CHEMICAL REACTION ENGINEERING CHEE 423

LABORATORY REPORT

Department of Chemical Engineering

McGill University

Michael Garibaldi (260353823)Zachary Morrow (260352999)

Pin Chun Lee (260212277)Chung Hyuk Lee (260413295)

Submitted to: Professor Anne- Marie Kietzig

Submitted on: 8 November 2013

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Introduction

Designing chemical reactors involves the consideration of many different variables and their consequential effect on the reaction(s) involved. Many of these considerations involve some assumption towards ideality, however real reactors do not operate in an ideal manner. To take the ideal chemical reactor case to describe the real reactor case, the measurement of residence times and their respective distributions are used in order to accomplish this transition. This laboratory experiment utilizes the residence time distribution method to diagnose problems in a stirred tank reactor and predict the conversion or effluent concentrations using the tanks in series (TIS) model.

Objective

The objective of this laboratory experiment is to employ the residence time distribution method to analyze and compare the residence time distribution function, cumulative distribution function, and mean residence time of a real laboratory scale reactor using the tanks in series model. These functions are studied to gain understanding of how real reactors are modeled.

Background

Residence time distribution Function &Method

The RTD function is a probability distribution function which characterizes quantitatively the amount of time various fluid elements have spent in a reactor.

Residence time distribution or RTD method is a method which measures residence times of an inert tracer which is injected into the reactor and the concentration of said tracer is measured in the effluent. These distributions give insight as to how the reactor operates, and helps diagnose any problems which may occur in the reactor. Conditions of concern are dead volume, and bypassing. There are two types of injection methods, step input method and pulse injection method. In this experiment the pulse method is utilized.

Pulse Input Experiment

In the pulse input experiment a tracer of known concentration and volume is injected instantly (ideally) into the reactor and the respective concentration is then measured in the effluent stream. A RTD can be established by the measurement of the concentration of the tracer at the outlet at time set intervals.

Tracer

A tracer is an inert material which is used in RTD experiments. The qualities of which tracer is used for which reactor is extremely important. A tracer must be chosen such that the tracer is easily detectable, have similar properties of the reacting and product species (but is itself not reactive), completely soluble in the solution, and not adsorb to the surfaces of the reactor.

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CSTR

Stirred tank reactors are very common reactors used extensively in industrial applications. It is primarily used for reacting material in the liquid phase. Its operation is normally operated at steady state. When you model these reactors and make the assumption of perfect mixing, they are called CSTRs or continuously stirred tank reactors which are the ideal version of stirred tanks.

Dead Volume

Dead volume describes the portion of reactor volume which does not change or interact at all with the stirring mechanism of the CSTR or interact with an incoming fluid or reactant, and does not exit the reactor. It essentially creates a perception of the volume of a reactor being smaller than it actually is. This phenomenon causes the time required for the tracer to completely leave the system to decrease, and therefore lower the mean residence time.

By-Passing

By-passing is a peculiarity in which some volume of the tracer by-passes the reactor entirely and immediately exits. However, the entire volume of tracer is not by-passed, some passes through the reactor at a lower concentration. This lowered concentration takes longer to exit the reactor. This will alter the RTD functions and increase the mean residence time.

Mean Residence Time

The mean residence time is essentially the average time a particle spends within a reactor. It helps to give insight into the cumulative distribution function and the tanks in series model.

Cumulative Distribution Function

The cumulative distribution function represents the fraction of effluent that has left the reactor at certain time. The function is the area under the RTD curve and its value approaches one as time increases meaning that all of the tracer has exited the reactor. It is essentially the percentage of tracer which has exited the reactor at a particular moment of time.

Tanks in Series Model

The tank in series model (T-I-S model) is used to describe non-ideal reactors and calculate the respective conversion from the RTD data. It is used to first generalize a reactor to a series of n ideal reactors and derive an equation to give forth the number of tanks in series that best fits the RTD data.

