles of A and of H and of H ₂ SO ₄ in the reactor and the concentration of A and of H ₂ SO ₄ as a function of time.
(b) Obtain an analytical solution for the number of moles of A, N_A , and the concentration of A, C_A , as a function of time. What are the concentrations of A and of and of H ₂ SO ₄ after 30 min?
(c) If the reaction is reversible with K_C = 2.0, plot the concentration of A and C as a function of time.
(d) How would your answers change if a 2 M solution of A were fed to a 2 M solution by H ₂ SO ₄ that had an initial volume of 100 dm ³ ?
(e) Rework this problem where A is fed at a concentration of 2 M and 5 dm³ /min to 200 dm of 2 M H ₂ SO ₄ . If the reaction is first order in A and zero-order in H ₂ O and the specific reaction rate is 0.05 min^-1 , what is the concentration of A after 30 min? If the reaction is reversible, with K_C = 2.0, plot the equilibrium conversion and the concentrations of A and C as a function of time.

(Hint: Try to use the mole balance expressed in terms of N_A .)

In many industrial processes where the conversion per pass through the reactor is low, it may be advantageous to use a recycle reactor (Figure CDP4-K). Here a significant portion of the exit stream is recycled back through the reactor. Calculate the overall conversion

that can be achieved in a 2-m ³ plug-flow reactor when the irreversible, isothermal first order gas-phase reaction

is carried out at 500°C and 5 m ³ feed of gas is recycled for every cubic meter of fresh.

Figure CDP4-K

Additional information:

[1st Ed. P4-28]

The elementary gas-phase isomerization reaction

is carried out in a packed-bed recycle reactor. The recycle ratio is 5 mol recycled per mole taken off in the exit stream. For a volumetric flow rate of 10 dm ³ /s through the reactor (Figure CDP4-L), the corresponding pressure gradient (assumed constant) in the reactor is 0.0025 atm/ m. The flow in the reactor is turbulent. What overall conversion can be achieved in a reactor that is 10 m in length and 0.02 m ² in cross-sectional area?

Additional information:

C _A0 =0.01 mol /dm ³ (P ₀ = 2 atm). The volumetric flow rate of fresh reactant to the reactor is ₀ = 10 dm ³ /s. The specific reaction rate is k = 0.25 s ^-1 .
[1st Ed. P4-22]

Figure CDP4-L

Consider the recycle reactor system shown in Figure CDP4-M, where the elementary irreversible gas-phase reaction is carried out isothermally at 570°C and 1 atm. A part of the condenser unit is heated to reactor temperature and recycled to reactor inlet. The pressure drop in the conduits can be assumed to be negligible. Estimate the reactor volume for 50% converstion of the feed F _A0 .

Additional information:

F _A0 = 1 kmol/h
F _B0 = 1 kmol/h
k = 100 m³ /kmolh
R = 4
Condenser exit temperature = 45°C
Vapor pressure of C at 45°C = 0.2 atm

Figure CDP4-M
(Courtesy of H. S. Shankar)

Radial-flow reactors can be used to good advantage for exothermic reactions with high heats of reaction. The high radial velocities at the entrance to the reactor are useful in reducing hot spots within the reactor. As fluid moves out into the reactor, the velocity, U, varies inversely with r:

where U ₀ is the velocity (m/s) at the inlet radius, R irreversible, gas-phase reaction R ₀. Consider the elementary, irreversible, gas-phase reaction

carried out in a radial-flow reactor similar to the one shown in Figure CDP4-N. Derive an equation for conversion as a function of radius carried out in a radial-flow reactor similar to the one shown in Figure P4-38.

(a) Derive an equation for conversion as a function of radius for isothermal operation neglecting pressure drop.
(b) Plot X as a function of r for the case when the pressure drop is significant with a 5 0.07 kg ^-1 .
(c) Vary the reaction and reactor parameter values and write a paragraph describing your findings. What parameter effects the results the most?

Figure CDP4-N

The growth of a bacterium, B, is to be carried out in excess nutrient:

The growth rate for this bacteria is best described by a logistic growth model:

Where C_B is the cell concentration (g/dm ³ ) and and C _MAX are constant rate law parameters.

(1) Plot the cell concentration as a function of time in a 10-dm ³ batch reactor.
(2) If the reaction is to be carried out in a 10-dm3 CSTR, what is the exit cell concentration for a volumetric flow rate of 0.5 dm ³ /h. Vary the volumetric flow rate to arive at a plot of exit cell concentration as a function of space time. On the same figure, plot the total numbr of cells exiting the reactor (i.e. C _B ) as a function of space time.

Additional information:
Initial cell concentration in feed 10 ^-6 g/dm ³ , = 0.5 h< ^-1 , C _Bmax = 5 x 10 ^-3 g/ dm ³ .

[2nd Ed. P4-35]

A bimolecular (elementary) second-order reaction, takes place in a homogeneous liquide system. Reactants and products are mutually soluble, and the volume change as a result of reaction is negligible. Feed to a tubular (plug-flow) reactor that operates essentially isothermally at 260°F consists of 210 lb/h of A and 260 lb/h of B. Total volume of the reactor is 5.33 ft ³ , and with this feed rate, 50% of compound A in the feed is converted. It is proposed that to increase conversion, a stirred reactor of 100-gal capacity be installed in series with, and immediately upstream of, the tubular reactor. If the stirred reactor operates at the same temperature, estimate the conversion of A that can be expected in the revised system; neglect the reverse reaction. Other available data include:

(Ans.: X = 0.68.) (California Professional Engineers Exam)
[2nd Ed. P4-11]

The irreversible liquid phase acid catalyzed reaction

is carried out in a semibatch reactor containing H ₂ SO ₄ . A is fed at a constant rate of 10 mol/min. The volumetric flow rate of liquid entering the semibatch reactor is 5 dm³ /min. The initial volume of a 3 M solution of H ₂ SO ₄ catalyst in the reactor is 100 dm³ (no A is present initially). The specific reaction rate is 0.05 min^-1 . The reaction is first order in A and zero-order in catalyst concentration.

(a) Use POLYMATH or MATLAB to determine both the number of moles of A in the tank and the concentration of A and of H ₂ SO ₄ as a function of time.
(b) Obtain an analytical solution for the number of moles of A, N_A , and the concentration of A, C_A , as a function of time. What is the concentration of A after 30 min? How many moles of A will there be in a tank after long times (i.e., )? (Ans.: N_A = 200 mol) Explain why the number of moles remains virtually constant at long times.
(c) Rework part (a) assuming the reaction is first order in H ₂ SO ₄ with k = 0.02 dm ³ / molmin.

(Hint: Do not try to use conversion in solving this problem!)