Determining Residence Time

Distributions in a CSTR CHE 4112

October 23, 2008

Shelli Schoolcraft

Sarah Wilkes

2

Executive Summary

The purpose of this experiment was to develop a mathematical model that predicts the ideal transient concentration and residence time distribution in the unit operations laboratory continuous stirred tank reactor. Data was collected using aqueous NaCl solutions of varying concentrations to compare to these ideal models. This experiment provides important information about the characteristics of the reactor and is useful in predicting the amount of time a particular solution has spent in the vessel. The negative step input approach with baffles was employed for this project. Fifteen total tests were performed with three initial concentrations of 0.45 M, 0.60 M, and 0.75 M. Each concentration was placed in the reactor tank and allowed to operate at mixing speeds of each 200, 400, 600, 800, and 1000 RPM. Sample solutions were taken from the effluent stream every twenty seconds and the conductivity was measured and recorded. This data was converted into concentration data and analyzed against the ideal models. It was found that by using baffles during the mixing process, the presence of dead zones and by-pass streams was greatly reduced. We concluded that the mixing speed shares an inverse correlation with concentration in that as speed increases, the concentration decreases more rapidly. By comparing the residence time distributions developed from collected data to the ideal model, we determined that the 0.60 M solution gave the least deviation. Therefore by

using this initial concentration, the results generated are more predictable by conventional

methods. We recommend further experimentation to verify these results. A single source of salt should be used and the reactor should be operated at mixing speeds no lower than 100 RPM. Slower speeds do not produce enough agitation and therefore a majority of the salt content settles at the bottom of the tank and is not present in the effluent stream.

3

Table of Contents Objective.......................................................................................................................................... 4 Background ...................................................................................................................................... 4 Overview .......................................................................................................................................... 4 Experimental Equipment ................................................................................................................. 4 Environmental Health and Safety .................................................................................................... 6 Method of Analysis .......................................................................................................................... 6 Results ............................................................................................................................................. 8 Propagation of Error ...................................................................................................................... 16 Recommendations......................................................................................................................... 16 Appendix: Experimental Data and Sample Calculations ............................................................... 18

4

Objective

This experiment develops a mathematical model to predict the ideal transient concentration in a continuous stirred tank reactor (CSTR). By developing this model, the residence time distribution (RTD) is created and compared to the ideal RTD for multiple impeller speeds.

Background A residence time distribution is one of the most informative characterizations for a reactor. The deviation of the experimental residence time distribution from the ideal residence time distribution allows engineers in industry to adjust the parameters of the reactor to provide the maximum desired output. Knowledge of the Bioflow III reactor in the unit operations laboratory provides a good basis for scale-up industrial reactor work.

Overview

Varying concentrations of an aqueous sodium chloride solution underwent mixing at multiple impeller speeds in the Bioflow III CSTR in the unit operations laboratory. Due to the transient nature of sodium chloride, the Sension 5 conductivity meter was utilized and the residence time distribution was determined. The results are then compared to the ideal residence time distribution model and the optimal impeller speed and solution concentration are established.

Experimental Equipment

The basic experimental setup consists of several components: the reactor, a stopwatch, three graduated cylinders, three buckets, a balance, baffles, and a conductivity meter. For this experiment an aqueous salt solution enters from the top of the reactor and is deflected by a horizontal plate in order to minimize vertical mixing effects. Baffles are utilized to increase turbulence through agitation and prevent the formation of vertices and dead zones. The fluid exits the bottom of the reactor where the effluent is collected for analysis.

5

Figure 1. Continuous Stirred Tank Reactor[1]

A conductivity meter, Figure 2, is utilized to measure the ion concentration in the gathered samples and these data are converted into sodium chloride concentrations. A stopwatch is used to establish amount of time between sample collections.

