ABSTRACT

The experiment was constructed to examine the effect of a pulse input and step change input in

a tubular flow reactor. Other than that, the aim of this experiment is to construct a residence

time distribution (RTD) function for the tubular flow reactor. The first step in running this

experiment was opened the valve V9 and switched on pump P1. The flow rate of de-ionized

water into the reactor R1 was controlled to constant at 700 mL/min at Fl-01. The water was

continued to flow until the conductivity of inlet and output were stable a low levels. The value

was 0.0. Then, V9 and P1 were closed and turned off. V11 and P2 were opened. P2 was

adjusted to give a constant flow rate of salt solution into R1 at 700 mL/min at Fl-02. The salt

solution was allowed to flow for one minute. Then, the timer was reset and restarts. This allowed

the time started at the average pulse input. V11 and P2 were closed and quickly opened V9 and

P1. For every 30 s, the conductivity of Q1 and Q2 were recorded until all the readings were

constant and approached the stable low level values. Then, a graph of outlet conductivity values

versus time was plotted to get C(t) curve. From the C(t) curve, graph of E(t) is plotted as the

function of time. This is the residence time distribution (RTD) function for the plug flow reactor.

In experiment 1, the conductivity increased at minute 1.0 at 0.2 ms/cm and reached the peak at

minute 2.0 with conductivity at 2.6 ms/cm. Then, the conductivity decreased as the time

increased until its reading is 0.0 ms/cm. It showed that the reactant was all used up in the

reactor. The time the materials spend in the reactor thus called the residence time. Same steps

as in the first experiment, a graph of conductivity versus time was plotted to find the C(t) curve.

Then, E(t) graph is plotted from the values obtained versus the time. The mean residence time,

tm was 2.128 min, second moment variance, σ2 was 25.89 and the third moment skewness, s3

was -0.232. In experiment 2 which is the step change input experiment, the graph of outlet

conductivity values versus time show that the curve start increasing rapidly only at 2.5 min and

constant at 3.5 min with 2.8 conductivity value. The mean residence time, tm was 22.263,

second moment variance, σ2 was 1556.452 and the third moment skewness, s3 was -89.222.

The value of E(t) is depends on the value of C(t).

1

INTRODUCTION

In chemical plant especially in chemical processes, reactor is one vital item needed for raw

materials to undergo a chemical reaction to form desired output or products. Thus, the operation

and design of the reactors play an important role for a success of processes in the industrial.

Depending on the nature of the feed materials and the output, the reactors are appeal in many

forms. Hence, proper approaches of understanding of the nature of the reactors work to handle

the system of reaction. Reactor is divided into two main types which is batch reactors and

continuous flow reactors.

Plug reactor is considered as one of the continuous flow reactors that almost always operated at

steady state (Fogler, 2006). It is also known as tubular reactor with its cylindrical pipe and

mostly used for gas-phase reactions. It also resembles batch reactors in providing initial driving

forces, which diminish as the reactions progress down the tube. The key assumption is that

tubular acts like plug where it mixed in the radial direction but not in the axial direction.

According to Fogler, the reactants are continually consumed as they flow down the length of the

reactor. The systems is said to be in the uniform velocity as in turbulent flow and no radial

variation in reaction rate.

Figure 1. Plug Flow Reactor

As the reactor suggested, there is no mixing of the medium along the length of the reactor since

PFR is an idealized reactor. However, there is no such reactor exist in reality because it is quite

impossible to have no mixing at all during the reaction. At most, the amount of mixing in the

reactor can be minimized.

In PFR, the residence time is the same as for all elements of fluids. However, there is no

interchange of material in the pug with the material in the other leading or following occurs.

2

There is no back mixing in the reactor. The advantages of PFR are that it has a high volumetric

unit conversion, long periods of time in operation without any maintenance. However, it is hard

to control the temperature of the reactors and the maintenance of the reactor is quite costly

compared to CSTR. As it flows down the tubular flow reactor, the residence time of the plug is a

function of it position in the reactor.

In this experiment, the reactor used is BP 101-B that has been designed for students in order to

perform an experiment on chemical reactions in liquid phase under isothermal and adiabatic

conditions. It comes with a jacketed plug flow reactor, individual reactant tank of feed and

pumps, temperature sensors and conductivity measuring sensor. For this purpose,

saponification reaction between ethyl acetate and sodium hydroxide can be performed.

