Residence time distribution of the electrodialyzer under electric field conditions

Desalination 342 (2014) 139–147

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Residence time distribution of the electrodialyzer under electric field conditions M. Turek, K. Mitko ⁎ Silesian University of Technology, Faculty of Chemistry, Department of Chemistry, Inorganic Technology and Fuels, ul. B. Krzywoustego 6, 44-100 Gliwice, Poland

H I G H L I G H T S • A method of measuring residence time of working electrodialyzer is proposed. • Mean residence time and its variance decrease as applied current density increase. • Hydrodynamics of the electrodialyzer is discussed.

a r t i c l e

i n f o

Article history: Received 27 May 2013 Received in revised form 14 November 2013 Accepted 16 November 2013 Available online 8 January 2014 Keywords: Electrodialysis Residence time distribution Velocity distribution Dispersion model Plate-and-frame module Scaling prevention

a b s t r a c t To avoid the membrane scaling during electrodialytic desalination of sparingly soluble salt solutions, the electrodialysis should be performed in a way that the vast majority of growing clusters is washed out from the module before the crystallization occurs (the residence time is lower than nucleation induction time). Such method requires the determination of residence time distribution (RTD) of the electrodialyzer. While available literature presents only RTD measured at no current conditions, with only water passing through the module, this study presents a method of measuring the RTD on the working electrodialyzer, when the applied current causes ion migration and electroosmotic water flux. The change in concentrate compartment RTD was confirmed for the linear flow velocity range of 0.38–0.84 cm/s and current density range of 0–673 A/m2. The hydrodynamic conditions inside the module were described using an open–open vessel dispersion model and it was shown that significant longitudinal dispersion is observed (on average D/uL = 0.08). The current density distribution along the electrodialyzer was observed and quantified and the question of velocity distribution along the electrodialyzer was tackled. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Brackish water desalination can be achieved with the electrodialysis (ED) or electrodialysis reversal (EDR)—an ED performed with periodic reversal of electrodes' polarity. To increase the efficiency and decrease the environmental impact of desalination, the concentrate produced during the desalination should have as small volume as possible. Simultaneously, to allow the further reuse of ED concentrate for the production of evaporated salt or as a sodium chloride source for membrane electrolysis [1], the brine concentration should be as high as possible. Because of these two reasons, high water recovery is desired in the electrodialytic desalination. However, high salinity of the concentrate may increase the risk of sparingly soluble salt crystallization on the membrane surface (scaling). Several methods of reduction of scaling risk were proposed. Korngold et al. [2,3] presented the idea of reducing EDR scaling probability by controlled precipitation of recirculated (to achieve high recovery ratio) concentrate, oversaturated with calcium sulfate. The relative supersaturation in recirculated brine after the ⁎ Corresponding author. Tel.: +48 32 2372805. E-mail address: [email protected] (K. Mitko). 0011-9164/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.desal.2013.11.042

precipitator was less than 10% and no gypsum crystallization inside the ED stack was observed. Higher concentration of brine stream probably would not be possible, because of the excess turbidity of precipitator effluent. Korngold et al. pointed that this indicates the presence of small, difficult to remove crystals, which accelerate the gypsum precipitation. The idea was applied in a pilot-scale desalination plant [4], which showed the possibility of stable work with brine gypsum oversaturation at EDR stack exit reaching 35%. Korngold et al. and Oren et al. [2–4] fed antiscalants to a BWRO node and used the RO retentate as the feed for the diluate compartment of the ED node. In their case, antiscalants would not be transported into the concentrate stream and would not be problematic if the concentrate was to be used in the controlled crystallization step. If, however, the BWRO retentate stream was to be used as a concentrate in the EDR unit to the crystallizer [5], there should be no antiscalants in the system. Authors' earlier research [5–9] showed that the high recovery ratio may be achieved not by concentrate recycle, but by the application of thin spacer (intermembrane distance 0.16–0.25 mm) having densely distributed membrane abutments. Such spacers allow different linear velocities of diluate and concentrate streams without membrane bulging. A single-pass EDR can be performed with relative CaSO4 saturation of factor 5.2 and LSI