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Analysis Protocol

1. There are two points of injection for the tracer experiment. One is a port on the inlet stream; the other is through a hole on the top of the CSTR. The first part of the experiment is concerned with the inlet injection port. Twenty milliliters of salt solution are measured in a syringe and injected into the inlet stream. The salt molecules then enter the CSTR with the inflow of water, where they are mixed. Each individual salt molecule spends a particular amount of time in the CSTR before it leaves in the outlet stream. Salt that leaves with the outflow passes a voltmeter wrapped around the outside of the outlet tube. The voltmeter measures the electrical resistance across the salt water solution inside the tube and returns an output in conductance, the inverse quantity of resistance.

2. Data from the CSTR experiment is recorded by the voltmeter in mmho. The voltmeter measures the conductivity of the solution at the outlet of the CSTR. The readings are transmitted to the chart recorder which creates a visual 2D plot of the change in voltage over time. The voltage is recorded in volts and time represented in centimeters.

3. The units of interest in this experiment are mmho and seconds; hence a conversion must be made from volts to mmho and from centimeters to seconds. The conversion formula for mmho is as follows,

[ mmho ]=10 [V ]+0.26 mmho .

Also of interest in defining the tanks in series model is time, in seconds. As time is measured in centimeters on the chart, the feed rate of the chart must be known. In this case, the conversion is as follows,

2cm=1 minute=60 seconds .

4. To acquire data from the chart in a reasonable manner, points must be taken and data recorded in intervals. Starting from the initiation towards the peak, data was taken fairly arbitrarily. Past the peak to the end of the experiment, data is taken in one centimeter intervals and recorded in a spreadsheet.

5. To attain values for conductance through the channel, the temperature of the fluid at the outlet of the CSTR must be known. The temperature of the water is recorded by a thermocouple placed directly at the outlet, in such a manner that it does not interfere with flow or create bubbles. The importance of the lack of bubbles across the voltmeter is to avoid sudden drops in conductivity due to the insulating properties of air. The temperature of the water at the CSTR outlet is recorded as being 15°C.

6. Once the conductance for the salt in water at 15°C is known at all time-points, it is next important to calculate the conductance at 22°C. The conductance at 22°C is necessary for subsequent

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concentration calculations. The conductance of the solution at 22°C is found through the following equation,

G22 °C=GT

[1−0.02 (22−T ) ]

Where T is the temperature of the water, in °C and G is the conductance, in mmho.

7. Next, the base conductance of the tap water is calculated so that there is a baseline for determining the concentration of NaCl tracer from the overall conductance. To do this, the same equation as above is applied, where GT is equal to the base conductance of the tap water at 15˚C.

8. Following the determination of the base conductance, the initial concentration of NaCl (or other conductive elements in the tap water) is found. This is done using the equation,

[ NaCl ]base=0.01004 × G22°C−0.00164

9. Now that the conductance of the tap water is known, the concentration of NaCl tracer flowing out of the CSTR at each time-point can be found. solution is known for all points at 22°C, the concentration of salt as a function of conductance is found using the same formula as above, except now the base concentration of [NaCl] is taken into account:

[ NaCl ]=0.01004 × G22°C−0.00164−[NaCl ]base

10. With this information known, it is then possible to construct a NaCl concentration versus time graph. Note that this graph has similarities with the original voltage versus chart distance/time graph, except that now the ordinate axis is in mol/L and the abscissa in seconds.

11. Following the C(t) vs. t concentration plot, the area under the curve is taken. As integration of the curve is cumbersome, the trapezoidal rule is used to approximate the definite integral from t = 0 to t = 1000 seconds. To accomplish this, points on the curve are taken in identical intervals; from each of these points, a straight line is drawn to the next point in the sequence. The resulting enclosed area resembles a trapezoid and thus the area may be found as follows,

Ai=∆ x2 ( f ( x i−1 )−f ( xi ))

Where A is the area, Δx is distance on the x-axis, f(x) is the y-coordinate and i represents the interval.To calculate the total area under the curve, the sum of these trapezoidal areas is taken, which is important for this experiment because this sum leads to the initial concentration of NaCl, or C0.