In addition, the injection of a tracer such as salt solution in the reactor feed can perform the

transient behavior in the PFR and the conductivity can be measured at the outlet of the reactor

as it indicate the progression of the tracer throughout the reactor.

OBJECTIVES

1. To examine the effect of a pulse input in a tubular flow reactor.

2. To examine the effect of a step change input in a tubular flow reactor.

3. To construct a residence time distribution (RTD) function for the tubular flow reactor.

3

THEORY

As fluid flows down the plug flow reactor, the fluid is mixed in the radial direction, but mixing

does not occur in the axial direction each plug of fluid is considered a separate entity as it flows

down the pipe. However, as the plug of fluid flows downstream, time passes. Therefore, there is

implicit time dependence even in steady-state PFR problems. However, because the velocity of

the fluid in the PFR is constant, time and downstream distance are interchangeable.

A given chemical reaction rate can be expresses in several ways either as the rate of

disappearance of the reactants (−r A) or the rate of formation of products (r A). For instant, in the

following reaction,

aA + bB cC + dD

Where;

A = B = reactants

C = D = products

a = b = c = d = stoichiometric coefficients

If the species A is considered as the reaction basis, then the rate of reaction can be represented

by the rate of disappearance of A. The rate of reaction can be also represented by the rate of

disappearance of another species such as –rB and the rate of formation of a product such as rC

and rD. They can be related in the following equation,

−r A

a=

−r B

b=rCc

=rD

d

The rate of chemical reaction is an intensive quantity and depends on temperature and

concentration. One of the most common forms of rate law equation is shown to be:

−r A=K AC Aa CB

b

The saponification of sodium hydroxide and ethyl acetate is represented by the following

stoichiometric equation:

4

C H 3COOC2H 5+NaOH→CH 3COONa+C2H 5OH

This reaction is found to be second order and practically irreversible. The rate of reaction is

represented in the following:

r=k CACBKmol / (m3 s)

Where:

k=k0 e(−ERT ) (Arrhenius law expression)

k 0=18.6×106m3/(kmol . s)

E=4.688×104 J /mol

The reactor’s residence time is defined as the reactor volume divided by the total feed flow

rates.

τ=V TFR

V 0

For a second order equimolar reaction with the same initial reactants concentration

(C ¿¿ A 0=CB0)¿ , the rate law is shown to be:

−r A=k C ACB=kC A02 =k CA 0

2 (1−X )2

V TFR=V 0

k CA 0

( X1−X

)

For Constant PFR volume, flow rate and initial concentrations, the reaction rate constant is

calculated by:

k=V 0

V TFRCA 0

( X1−X

)

The conversion (or fractional conversion), denoted X, is a frequently used measure of the

degree of reaction. It i s defined as

5

X = moles of a species that have reacted

molesof samespecies iniatially present

In pulse input, an amount of tracer is injected in one shot into the feed stream entering the

reactor in as short time as possible. The outlet concentration is then measured as a function of

time. The effluent concentration versus time curve is referred to as C(t) curve in residence time

distribution analysis.

The residence-time distribution of a reactor is a characteristic of the mixing that occurs in the

chemical reactor. In an ideal plug flow reactor, all the atoms of materials leaving the reactor

have been inside the reactor for exactly the same amount of time. The time the atoms spent in

the reactor is called the residence time of the atoms in the reactor.

By considering an injection of a tracer pulse for a single input and single output system, for a

small time increment, ∆t which is sufficiently small so that the concentration of tracer is

essentially constant during the time period, the amount of tracer C(t) exiting between time t and

t + ∆t is then

∆ N=C (t ) v ∆ t

Dividing by the total amount of material that was injected into the reactor, N0

∆ NN0

=vC (t)N 0

∆ t

For a pulse injection,

E (t )= vC (t)N 0

So that

∆ NN0

=E ( t )∆ t

6

In which E(t) is the residence time distribution function that describes in a quantitative manner

how much time different fluid elements have spent in the reactor. The quantity E(t)dt is the

fraction of the fluid exiting the reactor that has spent time t and t + ∆t inside the reactor.

