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On the rate of discharge of electrically charged hydrocarbon liquids
Chemical Engineering
Science, 1965, Vol. 20, pp. 923-930. Pergamon Press Ltd., Oxford. Printed in Great Britain.
On the rate of discharge of electrically .charged hydrocarbon liquids S. Bataafse Internationale
J. VELLENGA and A. KLINKENBERG
Petroleum
Maatschappij,
N.V. (Royal Dutch/Shell
Group), The Hague, Holland
(Received4 March 1965) Abstract--It is well known that the discharge of an electrically charged hydrocarbon liquid need not follow an exponential law. The present paper shows that this process can still be correctly described mathematically with the aid of Ohm’s Law. Recently a new theory for static relaxation has been put forward. It has been claimed that this “hyperbolic theory” better describes relaxation of charge from high resistivity fuels than is possible with the “ohmic theory”, by which is meant a theory based on the assumption of constant conductivity. However, Ohm’s Law does not require that the conductivity shall be constant. It is shown in the present paper that the hyperbolic theory is a special case of a more general treatment. Furthermore it is shown that by taking association and dissociation effects into account it is possible to describe certain types of relaxation curves which cannot be described by either the hyperbolic theory or the constant conductivity theory.
where D, is the component of the vector D normal to the element of surface, dA. This Dn is taken as positive in the outward direction. On the other hand, the rate of loss of charge from the element is equal to the surface integral of the current density J
BASIC CONCEPTS OIL products often acquire an electric charge during handling and from the point of view of safety it is important to know the rate at which such products lose their charge after operations have ceased. This mainly depends on the product in question and on the degree of charging. However the manner in which the product has become charged also plays a part. This will be further discussed in the following. A basic assumption in electric theory is the validity of Ohm’s Law. A general form of Ohm’s Law states that in a medium at rest and at a given point and time the current density J is proportional to the local field strength E
J = ICE
d(sV) - -= dt
By definition
where the factor z is known as the relaxation time. Therefore one has --=
(1)
The proportionality factor rc is called the conductivity and may be a function of both time and place. Now considering a small element of volume V, containing an electric charge sV, where s is the charge density, one can easily find the rate at which this element loses its charge. For, by definition, the charge enclosed within the surface of the element is equal to the surface integral of the dielectric displacement D sv =
D, dA
J,dA
Hence, if r can be taken constant over the surface of the element and ignoring transport of charge by diffusion or convection as required by the definition of Ohm’s Law, one obtains ds -= dt
--
s z
(2)
This differential equation will be basic in the following discussion. The above result is equally useful for a moving medium for convection effects can be ruled out
923
S. J. VELLENGA
and A. KLINKENBERG
by letting the element of volume move along with the liquid. This corresponds to the so-called Lagrangian description of motion in hydrodynamics. In the above a continuum has been implicitly assumed. This has considerably simplified the mathematical treatment and does not impose a great restriction in practice. In fact, the transport of electric charge through a liquid is usually described in terms of conductivity. It is also possible to describe the actual transport mechanism which consists in the bodily movement of electrically charged particles. If an electrically charged particle is placed in an electric field of strength E it will acquire a velocity c v=uE
current density in an electric field of strength E is J =C (CiZiFUi)E I
(3)
where c = n/N and where the index i refers to the i-th species. Comparing equations (1) and (3) it follows that the connection between the description in terms of conductivity and that in terms of movement of charged particles is K = ~ (ciZiFui)
(4)
The charge density of the liquid, s, can similarly be described as s = 7 (ci@) (5) For a binary monovalent electrolyte this reduces to
where the proportionality factor u is known as the mobility of the particle in the medium considered. If the charge is negative the same equation still applies if the sign is accounted for in the numerical value of U. Strictly speaking diffusion and convection should also be taken into account. However it is found that diffusion effects only become important in small apparatus or very near an interface where large concentration gradients occur so that in technical equipment the contribution of diffusion can usually be ignored [l]. Convection can be ruled out by the above-mentioned Lagrangian description. If, passing over to the macroscopic point of view, one places n electrically charged particles per unit of volume of an electrically indifferent fluid in an electric field of strength E, one finds that the quantity of charge passing per unit time through unit of area normal to the direction of the field is ziFv=z;FuE
where z is the number of elementary charges per particle, F is Faraday’s constant and N is Avogadro’s constant. This amounts to an electric current density. If the ‘charge happens to be negative there is a quantity of negative charge flowing in the negative direction which again represents a positive electric current. (In this case both the valency z and the mobility u have the negative sign). Thus, if there are a number of species of electrically charged particles, the
JC= (c’ + c-)uF
(assuming u+ = -u-
= u)
and s = (c’ - c-)F In a liquid at rest and in equilibrium or in a liquid under steady state conditions (e.g. one moving in a long line) the symbol ci on the right-hand side of equation (4) represents quantities which are constant. However this is no longer true under nonsteady state conditions. Charged particles or even whole species of charged particles may be removed or introduced through the interaction of the liquid with the surfaces past which it flows. This destroys the equilibrium between the association and dissociation reactions in the liquid. The net change in the rates of these reactions tends to restore the equilibrium but still the non-equilibrium conductivity may differ appreciably from the conductivity of the liquid at equilibrium. Even an uncharged liquid may therefore have a conductivity which differs from the equilibrium conductivity, namely when its ionic equilibrium has been recently disturbed. These effects have been found experimentally. In Shell publications the non-equilibrium conductivity is usually referred to as the in situ or efictive conductivity since it, and not the conductivity of the liquid at equilibrium, determines the rate of discharge of an electrically charged liquid P, 31. The right-hand side of equation (4) consists
924
On the rate of dischargeof electrically chargedhydrocarbonliquids
account for a charge in conductivity of
mostly of unknowns, certainly so in the nonequilibrium case. In order to be able to apply the insights which can be gained from the description in terms of charged particles one has to simplify drastically. Firstly, association and dissociation are not usually taken into account quantitatively. Secondly the discussion is almost invariably limited to two species of charged particles, having the same absolute value of the mobility, which is assumed constant, and with z = + 1 respectively. The latter assumption, that the ions are monovalent, is not a serious restriction in dilute solutions of low dielectric constant [4].
