Effect Pulse

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CONTINUOUS STIRRED TANK REACTOR in SERIES

ABSTRACT

On 24th of August 2005, an experiment was conducted . Title of the experiment was Continuous stirred tank reactor (CSTR) in series. Two experiments were conducted. The first one is the effect of step change input which is a step-change input would be introduced and the progression of the tracer will be monitored via the conductivity measurements in all three reactors and the second one is the effe effect ct of pu puls lsee input input whic whichh is a pu puls lsee input input woul wouldd be intr introd oduce ucedd an andd the the progression of the tracer will be monitored via the conductivity measurements in all three reactors too. Also we have to plot the graph of conductivity against time for all three reactors to the both experiments.

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INTRODUCTION

A stirred tank is the most fundamental of mixers and many common mixers from the Brabender mixer used in lab to a cup of coffee with a spoon can be considered a stirred tank under some set of approximation. Mixing in a stirred tank is complicated and not well described (Middleman p. 340-348) although the use of dimensionless numbers and comparison with literature accounts can lead to some predictive capabilities. Often stirred tanks are used as industrial reactors where a chemical component of a flow stream resides for some time in the tank and then proceeds on to other steps in a chemical process. The residence time distribution becomes a measure of the extent of a chemical reaction in this situation. For mixing one can sometimes assume a constant rate of strain in the stirred tank, Middleman p. 340-348, and the residence time distribution can then be used under this approximation, as a measure of the extent of mixing. Dead zones in a stirred tank for high viscosity fluids should be very familiar to anyone who has worked in a kitchen mixing dough with a hand mixer. Fluid motion in a stirred tank is confined to the immediate region of the mixer blades for high viscosity fluids. In the simplest approximation that a uniform extent extent of mixing occurs in the stirred tank, Middleman Middleman p. 301-306, 301-306, this is called called the "perfect mixer". Consider a stream of butene in cyclohexane cyc lohexane that is converted to butene epoxide by reaction with a peroxide in a CSTR. The flow rate through the tank is Q and the concentration of heptene is C 0. The tank is at steady state meaning the volumetric in-flow equals the volumetric out-flow. The tank volume is V. The ratio of butene epoxide to butene is governed by the temperature, catalyst concentration, effectiveness of the catalyst and the residence time in the reactor. A master curve in terms of conversion at constant conditions as a function of reaction time can easily be made in the lab.

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Then a calculation of residence time distribution in the reactor can be directly mapped, using the lab results, to conversion ratio for the desired product. If the butene epoxide is to be used in a second CSTR to produce the final produc productt the thenn this this conver conversio sionn ratio ratio become becomess the the input input concen concentra tratio tionn for for the second CSTR. Typically a synthetic chemical process will involve a number of CSTR's joined in this way. Then we need to determine the residence time distribution (RTD), f(t), for a perfect mixer to approximate the conversion for this CSTR. In order to determine the RTD, f(t), for the CSTR we consider a simpler situation where a concentration C 0 of a component in a flow stream Q flows into a tank of volume V. At an instant of time all of the concentration C 0 is tagged red so that it can be distinguished from the other reactant in the stirred tank. We then look for the red tagged reactant in the outflow stream to determine the residence time of the reactant in the tank. The amount of tagged material the has left the tank at time "t" is given by the cumulative residence time distribution function, F(t), F(t)C 0Q. This is related to the concentration concentration of tagged material in the effluent stream, QC(t),

so

Then F(t) is the response, efflux, of the system to a pulse of concentration in the influx.A material balance for the CSTR under the assumption of perfect mixing yields,

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with the starting condition that the concentration of the tagged component in the effluent is 0 at t = 0, C(t=0) = 0. The solution to this differential equation is, where V/Q = t is a kind of time constant for the system. Under the assumption of perfect

mixing, this time constant is the mean residence time for the CSTR, t = V/Q. The residence time distribution function is the derivative of the cumulative residence time distribution function,

The function has a value at t = 0 of 1 t =Q/V which decays exponentially with time. The function has a value at t = 0 because mixing is perfect, that is some material is instantaneously in the effluent at the instant material is introduced to the tank. Obviously this is not realistic. Nonetheless, the exponential approximation for a CSTR is a common assumption both in polymer processing and in the chemical process industry as a whole. It is widely used in a wide range of scientific fields as for a first approximation for quantities such as residence time in a lake or ocean or for an approximation of a drug or toxins residence in the human body since it depends only on the system volume and rate of dilution. The function can be modified for dead space using an effective volume rather than the actual volume of the system. Alternatively, tracer studies can be used to measure the mean residence time.