12. C0 comes into play when designing a residence time distribution (RTD) plot. It is as follows that, since the flow rate is kept constant,

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E ( t )=CNaCl ( t )

C 0

After calculating E(t) for all time-points, the RTD plot can be made by graphing all E(t) values versus their corresponding t values. This gives a probability density function which is depicted in the Results section.

13. Next of concern is the area under the E(t) curve. This is again evaluated using the trapezoidal rule to approximate the definite integral. The sum of the first two trapezoidal areas gives the second F(t) value on the cumulative RTD curve. The sum of the first three areas gives the third point and the sum of the first four areas, the fourth point. This is continued until the total sum of all areas has been attained, which represents the last F(t) point. Taking the very first point to be zero, a curve can be created from the data for F(t) vs. respective t values. The resulting plot is the cumulative residence time distribution function.

14. To find residence time, τ, the volume of the reactor and the flowrate must be known. V, the tank volume, in this case, is 625 mL. The flowrate, υ, is 169 mL/min and changes negligibly between the inlet and outlet. Thus υ = υ0. According to the equation,

τ=Vυ

Note that the nominal flow rate was determined according to the rotameter reading and the corresponding volumetric flow rate was found from the rotameter calibration curve for flow rates.

15. Also of importance is mean residence time, tm. It can be found according to the equation,

tm=∫ t × E(t)dt

Thus it is necessary to calculate tE(t) for all time-points by multiplying each time-point by its corresponding residence time distribution function. Once this has been done, tm is found and tabulated. Note that, if tm ≠ τ, this is not an ideal reactor and perfect mixing is not occurring.

16. The equation,

N= τ2

σ 2

estimates the theoretical number of effective stirred tanks within the single CSTR volume. Therefore, the only value remaining that must be calculated is the variance, σ2. It is found through a series of steps. The equation typically used to determine the variance is as follows:

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σ 2=∫0

∞

(t−t m )2 E ( t ) dt=∑0

∞

(t−tm )2 E (t)∆ t

To start, the sum of (t-tm)2E(t) is taken starting at the initial time-point. This gives a set of values that are then plotted on the y-axis of a new graph, with t on the abscissa. The trapezoidal area is then determined for each pair of time-points, according to:

[ (t 2−tm )2 E (t 2 )−( t1−tm )2 E (t 1) ]2

∗(t 2−t 1)=Area

The sum of these areas indicates the σ 2 value.

Next, τ2 is divided by σ 2, yielding N, the number of tanks.

17. Every single one of the preceding steps is repeated for the second injection site, the hole on top of the CSTR.

18. With both experimental models known, it is time to approximate the Tanks in Series Model for the CSTR.

19. Firstly, τ for the tanks in series is taken as τ from the experiment divided by the number of theoretical tanks, N, which is known from the calculations preceding this step.

20. To begin modeling with Tanks in Series, the theoretical residence time distribution is derived according to the relation,

En (t )= t n−1

(n−1 ) !τ i2∗e

−tτ i

E2(t) is calculated for all time-points from the start of the tracer experiment to the finish.

21. Next, the cumulative residence time distribution is calculated. The general form for finding F(t) with the Tanks in Series Model is as follows:

Fn ( t )=1−e−tτ i ∗(1+ t

τ i+ t 2

2 τ i2 +

t 3

3 τ i3 +…+ t n−1

(n−1 )! τ in−1 )

22. With E(t) and F(t) known at all time-points, next a plot is constructed for each with respect to t. These plots allow for a comparison between the theoretical and experimental models. Since it is being assumed that the Tanks in Series Model is the best fit for the CSTR, the plots for

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experimental E(t) and theoretical E(t), and experimental F(t) and theoretical F(t) should be identical.