10-3
$
x
10-g
m2 -I/s -_ lo-l2 &
= 1 pmho/m
and this is of the order of values found in practice. One can consider, as an extreme case, that all ions of one sign are removed so that the effective conductivity is wholly due to the ions constituting the net charge density. (An alternative physical counterpart of the above model would be a liquid with zero rest conductivity to which ions of one sign are added). The conductivity now becomes rc = cFu
THE EXPONENTIALLAW OF RELAXATION
However, even with these restrictive assumptions, For various developments are still possible. instance, if the conductivity is not very low it is quite usual and often permissible to ignore the difference between the effective conductivity and the rest conductivity. The theory based on this assumption has been referred to as the ohmic theory, which is unfortunate since the application of Ohm’s Law does not require that the conductivity shall be constant. In this case the rate of discharge is given by equation (2) with r = constant. Integrating this equation the following relation is obtained s = s0 exp( - t/r)
(6)
This exponential law of relaxation describes the rate of discharge of an electrically charged liquid if the assumption of a constant conductivity holds. THE HYPERBOLICLAW OF RELAXATION The assumption that the conductivity is constant does certainly not hold when the total number of charged particles is so small that the change in concentrations which is needed to create a net charge density (see equation 5) affects significantly the right-hand side of equation (4). Comparing equations (4) and (5) it is seen that the contribution of an ionic species to the charge density, multiplied by its mobility, gives a contribution to the conductivity. Thus, if all ions have a mobility of 10eg m’/Vs, a charge density of O+OOlC/m3 will
and is no longer constant. Since, by equation (5) the charge density s = cF and since by definition z = se&c the equivalent to equation (2) is ds A=-dt
us2 EEL
which on integration yields 1 -=EE+-=s
ut
1
t/k,
+
1
(7) 0
so
so
where r. is the relaxation time at t = 0 and so is the initial charge density. Since the graph of equation (7) is a hyperbola one can refer to this equation as the h_vperboliclaw of relaxation. It describes the rate of discharge of an electrically charged liquid if the conductivity is directly proportional to the charge density. Note that when this law applies the liquid can lose its charge slower as well as faster than would be predicted by the exponential law of relaxation which assumes the effective conductivity to be constant and equal to the rest conductivity. If all ions of one sign are removed then the actual rate of relaxation will be slower than predicted by the exponential law; if the charge is due to ions being added to a liquid of negligible conductivity then, c$ course, the rate of relaxation is faster than predicted. For this latter case, see Ref. [5]. GENERALIZEDHYPERBOLICLAW OF RELAXATION
As a more general case consider a liquid in which the difference between the concentrations of
925
S. J. VELLENGA and A.
KLINKENBERG
of the electric relaxation of a given liquid is to plot the logarithm of the charge density against time. From equations (2) or (6) it follows immediately that this graph should be a straight line if the exponential law of relaxation is to be valid and that the negative slope of this line is inversely rc = (c’u’ - c-u-)F proportional to the relaxation time. and the charge density In practice such graphs are often obtained by connecting the output of a charge density meter to s= (C’ - c_)F a logarithmic recorder and it is then sometimes Assuming an excess of positive particles and found that such graphs are curved. If the convex assuming that the number of negative particles side of the curve is turned towards the time-axis remains constant, one can write (curve b in Fig. 1 or Fig. 11 of Ref [6]) this means that the relaxation time r increases during the fc = su+ + c-(U’ - u_)F relaxation process, or, alternatively that the conHere the first term is proportional to the charge ductivity decreases. Such a case can sometimes be density and the second term is constant. Hence described by the generalized hyperbolic law of equation (2) now becomes relaxation or even by equation (7). However sometimes it happens that the concave side of the ds s[su+ + c-(u’ - u-)] curve is turned towards the time axis, either during ;i;=eeo the whole period of relaxation or for the first part Writing z- = EE~/c-(u~ - u-)F, the solution of only (curve c in Fig. 1 or Figs. 12 and 13 in Ref. the above equation is [6]). This means that the conductivity increases during the relaxation process, i.e. that electrically G[l-exp(G)] (8) charged particles are formed in excess to those which are being dissipated. Mathematically the situation is succinctly desFor U+ = -u- and with c- very small one finds Z- very large and equation (8) approaching equa- cribed by considering the second derivative of the tion (7). If on the other hand the difference between log(s) vs. time curve. If the convex side of the curve the concentrations of positive and negative parti- is turned towards the time axis then the slope of the curve increases, i.e. becomes less strongly negative, cles is very small then K is practically independent of s and hence equation (8) reduces to equation (6) and the second derivative is positive (see curve b). since one can then neglect the term with sso with If there is a point of inflexion then the second respect to the terms with s and so. Equation (8) derivative goes through zero. So far the second thus includes (6) and (7) as special cases and derivative has only been observed to go from could therefore be referred to as the generalized negative to positive (see curve d) but there seems to be no physical reason why the reverse should hyperbolic law of relaxation. It describes the rate of discharge of an electrically charged liquid if, not also occur. during relaxation, the conductivity is a linear a : r = constant b : T increases function of the charge density. It is of course c : T decreases recognized that the assumptions underlying the hyperbolic laws of relaxation represent conditions which are not often met in practice.
positive and negative particles is of the same order as the total concentration of charged particles. Furthermore one can easily extend the treatment to non-equal mobilities. In this case the conductivity becomes
l-l (:) =iexp -=+
ACTUAL RELAXATIONOF A CHARGEDLIQUID A convenient way of testing whether the exponential law of relaxation is an adequate description
Time FIG. 1.