Figure 2. Conductivity Meter[1]

6

Environmental Health and Safety Sodium chloride is the only potential chemical hazard. Protective eyewear should be worn at all times; however, if ocular exposure occurs, the eyewash station should be utilized. Also, the sodium chloride solution causes skin irritation and will burn if spilled on an open wound. Electrical hazards are present due to the use of electricity near liquids, so all valves should be tightly sealed before the experiment is initiated. The weight of the motor and cover also pose potential problems. Use caution when moving both.

Method of Analysis Theory

A mathematical model to predict ideal transient concentration in a CSTR is developed by using principles of a simple material balance. From the material balance, the ideal residence time distribution is derived. In order to create the experimental model, a negative step input method is utilized. This process is used instead of the positive step method due to the difficulty of keeping an initial tracer concentration in the feed stream. Accumulation = In – Out Eq. (1) Since the negative step was used, Co = 0, reducing Eq. 1 to:

i

i QCdt

dCV 0 t > 0 Eq. (2)

Where, Ci = Concentration of species i (mol/L) t = time (sec) Q = Volumetric flowrate of species i (L/sec) V = reactor volume (L) For this model to work, certain assumptions must be made. First, the ideal model assumes perfect mixing. There are no bypass streams or dead zones and the inlet concentration is instantly and completely mixed. The fluid stream must also maintain a constant density and temperature throughout the process. The final assumption is the system operates at steady-state and a single, irreversible reaction takes place. The rate expression follows a first order reaction. The integration of Eq. 2 yields the ideal concentration model:

/)( t

oi eCtC Eq. (3)

Where, Co = Initial concentration of species i (mol/L)

7

T = V/Q = Space time (sec) This equation gives the concentration of species i in the outlet stream at any time t. The residence-time distribution function, E(t), is given by Eq. 4:

0

)(

)()(

dttC

tCtE

i

i

Eq. (4)

By substituting Eq. 3 into Eq. 4 and solving, we obtain the following expression which describes the amount of time a tracer spends in the reactor:

/

)(te

tE

Eq. (5)

The ideal cumulative concentration distribution, F(t), is also practical when evaluating the residence time distribution, providing the percent of material that has a RTD of time t or less[2].

stepo

outi

C

CtF

,)( Eq. (6) [2]

By definition, E(t) = -dF(t)/dt for a negative step input. Therefore, by differentiating Eq. 6, we obtain the residence-time distribution function for a non-ideal CSTR.

stepo

outi

C

C

dt

dtE

,)( Eq. (7)

Experimental Method

Emergency Shutdown[1]

1. Turn OFF the CSTR ON/OFF switch. 2. Close the water flow valve attached to the CSTR water feed line. 3. Notify professors and students of potential hazards.

Operating Procedure[1]

1. Calibrate conductivity meter.

A. Make aqueous sodium chloride solution using graduated cylinder. B. Place probe in solution. Avoid probe contact with walls or bottom of container. C. Press CAL. which creates a reference standard.

8

D. Scroll to factory calibration settings and press enter. E. Repeat for multiple standard solutions. F. Plot concentration versus conductivity to obtain calibration.

2. Fill tank with known amount of solution. 3. Set impeller to desired speed. 4. Allow solution to mix thoroughly. 5. Place sample collection beaker under the reactor exit nozzle. 6. Open water control supply valve to begin water flow into CSTR. 7. Use the beaker and stopwatch method to record conductivity measurements with

respect to time using constant time intervals. To operate the conductivity meter: A. Place probe into sample, immersing the slot on the end of the probe. B. Shake the probe for five to ten seconds to remove bubbles. C. Press COND. to calculate conductivity. D. Record conductivity and time values. E. Use the concentration versus conductivity plot to record concentration values

8. Stop experiment once the conductivity values flat line near water conductivity value 9. After the run is complete, thoroughly clean the tank to remove any excess salt residue. 10. Repeat steps 2-9 for additional impeller speeds. 11. Repeat steps 1-10 for additional sodium chloride solutions.