23. For further comparison, it may be desirable to design RTD and cumulative RTD plots for a single CSTR. To do this, the same equations for E(t) and F(t) as above are used, except with N equal to 1. Thus the equations are reduced to,

E ( t )= 1τ i

2∗e−tτ i

¿ F (t )=1−e−tτ i

Note that, since there is only one theoretical tank in this instance, the τ obtained from the experiment is used for RTD calculations.

24. Lastly, the mole balance for the tracer solution must be performed. This is achieved by first multiplying the flow rate by the initial concentration of NaCl without the base concentration. Recall that the initial input of NaCl is found from the area under the [NaCl] vs. time curve. This yields a value in [mol mL L-1 min-1]. Dimensional analysis will yield results in [mol s-1].The balance is repeated for the inlet, the outlet for the inlet feed condition and the outlet for the direct tank feed condition.

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Results

Inlet Feed

Table 1: Raw & Transformed Data of Inlet Feed

cm V t (s) base [NaCl] (mol/L) tracer [NaCl] (mol/L) E(t) F(t)

0.15 0.022 4.5 1.40E-03 2.57E-03 2.49E-04 0.00E+00

0.2 0.2 6 2.33E-02 2.26E-03 1.88E-030.3 0.3 9 3.50E-02 3.39E-03 1.04E-02

0.35 0.32 10.5 3.74E-02 3.62E-03 1.56E-020.4 0.4 12 4.67E-02 4.52E-03 2.17E-02

0.75 0.479 22.5 5.59E-02 5.42E-03 7.39E-021.12 0.46 33.6 5.37E-02 5.20E-03 1.33E-012.12 0.389 63.6 4.54E-02 4.40E-03 2.77E-013.12 0.325 93.6 3.79E-02 3.67E-03 3.98E-014.12 0.27 123.6 3.15E-02 3.05E-03 4.99E-015.12 0.225 153.6 2.63E-02 2.54E-03 5.83E-016.12 0.188 183.6 2.19E-02 2.13E-03 6.53E-017.12 0.158 213.6 1.84E-02 1.79E-03 7.12E-018.12 0.13 243.6 1.52E-02 1.47E-03 7.60E-019.12 0.109 273.6 1.27E-02 1.23E-03 8.01E-01

10.12 0.09 303.6 1.05E-02 1.02E-03 8.35E-0111.12 0.075 333.6 8.76E-03 8.48E-04 8.63E-0112.12 0.061 363.6 7.12E-03 6.90E-04 8.86E-0113.12 0.051 393.6 5.95E-03 5.77E-04 9.05E-0114.12 0.045 423.6 5.25E-03 5.09E-04 9.21E-0115.12 0.036 453.6 4.20E-03 4.07E-04 9.35E-0116.12 0.03 483.6 3.50E-03 3.39E-04 9.46E-0117.12 0.025 513.6 2.92E-03 2.83E-04 9.55E-0118.12 0.02 543.6 2.33E-03 2.26E-04 9.63E-0119.12 0.019 573.6 2.22E-03 2.15E-04 9.69E-0120.12 0.016 603.6 1.87E-03 1.81E-04 9.75E-0121.12 0.012 633.6 1.40E-03 1.36E-04 9.80E-0122.12 0.01 663.6 1.17E-03 1.13E-04 9.84E-0123.12 0.009 693.6 1.05E-03 1.02E-04 9.87E-0124.12 0.008 723.6 9.34E-04 9.05E-05 9.90E-0125.12 0.007 753.6 8.17E-04 7.91E-05 9.93E-0126.12 0.006 783.6 7.00E-04 6.78E-05 9.95E-0127.12 0.004 813.6 4.67E-04 4.52E-05 9.96E-0128.12 0.003 843.6 3.50E-04 3.39E-05 9.98E-01

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29.12 0.002 873.6 2.33E-04 2.26E-05 9.98E-0130.12 0.001 903.6 1.17E-04 1.13E-05 9.99E-0131.12 0.001 933.6 1.17E-04 1.13E-05 9.99E-01

