A Linearized Corrosion Double-Layer Model for Laminar Flow

A metal/liquid interface corrosion-reaction model is developed for the flow electrification of low-conductivity liquids in metal pipes. In the propose...
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Ind. Eng. Chem. Res. 1996, 35, 3195-3202

3195

A Linearized Corrosion Double-Layer Model for Laminar Flow Electrification of Hydrocarbon Liquids in Metal Pipes H. Chen,† G. G. Touchard,‡ and C. J. Radke*,† Department of Chemical Engineering, University of California, Berkeley, California 94720, and Laboratorie de Physique et Mecanique des Fluides, Universite de Poitiers, Poitiers 86022, France

A metal/liquid interface corrosion-reaction model is developed for the flow electrification of lowconductivity liquids in metal pipes. In the proposed model, impurity anions participate in a corrosion reaction at the wall, leaving a net positive ion concentration in the diffuse electrical double layer. Convection of this positive charge constitutes the streaming current. Theoretical calculations for the convected space charge density demonstrate a velocity-dependent entrance effect that diminishes in pipes of larger radii, in agreement with experimental data for heptane in stainless steel pipes. Far downstream, the proposed model also correctly predicts that the convected space charge density falls with increasing pipe radius. As in previous work, the convected space charge density far downstream, (I/Qecb)∞, is found to be linear with the ζ-potential. However, the proposed model is self-consistent in that the ζ-potential arises as part of the calculation and is not an adjustable constant characteristic only of the metal/ hydrocarbon interface. In the entrance region, the convected space charge density is assumed to vary exponentially with axial position with the form (I/Qecb) ) (I/Qecb)∞ - A(I/Qecb)1 × exp(-Rσ), where A is a preexponential factor, (I/Qecb)1 is a deviation function, R is a characteristic eigenvalue, and σ is the dimensionless axial coordinate. With a known value of 79.8 µm for the solution Debye length, calculations show A to be 0.075, and (I/Qecb)1 and R to be 7.98 (10-4), 1.42 (10-4), and 2.60 (10-5) and 1.242, 2.425, and 2.938, respectively, for pipe radii of 0.24, 0.58, and 1.25 mm, respectively. Introduction When a hydrocarbon liquid flows through a metal pipe, an axial streaming current is generated, convected along the pipe, and spilled out into a collection vessel. This “flow electrification” phenomenon is illustrated in Figure 1. Since the liquid conductivity is very low, little backmigration of current occurs. Instead, a difference in potential arises between the pipe and the receiving vessel that drives a return current from the vessel back to the pipe, provided an electrical connection is available. At steady state, the streaming and wall currents match. Flow electrification raises explosion concerns in the petroleum industry, and extensive measurements have been made (Boumans, 1957; Klinkenberg and van der Minne, 1958; Gavis and Wagner, 1968; Koszman and Gavis, 1962; Gibbings, 1970; Touchard and Romat, 1981, 1982; Walmsley and Woodford, 1981; Touchard and Dumargue, 1983). Early research by Klinkenberg and van der Minne (1958) concluded that impurity ions in the hydrocarbon liquid give rise to flow electrification and that, for a linear velocity profile near the wall, the streaming current is directly proportional to the potential difference between the bulk liquid and the liquid immediately next to the wall, the ζ-potential. Later, Koszman and Gavis (1962) attempted to quantify this potential based on more fundamental properties. Still later, the mechanism of charge formation was found to depend strongly on the charge density of impurity ions at the inner tube wall and that flow electrification may be characterized by the convection of the equilibrium electrical double layer created by these ions at the pipe-liquid interface * To whom correspondence should be addressed. Phone: 510-642-5204. FAX: 510-642-4778. Email: radke@ cchem.berkeley.edu. † University of California. ‡ Universite de Poitiers.

S0888-5885(96)00021-8 CCC: $12.00

Figure 1. Schematic of steady flow electrification. Is and Iw denote the streaming and wall currents, respectively.

(Touchard and Romat, 1982; Touchard and Dumargue, 1983). Along with the parameters of tube radius and impurity ion diffusivity, Touchard and Dumargue (1983) were able to quantify the streaming current with a specified wall volume charge density in place of the ζ-potential. Meanwhile, Walmsley and Woodford (1981) attempted to explain the formation of the streaming current through differences in impurity anion and cation adsorption rates at the wall by defining a net wall flux for ion injection or loss. However, the actual injection or loss mechanism was not specified. To date, no selfconsistent theory has been proposed to explain flow electrification based on accepted electrochemical principles. Convected space charge density, or the ratio of the generated streaming current, I, to the volumetric flow rate, Q, has been measured as a function of flow velocity. Figure 2 displays, as a function of Reynolds number, experimental I/Q data for heptane in laminar flow through three stainless steel tubes, all 4 m in length but of varying diameter (Touchard and Romat, 1981). Two important features of laminar flow electrification are highlighted in this figure. First, I/Q depends inversely on velocity for the 0.24 mm radius tube, suggesting an entrance length of several meters (Tou© 1996 American Chemical Society

