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Capillary viscometry by perturbation of flow and composition
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Chemical Engineering Science, Vol. 53, No. 15, pp. 2665—2674, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0009–2509(98)000106–7 0009—2509/98/$—See front matter
Capillary viscometry by perturbation of flow and composition G. Mason,* B. A. Buffham, M. J. Heslop and B. Zhangs Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, U.K. (Received 4 November 1997; received in revised form 2 April 1998; accepted 2 April 1998) Abstract—A new technique for making viscosity measurements on gas mixtures is introduced. The composition and flowrate of a mixture flowing through a capillary tube are perturbed by adding a small stream of perturbation gas. This is usually a pure, individual component of the mixture. The pressure at the inlet of the capillary tube rises when the perturbation gas is added and this pressure increase is proportional to the flowrate change. Because there is empty volume between the point where the perturbation gas is added and the capillary tube, it is some time later that the pressure changes again when the composition of the gas flowing through the tube changes. This second pressure change is proportional to the change in viscosity. The ratio of these two steps of pressure is proportional to d ln k/dX where k is the viscosity and X is the i i mole fraction of component i. An apparatus has been developed which is capable of making measurements with suitable precision and some preliminary data for the nitrogen—argon system at 1.2 bar and 24°C are given. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Viscosity; gas viscometry; gas mixtures; physical properties; transport properties; viscometers. INTRODUCTION
Gas viscosities have been measured for over a century and there are data books listing collected values (Touloukian et al., 1975). Accurate measurements of the variation of gas viscosity with composition, pressure and temperature can give information about how molecules, and particularly pairs of different molecules, interact with each other (Chapman and Cowling, 1970). Viscosity is therefore an important parameter in thermophysical and transport-property models (Bird, 1994). Because of the precision required and the methods used, measuring viscosity can be a slow process. Nevertheless, most single-component viscosities are well documented; measurements on binary systems are quite common but only a few ternary and virtually no quaternary systems have been measured. We give here a new comparative method for measuring gas viscosities that is potentially fast, should work with multicomponent systems, and may well work at very high temperature. Touloukian et al. (1975), and Kestin and Wakeham (1988) give authoritative reviews of the theory and practice of the methods that are available to measure viscosity. The most-used methods for accurate
* Corresponding author. Tel.: 01509 222509; fax: 01509 223923; e-mail: [email protected]. sPresent address: Department of Chemical Engineering, Tianjin University, Tianjin, 300072, P.R. China.
measurement are oscillating-body viscometry and capillary-flow viscometry. The principles of both methods are well established and with modern techniques both can give very accurate values. Values obtained from both methods are mutually consistent except at very high temperatures where systematic errors may give differences. Because of their precision and ability to give absolute values, these methods would normally be the primary choice for viscosity measurement, even if they are slow. Viscosity is a dynamic variable and something in the apparatus has to move in order that viscosity can be measured; somehow the fluid must be sheared. In an oscillating body viscometer, an axially symmetric body performs torsional oscillations inside a stator filled with fluid. The oscillating body, usually a disc, is suspended from a taut wire and the period is often tens of seconds. The change in frequency and damping decrement depend only on the density and viscosity of the fluid and the torsional stiffness of the suspension system. In principle, both the viscosity and density of the fluid can be obtained from measurements of the frequency and decrement. However, density can be measured by other methods and so the oscillating-body method usually only measures viscosity. In a capillary viscometer, the gas flows through a capillary tube and the pressure drop and volumetric flowrate through the tube are measured. The usual problem is that, for the method to be absolute, the
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diameter of the tube has to be uniform and be known with great precision. A variant is the rolling-ball viscometer in which a ball rolls down inside a reasonably close-fitting tube. This method can give very reproducible results but the basic principles are poorly understood. The gap between the ball and tube is crescent-shaped with the ball rotating and these factors give a complex flow-velocity profile. The oscillating body and capillary viscometers can be used with gas mixtures as well as with pure components. All that is needed is a supply of gas of known composition. Obtaining this can sometimes be a problem. Both methods measure the absolute viscosity. However, if relative viscosities are acceptable then either method can be used in the ‘relative’ mode provided that a gas of known viscosity is available to determine the apparatus constant. We present here a method that is new in both concept and application. There are two new ideas. The first idea is that instead of measuring the gas viscosity directly we measure the gradient of the viscosity—composition function. We change the composition slightly and measure the small change in viscosity. If gradients are known across the composition range then integration can give values of the relative viscosity across the composition range. The attraction of this approach is that, all things being equal, measurement of differentials followed by integration is more accurate than direct measurement. Alternatively, the gradients can be measured with less accuracy than the direct values and still give equally good values of relative viscosity. The second idea involves the detail of the apparatus and the theory of laminar flow through tubes. We use a capillary tube first as a flow meter and then shortly after as a viscometer: when one response is divided by the other the gradient of viscosity with composition is obtained. The advantage of this idea is that the result of the experiment will not depend on the properties of the capillary tube. We therefore have a capillary tube viscometer which produces results that do not depend on the uniformity of the tube or, indeed, on knowing its diameter accurately. Also, only the properties of the gas in the capillary tube are measured, and so the tube can be heated to high temperature and the method should still work. The method is only relative because, and this is only one way of regarding it, the tube constant in the equation for laminar flow has been found by measuring the change in pressure caused by changing the flowrate through the tube. Normally, a relative viscometer would be calibrated using a gas of known viscosity. The same tube constant is used in the expression for the change in pressure caused by the viscosity change. But the composition change also depends on the flowrate change, so, when one pressure change is divided by the other, both the tube constant and flowrate change cancel. In absolute capillary viscometry it may be necessary to take account of secondary effects such as slip at the tube wall, kinetic energy loss and the
entrance length needed to establish laminar flow. In relative methods, including ours, these effects tend to cancel. THEORY OF THE METHOD
(i) Fundamentals Consider the schematic apparatus shown in Fig. 1. There are five components. The main stream of gas mixture enters the upstream capillary choke A at constant pressure. This choke is typically a length of small-bore capillary tube and develops most of the pressure drop in the apparatus. The gas flows through a valve B and enters the delay line C. The delay line is usually a length of 4 mm i.d. tubing and contains volume but produces very little pressure drop. After leaving the delay line the gas mixture passes through the downstream capillary choke E and escapes to atmosphere. About 1% of the pressure drop occurs in choke E. A pressure gauge D measures the pressure just upstream of the downstream capillary choke E. An experiment consists of allowing the gas to flow until the pressure indicated by gauge D is constant and then changing valve B to admit a small flow of perturbation gas. Typically, the perturbation gas is one of the individual components of the mixture and has a flowrate of about 1% of the total (or main) flowrate. The response of the pressure gauge (or transducer) is monitored. An experiment does not start until the gauge indicates a constant value. When the valve is switched and the perturbation flow enters the system, the pressure at the gauge rises and levels out. The viscosity of the gas in the downstream capillary is unchanged because the new mixture is traversing the delay line and has not yet reached the downstream capillary choke. Therefore, the pressure gauge is responding only to changes in flowrate in the downstream choke E. When, eventually, gas with the new composition reaches the downstream capillary choke,
Fig. 1. Schematic diagram showing the principles of operation of the apparatus. Initially valve B is set to pass the main gas stream. The resistance of choke A is much greater than that of the rest of the system and so determines the flowrate. In an experiment, the valve B is switched to add a small stream of perturbation gas to the main stream. The increase in flowrate in the downstream choke E is sensed by the pressure at the gauge D increasing. But, because of the presence of the delay line C, the gas in this choke still has the original composition. When gas of the new composition reaches choke E, the pressure shown by gauge D changes again.
