Heat transfer at gas evolving electrodes

Ekwahimica “da. Vol. 23. pi 1019-1022. @ Peqamon PressLtd. 1978. Printed in Great Britain.

0019_4686178/I001L1019 soz.c@p

HEAT TRANSFER AT GAS EVOLVING ELECTRODES HELMUT VCCT Facbbereicb Verfabreostecbuik da Techuischen Fachhochschule D-1000 Berlin 65, Germany (Received 6 January 1978; and infinalfinn

Berlin, Luxemburger Str. 10,

20 February

1978)

Abstract-The outlines of an available concept for heat transfer at gas evolving electrodes (Z&e-Magrini) are discussed. A new method for prediction of heat transfer on the basis of the analogy between heat and mass transfer is derived from known theoretical equations and tested in comparison to experimental data of various workers.

polarization required for corrosion protection[l3]. A further example is the estimation of surface temperatures of operating electrodes[14], especially in electrochemical machining where temperatures near the boiling point are possibIe[l5]. One method for prediction of heat transfer, developed by Zuher[ 161 for nucleate boiling and adag ted by Magrini[5] for gas evolving electrodes, is available, but problematic in application. A different and more general method will be presented in the following.

NOMENCLATURE thermal ditTusivity (m2/s) electrode area (m*) geometric factor (-)

deoarture diameter Cm) ditfusion coefficient (I&/S) acceleration of gravity (m/s’) temperature (Kj gasvolume flow rate (m”/s) number of bubbles (-) heat transfer coefficient (W/m’ K) mass transfer coefficient (m/s) coefficient of thermat expansion (K-I) density (kg/m3) dynamic viscosity (kg/m s) fractional surface coverage (- ) thermal conductivity (W/m K) kinematic viscosity (m*/s) gas hold.up (-) Nu Pr Re SC Sh

= = = = =

ad/i v/a V&/Av v/D bd/D

2. THE ZUBER-MAGBINI METHOD Zuber’s theory bases on the leading idea that in nucleate boiling is controlled by the hydrodynamic flow of the vapour-liquid mixture generated by the process. Heat transfer by natural convection may be described by heat transfer

Nusselt number Prandtl Reynolds number Schmidt number Sherwood number

Nu = t

= const (Gr Pr)1’3

where the Prandtl number is defined as usually by

Subscripts: G L w co

Pr

gas liquid electrode surface outside boundary layer

Gr =

on the mass transfer at

gas evolving elcktrodes, even less[l-81 on the heat transfer problem. This statement becomesevident with regard to the extensive investigation done in the analogous field of nucleate boiling heat and mass transfer. On the other hand there is an urgent need in practical methods to predict heat transfer in various fields of research and technology. One example of technical application is the prediction of the heat transfer coefficient of industrial electrodes which are simultaneously operated as cooling surfaces as occurs in chlorate cells[7,9] and pcrchlorate cells[lO]. This applies also to cooling surfaces which are operated as gas evolving cathodes inside[ll] or completely outside the ceIl[ 12-j only to ensure the minimal cathodic Sn .?j/l&D

~1 a

whereas the Grashof number in the usual definition is extended by a term which considers the etktive diminution of gravity caused by the gas hold-up in addition to the thermal expansion :

1. INTRODUCTION Rather little has been published

(1)

‘!!

p(T,

[

_ T,)

+

xGPL,w PL.rn 1

(3)

Validity of (1) is restricted to (GrPr) > 2.10’[17]. Zuber[16] described the controlling gas hold-up in the liquid close to the wall by different expressions, the most simplified form of which resulted in

when the waiting time (between the bubble departure and the beginning ofgrowth of the consecutive bubble) was considered zero. Zuber was able to show by comparison with experimental results for actual hub ble population densities that heat transfer in nucleate boiling can be interpreted on the basis of a hy-