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THEORY

Continuous Stirred Tank Reactors (CSTRs)

Figure 1

Type of Reactor Continuously

Characteristics Stirred Run at steady state with continuous flow of reactants and

Tank Reactor (CSTR)

products; the feed assumes a uniform composition throughout the reactor, exit stream has the same composition as in the tank

Kinds of Phases Usage

Advantages

Disadvantages

1.Continuous operation

1.Lowest

Present 1. Liquid phase

1.When agitation is

2. Gas-liquid rxns

required

3. Solid-liquid rxns 2.Series configurations for different concentration streams

conversion per unit 2.Good temperature control

volume

3.Easily adapts to two

2. By-passing and

phase runs

channeling possible

4. Good control

with poor agitation

5. Simplicity of construction

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6.Low operating (labor) cost 7.Easy to clean General Mole Balance Equation

Figure 2

Assumptions

1) Steady state therefore 2) Well mixed therefore r A is the same throughout the reactor

Rearranging the generation

In terms of conversion

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Reactor Sizing Given –r A as a function of conversion, –r A = f(X), one can size any type of reactor. The volume of a CSTR can be represented as the shaded areas in the Levenspiel Plot shown below:

Figure 3

Reactors in Series Given –r A as a function of conversion, , –r A = f(X), one can also design any sequence of reactors in series provided there are no side streams by defining the overall conversion at any point.

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Figure 4

Mole Balance on Reactor 1

Mole Balance on Reactor 2

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Given –r A = f(X) the Levenspiel Plot can be used to find the reactor volume

Figure 5 For a PFR between two CSTRs

Figure 6

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Given -r A = f (x), the Levenspiel Plot can be used to find the reactor volume

.

Figure 7

Tracer Analysis on the Transient Behaviors of Continuous Stirred-Tank in Series. Unlike the above, the tracer analysis will help to understand the transient behaviors of the continuous stirred tank reactor in series by having a step input or pulse of tracer component such as salts. The conductivity measurement will indicate the progression of the tracer throughout the stirred tank in series. CO

C1

C2

C3

Figure 8

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dCi/ dt = (C 1-1 – Ci ) / τ where τ = V/v and V = Tank Volume, v = volume flow rate, and C i= concentration in i thTank. The differential equations must be solved simultaneously. A real reactor will be modeled as a number of equality sized tanks-in-series. Each tank behaves as an ideal CSTR. The number of tanks necessary, n (our one parameter), is determined from the E(t) curve. For n tanks in series, E(t) is, E (t) = tn-1 e-t/τI (0 - 1)/ τIn Where, τI = τ/n It can be shown that tm = τ = nτI In dimensionless from θ = t / τ = t/τI

nθ = t / τI E (θ) = τ E (t) = n(n θ)n-1 e-nθ (n-1)

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σө2 = σ2

=

0∫

∞

( t – τ )2 E (t) d τ

τ2

σө2 = σ2


τ2

=

0∫

∞

( Ө - 1 ) 2 E ( Ө) d Ө

τ2 Carrying out the integration for the n tanks in series E(t). σө2 = σ2

= 1

τ2 n=

n

τ2 σ2

For a first order reaction, X = 1-

1

τi = τ

( 1 + τi k ) n

n

For reactions other than first order and for multiple reactions, the sequential equations must be solved. Vi = V / n Vi = vo ( C Ao – C A1) -rA1 Vi = vo

( C A1 – C A2) -r A2

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Vi = vo


( C A(n – 1) – C An ) -r An

Example For a second order reaction with n = 3, ( V1 = V2 = V3 = V / 3 ) V3 =

( C Ao – C A )

vo

kC A2 ( τ1 = τ2 = τ3 )

τ3 = kC A2 + C A - C A = 0 C A1 = -1 + √ 1+ 4 τ3 k C Ao 2 τ3 k Similarly, C A2 = -1 + √ 1+ 4 τ3 k C A1 2 τ3 k C A3 = -1 + √ 1+ 4 τ3 k C A2 2 τ3 k X = 1 - (C A3 / C A0) , τ3 = τ / 3

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Effect of Step Change in Input Concentration to the Concentration of Solute in Stirred Tank Reactors in Series. When a step change of solute concentration is introduced at the feed of tank 1, the tank in series will experience a transient behavior as of Figure 8 below. The response will be dependent on the residence time of each reactor in series.

figure 9a: step change input.