32.12 0.001 963.6 1.17E-04 1.13E-05 1.00E+00

33.12 0.001 993.6 1.17E-04 1.13E-05 1.00E+00

Sum 5.10E-02

Figure 1 : Concentration Distribution with Inlet Feed

0 100 200 300 400 500 600 700 800 900 10000.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

Time (s)

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Figure 2 : Inlet Feed Experiment RTD

0 100 200 300 400 500 600 700 800 900 10000.000

0.001

0.002

0.003

0.004

0.005

0.006

Time (s)

E(t)

Figure 3 : Inlet Feed Experiment Cumulative RTD

0 100 200 300 400 500 600 700 800 900 10000.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Time (s)

F(t)

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Direct Tank Feed

Table 2: Raw & Transformed Data for Direct Tank Feed

cm V t (s) base [NaCl] [NaCl] E(t) F(t)

0.3 0.1 9 1.40E-03 1.17E-02 1.16E-03 0.00E+00

0.35 0.2 10.5 2.33E-02 2.32E-03 2.61E-03

0.4 0.3 12 3.50E-02 3.47E-03 6.95E-03

0.45 0.4 13.5 4.67E-02 4.63E-03 1.30E-02

0.5 0.44 15 5.14E-02 5.09E-03 2.03E-02

0.7 0.468 21 5.46E-02 5.42E-03 5.19E-02

1.3 0.44 39 5.14E-02 5.09E-03 1.46E-01

2.3 0.369 69 4.31E-02 4.27E-03 2.87E-01

3.3 0.31 99 3.62E-02 3.59E-03 4.05E-01

4.3 0.259 129 3.02E-02 3.00E-03 5.04E-01

5.3 0.215 159 2.51E-02 2.49E-03 5.86E-01

6.3 0.18 189 2.10E-02 2.08E-03 6.55E-01

7.3 0.15 219 1.75E-02 1.74E-03 7.12E-01

8.3 0.123 249 1.44E-02 1.42E-03 7.59E-01

9.3 0.105 279 1.23E-02 1.22E-03 7.99E-01

10.3 0.088 309 1.03E-02 1.02E-03 8.33E-01

11.3 0.072 339 8.41E-03 8.34E-04 8.60E-01

12.3 0.06 369 7.00E-03 6.95E-04 8.83E-01

13.3 0.05 399 5.84E-03 5.79E-04 9.02E-01

14.3 0.043 429 5.02E-03 4.98E-04 9.19E-01

15.3 0.036 459 4.20E-03 4.17E-04 9.32E-01

16.3 0.03 489 3.50E-03 3.47E-04 9.44E-01

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17.3 0.025 519 2.92E-03 2.89E-04 9.53E-01

18.3 0.021 549 2.45E-03 2.43E-04 9.61E-01

19.3 0.019 579 2.22E-03 2.20E-04 9.68E-01

20.3 0.016 609 1.87E-03 1.85E-04 9.74E-01

21.3 0.011 639 1.28E-03 1.27E-04 9.79E-01

22.3 0.01 669 1.17E-03 1.16E-04 9.83E-01

23.3 0.009 699 1.05E-03 1.04E-04 9.86E-01

24.3 0.008 729 9.34E-04 9.26E-05 9.89E-01

25.3 0.007 759 8.17E-04 8.11E-05 9.91E-01

26.3 0.006 789 7.00E-04 6.95E-05 9.94E-01

27.3 0.005 819 5.84E-04 5.79E-05 9.96E-01

28.3 0.004 849 4.67E-04 4.63E-05 9.97E-01

29.3 0.003 879 3.50E-04 3.47E-05 9.98E-01

30.3 0.002 909 2.33E-04 2.32E-05 9.99E-01

31.3 0.002 939 2.33E-04 2.32E-05 1.00E+00

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Figure 4 : Concentration Distribution of Direct Tank Feed

0 100 200 300 400 500 600 700 800 900 10000.00

0.01

0.02

0.03

0.04

0.05

0.06

Time (s)

Conc

entr

ation

(mol

/L)