3196 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

Figure 4. Proposed corrosion mechanism for the origin of flow electrification. Figure 2. Measured convected space charge density for heptane in stainless steel tubes at ambient temperature.

and metal surface charge densities

(

qw ) q2 + 1 +

Figure 3. Electrical double-layer structure. The left side portrays the electrostatic potential profile, and the right side pictures the interface ionic structure.

chard and Romat, 1981). As shown, this entrance length diminishes in larger diameter tubes. Second, the convected charge density at zero velocity (which corresponds to I/Q for pipes of infinite length) also falls with increasing pipe radius. Our purpose is to quantify the effects shown in Figure 2 based on a metal-liquid interface corrosion-reaction model. Physicochemical Model From previous studies, it is apparent that unavoidable ionic impurity species play a crucial role in flow electrification (Boumans, 1957; Klinkenberg and van der Minne, 1958; Gavis and Wagner, 1968; Koszman and Gavis, 1962; Gibbings, 1970; Touchard and Romat, 1981, 1982; Walmsley and Woodford, 1981; Touchard and Dumargue, 1983). If the metallic wall acquires a charge, the solution ions distribute in a diffuse electrical double layer whose extent is characterized by the Debye length (Newman, 1991)

λb )

( ) kBT

∑i

e2

∫0RF2πr dr

δ q )R m

)

In eq 2, δ is the Stern plane thickness, r is the tube radial coordinate, R + δ is the tube radius, and F ) ∑izieci is the net local solution volume charge density. Since the whole system is electrically neutral, a charge equal to the negative of qw must exist in the diffuse double layer, as portrayed by the last term in eq 2. Provided that axial diffusion and migration are negligible compared to convection, flow of the entrained charges in the diffuse double layer constitutes the observed streaming current. By assuming a 1:1 impurity electrolyte, conductivity and diffusion coefficient measurements on the heptane used for Figure 2 yield an ionic diffusion coefficient of 1.55 × 10-9 m2/s and a calculated Debye length of almost 80 µm (Touchard and Romat, 1981), confirming a weak ionic character for the nonpolar liquid. From the definition of the Debye length, the bulk concentration cb of dissociated impurity ions in the liquid entering the tube is established to be 2.14 × 1014 ions/m3 (3.55 × 10-13 kmol/m3). In light of this, we propose the physical picture in Figure 4 for the origin of flow electrification. A very dilute impurity species, AB, partially to fully dissociates into ions in the liquid according to the equilibrium reaction

AB a A+ + B-

(1)

zi2cib

where  is the dielectric permittivity of the solution, kB is Boltzmann’s constant, e is the electron charge, zi is the valence of ionic species i, and cib is the bulk concentration of impurity ionic species i. As diagrammed in Figure 3, solution anions preferentially adsorb into the Stern plane immediately adjacent to the metal interface, giving rise to a surface charge density, q2. Since a surface charge density qm in the metal phase may also be present, it is useful to define a wall surface charge density from Gauss’s law, qw, to be the geometrically weighted sum of the adsorbed Stern plane

(3)

Although all three species may adsorb, it is assumed (similar to Klinkenberg and van der Minne, 1958) that the anion from the dissociation preferentially adsorbs at the metal wall. Subsequently, B- reacts with the metal to produce a neutral adsorbed species MeB

B- + Me a MeB + e-

1/2

(2)

(4)

In turn, MeB desorbs from the surface and partially dissociates into Me+ and B- ions. The amount of solution ions produced by this second dissociation reaction is considered negligible. The result of these two reactions is a net loss of negative charge from the liquid. Hence, there is a net positive charge in the flowing stream, giving rise to a wall charge and a streaming current, expressed as

Is )

∫0RFvz(r) 2πr dr

(5)

where vz is the axial laminar flow velocity. Equation 5 may be nondimensionalized to read

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 3197

∫01[C+(ξ) - C-(ξ)](1 - ξ2)ξ dξ

I )4 Qecb

(6)

where ξ ) r/R is the dimensionless tube radial coordinate and Ci ) ci/cb is the local dimensionless concentration of A+ or B- ions. The quadratic factor in eq 6 corresponds to the Hagen-Poiseuille parabolic velocity profile. There is essentially no axial flow of current in the metal pipe, and the electrostatic potential of the metal, φm, is taken as a constant (see Figure 3). To complete the electrochemical circuit, cations flowing out of the pipe undergo a reduction reaction to plate out on the walls of the collection vessel. Hence, flow electrification involves a corrosion/plating couple in which the pipe is corroded and the collection vessel is plated. Electrons generated by oxidation of the metal in eq 4 produce the return wall current experimentally observed to balance the streaming current at steady state (Touchard and Romat, 1981), Is ) Iw ) I. Due to mass-transfer and/or kinetic limitations, the wall current and the accompanying solution electrical double layer are distributed axially, as well as radially, in the entrance region of the tube. The wall current is related to N-,r, the radial flux of anions to the wall by

