Perturbation viscometry of gas mixtures. Addition and removal of finite perturbations

Chemical Engineering Science 55 (2000) 5747}5754

Perturbation viscometry of gas mixtures. Addition and removal of "nite perturbations G. Mason*, B. A. Bu!ham, M. J. Heslop, P. A. Russell, B. Zhang Department of Chemical Engineering, Loughborough University, Loughborough LE11 3TU, UK Received 23 July 1999; accepted 5 June 2000

Abstract Modi"cations to the theory of a new technique for making viscosity measurements on multicomponent gas mixtures are described. The new technique involves slightly altering the composition of a gas mixture #owing through a capillary tube by adding a small stream of perturbation gas. The perturbation gas is usually one of the components in the mixture and is consequently of known composition. The pressure at the inlet of the capillary tube rises when the perturbation gas is added and this pressure increase is proportional to the yowrate increase. A short time later, the pressure changes again when the composition of the gas #owing through the tube changes. This second pressure change is proportional to the viscosity change and can be an increase or a decrease. For in"nitesimally small perturbation #ow rates, the ratio of the second pressure step to the "rst is proportional to dln k/dX where dX is G G the change in the mole fraction of component i and k is the initial viscosity, and the ratio is independent of whether the perturbation stream is added or removed. However, when small "nite perturbations are made, there are systematic di!erences between the ratio of the two steps of pressure depending on whether the perturbation gas is added or removed. These di!erences are analysed theoretically and demonstrated experimentally using the argon}nitrogen system at 243C and at an absolute pressure of 1.32 bar.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Lamina #ow; Viscosity; Gas mixtures; Momentum transfer; Viscometry

1. Introduction In previous papers (Heslop, Bu!ham, Mason & Ireland, 1996; Heslop, Bu!ham & Mason, 1998; Mason, Bu!ham, Heslop & Zhang, 1998), details were given of a new method of measuring the gradient of the viscositycomposition function for binary gas mixtures. The method consists of altering the #ow of a gas mixture passing through a capillary tube by the addition of a very small stream of perturbation gas (usually one of the individual components of the mixture). The pressure at the upstream end of the capillary tube rises as the perturbation is added, and then changes again a short time later when the change in viscosity reaches the capillary tube. The ratio of the sizes of these two pressure changes

* Corresponding author. Tel.: #44-(0)1509-263171; fax: #44(0)1509-223923. E-mail address: [email protected] (G. Mason).  Present address: Department of Chemical Engineering, Tianjin University, Tianjin 300072, People's Republic of China.

gives *k/k *X , where k is the pre-perturbation viscosity G and *k and *X are the changes in the viscosity and mole G fraction X of component i caused by the perturbation. G The viscosity relative to one of the pure components vs. composition function over the composition range is obtained by numerical integration starting with either of the two pure components. The method has several advantages, the main one being that the gradient rather than the actual function is measured and this means that, after integration, the results should be of considerable accuracy. In an earlier paper (Mason et al., 1998), the theory was derived for the case where the perturbation step size is in"nitesimal. Experiments were reported for the addition of perturbation gas to nitrogen}argon mixtures. In practice, it is not possible to make perturbations that are small enough to be in"nitesimal because very small perturbations give responses that are too small to be measured accurately enough. There must be a compromise between accurate measurements and the approximations in the theory. In this paper, we develop the theory of the method in more detail and derive results for small, "nite

0009-2509/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 1 7 9 - 2

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G. Mason et al. / Chemical Engineering Science 55 (2000) 5747}5754

perturbation step sizes. The theory shows that there will be systematic di!erences between the results of adding the perturbation gas and of removing it. These di!erences are con"rmed by experiment. The experiments also show that the method is surprisingly robust and, perhaps, has the potential to be developed into a standard method of measuring the viscosity of multicomponent gas mixtures.