Figure 9b: transient response of tank in series to the step input.

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Effect of Pulse in Input Concentration to the Concentration of solute in Stirred Tank in Series. When a pulse input of solute concentration is introduced at the feed of tank 1, the transient behavior will be different than the step change input due to the diminishing concentration from the input after pulsing.

figure 10a: pulse input.

Figure 10b: Transient response of tank in series to the pulse input.

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OBJECTIVE

To determine; •

The effect of step changes in input concentration of solute.

•

The effect of pulse or residence time in input concentration of solute curve.

APPARATUS

1. Distillation water 2. Sodium Chloride 3. Continuous reactor in series 4. Stirrer system 5. Feed tanks 6. Waste tank 7. Dead time coil 8. Computerize system

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APPARATUS DISCRIPTION

Figure 11 Before operating the unit, we must familiarize ourselves with the unit. Please refer to figure 1 to understand the process. The unit consists of the followings: Reactors Three reactors made of borosilicate glass, each having approximately 2 liters capacity. Each reactor is fitted with variable speed stirred mounted on the top plate. Temperature and conductivity sensors are provided for each reactor. Flows between vessels are by gravity. Overflow tubes are provided for the 2 nd and 3rd reactor. 17

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Stirred System Variable speed stirred system with digital display consisting of a motor and a shaft with impellers made of stainless steel. Speed adjustment by means of a speed controller knob on each stirrer. Feed Tanks Two 15-L cylindrical tanks made of stainless steels are provided with the unit. Each tank has a feed pump to transfer the liquid from feed tank to the reactors. Each tank is fitted with a level switch to protect the pumps from dry on. Waste Tank A rectangular 50-L waste tank made of stainless steel is provided at the bottom of the equipment. Dead Time Coil Material: 3/8” stainless steel tubing Volume: approx. 200 ml Instrumentations 1) Flow meter Range: 0 to 500 ml/min Output: 0 to 5 VCD Display: LCD digital display 2) Conductivity Meter Sensor Range: 0 to 200 mS/cm No. of sensors: 4 (CT1, CT2, CT3, CT4) Output: 4 to 20 mA Display: conductivity controller with digital display for each sensor mounted on the control panel.

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3) Temperature Sensor No. Of sensors: 3 (TT1, TT2, TT3) Sensor type: RTD 4) Data Acquisition System The Data Acquisition System consists of a personal computer, ADC modules and

instrumentations for measuring the process parameters. A

flow meter with 0 to 5

VCD output signal is supplied for feed flow rate

measurement. Conductivity

sensors with controller are provided for

monitoring the tracer concentration in

each reactor. The ADC modules into

digital signals will convert all analog signals

from the sensors before being

sent to the personal computer for display and manipulation. Equipment Description The system consists of three agitated, glass reactor vessels connected in series, two feed tanks, two variable through put feed pumps, variable speed agitators, fixed height overflow, and an electrical conductivity meter. Two feed systems are provided. Each feed system consists of a feed tank, a variable speed pump, and a mixer (tee-type) at the first tank inlet. The feeds flow through each tank where a constant liquid holdup (steady-state operation). Chemicals The conductive component with be potassium chloride at low concentrations.

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EXPERIMENTAL PROCEDURES

Experiment A: The effect of Step Change In this experiment a step change input would be introduced and the progression of the tracer will be monitored via the conductivity measurement in all the three reactors. 1. The two of three tanks (tank 1 and tank 2) was filled up with 20L feeds deionized water. 2. 300 g of sodium chloride in tank 1 was dissolved. Make sure the salts dissolve entirely and the solution is homogenous. 3. The three way valve (V3) was setting to position 2 so that the deionized water from tank 2 will flow into reactor 1. 4. The pump 2 is switch on to fill up all three reactors with deionized water. 5. The flow rate (Fl1) was set to 150 ml/min by adjusting the needles valve (V4). Do not use too high flow rate to avoid the over flow. Make sure no air bubbles trapped in the piping. The stirrers 1, 2 and 3 were switch on. 6. The deionized water was continued pumped for about 10 minute until the conductivity readings for all three reactors are stable at low values. 7.