Iw )

∫0

L

eN-,r(R,z) 2πR dz

(7)

where z is the axial tube coordinate. Near the tube entrance, a large wall current density, eN-,r(R,z), is present due to the high concentration and radial flux of B-. Far downstream, however, the wall flux of Bdrops to zero as the corrosion reaction reaches equilibrium with the solution ions. Beyond this location, the streaming current and electrical double layer are independent of pipe length. The value of the streaming current in this situation is denoted as I∞. In Figure 2, (I/Q)∞ corresponds to the convected charge density in the limit of zero Reynolds number. Figure 5 plots (I/ Q)∞ from Figure 2 in open squares, along with low Reynolds number I/Q measured by Touchard and Dumargue (1983) in filled squares, against the dimensionless pipe radius R/λb. The fully developed streaming current varies inversely by approximately the square of the tube radius (Touchard and Dumargue, 1983), as illustrated by the dashed line drawn in Figure 5. The problem may now be divided into two parts: the fully developed, far-downstream current corresponding to the zero Reynolds number intercepts in Figure 2 and the axially evolving streaming current corresponding to the velocity dependence in Figure 2. Each part receives separate treatment below. Fully Developed Streaming Current By removing negative charges from the liquid, the corrosion reaction in eq 4 demands a compensating wall surface charge density. Thus, qw (or equivalently the ζ-potential) is not an independent parameter but must be solved as part of the flow electrification model. In the fully developed section of the pipe, all solution species exhibit zero radial flux, and the ionic species are thus distributed radially to achieve equilibrium with the wall charge. However, the equilibrium value of the charge is unknown. Because of the overall reaction scheme portrayed in Figure 4, the concentrations of solution species change axially along the pipe. Thus, to establish the fardownstream wall surface charge density, and hence the streaming current I∞, we write atom balances on the

Figure 5. Measured far-downstream convected space charge density with cb ) 2.14 × 1014 m-3.

impurity species A and B in all molecular forms over a control volume between the tube inlet where C+ ) C) 1 and the far-downstream position where species concentrations no longer change axially. These read in order

2〈CAB + C+(ξ) (1 - ξ2)〉 ) 1 + CAB0

(8)

2〈[CAB + C-(ξ) + C](1 - ξ2)〉 ) 1 + CAB0

(9)

where CAB ) cAB/cb, C ) cMeB/cb, diagonal braces 〈 〉 denote radial cross-section averaging, and the superscript 0 denotes the inlet value. Thus, in eq 8 A atoms must be accounted for in the species AB and A+; similarly in eq 9 B atoms appear in species AB, B-, and MeB. Aside from the mass balances, the PoissonBoltzmann equation in linearized form for small electrical potentials is necessary

[ ] ()

R 2 1 d dΦ ξ + [D0 - S0Φ] ) 0 ξ dξ dξ λb

(10)

In eq 10, Φ ≡ e(φ - φ0)/kBT is the reduced electrostatic potential difference between a radial position and the centerline, while D0 ≡ (C+,0 - C-,0)/2 and S0 ≡ (C+,0 + C-,0)/2 are the difference and the sum of the centerline anion and cation concentrations, respectively, with the subscript zero denoting the centerline value. In cylindrical geometry, the solution to the linearized PoissonBoltzmann equation involves modified Bessel functions In of order n (Newman, 1991)

Φ(ξ) )

( [ ])

RxS0 D0 1 - I0 ξ S0 λb

(11)

Again, the anion and cation solution concentrations far downstream vary with radius only and obey the linearized Boltzmann distribution

[ ( [ ])]

C( ) C(,0[1 - Φ] ) C(,0 1 -

RxS0 D0 1 - I0 ξ S0 λb

(12)

Conversely, the concentration of neutral species MeB outside the entrance region is constant across the tube cross section and obeys the equilibrium relation of eq 4. For simple adsorption isotherms, such as multicomponent Langmuir, eq 4 can be expressed as

KS )

[

]

-e(φm - φ2) C(ξ)1) exp k BT C-(ξ)1)

(13)

3198 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

where KS is the equilibrium constant that includes the standard Gibbs adsorption free energies of B- and MeB in addition to the standard Gibbs reaction free energy of eq 4. The exponential factor accounts for the electrostatic energy change for the charge-transfer reaction between metal ions Me+ at the metal surface and the adsorbed anions B- at the Stern plane. Appendix A demonstrates that, because the Stern plane thickness is miniscule compared to the tube radius, the exponential term in eq 13 reduces to unity. Further, since C-(1) follows from eq 12, we utilize eq 13 in the form

KS )

C C-,0(1 + Φ2)