2. Theory of the method 2.1. Basic equations Consider the schematic apparatus diagram shown in Fig. 1. There are two capillary chokes. The upstream choke A provides almost all of the pneumatic resistance and consequently the addition of a small #ow of perturbation gas downstream of that choke will hardly a!ect the gas #owrate through it. Fig. 1 shows the switching valve B set to not admit perturbation gas. An experiment is started by switching the valve to add perturbation gas to the main #ow. The #owrate in the measuring choke E is increased and this is indicated by the pressure at pressure gauge D rising. Because of the presence of the delay line C, the composition of the gas passing through the measuring choke remains at the original value. The viscosity also remains at its original value and the combination of the pressure gauge D and choke E acts as a #owmeter. Eventually, gas of changed composition reaches choke E and the pressure at gauge D changes again because the viscosity change alters the choke's resistance. The combination of the pressure gauge D and choke E now becomes a viscometer. The analysis will focus on the downstream choke E. For laminar #ow of an ideal gas through a capillary tube with inlet pressure p and outlet pressure

P the equation governing #ow is (Mason et al., .0 1998) (p!P )"2KkMR¹, .0

(1)

where K is a constant for the system, M is the molar #owrate, R is the gas constant and ¹ is the absolute temperature. The outlet pressure may be above atmospheric pressure if, for example, it is set with a back pressure regulator (hence the subscript BPR). All pressures are absolute. The squared terms arise because the gas in the choke is compressed and is denser at the inlet than at the outlet. Now, consider the mixing valve. If the main stream entering the mixing valve contains a mole fraction X of G component i and the perturbation stream has a #owrate m and composition X2, then the addition of the perturbaG tion stream will produce a molar composition change in the stream #owing through the measuring choke of m *X " (X2!X). G M#m G G

(2)

Usually m will be small compared to M; typically the ratio m/M will be about 0.01. 2.2. Diwerential perturbations Before the perturbation gas is added, the pressure indicated by the pressure gauge is p . This pressure  changes to p after the perturbation gas is added and to  p after the viscosity change has passed the capillary. We  showed in the earlier paper (Mason et al., 1998) that the ratio of the change in pressure (p !p ) at the capillary   inlet caused by the change in viscosity (*k) to the pressure change (p !p ) caused by the addition of the   perturbation of #owrate m, is related to the small changes of viscosity and composition by 1 *k p !p 1  "  . k *X p !p X2!X G G   G

(3)

Eq. (3) applies only for small perturbations. In the limit when *k and *X become very small, a di!erential equaG tion is obtained and *k dk dln k P " . k *X k dX dX G G G Fig. 1. Schematic diagram of the basic apparatus. The high-resistance choke A sets the #ow. The low-resistance choke E enables the pressure gauge D to detect changes in #owrate and viscosity. The delay line C (empty tube) separates in time the response of the pressure gauge to the changes in #owrate and viscosity. In an experiment, the small stream of perturbation gas is added and the responses of the pressure gauge (one for #owrate change and the other for viscosity change) are measured. Further measurements are made when the #ow of perturbation gas is cut-o!.

(4)

In an experiment, we measure the two pressure changes p !p and p !p . X2 and X are both com    G G positions of input streams and will be known. Generally, a pure component will be used as perturbation gas because its composition is certain. Thus, the gradient *k/k *X is determined from the ratio of p !p G   to p !p and knowledge of X2 and X. No other   G G variable need be known. Because the ratio of the pressure

G. Mason et al. / Chemical Engineering Science 55 (2000) 5747}5754

di!erences is required, the pressures do not have to be absolute and can be in arbitrary units. 2.3. Finite perturbations and adding and removing perturbation gas In the experiments reported previously (Mason et al., 1998), we recorded the changing pressure just upstream of a capillary tube when a small stream of perturbation gas was added. The inverse experiment can be performed when the perturbation gas is removed from the much larger stream of carrier gas; that is to say, addition of perturbation gas is stopped. A typical pressure record is shown in Fig. 2: it contains the steps associated with the addition and removal of perturbation gas. Eq. (1) is the basic equation linking the absolute pressures with the appropriate #owrate and viscosity. In the analysis already reported, we di!erentiated Eq. (1) in order to work out the logarithm of the gradient of the viscosity with composition from the two pressure steps p !p and   p !p .   Di!erentiation implies in"nitesimally small steps but in experiments the perturbation has to be "nite, otherwise there is nothing to measure. For this reason, we used a ratio of perturbation gas #owrate, m, to the main gas #owrate, M, of about one to a hundred in our previous experiments, but we will see later that this is not small

