Research ArticleModeling Residence Time Distribution (RTD) Behavior in aPacked-Bed Electrochemical Reactor (PBER)

Sananth H. Menon ,1 G. Madhu,2 and Jojo Mathew1

1Ammonium Perchlorate Experimental Plant, Vikram Sarabhai Space Centre, ISRO, Aluva, India2School of Engineering, Cochin University of Science & Technology, Kochi, India

Correspondence should be addressed to Sananth H. Menon; sananthisro@gmail.com

Received 28 July 2018; Revised 15 December 2018; Accepted 19 January 2019; Published 26 February 2019

Academic Editor: Deepak Kunzru

Copyright © 2019 SananthH.Menon et al.+is is an open access article distributed under theCreative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

+is paper focuses on understanding the electrolyte flow characteristics in a typical packed-bed electrochemical reactor usingResidence Time Distribution (RTD) studies. RTD behavior was critically analyzed using tracer studies at various flow rates,initially under nonelectrolyzing conditions. Validation of these results using available theoretical models was carried out.Significant disparity in RTD curves under electrolyzing conditions was examined and details are recorded. Finally, a suitablemathematical model (Modified Dispersed Plug Flow Model (MDPFM)) was developed for validating these results underelectrolyzing conditions.

1. Introduction

It is known that a packed-bed electrochemical reactor havingparticulate electrodes can provide a relatively large electrodesurface when compared with a conventional flat-electrodeconfiguration. Consequently, this packed-bed electrolyzerwill be remarkably useful when dealing with low reactantconcentrations or slow reactions [1–4]. +ey also find abetter alternate for large-scale manufacturing of basicchemicals and intermediates as well as for the removal ofharmful or toxic chemicals from gas or liquid streams [5].

Flow behavior of electrolyte through these reactors viaRTD studies has been one of the key components in un-derstanding its vessel hydrodynamics. In an experimentalstudy of residence-time distribution, flow elements aretagged by a tracer (colored, radioactive, etc.) and the vari-ation of tracer concentration in the exit stream with timeis measured. +e injection of tracer into the flow streamis frequently done in such a manner that it can be wellapproximated by a delta function or a thin finite widthpulse. +e tracer concentration distribution at the exit(called also the tracer output signal) has a characteristicshape depending upon the relative strength of dispersionand on the location of tracer injection and detection.

Developing a suitable theoretical model justifying RTDbehavior has been an onus among the engineers for quite along period of time. Not surprisingly, various studies werereported exhibiting peculiar flow behaviors in variety ofsystems. Saravanathamizhan et al. [6] provided a three-parameter model to describe the electrolyte flow in con-tinuous stirred tank electrochemical reactor (CSTER) con-sisting bypass, active, and dead zones with exchange flowbetween active and dead zones. +e authors validated themodel for the effluent color removal inside a typical CSTER.Atmakidis and Kenig [7] conducted a numerical analysis ofdispersion in packed beds and developed an RTD modelusing CFD modeling. Benhabiles et al. [8] conducted theexperimental study of photo catalytic degradation of anaqueous solution of linuron in a tubular type reactor andused RTD data for investigating the malfunction of thephoto reactor. Martin [9] showed that ETIS (extension totanks in series) model in tandem with the reactor networkstructure is a versatile method of describing the charac-teristics of a small but diverse group of reactors.

Earlier studies had shown that conventional models [10]like open dispersion models, small dispersion models, andtanks in series models can explain with lot of clarity thebehavior of electrolyte inside a typical packed-bed reactor

HindawiInternational Journal of Chemical EngineeringVolume 2019, Article ID 7856340, 9 pageshttps://doi.org/10.1155/2019/7856340

mailto:sananthisro@gmail.comhttp://orcid.org/0000-0002-6561-413Xhttps://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/7856340

under nonelectrolyzing conditions. However, not manystudies were reported in the literature regarding the ap-plicability of a suitable model in a packed bed reactor op-erating under electrolyzing conditions. Many of thesemodels fail to explain the recirculation flow expected insidesuch reactor due to obvious gas evolution around theparticulate electrodes. In the initial phase of present study,we made a detailed interpretation of electrolyte flow inside apacked-bed electrochemical bed reactor under non-electrolyzing conditions using an experimental RTD anal-ysis. Applicability of available theoretical models was alsocarried out to strengthen the experimental findings. At alater part of the study, similar RTD studies were repeated atanalogous flow conditions but operated under electrolyzingenvironment, for getting comparative flow behaviors withand without electrolyzing environment. A Modified Dis-persed Plug Flow Model (MDPFM) was developed to val-idate the variation in such flow circumstances.