1019

HELMUT

1020

drodynamic flow by free convection. An application in industrial practice is, however, obstructed by the need of bubble population data which are usually unknown and cannot yet be predicted with satisfactory reliability, since the quantitative influence of surface condition is largely unknown. It is obvious that Zuber’s model should also be useful for gas evolving electrodes. It was Magrini[S] who applied Zuber’s result to correlate heat transfer data, obtained at a heated cylinder (42mm dia) operating as cathode in strongly diluted sulfuric acid. The difficulty to determine the gas hold-up in absence of bubble population data was circumvented by estimations for the cylinder, the application of which cannot simply be generalized. After all, the particular modifications were of such a kind that good agreement with experimental data could be stated. 3. A METHOD BASING TRANSFER

ON MASS

EQUATIONS

Quite a different alternative to correlate heat transfer data at gas evolving electrodes exists in referring to the analogy between heat and mass transfer. Mass transfer equations for gas evolving electrodes are known. Transformation into heat transfer equations can be carried out by simple replacement of dimensionless groups by the corresponding heat transfer groups. The mass transfer equations derived by Ibl and published by Venczel and Ibl[lS-211 can be put into a dimensionless form[22], Sh = $

(Re Sc)“.5(1 - O)“.sF (5)

where

Sh=!!! D

Re=V,d Av SC = ;

The departure diameter d is the diameter of a sphere with the mean volume of the departing bubble, 8 is the fractional surface coverage as used by Venczel[lS], and C is a factor considering the geometry of the real bubble, C = 4 for the hemispherical, C = 8 for the spherical bubble. A different model for mass transfer was established by Vogt and resulted in a further dimensionless equation which was compared with various experimental resuhs[22,23] : Sh = 0.93 Re”.’ SC’.~*’ Formal replacement of the dimensionless groups results in heat transfer equations for gas evolving electrodes,

* Owing to a misinterpretation of the bubble radius used by Ibl the numerical factor quoted in[22] differs from the value given here.

VCGT

2.76 Nu = c”3 (Re Pr)0.5(l - 0)“.5

(ID)

Nu = 0.93 Re0,5 Pro.4*7

(II)

and

The Reynolds and the Prandtl number are already defined by (7) and (2). For application of (10) the correlation (in default of a better one) for the fractional surface coverage 6 = 0.5 Re”,‘s

(12)

is proposed, derived from experimental results in nucleate hydrogen evolution from acid electrolyte solution at various electrode materials, carried out by Venczel[18,24]. It must be emphasized that (10) and (11) only represent heat transfer free of external (forced or natural) convection. Remarkable superposed convective heat transfer must, if necessary, be taken into separately by use of appropriate account methods[25,26].

4.

COMPARISON WITH EXPERIMENTAL RESULTS

To test (10) and (1 I) the few available experimental data are used. Mixon et a/[11 carried out measurements of heat transfer in nucleate boiling and at lower temperatures with superposed electrochemical hydrogen nucleate evolution in order to simulate the phenomena of vaporization independent of temperature. At high temperature differences (?“, - T,) between wall and bulk of liquid heat transfer caused by nucleate gas evolution was interfered by nucleate boiling. Another superposition results from the natural convection caused by thermal expansion as discussed above, occurring even at low temperature differences. Both influences decrease with decreasing temperature difference and can be eliminated by extrapolation to zero temperature difference, delivering a condition which corresponds to the assumptions of the models which (10) and (11) are based upon. Results are represented in Fig. 1. As adilute aqueous solution was used (0.1 and 0.5 N NaOH), properties of water were approximately employed. The departure diameter of the bubble was set d = 50pm in agreement with experimental investigations by Janssen and Hoogland[27] and Rot&r et a([8]. Magrini’s experimental data[5, Figs 3 and 4] of the heat transfer coefficients at a horizontal cylindrical cathode scatter considerably, thus preventing a reliable extrapolation to zero temperature difference. However, heat transfer in purely natural convection (with zero current) was experimentally determined, too, the results of which are in good agreement with data calculated from known equations. To get data which apply to gas evolving electrodes without influence of superposed natural bulk convection, as represented by (10) and (11) a method proposed earlier[26] was used. The temperature of the liquid bulk was assumed to WC in accordance with earlier experiments of Magrini[Z] with vertical electrodes. Properties of the electrolyte solution (0.1 N HsSO,)