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the pressure gauge reading alters again. It will rise if the gas viscosity has increased, and fall if it has decreased. Because the flowrate is now constant, the pressure gauge is now responding only to changes in viscosity. Consequently, the pressure gauge reading can be regarded as the output of a capillary meter which has acted first as a flowmeter and second as a viscometer. It is the delay line that separates these two responses. Without the delay line the middle plateau region of the pressure-gauge response would not be observed and the response could not be separated into two effects. (ii) Analysis Let us analyse the behaviour of the apparatus shown in Fig. 1 when the stream of perturbation gas is added. The salient features of the expected response are shown in Fig. 2. The steps in the pressure gauge record can be identified as three plateaus, p , p , and 0 1 p . Initially no perturbation gas is added and the 2 pressure gauge reads p . At time zero the small stream 0 of perturbation gas of known composition is added. This immediately increases the flowrate in the downstream capillary choke but does not yet change the composition. The pressure indicated on the pressure gauge rises to p and the increase is only caused by the 1 flowrate change in the downstream capillary choke. Later, when the composition front has traversed the delay line, the composition change reaches the downstream capillary choke. The change in viscosity causes, say, the pressure to rise (say because sometimes it falls) to p . When the capillary choke acts as 2 a flowmeter p !p is proportional to the increase in 1 0 flowrate and when it later acts as a viscometer p !p 2 1 is proportional to the change in viscosity. The change in pressure p with distance z for z laminar flow of a fluid through a section of tube is given by dp z"!K kQ c dz
(1)
where Q is the volumetric flowrate, k is the viscosity and K is a constant depending on the characteristics c of the tube. If the gas is ideal then MR¹ Q" (2) p z M is the molar flowrate of gas, p is the pressure at z distance z, ¹ is the absolute temperature, and R is the gas constant. Consequently the longitudinal pressure gradient can be expressed as dp p z"!K kMR¹. z dz c
(3)
The viscosity of a gas depends primarily on temperature. At low pressure it is almost independent of pressure and so the changes in pressure in the downstream capillary choke (which really are very small) do not cause a detectable change in viscosity. As k is constant eq. (3) can be integrated along the whole
Fig. 2. Main features of the experimental response. At time zero, a perturbation gas of known composition is added to the main stream and the pressure indicated by the gauge rises from p to p . During the time that the composition front 0 1 remains in the delay line, the pressure remains at p . The 1 increase in flowrate caused by the addition of perturbation gas is proportional to p !p . When the front reaches the 1 0 capillary tube, the change of viscosity caused by composition change causes a further response and the pressure changes from p to p . This may be a rise or fall depending on 1 2 whether the viscosity increases or decreases. The viscosity change is proportional to p !p . 2 1
length of the capillary tube to give p2!P2 "2KkMR¹. (4) BPR where K"K ¸, ¸ is the length of the tube and p is the c pressure at the inlet of the downstream capillary choke. The squared terms arise because the gas in the choke is compressed, being denser at the inlet than the outlet. P is the outlet pressure of the downstream BPR capillary choke. Figure 1 shows this pressure to be atmospheric pressure, but it may be above atmospheric pressure if, for example, it is set to a constant pressure with a back-pressure regulator (hence the subscript BPR). If some changes in flowrate and viscosity have occurred such that the inlet pressure of the capillary tube has risen by dp in response to a change in flowrate dM and a change in viscosity dk then eq. (4) gives (p#dp)2!P2 "2K(k#dk)(M#dM)R¹. (5) BPR Expanding and subtracting eq. (4) gives 2pdp#(dp)2"2K(kdM#Mdk#dMdk)R¹. (6) For small changes, the second-order terms (dp)2 and dMdk are small and so pdp"K(kdM#Mdk)R¹
(7)
or 2pdp dM dk " # . (8) M k p2!P2 BPR Equation (8) is the basic equation for our analysis. We can now use it twice, once for each of the two steps in pressure in Fig. 2. Since there is no viscosity change
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(dk"0) for the first step in Fig. 2, eq. (8) becomes 2p (p !p ) dM 0 1 0" . (9) (p2!P2 ) M 0 BPR For the second step in Fig. 2, there is no change in flowrate (dM"0) but only a change in viscosity so eq. (8) becomes 2p (p !p ) dk 1 2 1" . (p2!P2 ) k 1 BPR Taking the ratio of eq. (9) to eq. (10) generates
(10)
p (p !p ) p2!P2 M dk 1 2 1 0 BPR" . (11) p (p !p ) p2!P2 k dM 0 1 0 1 BPR If we choose the capillary and the pressure step sizes such that p +p +p and p2!P2 +p2!P2 , 0 1 2 0 BPR 1 BPR then M dk p !p 2 1" . (12) k dM p !p 1 0 If the main stream has a composition of X0 and the i perturbation stream has a flowrate of dM and composition XT , the addition of the perturbation stream i will produce a composition change in the main stream of dM dX " (XT!X0). i M#dM i i
(13)
We use the subscript i to denote any one of the components in the mixture. Equation (13) comes from a simple mass balance. If we eliminate dM between eqs. (12) and (13) then we can arrive at an equation giving the change in k in terms of the change in X and the size of the two measured steps in pressure. i Also dM is negligible relative to M for differentialsized steps. The final working equation is 1 dk p !p 1 1 " 2 . (14) k dX p !p (XT!X0) i 1 0 i i In the limit when dk and dX become very small i dk/kdX P(dk/dX)/k and a differential equation is i obtained. 1 dk 1 dk d ln k p !p 1 1 lim " " " 2 . k dX k dX dX p !p (XT!X0) dXi?0 i i i 1 0 i i (15) In an experiment we measure the two pressure changes p !p and p !p . XTand X0 are both 1 0 2 1 i i input streams and will be of known composition. Generally, a pure component will be used as perturbation gas and its composition is certain. Thus, the gradient d ln k/dX is determined from the ratio of i p !p to p !p and knowledge of XTand X0. No 2 1 1 0 i i other variable need be known. In fact, a more detailed analysis shows that the tube does not have to be uniform and the gas need not be ideal. The method therefore only depends on how accurately the two measurements of the change in pressure can be made.
(iii) Sensitivity to random errors Although this method is elegant in principle it is still subject to the effects of random and systematic errors and, as we shall see, these have the potential to be serious. For example, let us aim for a precision of about 1% in d ln k/dX . For many gas mixtures the i variation of viscosity with composition is not large and, typically, Dd ln k/dX D is about 0.1 for a binary i mixture. So of the two steps in Fig. 2, p !p will 2 1 usually be about one-tenth the size of p !p . As 1 0 a result, the same-sized error in measuring the pressure will cause a ten times greater proportional error in p !p than in p !p . So, ideally, we would like 2 1 1 0 to make p !p as large as possible. However, be2 1 cause we need to keep the perturbation relatively small dX will be, at most, 0.01 in an actual experii ment and so the main gas flowrate M will be over 100 times greater than the perturbation flowrate dM. We aim to measure dM (via p !p ) to 1% accuracy and 1 0 so the main flowrate M will have to remain stable to 1 part in 104. But p !p is one-tenth of p !p and 2 1 1 0 so an error of 1% in (p !p )/(p !p ) will result if 2 1 1 0 the main flow drifts by 1 part in 105 in the course of an experiment. Therefore, very special attention will have to be paid to keeping the main flow constant. Temperature plays a part here too. For example, if the main flow is produced by a fixed pressure drop across a fixed capillary tube (the upstream capillary choke), the absolute temperature will have to be constant to 1 part in 105—that is 0.003°C—because the pressure drop is proportional to volumetric flow rather than to molar flow. The