Shutdown Procedure[1]

1. Turn the CSTR impeller speed setting to 0 RPM. 2. Turn OFF the CSTR ON/OFF switch. 3. Close the water control valve on the wall to the right of the CSTR. 4. Close the water control regulator on the wall to the right of the CSTR.

5. Close the water flow valve attached to the CSTR feed line.

Data Processing

In order to establish the ideal RTD, concentration curve, and F curve, Eqs. 2 and 4 are evaluated and plotted as a function of time. Ci is identified by adding a known quantity of NaCl tracer to the reactor. V is identified by adding a known amount of water into the reactor tank and Q is identified by the stop watch and beaker method. Once the base line is established, the experimental concentration curves are created by taking effluent samples from the outlet at known times and measuring NaCl conductivity. Again Eqs. 2 and 4 are utilized to produce concentration curves with respect to time. Each set of data for a given sodium chloride solution is compared to the ideal curve.

Results The initial conductivity readings for each known concentration were used to develop a conductivity meter calibration chart. The result, Figure 3, was used to convert the conductivity data into workable concentrations. These concentrations represent the amount of NaCl present in the reactor at any given time.

9

Figure 3. Conductivity Meter Calibration

The consequent concentration values were plotted over time for each solution/mixing speed combination. An ideal curve was also calculated using Eq. 4 and plotted on the same graph. Sample calculations can be viewed in the Appendix section of this report. In nearly all trials, the solutions exhibited a faster exponential rate of reduction as the mixing speed was increased. This is contrary to the findings of a similar study performed by Hamit et. al in which no baffles were used.[1] The inverse correlation between concentration and mixing speed supports our predictions, suggesting that baffles help reduce the presence of dead zones and vortices. The 0.60 M solution (150 g NaCl), in particular, generated the results that most closely matched the ideal curve. These data can be viewed in Figures 4-6.

10

Figure 4. 0.45 M Concentration Data

11

Figure 5. 0.60 M Concentration Data

12

Figure 6. 0.75 M Concentration Data

In accordance with Eq. 6, the negative derivatives of the best-fit line equations for each mixing speed were taken to obtain the residence time distribution function, E(t). Time values were plugged into each equation to calculate the RTD. The E values for nearly every trial are above the ideal curve until about 200 seconds, and then continue below the curve until it converges at 700 sec. The equations are tabulated in Table 1, while the corresponding RTD graphs are represented in Figures 7-9.

13

C(t) E(t)

Mixing

Speed

(RPM) Ideal

0.45

M

0.60

M

0.75

M Ideal 0.45 M 0.60 M 0.75 M

200

e-0.004t

e-0.005t e-0.003t e-0.004t

0.004e-

0.004t

0.005e-

0.005t

0.003e-

0.003t

0.004e-

0.004t

400 e-0.006t e-0.004t e-0.005t

0.006e-

0.006t

0.004e-

0.004t

0.005e-

0.005t

600 e-0.006t e-0.004t e-0.005t

0.006e-

0.006t

0.004e-

0.004t

0.005e-

0.005t

800 e-0.005t e-0.004t e-0.006t

0.005e-

0.005t

0.004e-

0.004t

0.006e-

0.006t

1000 e-0.006t e-0.004t e-0.006t

0.006e-

0.006t

0.004e-

0.004t

0.006e-

0.006t

Table 1. C(t) and E(t) Equations

Figure 7. 0.45 M RTD

14

Figure 8. 0.60 M RTD

15

Figure 9. 0.75 M RTD

It is obvious that the 0.60 M solution RTD once again gave results closest to the ideal calculations. The curves are also more consistent at mixing speeds above 200 RPM, which, based on these findings, leads to more accurate predictability. One experimental run was performed at a mixing speed of 25 RPM in an attempt to produce a concentration vs. time curve that had an observably larger difference than any of Figures 4-6. This mixing speed, however, proved to be too insignificant to obtain any reliable data. There was not enough agitation to thoroughly mix the amount of salt, and therefore the majority settled at the bottom of the tank. The salt began to dissolve around 200 seconds. At this point, the NaCl is distributed more evenly throughout the vessel, leading to more reliable readings. Figure 10 shows the data collected from this trial.