The values of conductivity were recorded at t 0.

8. T he pump 2 was switch off after 5 minutes. The valve (V3) was switch to position 1 and the pump 1 was switch on. The timer was started. 9.

The conductivity values for each reactor were recorded every three minutes.

10. Record the conductivity values were continued until reading for the three reactors were closed to the starting value recorded. 11. Pump 2 was switch off and the valve (V4) was closed. 12. All liquids in reactors were drained by opening valves V5 and V6.

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Experiment 2: The effect of pulse input. In this experiment a pulse input would be introduced and the progression of the tracer will be monitored via the conductivity measurements in all three reactors. 1. The two of three tanks (tank 1 and tank 2) was filled up with 20L feeds deionized water. 2. 300 g of sodium chloride in tank 1 was dissolved. Make sure the salts dissolve entirely and the solution is homogenous. 3. The three way valve (V3) was setting to position 2 so that the deionized water from tank 2 will flow into reactor 1. 4. The pump 2 is switch on to fill up all three reactors with deionized water. 5. The flow rate (Fl1) was set to 150 ml/min by adjusting the needles valve (V4). Do not use too high flow rate to avoid the over flow. Make sure no air bubbles trapped in the piping. The stirrers 1, 2 and 3 were switch on. 6. The deionized water was continued pumped for about 10 minute until the conductivity readings for all three reactors are stable at low values. 7.

The values of conductivity were recorded at t 0.

8. Switch off pump 2 after 5 minutes. The valve (V3) was switch to position 1 and switch on pump 1. The timer was started. 9. Let the pump 1 to operate for 5 minute, and then switch off pump 1. Switch the three ways valve (V3) back to position 2. The pump 2 was switch on. 10. The conductivity values for each reactor were recorded every three minutes. 11. Record the conductivity values were continued until reading for the three reactors were closed to the starting value recorded. 12. Pump 2 was switch off and the valve (V4) was closed. 13. All liquids in reactors were drained by opening valves V5 and V6.

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RESULTS

Experiment 1: The effect of step change input. FT: 139.8 ml / min Time (min) 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 88.1

TT1: 25.5 oC TT2: 25.4 oC TT3 25.3 oC

QT1(mS/cm) 0.0126 1.3219 2.1260 2.6304 3.0477 3.3034 3.5090 3.6304 3.7146 3.5343 3.8813 3.9561 4.0157 3.9349 3.8083 4.0114 4.0294 3.9380 4.0201 3.9122 4.0844 4.0220 4.1531 4.1140 4.0197 4.0778 4.0297 3.9986 4.1068 4.0198

QT2(mS/cm) 0.0389 0.1109 0.6421 0.8866 1.3243 1.9302 2.2321 2.5395 2.8006 3.0621 3.2544 3.4134 3.4241 3.5468 3.5900 3.7752 3.7211 3.7843 3.8207 3.8028 3.7139 3.8698 3.9130 3.9109 3.9402 3.8673 3.9166 3.8304 3.9662 3.8499

QT3(mS/cm) 0.0686 0.0000 0.0000 0.1946 0.3556 0.6332 1.0517 1.2374 1.6147 1.8299 2.0741 2.3752 2.6279 2.6996 2.9115 2.9957 3.1716 3.2389 3.3472 3.2233 3.3454 3.5177 3.3882 3.4497 3.5086 3.4742 3.6840 3.5867 3.5754 3.5305

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Experiment 2: The effect of pulse input. FT: 0.2 ml / min Time (min) 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78

TT1: 24.9 oC TT2: 24.9 oC TT3 24.8 oC QT1(mS/cm) 1.3767 1.4100 1.1217 0.8780 0.6144 0.4597 0.3506 0.2307 0.1596 0.1578 0.0934 0.1610 0.0319 0.0513 0.0596 0.0056 0.0011 0.0553 0.0000 0.0515 0.0636 0.0185 0.0000 0.0094 0.0000 0.0395 0.0663