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of +2.3 [9]. To determine the maximal saturation conditions in the ED process, the appearance of self-growing crystal nuclei was assumed to be a limiting point of the process. The ED should be performed in a manner prohibiting the creation of such crystal nuclei inside the ED module, meaning that the residence time of majority of the particles should be smaller than the nucleation induction time. In order to calculate how long exactly the particles spent inside the electrodialyzer, the residence time distribution (RTD) [10] has to be determined. The experimentally measured tracer concentration is not a residence time of particles leaving sole reactor, but a RTD of the whole detection system, including tracer injector, reactor and tracer detector. Usually, if the detector and injector residence times are very short compared to reactor residence time, the influence of detector can be neglected. This however is not the case in some of the experimental set-ups with relatively small mean residence time of the membrane modules. From the mathematical point of view, a measured RTD signal is a convolution of a reactor RTD and RTD of all other parts of the system, so the measured signals should be deconvoluted first. For instance, Miskiewicz et al. [11] have used radioactive tracers and fluorescein to investigate hydrodynamic conditions inside ultrafiltration modules, but scintillation probes placed along the membrane were used as the detector, and the measured RTD could be assumed to be identical with RTD of the membrane module. This was not the case for Dydo et al. [12], who used ion-selective electrode immersed in the outlet stream, so the module RTD had to be recalculated with a Kalman filter algorithm. Roth et al. [13–15] have investigated the RTD of reverse osmosis (RO) module. They have used the RTD experiments to determine which tracer was inert to the membrane surface and used the Kalman filter method for the deconvolution of RTD signals. Later they investigated the influence of membrane fouling on RTD. Yang et al. [16] and Hasson et al. [17] have used the RTD experiments for fouling detection in reverse osmosis and showed that new dead zones may be created by wearing out of the membrane, a phenomenon which was also observed in electrodialysis [18]. The problem with determination of RTD of membrane module is the change of linear flow velocity along the membrane caused by the water transport across membranes. In case of pressure-driven membrane processes, the collection of permeate causes the retentate solution to slow down. In case of electrodialysis, the main reason for the water transport is the electroosmotic flow—ions are transported with their solvation water. Osmosis and hydraulic pumping can also influence the linear flow velocity. As the current density increases, the electroosmotic transport also increases, causing the increase in linear flow velocity in the concentrate and decrease in the diluate. In many cases the flow changes can be, and are, neglected. If the linear flow velocity changes are negligible, then RTD experiments can be performed in a non-working condition, when only water is flowing through the electrodialyzer and no current is applied. However, if a high water recovery is to be achieved, it is necessary to apply thin intermembrane

spacers and low concentrate velocity. In such conditions, the concentrate linear flow velocity at the end of the concentrate compartment (later called “the terminal velocity”) can be up to 50% higher than the concentrate linear flow velocity at the beginning of the concentrate compartment (“initial velocity”) [9], making RTD experiments performed in a non-working condition unreliable. In this paper the RTD of the electrodialyzer was investigated in working conditions. 2. Experimental The measurement setup consisted of feed tank, pumps, injector – a six-way valve with 12 cm sample loop of 1 mL volume, electrodialyzer, reactor and detector – see Fig. 1. The electrodialyzer consisted of four pairs of alternately placed Neosepta ACS and CMS ion-exchange membranes (see Fig. 2). Concentrate and diluate compartment spacers were made of 0.26 mm thick net sealed with silicone caulk. The effective membrane length was 42 cm, channel width was 2 cm. Diluate and concentrate entrances were 2.5 cm long and 5 mm wide. Electrolytic compartment spacer was 0.5 cm thick cored Plexiglas, perforated in order to both allow electric current flow, and to support the membrane. The segmented electrodes were made of platinum-coated titanium and the applied current was measured on each of the five electrode's segments, allowing the observation of a current density distribution. Each of the electrode's segments was 8 cm long, with 0.5 cm non-conducting space between them. The central points of the electrodes, used for current distribution calculations in Section 3.2, were located 4, 12.5, 21, 29.5 and 38 cm from the beginning of effective membrane length. Linear flow velocities in the concentrate compartment varied from 0.38 to 0.84 cm/s (see Appendix B) and were relatively low, but probably similar changes in residence time could be observed with higher velocities on longer, industrial-scale electrodialyzers. Diluate initial velocity (that is the linear flow velocity and the beginning of the compartment) was 6 times higher than concentrate initial velocity to ensure the conditions allowing high supersaturation of the concentrate. The electrodialyzer was operated in a single pass, counter-current mode with respect to diluate and concentrate, while the electrode rinsing solution was circulated. Anolyte and catholyte were allowed to mix in an open tank. The experimentally determined residence time distribution (RTD) of the whole system, EEXP, is a convolution of the electrodialyzer RTD, EED, and the RTD of every other part of the system, EDET [13,14,19] or in other words, the RTD of the system in which the electrodialyzer has been bypassed: EEXP ðt Þ ¼ ðEED  EDET Þðt Þ:

ð1Þ

Both the RTD of the whole system and the RTD of the system with bypassed electrodialyzer were determined experimentally. The functions describing the behavior of EDET at varying outlet flow rates

Mixed reagent tank

Injector Feed tank

Concentrate

Diluate

Fig. 1. Experimental set-up with the working electrodialyzer.