5749

enough to be in"nitesimally small. That this step is "nite, even though small, makes a perceptible di!erence in the interpretation of some of our results, particularly when the perturbation stream is removed. We now give the analysis for small "nite perturbations. Eq. (1) is exact and is the core of the analysis. Initially, the pressure at the transducer is p , the #ow rate is M, the viscosity is k and (p !P )"2KkMR¹. (5)  .0 After the perturbation #ow has been added, the pressure at the pressure gauge rises to p and the #owrate to  M#m, but, for this "rst step, the viscosity in the downstream capillary choke remains at k. Hence, (p !P )"2Kk(M#m)R¹. (6)  .0 After the viscosity change has passed through the delay line and arrived at the downstream capillary choke the pressure becomes p and the viscosity k#*k. So, for this  second step (p !P )"2K(k#*k)(M#m)R¹.  .0 Subtracting Eq. (5) from Eq. (6) gives

(7)

p !p "2KR¹km,   which factorises to

(8)

(p !p )(p #p )"2KR¹km. (9)     If we repeat this procedure for Eqs. (6) and (7) we obtain (p !p )(p #p )"2KR¹ *k(M#m).     Now, dividing Eq. (10) by Eq. (9) gives

Fig. 2. An actual record of pressure when a perturbation of a small #ow of argon was added to, and subsequently removed from, a main #ow with an argon mole fraction of 0.2915. The initial pressure is p . After  the argon perturbation has been added and before the composition change has reached the downstream choke the pressure is p . It rises to  p when the change in viscosity reaches the downstream choke. When  the perturbation #ow is cut-o! the pressure drops of p and then the  change in viscosity takes it to p . If the temperature in the apparatus  remains stable then p will equal p . Two-hundred data points were   averaged on each plateau. The average values, together with minimum and maximum values, are listed in Table 1.

(10)

p !p p #p M#m *k    " "R . (11)  p !p p #p m k     Here R is an experimental quantity because p , p ,    and p in the ratios on the left-hand side can all be  measured. This expression di!ers from that given in the paper by Mason et al. (1998) by the presence of the term (p #p )/(p #p ). Because all of the p's are about one     bar with the di!erences p !p and p !p being typi    cally fractions of a millibar the ratio (p #p )/(p #p )     is usually unity to within a few parts per million and can be ignored. We can repeat this process for removing the perturbation stream. Now, the pressure changes from p to p as   the #owrate is reduced from M#m to M with a viscosity in the capillary choke of k#*k, and from p to p as the   viscosity changes back from k#*k to k. R is the ratio  when removing the perturbation gas p !p p #p M *k    " "R .  p !p p #p m (k#*k)    

(12)

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G. Mason et al. / Chemical Engineering Science 55 (2000) 5747}5754

Again, the term involving the pressure sums (p #p )/   (p #p ) is virtually unity and can usually be ignored.   Taking the ratio of Eqs. (11) and (12) gives




R m  " 1# R M 






*k 1# , k

(13)

which shows that R will not be exactly the same as  R , but will depend on the relative size of the perturba tion and the relative change in viscosity. 2.4. Goal function and composition Since R and R are not the same, we need to look   ahead to "nd what functions we will need in processing the experimental results. In the in"nitesimal analysis there is no problem, because the experimental results give values of dln k/dX at several compositions and the value G of k at composition X can be found by integrating these G values from X "0 to X . That analysis, for in"nitesimalG G ly small steps, gives a natural way forward for small, but not too large, perturbations. When making the measurements, we are considering two points (X , k) and G (X #*X , k#*k) and the gradient of the chord beG G tween them. In terms of the logarithm function, this gradient is given by *ln k ln(k#*k)!ln k " , (X #*X )!X *X G G G G

(14)

which rearranges to *ln k ln(1#*k/k) " . *X *X G G





3. Experimental apparatus and results



*k 1 *k *ln k " 1! . *X k *X 2 k G G

(16)

