i = 6 1 Jn(x) = Bessel function, first kind, order n k = slope of equilibrium isotherm, defined by eq 5 k , = mass transfer coefficient (calculated from eq 29) L = length of packed bed Le = Lewis number, cue/% = Sc/Pr Nu, = particle Nusselt number, Rskc/Dp P a = PeM(kp,h, + hp)/Sh PeH = Peclet number for heat, R2w/cue PeM = Peclet number for mass, R2w/D Pep = particle Peclet number for mass, RS2w/Dp P r = Prandtl number, v / a e q = solids composition, mol/gr of compressed solid r = cylindrical radial coordinate for packed bed r, = spherical radial coordinate inside solid particle R = columnradius R, = radius of adsorbent particles Ro = dispersion dissipative term defined by eq 25 RM = film mass transfer dissipative term, defined by eq 26 Rp = pore mass transfer dissipative term defined by eq 47 Sc = Schmidt number, v / Sh = Sherwood number, kcaR2/tbD Sh, = particle Sherwood number, kcaRs2/cbDp t = time T = local bed temperature AT = amplitude of applied temperature u = interstitial fluid velocity WL1 = parameter defined by eq 22b x = axial coordinate of bed zL1 = defined by eq 22a Z R = Real(Z+1) ZI = Imag ( Z + I )
Greek Symbols = effective thermal diffusivity of bed and fluid @ = (1+ HD)/2 t b = bulk voidageof bed t p = pore voidage of solid particles ole
T=vTG
d=zz
A+, = h = p,h,ATa;' v = kinematic viscosity [ = dimensionless cyclindrical coordinate, r / R . ^
Fs = dimensionless spherical coordinate, re/Rs
mP)
= com ressed solid density, g/cm3 = w = frequency of oscillation ps
T+1
Literature Cited Aris. R.. Amundson, N. R.. "Mathematical Methods in Chemical Engineering", Vol. 2, Prentice-Hall Inc.. 1973. Baker, B . , Pigford. R . L.. lnd. Eng. Chem., Fundam.. 10, 283 (1971). Bird, R. B., Stewart, W. E.. Lightfoot, E. N., "Transport Phenomena", p 647, Wiley, New York, N.Y.. 1960. Jenczewski, T. J., Myers, A . L., lnd. Eng. Chem.. Fundarn.. 9 , 216 (1970). Jones, P. J., B. E. Thesis, University of Queensland, 1974. Perry, R. H., Chilton, C. H., Kirkpatrick. S. D.. Ed., "Chemical Engineers' Handbook," 4th ed, p 5-51, McGraw-Hill, New York, N.Y.. 1963. Rice, R. G., Ind. Eng. Chem., fundam.. 12, 406 (1973). Rice, R. G., Foo, S. C., Ind. Eng. Chem.. Fundarn., 13, 396 (1974). Rice, R. G.. Mackenzie. M.. Ind. Eng. Chem., Fundam., 12, 486 (1973). Sweed, N. W. "Progress in Separation and Purification", Vol. 4, Wiley. New York, N.Y.. 1971. Sweed, N. W., Gregory, R. A.,A.I.Ch.E.J., 17, 171 (1971). Sweed, N. W., Wilhelm. R. H.. Ind. Eng. Chern.. fundarn.. 8, 221 (1969).
Received for review June 19, 1974 Accepted March 5, 1975
A New Probe and Circuit for Measuring Electrolyte Conductivity Soon Jai Khang and Thomas J. Fitzgerald* Department of Chemical Engineering, Oregon State University, Corvallis. Oregon 9733 1
A new probe design used in an operational amplifier circuit gives linear response to changes in electrolyte concentration and is not subject to errors from stray current effects, thus allowing multiple simultaneous conductivity measurements in the same vessel, a s well as the presence of other metal surfaces in contact with the electrolyte solution.
The response to a tracer impulse injection can give useful information about a system. For example, in a flow system the tracer impulse response is the residence time distribution of molecules in the exit stream and is useful in formulating a dynamic model of the system. A good tracer must not interfere with the behavior of the system, should be easily measured a t any point in the system or in the output stream, and should be easy and safe to handle. Electrolyte solutions fulfill all the above requirements in many liquid flow systems. Methods of measuring concentrations of electrolyte solutions by means of conductivity cells and alternating current were developed in late 19th century. Parker (1923) first noted that the cell constant of a conductivity cell changes during measurement. Later, Jones et al. (1928) carefully studied this effect and refined the techniques of measurement of the conductivity of uniform solutions. 208
Ind. Eng. Chem., Fundam.. Voi. 14, No. 3, 1975
Their major contribution was the platinization technique to reduce polarization on the surface of electrodes. More recently, methods for measuring time-varying local concentrations have interested investigators in various fields. Kramers et al. (1953) reported the first probes, shown in Figure l a , which were employed to detect local concentration variations in a mixing tank. Prausnitz and Wilhelm (1956) and Clements and Schnelle (1963) made probes with two exposed platinum wires (Figure l b ) . The probe configurations were further changed to measure concentrations in a much smaller volume element. Lamb et al. (1960) designed a probe which consisted of a point electrode and a wire ring electrode (Figure IC). Since the surface area of a point electrode is much smaller than that of a wire ring electrode, the current flow, and thus the conductance through the probe, is mainly proportional to the concentration near the point electrode. Reith (1965)
RD
Cylindrically electrode
coiled wire electrode
\
Figure
1. Variety of probe
configurations.