2. Experimental Details

Experiments were carried out in the packed-bed electro-chemical reactor schematically presented in [1]. +e cellwas of cylindrical geometry and was made of high-densitypolyethylene (HDPE); the overall dimensions were 0.178m(ID)× 0.30m (H). Particle size distribution of these particlesobtained using sieve analysis and its image analysis usingzoom stereoscopic microscope are mentioned in Tables 1and 2, respectively. Shape-scrapped lead dioxide particles(about 3.5 Kg) were thoroughly cleaned using DI water andclosely packed in the electrolyzer up to 5 cm height. Per-forated polypropylene supports (which also serves as dis-tributor) with nylon mesh filter were used at both ends forensuring rigid and leak proof packing. For carrying out RTDstudies, methylene blue having a concentration of 40/80 ppmwas selected as tracer. Tracer was injected to cell closer toits inlet at about 2 cm from HDPE body through a Teeprovided at the feed bottom. Flow medium used was waterfor conducting studies in the nonelectrolytic mode. On thecontrary, sodium chlorate solution having 5Kg/m3 con-centrations was chosen for studies in the electrolytic mode.Flow rates of the medium varied from 3.33×10−5m3/sec to1.33×10−4m3/sec for understanding the variation in RTDbehavior. A DC current of 15A was fed into the electrolyzer(under electrolytic mode) using as Rectifier having 200A,60V specification. A double-beam UV spectrometer wasused for estimating the transient variation in concentrationof tracer in effluent, indirectly by measuring the color in-tensity. Figures 1 and 2 show the schematic experimentalsetup.

3. Modeling RTD Behavior

3.1. Open Dispersion Model. +is model predicts that elec-trolyte flow in the PBER is undisturbed at the inlet andoutlet. +e fundamental mathematical form is

zC

zθ�

D

u · L

z2C

zz2−

zC

zz. (1)

Boundary conditions from [11] are as follows.For open system, at entrance, FT(0−, t) � FT(0+, t). +at

is, −D(zCT/zz)−z�0 + UCT(0

−, t) � −D(zCT/zz)+z�0 + UCT

(0+, t) or CT(0−, t) � CT(0+, t).At the exit,

CT 0−, t( ) � CT 0

+, t( ,

−DzCT

zz

−

z�L

+ UCT L−, t( ) � −D

zCT

zz

+

z�L

+ UCT L+, t( .

(2)

Analytical solution of (1) from [10] is as follows:

E(θ) �C

Cd�

1����������4π(D/u · L)

· exp−(1− θ)2u · L

4θ · D .

(3)

3.2. Model Predicting Small Extent of Dispersion (SmallDispersion). For small extent of dispersion, the spreadingtracer curve does not change its shape as it passes themeasuring point. +is yields a symmetric curve and ana-lytical solution is as follows:

E(θ) �C

Cd�

1����������4π(D/u · L)

· exp−(1− θ)2u · L

4 · D .

(4)

3.3. Tanks in Series Model. +is model predicts that elec-trolyte flow in PBER is discretised into equal sized hypo-thetical CSTR’s. +e number of tanks in series nT describesthe dispersion with nT �1 representing infinite dispersionand being equivalent to Pe� 0. Analytical solution which isalso the definition of Erlang distribution is as follows:

Table 1

Particle size distributionSize range Weight (%)>710 µ 87.53500–710 µ 9.46355–500 µ 2.28300–355 µ 0.27

E(θ) �C

Cd�

nnTT

nT − 1( !θnT−1( )

T e−nTθ

. (5)

4. Results and Discussion

4.1. RTD Curves at Various Flow Rates. RTD behavior ofPBER under various electrolyte flows is depicted inFigures 3(a)–3(d). Table 3 shows the calculated 1st and 2ndmoment about mean. Reasonably good flow was observedthrough the reactor at 3.33×10−5m3/sec, as the mean timeinterval falls at the right place. As the flow rate is increased, thecurve shifts towards the left indicating the presence of earlytime mean. +is observation along with long tail indicates thepresence of stagnant backwaters. +is can be ascertained bycomparing the space time under each flow rate with theobserved mean from graph [10]. Table 4 indicates that per-centage difference predominantly increases at higher flowssubstantiating the presence of stagnant regions.