pressure gauge will also have to be stable and have a resolution better than 1 part in 105. The upstream and downstream pressures similarly need to be kept stable to the same standard. All of this is true for the apparatus shown schematically in Fig. 1. We will see later that a differential apparatus design overcomes many of these potential difficulties. In order to calculate d ln k/dX we have to divide i (p !p )/(p !p ) by XT!X0. If the compositions 2 1 1 0 i i of the main flow and perturbation flow gases are known to a precision of, say, 0.1% mole fraction, which is quite accurate, then XT!X0 would be i i known to about 1 part in 1000 if XT"1 and X0"0. i i However, X0 has to be varied across the range 0 to 1. i If XT"1 and X0"0.9 then XT!X0 is only known i i i i to about 1 part in 100. We can conclude that the most precise results will be obtained when XT and X0 differ i i by at least 0.5 and the results will be very poor when XT and X0 are almost the same. Using a pure comi i ponent as perturbation gas makes XT either 0 or i 1 and there is no error in its composition. There are several techniques that can be used to overcome some of the difficulties caused by having to keep the main gas flow constant to precise limits. The first is to borrow an idea from chromatography and use a second (or reference) stream which is closely similar to the system shown in Fig. 1, but without the valve B. In essence, this means duplicating the upstream capillary choke, the delay line and the
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downstream capillary choke. The reference stream now provides a similar pressure to the measurement stream pressure and the pressure gauge can be replaced by a differential pressure gauge (or transducer). For our 1% accuracy the differential pressure gauge need only detect to 1 part in 1000 instead of 1 part in 100,000 before. Also, provided the temperature throughout the reference stream is kept the same as the temperature throughout the measurement stream, the effect of temperature changes disappears. Keeping temperatures locally the same can be done relatively easily by twisting the capillary tubes and also the delay lines, together. However, the main benefit of the differential design is that the pressures upstream and downstream are common to both the measurement and reference streams and any small changes in pressure of both chokes caused, say, by drift in a pressure regulator affect both streams together and cause little change at the differential pressure transducer. This differential design overcomes the difficulty of maintaining the upstream and downstream pressures constant to 1 part in 100,000. EXPERIMENTAL APPARATUS
In order to apply the theory of the method many technical details have to be settled before a working machine can be made. We were fortunate in that we observed the effects described in the THEORY OF THE METHOD, above, whilst working with a novel analytical chromatograph (Buffham et al., 1993) and an apparatus used to measure binary gas adsorption (Mason et al., 1996, 1997) and ternary gas adsorption (Heslop et al., 1997). The double-sided system common to gas chromatographs was employed in these devices. The advantages of this particular feature have already been explained. We therefore started with an apparatus that partially worked and gave measurable effects and which only had to be progressively developed. By using an approximately 1 bar pressure drop across the upstream capillary chokes a main gas flowrate of about 25 ml/min was produced. Such a modest main gas flowrate makes the flow of perturbation gas very low (usually less than 0.25 ml/min) and such a low flow cannot conveniently be produced by a fixed pressure drop across a capillary tube. However, only a stability of 1 part in 1000 is required. We have found that Porter Instruments Type VCD 1000 mass flow controllers can give these low flows consistently provided that flow through the controller is never switched off. Their dynamic response at low flowrates is very sluggish. We chose to work with the nitrogen—argon system at ambient conditions whilst developing the apparatus. The purity of the nitrogen and argon was specified by the supplier to be 99.998%. (i) Sensitivity of the pressure transducer Only tiny changes in differential pressure are observed and so a very sensitive differential pressure transducer is employed. We used a Furness Controls
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type FCO 40 with a sensitivity of $10 mm H O/volt. 2 These transducers were originally designed for measuring pressures developed by pitot tubes in wind tunnels. They are of diaphragm type and are surprisingly rugged being able to withstand relatively massive pressure overload caused by accidental major leaks in the apparatus. A typical voltage change of the output signal produced by adding the perturbation stream was about 40 mV, corresponding to a pressure difference of about 0.4 mm H O. The transducer can 2 easily detect less than one-thousandth of this pressure. The voltage signal was digitised with an analogue to digital converter and recorded using a personal computer. (ii) Pressure and back-pressure regulators A pressure regulator contains a valve which moves to keep the outlet pressure constant even though the outlet flowrate changes. A back-pressure regulator performs a similar function but this time keeps the inlet pressure constant. Normally both types of regulator use atmospheric pressure as the reference and set the pressure by compressing a spring. A problem is that any fluctuations in atmospheric pressure are transmitted to the regulated pressure and, in our apparatus, this can cause pressure noise at the pressure transducer (Heslop et al., 1996). Because of the design of the apparatus the back-pressure regulators, and particularly the outlet regulator, cause the greatest effect. To avoid this form of pressure noise we modified commercial back-pressure regulators by sealing off the spring cavities, but still keeping the spring adjustment. These back-pressure regulators are termed BPR(M). (iii) Setting the main gas flow Figure 3 shows a schematic flowsheet of our final apparatus for the nitrogen—argon system based on the theoretical and practical considerations which have already been described. The main gas mixture is made by blending nitrogen and argon in a gas mixing machine of our own design. The gas mixing machine has a constant outlet pressure which is set by one of the special back-pressure regulators (BPR-1(M)). This pressure is applied upstream of the two upstream capillary chokes. The main gas mixture passes through both chokes, one gas stream flowing on through the measurement side and the other through the reference side. Most of the pressure drop (it was about 1.1 bar) in each stream occurs across the upstream chokes, which are 150 cm lengths of 0.25 mm i.d. (1/16 inch o.d.) stainless-steel tubing. The two needle valves NV-1 and NV-2 are used for balancing the flows in the measurement and reference sides when there is no perturbation gas flowing into the measurement side. Unfortunately, changing needle valve settings adds to (or reduces) the volume in the valves and this will temporarily change the flowrate through the downstream capillary choke. Placing the valves upstream of the upstream capillary chokes reduces the effect of these volume changes and makes balancing the flows very much easier.
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Fig. 3. Schematic flowsheet of experimental apparatus. BPR-1 (M) and BPR-2 (M)"Modified backpressure regulators based on Porter Instruments 9000 BPR; BPR-3"Porter Instruments 9000 backpressure regulator; DPT"Furness Controls FCO 40 differential pressure transducer; MEVA"»ICI ETMA micro-electric two-position valve actuator; MFC-1, MFC-2 and MFC-3"Porter Instruments VCD 1000 flow controller; MFC-4"Condyne 202 flow controller; NV-1 and NV-2"Brooks Instruments needle valves; PR-1, PR-2 and PR-3"Porter Instruments 8286 pressure regulators; 3PSV-1 and 3PSV-2 and 3PSV-3"»alco UWP three-port switching valves; 4PSV-1 and 4PSV-2"»alco UWP four-port switching valves; 3WV-1, 3WV-2 and 3WV-3"SSI 02-0182 three-way valves.