This is our goal function and we need to "nd the righthand side in terms of R for the addition perturbation  and in terms of R for the removal perturbation. Both  addition and removal perturbations should give the same value for *ln k/*X . Any di!erences will be due to experiG mental errors in R and R .   The value of *X for both perturbations is the same G (given by Eq. (2)). Using Eqs. (11), (16) and (2) gives the logarithmic viscosity di!erence ratio in terms of R 








(15)

Expanding the logarithm term as a power series in *k/k and truncating it before the second-order term gives




and using Eqs. (12), (16) and (2) gives it in terms of R  *ln k R m  " 1# *X (X2!X) M G G G mR M!2mR   . ; 1# (18) 2(M!mR ) M!mR   For the experiments (Mason et al., 1998) using mixtures of nitrogen and argon, the value of R was approxim ately 0.1 and m/M was about 0.01. With these values the bracketed [ ] terms in Eqs. (17) and (18) are both unity within 0.1% and can be ignored. Only the (1#m/M) term in Eq. (18) is signi"cant as a 1% correction. This is within the sensitivity of the experiments. The ratio of the two ratios (R /R ) is thus 1.01 and so we expect the   experimental value of R to be always about 1.01 times  the size of the ratio R . The size of the perturbation  would have to be very small indeed to eliminate this e!ect. The gradient of the chord *ln k/*X has been calG culated between the points (X , k) and (X #*X , G G G k#*k). X is the composition at the initial point which, G in an experiment, relates to the main #ow composition X. To analyse the experimental results we need to know G the composition at which dln k/dX equals the gradient G of the chord *ln k/*X as this will not be exactly equal to G the initial #ow composition but will lie somewhere between X and X #*X . In fact, to a close approximaG G G tion for small perturbations it is half-way between at a composition of X #*X and the main #ow composiG  G tion must be adjusted to X #*X when the experiG  G mental results are analysed.

*ln k R mR   " 1! *X (X2!X) 2(M#m) G G G



(17)

3.1. Apparatus An apparatus built on the schematic concept of Fig. 1 would be prone to experimental error (Mason et al., 1998). Outside in#uences can be much reduced by using a twin apparatus in which one channel acts as a reference. This idea is shown in Fig. 3. A further modi"cation shown in Fig. 3 is that the gas mixture passes through a further delay line and back-pressure regulator before leaving the apparatus. The bene"t of the twin-channel design is that the environmental #uctuations tend to a!ect the two sides equally and their e!ects are eliminated by making a di!erential measurement. The purpose of the back-pressure regulator is to protect the apparatus from atmospheric pressure #uctuations. Adding the back-pressure regulator itself has an untoward e!ect. The back pressure maintained by the regulator at a given setting depends slightly on the gas viscosity. Adding the extra delay line postpones the arrival of gas of

G. Mason et al. / Chemical Engineering Science 55 (2000) 5747}5754

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Table 1 Typical values from an experimental run

Fig. 3. Schematic diagram of the complete apparatus showing how the two sides are connected. Small changes in the upstream and downstream pressures and also in the temperature of the upstream chokes a!ects both sides of the apparatus in a similar way and thus, in the steady state, the pressure indicated by the pressure transducer in a!ected to a much smaller degree. The dynamic response of each side of the apparatus depends upon the volume of each side of the apparatus. These volumes have to be closely matched if the overall dynamic e!ects produced, say, by a change in downstream pressure, are to be minimised.

di!erent composition at the back-pressure regulator long enough for an experiment to be "nished. A full description of the apparatus and account of possible experimental errors has been given previously (Mason et al., 1998). The complete system is much more complex than the simpli"ed diagram shown in Fig. 3. The output of the di!erential pressure transducer was logged directly. After the system has been properly balanced, carrying out a run is fairly quick. We recorded a complete cycle of the addition and removal of perturbation gas. This linked the addition of the perturbation gas, and the subsequent viscosity step, with the removal of the perturbation gas and its viscosity step all in one continuous data "le. This single record allows R and R to   be determined for one perturbation gas at a particular composition of the main #ow. Any drift in the baseline gives an indication of whether the temperature and pressure have undergone any change during the course of a run. A run took between 5 and 10 min. Measurement of the #owrates in the gas blender (to obtain the composition of the main #ow) took a further 10 min or so. The small perturbation gas #owrates (typically less than 0.2 mL/min) were mostly calculated from the pressure di!erence p !p and comparison with perturbations   for which the #owrates were checked with a low-#ow soap "lm meter. 3.2. Results A typical example of the data record of the variation of pressure with time has been shown in Fig. 2. The steps in the record caused by both #owrate and viscosity changes are relatively noise-free for adding and removing the perturbation gas. The data records were analysed using a separate computer program. The program calculates the average of