QT2(mS/cm) 0.3259 0.6906 0.8530 0.8989 0.8570 0.7494 0.7071 0.6185 0.4772 0.4298 0.3096 0.2545 0.2005 0.2184 0.1412 0.0925 0.1384 0.0898 0.0377 0.0229 0.0367 0.0173 0.0000 0.0669 0.0315 0.0000 0.0000

QT3(mS/cm) 0.2262 0.2822 0.3149 0.4877 0.5634 0.6211 0.7105 0.6560 0.5734 0.5716 0.5631 0.4254 0.3905 0.3477 0.3184 0.2914 0.1847 0.2365 0.1309 0.1486 0.1445 0.0804 0.1223 0.0644 0.0673 0.0518 0.0561

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GRAPH: EXPERIMENT 1

Conductity change with time for each reactor in step change 5 4 y t i v 3 i t c u d n 2 o C

QT1 QT2 QT3

1 0 0

50

100

Time

Graph 1

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GRAPH: EXPERIMENT 2

Conductivity change with time for each reactor in pulse change 2

y t i v i t c u d n o c

1.5 QT1 QT2 QT3

1 0.5 0 0

10 20 30 40 50 60 70 80 time Graph 2

SAMPLE OF CALCULATION

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No sample of calculation involve. DISCUSSIONS

The Continuous Flow Stirred Tank Reactor (CSTR) is probably the easiest way to transfer a batch process to a continuous one since all engineering and scaleup data can be used from the batch process. The very broad residence time distribution (RTD) profile

however can pose some sever complications, therefore

the pitfalls are mentioned first A minor problem is that the lower volumetric reaction rate results in a lower productivity ,

this can easily be compensated by the reduction of waiting times

(filling, heating, cooling, emptying of the batch reactor) and a more constant product quality. The effect on the selectivity of reactions can either be good or bad. In the case of production of homogeneous copolymers for example, the CSTR can be an ideal reactor since composition drift is absent during steady state operation. The main reason that a series of CSTRs is not suited for the entire emulsion polymerisation process is that the first stage of the process, the particle nucleation , is very

sensitive towards residence time distribution.

First of all, compared to the batch process, the steady state operation results in more constant product properties, an improved energy consumption (the heat of reaction can be used to heat feed streams) and a higher productivity through the reduction of inactive periods (filling, heating, cooling, emptying). The control over local mixing independently from net flow is one of the key advantages compared to a traditional tubular reactor. The ideal mixing in a single vessel can, besides all the negative effects mentioned above, also have its positive effects. For the production of homogeneous copolymers, the steady state concentrations in each CSTR make it a lot easier to calculate and feed additional monomer streams to each CSTR compared to the time-based addition profile for the semi-batch process. Because

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no backmixing is possible from downstream CSTRs, the reaction conditions can be changed sharply in subsequent reactors, without having a (negative) effect on the reactors upstream. In the first experiment of the effect of step change, the values of the conductivity for those three stirrers were recorded in each three minutes. From the graph we can see that the concentration of the deionized water was increasing with time. The concentration of the tank 1 is higher than the others. This is because; the 300 g of sodium chloride was dissolved firstly in tank 1. Then, the deionized water that was dissolved to the sodium in tank 1 was pumped to other one by one which was full with deionized water. Though the concentration is uniform in each reactor, nevertheless there are changes in concentration as fluid moves on from 1st reactor to others. After 93 minutes, the concentration in the 3 reactors becomes stable which is equivalent for each reactor. The resulted was shown in graph 1. In the second experiment, the purpose on running this experiment was to determine the concentration response to pulse change. Same as before, the conductivity was recorded for those reactors in each 3 minutes. From the graph, we can see that the concentration of the deionized water was increasing with time. After 87 minute mixing, the concentration reading of the three stirrers reached a stable at low value which is equivalent for each stirrer. This situation above is one of the draw back of using cstr in series if compared to single STR. Although the series STR requires a lower reactor volumes but it take longer to reach the conversion and it also requires more stirrer than the single STR.

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CONCLUSIONS

From the graph and observation, it can be concluding that; o

Both first experiment and second experiment has the graph pattern as the standard graph (figure 9 and 10).

o

For the first experiment, it takes more time to achieving equilibrium in concentration which rather it can be said that the concentration on the reactor are directly proportional with time .

o

Time measuring very important because it is can affect the conductivity.