Reactor

Detector

M. Turek, K. Mitko / Desalination 342 (2014) 139–147

141

Fig. 2. Membrane arrangement inside the electrodialyzer. C—cation-exchange membrane, A—anion-exchange membrane, DIL—diluate, CON—concentrate, ELE—electrode rinsing solution.

were established by performing experiments on the system without the electrodialyzer, and when pipes connected to electrodialyzer concentrate inlet and outlet nozzles were directly hooked up. Next, the EEXP was determined at known outlet flow rate by performing experiments on the system with the electrodialyzer. Next, EDET was calculated using previously established functions, with volumetric flow rate same as in the EEXP determination experiments. Finally, Gauss– Newton least squares method was used to find the EED, which convolution with calculated EDET would produce EEXP. The stimulus–response experiments were performed by injecting tracer three times with known time interval between injections, waiting until the signal measured by the detector drops to baseline and repeating the measurements at another current density. Experimental data were smoothed using a low-pass frequency filter and always normalized before further calculations. Since electrodialysis is based on the ion transport, an inert, non-ionic tracer is preferable in case of RTD experiments in working electrodialyzer. Tracer of high molecular weight (i.e. dye) could be adsorbed on the membrane surface or be unreliable in predicting the behavior of solutions containing mostly small, light substances (i.e. inorganic ions). The tracer for RTD experiments in working conditions should be a small, non-ionic substance which does not cause the degradation of ion-exchange membranes. A 1 mL of 0.01 M formaldehyde solution was used as a tracer. The formaldehyde detection was based on the reaction described by Qiong et al. [20] – a liquid leaving the electrodialyzer was mixing with the stream of analytical solution (2 M ammonium acetate and 0.025 M methyl acetoacetate) of volumetric flow of 4.87 mL/min, next it passed through the reactor – a 3 m long pipe – and then the transmittance (later recalculated to absorbance) of the reaction product was measured with Panorama Fluorat-02 fluorometer. All used chemicals were POCh reagents of analytical grade. The working medium was sodium chloride solution of conductivity 60 mS/cm. Since the flow of analyte varied between the series, the linear dependence of peak area on formaldehyde concentration was tested in the range of 0.1 M–1.28 ⋅ 10−5 M at both largest and smallest volumetric flows. The upper linearity level was found to be 1.08 ⋅ 10− 2 M, limit of quantification (LOQ) was found to be 1.93 ⋅ 10− 5 M and limit of detection (LOD) was found to be 5.79 ⋅ 10− 6 M.

3. Results and discussion 3.1. The detector function The shape of the RTD curves can be described by a variety of mathematical functions [21]. To find the mathematical function which best describes the shape in this particular set of experiments, they were confronted with the single peaks observed with tracer injection into the system with and without the electrodialyzer present. The tested models were: log-normal probability distribution, commonly used in chromatography and flow injection analysis:

f ðt; t K ; m; sÞ ¼

  2  lnðt−t K Þ−m pffiffiffi exp − s 2 pffiffiffiffiffiffi ; t Nt K ðt−t K Þs 2π

ð2Þ

f ðt; t K ; m; sÞ ¼ 0; t ≤t K

where t K is a time between the tracer injection and the first appearance of the RTD signal at the detector, m and s are mean and standard deviation of the natural logarithm of independent variable (time). The second model was bimodal distribution consisting of two log-normal distributions, a modification of log-normal which was used for better explanation of the signal tail:

Eðt; t K ; m; sÞ ¼

2

1

f ðt; t K ; m; sÞ þ f ðt; 1:49t K ; 1:2m; 3:73sÞ 3 3    s2 Eðt; t K ; m; sÞ: Eðθ; t K ; m; sÞ ¼ t K þ exp m þ

ð3Þ

2

The function parameters 23, 13, 1.49, 1.2 and 3.73 were chosen empirically to assure best possible fit to the experimental data. Another

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equation was an exponentially-modified Gaussian distribution [16,21]: Eθ ¼ t

3

C ðt Þ

ð4Þ

Z∞

C ðt Þ 0

Flow patterns

where C(t) is the function describing change of outlet tracer concentration in time, with the parameters w, xC, and t0, given as: "

t−x w  er f pffiffiffi c − pffiffiffi 2t 0 2w

!