4.2. Modeling RTD Behavior. Let QR be the flow of elec-trolyte through the bed and VR be the volume of PBER.+entank residence time TR � VR/QR. All time domains wereconverted to dimensionless θ mode, where θ � t/TR. Fromthe respective exit age distribution curves, σ2 and tmean areestimated.

+en,

σ2θ �σ2

t2mean. (6)

Peclet no. can be found out using the following equation:

σ2θ � 2 × D/(u · L)− 2 ×(D/(u · L))2

× 1− e(−u·L/D) . (7)

+e aforementioned parameters were inserted in re-spective modeling equations mentioned in equations (2) and(3) to get the predicted Eθ values using the Open dispersionmodel and model predicting small extent of dispersion,respectively. For tanks in series model, following equationswere used:

Pe � 2 nT − 1( ,

Eθ �n

nTT

nT − 1( ! · θn−1T · e

−nT ·θ,

(8)

where nT represents the number of tanks in series.Using these parameters and respective model equations,

Eθ was analytically determined under various flow rates.Figure 4 shows the graphical representation of these models.

From Figures 4(a)–4(d), it is quite explicit that RTDbehavior of PBER can be well approximated by the Opendispersion model and model predicting small extent ofdispersion. +ough Tank in Series model could predict athigher flow rates, significant disparities could be seen atlower flows. In order to estimate the extent of dispersion,Vessel Dispersion number (D/u·L) was determined fromequation (7), and the same was plotted under various flowrates in Figure 5. Table 4 shows the calculated values of D/u·L. It is observed that D/u·L values increases till the flowreaches 6 LPM and beyond which it decreases. It shows thataxial dispersion coefficients competes for their prominencewith obvious backmixing owing to recirculation flows ob-served in high flow rates. +is is the probable reason fordecrease in D/u·L values under high flow rates.

4.3. RTD Curves under Electrolyzing Conditions. Figure 6shows RTD behavior under electrolyzing conditions at

+

–

R

I

E

T

P

D

Zoomed view

Figure 1: Schematic sketch of experimental setup. P: PBER; T: tracer in; I: influent stream; E: effluent stream; R: rectifier; D:spectrophotometer.

Tracer in

Effluent out

Electrolyser

Flow medium

Figure 2: Experimental setup.

International Journal of Chemical Engineering 3

Table 3

Sl no. Flow rate (m3/sec) 1st moment (sec) tmean � ∞0 t · E(t) dt 2

nd moment about mean (sec2) σ2 � ∞0 (t− tmean)2E(t) dt

1 3.33×10−5 141.48 32572 6.67×10−5 65.05 2339.53 1× 10−4 41.32 11094 1.33×10−4 22.35 153.33

Table 4

Sl no. Flow rate (m3/sec) Space time (τ) (sec) tmean (observed from Figure 3) (sec) Difference (τ − tmean) (%)1 3.33×10−5 150 141.48 62 6.67×10−5 75 65.05 10.643 1× 10−4 50 41.32 214 1.33×10−4 37.5 22.35 67.78

0.000

0.002

0.003

0.006

0.0050.006

0.0060.006

0.004

0.0040.004

0.003

0.0000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0 50 100 150 200 250 300 350

Exit

age d

istrib

utio

n

Time (sec)

(a)

0.000

0.009

0.011

0.009

0.0050.005

0.003

0.004

0.0020.002

0.0000

0.002

0.004

0.006

0.008

0.01

0.012

0 50 100 150 200 250

Exit

age d

istrib

utio

n

Time (sec)

(b)

0.000

0.015

0.014

0.010

0.006

0.0020.0000.0010.001 0.0000

0.0020.0040.0060.008

0.010.0120.0140.0160.018

0 50 100 150 200

Exit

age d

istrib

utio

n

Time (sec)

(c)

0

0.0325

0.0375

0.023750.02

0.00250 0 0 0 0

–0.0050

0.0050.01

0.0150.02

0.0250.03

0.0350.04

0.045

0 20 40 60 80 100 120Time (sec)

Exit

age d

istrib

utio

n

(d)

Figure 3: RTD behavior under various electrolyte flow rates. (a) At 3.33×10−5m3/sec. (b) At 6.67×10−5m3/sec. (c) At 1× 10−4m3/sec.(d) At 1.33×10−4m3/sec.