A separate stream of main gas flow mixture, the flow of which is set by a mass flow regulator, goes to purge unwanted perturbation gas. (More about this later). (iv) Delay lines and downstream capillary chokes Downstream of the upstream capillary chokes are the delay lines, which are 100 cm lengths of 4 mm i.d. (1/4 inch o.d.) nylon tubing, followed by the two downstream capillary chokes which are 50 cm lengths of 0.76 mm i.d. (1/16 inch o.d.) stainless-steel tubing. There was a pressure drop of about 0.1 mbar along the delay lines and a pressure drop of about 15 mbar along the downstream capillary chokes. The outlets of the two capillaries are combined together and the combined streams pass through a further delay line before reaching the final back-pressure regulator. The pressure set by any back-pressure regulator is very slightly affected by both flowrate and gas viscosity. The final delay line retains sufficient volume of gas to keep the composition at the outlet backpressure regulator (BPR 2(M)) constant during an experiment. Also the purge stream, which is set by the mass flow regulator, and into which the perturbation stream flows when it is not being added to the measurement side, joins before the third delay line and this arrangement keeps the flow through the back-pressure regulator constant when the perturbation gas valve is switched. The result is that the pressure set by the outlet back-pressure regulator is kept
constant to close limits because neither the flow nor the composition changes during the course of an experimental run. The differential pressure transducer measures the pressure difference between the measurement side and the reference side at the positions before the two downstream capillary chokes. (v) Setting the perturbation flows The flowrates of the perturbation gases are adjusted to be between 0.5% and 1% of that of the main gas flow by pressure regulators PR-1, PR-2 or PR-3 and mass flow controllers MFC-1, MFC-2 or MFC-3, respectively. A mass flow controller maintains a pressure drop across a flow restrictor by means of a selfadjusting valve. In principle, the flowrate delivered by the controller can be set in two ways; the flow restrictor can be adjusted (by making it a needle valve) or the pressure drop can be altered by compressing a spring. Porter mass flow controllers are the latter type and use a sintered metal body as a fixed flowrestrictor and have a spring to adjust the differential pressure. There are three mass flow controllers, one for each perturbation gas, so that each one can run with a constant flowrate and gas composition. The perturbation gas is selected by using two four-way valves 4PSV-1, 4PSV-2 and one three-way valve 3PSV-1. In Fig. 3, the selected perturbation gas is the mixture of nitrogen and argon. The flowrate of perturbation gas can be measured from one port of
Viscometry by flow and composition
3PSV-1. The selected perturbation gas is added or removed by switching the three-port valve, 3PSV-2, which is actuated by a VICI ETMA micro-electric two-position valve actuator. The valve plug was originally turned manually, but we have found that the electric actuation produced more uniform rotation. When the perturbation gas is added, it goes to point B through the valve 3PSV-2, and is then added to the stream of gas flowing through the measurement side of the apparatus. When the perturbation is removed (as in the position shown in Fig. 3) it goes through 3PSV-2 and reaches point C, where it is combined with the purge flow of gas mixture. The purpose of this arrangement is to exactly match the conditions in the measurement stream with the conditions in the purge stream. The valve 3PSV-2 always transfers a small volume from one stream to the other when it is switched over. Matching conditions at each side of the valve avoids surges in flow and diffusion effects in the valve and helps to generate a sharp concentration front. The purge stream flows through a third downstream length of capillary tube to reach point A. The two streams of main gas flow at point A are joined in front of the third delay line. As mentioned already, this arrangement maintains the constant flowrate and composition requirement in BPR-2(M). Special fittings are used with valve 3PSV-2 which bring the main gas flow into the valve and then out again through a concentric tube. The arrangement minimises the dead volume in the pipework that occurs with the valve. The fittings are more fully described by Addison et al. (1994) who used them to produce dispersion-free fronts. Because the flowrate of the perturbation gas is very small, it takes days to purge the gas in the mass flow regulators. It even takes hours to displace gas in the tubes after changing from one perturbation gas to another. On the other hand, keeping the perturbation flowrate constant is very important and adjustments to mass flow controllers MFC-1, MFC-2 or MFC-3 must be minimised because they take many hours to settle down again. Ideally they should be left alone. This is why the SSI 02-0182 three-way valves are combined with lengths of high resistance capillary tube (about the same as the upstream capillary chokes) and connected to valves which by-pass the mass flow regulators. Opening a valve flushes the pipework with new perturbation gas. This operation is done with the perturbation gas going into the purge stream and so the measurement stream is not disturbed. These modifications introduce no dead volume and enable the perturbation gases to be changed in a few minutes. This speeds up the experimental process greatly. Before adding the perturbation gas the pressure at both sides of the perturbation stream addition valve have to be exactly equal, otherwise a sharp deviation in pressure occurs when the perturbation valve is turned. Because they are all connected together just before the third delay line, both main gas flows and the purge flow plus the perturbation gas flow all have
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the same outlet pressure. Slight adjustment of the purge flow using MFC-4 changes the pressure drop through the third downstream capillary choke and alters the pressure at the perturbation stream addition valve. Adjustment is carried out until turning the perturbation stream addition valve 3PSV-2 gives an even step change in the response of the pressure transducer with no overshoot and no undershoot. (vi) Gas mixtures The gas blender has two main roles. First, it enables the main-flow composition to be easily and quickly set to approximately to the desired value. It maintains this composition constant even when the pressure downstream changes. Second, it enables the approximately set main-flow composition to be measured. Because of the blender’s design, it is possible to shut off the flow of either of the components and for the blender to continue to deliver the other component at the same flowrate as it was previously contributed to the mixture. This feature enables the composition of the mixture delivered by the blender to be precisely determined by measuring the individual flowrates of the gases making up the mixture with a soap-film meter. The flowrates were measured by timing the passage of the soap film between particular graduation marks. Mole fractions thus determined do not depend on the volumetric calibration of the meter tube. Determining the gas mixture composition has been a problem in work on the viscosity of gas mixtures and in some work has been the greatest source of uncertainty (see, for example, Kestin and Leidenfrost, 1959; and Iwasaki and Kestin, 1963). Kestin’s team overcame the problem by weighing the gases into a bottle (Kestin et al., 1966). In more recent work, Matthews et al. (1982) used a gas-density balance. Our use of the blender has the advantage that the composition may be changed quickly. In other work we had come to distrust the analysis of the nominal 50% argon—50% nitrogen mixture we used as one of the perturbation gases. A null experiment comparing the ‘50—50’ mixture with gas from the blender showed that the mixture composition was 47 mole% argon. In the null experiment the ‘50—50’ mixture was used as the perturbation gas and the main gas stream was provided by the gas blender. The composition of the main gas stream was adjusted until adding perturbation gas produced no discernible viscosity step. The ‘47% argon’ main gas stream did exactly this and this composition has been used in the analysis of our results. (vii) ¹emperature control The entire apparatus is not in a temperature-controlled bath. This may seem surprising but it was done for convenience because it eased making changes and developments to the pneumatic system. The gradient of the relative viscosity with composition curve depends much less strongly on temperature than the viscosity—composition function. However, it did prove necessary to enclose the apparatus in a closed
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metal cabinet so that it was kept at constant, but not controlled, temperature. The box isolated the apparatus from rapid changes in the environment. We believe that radiation heat transfer causes most thermal noise in the apparatus, and this is why the simple metal cabinet worked so well. EXPERIMENTAL RESULTS
(i) Data acquisition All the development work was carried out using a chart recorder to monitor the output from the differential pressure transducer. However, actual data acquisition of the pressure changes was carried out using a 16 bit analogue-to-digital converter and ¼orkBench PC software to digitise the signal from the transducer 25 times a second. When the system is properly balanced, carrying out a run is fairly quick. We found it useful to record a complete cycle of the addition and removal of perturbation gas. The cycle links the addition of the perturbation gas, and the subsequent viscosity step, followed by the removal of the perturbation gas and its viscosity step all in one continuous data file. A run took about 10 min. 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 typical data record is shown in Fig. 4. This is a direct plot of the raw digitised data file. The noise is lost in the thickness of the plotline. In Fig. 4 the variation of the differential pressure transducer signal for both the addition and removal of the perturbation gas are shown. The line AE is the baseline of the signal. After the addition of the perturbation stream, the voltage signal increases rapidly indicating the additional flow of the perturbation gas. The record now becomes horizontal, B, whilst the composition front caused by adding the perturbation gas is moving through the delay line. When the front arrives at the downstream capillary the resulting change of viscosity changes the signal from B to C and the system reaches a new steady state. The change between B and C is not as steep as that between A and B because there is dispersion in the delay line (Addison et al., 1994). The perturbation gas is removed after the vertical broken line and the response is repeated in the opposite direction. The character of the noise is shown in Fig. 5. Plotted in Fig. 5 relative to arbitrary datum points are 10 s stretches of plateaus A, B and C in Fig. 4. The stretches are 10—20 s, 60—70 s and 110—120 s and each stretch contains 250 points. The resolution of the digitisation is just sufficient to reveal the short-term noise. (ii) Data analysis The plateaus of both flowrate and viscosity are relatively free of noise. The data records were analysed using a separate computer program. The program calculates the average of several hundred data points in two parts of the plateau region of each step
Fig. 4. A typical experimental result. The mole fraction of argon in the main gas flow is 0.686 and nitrogen is used as the perturbation gas. This is a computer-generated plot of the original raw digitised data file. The noise is lost in the thickness of the plotline. The noise is displayed in Fig. 5 where the boxed stretches are plotted much enlarged. See also Table 1.