Pressure

Average

Minimum

Maximum

p  p  p  p  p 

10.609 47.771 54.498 17.335 10.640

10.593 47.750 54.488 17.322 10.624

10.620 47.795 54.510 17.344 10.657

In this case the voltage values from the transducer (in mV) are for each plateau in Fig. 2. The main #ow composition was 0.2915 mole fraction argon. The perturbation gas was pure argon. The ratio of the #ow of perturbation gas to the main #ow (m/M) was 6.19;10\ and the adjusted composition (X #*X ) was 0.2937. Addition    of perturbation gas: p !p "37.162, p !p "6.727 mV, and     R "(p !p )/(p !p )"0.18102. Removal of perturbation gas:      p !p "!37.163, p !p "!6.695 mV and R "(p !p )/        (p !p )"0.18015. Ratio of two values R /R "1.0048.    

100 data points in two parts of each of the "ve plateau regions, together with the maximum and minimum values in the two blocks of each plateau. These maximum and minimum values give an indication of the extremes of noise in the signal. Values of each step for the data in Fig. 2 are listed in Table 1. The values shown in Table 1 con"rm that the value of (p !p )/(p !p ) obtained when adding the perturba    tion stream is not exactly the same as the value of (p !p )/(p !p ) observed while removing the per    turbation system. This di!erence is expected for "nitesized perturbations and the theory has been given above. For argon}nitrogen mixtures the variation of viscosity with composition is relatively small and so the ratio of R }R is almost exactly (1#m/M). Using R in    Eq. (17) gives *ln k/*X "0.2554. Using R in Eq.   (18), gives *ln k/*X "0.2560. These values di!er by  0.25%, which is close agreement. We made measurements of R and R across the   composition range for the argon}nitrogen system at 243C (the laboratory temperature) using three di!erent perturbation gases, pure argon, pure nitrogen, and a commercial bottled mixture of 47% argon and 53% nitrogen. The correction for the e!ect of adding and removing the perturbation gas (Eqs. (17) and (18)) gave the results shown in Table 2. The results are also plotted graphically in Fig. 4. The results shown in italics in Table 2 are not plotted in Fig. 4. When nitrogen and argon are added as perturbations to an argon}nitrogen mixture the composition appropriate to the gradient which is being measured is not X (the composition of the unperturbed main #ow) but  that of the composition midway between the two extremes of composition X and X #*X . The sign of    *X for the nitrogen perturbation is opposite to the sign  of *X for the argon perturbation and so, even though 

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G. Mason et al. / Chemical Engineering Science 55 (2000) 5747}5754

Table 2 Summary of the experimental values obtained Adjusted main #ow composition, X #*X   

Ratio of perturbation to main #ow, 1000 m M

Logarithmic viscosity *ln k gradient, *X 

0.0032 0.1246 0.1890 0.2937 0.3984 0.4940 0.6027 0.6870 0.7948 0.7980 0.8818 0.8858

6.35 5.95 6.06 6.19 6.32 6.53 6.67 7.52 7.62 7.28 7.02 7.39

0.2971 0.2808 0.2711 0.2554 0.2453 0.2323 0.2192 0.2123 0.1984 0.1993 0.1882 0.1877

0.2980 0.2804 0.2718 0.2560 0.2472 0.2335 0.2205 0.2115 0.1992 0.2009 0.1883 0.1909