used a similar probe (Figure le), and Gibson and Schwartz (1963) employed a single point electrode (Figure Id) using the grounded metal walls of the vessel as a second electrode. I t is well established that the electrode material should be platinum to avoid polarization. The majority of authors employed platinized platinum wires for electrodes. Holmes et al. (1964) and Voncken et al. (1965) used copper rings and it is not clear how they avoided errors due to polarization effects. Figure 2 shows the conventional circuits used by the previous authors. In each case RP is the resistance of the probe, R , is the source impedance for the ac signal, and'R, is a shunt resistance. As we will see in the following section, none of the circuits reported in the literature gives voltage outputs which are proportional to the conductance of the probe. The circuits in Figures 2a and 2b are commonly used for large conductance variations and the Wheatstone bridge circuit shown in Figure 2c is primarily used for very small conductance variations such as concentration fluctuations in a turbulent field. Clements and Schnelle (1963) adopted the balanced Wheatstone bridge circuit for large variation of concentrations and calibrated the output voltage to measure concentration. Table I summarizes probe and circuit designs by the previous investigators. The Circuit Problems with Conventional Methods The nonlinearity of the conductance measuring circuits that are described in the literature has been mentioned; a more serious problem for many applications is the susceptibility of the conductance probes to stray currents which may be present.
t T
(C)
Figure 2. Conventional circuits for measuring conductivity showing stray current paths. In order to understand this effect, we must consider not only the resistance between electrodes but also the resistance from each electrode to the bulk fluid. Figure 2 shows the conventional circuits with this modification. The resistance from the larger electrode to the bulk solution is labeled as RL and the-resistance from the smaller electrode to the bulk solution is labeled as R H .The stray current source in each case is shown as a battery connected from RL and R Hto ground. (a) Internal Battery Effects. In most practical cases, the vessel in which probes are inserted contains metals other than platinum. Different metals and an electrolyte solution form internal batteries. This battery effect creates a potential between the probe electrodes and other metal parts and can result in biased output voltage el (Figure 2 ) . Since the effective resistance and the emf of internal batteries depend on local concentrations. the biased potential fluctuates and produces noise which cannot be separated from the true response. To minimize this effect, the following conditions are required: RI. >> R , for circuit a, and RH >> R , for circuits a. b, and c. The first condition can be met by using a low impedance ac source, but the second condition depends on probe geometry and
Table I . Summary of Probe and Circuit Designs by Previous Investigators ~
~~~~~~
Authors
~~~
Probe
Kramers et al. (1953) Prausnitz and Wilhelm (1956)
Figure l a Figure l b
Lamb et al. (1960)
Figure I C ,d
Gibson and Schwarz (1963)
Figure I d
Clements and Schnelle (1963)
Figure l b
Holmes et al. (1964) Voncken, et ai. (1965) Reith (1965)
Three large concentric rings Figure l e
Clements (1968)
Not given
Materials of probe Platinum wires Platinum wires (platinized) Epoxy seal Platinum wires (platinized) Epoxy seal Platinum wire (platinized) Epoxy seal Platinum wires (platinized) Sealed in a nylon or glass tube Copper Platinum wire (platinized) Glass seal Not given
Basic circuit
Carrier frequency, Hz
Figure 2c Figure 2a
l k
Figure 2a
15 k
Figure 2b, c
25 k
Figure 2c
60
Not given Figure 2c
Not
Figure 2a
l k
given 13.3 k
Ind. Eng. Chem.. Fundam.. Vol. 14, No. 3. 1975
209
R.
