Химия и химические технологии/5. Фундаментальные проблемы создания новых материалов и технологий

Gnatko E.N., Omelchenko A.M., Kravets V.I.

Ukrainian State Chemico-Technological University, Dnieprpotrovsk, Ukraine

HYDRODYNAMIC REGIME AND MASS TRANSFER COEFFICIENT OF THE CONCENTRIC CYLINDRICAL FLOWING ELECTROLYZER WITH STATIONARY ELECTRODES

In the diffusion-controlled electrochemical reactions, the maximal electric current is determined by the mass transfer coefficient. A larger mass transfer coefficient leads to the higher current efficiency, due to alleviation of the side reactions, and lower thermal loses. The main objectives of this publication are to theoretically evaluate the mass transfer coefficient and, sequentially, the maximal diffusion-limited current in the concentric cylindrical electrolyzer flowing with stationary electrodes.

The maximal diffusion-limited current is given by the equation

(1),

where I_L is the current limited by convective diffusion, z is the valency, F is the Faraday constant, k_m is the mass transport coefficient, which is the complex function of the reactant’s diffusion coefficient, thickness of unstirred layer of electrolyte and physical characteristics of electrolyte, A is the effective electrode area, and c_o is the bulk concentration of the reactant.

From Eq. (1), the maximal value of diffusion-limited current does not depend on voltage meaning the reaction rate will attain the maximum at certain voltage and further increase of voltage will just cause non-productive loses of energy.

Due to the uncertainty in the effective electrode area, the product of k_m and A, so-called k_mA factor, is usually calculated by Eq. (1) using experimentally measured value of limited current. The disadvantage of this method consists in that the determined value is limited to the given electrochemical reactor.

In more general manner, the mass transfer coefficient could be theoretically evaluated of on the basis of analysis of hydrodynamic regime (laminar or turbulent) of the flow inside the electrolyzer and its geometry. That value of mass transfer coefficient could be assumed for other similar electrolyzers.

The mass transfer coefficient is best expressed in terms of dimensionless groups. The Sherwood number (Nusselt number), characterizing mass transfer, can be used for calculation of the average mass transfer coefficient k_AV by formula:

(2),

where Sh_AV is the average Sherwood number for entire anode or cathode chambers, D is the diffusion coefficient of the reactant, and d_h is the hydraulic diameter of the electrolyzer. For the concentric cylindrical reactor d_e is calculated by formula [1]:

(3),

where d_i and d_o are the inner diameter of the outside cylinder and outside diameter of the inner cylinders, respectively. In our case, d_h are 0.24 cm and 0.21 cm for the anode and cathode chambers, respectively.

The Sherwood number depends on the hydrodynamic regime inside the reactor and its geometry, e.g. on the length of the electrodes. The length of the electrodes L in our study is 21 cm. Thus, they can be considered as long ones, i.e., having a length more than 20d_h (in our case, 4.0 – 4.6 cm). For the long electrodes, the condition of constant mass flux at the electrode surface is applicable [1]. For the concentric cylindrical electrolyzer with long stationary electrodes and fully developed laminar flow, the Sherwood number depends on the aspect ratio r = d_o/d_i [1]. Data for Sh_AV as a function of r are given in Table 1.

Table 1. Variation of Sh_AV with aspect ratio in a long annulus [1]

r	0.8	0.6	0.4	0.2	0.1
ShAV	5.580	5.912	6.583	8.499	11.910

The Reynolds number characterizes hydrodynamic regime, i.e., laminar or turbulent, of the flow inside the electrolyzer. It depends on the flow rate, geometry of the reactor and physical properties of fluid. Experimentally shown that for the Reynolds number about 2400 and below the flow of fluid through a tube is laminar, and turbulence occurs if the Reynolds number is about 3000 and above. The average Reynolds number Re_AV for entire electrochemical reactor can be calculated using equation [1]:

(4),

where U_AV is the average linear flow rate in the reactor, cm s^-1, and n is the kinematic viscosity of fluid in the reactor, cm² s^-1. For the calculations, we used n = 10^-2 cm²/s, which is characteristic for the most aqueous solutions.