By differentiation and integration

d N=vC (t)dt

N0 ¿∫0

∞

vC ( t )dt

The volumetric flow rate v is constant, so E(t) can be define as

E ( t )= C (t )

∫0

∞

C (t )dt

For step change, a constant rate of tracer addition to a feed that is initiated at time t = 0. Before

this time no tracer was added to the feed. Thus, we have

C0(t) = 0 t < 0

C0(t) = C0 t ≥ 0

Because the inlet concentration is a constant with time, Co, it can be take outside the integral

sign,

Cout=¿C0∫0

t

E (t ' )dt '

Dividing by C0 yields

¿ ] step = ∫0

t

E (t ' )d t '=F (t)

F (t )=[Cout

C0]step

7

We differentiate this expression to obtain the RTD function E(t):

E (t )= ddt [C (t )

C0 ]step

APPARATUS

1) Soltec Tubular Flow Reactor instrument

2) Solution 0.025M Sodium Chloride and De-ionized water

3) Stopwatch

8

Figure 2Tubular Flow Reactor (SOLTEQ model :BP 101-B)

PROCEDURES

Operating procedures

General start-up procedures

1. All the valves are initially except for valve V7.

2. 20 liter of salt solution (NaCl; 0.025M) was prepared.

3. The feed tank was filled with the NaCl solution.

9

4. The power was turn on from the control panel.

5. The water de-ionizer was connected to the laboratory water supply. Valve V3 was

opened and filled with the de-ionized water. Then, the valve V3 closed.

6. Valve V2 and V10 was opened before pump P1 switched on. P1 flow controller was

adjusted to obtain a flow rate of approximately 700 ml/min at flow meter FT-01. The

conductivity displays were observed at low value before valves V10 was closed and

pump P1 was switched off.

7. Valves V6 and V12 were opened with pump P2 was switched on. The flow controller

was also adjusted to obtain a flow rate of approximately 700ml/min at flow meter FT-02.

After the flow rate was obtained valves V12 was closed and pump P2 was switched off.

General Shut-Down Procedure

1. Both pump P1, P2 and P3 were switched off and valve V2 and V6 switched off.

2. The heater then switched off.

3. The water keeps circulating through the reactor while the stirrer motor was running to

allow the water jacket to cool down to room temperature.

4. The power control then switched off.

Experiment 1 : Pulse input in a tubular flow reactor

1. Valve V9 was opened and pump P1 was switched on.

2. Pump P1 flow controller was adjusted to give a constant flow rate of de-ionized water

into the reactor R1 at approximately 700 ml/min at FT-01.

3. The de-ionized water was allowed to flow through the reactor until the inlet (QI-01) and

outlet (QI-02) conductivity values are stable at low levels. The conductivity value was

recorded.

10

4. Valve V9 was closed and pump P1 was switched off.

5. Valve V11 was opened and pump P2 was switched on and the timer start

simultaneously.

6. Pump P2 flow controller was adjusted to give a constant flow rate of salt solution into the

reactor R1 at 700 ml/min at FT-02.

7. The salt solution was allowed to flow for 1 minute, then reset and starts the timer. This

will start the time at the average pulse input.

8. Valve V11 was closed and pump P2 was switched off before valve V9 was quickly

opened and pump P1 was switched on.

9. The de-ionized water flow rate was make sure to always maintained at 700 ml/min by

adjusting P1 flow controller.

10. The conductivity value at both inlet (QI-01) and outlet (QI-02) was recorded at regular

intervals of 30 seconds.

11. The conductivity value was recorded until the reading at both inlet and outlet are almost

constant and approach the stable low level values.

Experiment 2 : Step change input in a tubular flow reactor

1. The general start-up procedure was performed.

2. Valve V9 was opened and pump P1 was switched on.

3. Pump P1 flow controller was adjusted to give a constant flow rate of de-ionized water

into the reactor R1 at approximately 700 ml/min at FT-01.

11

4. The de-ionized water was allowed to flow through the reactor until the inlet (QI-01) and

outlet (QI-02) conductivity values are stable at low levels. The conductivity value was

recorded.

5. Valve V9 was closed and pump P1 was switched off.

6. Valve V11 was opened and pump P2 was switched on and the timer start

simultaneously.