926
Possible forms of relaxation
curves.
On the rate of discharge of electrically charged hydrocarbon liquids RELAXATION WITH ASSOCIATIONAND
The above leads to the question whether it is not possible to draw association and dissociation quantitatively into account. An analysis based upon the simplest reaction mechanism imaginable is given in the Appendix. The analysis is not meant to provide a theory which can immediately be put into practice since basic data are lacking anyway. It is only meant to demonstrate that an association/dissociation mechanism can account for some types of charge decay curves occurring in practice and this is found to be indeed the case. CONCLUSIONS 1. Ohm’s Law can describe correctly the observed behaviour of petroleum products losing electric charge. 2. Conductivity (the constant of proportionality in Ohm’s Law) is not only a property of the product but also of the condition of charging. Conductivity need not be constant in time. 3. A charged product may lose its charge slower as well as faster than would be indicated by its conductivity in uncharged condition. 4. Taking account of association and dissociation makes it possible to describe relaxation curves which contain a point of inflection. APPENDIX.
+
DISSOCIATION positive outward direction is
RELAXATION WITH ASSOCIATION AND DISSOCIATION
Consider a small element of volume containing a concentration c+ of positive ions and c- of negative ions both monovalent. The total charge contained in the volume is (c* - c-)FV and the relationship between this quantity and the dielectric displacement is
c+uE dA = !-f! D,,!dA E&O P c+uFV =Y(c+
-c-)
Similarly, for the negative ions c-uFV
(c’ - c-1
y-&-
Furthermore it is assumed that the rate of dissociation is proportional to the concentration of the undissociated molecules, C, and that the rate of association is proportional to the product’of the ion concentrations. Finally it is assumed that in equilibrium c+ = c- = c. Let also 1-=- 2cuF z
E&O
where z is the relaxation time of the liquid in equilibrium. Now the following two material balances can be set up for the positive and negative ions respectively dc+ dt=dc-&-=+
c’(c’
- c-) 2ct
c-(c+ - c-) 2cz
+ klC - k,c+c(92 + klC - k,c+c-
Introducing dimensionless variables: c+/c = X; c-/c = y; T = t/z and writing k,C/c = k,c = a (since in equilibrium klC = k&), one obtains dx = -+x(x dT
- y) + a(1 - x-y) = a - $x2 - (a - $)xy
dy = +*y(x dT
(10) - y) + a(1 - xy) =a - 3~’ - (a - t)xy
(c’ - c-)FV =
$
D, dA
Volume V will be assumed to be sufficiently small to warrant the assumption that c+ and c- are constant over its surface. Then the flow of positive ions across the boundary of the volume in the
with the boundary conditions x = x0 and y = y. for T = 0. Without loss of generality one can take x0 > yo. It is easily shown that
927
4x - Y>
-=
dT
-3(x--Y)(x+Y)
(10
S. J. VELLENGA and A. KLINKENBERG d(xy) - = a(l - x y ) ( x + y) dT d(x + y) dT-2a-½(x+y)2-2(a-1)xy
= 2a(1 - x y ) - ½(x - y)2
(12)
d In (x - y) (13) (13a)
W r i t i n g the charge density in terms of x and y one finds s = (c + - c - ) F = (x - y)cF
(14)
H e n c e the second derivative of the logarithm of s is, u s i n g equations (11) and (13a) d 2 In (s) dT 2
1 d(x + y) 2 dT = ¼(x -- y)2 _ a(1 - x y )
xoy o > 1. Substituting this relation in equation (11) one finds
(15)
dT
= --½[(x - y)2 + 4(1 -T- A ( x - y)2,)]1/2
(17)
In this differential e q u a t i o n the variables ( x - y) and T have been separated. Nevertheless the integration c a n n o t be readily performed except for certain special values of a. These will now be considered. Special cases
1. a = 0. This m e a n s that dissociation is u n i m portant. From e q u a t i o n (16) it follows immediately that x y = 1 -T- A = XoYo (18)
This e q u a t i o n shows that the second derivative of the logarithm of s with respect to time can indeed be negative as well as positive so that the present m o d e l describes observed behaviour more accurately in this respect than either the exponential or the hyperbolic law. If one writes z = eeo/(C+ + c - ) u F instead of e%/2cuF it is easily s h o w n that equation (11) is equivalent to e q u a t i o n (2). Therefore the exponential law is a special case of the present solution, viz. the case that (x + y) is constant. Of course this condition only expresses that the conductivity is constant. The generalized hyperbolic law is not a special case of the present p r o b l e m because it is assumed that u ÷ = u- = u. A p p l y i n g this assumption in the derivation of the hyperbolic law results in a n o t h e r special case of the present solution, viz. the case that y is constant, which corresponds to the requirement that the concentration of the negafive ions shall not change. Reverting to equations (11) and (12) a further relation can be obtained by combining these two equations and integrating:
2. a very large. This m e a n s that dissociation is very r a p i d so that at every i n s t a n t
(1 -- x y ) = +_A(x -- y)2a
x y = xoy o = 1
(16)
where A is a positive c o n s t a n t of integration which can be f o u n d by applying the initial conditions to the a b o v e equation. The positive sign should be t a k e n when xoyo < 1 and the negative sign when 928
The solution of (17) is now (x - y) = 2x/(XoYo) cosech[x/(XoYo)T + B]
(19)
from which follows (x + y) = 2x/(XoYo) coth[x/(XoYo)T + B]
(20)
x = x/(XoYo) coth ½[x/(XoYo)T + B]
(21)
Y = ~/(XoYo) tgh ½[~/(XoYo)T + B]
(22)
where n ,--w---7--:,, -+1 -
B=
~/(xo/Yo)- 1
/
- 2 t g h - , / ( ,-o1 N\Xo/
(23)
Since the right-hand side of equation (15) is always positive there c a n n o t be a p o i n t of inflection in the logarithmic decay curve and the curve will always turn its convex side towards the time axis. Writing (x - y) = s/cF and (Xo - Yo) = So/CF it follows from equation (19) that s So
sinh B sinh[~/(XoYo)t/r + B]
(24)
Substituting this result in equations (21) and (22) it follows that x = coth ½(T + B) (25) y = tgh ½(T + B)
(26)
On the rate of discharge of electrically charged hydrocarbon liquids
The logarithmic decay curve will again always be convex towards the time axis. 3. a = 3. Equations (10) can now readily be solved. Assuming x,, > yO, there are still three cases : E. x,>
y, > 1
III.