6.28 6.38 6.57 6.67 6.87 7.06 7.21 7.36 7.46 7.30

0.2856 0.2743 0.2611 0.2462 0.2374 0.2243 0.2161 0.2055 0.1976 0.1869

0.2842 0.2719 0.2610 0.2447 0.2376 0.2232 0.2160 0.2050 0.1969 0.1875

5.82 5.91 6.03 6.12 6.26 6.48 6.65 6.78 6.83 6.96 7.33

0.2978 0.2854 0.2741 0.2605 0.2575 0.2473 0.2297 0.2157 0.2059 0.1988 0.1881

0.2992 0.2849 0.2742 0.2587 0.2619 0.2458 0.2250 0.2138 0.2064 0.1972 0.1865

0.1216 0.1860 0.2925 0.3952 0.4776 0.5942 0.6834 0.7911 0.8781 0.9964 0.0014 0.1230 0.1874 0.2920 0.3967 0.4923 0.6010 0.6958 0.7929 0.8800 0.9981

Perturbation gas composition, X2 

1.0000

0.0000

0.4700

Perturbation Added Removed

The main #ow composition, X , has been adjusted to allow for the  measured viscosity gradient being at a composition slightly di!erent from the main #ow composition. The ratio of the size of the perturbation #ow to the main #ow varies systematically because the main #ow is a function of viscosity. The logarithm viscosity gradient values are given for adding and removing perturbation gas and also for three di!erent perturbation gases. The most accurate values are indicated in bold. For them "X2 !X " is greater than 0.5. The values in italics are   for almost duplicate points. They are not included in either Fig. 4 or the data processing.

there is a single main #ow composition, the values of *ln k/*X from the two perturbations are for composi tions that di!er slightly from each other and the main #ow composition. For example, with a main #ow composition of 50% argon and perturbations of pure argon and pure nitrogen, the appropriate compositions are 50.25 and 49.75%, respectively.

Fig. 4. Experimental values of the logarithmic viscosity derivative *ln k/*X for nitrogen}argon mixtures at 243C and at an absolute  pressure of 1.32 bar. There are six lines on this graph. Note that the very close agreement for the points for adding and removing perturbation gas. The only error in these points is the measurement of the sizes of the pressure steps and the correction, using the theory, for the di!erence between adding and removing the perturbation stream. There is less good agreement between the results for perturbations of di!erent compositions. This is almost certainly due to the factor (X2 !X ). When   the composition of the perturbation #ow X2 is similar to the composi tion of the main #ow X then the proportional error in the di!erence  can be considerable. Solid symbols are used for adding perturbation gas and hollow symbols represent removing perturbation gas. (䉱) indicates results using pure argon as perturbation gas, (䉲) indicates pure nitrogen and (䉬) indicates a gas mixture (0.47 mol fraction argon) as perturbation gas.

3.3. Interpretation The magnitude of the errors in the measurement of gradient can be estimated from Table 2. The values of *ln k/*X for adding and removing perturbation gas  are, after correction, usually very similar, mostly di!ering by less than 1%. However, for these values X2 !X is   a constant factor and the comparison of adding and removing a particular perturbation is really only a test of the precision of maintaining #ows and of measuring the voltages. When we compare the values of *ln k/*X for  similar values of the main gas #ow composition but with diwerent perturbation gases the consistency is not as good. For this comparison X2 !X is not a constant  