0

0.5

1

1.5

2

2.5

3

0.00 0.13 0.26 0.39 0.52 0.65 0.77 0.90 1.03 1.16 1.29 1.55 1.94

Calc E(θ)Open dispersion model

Tanks in series modelSmall dispersion model

θ

Eθ

(a)

Eθ

θ

00.10.20.30.40.50.60.70.80.9

0.00 0.27 0.53 0.80 1.07 1.33 1.60 1.87 2.13 2.932.40

Real behaviourOpen dispersion model

Tanks in seriesSmall dispersion

(b)

Figure 4: Continued.

4 International Journal of Chemical Engineering

various flow rates. Table 5 represents the values of vesseldispersion number for various flow rates. Table 6 shows thecalculated 1st and 2nd moment about mean. In general, anaxial dispersion effect gets diminished under the electro-lyzing mode at intermediate flow rates. +is is primarilybecause nonideal axial flow currents gets disturbed by theback flow currents generated by gases which are inevitablyproduced at the electrode surface under electrolyzing con-ditions. Recirculation, channeling, and short circuit flowsobserved under certain flow conditions got totally elimi-nated under electrolyzing mode which may obviously be due

to adequate backmixing of electrolyte between the particles,contributed by the gases generated around these electrodeparticles.

4.4. Modified Dispersed Plug Flow Model (MDPFM) underElectrolyzing Conditions. Refer to Appendix for detailedanalytical treatment.

+is model assumes that a fraction of recirculation flow(α) generated due to gaseous evolution around the electrodesflows back to the conventional platform of dispersed plugflow. It is pictorially denoted in Figure 7.

Exit age distribution Eθ as per dispersed plug flowmodel� 1/

�������������������(4πPe−1) × e−[(1−θ)2Pe/4]

.

+us, as per the above model,

C1′(t)C0

�1

��������

4πPe−1( × e

− 1−t/TR( )2Pe/4[ ],

C1′(t) �C0��������

4πPe−1( × e

− 1−t/TR( )2Pe/4[ ].

(9)

4.4.1. Taking Material Balances.

Q + αQ � Q(1 + α). (10)

At junction point M,

Q(1 + α)C1′ � Q · C1 + αq · δ(t � 0). (11)

Taking Laplace transform across (11),

Q · C1(s) + αq � Q(1 + α)C1′(s). (12)

We know

C1′(s) � ∞

0e−st

C1(t) dt. (13)

Using standard results from integration,

Eθ

0.000.100.200.300.400.500.600.700.800.90

0.00 0.40 0.80 1.20 1.61 2.01 2.41 2.81 3.21 3.61

Real behaviourOpen dispersion model

Tanks in seriesSmall dispersion

θ

(c)

Eθ

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.00 0.27 0.53 0.80 1.07 1.33 1.60 1.87 2.13 2.40 2.67θ

Real behaviourOpen dispersion model

Tanks in seriesSmall dispersion

(d)

Figure 4: Modeling RTD behavior under various electrolyte flow rates. (a) At 3.33×10−5m3/sec. (b) At 6.67×10−5m3/sec. (c) At1× 10−4m3/sec. (d) At 1.33×10−4m3/sec.

1.33 × 10E (–4) m3/sec

6.67 × 10E (–5) m3/sec

3.33 × 10E (–5) m3/sec

3.33 × 10E (–5) m3/sec

1 × 10E (–4) m3/sec

0.009

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

01

6.67 × 10E (–5) m3/sec1 × 10E (–4) m3/sec1.33 × 10E (–4) m3/sec

Figure 5: Plot of D/u · L for various flow rates.

International Journal of Chemical Engineering 5

C1(s) � (1 + α)C0��������

4πPe−1( × TR

�����πPe−1

× erfc����Pe−1

· TR s + 0.5 ×

PeTR

× e(1/Pe)·TR2 s2+s·Pe/TR( )[ ] −

αqQ

.