xg > 1 > yo
1 > x() > yo
x = coth $(T + P’) y = tgh +(T + Q”) where Q” = 2 tgh- ’ y,
Equation (15) now becomes dT=
(x + y) = 2 coth (T + R) (x - y) = 2 cosech(T + R)
Ii.
x0 + y, < 2
(x + y) = 2 tgh(T + R’) (30) (x - y) = 2Q’ sech(T + R’)
d2 In (s) = )(x + y)2 - 1 dT2
(27)
x = tgh $(T + P”) where P” = 2 tgh-’ x0 y=tgh+(T+Q”)
d2 In (s) = $(x2 + y2 - 2)
x0 + y. > 2
where the constants of integration, R, R’, Q and Q’ can be found from the initial conditions. Equation (15) now becomes
x = coth j(T + P’) where P’ = 2 coth-’ x0
y = coth +(T + Q’) where Q’ = 2 coth-’ y0 II.
I.
(28)
Since coth z is larger than 1 for any value of z it follows from equations (27) and (28) that case I corresponds to a logarithmic decay curve which is shaped like curve b in Fig. 1. By similar reasoning it is seen that case III corresponds to curve c in Fig. 1. The logarithmic decay curve for case II may have a point of inflexion, namely when the equation (xZ+yZ)=2 (29)
Since coth z is larger than 1 for any value of z it follows from the solution for x -I- y and equation (31) that case I corresponds to a logarithmic decay curve which is shaped like curve b in Fig. 1. Similar reasoning shows that case II corresponds tocurvecinFig. 1. Ifx, + y, = 2thenx + y = 2 and the right-hand side of (31) reduces to zero. Hence the decay curve is a straight line. 5. a = 2. If one substitutes (x - Y)~ = z and writes (1 + A) = M2 it will be seen that equation (17) can be solved for (x - y). With equation (11) one can then find (x f y) which completes the solution. There are three cases: I. M2 > 0
(x - Y)~ = M cosh(2;
(x + Y) = II.
has a solution for a real and positive value of T. Substituting the solutions for x and y into equation (29) the latter reduces to
M2 < 0
Write:
(x - y)2 =
sinh +(T + P’) = cash f(T + Q”) which can easily be solved by writing out the hyperbolic functions in terms of exponentials. It follows that the condition for real and positive values of T is that P > Q” or xoyo < 1. If xoyo = 1 then P’ = Q” from which follows that xy = 1. Thus for x,y, >, 1 the logarithmic decay curve is shaped like curve b. 4. a = 1. Now equation (13) becomes easy to solve for x + y. Having obtained x + y one can find x - y by means of equation (11). There are two cases.
2M sinh(2T + S) M cosh(2T + S) - 1 S N sinh(2T + S’) - 1
M2 = - N2
(x + y) = III.
+ S) _ I
M2 = 0
(x - y)2 =
(x + Y) =
2N cosh(2T + S’) N sinh(2T + S’) - 1 S exp(2T + s”) - 1
2 exp(2T + S”) exp(2T + S”) - 1 (32)
Equation (15) now becomes
929
I. II.
d2 In(s)
F=
2M cosh(2T + S) - 2M2 [M cosh(2T + S) - 11’ 2N sinh(2T + S’) + 2N2 = [N sinh(2T + S’) - 112
(33)
S. J. VELLENGAand A. KLINKENBERG
III.
exp(2T + SN) = Iexp(2T + S) - 112
The values for the constants of integration, and S” can be found from the initial conditions. Inspection of (33) shows that cases II and III correspond to decay curves which are shaped like curve c of Fig. 1 but that case I may have a point of inflexion and may then correspond to curve d of Fig. 1. Closed solutions may be found for a few other cases but these solutions become increasingly complex and at present do not seem to offer more insight. M, N, S, S’
NOTATION
2
kmol/m3 Concentration Concentration of undissociated ions kmol/ma Dielectric displacement C/m2 Field strength V/m Faraday’s constant, 96.5 x 106 C/km01 equivalent A/ma Current density 3 Particle concentration Avogadro’s constant, 6.02 x 10z8 gal-1 C/m3 Charge density set Time Reduced time, see equation (10) Mobility of ions m2/Vs Velocity of ions m/s m3 Volume Reduced ionic concentration, see (10) Z Valency & Relative dielectric constant Absolute dielectric constant of EO vacuum, 8.854 x lo-12 AslVm K Conductivity A/Vm = mho/m SeC 7 Relaxation time The rationalized Giorgi system is used throughout.
111 GAVIS J., Chem. Engng. Sci. 1964 19, 237. A., Advances in Petroleum Chemistry and Refining, (Edited by MCKETTAJ. J.) Vol VIII, Chap. 2. InterPI KL~NKENBERG
science, 1964. [31 WINTERE. F., J. Roy. Aeronaut. Sot. 1962 66,429. 141 GAVISJ. and KOSZMANI., Development of charge in low conductivity liquids flowing past surfaces. Report to American Petroleum Institute, June 1961. BUSTINW. M., KOSZMANI. and TOBYEI. T., Hydrocarbon Process. 1964 43,209. ;i; Electrostatic discharges in aircraft fuel systems, Phase II. Report No. 355 of the Aviation Fuel, Lubricant and Equipment Research Committee of the Coordinating Research Council, Inc. September 1961.