7. The conductivity value at both inlet (QI-01) and outlet (QI-02) was recorded at regular

intervals of 30 seconds.

8. The conductivity value was recorded until the reading at both inlet and outlet are almost

constant.

RESULT

Flow rate: 700 mL/min

Experiment 1: Pulse Input in a Tubular Flow Reactor

Time (s) Time (min) Pulse Input

12

Conductivity (ms/cm)

Input, Q1 Output, Q2

0.0 0.0 0.0 0.0

30.0 0.5 0.4 0.0

60.0 1.0 0.1 0.2

90.0 1.5 0.1 2.3

120.0 2.0 0.0 2.6

150.0 2.5 0.0 0.5

180.0 3.0 0.0 0.2

210.0 3.5 0.0 0.1

240.0 4.0 0.0 0.0

Experiment 2: Step Change Input in a Tubular Flow Reactor

Time (s) Time (min)

Step Change Input

Conductivity (ms/cm)

Input, Q1 Output, Q2

13

0.0 0.0 0.0 0.0

30.0 0.5 4.4 0.0

60.0 1.0 4.7 0.0

90.0 1.5 4.8 0.0

120.0 2.0 4.9 0.0

150.0 2.5 4.9 2.0

180.0 3.0 4.9 2.0

210.0 3.5 4.9 2.9

240.0 4.0 4.9 2.9

270.0 4.5 4.8 3.0

300.0 5.0 4.8 3.0

330.0 5.5 4.8 3.1

360.0 6.0 4.7 3.1

390.0 6.5 4.6 3.2

420.0 7.0 4.6 3.2

450.0 7.5 4.5 3.2

CALCULATIONS

Experiment 1: Pulse Input in a Tubular Flow Reactor

14

∫0

∞

C ( t )dt = area under the graph of C(t)dt.

∫0

∞

C (t )dt=∫0

4.0

−0.442 t2+1.718 t−0.2758dt

¿∫0

0.5

−0.442t 2+1.718 t−0.2758dt+∫0.5

1.0

−0.442 t2+1.718 t−0.2758dt+∫1.0

1.5

−0.442 t 2+1.718 t−0.2758 dt+∫1.5

2.0

−0.442t 2+1.718 t−0.2758dt+∫2.0

2.5

−0.442 t2+1.718 t−0.2758dt+∫2.5

3.0

−0.442 t 2+1.718 t−0.2758 dt+∫3.0

3.5

−0.442t 2+1.718 t−0.2758dt+∫3.5

4.0

−0.442 t 2+1.718 t−0.2758dt

¿0.0584+0.3775+0.6445+0.684+0.5488+0.3155−0.028 4

¿2.6003

E ( t )= C( t)

∫0

∞

C (t )dt¿

C( t)2.6003

When C (t) = 0.2

¿0.22.6003

= 0.0769

Mean residence time, tm = ∫0

∞

tE (t )dt

tm=∫0.0

0.5

0.0dt+∫0.5

1.0

0.0dt+∫1.0

1.5

0.077dt+¿∫1.5

2.0

1.328dt+∫2.0

2.5

2.000dt+∫2.5

3.0

0.481dt+¿∫3.0

3.5

0.231dt+∫3.5

4.0

0.135dt ¿¿

¿0.000+0.000+0.0385+0.664+1.000+0.241+0.116+0.068+0.000

¿2.128 min

Second moment, Variance, σ2 = ∫0

∞

¿¿

When t = 1.0 ;

15

¿

¿0.098

∫0

∞

¿¿

¿25.89

Third moment, Skewness, s3 = 1

σ32

∫0

∞

¿¿

¿1

4.212(0.055)

¿0.232

Time t, (min) 0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000

C(t) (ms/cm) 0.000 0.000 0.200 2.300 2.600 0.500 0.200 0.100 0.000

E (t) 0.000 0.000 0.077 0.885 1.000 0.192 0.077 0.038 0.000

tE (t) 0.000 0.000 0.077 1.328 2.000 0.481 0.231 0.135 0.000

tm 0.000 0.000 0.039 0.664 1.000 0.241 0.116 0.068 0.000

(t –tm )2 4.528 2.650 1.272 0.394 0.016 0.138 0.760 1.882 3.504

Variance, σ2 0.000 0.000 0.049 0.174 0.008 0.013 0.029 0.036 0.000

1

σ32

0.000 0.000 4.212 3.712 37.38

4

25.97

4

14.23

0

12.10

0

0.000

Skewness,s3 0.0 0.0 0.232 0.405 0.038 0.128 0.363 6.01 0.000

Experiment 2 : Step change input in a tubular flow reactor

16

E ( t )= ddt (C ( t )