RECOMMENDATIONS

During the experiment there are some problem occur, so these is the list of the recommendation that can be considered if we want to produce better results; o

Firstly all the basic procedure and maintenance must be followed.

o

Secondly, the instrument must have certain program which can automatically record every 3 minutes.

o

The time for doing the experiment must be extended because it takes time to achieved equivalent data.

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REFERENCES

i.

Levenspiel Octave, Department of Chemical Engineering Oregon State University, Chemical Reaction Engineering Third Edition, John Wiley &

Sons, 1999. ii.

Schmidt, Lanny D., The Engineering of Chemical Reactions . New York: Oxford University Press, 1998.

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APPENDICES CE 442

DESIGN EXAMPLE - Application of Chemical Reactor Theory -

A chemical manufacturing facility is planning on a product modification that will result in a change in wastewater character. Of significance is the anticipated presence of phenol, an organic chemical that has not been treated by the company's wastewater treatment facility. Consequently, the ability of the existing facility to “treat” phenol is in question. In order to predict the existing reactor performance, kinetic and dispersion information is needed. Reaction kinetic data have been collected for the chemical destruction of phenol through batch reactor tests (Table 1). Table 1. Batch reactor reaction kinetic data. Time Phenol (min) (mg/L)

0 10 20 30 40 60 80

100 55 22 8 5 0.82 0.15

The phenol-bearing wastewater flow is projected to be 0.45 MGD with an anticipated phenol concentration of 95 mg/L. The reactor ( a chemical oxidation process) has a total liquid volume of 25,000 gal. (L=35 ft., W=12 ft., H=8 ft.). A reactor dispersion analysis has been performed by injecting an impulse of non-reactive material into the reactor feed stream. The flow rate was monitored during the dispersion study and was determined to be 0.45 MGD. The reactor dye trace data are presented in Table 2. The phenol discharge permit limit has been set at 0.1 mg/L by the state regulatory board. Estimate the effluent concentration of phenol based on the available information. Determine what might be done to meet the discharge limit. It would be desirable to maintain the existing reactor due to obvious economic reasons. However, any reasonable design option may be considered. Justify your final design recommendation. Show all calculations in a clear and concise fashion. State all assumptions that you make in the design process. It should be assumed that the kinetic relationship (Table 1) are fixed and cannot be altered. That is, the reaction kinetic data represents optimum conditions.

Table 2. Field scale reactor dispersion data. t Conc (min) (mg/L)

0 44.75 46 46.92

0 0.23 0.32 0.36

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_______________________________________ 48 48.5 49 49.33 50 50.75 51.67 52.75 54 56 58 60 62 64 66 68 70 75 79 83 98 113 128


0.82 1.24 2.15 2.61 2.98 2.93 2.98 2.89 2.84 2.89 2.84 2.52 2.52 1.92 1.79 1.6 1.33 1.28 1.14 1.1 0.41 0.39 0.27

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Solution

The solution approach is based on utilizing the kinetic information (Table 1) and dispersion information (Table 2) to estimate existing reactor performance and to develop alternatives for upgrading the treatment process. The solution approach is presented below. 1.

determine reaction rate and order, 2

σ θ

= σ2/tbar 2),

2.

determine the normalized variance (

3. 3. 4.

determine the number (N) of CSTR's in series that simulates the calculated dispersion ( estimate the existing reactor effluent concentration, and develop alternative reactor design modifications.

2

σ θ

= 1/N),

Reaction rate constant and order. Reaction rate and order is determined using the data in Table 1 of the problem

statement. The general rate expression (equation 1) is applied in the analysis process.

dC dt

= −kC

n

(1)

where k = reaction rate constant n = reaction order After assuming a reaction order, equation 1 is integrated and algebraically manipulated such that a general straight line relationship (equation 2) is developed.

y = mx + b

(2)

For example, if we assume n=0, the following solution applies. C t

= −kt + C 0

dC dt

(3)

= −kC 0 = −k

dC = −kdt C t

t

C 0

0

∫ dC = −k ∫ dt

C t − C 0 = −kt Equation 3 is a straight line relationship and indicates that if the reaction data is truly zero order, a plot of C t vs. t would result in a straight line. If, however, the data is not linear, another order must be assumed and the process repeated. For example, assuming n=1 yields the following solution to equation 1.

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Effect Pulse

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