1021

Heat transfer at gas evolving electrodes 5. CORRECTIONOF DATA FOR SMALL REYNOLDS NUMBERS

I

II

IOF Reynolds number, Re

!I

10-z

Fig. 1. Comparison of experimental data of various workers with thearetical heat transfer equations. (a) Equation (10) for the hemispherical [upper) and spherical bubble (lower line). (b) Equation (11). Mixon et uI[ I] : 0 Magrini[S]: D-AT = IOK m-AT = 18K MacMullin et a![63 : V Rou6ar et a@] : l-Re’ = 187 A -Re’ = 313 A-Re’ = 747.

were approximately considered to be those of water. The results are also plotted in Fig. 1. Heat transfer data in different cells with molten KF ,2HF as electrolyte were reported by MacMullin et a@]. The results, taken from Table 7[6] are also shown in Fig. 1, using properties as indicated by the authors; the Prandtl number, however, was calculated by estimating the ratio tr/Pr with the method given by PaImer[28], basing on Weber’s[29] prediction rule. In the cells used for measurements the cathode area was not identical with the heat transfer area. Extrapolation procedures had to be applied which, apart from the uncertainty of the properties used for recalculation, give rise to consider the quantitative evaluation of the results doubtful. Roubar et a\[81 investigated heat transfer at gas evolving electrodes for the case of superposed forced convection through a rectangular channel with varying cds and Reynolds numbers of the duct flow. Hydrogen was evolved out of 0.5 N KOH solution at 25°C. As can be read from a graph [S, Fig. 61, the Reynolds numbers of the duct flow seem not to affect heat transfer. Introducing the equivalent diameter[30J into the Reynolds number, calculation makes evident that even the highest forced flow rate is not effective enough to interfere remarkably with the heat transfer by nucleate gas evolution. Hence, a superposition of both heat transfer mechanisms need not be considered. Heat transfer results are plotted in Fig. 1. Properties were used as given by Rot&r et nI[8]. Serious objections exist in regard to an evaluation of the experimental data obtained by Bhand et aI[4] at a thin horizontal wire electrode having a dia of only 0.1 mm, which corresponds to the order of magnitude of the departure diameters of the evolved bubbles.

Conspicuously, a view on Fig. 1 shows that the data of the more elaborate experiments carried out by Mixon et e/Cl] and by Rougar et al[8] suggest a smaller slope as compared to (10) and (ll), rather Sh _ R@ instead of Sh _ Re”‘. This behaviour can be made plausible for both experimental series. Under the conditions used by RouSar et a[[81 the influence of forced convection is negligible but an estimation shows that heat transfer due to natural convection for low gas fluxes is of the order of magnitudeofheat transfer due to gas evolution. In fact, a combined heat transfer is experimentally determined which is greater than described by the equation for gas evolution alone. As the Reynolds number of gas evolution increases the influence of natural convection goes down. Hence, deviation of the experimental values from the theoretical tine (for puregas evolution) is most prominent for low Reynolds numbers. The Mixon values employed iu Fig. 1 were obtained by linear extrapolation to zero temperature difference for the reasons as discussed above, Fig. 2, dashed line. A linear extrapolation is objectionable for low gas fluxes, because the real curves should be expected to approach graduaily the natural convection curve with no gas evolution. An extrapolation more appropriate to the physical reality would lead to smaller heat transfer coefficients as shown by the dotted line. This influence decreases as the gas evolution increases. Hence, the uncorrected experimental data of both experimental series usedin Fig. 1 pretend higher values for low Reynolds numbers. A regression line through the values corrected for pure heat transfer due to gas evolution would show a slope nearer to the theoretical lines.

0

20

40

Temperature

6G

difference,

80

100

120

IT,-T,)/“F

Fig. 2. Natural convective heat transfer at various cds of superposed gas evolution reaction. According to Mixon et crf[l], redrawn. A-235 A/m= D-148 A/m* e-88 A/m’ a-59 A/m* a-29.5 A/m= O--O A/m2.