Fig. 5. The character of the short-term noise is shown here by plotting the 10—20 second (A), 60—70 s (B) and 110—120 s (C) stretches of plataus A, B and C in Fig. 4 relative to arbitrary datum points. Each of these stretches contains 250 points. The resolution of the digitisation is just sufficient to reveal the short-term noise.
and records the maximum and minimum values in the ranges examined. The maximum and minimum values give an indication of the extremes of noise in the signal. Values of each step for the data in Fig. 4 are listed in Table 1. Close examination shows that the pressure step sizes obtained when the perturbation stream is added (p !p "43.278 and p !p "!6.411) are not 1 0 2 1 exactly the same as those observed (p !p " 1 0 !43.255 and p !p "6.364) when the perturbation 2 1 stream is removed. The reason is that the viscosity of the gas in the downstream capillary tubes is different
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Table 1. Typical transducer voltage values (in mV) for each plateau in Fig. 4 Plateau
Average
Minimum
Maximum
A B C D E
11.559 54.837 48.426 5.171 11.535
11.545 54.813 48.413 5.154 11.508
11.572 54.858 48.441 5.200 11.554
for the two steps: it is k for adding the perturbation stream and k#dk when removing the perturbation stream. The analysis of this systematic difference will be given in a subsequent paper. We made measurements across the composition range for the nitrogen—argon system at 24°C (the laboratory temperature). Three perturbation gases were used, pure argon, pure nitrogen, and a commercial bottled mixture of approximately 50% argon and 50% nitrogen. The results are shown in Fig. 6. (iii) Comparison with published results We would like to compare our results for the nitrogen—argon system with published results. There are no published direct measurements of the gradient of viscosity with composition, but there are well-established values for the viscosities of the single components, nitrogen and argon. For us to find the viscosity of argon relative to nitrogen we need to integrate eq. (15)
P
P
1 p !p 1 1 1 dk 2 1 dX " dX A3 A3 p !p (XT !X0 ) k dX 0 1 0 A3 A3 0 A3
P
"
1 d ln k k dX "ln A3 A3 dX k 0 A3 N2
(16)
using the experimental values for d ln k/dX shown A3 in Fig. 6. Figure 6 shows that d ln k/dX varies alA3 most linearly with argon mole fraction, so the missing infinite-dilution (‘pure-component’) values of d ln k/ dX were estimated by linear extrapolation from the A3 last or first pair of points, as appropriate. Integration was performed numerically using the trapezium rule for unequal intervals
Fig. 6. Experimental values of the logarithmic viscosity derivative, d ln k/dX , for nitrogen—argon mixtures at 24°C. A3 These results were obtained (.) by adding argon perturbation gas, (m) by adding nitrogen perturbation gas and (r) by adding mixed perturbation gas.
different. However, it should be realised that the error in d ln k/dX is not constant across the composition A3 range. The best results are obtained, and the need to extrapolate avoided, by using nitrogen perturbations into argon-rich mixtures and argon perturbations into nitrogen-rich mixtures. If these values are used then the ratio becomes 1.2670. Matthews et al. (1982) give the ratio of the viscosities as 1.2718 at 298.18 K which is close to our temperature. Their ratio and ours differ by about 0.5%. Clarke and Smith (1968) give a smoothed value of 1.2699 at 297 K, which is in between. CONCLUSIONS
CA B A BD
k 1 N~1 ln A3" + k 2 N2 n/1 #
d ln k dX A3 n`1
d ln k [(X ) !(X ) ], (17) A3 n`1 A3 n dX A3 n
where n indicates the nth of a set of N experimental values. No attempt was made to smooth the data before integration. The viscosity of argon relative to nitrogen was found to be 1.2630 using the argon perturbation, 1.2700 using the nitrogen perturbation and 1.2734 using the nitrogen—argon mixture as the perturbation gas. These values are all slightly
A capillary viscometer has been developed that can measure the gradient of the viscosity—composition curve of a gas with a precision of about 1%. The theory of the method using differential-sized perturbations is confirmed by the close agreement between the viscosity of argon relative to nitrogen obtained by integrating the experimental results and previously published experimental values. The method is relatively fast and only involves measuring pressure differences. It seems likely that the technology can be developed to enable measurements to be made at more extreme conditions than we have used. The easiest further development might be to
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measure gas relative-viscosities at high temperature: the viscometric tubes can be at elevated temperature and the pressure transducer at ambient conditions. High-pressure measurements pose no difficulty in principle, but the transducer must be capable of sensing small pressure differences while being capable of withstanding high pressure. It must be mounted in a pressurised case or must itself be sufficiently rugged. Corrosive gases could be measured by purging or flexibly sealing the transducer connections or by making the transducer of sufficiently resistant materials. Acknowledgements This work was supported by the Engineering and Physical Sciences Research Council and by the Department of Chemical Engineering, Loughborough University. NOTATION
K Kc ¸ M p
pz P BPR Q R ¹ Xi z
Kc ¸ viscometer constant length of capillary tube molar flowrate in viscometric capillary tube pressure at viscometric capillary tube entrance; p at the start of an experiment, 0 p1 after the perturbation has caused the flowrate to change, p2 when the composition in the viscometric capillary has changed pressure at a distance z from capillary tube entrance system exhaust pressure set with a backpressure regulator volumetric flowrate the gas constant temperature mole fraction of ith species; X0i in main stream, XT in perturbation stream i distance from capillary tube entrance
Greek letter k viscosity REFERENCES
Addison, P. A., Buffham, B. A., Mason, G. and Yadav, G. D. (1994) Gas-phase dispersion assessed from tracer holdup measurements on a packed bed: theory, apparatus and experimental test (or How to measure dispersion with a pressure gauge). Chem. Engng Sci. 49, 561—572.
Bird, G. A. (1994) Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford. Buffham, B. A., Mason, G. and Meacham, R. I. (1993) Absolute gas chromatography. Proc. Roy Soc. ¸ond. A 440, 291—301. Chapman, S. and Cowling, T. G. (1970) ¹he Mathematical ¹heory of Non-ºniform Gases, Cambridge University Press, Cambridge. Clarke, A. G. and Smith, E. B. (1968) Low temperature viscosities of argon, krypton and xenon. J. Chem. Phys. 48, 3988—3991. Heslop, M. J., Buffham, B. A., Mason, G. and Ireland, N. (1996) New method to estimate binary gasmixture viscosities. Institution of Chemical Engineers Research Event, Leeds. Vol. 2, pp. 934—936. Heslop, M. J., Buffham, B. A. and Mason, G. (1997) Simultaneous ternary adsorption isotherms from micro-plant flowrate measurements. Institution of Chemical Engineers Jubilee Research Event, Nottingham. Vol. 2, pp. 1129—1132. Iwasaki, H. and Kestin, J. (1963) The viscosity of argon—helium mixtures. Physica 29, 1345—1372. Kestin, J., Kobayashi, Y. and Wood, R. T. (1966) The viscosity of four binary, gas mixtures at 20°C and 30°C. Physica 32, 1065—1089. Kestin, J. and Leidenfrost, W. (1959) The effect of pressure on the viscosity of N —CO mixtures. 2 2 Physica 25, 525—535. Kestin, J. and Wakeham, W. A. (1988) ¹ransport Properties of Fluids: ¹hermal Conductivity, »iscosity and Diffusion Coefficient, pp. 73—148. Hemisphere, New York. Mason, G., Buffham, B. A. (1996) Adsorption isotherms from composition and flow-rate transient times in chromatographic columns. II. Effect of pressure changes. Proc. Roy. Soc. ¸ond. A 452, 1287—1300. Mason, G., Buffham, B. A. and Heslop, M. J. (1997) Gas adsorption isotherms from composition and flow-rate transient times in chromatographic columns. III. Effect of gas viscosity changes. Proc. Roy. Soc. ¸ond. A, 453, 1569—1592. Matthews, G. P., Schofield, H., Smith, E. B. and Tindell, A. R. (1982) Viscosities of gaseous argon—nitrogen mixtures. J. Chem. Soc. Faraday ¹rans. I, 78, 2529—2534. Touloukian, Y. S., Saxena, S. C. and Hestermans, P. (1975) ¹hermophysical Properties of Matter: ¹he ¹PRC Data Series, Vol. 11, »iscosity. IFI/Plenum Press, New York.