Figure 5 : RTD of Direct Tracer Feed Experiment

0 100 200 300 400 500 600 700 800 900 10000.000

0.001

0.002

0.003

0.004

0.005

0.006

Time (s)

E(t)

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Figure 6 : Cumulative RTD of Direct Tank Feed Experiment

0 100 200 300 400 500 600 700 800 900 10000.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Time (s)

F(t)

TIS Model

Table 3: RTD & Cumulative RTD of 2TIS and an Ideal CSTR

TIS Ideal (1 CSTR)

E(t) F(t) E(t) F(t)

3.51E-04 8.01E-04 4.33E-03 2.01E-024.62E-04 1.41E-03 4.27E-03 2.67E-026.74E-04 3.12E-03 4.16E-03 3.97E-027.76E-04 4.21E-03 4.10E-03 4.62E-028.75E-04 5.44E-03 4.04E-03 5.26E-021.49E-03 1.80E-02 3.68E-03 9.64E-022.02E-03 3.76E-02 3.33E-03 1.41E-012.91E-03 1.13E-01 2.54E-03 2.49E-013.27E-03 2.07E-01 1.94E-03 3.44E-013.30E-03 3.06E-01 1.48E-03 4.27E-013.13E-03 4.03E-01 1.13E-03 5.00E-012.85E-03 4.93E-01 8.61E-04 5.63E-012.53E-03 5.73E-01 6.57E-04 6.18E-012.20E-03 6.44E-01 5.02E-04 6.66E-011.89E-03 7.06E-01 3.83E-04 7.09E-011.60E-03 7.58E-01 2.92E-04 7.45E-011.34E-03 8.02E-01 2.23E-04 7.78E-011.11E-03 8.39E-01 1.70E-04 8.06E-01

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9.21E-04 8.69E-01 1.30E-04 8.30E-017.56E-04 8.94E-01 9.90E-05 8.52E-016.18E-04 9.15E-01 7.56E-05 8.71E-015.03E-04 9.31E-01 5.77E-05 8.87E-014.07E-04 9.45E-01 4.40E-05 9.01E-013.29E-04 9.56E-01 3.36E-05 9.14E-012.65E-04 9.65E-01 2.56E-05 9.25E-012.13E-04 9.72E-01 1.95E-05 9.34E-011.70E-04 9.78E-01 1.49E-05 9.42E-011.36E-04 9.82E-01 1.14E-05 9.50E-011.09E-04 9.86E-01 8.69E-06 9.56E-018.65E-05 9.89E-01 6.63E-06 9.62E-016.87E-05 9.91E-01 5.06E-06 9.67E-015.45E-05 9.93E-01 3.86E-06 9.71E-014.32E-05 9.95E-01 2.94E-06 9.74E-013.42E-05 9.96E-01 2.25E-06 9.78E-012.70E-05 9.97E-01 1.71E-06 9.80E-012.13E-05 9.97E-01 1.31E-06 9.83E-011.68E-05 9.98E-01 9.98E-07 9.85E-011.32E-05 9.98E-01 7.62E-07 9.87E-011.04E-05 9.99E-01 5.81E-07 9.89E-01

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Figure 7 : RTD of 1 Ideal CSTR

0 100 200 300 400 500 600 700 800 900 10000.000

0.001

0.002

0.003

0.004

0.005

Time (s)

E(t)

Figure 8: RTD of 2 TIS Model

0 100 200 300 400 500 600 700 800 900 10000.000

0.001

0.002

0.003

0.004

Time (s)

E(t)

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Figure 9: Cumulative RTD of 1 Ideal CSTR

0 100 200 300 400 500 600 700 800 900 10000.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Time (s)

F(t)

Figure 8 : Cumulative RTD of 2 TIS Model

0 100 200 300 400 500 600 700 800 900 10000.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Time (s)

F(t)

Mole Balance

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MOL BALANCEIn 0.0400 mol

Inlet Feed Out 0.0291 molDirect Tank Feed Out 0.0284 mol

Discussion

The tracer is injected to two different locations. Maximum concentration for the direct tank feed is reached 1.5 seconds faster than that of the inlet feed. However, the maximum concentration is slightly smaller for the direct tank feed. This demonstrates how the location of inlet (and perhaps outlet) will affect the tracer measurement. The inlet and outlet location can be a cause of deviation from ideality for stirred-tank reactors. For the calculation, it is assumed that the temperature of water remained constant throughout the experiment.