1
I
+I
I
i
Figure 3. Cross-talk effect of double measuring circuit.
cannot be met by any of the probes reported in the literature. For the probes shown in Figure l, the resistance from the smaller electrode to the larger electrode is of the same order of magnitude as the resistance from the smaller electrode to the bulk fluid. Probe (a) used by Kramers et al. comes closest to a good design since the coiled outer electrode shields much of the smaller electrode from the bulk fluid; the tip of the probe however is not well shielded from stray currents. (b) Cross-Talk Effects. In many cases we need simultaneous conductance measurements in a vessel, requiring that multiple probes be positioned a t different places in an electrolyte solution. Independent measurements are required, but the presence of one probe may interfere with the measurement of another one. We call this behavior “cross-talk”. All conventional circuits give this effect. Figure 3 shows one example. Probes 1 and 2 are connected to the same ac power source and interfere with each other through the conductance of the bulk solution. To minimize the interference, the following conditions are needed.
RD1>> R,; RD2>> Ri; RH1
>>
RDi; RH2
>>
RP2
The first two inequalities can be satisfied by using a low impedance ac source, but the two latter inequalities cannot be satisfied using the conventional probes shown in Figure 1. ( c ) Nonlinearity. Let us consider the conventional circuits shown in Figure 2 neglecting the complications which would be caused by internal battery or cross-talk effects. For (2a) and (2b) the output voltage e l across the resistance R, is
R
4 R, eo
Figure 4. Operational amplifier circuits for linear conductivity
measurements. Eliminating Nonlinearity Two relatively simple circuits involving high-gain dc amplifiers (operational amplifiers) can be used to obtain an output voltage which is linearly related to the conductance of the conductivity cell. They are shown in Figure 4. The behavior of these circuits is readily calculated using the (reasonable) assumptions that: (1) no current flows into either input terminal of the amplifier, and ( 2 ) the amplifier gain is so high that the potential difference across the amplifier input is essentially zero. With these assumptions we find that el = % e , f o r circuit 4a R,
and el = (1
+
2)
e , for circuit 4b
4a has the advantage that the output voltage is strictly proportional to the measured cell conductivity; 4b has the somewhat more important advantages that one of the electrodes can be grounded and a low impedance ac source is not needed. Eliminating Stray Current Effects
relatively low value to prevent a large current flow which might cause the formation of bubbles a t the probe. The requirement that R, be small and that eo not be large means that the magnitude of el will be very small and will require amplification before it can be measured. Now consider the other conventional circuit (a Wheatstone bridge), shown in Figure 2c. The voltage output el is
We have noted that in order to eliminate internal battery effects and cross talk in conventional circuits, we should have a probe so designed that the resistance from the bulk fluid to the smaller electrode ( R H )is orders of magnitude greater than the resistance between the large probe and the smaller probe (Rp). This same condition is required for the linear conductivity circuits as can be seen from Figure 5, which shows the effect of an internal battery and the effect of cross-talk on the linear conductance circuit of Figure 4b. It is apparent that if RH >> Rp then the internal battery is effectively shunted to ground through R L , and has no effect on the output voltage e l . Similarly, if R H >> ~ RP~, and R H >> ~ Rpl, then the outputs el and e2 will not interact, and hence there will be no cross talk.
To have approximate linearity between e l and l / R p , we must balance the bridge circuit. Assuming R1 = Rz and R , s Ra, the equation above becomes
Design of the Probe
= R,
thus el is not proportional to the conductance l / R p unless RP is very much bigger than R,. Now eo is limited to a
The linearity holds only for small variations of R p from the resistance Ra. Therefore, the bridge circuit has limited applications for small concentration variations around a fixed mean value. 210
Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
The resistance from the smaller electrode to the bulk fluid can be made very large (compared to the intraprobe resistance) by making the larger probe in the form of a cage which encloses the smaller one. The effect of the cage is demonstrated below for a similar (but simpler) geometry. Consider a single conducting rod (the smaller electrode) surrounded by a bundle of n rods (the larger electrode)
Y
X
(b) Figure 3. Effect of a n internal battery and the effect of cross-talk on the linear conductivity circuit of Figure 4b.
Figure 6. Two dimensional probe configuration with outer electrode in the form of a cage.
where each of radius rl a t a distance p 1 from the center, with the entire bundle surrounded by a cylinder a t distance p 2 from the center as shown in Figure 6. The resistance from the center to the parallel bundle a t p 1 is analogous to the probe resistance R , and the resistance from the center to the outside cylinder is analogous to RH, while the resistance from the bundle at p1 to the cylinder a t p 2 is analogous to Rr,. Suppose the center is at a potential eo, the bundle at a potential el, and the outer shell is a t a potential e2; then the current flow from the center, io, can be portioned thus
il
is the current flow to each rod at
p1.
Aligning the
x axis as shown in Figure 6 and assuming rl and ro are
sufficiently small so that the potential fields around the origin and the rods remain essentially circular, potentials a t x = ro, x = p 1 and x = p 2 are
e, = -
2n In ( p z )
Thus, for ro, rl