G. Mason et al. / Chemical Engineering Science 55 (2000) 5747}5754

factor because X2 is either zero, or unity, or 0.47 mol  fraction argon (the three di!erent perturbation gas compositions). There is very little error in the perturbation gas composition when nitrogen and argon are used as perturbation gases because they are both pure gases. Consequently, any di!erences between values of *ln k/ *X have to be caused by an error in the main #ow  composition X . This conclusion is borne out by careful  study of Fig. 4. Three perturbation gases were used and across the graph the results from two perturbation gases agree quite closely. The errors are greatest when the composition of the perturbation X2 is nearly the same  as the carrier composition X because then the propor tional error in the di!erence X2 !X can be quite   large. The results of two repeat experiments (indicated by italics in the top panel of Table 2) give an idea of the experimental reproducibility in unfavourable circumstances. The most consistent results are when "X2 !X " is   greater than about 0.5. The viscosities of argon and nitrogen have been measured by other workers. We can compare our results with theirs by "nding the ratio of the argon viscosity relative to that of nitrogen. There are various strategies that can be used to do this. The most straightforward is to numerically integrate the values of *ln k/*X across the com position range in the way described previously (Mason et al., 1998) and thus "nd k /k  . The results in italics in  , Table 2 were not incorporated into the numerical integration for argon perturbation because the extrapolation to argon mole fraction of unity would have been compromised. There are six sets of data in Table 2 and six lines on Fig. 4. Table 3 gives the six values for k /k  ob , tained from the integration of the lines. All of the values of k /k  in Table 3 are in quite good  , agreement, even though the gradients shown in Fig. 4 appear to have considerable error. The reason for the good agreement is that the measurements are of gradients and subsequent integration reduces the errors. However, if we ignore the values for which X2 !X   produces the large errors and con"ne attention to the most accurate points (shown bold in Table 2) we get a value for k /k  of 1.2669 when adding the perturba , tion gas and 1.2672 when removing it. The ratio of the viscosity of argon to the viscosity of nitrogen has been measured by many workers, but no results for 243C are

Table 3 Values of k /k  obtained from integration of the six data sets shown  , in Fig. 4. There were three di!erent perturbation gas compositions and results were obtained for the addition and removal of each of the perturbation gases Perturbation

Argon

Nitrogen

Ar}N mixture 

Adding Removing

1.2630 1.2637

1.2697 1.2688

1.2731 1.2721

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available and so, for comparison, other workers' results have to be interpolated. The results of Hellemans, Kestin and Ro (1972) obtained using an oscillating-disc viscometer give a value of 1.2707 for the viscosity ratio at 243C. Using a capillary viscometer Clarke and Smith (1968) obtained results which give a ratio of 1.2699. Similarly from Dawe and Smith (1970) the result is 1.2721 and from Matthews, Scho"eld, Smith and Tindell (1982) the result is 1.2707. Our averaged value of 1.2670 is consistently lower but the maximum di!erence is only 0.4%. It should be remembered that our apparatus was not thermostatically controlled because these were preliminary experiments and because it was known that the gradient of viscosity with composition does not vary much with temperature. The viscosity ratio of the pure components varies by about 0.001 deg\ centigrade and so an error of a few degrees is enough to bring our results within those of other workers.

4. Conclusions The theory for measuring the gradient of the viscositycomposition function for "nite changes in composition has been given. The major correction is for the size of the added perturbation stream relative to the main #ow of gas. This correction is only needed for results obtained when removing the perturbation stream. In a series of experiments using three di!erent perturbation gases, each being added and removed, the results were all in reasonable agreement. It is evident from the results that the major error in the experiments arises from the lack of precision with which the main #ow gas composition is known. Kestin, Kobayashi and Wood (1966) and Iwasaki and Kestin (1963) also had this problem when they made direct measurements of the viscosity of gas mixtures and almost all of their error was due to the imprecision of the gas compositions. Indeed, it could be concluded, that it might be better to determine the composition of the main #ow by making two measurements of *ln k/*X , one using argon and the other using G nitrogen as perturbation gas whilst keeping the composition of the main #ow constant. The value of X, the main G #ow composition, would be the one that makes the two values of *ln k/*X the same. We cannot do this with our G results because the main #ow composition was often altered between changing the perturbation gas compositions and it could not be exactly reset.

Notation K ¸ m p

constant length of capillary tube molar #ow rate of perturbation gas pressure

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p  p  p  p  p  P .0 M R R ¹ X

G. Mason et al. / Chemical Engineering Science 55 (2000) 5747}5754

pressure at gauge before perturbation #ow is added pressure at gauge after perturbation #ow is added pressure at gauge after viscosity change has occurred pressure at gauge after perturbation #ow is removed pressure at gauge after viscosity change has occurred "xed pressure set with a back-pressure regulator molar #ow rate of carrier gas the gas constant ratio of viscosity pressure change to #ow pressure change temperature mole fraction

Greek letters k viscosity * a small, but "nite, size di!erence Subscripts i species i add adding perturbation rem removing perturbation Superscripts 0 carrier gas T perturbation gas

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