C0)

Where C0 = final constant value of outlet conductivity

= 3.2

C (t ) = -0.091t2 + 1.2122t – 0.7569

When : t = 0

C (t) = - 0.7569

C(t )C0

= −0.75693.2

= -0.237

Mean residence time, tm = ∫0

∞

tE (t )dt

¿∫0.0

0.5

0dt+∫0.5

1.0

−0 .027dt+∫1.0

1.5

0 .114 dt∫1.5

2.0

0 .402dt+∫2.0

2.5

0.816dt+∫2.5

3.0

1.333dt+∫3.0

3.5

1.932dt+∫3.5

4.0

2.594 dt+∫4.0

4.5

3.296dt+∫4.5

5.0

4.014 dt+∫5.0

5.5

4.735dt+∫5.5

6.0

5.429dt+∫6.0

6.5

6.078dt+∫6.5

7.0

6.656dt+∫7.0

7.5

7.154 dt

¿0−0.014+0.057+0.201+0.408+0.666+0.966+1.297+1.648+2.007+2.368+2.714+3.039+3.328+3.577

¿22.263

Second moment, Variance, σ2 = ∫0

∞

¿¿

When t = 0.5 ;

¿

¿−25.576

17

∫0

∞

¿¿

= 1556.452

Third moment, Skewness, s3 = 1

σ32

∫0

∞

¿¿

¿ 1

39.4532

(278.305−547.960−1199.431−1697.2325−2057.102−2301.589−2447.344−2509.653−2499.677−2435.953−2324.569−2178.620−2005.334−1816.94)

= -89.222

Time,

t(min)

E (t) tm tE ( t )dt ( t - tm )3

E(t)

Variance,

σ2

Skewness,s3

0.0 -0.237 0.000 0.000 0.000 0.000 0.000

0.5 -0.054 -0.135 -0.027 556.609 -25.576 2.246

1.0 0.114 0.057 0.114 -1095.919 51.541 -4.423

1.5 0.268 0.201 0.402 -2398.861 115.535 -9.681

2.0 0.408 0.408 0.816 -3394.465 167.520 -13.699

2.5 0.533 0.666 1.333 -4114.204 208.177 -16.604

3.0 0.644 0.966 1.932 -4603.177 238.965 -18.577

3.5 0.741 1.297 2.594 -4894.688 260.869 -19.754

4.0 0.824 1.648 3.296 -5019.305 274.835 -20.257

4.5 0.892 2.007 4.014 -4999.353 281.448 -20.176

5.0 0.947 2.368 4.735 -4871.905 282.217 -19.662

5.5 0.987 2.714 5.429 -4649.137 277.345 -18.763

6.0 1.013 3.039 6.078 -4357.239 267.923 -17.585

6.5 1.024 3.328 6.656 -4010.667 254.436 -16.186

7.0 1.022 3.577 7.154 -3633.880 238.084 -14.666

7.5 1.005 3.769 7.538 -3233.627 219.036 -13.050

18

DISCUSSION

The experiment is to examine the effect of a pulse input and step change input in a tubular flow

reactor and to construct a residence time distribution (RTD) function for the tubular flow reactor.

Both inlet and outlet conductivity values was recorded at regular intervals of 30 seconds until all

the reading are almost constant and approach the stable low level values. The de-ionized water

flow rate is maintained at 700 ml/min.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

1.2

f(x) = − 0.170021645021645 x² + 0.66078658008658 x − 0.106006060606061

Outlet Conductivity values vs time

Time (min)

Cond

uctiv

ity (m

s/ cm

)

For the first experiment which is to examine the effect of a pulse input in a tubular flow reactor,

the salt solution was allowed to flow inside the reactor for 1 minute only. According to the graph

above, the outlet conductivity, Q2 value started to rise at minute 1.0 which is 0.2 ms/cm to its

peak at minute 2.0 which is 2.6 ms/cm. Then, it drastically dropped to 0.5 ms/cm at minute 2.5

before decreased uniformly to 0.2, 0.1 and 0.0 ms/cm at 3.0, 3.5 and 4.0 respectively. The

increased show the maximum reaction and the decreased show that the amount of reactant in

the reactor started to decrease until there is no reactant left because it all has been used up in

the reaction.