The E(t) graphs obtained when the tracer was injected into the tank inlet location and the injection hole have a very similar trend to that of an ideal CSTR E(t) graph. The biggest difference is that concentration starts at its highest peak for the ideal reactor, while for the experimental data it takes approximately 20 seconds. This is because an ideal CSTR assumes that as soon as the tracer enters the reactor, perfect mixing occurs, and the concentration everywhere inside the reactor and the outlet becomes homogeneous. This is not the case for a real reactor, where it takes some time for the tracer to be mixed inside the reactor, and to travel to the outlet where the concentration is measured. Also, the maximum E value is much higher for the ideal CSTR for the same reason; much of the tracer leaves the reactor at time zero for the ideal reactor, while for non-perfect mixing it does not. Because of this, the whole tracer amount leaves the reactor faster for the ideal reactor than the real reactor.

The F(t) graphs also looks very similar to that of an ideal CSTR, except that the real response converges to 1 faster than the ideal reactor responses. The steepness suggests that dead volumes are present in the real reactor. The fluid inside will pass through the reactor more quickly than the perfectly operated tank, because of the dead volumes present in the system. There are no particular signs of by-passing in the distributions. Some of the assumptions made in the calculation of ideal reactor are that mixing is perfect, and the volumetric flow rate is constant.

The residence time distribution graphs for the real reactors follow the tank-in-series model trend. N was calculated to be small (about 2), and this implies that two CSTRs in series can model the real reactor. The E(t) graph trend is very similar to that of the real reactors, except it takes a little more time to reach the maximum E value. However, the F(t) graph seems slightly different, where the curve switches direction around 50 seconds. When the number of tanks increases for this model, this graph will look more and more like a step function. This maybe because the calculated N values for the TIS model is relatively small and is almost like a single CSTR. Some of the assumptions made in the calculation of T.I.S. are that all sizes of tanks are equal, and the volumetric flow rate is constant throughout. This means that the residence time for the reactor is also identical under both conditions.

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The tracer balance shows a small decrease in the tracer that is leaving than what has entered the system. The reactor has another outlet that maintains the volume inside the reactor to 625mL. When 20mL of the tracer is added, it increases the volume and some of the tracers leave the tank through the outlet. Therefore, it is projected that some of the tracer is inevitably lost.

Sources of error for this experiment may be the fact that the tracer is not entirely injected at once, but is pushed through a syringe. Instead of an ideal pulse experiment, the tracer is entering the reactor over a time period of a few seconds. This may have caused the 20 seconds lag to reach the highest concentration measurement. Also, trapezoidal method does not give the exact area under a curve, and this may have contributed to the deviation from ideality, as the sum of all of the trapezoidal areas is only an approximate value for the actual area.

Conclusion

The residence time distribution and the cumulative time distribution graphs greatly help in analyzing and identifying sources of error. By comparing the real reactor distribution graphs with an ideal reactor, causes for the deviation from ideality can be found, like how dead volume is identified in the discussion section. There was also evidence of bypassing, as the concentration of tracer at the outlet quickly increased after injection only to fall seconds later. Also, the TIS model helps to explain the residence time distribution of a real reactor in terms of ideal tanks in series, which are much more easily solved. E(t) and F(t) curves of 2 ideal tanks are used to model the non-ideality of the real reactor. This powerful tool assists to gain better understanding of a real reactor behavior in order to predict the performance and aid in design. As the derived model for two tanks in series suggests, the response curve for a tracer pulse injection follows theory very closely.