Chemical Engineering Science, Vol. 53, No. 15, pp. 2665—2674, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0009–2509(98)000106–7 0009—2509/98/$—See front matter
Capillary viscometry by perturbation of flow and composition G. Mason,* B. A. Buffham, M. J. Heslop and B. Zhangs Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, U.K. (Received 4 November 1997; received in revised form 2 April 1998; accepted 2 April 1998) Abstract—A new technique for making viscosity measurements on gas mixtures is introduced. The composition and flowrate of a mixture flowing through a capillary tube are perturbed by adding a small stream of perturbation gas. This is usually a pure, individual component of the mixture. The pressure at the inlet of the capillary tube rises when the perturbation gas is added and this pressure increase is proportional to the flowrate change. Because there is empty volume between the point where the perturbation gas is added and the capillary tube, it is some time later that the pressure changes again when the composition of the gas flowing through the tube changes. This second pressure change is proportional to the change in viscosity. The ratio of these two steps of pressure is proportional to d ln k/dX where k is the viscosity and X is the i i mole fraction of component i. An apparatus has been developed which is capable of making measurements with suitable precision and some preliminary data for the nitrogen—argon system at 1.2 bar and 24°C are given. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Viscosity; gas viscometry; gas mixtures; physical properties; transport properties; viscometers. INTRODUCTION
Gas viscosities have been measured for over a century and there are data books listing collected values (Touloukian et al., 1975). Accurate measurements of the variation of gas viscosity with composition, pressure and temperature can give information about how molecules, and particularly pairs of different molecules, interact with each other (Chapman and Cowling, 1970). Viscosity is therefore an important parameter in thermophysical and transport-property models (Bird, 1994). Because of the precision required and the methods used, measuring viscosity can be a slow process. Nevertheless, most single-component viscosities are well documented; measurements on binary systems are quite common but only a few ternary and virtually no quaternary systems have been measured. We give here a new comparative method for measuring gas viscosities that is potentially fast, should work with multicomponent systems, and may well work at very high temperature. Touloukian et al. (1975), and Kestin and Wakeham (1988) give authoritative reviews of the theory and practice of the methods that are available to measure viscosity. The most-used methods for accurate
* Corresponding author. Tel.: 01509 222509; fax: 01509 223923; e-mail: [email protected]. sPresent address: Department of Chemical Engineering, Tianjin University, Tianjin, 300072, P.R. China.
measurement are oscillating-body viscometry and capillary-flow viscometry. The principles of both methods are well established and with modern techniques both can give very accurate values. Values obtained from both methods are mutually consistent except at very high temperatures where systematic errors may give differences. Because of their precision and ability to give absolute values, these methods would normally be the primary choice for viscosity measurement, even if they are slow. Viscosity is a dynamic variable and something in the apparatus has to move in order that viscosity can be measured; somehow the fluid must be sheared. In an oscillating body viscometer, an axially symmetric body performs torsional oscillations inside a stator filled with fluid. The oscillating body, usually a disc, is suspended from a taut wire and the period is often tens of seconds. The change in frequency and damping decrement depend only on the density and viscosity of the fluid and the torsional stiffness of the suspension system. In principle, both the viscosity and density of the fluid can be obtained from measurements of the frequency and decrement. However, density can be measured by other methods and so the oscillating-body method usually only measures viscosity. In a capillary viscometer, the gas flows through a capillary tube and the pressure drop and volumetric flowrate through the tube are measured. The usual problem is that, for the method to be absolute, the
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diameter of the tube has to be uniform and be known with great precision. A variant is the rolling-ball viscometer in which a ball rolls down inside a reasonably close-fitting tube. This method can give very reproducible results but the basic principles are poorly understood. The gap between the ball and tube is crescent-shaped with the ball rotating and these factors give a complex flow-velocity profile. The oscillating body and capillary viscometers can be used with gas mixtures as well as with pure components. All that is needed is a supply of gas of known composition. Obtaining this can sometimes be a problem. Both methods measure the absolute viscosity. However, if relative viscosities are acceptable then either method can be used in the ‘relative’ mode provided that a gas of known viscosity is available to determine the apparatus constant. We present here a method that is new in both concept and application. There are two new ideas. The first idea is that instead of measuring the gas viscosity directly we measure the gradient of the viscosity—composition function. We change the composition slightly and measure the small change in viscosity. If gradients are known across the composition range then integration can give values of the relative viscosity across the composition range. The attraction of this approach is that, all things being equal, measurement of differentials followed by integration is more accurate than direct measurement. Alternatively, the gradients can be measured with less accuracy than the direct values and still give equally good values of relative viscosity. The second idea involves the detail of the apparatus and the theory of laminar flow through tubes. We use a capillary tube first as a flow meter and then shortly after as a viscometer: when one response is divided by the other the gradient of viscosity with composition is obtained. The advantage of this idea is that the result of the experiment will not depend on the properties of the capillary tube. We therefore have a capillary tube viscometer which produces results that do not depend on the uniformity of the tube or, indeed, on knowing its diameter accurately. Also, only the properties of the gas in the capillary tube are measured, and so the tube can be heated to high temperature and the method should still work. The method is only relative because, and this is only one way of regarding it, the tube constant in the equation for laminar flow has been found by measuring the change in pressure caused by changing the flowrate through the tube. Normally, a relative viscometer would be calibrated using a gas of known viscosity. The same tube constant is used in the expression for the change in pressure caused by the viscosity change. But the composition change also depends on the flowrate change, so, when one pressure change is divided by the other, both the tube constant and flowrate change cancel. In absolute capillary viscometry it may be necessary to take account of secondary effects such as slip at the tube wall, kinetic energy loss and the
entrance length needed to establish laminar flow. In relative methods, including ours, these effects tend to cancel. THEORY OF THE METHOD
(i) Fundamentals Consider the schematic apparatus shown in Fig. 1. There are five components. The main stream of gas mixture enters the upstream capillary choke A at constant pressure. This choke is typically a length of small-bore capillary tube and develops most of the pressure drop in the apparatus. The gas flows through a valve B and enters the delay line C. The delay line is usually a length of 4 mm i.d. tubing and contains volume but produces very little pressure drop. After leaving the delay line the gas mixture passes through the downstream capillary choke E and escapes to atmosphere. About 1% of the pressure drop occurs in choke E. A pressure gauge D measures the pressure just upstream of the downstream capillary choke E. An experiment consists of allowing the gas to flow until the pressure indicated by gauge D is constant and then changing valve B to admit a small flow of perturbation gas. Typically, the perturbation gas is one of the individual components of the mixture and has a flowrate of about 1% of the total (or main) flowrate. The response of the pressure gauge (or transducer) is monitored. An experiment does not start until the gauge indicates a constant value. When the valve is switched and the perturbation flow enters the system, the pressure at the gauge rises and levels out. The viscosity of the gas in the downstream capillary is unchanged because the new mixture is traversing the delay line and has not yet reached the downstream capillary choke. Therefore, the pressure gauge is responding only to changes in flowrate in the downstream choke E. When, eventually, gas with the new composition reaches the downstream capillary choke,
Fig. 1. Schematic diagram showing the principles of operation of the apparatus. Initially valve B is set to pass the main gas stream. The resistance of choke A is much greater than that of the rest of the system and so determines the flowrate. In an experiment, the valve B is switched to add a small stream of perturbation gas to the main stream. The increase in flowrate in the downstream choke E is sensed by the pressure at the gauge D increasing. But, because of the presence of the delay line C, the gas in this choke still has the original composition. When gas of the new composition reaches choke E, the pressure shown by gauge D changes again.