(14)

4.4.2. Finding C1(t) by Taking Inverse Laplace Transform ofC1(s). Comparing from the standard form of results for

inverse Laplace transform for product form of exponentialand error function,

L−1

ek2·s2

× erfc(ks) k> 0 �1

(k��π

√)

· e(−(t2/4k2))

,

C1(t) � (1 + α) ·C0��������

4πPe−1( · TR

��������

π · Pe−1(

·1

(k ·��π

√)

· e−Pe

· e(−(t2/4k2))

· e− Pe/2TR( )( )t −

αqQ

.

(15)

Rearranging (15),

C1(t) � 0.5(1 + α)C0 ·TR

k��π

√ · e(−0.5Pe(θ+θ2/2)) −

αqQ

. (16)

Further simplifying and putting residence time distri-bution function as E(θ) � C1(t)/C0 (from equation (A.13)),

E(θ) � (1 + α) ·e−Pe

2 ·√πPe−1( × e

(−0.5Pe(θ+θ2/2)) −αq

QC0.

(17)

4.5. Validation ofMDPFM. By putting α� 0.5, equation (17)was used to predict the values of E(θ) under various flowrates. Figure 8 represents the comparison with the actualbehavior.

0.000

0.012

0.0130.012

0.008

0.0030.002

0.000 0.000 0.0000.000 0.000 0.000

–0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 50 100 150 200 250 300 350

Exit

age d

istrib

utio

n

Time (sec)

(a)

0.000 0.001

0.015

0.013

0.009

0.0060.005

0.0030.000 0.000 0.000

–0.0020

0.0020.0040.0060.008

0.010.0120.0140.0160.018

0 50 100 150 200 250

Exit

age d

istrib

utio

n

Time (sec)

(b)

0.000

0.019

0.015

0.009

0.0040.003

0.000 0.000 0.000 0.000

–0.005

0

0.005

0.01

0.015

0.02

0.025

0 20 40 60 80 100 120 140 160 180 200

Exit

age d

istrib

utio

n

Time (sec)

(c)

0.000

0.1050.094

0.083

0.055

0.041

0.0130.003 0.000 0.000 0.000

–0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0 20 40 60 80 100 120Time (sec)

Exit

age d

istrib

utio

n

(d)

Figure 6: RTD behavior under various electrolyte flow rates. (a) At 3.33×10−5m3/sec. (b) At 6.67×10−5m3/sec. (c) At 1× 10−4m3/sec.(d) At 1.33×10−4m3/sec.

Table 5

Sl no. Flow rate(m3/sec)σ2θ

(from graph)D/u·L

(from equation (7))1 3.33×10−5 0.0024 0.00122 6.67×10−5 0.016 0.00813 1× 10−4 0.0178 0.0094 1.33×10−4 0.0027 0.00136

Table 6

Sl no. Flow rate(m3/sec)1st moment (sec)

tmean � ∞0 t · E(t) dt

2nd moment aboutmean (sec2) σ2 �

∞0 (t− tmean)

2E(t) dt

1 3.33×10−5 64.13 775.712 6.67×10−5 59.2 7413 1× 10−4 42.38 5444 1.33×10−4 27 217

6 International Journal of Chemical Engineering

Equation (17) shows the complimentary influences oftwo important terms e−0.5Pe(θ+θ2/2) and αq/QC0. I term beingthe dispersion effects due to nonidealities in flow dynamicsand II term shows the back flow effects owing to the liquidrecirculation aided by gas evolution under electrolyzingmode. Understandably, from the above graphs, it is clear thatnonidealities in flow currents get diminished by the backflow currents generated by gases which are inevitably pro-duced at the electrode surface under electrolyzing condi-tions. From Figures 8(a) to 8(d), it is clear that MDPFM

predicts RTD behavior of PBER under electrolyzingconditions.