16

Figure 10. 0.45 M Concentration Data at 25 RPM

Propagation of Error The increase in ideality from the 0.45 M to the 0.60 M mixtures can be justified by the increase in solution saturation. The 0.75 M mixture, however, does not follow this trend. A possible explanation is that a different box of salt was used to make this concentration, creating a significant source of error. There were many other potential sources for error in this experiment. The liquid level in the tank may have varied slightly for each run, allowing for differences in initial concentrations. The bucket and stopwatch method, which was used to calculate flowrates, is inherently inaccurate. Also, the outlet stream is located high on the vessel so that any NaCl that may have settled near the bottom of the tank will not show up in the effluent.

Recommendations We recommend this experiment be performed again with a single a single source of salt. If the results indicate that higher concentrations more closely mimic ideal conditions, then it can be concluded the solutions of 0.60 M and higher are optimal to run in this particular CSTR. It should

17

also be noted that unless the outlet location is lowered on the tank, it is not recommended to operate the CSTR at speeds lower than 100 RPM.

References [1] Hamit, Josh, Daniel Meysing, and Afshan Samli. Development of a mathematical model in a Continuous Stirred Tank Reactor. September 18, 2008. [2] Fogler, H. Scott. Elements of Chemical Reaction Engineering: 4th Edition. P. 869-878 [3] “Sodium Chloride MSDS.” ScienceLab. Http://sciencelab.com/msds.php?msdsid=9927593. 29 Oct. 2008

18

Appendix: Experimental Data and Sample Calculations

100 g Concentration (M)