1022

HELMUTVOGT 6. CONCLUSION

By

and large, the correlation of the available data by the two equations (10) and (11) is satisfactory, at least for the experiments of Mixon[l] and Rouiatfl] and with special regard to the fact that the heat transfer equations (as well as the corresponding mass transfer equations) are entirely theoretical. Altogether, concordance is far from being excellent and much poorer than for correlations of single-phase heat transfer data. The reason must be sought in the severe complexity of the microevents at gas evolving electrodes and the resulting inadequacy of the simplification of the models used for deriving (10) and (11). On the other hand, such heat transfer experiments are rather difficult to be carried out and evaluated properly. Additional expe rimental work dedicated to this very task must be done in the future. Nevertheless, (10) and (11) prove suitable to serve as first general design equations for gas evolving electrodes.

REFERENCES 1. F. 0. Mixon, W. Y. Chon and K. 0. Beatty, Chem. Engng

2. 3. 4.

5. 6. 7.

Prog. 55 (lo), 49 (1959). Chem. Engng Prog. Symp. Ser. No. 30, 56, 75 (1960). U. Magrini, Tertnorecnica 18, 339 (1964). IJ. Ma&i. Termofecnico IS, 578 (1964). S. C. Biand, G. V. Patgaonkar and D. V. Gogate, Int. J. Heat Muss Transfer 8, 111 (1965). U. Magrini, Ccl/o& 37, 125 (1966). R. B. MacMullin, K. L. Mills and F. N. Ruehlen, J. electrochem. See. 118, IS82 (1971). I. RouBar and V. Cezner, Electrochim. Acta 20,289 (1975).

8. I. RouSar, J. KaEin, E. Lippert, F. $mirous and V. Cezner, Electrochin. Actn 20,295 (1975). 9. H. Vogt, UllmannsEncyklopiidieder technischenChemie. 4th ed. Vol. 9, pp. 553-565. Verlag Chemie: Weinheim (1975). 10. J. C. Schumacher, Perch/orates: Their Properties, Mnnufacture,and Uses, p. 81. Reinhold : New York (1960). 11. W. A. Miiller, Ullmunns Encyklopiidie der tee/m&en Chemie, 3rded. Vol. 5, p. 534. Urban und Schwarzenberg: Munich (1954). 12. Private communication Dr. G. Wagner, Laufenburg. 13. V. I. G&burg, M. A. Mel’nikov, Soviet Chem. Ind. 6,414 (1971). 14. J. Miiller, Chemie-lngr-Tech. 49, 326 (1977). 15. W. G. Clark, J. A. McGeough, .I. oppl. Electrochem. 7,277 11977). 16. h. Ziber, lat. .I. HeatMossTran$er 6, 53 (1963). 17. S. S. Kutateladze. V. M. Borishanskii. A. Concise Encyclopedia 01 Heat Transfer, p. 168. Pergamon Press, Oxford (1966). 18. J. Venczel, Cuberden Sto&ansport an gasentwtckelnden Elektroden. Diss. ETH Ziirich (1961). 19. N. Ibl, Chemie-lngr-Tech, 35, 353 (1963). 20. N. Ibl, J. Venczel, MetaNoberjiiche 24, 365 (1970). 21. N. Ibl, E. Adam, J. Venczel, E. Schalch, Chemie-IngrTech. 43,202 (1971). 22. H. Vogt, Ein Beitrag zum Stofftibergang an gasentwikelmien Elektroden. Diss. Univ. Stuttgart (1977). 23. K. Stephan,H. Vogt, Electrochim. Acta, in press. 24. J. Venczel, Electrochim. Acra 15, 1909 (1970). 25. M. D. Birkett and A. Kufin, Electrochim. Acto 22, 1427 (1977). 26. H. Vogt, E&whim. Acta 23, 203 (1978). 27. I.. J. J. Janssen, J. G. Hoogland, Electrochim. Acta 18,543 (1973). 28. G. Palmer, Ind. Engng Chem. 40, 89 (1948). 29. H. F. Weber, Ann. Phys. Cbemie 46, 316 (1880). 30. Private communication by Prof. RouSar, 7 Oct. (1977).