(14)

where in the fully developed region, C(1) ) C∞, a constant independent of the radial coordinate. Because of the simplicity of eq 14, it is not necessary to consider separately the surface charge densities at the Stern plane and at the metal surface. Similarly, partial dissociation of the impurity species AB is characterized by the equilibrium constant KAB such that

KAB ) C+C-/CAB

(15)

Since the initial concentration of AB is equal to cb2/KAB, the total initial concentration of neutral and ionic impurity species cT0 may be defined as

cT0 cb )1+ cb KAB

(16)

The ratio of cb to cT0 is the fraction of dissociation. Because cb is fixed from the experimental conductivity measurements, different values of the dissociation equilibrium constant KAB correspond to different total inlet impurity concentrations. Substitution of eqs 11, 12, 14, and 15 into the atom balances of eqs 8 and 9 followed by radial integration over the tube cross section yields, in turn, two nonlinear algebraic relations in the unknowns S0 and D0 (or equivalently in C(,0)

[ ( ) ]( ) ( )( )( ) [ ]

S0 1 -

D0 S0

2

1+

cbS0 + KAB

RxS0 D0 D0 λb cb 1+ I2 - 1 ) 0 (17) S0 S0 R λb KAB 2

8

and

[ ( ) ]( ) ( )( )( ) [ ] ( )( [ ])

S0 1 -

D0 S0

2

1+

cbS0 KAB

RxS0 D0 D0 λb 1I2 + 8 S0 S0 R λb 2

KSS0 1 -

D0 D0 D0 RxS0 1+ - I S0 S0 S0 0 λb

-

cb -1)0 KAB (18)

which are solved by Newton iteration as functions of the three problem parameters: the corrosion equilibrium constant KS, the dimensionless dissociation equilibrium constant KAB/cb, and the ratio of the tube radius to the Debye length, R/λb. Equation 6 then allows calculation of the far-downstream dimensionless charge density convected, (I/Qecb)∞

( ) I Qecb

∞

() [ ]

RxS0 D0 λb I2 S0 R λb 2

) 16

(19)

Finally, the far-downstream wall surface charge density may be calculated from Gauss’s law, qw ) (dφ/dr)R, and eq 11

[ ]

RxS0 qw D0 )I1 2ecbλb λb xS0

(20)

Thus, once D0 and S0 are known from eqs 17 and 18, the wall surface charge density and the ζ-potential, defined here as φ2 - φ0, follow in a self-consistent manner. An asymptotic analysis of eq 6 for large R/λb uncovers the thin double-layer result of Klinkenberg and van der Minne (1958) that I/Q ) -8ζ/R2. The Klinkenberg and van der Minne equation is deceptively simple. However, it lacks explanation for the electrochemical origin of the ζ-potential. According to the present corrosion model, the ζ-potential and the wall surface charge density qw are not constants, characteristic only of the metalsolution interface, but rather vary with system parameters, and in particular with the tube radius, as discussed below. Figures 6 and 7 graph calculated corrosion model values of the fully developed ζ-potential and the wall surface charge density, qw, respectively, as functions of R/λb, for a fixed dissociation equilibrium constant KAB and for several corrosion equilibrium constants KS. Both the ζ-potential and the wall surface charge are negative, as demanded by the corrosion reaction. In Figure 6, the magnitude of the ζ-potential rises, eventually to a unity plateau as R/λb increases. This rise is a strong function of increasing KS up to values of 103. As the tube radius increases with a fixed λb, the relative extent of the diffuse double layer shrinks and solution cations compact more closely to the surface. By electroneutrality, the concentration of cations outside the double layer must fall to match the loss of anions due to reaction. These two effects demand a more negative potential drop across the diffuse double layer. Likewise, as more B- converts to MeB at larger KS, the diffuse double layer contains more net positive charge and therefore exhibits a more negative potential difference. The linearized Poisson-Boltzmann equation (10) is valid only for potential values of |Φ2| , 1, which is not the case for large R/λb values in Figure 6. As noted by the dashed portions of the curves in Figure 6 (and 7), our linearized corrosion flow electrification model is strictly valid only for small R/λb, especially when KS is large. Hence, the asymptotic limit of -Φ2 ) 1 in Figure 6 is suspect physically. Figure 7 demonstrates the remarkably small wall surface charge densities that occur during flow electrification. Typical charge densities on the surface are 1 in every 100 µm2. As with the ζ-potential, qw increases in magnitude with increasing KS, and for the same reason. The nonmonotonic behavior of qw for larger KS falls outside the range of the linearized PoissonBoltzmann equation and is thus likely aphysical. Figures 8 and 9 display as solid lines the calculated far-downstream dimensionless convected charge density, (I/Qecb)∞, as a function of R/λb for several values of KS. Figure 8 corresponds to a dissociation equilibrium constant KAB of 1013 m-3 (KAB/cb ) 0.0467), whereas Figure 9 reflects a KAB of 1015 m-3 (KAB/cb ) 4.67). These figures dramatically confirm the experimental

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 3199

Figure 6. Calculated far-downstream nondimensional ζ-potential with cb ) 2.14 × 1014 m-3.