# þ1

ð5Þ

Bimodal

and t is the mean residence time of the reactor in question. Another tested model was axial dispersion [16,19]: " # 1 ð1−θÞ2 : Eθ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp − 4θðD=uLÞ 4πðD=uLÞ

Axial dispersion

Eθ

2

A w t−xc exp − C ðt Þ ¼ t0 2t 0 2t 20

!

Log−normal

2

EMG

1

Experimental

ð6Þ

The presented equations were reformulated into E(θ) and fitted to experimentally measured, smoothed and normalized peaks. The lognormal distribution correctly predicted the E(θ) values in the θ range from 0 to 1, however it underestimated the peak tail. The exponentially modified Gaussian model overestimated the early parts of tail and underestimated the tail at large θ values. The axial dispersion model did not predict correctly the peak maximum. The bimodal distribution model most accurately fitted the experimental data. An example comparison is shown in Fig. 4. The log-normal distribution had an advantage of being an internal function of the scripting language used in calculations [22] and thus in practice it was much faster to calculate the convolution of log-normal and bimodal functions (the latter being simple sum of internal functions) than axial dispersion, exponentially modified Gaussian or other functions we have tried to implement. The bimodal distribution fitted the residence time experimental data of the system both with and without the electrodialyzer. Because of that, it was assumed that the flow conditions inside the detector are the reason why the measured distribution showed bimodality. The bimodality may be a result of not perfectly instantaneous injection of tracer—as the six-way valve was switched, the whole sample loop was “inserted” into the electrodialyzer feed stream, which was probably a cause of longer tail in experimentally-measured distributions. However, because the tracer injection was always the same, either with or without the electrodialyzer connected into the system, it should not affect the deconvoluted RTD of the electrodialyzer. That is why the RTD of the whole system (injector, electrodialyzer, reactor, detector), EEXP and the RTD of the system with bypassed electrodialyzer (injector, reactor, detector), EDET, were described by a bimodal distribution (Eq. (3)), whereas the RTD of the sole electrodialyzer, ERTD, was described by the

0 0

1

2

3

4

θ Fig. 4. An exemplary comparison of how tested equations explained the experimentallyobserved peak.

log-normal distribution f(t) (Eq. (2)):   EED ðt Þ ≡ f t; t K;ED ; mED ; sED

ð7Þ

  EDET ðt Þ ≡ g t; t K;DET ; mDET ; sDET

ð8Þ

  EEXP ðt Þ ≡ g t; t K;EXP ; mEXP ; sEXP :

ð9Þ

The tK parameter of functions f and g is a time-shift and should by definition be dependent on the reciprocal of volumetric flow rate,V.˙ Parameters m and s are the mean and standard deviation of the natural logarithm of time, so the m and squared s should be dependent on the natural logarithm of volumetric flow. To calculate the detector function, EDET, a set of RTD experiments with the bypassed electrodialyzer was performed with varying volumetric flow of the feed and constant analytical reagent flow. The overall volumetric flows were 8.19, 9.32, 10.33, 13.25, 15, 18.48, 20.24 and 30 mL/min. The results are presented in Fig. 3. The established detector functions are presented in Table 1.

0.150

400 5.25

0.125 300

s2

m

tK

5.00 0.100

4.75

200

0.075 4.50 100

0.050

4.25 0.050

0.075

1/V

0.100

0.125

2.5

3.0

ln(V) Fig. 3. Dependence of tK, m and s2 parameters of the detector function on volumetric flow rate,V.˙

2.5

3.0

ln(V)

M. Turek, K. Mitko / Desalination 342 (2014) 139–147

143

Table 1 The established functions describing detector RTD parameters. Variables Dependent

Independent

tK,DET mDET (sDET)2

1 V˙

  ln V˙   ln V˙

3.2. Current density distribution Thanks to the segmented electrode, a current density distribution was observable. The current density showed a quadratic relation with the membrane length. Based on the current values measured at positions 1–5, presented in Appendix A (electrode center at 4, 12.5, 21, 29.5 and 38 cm from the diluate inlet/concentrate outlet) a quadratic equation describing the dependence of measured current, I[A], on position, x[cm], and overall current density, i[A/m2], was established as: 2