In general, the average linear flow rate could vary for anode and cathode chambers due to their different volumes. The average linear flow rate and the retention time t_R inside the reactor are given by formulas:

and (5),

where V is the volume of the anode or cathode chamber, cm³, and f is the volumetric flow rate, cm³ s^-1.

The calculated values of the average linear flow rates through the cathode and anode chambers and the Reynolds numbers are presented in Table 2. The calculations were based on the following geometric parameters: anode: length – 21 cm, surface area – 52.8 cm²; cathode: length – 21 cm, inner, i.e., electrochemically active, surface area – 88.8 cm²; anode chamber: volume – 7.3 cm³; cathode chamber: volume – 8.4 cm³.

It can be seen that flow inside the electrolyzer is laminar up to the practically used volumetric flow of 2000 cm³/min, and values of Sh_AV presented in Table 1 are applicable for evaluation of I_LAV. Since the aspect ratio r in our case is ~0.8, the value of Sh_AV = 5.580 was used for calculations.

The value of the maximal diffusion-limited current, averaged over entire chamber I_LAV, was calculated by formula:

Table 2. The Reynolds numbers for the FEM3 reactor for the various flow rates.

Volumetric flow rate, cm^3/min	Average linear flow rate, cm/s		Reynolds number
	Cathode chamber	Anode chamber	Cathode chamber	Anode chamber
100	4.15	5.04	85.40	116.02
200	8.29	10.08	170.81	232.04
300	12.43	15.13	256.21	348.06
400	16.58	20.18	341.62	464.07
500	20.72	25.22	427.02	580.09
600	24.87	30.27	512.42	696.11
700	29.02	35.31	597.83	812.13
800	33.17	40.35	683.23	928.15
900	37.31	45.40	768.64	1044.17
1000	41.46	50.44	854.04	1160.19
1100	45.60	55.49	939.44	1276.21
1200	49.75	60.53	1024.85	1392.22
1300	53.90	65.57	1110.25	1508.24
1400	58.04	70.62	1195.66	1624.26
1500	62.19	75.66	1281.06	1740.28
1600	66.33	80.71	1366.46	1856.30
1700	70.48	85.75	1451.87	1972.32
1800	74.62	90.80	1537.27	2088.34
1900	78.77	95.84	1622.68	2204.36
2000	82.92	100.89	1708.08	2320.37

(6)

where t_r is the transport number of the reacting species. In general, t_r is about 0.9 for the reacting species in both the anode and cathode chambers. For the typical bulk concentration 0.118 M, maximal diffusion-limited current for the anode is calculated to be 2.8 A. Since water molecules, which are abundant in the solution, are

discharged on the cathode, the kinetics of the cathode reaction is not diffusion-limited. Therefore, an overall maximal current could be evaluated as ~ 3A, since both chambers are connected in series. This value is at the same order of magnitude with the experimentally observed currents in our electrolyzer.

Conclusions

1. The flow inside the electrolyzer is probably laminar in the range of the usually employed volumetric flows of 300 – 600 ml/min.

2. Due to the laminar character of the flow the mass transfer coefficient does not depend on the volumetric flow rate. In this case, increasing of the flow rate will lead to decrease of the retention time and reduce sodium chloride transformation.

Theoretical lowest level of the volumetric flow rate should be estimated on the basis of actual saline concentration inside the electrolyzer and the Faraday’s law.

3. To increase the effectiveness of the mass transfer, the usage of some turbulence-creating devices should be considered during the design of a new electrochemical reactor.

References

1. Pickett, D. J. Electrochemical reactor design. Second edition, 1-529. 1979. Amsterdam, Elsevier. Chemical engineering monographs. V. 9 (Editor Churchill, S.W.)