Time

(sec) Ideal 200 RPM 400 RPM 600 RPM 800 RPM 1000 RPM

0 1.000 1.000 1.000 1.001 1.000 1.000

20 0.930 0.912 0.856 0.894 0.872 0.865

40 0.864 0.817 0.785 0.790 0.761 0.747

60 0.803 0.725 0.690 0.697 0.702 0.626

80 0.746 0.637 0.611 0.619 0.630 0.563

100 0.694 0.571 0.532 0.541 0.562 0.496

120 0.645 0.505 0.467 0.475 0.501 0.425

140 0.599 0.447 0.414 0.423 0.447 0.366

160 0.557 0.395 0.357 0.366 0.403 0.320

180 0.518 0.346 0.313 0.321 0.354 0.278

200 0.481 0.310 0.276 0.283 0.315 0.241

220 0.447 0.275 0.240 0.248 0.283 0.211

240 0.416 0.243 0.212 0.218 0.252 0.183

260 0.387 0.216 0.187 0.193 0.225 0.161

280 0.359 0.192 0.164 0.180 0.202 0.141

300 0.334 0.171 0.145 0.149 0.180 0.123

320 0.310 0.150 0.128 0.132 0.162 0.107

340 0.289 0.135 0.113 0.116 0.144 0.096

360 0.268 0.119 0.099 0.103 0.129 0.084

380 0.249 0.106 0.089 0.092 0.116 0.075

400 0.232 0.095 0.079 0.080 0.104 0.066

420 0.215 0.084 0.070 0.072 0.093 0.059

440 0.200 0.076 0.063 0.065 0.085 0.053

460 0.186 0.068 0.056 0.058 0.076 0.047

480 0.173 0.061 0.050 0.052 0.069 0.042

500 0.161 0.055 0.045 0.047 0.063 0.040

520 0.149 0.050 0.041 0.042 0.057 0.036

540 0.139 0.046 0.038 0.040 0.052 0.033

560 0.129 0.041 0.036 0.037 0.048 0.031

580 0.120 0.040 0.032 0.033 0.044 0.028

600 0.112 0.037 0.030 0.031 0.040 0.026

620 0.104 0.034 0.028 0.029 0.038 0.025

640 0.096 0.031 0.026 0.026 0.036 0.023

660 0.090 0.029 0.024 0.025 0.033 0.022

680 0.083 0.028 0.023 0.024 0.031 0.021

700 0.077 0.025 0.022 0.022 0.029 0.020

720 0.072 0.024 0.021 0.021 0.027 0.020

740 0.067 0.022 0.020 0.020 0.025 0.018

760 0.062 0.021 0.019 0.019 0.024 0.018

780 0.058 0.021 0.018 0.018 0.023 0.017

Table 2. 0.45 M Concentration Data

19

150 g Concentration (M)

Time

(sec) Ideal 200 RPM 400 RPM 600 RPM 800 RPM 1000 RPM

0 1.000 1.001 0.999 1.000 1.001 0.999

20 0.930 0.952 0.935 0.921 0.902 0.922

40 0.864 0.882 0.870 0.860 0.853 0.855

60 0.803 0.822 0.815 0.799 0.791 0.794

80 0.746 0.767 0.758 0.745 0.736 0.734

100 0.694 0.710 0.691 0.692 0.682 0.683

120 0.645 0.662 0.655 0.646 0.632 0.633

140 0.599 0.613 0.611 0.598 0.580 0.584

160 0.557 0.570 0.565 0.557 0.538 0.542

180 0.518 0.530 0.526 0.517 0.497 0.503

200 0.481 0.493 0.488 0.480 0.459 0.465

220 0.447 0.458 0.456 0.446 0.423 0.430

240 0.416 0.428 0.422 0.414 0.390 0.399

260 0.387 0.395 0.394 0.383 0.360 0.369

280 0.359 0.369 0.364 0.355 0.332 0.341

300 0.334 0.341 0.337 0.327 0.306 0.316

320 0.310 0.320 0.312 0.304 0.281 0.291

340 0.289 0.297 0.290 0.281 0.259 0.269

360 0.268 0.279 0.268 0.261 0.239 0.250

380 0.249 0.262 0.249 0.241 0.219 0.230

400 0.232 0.244 0.230 0.224 0.203 0.216

420 0.215 0.229 0.215 0.209 0.188 0.199

440 0.200 0.214 0.199 0.193 0.173 0.183

460 0.186 0.203 0.185 0.179 0.159 0.168

480 0.173 0.191 0.172 0.165 0.148 0.158

500 0.161 0.180 0.160 0.153 0.136 0.146

520 0.149 0.170 0.149 0.142 0.126 0.135

540 0.139 0.160 0.139 0.132 0.117 0.125

560 0.129 0.152 0.122 0.122 0.107 0.116

580 0.120 0.144 0.117 0.114 0.099 0.107

600 0.112 0.138 0.109 0.106 0.092 0.100

620 0.104 0.131 0.103 0.098 0.085 0.092

640 0.096 0.125 0.097 0.091 0.079 0.086

660 0.090 0.120 0.090 0.085 0.073 0.080

680 0.083 0.114 0.084 0.079 0.068 0.074

700 0.077 0.110 0.078 0.074 0.062 0.070

720 0.072 0.106 0.073 0.069 0.058 0.065

740 0.067 0.102 0.068 0.064 0.054 0.060

760 0.062 0.098 0.064 0.060 0.050 0.056

780 0.058 0.095 0.060 0.056 0.047 0.053

Table 3. 0.60 M Concentration Data

20

200 g Concentration (M)