Iðx; iÞ ¼ ðA0 i þ A1 Þx þ ðA2 i þ A3 Þx þ A4 i þ A5

ð10Þ

where Ai were empirical coefficients given as: A0 = − 1.38 ⋅ 10−6, A1 = − 2.19 ⋅ 10−5, A2 = 6.52 ⋅ 105, A3 = 4.36 ⋅ 10− 4, A4 = 1.12 ⋅ 10− 3 and A5 = 4.09 ⋅ 10− 3. The established model allowed one to explain majority of the observed points, assuming 10% error margin. The quadratic distribution of current density along the membrane was observed in previous studies by Tanaka [23], however in his case the maximum current was not observed in the middle of the electrodialyzer and the electrodialyzer was not operated in a singlepass mode, but recycled until it reached constant concentration and the current. It's unclear why there is a maximum, presumably the combination of concentrate, diluate and membrane electric resistance changes along the membrane caused by the concentration and velocity changes were the reason why the middle point had the lowest overall resistance. 3.3. RTD of the electrodialyzer Nine series of experiments at different concentrate inlet flow velocities were performed to ensure that the influence of current density on residence time distribution would be confirmed. Mean residence time and its variance were calculated based on the parameters tK, m and s of the electrodialyzer RTD:   2 s t ¼ t K;ED þ exp mED þ ED

ð11Þ

      2 2 2 σ ¼ exp sED −1 exp 2mED þ sED :

ð12Þ

2

Slope

Intercept

R2

3470 −0.907 6.89 ⋅ 10−2

−20.6 7.63 0.289

0.998 0.97 0.97

washed out if the experimentally determined RTD results were to be used in designing the electrodialysis of sparingly soluble salt solutions. The mean residence time, its variance and cumulative distributions of the concentrate compartment drop as the current density increases. This was an expected behavior, since the water transport across the membranes results in increasing linear flow velocity along the electrodialyzer. As it turned out, the change in residence time was significant, especially looking from the point of scaling prevention in the electrodialysis of sparingly soluble salt solutions performed at high current density, when the main interest is not the mean residence time, but its cumulative distribution, as the cause of scaling may be the relatively small number of growing clusters, which are withheld somewhere in the concentrate compartment (the tail of the RTD). The exception is series no. 6, which did not show the expected behavior and results at 0 A/ m2 and 102 A/m2 in series 1. These points were excluded from calculations of velocity distribution along the electrodialyzer (Section 3.5). In the previous research [9] a method of scaling risk assessment was proposed, which was based on comparing the change in sparingly soluble salt nucleation induction time and residence time along the effective membrane length. The results of RTD experiments in working electrodialyzer suggest that if the mentioned method is applied, the RTD should not be measured at non-current conditions, as was done in the mentioned paper, but using a working electrodialyzer. The decreased concentrate residence time should decrease the area enclosed by residence time and nucleation induction curves, allowing the work with higher supersaturation of the concentrate. On the other hand, when electrodialysis with univalent permselective membranes is used to concentrate brine at high current density [1], the sodium chloride is continuously removed from the diluate compartment, but calcium, magnesium and sulfates are not. As the diluate is depleted from the sodium chloride, the ionic strength of the solution drops, increasing the supersaturation of calcium sulfate and causing a scaling risk. The results showing a decrease in concentrate residence time suggest that the diluate residence time increases, as the water is transported from the diluate to the concentrate together with sodium chloride. Increasing diluate residence time increases the scaling risk in the diluate compartment, as calcium sulfate solution spends longer time in the electrodialyzer. 3.4. Hydrodynamic conditions

The calculated mean residence times and their variances are shown in Appendix B. These include times required to travel not only along the effective length of the membrane, but also through inlet and outlet of the electrodialyzer—this theoretical mean residence time in the inlet and outlet nozzles, t C , was calculated as:   1 1 t C ¼ 2:615 þ u0 uL

ð13Þ

where u0 and uL are the initial and terminal velocities (that is linear flow velocities at the beginning and end of the compartment) of the electrodialyzer (in cm/s). The value of 2.615 was calculated by dividing the inlet and outlet volume by the concentrate channel cross-section. Such time was added to the theoretical mean residence time of the 42 cm long concentrate compartment. The times required for 99% of tracer particles to pass the electrodialyzer are presented in Fig. 5—the value assumed to correspond to the vast majority of growing clusters