930
Science, 1965, Vol. 20, pp. 923-930. Pergamon Press Ltd., Oxford. Printed in Great Britain.
On the rate of discharge of electrically .charged hydrocarbon liquids S. Bataafse Internationale
J. VELLENGA and A. KLINKENBERG
Petroleum
Maatschappij,
N.V. (Royal Dutch/Shell
Group), The Hague, Holland
(Received4 March 1965) Abstract--It is well known that the discharge of an electrically charged hydrocarbon liquid need not follow an exponential law. The present paper shows that this process can still be correctly described mathematically with the aid of Ohm’s Law. Recently a new theory for static relaxation has been put forward. It has been claimed that this “hyperbolic theory” better describes relaxation of charge from high resistivity fuels than is possible with the “ohmic theory”, by which is meant a theory based on the assumption of constant conductivity. However, Ohm’s Law does not require that the conductivity shall be constant. It is shown in the present paper that the hyperbolic theory is a special case of a more general treatment. Furthermore it is shown that by taking association and dissociation effects into account it is possible to describe certain types of relaxation curves which cannot be described by either the hyperbolic theory or the constant conductivity theory.
where D, is the component of the vector D normal to the element of surface, dA. This Dn is taken as positive in the outward direction. On the other hand, the rate of loss of charge from the element is equal to the surface integral of the current density J
BASIC CONCEPTS OIL products often acquire an electric charge during handling and from the point of view of safety it is important to know the rate at which such products lose their charge after operations have ceased. This mainly depends on the product in question and on the degree of charging. However the manner in which the product has become charged also plays a part. This will be further discussed in the following. A basic assumption in electric theory is the validity of Ohm’s Law. A general form of Ohm’s Law states that in a medium at rest and at a given point and time the current density J is proportional to the local field strength E
J = ICE
d(sV) - -= dt
By definition
where the factor z is known as the relaxation time. Therefore one has --=
(1)
The proportionality factor rc is called the conductivity and may be a function of both time and place. Now considering a small element of volume V, containing an electric charge sV, where s is the charge density, one can easily find the rate at which this element loses its charge. For, by definition, the charge enclosed within the surface of the element is equal to the surface integral of the dielectric displacement D sv =
D, dA
J,dA
Hence, if r can be taken constant over the surface of the element and ignoring transport of charge by diffusion or convection as required by the definition of Ohm’s Law, one obtains ds -= dt
--
s z
(2)
This differential equation will be basic in the following discussion. The above result is equally useful for a moving medium for convection effects can be ruled out
923
S. J. VELLENGA
and A. KLINKENBERG
by letting the element of volume move along with the liquid. This corresponds to the so-called Lagrangian description of motion in hydrodynamics. In the above a continuum has been implicitly assumed. This has considerably simplified the mathematical treatment and does not impose a great restriction in practice. In fact, the transport of electric charge through a liquid is usually described in terms of conductivity. It is also possible to describe the actual transport mechanism which consists in the bodily movement of electrically charged particles. If an electrically charged particle is placed in an electric field of strength E it will acquire a velocity c v=uE
current density in an electric field of strength E is J =C (CiZiFUi)E I
(3)
where c = n/N and where the index i refers to the i-th species. Comparing equations (1) and (3) it follows that the connection between the description in terms of conductivity and that in terms of movement of charged particles is K = ~ (ciZiFui)
(4)
The charge density of the liquid, s, can similarly be described as s = 7 (ci@) (5) For a binary monovalent electrolyte this reduces to
where the proportionality factor u is known as the mobility of the particle in the medium considered. If the charge is negative the same equation still applies if the sign is accounted for in the numerical value of U. Strictly speaking diffusion and convection should also be taken into account. However it is found that diffusion effects only become important in small apparatus or very near an interface where large concentration gradients occur so that in technical equipment the contribution of diffusion can usually be ignored [l]. Convection can be ruled out by the above-mentioned Lagrangian description. If, passing over to the macroscopic point of view, one places n electrically charged particles per unit of volume of an electrically indifferent fluid in an electric field of strength E, one finds that the quantity of charge passing per unit time through unit of area normal to the direction of the field is ziFv=z;FuE
where z is the number of elementary charges per particle, F is Faraday’s constant and N is Avogadro’s constant. This amounts to an electric current density. If the ‘charge happens to be negative there is a quantity of negative charge flowing in the negative direction which again represents a positive electric current. (In this case both the valency z and the mobility u have the negative sign). Thus, if there are a number of species of electrically charged particles, the
JC= (c’ + c-)uF
(assuming u+ = -u-
= u)
and s = (c’ - c-)F In a liquid at rest and in equilibrium or in a liquid under steady state conditions (e.g. one moving in a long line) the symbol ci on the right-hand side of equation (4) represents quantities which are constant. However this is no longer true under nonsteady state conditions. Charged particles or even whole species of charged particles may be removed or introduced through the interaction of the liquid with the surfaces past which it flows. This destroys the equilibrium between the association and dissociation reactions in the liquid. The net change in the rates of these reactions tends to restore the equilibrium but still the non-equilibrium conductivity may differ appreciably from the conductivity of the liquid at equilibrium. Even an uncharged liquid may therefore have a conductivity which differs from the equilibrium conductivity, namely when its ionic equilibrium has been recently disturbed. These effects have been found experimentally. In Shell publications the non-equilibrium conductivity is usually referred to as the in situ or efictive conductivity since it, and not the conductivity of the liquid at equilibrium, determines the rate of discharge of an electrically charged liquid P, 31. The right-hand side of equation (4) consists
924
On the rate of dischargeof electrically chargedhydrocarbonliquids
account for a charge in conductivity of
mostly of unknowns, certainly so in the nonequilibrium case. In order to be able to apply the insights which can be gained from the description in terms of charged particles one has to simplify drastically. Firstly, association and dissociation are not usually taken into account quantitatively. Secondly the discussion is almost invariably limited to two species of charged particles, having the same absolute value of the mobility, which is assumed constant, and with z = + 1 respectively. The latter assumption, that the ions are monovalent, is not a serious restriction in dilute solutions of low dielectric constant [4].