Figure 7. Calculated far-downstream wall charge area density with cb ) 2.14 × 1014 m-3.

Figure 8. Linearized corrosion model for far-downstream streaming current with 4.5% dissociation of the impurity species and cb ) 2.14 × 1014 m-3.

trend of a decreasing equilibrium streaming current I∞ for larger tube radii. The reason is that increased R/λb values lead to more negative ζ-potentials (cf. Figure 6) and hence less conversion of B- to MeB, in obedience to the corrosion-reaction equilibrium expression in eq 14. In addition, a large R/λb confines the region of net solution volume charge density close to the wall in the slower portions of the parabolic velocity profile. Conversely, larger KS yields higher equilibrium conversions up to an asymptotic limit that depends on KAB. As seen by comparing Figures 8 and 9, there is a strong dependence of (I/Qecb)∞ on the impurity dissociation equilibrium constant between the values of 1013 and 1015 m-3. Outside these limits, little further change is seen. These two values of KAB correspond to 4.5 and 82.3% dissociation of the impurity AB species, respectively. Lower values of KAB thus lead to higher streaming currents because there is a reservoir of additional anions available in the undissociated species AB. The

Figure 9. Linearized corrosion model for far-downstream streaming current with 82.3% dissociation of the impurity species and cb ) 2.14 × 1014 m-3.

converse reasoning holds for large values of KAB. With complete dissociation of AB (i.e., KAB g 1017), the largest possible dimensionless convected charge density is unity. This limit reflects complete conversion of all impurity anions in the flowing hydrocarbon. Open and filled squares in Figures 8 and 9 give the experimental data for (I/Qecb)∞ of Touchard and Romat (1981) and Touchard and Dumargue (1983) (see Figure 5). Agreement between theory and experiment is not quantitative. Clearly, small fractional dissociation of the impurity species AB more closely represents the data. However, even with the small fractional dissociations of the impurity species (i.e., small KAB) and the largest conversions of the corrosion reaction (i.e., large KS), we underestimate the measured data. Several explanations are possible. First, the partially dissociating impurity species may not be symmetric but rather a 1:2 or 2:1 electrolyte, although Klinkenberg and van der Minne (1958) argue that this is unlikely. Second, linearization of the Poisson-Boltzmann equation is not adequate, and a full nonlinear solution is required. Using the simple, thin double-layer approximation (i.e., R/λb g 10) of Klinkenberg and van der Minne (1958), we estimate the ζ-potential from Figure 2 to be -100 mV. This is well outside the linear region of the Boltzmann distribution. The calculations of Touchard and Dumargue (1983) also confirm the necessity to consider the full nonlinear Poisson-Boltzmann equation. Nevertheless, even the linearized form of the proposed corrosion model for flow electrification contains the essential features of the measured fully developed streaming currents in the laminar regime. Entrance Region Streaming Current For short metal tubes, measured streaming currents increase with increased pipe length until the fully developed length is reached (Touchard and Romat, 1982). Hydrodynamic entrance lengths for development of the parabolic velocity profile (i.e., L/R e 0.02Re) are much smaller than measured flow electrification entrance lengths. Thus, the origin of the flow electrification entrance length lies in the axial development of the wall charge and the concomitant diffuse double layer. According to the proposed corrosion model, any masstransfer and/or kinetic resistances to the corrosion reaction and the net depletion of anions from the liquid may originate an entrance length. Thus, resistances to diffusion of B- to and MeB away from the wall or finite kinetics of B- adsorption at the wall, of MeB desorption from the wall, and of the wall corrosion reaction in eq 4 are candidates to account for an entrance length.

3200 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

(1 - ξ2)

Figure 10. Mass-transfer-limited Graetz solution with cb ) 2.51 × 1015 m-3.

Previously, Touchard et al. (1996) proposed a reaction kinetic resistance. Here, we reexamine the issue. The most explicit resistance is that of diffusion of reacting species to and from the wall. If we ignore for the moment the complication of migration and if we assume that the impurity species is fully dissociated and instantaneously reacts completely at the wall so that (I/Qecb)∞ is unity, then the well-known Graetz solution for laminar mass transfer in the entrance region of a cylindrical pipe applies (Graetz, 1883; Newman, 1991)

[(

where axial diffusion is neglected compared to convection since the Peclet number in the experiments of Figure 2 is on the order of 104. Implicit in eq 22 and below is the assumption that D+ ) D- ) D. The second term on the right of this expression reflects the electrical migration flux. For simplicity in the entrance-region analysis we assume the impurity species to be completely dissociated so there is no undissociated AB species in the fluid stream. Provided again that the Peclet number is large, Poisson’s equation in the entrance region reads