19

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

1.2

E(t) versus time (min)

time (min)

E(t)

This graph is plotted to obtained C(t) curve. It is observed that the materials leaving the reactor

spend between 1.5 min to 2.5 min in the reactor. It is also known as the residence time. From

C(t), E(t) curve was obtained which show the residence time distribution (RTD) function for the

plug flow reactor. It describes the fraction of fluid exiting the reactor that has spent between t

and t + dt inside the reactor. The mean residence time, tm was 2.128 min, second moment

variance, σ2 was 25.89 and the third moment skewness, s3 was 0.232.

20

For experiment 2 which is experiment to examine the effect of a step change input in a tubular

flow reactor. In this experiment, the conductivity started to rise at minute 2.5. The conductivity

started to rise drastically from 2.0 ms/cm to 2.8 ms/cm before it started to rise uniformly and

constant from minute 3.0 min to 7.5 min at conductivity of 2.8 ms/cm until 3.2 ms/cm. A graph of

conductivity versus time is plotted to obtain the C(t) curve which is the area under the graph as

shown in graph below.

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

3.5

f(x) = − 0.0910364145658263 x² + 1.21218487394958 x − 0.75686274509804

Outlet Conductivity versus time

time, min

Cond

uctiv

ity, m

s/cm

21

After obtained the value of C(t), a graph of E(t) as a function of time is plotted. This is the

residence time distribution (RTD) function for the plug flow reactor.

0 1 2 3 4 5 6 7 8

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

E(t) versus time

time (min)

E(t)

The mean residence time is 22.263 min and for the second moment variance, σ2 is 1556.452

and the third moment skewness, s3 is -89.222.

CONCLUSION

In experiment 1, based on the graph plotted it is observed that the outlet conductivity value is

the highest in the middle of the time frame which indicated that the reactant leaving the reactor

spent most of it time inside the reactor. As the amounts of the reactants are increases so does

the pulse input is also increases. For experiment 2, the outlet conductivity values shown no

changes until minute 2.5 where the conductivity suddenly rose to 2.0 ms/cm. Then, it increased

steadily until it reached constant value which is 3.2 ms/cm at minute 7.5. Graphs of outlet

conductivity versus time were plotted for both experiments to calculate the C(t) curve which

gave the value of E(t). The E(t) value was then used to calculate the mean residence time, tm,

second moment variance, σ2 and third moment skewness, s3 for both experiment. The

objectives of the experiment were achieved.

22

RECOMMENDATIONS

There are few recommendations that can be made to improve the result of the experiment. First,

make sure that the de-ionized water and salt solution flow rate is the same and maintained

throughout the experiment. Make sure the salt solution to flow for 1 minute before the timer is

reset and restart. The valve must be control quickly and properly as this will affect the flow and

time of the reactant in the reactor. Reduce the error in taking the conductivity reading. Do not

run the experiment too long to get the stable values because the solution in the tank will run out.

REFERENCE

Reaction kinetics studies in a plug flow reactor. Background and Theory. Retrieved November

19, 2013 from http://solve.nitk.ac.in/dmdocuments/Chemical/theory_plugflow.pdf

Residence time distribution. Retrieved November 19, 2013 from

http://en.wikipedia.org/wiki/Residence_time_distribution

Chemical reactors. Retrieved from November 19, 2013 from

http://www.essentialchemicalindustry.org/processes/chemical-reactors.html

Mole balances. Retrieved November 19, 2013 from

http://ptgmedia.pearsoncmg.com/images/0130473944/samplechapter/0130473944_ch01.pdf

Heterogeneous And Homogeneous Reactors. Retrieved November 19, 2013 from

http://www.indiamart.com/ss-fabrication/heterogeneous-and-homogeneous-reactor.html

Fogler, Scott H. Elements of Chemical Reaction Engineering. 4th ed. Englewood Cliffs,

NJ:Prentice-Hall, 2006.

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