10-3
$
x
10-g
m2 -I/s -_ lo-l2 &
= 1 pmho/m
and this is of the order of values found in practice. One can consider, as an extreme case, that all ions of one sign are removed so that the effective conductivity is wholly due to the ions constituting the net charge density. (An alternative physical counterpart of the above model would be a liquid with zero rest conductivity to which ions of one sign are added). The conductivity now becomes rc = cFu
THE EXPONENTIALLAW OF RELAXATION
However, even with these restrictive assumptions, For various developments are still possible. instance, if the conductivity is not very low it is quite usual and often permissible to ignore the difference between the effective conductivity and the rest conductivity. The theory based on this assumption has been referred to as the ohmic theory, which is unfortunate since the application of Ohm’s Law does not require that the conductivity shall be constant. In this case the rate of discharge is given by equation (2) with r = constant. Integrating this equation the following relation is obtained s = s0 exp( - t/r)
(6)
This exponential law of relaxation describes the rate of discharge of an electrically charged liquid if the assumption of a constant conductivity holds. THE HYPERBOLICLAW OF RELAXATION The assumption that the conductivity is constant does certainly not hold when the total number of charged particles is so small that the change in concentrations which is needed to create a net charge density (see equation 5) affects significantly the right-hand side of equation (4). Comparing equations (4) and (5) it is seen that the contribution of an ionic species to the charge density, multiplied by its mobility, gives a contribution to the conductivity. Thus, if all ions have a mobility of 10eg m’/Vs, a charge density of O+OOlC/m3 will
and is no longer constant. Since, by equation (5) the charge density s = cF and since by definition z = se&c the equivalent to equation (2) is ds A=-dt
us2 EEL
which on integration yields 1 -=EE+-=s
ut
1
t/k,
+
1
(7) 0
so
so
where r. is the relaxation time at t = 0 and so is the initial charge density. Since the graph of equation (7) is a hyperbola one can refer to this equation as the h_vperboliclaw of relaxation. It describes the rate of discharge of an electrically charged liquid if the conductivity is directly proportional to the charge density. Note that when this law applies the liquid can lose its charge slower as well as faster than would be predicted by the exponential law of relaxation which assumes the effective conductivity to be constant and equal to the rest conductivity. If all ions of one sign are removed then the actual rate of relaxation will be slower than predicted by the exponential law; if the charge is due to ions being added to a liquid of negligible conductivity then, c$ course, the rate of relaxation is faster than predicted. For this latter case, see Ref. [5]. GENERALIZEDHYPERBOLICLAW OF RELAXATION
As a more general case consider a liquid in which the difference between the concentrations of
925
S. J. VELLENGA and A.
KLINKENBERG
of the electric relaxation of a given liquid is to plot the logarithm of the charge density against time. From equations (2) or (6) it follows immediately that this graph should be a straight line if the exponential law of relaxation is to be valid and that the negative slope of this line is inversely rc = (c’u’ - c-u-)F proportional to the relaxation time. and the charge density In practice such graphs are often obtained by connecting the output of a charge density meter to s= (C’ - c_)F a logarithmic recorder and it is then sometimes Assuming an excess of positive particles and found that such graphs are curved. If the convex assuming that the number of negative particles side of the curve is turned towards the time-axis remains constant, one can write (curve b in Fig. 1 or Fig. 11 of Ref [6]) this means that the relaxation time r increases during the fc = su+ + c-(U’ - u_)F relaxation process, or, alternatively that the conHere the first term is proportional to the charge ductivity decreases. Such a case can sometimes be density and the second term is constant. Hence described by the generalized hyperbolic law of equation (2) now becomes relaxation or even by equation (7). However sometimes it happens that the concave side of the ds s[su+ + c-(u’ - u-)] curve is turned towards the time axis, either during ;i;=eeo the whole period of relaxation or for the first part Writing z- = EE~/c-(u~ - u-)F, the solution of only (curve c in Fig. 1 or Figs. 12 and 13 in Ref. the above equation is [6]). This means that the conductivity increases during the relaxation process, i.e. that electrically G[l-exp(G)] (8) charged particles are formed in excess to those which are being dissipated. Mathematically the situation is succinctly desFor U+ = -u- and with c- very small one finds Z- very large and equation (8) approaching equa- cribed by considering the second derivative of the tion (7). If on the other hand the difference between log(s) vs. time curve. If the convex side of the curve the concentrations of positive and negative parti- is turned towards the time axis then the slope of the curve increases, i.e. becomes less strongly negative, cles is very small then K is practically independent of s and hence equation (8) reduces to equation (6) and the second derivative is positive (see curve b). since one can then neglect the term with sso with If there is a point of inflexion then the second respect to the terms with s and so. Equation (8) derivative goes through zero. So far the second thus includes (6) and (7) as special cases and derivative has only been observed to go from could therefore be referred to as the generalized negative to positive (see curve d) but there seems to be no physical reason why the reverse should hyperbolic law of relaxation. It describes the rate of discharge of an electrically charged liquid if, not also occur. during relaxation, the conductivity is a linear a : r = constant b : T increases function of the charge density. It is of course c : T decreases recognized that the assumptions underlying the hyperbolic laws of relaxation represent conditions which are not often met in practice.
positive and negative particles is of the same order as the total concentration of charged particles. Furthermore one can easily extend the treatment to non-equal mobilities. In this case the conductivity becomes
l-l (:) =iexp -=+
ACTUAL RELAXATIONOF A CHARGEDLIQUID A convenient way of testing whether the exponential law of relaxation is an adequate description
Time FIG. 1.
926
Possible forms of relaxation
curves.