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Sample Calculations

1. Flow rate: converting rotameter reading to flow rateυ=5.4∗30+7=169 mL /min

2. Conductance: conversion from [V] to [mmho]: at t = 4.5 seconds (inlet feed)[ mmho ]=10∗(0.022 V )+0.26 mmho

[mmho] = 0.48 mmho = GT

3. Conductance: conversion from GT to G22°C: at t = 4.5 seconds (inlet feed)

G22 °C=0.48 mmho

[1−0.02 (22℃−15℃ ) ]=0.56 mmho

4. Tracer concentration: finding [NaCl]base (inlet feed)G22=0.26 mmho /¿

[ NaCl ]base=0.01004 (0.302 mmho )−0.00164=1.40 ×10−3 mol

5. Tracer concentration: [NaCl] at t = 4.5 seconds (inlet feed)[ NaCl ]=0.01004 (0.56 mmho )−0.00164−1.4 × 10−3=2.6× 10−3 mol

6. Concentration curve: area under the curve at t = 4.5 seconds using trapezoidal sums rule (inlet feed)

A1=6.0 s−4.5 s

2∗(2.3 ×10−2mol−2.6 ×10−3 mol )=1.9× 10−2 mol ∙ s

7. Residence time distribution curve: finding E(t) at t = 4.5 seconds (inlet feed)

E (t )=1.9× 10−2 mol ∙ s10.3 mol ∙ s

=2.5 × 10−3

8. Cumulative residence time distribution curve: finding F(t) at t = 4.5 seconds (inlet feed)F (t )=0+1.9× 10−3=1.9 ×10−3

9. Mean residence time: calculating tm (inlet feed)

First solve for all tE(t), i.e. at t = 6.0 secondstE (t )=6.0∗2.26 × 10−3=1.88 ×10−3

Then, plot all tE(t) values on ordinate axis and all corresponding t values on abscissa. Take the trapezoidal area for each pair of points, e.g.,

∑ Areas=t m=171 seconds

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10. Number of theoretical CSTRs: determining variance (inlet feed)At t = 4.5 seconds,

(t−tm )2 E (t )=(4.5−171 )2 (2.5 ×10−4 )=6.9After solving this for all time-points, plot this set of values on the ordinate axis. Plot t on

the abscissa. To get ∑0

∞

( t−tm )2 E (t ) ∆ t, take trapezoidal area for all segments as follows,

[ (t2−tm )¿¿2E (t 2 )−(t 1−t m )2 E(t 1)] /2∗(t 2−t 1)=[ (6.0−171 )2−(4.5−171 )2] /2∗(6.0−4.5)=51.4 ¿

The sum of all of these areas is equal to the variance, or,σ 2=∑ Areas=2.4 ×104

11. Number of theoretical CSTRs: determining N (inlet feed)

N= τ2

σ 2=(221.9 s )2

2.4 ×104 =2.02≅ 2tanks∈series

12. Tanks in Series Model Approximation: finding τi

τ i=τN

=221.89 s2

=110.95 s

13. Tanks in Series Model Approximation: finding E(t) at t = 4.5 seconds (inlet feed)

E2 ( t )= ( 4.5 s )(110.95 s )2

∗e−4.5s

110.95 s=3.5 ×10−4

14. Tanks in Series Model Approximation: finding F(t) at t = 4.5 seconds (inlet feed)

F2 (t )=1−e−( 4.5s )110.95 s∗(1+ 4.5 s

110.95 s )=1.4 ×10−3

15. Mole balance:

Inlet: ¿moles=

(2 molL ) (20 mL )

1000 mL=0.04 mol

Outlet (inlet tracer feed): ¿moles=¿

¿moles=(10.3 mol ∙ s )∗169( mLmin )( 1 L

1000mL )(1 min60 s )=0.029 mol

Outlet (direct tank feed): ¿moles=¿

¿moles=(10.1mol ∙ s )∗169( mLmin )( 1 L

1000 mL )( 1 min60 s )=0.028 mol

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