5. Conclusion

RTD behavior of PBER was studied in various flow rates withand without electrolysis. Conventional dispersion modelscould explain the RTD behavior when PBER is operatedwithout electrolysis. D/u·L values were determined by fittingin the dispersion model to assess the quantum of axial

0

0.5

E(θ)

1

1.5

2

2.5

0.00 0.13 0.26 0.39 0.52 0.65 0.77 0.90 1.03 1.16 1.29 1.55 1.94

Actual valuesMDPFM

θ

(a)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.27 0.53 0.80 1.07 1.33 1.60 1.87 2.13 2.40 2.93

E(θ)

θ

Actual valuesMDPFM

(b)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.40 0.80 1.20 1.61 2.01 2.41 2.81 3.21 3.61

Actual valuesMDPFM

E(θ)

θ

(c)

θ

0.000.501.001.502.002.503.003.504.004.50

0.00 0.27 0.53 0.80 1.07 1.33 1.60 1.87 2.13 2.40 2.67

Actual valuesMDPFM

E(θ)

(d)

Figure 8: Prediction of E(θ) using MDPFM. (a) At 3.33×10−5m3/sec. (b) At 6.67×10−5m3/sec. (c) At 1× 10−4m3/sec. (d) At1.33×10−4m3/sec.

Dispersed plug flowq

M

Q (1 + α)C1′QC0

QC1

αQ

Figure 7: MDPM conceptual block diagram.

International Journal of Chemical Engineering 7

dispersion inside the packed bed. Modified Dispersion PlugFlow Model (MDPFM) was developed to predict the resi-dence time distribution function (E(θ)) under electrolyzingconditions. Finally, the model was validated in RTD studiesusing PBER operated under electrolyzing mode at variousflow rates.

Nomenclature

Q: Volumetric flow rateC0: Initial concentrationQ: Tracer quantityα: Fraction of back flowC1′: Concentration immediately after the dispersed

plug flow regionC1: Final output concentrationTR: Tank residence timeT: Time elapsed after injection of tracerθ: t/TRδ(t): Dirac delta functionE(θ) or Eθ: Residence time distribution functionD: Dispersion coefficientC: Concentration of species at any instance,

u-velocity of electrolyteu: Velocity of electrolyteL: Length of bedQ: Bulk flow rate through the bedPe: Peclet no. u·L/DnT: No. of tanks in seriesσ2: Varianceα2θ: Variance (dimensionless)FT: Mass flow rate of tracerCT: Concentration of tracerU: Velocity.

Appendix

A. Dispersed Plug Flow Model Equation

Exit age distribution Eθ as per dispersed plug flowmodel� 1/

��������(4πPe−1)

× e−[(1−θ)2Pe/4].

+us, as per the above model,C1′(t)

C0�

1��������

4πPe−1( × e

− 1−t/T2R( )Pe/4[ ],

C1′(t) �C0��������

4πPe−1( × e

− 1−t/T2R( )Pe/4[ ].

(A.1)

Q + αQ � Q(1 + α). (A.2)

A.1. Taking Material Balances

At junction point M,

Q(1 + α)C1′ � Q · C1 + αq · δ(t � 0). (A.3)

Taking Laplace transform across (A.3),

Q · C1(s) + αq � Q(1 + α)C1′(s). (A.4)

We know,

C1′(s) � ∞

0e−st

C1(t) dt

� ∞

0e−st

×C0��������

4πPe−1( × e

− 1−t/T2R( )Pe/4[ ] dt

� ∞

0e−st

×C0��������

4πPe−1( × e

− −Pe/4T2R t2−2tTR+T2R( )[ ] dt

�C0��������

4πPe−1( ×

∞

0e− Pe/4TR2( )t2+ s+Pe/4TR( )t+(Pe/4)[ ] dt.

(A.5)

From the standard form of integration results for ex-ponential function,

∞

0e−(ax2+bx+c)

dx �12

×

�������������πa

e((b2−4ac)/4a)

× erfcb

2��a

√ ,

(A.6)

where erfc(p) � 2/��π

√∞0 e−x2 dx, a � Pe/4 · T2R, b � s+

(Pe/2TR), and c � Pe/4.Hence the solution of the integrand in (A.5) is

I1 �12

��������π

Pe/4T2R(

× es+0.5Pe/TR( )2−Pe/T2R/Pe/T

2R[ ]

× erfc s + 0.5Pe/TR( /����(Pe)

TR .

(A.7)

By simplifying,

I1 � TR

�������

πPe−1(

× e(1/Pe)T2R s2+s·Pe/TR[ ][ ]

× erfc����Pe−1

· TR s + 0.5Pe/TR( .

(A.8)

+us from (A.5), C1′(s) � C0/��������(4πPe−1)

× I1.