Theoretical space–time value is given by following equation [19]: τ¼

L V ¼ u V˙

ð14Þ

where V is the vessel volume and V˙ is the volumetric flow rate. Mean residence times obtained through the experiment are higher than theoretical space–time values, which suggest an open–open type vessel with large deviation from plug flow, described as [19]: θE;oo ¼

2

σ θ;oo ¼

t E;oo τ þ tC σ 2t;oo t

2

¼1þ

¼

2 Pe

2 8 þ Pe Pe2

ð15Þ

ð16Þ

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M. Turek, K. Mitko / Desalination 342 (2014) 139–147

340 320

420

280

t99 [s]

300

t99 [s]

t99 [s]

320

300

380

260

360

280 0

200

400

600

400

0

2

200

400

600

0

2

i [A/m ], Series 1

100

200

300

400

2

i [A/m ], Series 4

i [A/m ], Series 7 350

300 375

330

t99 [s]

t99 [s]

t99 [s]

280 350

290

260

325

310

270

300 0

200

400

600

0

2

200

400

600

0

2

i [A/m ], Series 2

400

600

2

i [A/m ], Series 5

300

200

i [A/m ], Series 8

500

250

275

250

t99 [s]

t99 [s]

t99 [s]

450 240

400 230

225 350 0

200

400 2

i [A/m ], Series 3

600

0

100

200

300

400

2

i [A/m ], Series 6

0

200

400

600

2

i [A/m ], Series 9

Fig. 5. Change in time required for 99% of tracer particles [s] to leave the electrodialyzer with increasing current density [A/m2].

where t E;oo and σ2t,oo are the mean residence time and residence time variance calculated from the experimental data as described earlier in Eqs. (11) and (12), τ is the electrodialyzer theoretical space–time and t C is the mean residence time in the electrodialyzer inlet and outlet. The Péclet numbers obtained from Eq. (16) were used for the calculation of longitudinal dispersion coefficients:

DL ¼

uL Pe

ð17Þ

where u is the linear flow velocity and L is the effective length of the membrane. Results are presented in Table 2.

3.5. Velocity distribution Another question was the distribution of linear flow velocity along the effective membrane length. Suppose a liquid flows through the electrodialyzer of length L with velocity u. Then the mean time, equal to space–time, required for the liquid to pass from the inlet to the outlet can be given as: ZL τ¼ 0

dl uðlÞ

ð18Þ

if u(l) = const then the equation reduces to Eq. (14). With the

M. Turek, K. Mitko / Desalination 342 (2014) 139–147 Table 2 Péclet numbers and longitudinal dispersion coefficients in the non-current conditions. Series

u[cm/s]

Pe

DL[cm2/s]

1 2 3 4 5 6 7 8 9

0.51 0.38 0.60 0.45 0.57 0.35 0.39 0.48 0.66

13.28 11.81 11.09 12.21 12.89 9.13 11.41 12.05 10.29

1.60 1.35 2.28 1.53 1.86 1.60 1.44 1.69 2.68

145

Table 3 Comparison of Pearson's correlation, covariance, mean percentage error (MPE), mean absolute error (MAE) and root mean square deviation (RMSD) between observed and predicted mean residence times (N = 27). Model

Eq. (20)

Eq. (24)

Eq. (26)

Eq. (28)

r Covariance MPE [%] MAE RMSD

0.904 327 −5.6 7.2 8.9

0.905 319 −8.3 9.3 11.1

0.899 336 −2.8 6.4 7.6

0.905 361 −1.1 6.3 7.1

And if the increase in linear flow velocity is parabolic: 2

assumption of the linear flow velocity increasing linearly along the electrodialyzer: uðlÞ ¼ al þ b uðl ¼ 0Þ ¼ u0 uðl ¼ LÞ ¼ uL

ð19Þ

then the term for mean residence time can be analytically solved:   L u ð20Þ τ¼ ln L : u0 uL −u0 The change in Péclet number was assumed to be negligible throughout each series—there was no clear pattern of how it is changing with linear flow velocity. Thus, a change in mean residence time due to the water transport increasing the linear flow velocity can be predicted based on equation created by combining Eqs. (20) and (15):  1þ

t¼

  2 L u tC þ ln L : Peði ¼ 0Þ uL −u0 u0

ð21Þ

Then the expected mean residence time change was calculated as: d¼

t ði ¼ 0Þ−t ðiÞ : t ði ¼ 0Þ

ð22Þ

However the concentrate velocity distribution does not have to be necessarily linear. In fact, the quadratic current density distribution along the membrane length, as observed in Eq. (10), indicates that the electroosmotic transport of water along the membrane is not constant, which would contribute to the changes in linear flow velocity of the concentrate. Three other models of flow velocity increase were tested. If the velocity increases corresponding to square root function: pffi uðlÞ ¼ a l þ b uðl ¼ 0Þ ¼ u0 uðl ¼ LÞ ¼ uL