On the rate of discharge of electrically charged hydrocarbon liquids RELAXATION WITH ASSOCIATIONAND
The above leads to the question whether it is not possible to draw association and dissociation quantitatively into account. An analysis based upon the simplest reaction mechanism imaginable is given in the Appendix. The analysis is not meant to provide a theory which can immediately be put into practice since basic data are lacking anyway. It is only meant to demonstrate that an association/dissociation mechanism can account for some types of charge decay curves occurring in practice and this is found to be indeed the case. CONCLUSIONS 1. Ohm’s Law can describe correctly the observed behaviour of petroleum products losing electric charge. 2. Conductivity (the constant of proportionality in Ohm’s Law) is not only a property of the product but also of the condition of charging. Conductivity need not be constant in time. 3. A charged product may lose its charge slower as well as faster than would be indicated by its conductivity in uncharged condition. 4. Taking account of association and dissociation makes it possible to describe relaxation curves which contain a point of inflection. APPENDIX.
+
DISSOCIATION positive outward direction is
RELAXATION WITH ASSOCIATION AND DISSOCIATION
Consider a small element of volume containing a concentration c+ of positive ions and c- of negative ions both monovalent. The total charge contained in the volume is (c* - c-)FV and the relationship between this quantity and the dielectric displacement is
c+uE dA = !-f! D,,!dA E&O P c+uFV =Y(c+
-c-)
Similarly, for the negative ions c-uFV
(c’ - c-1
y-&-
Furthermore it is assumed that the rate of dissociation is proportional to the concentration of the undissociated molecules, C, and that the rate of association is proportional to the product’of the ion concentrations. Finally it is assumed that in equilibrium c+ = c- = c. Let also 1-=- 2cuF z
E&O
where z is the relaxation time of the liquid in equilibrium. Now the following two material balances can be set up for the positive and negative ions respectively dc+ dt=dc-&-=+
c’(c’
- c-) 2ct
c-(c+ - c-) 2cz
+ klC - k,c+c(92 + klC - k,c+c-
Introducing dimensionless variables: c+/c = X; c-/c = y; T = t/z and writing k,C/c = k,c = a (since in equilibrium klC = k&), one obtains dx = -+x(x dT
- y) + a(1 - x-y) = a - $x2 - (a - $)xy
dy = +*y(x dT
(10) - y) + a(1 - xy) =a - 3~’ - (a - t)xy
(c’ - c-)FV =
$
D, dA
Volume V will be assumed to be sufficiently small to warrant the assumption that c+ and c- are constant over its surface. Then the flow of positive ions across the boundary of the volume in the
with the boundary conditions x = x0 and y = y. for T = 0. Without loss of generality one can take x0 > yo. It is easily shown that
927
4x - Y>
-=
dT
-3(x--Y)(x+Y)
(10
S. J. VELLENGA and A. KLINKENBERG d(xy) - = a(l - x y ) ( x + y) dT d(x + y) dT-2a-½(x+y)2-2(a-1)xy
= 2a(1 - x y ) - ½(x - y)2
(12)
d In (x - y) (13) (13a)
W r i t i n g the charge density in terms of x and y one finds s = (c + - c - ) F = (x - y)cF
(14)
H e n c e the second derivative of the logarithm of s is, u s i n g equations (11) and (13a) d 2 In (s) dT 2
1 d(x + y) 2 dT = ¼(x -- y)2 _ a(1 - x y )
xoy o > 1. Substituting this relation in equation (11) one finds
(15)
dT
= --½[(x - y)2 + 4(1 -T- A ( x - y)2,)]1/2
(17)
In this differential e q u a t i o n the variables ( x - y) and T have been separated. Nevertheless the integration c a n n o t be readily performed except for certain special values of a. These will now be considered. Special cases
1. a = 0. This m e a n s that dissociation is u n i m portant. From e q u a t i o n (16) it follows immediately that x y = 1 -T- A = XoYo (18)
This e q u a t i o n shows that the second derivative of the logarithm of s with respect to time can indeed be negative as well as positive so that the present m o d e l describes observed behaviour more accurately in this respect than either the exponential or the hyperbolic law. If one writes z = eeo/(C+ + c - ) u F instead of e%/2cuF it is easily s h o w n that equation (11) is equivalent to e q u a t i o n (2). Therefore the exponential law is a special case of the present solution, viz. the case that (x + y) is constant. Of course this condition only expresses that the conductivity is constant. The generalized hyperbolic law is not a special case of the present p r o b l e m because it is assumed that u ÷ = u- = u. A p p l y i n g this assumption in the derivation of the hyperbolic law results in a n o t h e r special case of the present solution, viz. the case that y is constant, which corresponds to the requirement that the concentration of the negafive ions shall not change. Reverting to equations (11) and (12) a further relation can be obtained by combining these two equations and integrating:
2. a very large. This m e a n s that dissociation is very r a p i d so that at every i n s t a n t
(1 -- x y ) = +_A(x -- y)2a
x y = xoy o = 1
(16)
where A is a positive c o n s t a n t of integration which can be f o u n d by applying the initial conditions to the a b o v e equation. The positive sign should be t a k e n when xoyo < 1 and the negative sign when 928
The solution of (17) is now (x - y) = 2x/(XoYo) cosech[x/(XoYo)T + B]
(19)
from which follows (x + y) = 2x/(XoYo) coth[x/(XoYo)T + B]
(20)
x = x/(XoYo) coth ½[x/(XoYo)T + B]
(21)
Y = ~/(XoYo) tgh ½[~/(XoYo)T + B]
(22)
where n ,--w---7--:,, -+1 -
B=
~/(xo/Yo)- 1
/
- 2 t g h - , / ( ,-o1 N\Xo/
(23)
Since the right-hand side of equation (15) is always positive there c a n n o t be a p o i n t of inflection in the logarithmic decay curve and the curve will always turn its convex side towards the time axis. Writing (x - y) = s/cF and (Xo - Yo) = So/CF it follows from equation (19) that s So
sinh B sinh[~/(XoYo)t/r + B]
(24)
Substituting this result in equations (21) and (22) it follows that x = coth ½(T + B) (25) y = tgh ½(T + B)
(26)
On the rate of discharge of electrically charged hydrocarbon liquids
The logarithmic decay curve will again always be convex towards the time axis. 3. a = 3. Equations (10) can now readily be solved. Assuming x,, > yO, there are still three cases : E. x,>
y, > 1
III.