[ ] ()

1 ∂ ∂Φ 1 R 2 ξ + [C+ - C-] ) 0 ξ ∂ξ ∂ξ 2 λb

Qecb

(23)

It is not possible to impose the Boltzmann distribution, since the diffuse electrical double layer in the entrance region is not at equilibrium. Equations 22 and 23 must now be solved subject to appropriate initial and boundary conditions. At the pipe centerline we impose symmetry

∂U/∂ξ ) 0 I

)]

∂Ci ∂Ci 1 ∂ ∂Φ ) + ziCi ξ ∂σ ξ ∂ξ ∂ξ ∂ξ i ) A+, B-, MeB (22)

at ξ ) 0

(24)

∞

)1-

∑ Mk exp(-Rk2σ)

(21)

k)1

where σ ) Dz/2〈v〉R2 is a dimensionless axial coordinate, or equivalently the inverse of the Graetz number, with 〈v〉 representing the average axial flow velocity and D representing the molecular diffusivity. We adopt the measured diffusivity (Touchard and Romat, 1981) of D+ ) D- ) D ) 1.55 × 10-9 m2/s for the impurity ions in the heptane liquid of Figure 2. Symbols Mk and Rk are the tabulated coefficients and eigenvalues, respectively, of the Graetz problem (Newman, 1991). Far downstream, the dimensionless convected space charge density asymptotes to unity, consistent with KS, KAB, and λb all approaching infinity in the corrosion model. At the entrance to the pipe, the convected space charge density falls to zero. Figure 10 compares eq 21 to the experimental data of Figure 2. Excellent agreement is seen for the 0.24 mm radius tube, partly due to the adjustment of cb to 2.51 × 1015 ions/m3 to fit the zero Reynolds number value of I/Q. Agreement is poor for the 0.58 and 1.25 mm tubes both for the fully developed streaming current (i.e., the zero Re intercept in Figure 10) and for the entrance-region behavior. The reason, of course, is the neglect of any electrical double-layer phenomena in the Graetz analysis. Nevertheless, Figure 10 reveals the important conclusion that the entrance length in laminar flow electrification is most likely due to masstransfer resistances. Thus, we take the corrosion reaction at the wall to be in equilibrium so that eq 14 applies even in the entrance region. Equation 21 also teaches that the Reynolds number plays no direct role in laminar flow electrification. Rather, the pertinent scaling variable is the inverse Graetz number: σ ) Dz/ 2〈v〉R2. To extend the Graetz analysis, we include electrical migration in the convected diffusion equation for all ionic solution species

where the symbol U(σ,ξ) stands generically for the variables C+, C-, C, and Φ. At the wall, there is no flux of the cations A+

∂C+ ∂Φ + C+ )0 ∂ξ ∂ξ

at ξ ) 1

(25)

whereas the fluxes of B- and MeB balance

∂C∂Φ ∂C - C+ )0 ∂ξ ∂ξ ∂ξ

at ξ ) 1

(26)

Finally, equilibrium between B- and MeB is demanded at the wall so that eq 14 applies. At the entrance to the pipe, we note that

C+ ) C- ) 1 and C ) Φ ) 0

at ζ ) 0 (27)

where we assume that no current enters the tube. Mathematical solution of the entrance-region equations thus presents a difficult numerical problem. Rather than attempt a full numerical solution, we investigate the behavior of the entrance region as it asymptotes to the fully developed section of the pipe. We seek a solution to eqs 22 and 23 of the following form:

U(σ,ξ) ) U∞(ξ) - AU1(ξ) exp(-Rσ)

(28)

This expression is suggested both from the Graetz analysis and from the empirical observation of Touchard and Romat (1981) that the wall volume charge density Fw ) F(ξ)1) should increase exponentially as a ratio of axial distance to average flow velocity. This ratio is confirmed in the inverse Graetz number σ. The prefactor A denotes an adjustable scaling constant, while R is an eigenvalue characteristic of the far-downstream conditions. The functions U∞(ξ) are known from the fully developed model described above. However, R must be ascertained as part of the problem. Thus, eqs

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 3201

mon experimental observation of an entrance region where the wall charge, wall current, and electrical double layer all evolve in axial distance. Further, we correctly predict a fully developed section of the pipe in which the convected space charge density falls with increasing pipe radius. The proposed model is selfconsistent in that the wall charge density and the ζ-potential arise naturally as part of the calculation and need not be specified independently. Acknowledgment

Figure 11. Linearized corrosion model for flow electrification in the entrance region with cb ) 2.51 × 1015 m-3, KS ) 0.75, and A ) 0.075, where ((I/Qecb)1, R) ) (7.98 × 10-4, 1.242) for the 0.24 mm tube, (1.42 × 10-4, 2.425) for the 0.58 mm tube, and (2.60 × 10-5, 2.938) for the 1.25 mm tube.