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the pressure gauge reading alters again. It will rise if the gas viscosity has increased, and fall if it has decreased. Because the flowrate is now constant, the pressure gauge is now responding only to changes in viscosity. Consequently, the pressure gauge reading can be regarded as the output of a capillary meter which has acted first as a flowmeter and second as a viscometer. It is the delay line that separates these two responses. Without the delay line the middle plateau region of the pressure-gauge response would not be observed and the response could not be separated into two effects. (ii) Analysis Let us analyse the behaviour of the apparatus shown in Fig. 1 when the stream of perturbation gas is added. The salient features of the expected response are shown in Fig. 2. The steps in the pressure gauge record can be identified as three plateaus, p , p , and 0 1 p . Initially no perturbation gas is added and the 2 pressure gauge reads p . At time zero the small stream 0 of perturbation gas of known composition is added. This immediately increases the flowrate in the downstream capillary choke but does not yet change the composition. The pressure indicated on the pressure gauge rises to p and the increase is only caused by the 1 flowrate change in the downstream capillary choke. Later, when the composition front has traversed the delay line, the composition change reaches the downstream capillary choke. The change in viscosity causes, say, the pressure to rise (say because sometimes it falls) to p . When the capillary choke acts as 2 a flowmeter p !p is proportional to the increase in 1 0 flowrate and when it later acts as a viscometer p !p 2 1 is proportional to the change in viscosity. The change in pressure p with distance z for z laminar flow of a fluid through a section of tube is given by dp z"!K kQ c dz
(1)
where Q is the volumetric flowrate, k is the viscosity and K is a constant depending on the characteristics c of the tube. If the gas is ideal then MR¹ Q" (2) p z M is the molar flowrate of gas, p is the pressure at z distance z, ¹ is the absolute temperature, and R is the gas constant. Consequently the longitudinal pressure gradient can be expressed as dp p z"!K kMR¹. z dz c
(3)
The viscosity of a gas depends primarily on temperature. At low pressure it is almost independent of pressure and so the changes in pressure in the downstream capillary choke (which really are very small) do not cause a detectable change in viscosity. As k is constant eq. (3) can be integrated along the whole
Fig. 2. Main features of the experimental response. At time zero, a perturbation gas of known composition is added to the main stream and the pressure indicated by the gauge rises from p to p . During the time that the composition front 0 1 remains in the delay line, the pressure remains at p . The 1 increase in flowrate caused by the addition of perturbation gas is proportional to p !p . When the front reaches the 1 0 capillary tube, the change of viscosity caused by composition change causes a further response and the pressure changes from p to p . This may be a rise or fall depending on 1 2 whether the viscosity increases or decreases. The viscosity change is proportional to p !p . 2 1
length of the capillary tube to give p2!P2 "2KkMR¹. (4) BPR where K"K ¸, ¸ is the length of the tube and p is the c pressure at the inlet of the downstream capillary choke. The squared terms arise because the gas in the choke is compressed, being denser at the inlet than the outlet. P is the outlet pressure of the downstream BPR capillary choke. Figure 1 shows this pressure to be atmospheric pressure, but it may be above atmospheric pressure if, for example, it is set to a constant pressure with a back-pressure regulator (hence the subscript BPR). If some changes in flowrate and viscosity have occurred such that the inlet pressure of the capillary tube has risen by dp in response to a change in flowrate dM and a change in viscosity dk then eq. (4) gives (p#dp)2!P2 "2K(k#dk)(M#dM)R¹. (5) BPR Expanding and subtracting eq. (4) gives 2pdp#(dp)2"2K(kdM#Mdk#dMdk)R¹. (6) For small changes, the second-order terms (dp)2 and dMdk are small and so pdp"K(kdM#Mdk)R¹
(7)
or 2pdp dM dk " # . (8) M k p2!P2 BPR Equation (8) is the basic equation for our analysis. We can now use it twice, once for each of the two steps in pressure in Fig. 2. Since there is no viscosity change
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(dk"0) for the first step in Fig. 2, eq. (8) becomes 2p (p !p ) dM 0 1 0" . (9) (p2!P2 ) M 0 BPR For the second step in Fig. 2, there is no change in flowrate (dM"0) but only a change in viscosity so eq. (8) becomes 2p (p !p ) dk 1 2 1" . (p2!P2 ) k 1 BPR Taking the ratio of eq. (9) to eq. (10) generates
(10)
p (p !p ) p2!P2 M dk 1 2 1 0 BPR" . (11) p (p !p ) p2!P2 k dM 0 1 0 1 BPR If we choose the capillary and the pressure step sizes such that p +p +p and p2!P2 +p2!P2 , 0 1 2 0 BPR 1 BPR then M dk p !p 2 1" . (12) k dM p !p 1 0 If the main stream has a composition of X0 and the i perturbation stream has a flowrate of dM and composition XT , the addition of the perturbation stream i will produce a composition change in the main stream of dM dX " (XT!X0). i M#dM i i
(13)
We use the subscript i to denote any one of the components in the mixture. Equation (13) comes from a simple mass balance. If we eliminate dM between eqs. (12) and (13) then we can arrive at an equation giving the change in k in terms of the change in X and the size of the two measured steps in pressure. i Also dM is negligible relative to M for differentialsized steps. The final working equation is 1 dk p !p 1 1 " 2 . (14) k dX p !p (XT!X0) i 1 0 i i In the limit when dk and dX become very small i dk/kdX P(dk/dX)/k and a differential equation is i obtained. 1 dk 1 dk d ln k p !p 1 1 lim " " " 2 . k dX k dX dX p !p (XT!X0) dXi?0 i i i 1 0 i i (15) In an experiment we measure the two pressure changes p !p and p !p . XTand X0 are both 1 0 2 1 i i input streams and will be of known composition. Generally, a pure component will be used as perturbation gas and its composition is certain. Thus, the gradient d ln k/dX is determined from the ratio of i p !p to p !p and knowledge of XTand X0. No 2 1 1 0 i i other variable need be known. In fact, a more detailed analysis shows that the tube does not have to be uniform and the gas need not be ideal. The method therefore only depends on how accurately the two measurements of the change in pressure can be made.
(iii) Sensitivity to random errors Although this method is elegant in principle it is still subject to the effects of random and systematic errors and, as we shall see, these have the potential to be serious. For example, let us aim for a precision of about 1% in d ln k/dX . For many gas mixtures the i variation of viscosity with composition is not large and, typically, Dd ln k/dX D is about 0.1 for a binary i mixture. So of the two steps in Fig. 2, p !p will 2 1 usually be about one-tenth the size of p !p . As 1 0 a result, the same-sized error in measuring the pressure will cause a ten times greater proportional error in p !p than in p !p . So, ideally, we would like 2 1 1 0 to make p !p as large as possible. However, be2 1 cause we need to keep the perturbation relatively small dX will be, at most, 0.01 in an actual experii ment and so the main gas flowrate M will be over 100 times greater than the perturbation flowrate dM. We aim to measure dM (via p !p ) to 1% accuracy and 1 0 so the main flowrate M will have to remain stable to 1 part in 104. But p !p is one-tenth of p !p and 2 1 1 0 so an error of 1% in (p !p )/(p !p ) will result if 2 1 1 0 the main flow drifts by 1 part in 105 in the course of an experiment. Therefore, very special attention will have to be paid to keeping the main flow constant. Temperature plays a part here too. For example, if the main flow is produced by a fixed pressure drop across a fixed capillary tube (the upstream capillary choke), the absolute temperature will have to be constant to 1 part in 105—that is 0.003°C—because the pressure drop is proportional to volumetric flow rather than to molar flow. The pressure gauge will also have to be stable and have a resolution better than 1 part in 105. The upstream and downstream pressures similarly need to be kept stable to the same standard. All of this is true for the apparatus shown schematically in Fig. 1. We will see later that a differential apparatus design overcomes many of these potential difficulties. In order to calculate d ln k/dX we have to divide i (p !p )/(p !p ) by XT!X0. If the compositions 2 1 1 0 i i of the main flow and perturbation flow gases are known to a precision of, say, 0.1% mole fraction, which is quite accurate, then XT!X0 would be i i known to about 1 part in 1000 if XT"1 and X0"0. i i However, X0 has to be varied across the range 0 to 1. i If XT"1 and X0"0.9 then XT!X0 is only known i i i i to about 1 part in 100. We can conclude that the most precise results will be obtained when XT and X0 differ i i by at least 0.5 and the results will be very poor when XT and X0 are almost the same. Using a pure comi i ponent as perturbation gas makes XT either 0 or i 1 and there is no error in its composition. There are several techniques that can be used to overcome some of the difficulties caused by having to keep the main gas flow constant to precise limits. The first is to borrow an idea from chromatography and use a second (or reference) stream which is closely similar to the system shown in Fig. 1, but without the valve B. In essence, this means duplicating the upstream capillary choke, the delay line and the
Viscometry by flow and composition