Time (sec) Ideal 200 RPM 400 RPM 600 RPM 800 RPM 1000 RPM

0 1.000 0.999 1.000 1.000 1.000 1.000

20 0.930 0.919 0.907 0.910 0.833 0.890

40 0.864 0.834 0.821 0.825 0.762 0.790

60 0.803 0.778 0.743 0.737 0.697 0.701

80 0.746 0.718 0.672 0.664 0.636 0.622

100 0.694 0.660 0.609 0.603 0.580 0.552

120 0.645 0.611 0.551 0.545 0.529 0.490

140 0.599 0.561 0.499 0.492 0.461 0.435

160 0.557 0.516 0.451 0.441 0.417 0.386

180 0.518 0.471 0.409 0.399 0.377 0.343

200 0.481 0.432 0.370 0.360 0.339 0.304

220 0.447 0.399 0.335 0.325 0.305 0.270

240 0.416 0.365 0.303 0.294 0.273 0.240

260 0.387 0.332 0.275 0.267 0.243 0.213

280 0.359 0.307 0.249 0.240 0.216 0.189

300 0.334 0.281 0.225 0.215 0.191 0.168

320 0.310 0.255 0.204 0.194 0.188 0.150

340 0.289 0.233 0.185 0.175 0.167 0.133

360 0.268 0.214 0.167 0.157 0.147 0.118

380 0.249 0.197 0.152 0.142 0.129 0.105

400 0.232 0.181 0.137 0.127 0.112 0.094

420 0.215 0.165 0.124 0.114 0.096 0.083

440 0.200 0.152 0.113 0.103 0.092 0.074

460 0.186 0.139 0.102 0.092 0.089 0.066

480 0.173 0.128 0.093 0.083 0.077 0.059

500 0.161 0.117 0.084 0.074 0.065 0.053

520 0.149 0.108 0.076 0.066 0.055 0.047

540 0.139 0.099 0.069 0.059 0.049 0.042

560 0.129 0.091 0.063 0.053 0.046 0.038

580 0.120 0.084 0.057 0.049 0.038 0.034

600 0.112 0.077 0.052 0.044 0.036 0.030

620 0.104 0.068 0.047 0.039 0.034 0.027

640 0.096 0.064 0.043 0.035 0.031 0.024

660 0.090 0.060 0.039 0.031 0.028 0.022

680 0.083 0.055 0.035 0.029 0.025 0.020

700 0.077 0.051 0.032 0.026 0.022 0.018

720 0.072 0.047 0.029 0.023 0.019 0.016

740 0.067 0.044 0.027 0.021 0.016 0.015

760 0.062 0.040 0.024 0.020 0.014 0.013

780 0.058 0.037 0.022 0.018 0.012 0.012

Table 4. 0.75 M Data

21

100 g

Time (sec) Ideal 25 RPM

0 1.000 0.102

20 0.930 0.036

40 0.864 0.048

60 0.803 0.053

80 0.746 0.057

100 0.694 0.053

120 0.645 0.059

140 0.599 0.061

160 0.557 0.061

180 0.518 0.062

200 0.481 0.062

220 0.447 0.059

240 0.416 0.057

260 0.387 0.054

280 0.359 0.050

300 0.334 0.046

320 0.310 0.044

340 0.289 0.041

360 0.268 0.040

380 0.249 0.037

400 0.232 0.035

420 0.215 0.033

440 0.200 0.031

460 0.186 0.030

480 0.173 0.029

500 0.161 0.027

520 0.149 0.026

540 0.139 0.025

560 0.129 0.024

580 0.120 0.023

600 0.112 0.022

620 0.104 0.022

640 0.096 0.022

660 0.090 0.021

680 0.083 0.021

700 0.077 0.020

720 0.072 0.020

740 0.067 0.020

760 0.062 0.019

780 0.058 0.019

Table 5. 0.45 M Slower Mixing Speed

22

Sample Calculations Conductivity to Concentration Conversion: From calibration chart,

C856.33

Therefore, for = 9.95 mS/cm,

MC 294.0856.33

95.9

856.33

Ideal C(t) Curve: From Eq. 2,

// )()( t

o

it

oi eC

tCeCtC

Where,

s

s

mL

mLQV 6.273

833.20

5700/

Ideal E(t) Curve: From Eq. 4,

/

)(te

tE

Where,

s6.273 And therefore,

6.273)(

6.273/tetE