ð23Þ

then after integration, τ becomes: τ¼

 

2L u0 u 1− ln 0 : uL −u0 uL uL −u0

ð24Þ

Another model was formulated as: 2

uðlÞ ¼ al þ b uðl ¼ 0Þ ¼ u0 : uðl ¼ LÞ ¼ uL

ð25Þ

After integration: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L uL −u0 : τ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arctan u0 u0 ðuL −u0 Þ

ð26Þ

uðlÞ ¼ al þ bl þ c : uðl ¼ 0Þ ¼ u0 uðl ¼ LÞ ¼ uL

ð27Þ

Then after integration τ becomes:  2 τ ¼ pffiffiffiffiffiffiarctan C0

pffiffiffiffiffiffi  L C0 u0 þ uL −aL

 2 −5 u −u −aL2 C 0 ¼ 4au0 − L 0 a ¼ 9:07  10 :

ð28Þ

L

All of the four models taken under considerations were compared with mean residence times obtained at current conditions of the electrodialyzer (series 6 and experimental point at 102 A/m2 from series 1 were excluded). Parameter a = 9.07 ⋅ 10− 5 of Eq. (28) was chosen after comparison of values predicted by model with those measured experimentally. The results are presented in Table 3. Eq. (28) showed the smallest mean percentage error (MPE), mean absolute error (MAE) and root mean squared deviation (RMSD). Although the differences between those models are not huge and we cannot state decisively that one is overwhelmingly better than the others, the fact that one fits experimental data better and is to be expected knowing the current density distribution indicates that it may be the correct one. The question of linear velocity distribution is a matter of another, ongoing study. All that can be said at this moment is that linear flow velocity might not increase linearly along the effective membrane length inside the concentrate compartment. 4. Conclusions A method for the investigation of residence time distribution of the particles leaving working electrodialyzer was proposed. It was confirmed that electroosmotic water transport, which changes the linear flow velocity along the electrodialyzer, changes the hydrodynamic conditions in the concentrate compartment. The change in residence time will affect the scaling probability as calculated in previous research. A relatively high residence time variance suggests that in case of plateand-frame modules equipped with thin intermembrane spacers, working in counter-current mode with different linear flow velocities of diluate and concentrate, the flow is nowhere near the often assumed plug flow. The results show that the electrodialyzer used in the experiments is an open–open type vessel with large deviation from the plug flow ( uLD was on average 0.08). The change in mean residence time with increasing current density suggests that the velocity may not increase linearly along the concentrate compartment, contrary to our previous assumptions [9]. The method for electrodialyzer RTD determination under applied current requires however additional verification, as we are unsure whether the supersaturated calcium sulfate solution flowing through the concentrate compartment would behave in a similar way to formaldehyde. Our main goal is to come up with a scaling prediction method, so the test will be to use the RTD data obtained with the formaldehyde tracer, calculate the scaling probability

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according to previously presented methodology [9] and check if the scaling can really be avoided. Such an experiment is a subject of ongoing study.

List of symbols d expected mean residence time change longitudinal dispersion coefficient [cm2/s] DL E(t) residence time distribution I current [A] i current density [A/m2] L effective membrane length [cm] m mean of natural logarithm of independent variable MAE mean absolute error MPE mean percentage error [%] Pe Péclet number r Pearson's correlation coefficient RMSD root mean squared deviation s standard deviation of natural logarithm of independent variable t time [s] t mean residence time [s] tC mean residence time of inlet and outlet nozzles [s] time between tracer injection and signal appearance at detectK tor [s] linear flow velocity at the compartment entrance [cm/s] u0 linear flow velocity at the compartment exit [cm/s] uL V reactor volume V˙ volumetric flow [mL/min]

Greek symbols residence time variance [s2] σ2 τ reactor space–time [s] θ dimensionless time relative to mean residence time