14 and 22-28 are combined with the restriction that (Newman, 1973)

dR/dξ ) 0

(29)

and solved simultaneously by finite differences and Newton iteration for the deviation variables U1(ξ) and R. Appendix B gives more detail on the numerical procedures. The local streaming current follows from eq 6. Figure 11 compares the asymptotic entrance-region analysis of the proposed corrosion model to the experimental data of Figure 2. The bulk concentration of completely dissociated impurity ions has been adjusted for the 0.24 mm radius tube, due to the inadequacy of the linearized Boltzmann distribution to predict quantitatively the experimental fully developed streaming currents (cf. Figures 8 and 9). Excellent agreement is seen for the 0.58 and 1.25 mm tubes in that the fully developed convected space charge density diminishes correctly for larger R and a much smaller velocity dependence is correctly predicted for larger R. Entrance behavior at the 0.24 mm tube is not sterling but is easily understood as eq 28 is not valid close to the tube inlet. Many eigenvalues are needed in eq 21 to capture the correct curvature of the experimental I versus 〈v〉 data for the 0.24 mm tube (cf. Figure 10). Figure 11 lends strong credence to the basic validity of the corrosionreaction model and to diffusion resistance as the origin of flow electrification entrance behavior. A deficiency of the present corrosion model is the linear approximation to the Boltzmann distribution. A full nonlinear analysis is warranted but involved. The wall corrosion reaction demands that metal corrosion products appear in the effluent stream. No direct evidence for this assertion is currently available and is required to confirm the basic electrochemical picture of the model (see Figure 4). Direct measurement of any dissolved metal will be difficult because of the extremely small species concentrations expected in flow electrification. Conversely, it may be possible to observe the plating out reaction in the collection vessel. Conclusions A physicochemical corrosion model is proposed to explain the origin of electrification of low-conductivity liquids flowing in metal pipes. In essence, a streaming current is generated by dissolution of the metal pipe. With this electrochemical picture and the assumption of mass-transfer control, we correctly predict the com-

H.C. performed this work as an undergraduate researcher at the University of California. C.J.R. thanks the University of Poitiers for a summer visiting scholar appointment. We thank John Newman for helpful comments. Nomenclature A ) scaling constant in eq 28 cb ) bulk concentration of dissociated 1:1 impurity ions, m-3 ci ) species i concentration, m-3 cib ) concentration of bulk species i, m-3 C ) cMeB/cb, reduced MeB concentration Ci ) ci/cb, reduced species concentration Di ) species i diffusion coefficient, m2/s D0 ) (C+,0 - C-,0)/2, difference in the centerline anions and cation-reduced concentrations e ) charge on electron, C I ) current, A kB ) Boltzmann’s constant, J/K KAB ) equilibrium constant for the dissociation reaction, m-3 KS ) equilibrium constant for the corrosion reaction L ) tube length, m Mk ) coefficient of Graetz solution Ni,r ) radial flux of species i, m-2 s-1 Q ) volumetric flow rate, m3/s q ) charge per area, C/m2 R ) tube radius less the Stern plane thickness, m r ) radial coordinate, m Re ) Reynolds number based on a kinematic viscosity of 4.45 (10-7) m2/s for heptane S0 ) (C+,0 + C-,0)/2, sum of the centerline anions and cation-reduced concentrations T ) absolute temperature, K U ) generic variable for C+, C-, C, and Φ vz ) axial velocity, m/s 〈v〉 ) average axial velocity, m/s z ) axial coordinate, m zi ) valence of ionic species i Greek Letters R ) eigenvalue of entrance-region problem Rk ) eigenvalue of Graetz solution δ ) Stern plane thickness, m  ) dielectric permittivity of the solution, F/m φ ) potential, V Φ ) e(φ - φ0)/kBT, reduced potential λb ) Debye length, m F ) net volume charge density, C/m3 ξ ) r/R, dimensionless radial coordinate ζ ) potential difference between bulk liquid and Stern plane, V σ ) Dz/2〈v〉R2, inverse Graetz number Subscripts b ) bulk i ) species m ) metal s ) streaming

3202 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

dCi,1 dΦ∞ dΦ1 dC1 + ziCi,1 + ziCi,∞ + )0 dξ dξ dξ dξ at ξ ) 1 and i ) B+ (B4)

w ) wall 0 ) centerline 1 ) deviation variables in eq 28 2 ) Stern plane + ) cation - ) anion ∞ ) far downstream Superscripts 0 ) initial

Appendix A. Exponential Factor in the Corrosion-Reaction Equilibrium Constant In this appendix we explore the magnitude of the exponential factor in eq 13 of the text. Inside the Stern plane, the net volume charge density is zero. Thus, the Poisson equation is

1 d dφ r )0 r dr dr

( )

RereR+δ

(A1)