From (A.4),

C1(s) � (1 + α)C0��������

4πPe−1( × TR

�����πPe−1

× erfc����Pe−1

· TR

s + 0.5 · PeTR

× e(1/Pe)·T2R s2+s·Pe/TR( )[ ] −

αqQ

.

(A.9)

A.2. Finding C1(t) by Taking Inverse LaplaceTransform of C1(s)

Compared with the standard form of results for inverseLaplace transform for product form of exponential and errorfunction,

8 International Journal of Chemical Engineering

L−1

ek2·s2

× erfc(ks) , k> 0 �1

(k��π

√)

· e(−(t2/4k2))

.

(A.10)

Value of complimentary error function in (A.9) is n �����Pe−1

√· T2R · (s + 0.5 · Pe/TR) and n

2 � Pe−1T2R(s2 + s · Pe/

TR + Pe2/4T2R).If we modify the exponential term in (A.9) as

e(1/Pe)·T2R ·(s2+(Pe/TR)s+Pe2/4T

2R) · e−Pe, this term will be of the

standard form ek2·s2 × erfc(ks) having k �����Pe−1

√· TR.

+us, for a function, F(s) � ek2·(s+0.5Pe/TR)2 × erfc(k·(s + 0.5Pe/TR)), F(t) as per the standard solution men-tioned above, F(t) � 1/(k ·

��π

√) · e(−t2/4k2)· e(−0.5Pe/TR)t · e−Pe.

+us,

C1(t) � (1 + α).C0��������

4πPe−1( · TR ·

��������π · Pe−1( )

·1

(k ·√π)· e−Pe

· e(−t2/(4k2))

· e−Pe/2TR( )t −

αqQ

.

(A.11)

Rearranging (A.11),

C1(t) � 0.5(1 + α)C0 ·TR

k��π

√ · e−Pe

× e(−0.5Pe(θ+θ2/2)) −

αqQ

.

(A.12)

Further simplifying and putting residence time distri-bution function as E(θ) � C1(t)/C0,

E(θ) � (1 + α) ·e−Pe

2 ·�����πPe−1

√

× e(−0.5Pe(θ+θ2/2)) −

αqQC0

.

(A.13)

Data Availability

No data were used to support this study. All essential for-mulae are mentioned in Annexure.

Conflicts of Interest

+e authors declare that they have no conflicts of interest.

Acknowledgments

+e authors gratefully acknowledge the Vikram SarabhaiSpace Research Centre, Trivandrum, and Cochin Universityof Science and Technology, Cochin, for supporting thiswork.

References

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[2] J. S. Newman and C. W. Tobias, “+eoretical analysis ofcurrent distribution in porous electrodes,” Journal of theElectrochemical Society, vol. 109, no. 12, pp. 1183–1191, 1962.

[3] T. Doherty, J. G. Sunderland, E. P. L. Roberts, andD. J. Pickett,“An improved model of potential and current distributionwithin a flow-through porous electrode,” Electrochimica Acta,vol. 41, no. 4, pp. 519–526, 1996.

[4] N. M. S. Kaminari, M. J. J. S. Ponte, H. A. Ponte, andA. C. Neto, “Study of the operational parameters involved indesigning a particle bed reactor for the removal of lead fromindustrial wastewater—central composite design methodol-ogy,” Chemical Engineering Journal, vol. 105, no. 3,pp. 111–115, 2005.

[5] G. Eigenberger and W. Ruppel, Catalytic Fixed-Bed Reactors,Wiley, New York, NY, USA, 2008.

[6] R. Saravanathamizhan, R. Paranthaman, N. Balasubramanian,and C. A. Basha, “Residence time distribution in continuousstirred tank electrochemical reactor,” Chemical EngineeringJournal, vol. 142, no. 2, pp. 209–216, 2008.

[7] T. Atmakidis and E. Y. Kenig, “Numerical analysis of resi-dence time distribution in packed bed reactors with irregularparticle arrangements,” Chemical Product and Process Mod-eling, vol. 10, no. 1, pp. 17–26, 2015.

[8] O. Benhabiles, N. Chekir, and W. Taane, “Determining of theresidence time distribution in CPC reactor type,” EnergyProcedia, vol. 18, pp. 368–376, 2012.

[9] A. D. Martin, “Interpretation of residence time distributiondata,” Chemical Engineering science, vol. 55, no. 23,pp. 5907–5917, 2000.

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