EXP Oo

system with electrodialyzer open–open vessel

Acknowledgment Series uL[cm/s] 1

2

3

4

5

6

7

8

9

Subscripts Series i[A/m2] I[A] at given position along the membrane [cm] DET system without electrodialyzer 4 12.5 21 29.5 ED electrodialyzer 1

2

3

4

5

6

7

8

9

93 249 485 653 170 336 501 668 174 334 506 673 173 309 507 668 223 444 664 102 219 444 147 220 441 220 445 657 220 447 665

0.128 0.339 0.637 0.848 0.224 0.418 0.646 0.866 0.182 0.470 0.712 0.908 0.279 0.563 0.835 1.074 0.344 0.690 1.026 0.145 0.298 0.580 0.176 0.309 0.625 0.311 0.631 0.937 0.309 0.567 0.784

0.168 0.475 0.888 1.155 0.319 0.604 0.885 1.154 0.353 0.582 0.882 1.183 0.259 0.544 0.786 1.089 0.372 0.736 1.069 0.179 0.368 0.750 0.228 0.397 0.800 0.402 0.806 1.117 0.374 0.769 1.090

0.168 0.456 0.883 1.181 0.314 0.606 0.908 1.187 0.329 0.593 0.897 1.185 0.333 0.654 0.970 1.274 0.415 0.839 1.277 0.193 0.409 0.822 0.426 0.430 0.855 0.429 0.871 1.298 0.432 0.905 1.373

0.173 0.432 0.894 1.223 0.314 0.636 0.938 1.280 0.318 0.604 0.918 1.225 0.312 0.310 0.906 1.189 0.397 0.794 1.199 0.180 0.413 0.820 0.223 0.392 0.787 0.392 0.799 1.226 0.401 0.830 1.277

0.51 0.53 0.57 0.61 0.65 0.38 0.43 0.46 0.50 0.55 0.60 0.65 0.69 0.74 0.76 0.45 0.48 0.53 0.57 0.61 0.57 0.62 0.67 0.73 0.35 0.36 0.44 0.49 0.39 0.44 0.45 0.51 0.48 0.54 0.60 0.65 0.66 0.72 0.78 0.84

i[A/m2]

t ½s

0 93 249 485 653 0 170 336 501 668 0 174 334 506 673 0 173 309 507 668 0 223 444 664 0 102 219 444 0 147 220 441 0 220 445 657 0 220 447 665

118 ± 128 ± 112 ± 99 ± 98 ± 141 ± 135 ± 131 ± 121 ± 124 ± 96 ± 89 ± 91 ± 92 ± 82 ± 122 ± 114 ± 111 ± 111 ± 108 ± 111 ± 104 ± 99 ± 98 ± 119 ± 144 ± 144 ± 139 ± 156 ± 154 ± 147 ± 139 ± 127 ± 116 ± 111 ± 106 ± 87 ± 90 ± 87 ± 88 ±

σ2[s2] 1 3 1 3 3 2 1 2 1 3 4 1 1 2 3 4 1 1 3 3 1 1 2 4 3 7 7 4 3 5 1 2 3 2 3 2 1 1 2 1

2729 ± 3065 ± 2385 ± 2159 ± 2016 ± 4507 ± 3512 ± 3353 ± 2909 ± 2838 ± 2261 ± 1678 ± 1679 ± 1594 ± 1486 ± 3237 ± 3637 ± 2892 ± 2862 ± 2532 ± 2505 ± 2298 ± 2475 ± 2031 ± 4459 ± 8266 ± 8266 ± 5960 ± 5759 ± 5476 ± 5728 ± 4310 ± 3566 ± 2872 ± 2728 ± 2377 ± 2043 ± 1847 ± 1696 ± 1666 ±

73 120 14 28 104 120 9 73 299 321 251 64 37 148 253 406 294 312 11 182 233 61 196 166 681 1575 1575 998 438 442 1160 302 227 262 269 274 54 61 191 30

38 0.146 0.392 0.773 1.075 0.261 0.562 0.833 1.122 0.281 0.553 0.845 1.153 0.268 0.522 0.765 0.985 0.344 0.672 1.006 0.156 0.353 0.757 0.179 0.324 0.635 0.317 0.634 0.937 0.335 0.683 1.062

This work was partially financed by the Polish National Science Center upon Decision No. DEC-2012/05/N/ST8/02951. Appendix A. Current measured at each of the electrode segments

Appendix B. Applied current density, concentrate terminal velocity, mean residence time and its variance calculated for each experimental series

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