In this appendix we rescale the electrostatic potential such that Φ ) e(φ - φ2)/kBT, giving Φ(1) ) 0. Additionally, C1(1) ) KSC-,1(1). We cannot impose the initial conditions because of eq 28. This equation set along with eq 29 of the text are central finite differenced and solved by Newton iteration following the procedures of Newman (1973). Since typical differences in centerline concentrations are small (D0 e 0.1), the initial guesses of cation and anion concentrations, C+,1 and C-,1, start at unity at the centerline. The guesses for C+,1 and C-,1 then increase and decrease linearly with position to the wall, respectively. The initial guess of R is also unity, as we desire the smallest possible value of R for the given parameters. Calculations require about four to five iterations and time on the order of hours on an IBM Powerstation 320 computer with a mesh size of 0.005.

with the conditions

φ ) φ2

dφ qw ) and dr 

Literature Cited

at r ) R

(A2)

Analytic solution of eqs A1 and A2 followed by substitution of eq 20 to replace qw yields

() [ ] ( )

RxS0 e(φm - φ2) δ R D0 )I ln 1 + k BT λb S 1 λb R x 0

(A3)

For R/λb ) 15, KAB/cb ) 5, KS ) 103, and δ/R ) 5 × 10-7 (i.e., R ) 1.5 mm and δ ) 5 Å), we find that e(φm - φ2)/ kBT e O(0.1). Since the parameter values chosen in this assignment provide the largest value of the exponential factor, we neglect its contribution in the proposed corrosion model for flow electrification. Appendix B. Entrance Region Numerics The asymptotic form of eq 28 is substituted into eqs 22-26 to yield a set of nonlinear ordinary differential equations in the variables U1(ξ). The first term in eq 28, U∞(ξ), reflects the linearlized far-downstream solution for which analytical expressions in terms of Bessel functions are given in the text. For the solution species we find that

[(

)]

dCi,1 dΦ∞ dΦ1 1 ∂ + ziCi,1 + ziCi,∞ ξ + ξ ∂ξ dξ dξ dξ 2 + i ) A , B-, MeB (B1) R(1 - ξ )Ci,1 ) 0 Next, Poisson’s equation reads

[ ] ()

dΦ1 1 R 2 1 d ξ + [C+,1 - C-,1] ) 0 ξ dξ dξ 2 λb

(B2)

Boumans, A. A. Streaming Current in Turbulent Flows and Metal Capillaries. Physica 1957, XXIII, 1007-1055. Gavis, J.; Wagner, J. P. Electrical charge generation during flow of hydrocarbons through microporous media. Chem. Eng. Sci. 1968, 23 (4), 381-391. Gibbings, J. C. Electrostatic charging in the laminar flow in pipes of varying length. J. Electroanal. Chem. 1970, 25, 497-504. Graetz, L. Ueber die warmeleitungsfahigkeit von flussigkeiten. Ann. Phys. Chem. 1883, 18, 79-94. Klinkenberg, A.; van der Minne, J. L. Electrostatics in the Petroleum Industry (The Prevention of Explosion Hazards); Elsevier: New York, 1958. Koszman, I.; Gavis, J. Development of charges in low conductivity liquids flowing past surfaces. Chem. Eng. Sci. 1962, 17, 10131023. Newman, J. The Fundamental Principles of Current Distribution and Mass Transport in Electrochemical Cells. In A. J., Bard, ed.; Electroanalytical Chemistry; Marcel Dekker: New York, 1973; Vol. 6, pp 189-352. Newman, J. Electrochemical Systems; Prentice-Hall: Englewood Cliffs, NJ, 1991. Touchard, G. G.; Romat, H. Electrostatic charges convected by flow of a dielectric liquid through pipes of different length and different radii. J. Electrost. 1981, 10, 275-281. Touchard, G. G.; Romat, H. Mechanism of charge formation in the double layer appearing at a hydrocarbon liquid-metal interface. J. Electrost. 1982, 12, 377-382. Touchard, G. G.; Dumargue, P. Streaming current in stainless steel and nickel pipes for heptane and hexane flows. J. Electrost. 1983, 14, 209-223. Touchard, G. G.; Patzek, T. W.; Radke, C. J. A physicochemical explanation for flow electrification in low conductivity liquids in contact with a corroding wall. To appear in IEEE Trans. Ind. Appl. 1996, 32 (5), Sept/Oct. Walmsley, H. L.; Woodford, G. The generation of electric currents by the laminar flow of dielectric liquids. J. Phys. D 1981, 14, 1761-1782.

At the centerline, ξ ) 0 and dU1/dξ ) 0, while at the wall we have that

Received for review January 23, 1996 Revised manuscript received April 24, 1996 Accepted April 27, 1996X

dCi,1 dΦ∞ dΦ1 + ziCi,1 + ziCi,∞ )0 dξ dξ dξ at ξ ) 1 and i ) A+ (B3)

IE960021V

and

X Abstract published in Advance ACS Abstracts, August 15, 1996.