xg > 1 > yo
1 > x() > yo
x = coth $(T + P’) y = tgh +(T + Q”) where Q” = 2 tgh- ’ y,
Equation (15) now becomes dT=
(x + y) = 2 coth (T + R) (x - y) = 2 cosech(T + R)
Ii.
x0 + y, < 2
(x + y) = 2 tgh(T + R’) (30) (x - y) = 2Q’ sech(T + R’)
d2 In (s) = )(x + y)2 - 1 dT2
(27)
x = tgh $(T + P”) where P” = 2 tgh-’ x0 y=tgh+(T+Q”)
d2 In (s) = $(x2 + y2 - 2)
x0 + y. > 2
where the constants of integration, R, R’, Q and Q’ can be found from the initial conditions. Equation (15) now becomes
x = coth j(T + P’) where P’ = 2 coth-’ x0
y = coth +(T + Q’) where Q’ = 2 coth-’ y0 II.
I.
(28)
Since coth z is larger than 1 for any value of z it follows from equations (27) and (28) that case I corresponds to a logarithmic decay curve which is shaped like curve b in Fig. 1. By similar reasoning it is seen that case III corresponds to curve c in Fig. 1. The logarithmic decay curve for case II may have a point of inflexion, namely when the equation (xZ+yZ)=2 (29)
Since coth z is larger than 1 for any value of z it follows from the solution for x -I- y and equation (31) that case I corresponds to a logarithmic decay curve which is shaped like curve b in Fig. 1. Similar reasoning shows that case II corresponds tocurvecinFig. 1. Ifx, + y, = 2thenx + y = 2 and the right-hand side of (31) reduces to zero. Hence the decay curve is a straight line. 5. a = 2. If one substitutes (x - Y)~ = z and writes (1 + A) = M2 it will be seen that equation (17) can be solved for (x - y). With equation (11) one can then find (x f y) which completes the solution. There are three cases: I. M2 > 0
(x - Y)~ = M cosh(2;
(x + Y) = II.
has a solution for a real and positive value of T. Substituting the solutions for x and y into equation (29) the latter reduces to
M2 < 0
Write:
(x - y)2 =
sinh +(T + P’) = cash f(T + Q”) which can easily be solved by writing out the hyperbolic functions in terms of exponentials. It follows that the condition for real and positive values of T is that P > Q” or xoyo < 1. If xoyo = 1 then P’ = Q” from which follows that xy = 1. Thus for x,y, >, 1 the logarithmic decay curve is shaped like curve b. 4. a = 1. Now equation (13) becomes easy to solve for x + y. Having obtained x + y one can find x - y by means of equation (11). There are two cases.
2M sinh(2T + S) M cosh(2T + S) - 1 S N sinh(2T + S’) - 1
M2 = - N2
(x + y) = III.
+ S) _ I
M2 = 0
(x - y)2 =
(x + Y) =
2N cosh(2T + S’) N sinh(2T + S’) - 1 S exp(2T + s”) - 1
2 exp(2T + S”) exp(2T + S”) - 1 (32)
Equation (15) now becomes
929
I. II.
d2 In(s)
F=
2M cosh(2T + S) - 2M2 [M cosh(2T + S) - 11’ 2N sinh(2T + S’) + 2N2 = [N sinh(2T + S’) - 112
(33)
S. J. VELLENGAand A. KLINKENBERG
III.
exp(2T + SN) = Iexp(2T + S) - 112
The values for the constants of integration, and S” can be found from the initial conditions. Inspection of (33) shows that cases II and III correspond to decay curves which are shaped like curve c of Fig. 1 but that case I may have a point of inflexion and may then correspond to curve d of Fig. 1. Closed solutions may be found for a few other cases but these solutions become increasingly complex and at present do not seem to offer more insight. M, N, S, S’
NOTATION
2
kmol/m3 Concentration Concentration of undissociated ions kmol/ma Dielectric displacement C/m2 Field strength V/m Faraday’s constant, 96.5 x 106 C/km01 equivalent A/ma Current density 3 Particle concentration Avogadro’s constant, 6.02 x 10z8 gal-1 C/m3 Charge density set Time Reduced time, see equation (10) Mobility of ions m2/Vs Velocity of ions m/s m3 Volume Reduced ionic concentration, see (10) Z Valency & Relative dielectric constant Absolute dielectric constant of EO vacuum, 8.854 x lo-12 AslVm K Conductivity A/Vm = mho/m SeC 7 Relaxation time The rationalized Giorgi system is used throughout.
111 GAVIS J., Chem. Engng. Sci. 1964 19, 237. A., Advances in Petroleum Chemistry and Refining, (Edited by MCKETTAJ. J.) Vol VIII, Chap. 2. InterPI KL~NKENBERG
science, 1964. [31 WINTERE. F., J. Roy. Aeronaut. Sot. 1962 66,429. 141 GAVISJ. and KOSZMANI., Development of charge in low conductivity liquids flowing past surfaces. Report to American Petroleum Institute, June 1961. BUSTINW. M., KOSZMANI. and TOBYEI. T., Hydrocarbon Process. 1964 43,209. ;i; Electrostatic discharges in aircraft fuel systems, Phase II. Report No. 355 of the Aviation Fuel, Lubricant and Equipment Research Committee of the Coordinating Research Council, Inc. September 1961.
930