downstream capillary choke. The reference stream now provides a similar pressure to the measurement stream pressure and the pressure gauge can be replaced by a differential pressure gauge (or transducer). For our 1% accuracy the differential pressure gauge need only detect to 1 part in 1000 instead of 1 part in 100,000 before. Also, provided the temperature throughout the reference stream is kept the same as the temperature throughout the measurement stream, the effect of temperature changes disappears. Keeping temperatures locally the same can be done relatively easily by twisting the capillary tubes and also the delay lines, together. However, the main benefit of the differential design is that the pressures upstream and downstream are common to both the measurement and reference streams and any small changes in pressure of both chokes caused, say, by drift in a pressure regulator affect both streams together and cause little change at the differential pressure transducer. This differential design overcomes the difficulty of maintaining the upstream and downstream pressures constant to 1 part in 100,000. EXPERIMENTAL APPARATUS
In order to apply the theory of the method many technical details have to be settled before a working machine can be made. We were fortunate in that we observed the effects described in the THEORY OF THE METHOD, above, whilst working with a novel analytical chromatograph (Buffham et al., 1993) and an apparatus used to measure binary gas adsorption (Mason et al., 1996, 1997) and ternary gas adsorption (Heslop et al., 1997). The double-sided system common to gas chromatographs was employed in these devices. The advantages of this particular feature have already been explained. We therefore started with an apparatus that partially worked and gave measurable effects and which only had to be progressively developed. By using an approximately 1 bar pressure drop across the upstream capillary chokes a main gas flowrate of about 25 ml/min was produced. Such a modest main gas flowrate makes the flow of perturbation gas very low (usually less than 0.25 ml/min) and such a low flow cannot conveniently be produced by a fixed pressure drop across a capillary tube. However, only a stability of 1 part in 1000 is required. We have found that Porter Instruments Type VCD 1000 mass flow controllers can give these low flows consistently provided that flow through the controller is never switched off. Their dynamic response at low flowrates is very sluggish. We chose to work with the nitrogen—argon system at ambient conditions whilst developing the apparatus. The purity of the nitrogen and argon was specified by the supplier to be 99.998%. (i) Sensitivity of the pressure transducer Only tiny changes in differential pressure are observed and so a very sensitive differential pressure transducer is employed. We used a Furness Controls
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type FCO 40 with a sensitivity of $10 mm H O/volt. 2 These transducers were originally designed for measuring pressures developed by pitot tubes in wind tunnels. They are of diaphragm type and are surprisingly rugged being able to withstand relatively massive pressure overload caused by accidental major leaks in the apparatus. A typical voltage change of the output signal produced by adding the perturbation stream was about 40 mV, corresponding to a pressure difference of about 0.4 mm H O. The transducer can 2 easily detect less than one-thousandth of this pressure. The voltage signal was digitised with an analogue to digital converter and recorded using a personal computer. (ii) Pressure and back-pressure regulators A pressure regulator contains a valve which moves to keep the outlet pressure constant even though the outlet flowrate changes. A back-pressure regulator performs a similar function but this time keeps the inlet pressure constant. Normally both types of regulator use atmospheric pressure as the reference and set the pressure by compressing a spring. A problem is that any fluctuations in atmospheric pressure are transmitted to the regulated pressure and, in our apparatus, this can cause pressure noise at the pressure transducer (Heslop et al., 1996). Because of the design of the apparatus the back-pressure regulators, and particularly the outlet regulator, cause the greatest effect. To avoid this form of pressure noise we modified commercial back-pressure regulators by sealing off the spring cavities, but still keeping the spring adjustment. These back-pressure regulators are termed BPR(M). (iii) Setting the main gas flow Figure 3 shows a schematic flowsheet of our final apparatus for the nitrogen—argon system based on the theoretical and practical considerations which have already been described. The main gas mixture is made by blending nitrogen and argon in a gas mixing machine of our own design. The gas mixing machine has a constant outlet pressure which is set by one of the special back-pressure regulators (BPR-1(M)). This pressure is applied upstream of the two upstream capillary chokes. The main gas mixture passes through both chokes, one gas stream flowing on through the measurement side and the other through the reference side. Most of the pressure drop (it was about 1.1 bar) in each stream occurs across the upstream chokes, which are 150 cm lengths of 0.25 mm i.d. (1/16 inch o.d.) stainless-steel tubing. The two needle valves NV-1 and NV-2 are used for balancing the flows in the measurement and reference sides when there is no perturbation gas flowing into the measurement side. Unfortunately, changing needle valve settings adds to (or reduces) the volume in the valves and this will temporarily change the flowrate through the downstream capillary choke. Placing the valves upstream of the upstream capillary chokes reduces the effect of these volume changes and makes balancing the flows very much easier.
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Fig. 3. Schematic flowsheet of experimental apparatus. BPR-1 (M) and BPR-2 (M)"Modified backpressure regulators based on Porter Instruments 9000 BPR; BPR-3"Porter Instruments 9000 backpressure regulator; DPT"Furness Controls FCO 40 differential pressure transducer; MEVA"»ICI ETMA micro-electric two-position valve actuator; MFC-1, MFC-2 and MFC-3"Porter Instruments VCD 1000 flow controller; MFC-4"Condyne 202 flow controller; NV-1 and NV-2"Brooks Instruments needle valves; PR-1, PR-2 and PR-3"Porter Instruments 8286 pressure regulators; 3PSV-1 and 3PSV-2 and 3PSV-3"»alco UWP three-port switching valves; 4PSV-1 and 4PSV-2"»alco UWP four-port switching valves; 3WV-1, 3WV-2 and 3WV-3"SSI 02-0182 three-way valves.
A separate stream of main gas flow mixture, the flow of which is set by a mass flow regulator, goes to purge unwanted perturbation gas. (More about this later). (iv) Delay lines and downstream capillary chokes Downstream of the upstream capillary chokes are the delay lines, which are 100 cm lengths of 4 mm i.d. (1/4 inch o.d.) nylon tubing, followed by the two downstream capillary chokes which are 50 cm lengths of 0.76 mm i.d. (1/16 inch o.d.) stainless-steel tubing. There was a pressure drop of about 0.1 mbar along the delay lines and a pressure drop of about 15 mbar along the downstream capillary chokes. The outlets of the two capillaries are combined together and the combined streams pass through a further delay line before reaching the final back-pressure regulator. The pressure set by any back-pressure regulator is very slightly affected by both flowrate and gas viscosity. The final delay line retains sufficient volume of gas to keep the composition at the outlet backpressure regulator (BPR 2(M)) constant during an experiment. Also the purge stream, which is set by the mass flow regulator, and into which the perturbation stream flows when it is not being added to the measurement side, joins before the third delay line and this arrangement keeps the flow through the back-pressure regulator constant when the perturbation gas valve is switched. The result is that the pressure set by the outlet back-pressure regulator is kept
constant to close limits because neither the flow nor the composition changes during the course of an experimental run. The differential pressure transducer measures the pressure difference between the measurement side and the reference side at the positions before the two downstream capillary chokes. (v) Setting the perturbation flows The flowrates of the perturbation gases are adjusted to be between 0.5% and 1% of that of the main gas flow by pressure regulators PR-1, PR-2 or PR-3 and mass flow controllers MFC-1, MFC-2 or MFC-3, respectively. A mass flow controller maintains a pressure drop across a flow restrictor by means of a selfadjusting valve. In principle, the flowrate delivered by the controller can be set in two ways; the flow restrictor can be adjusted (by making it a needle valve) or the pressure drop can be altered by compressing a spring. Porter mass flow controllers are the latter type and use a sintered metal body as a fixed flowrestrictor and have a spring to adjust the differential pressure. There are three mass flow controllers, one for each perturbation gas, so that each one can run with a constant flowrate and gas composition. The perturbation gas is selected by using two four-way valves 4PSV-1, 4PSV-2 and one three-way valve 3PSV-1. In Fig. 3, the selected perturbation gas is the mixture of nitrogen and argon. The flowrate of perturbation gas can be measured from one port of
Viscometry by flow and composition
3PSV-1. The selected perturbation gas is added or removed by switching the three-port valve, 3PSV-2, which is actuated by a VICI ETMA micro-electric two-position valve actuator. The valve plug was originally turned manually, but we have found that the electric actuation produced more uniform rotation. When the perturbation gas is added, it goes to point B through the valve 3PSV-2, and is then added to the stream of gas flowing through the measurement side of the apparatus. When the perturbation is removed (as in the position shown in Fig. 3) it goes through 3PSV-2 and reaches point C, where it is combined with the purge flow of gas mixture. The purpose of this arrangement is to exactly match the conditions in the measurement stream with the conditions in the purge stream. The valve 3PSV-2 always transfers a small volume from one stream to the other when it is switched over. Matching conditions at each side of the valve avoids surges in flow and diffusion effects in the valve and helps to generate a sharp concentration front. The purge stream flows through a third downstream length of capillary tube to reach point A. The two streams of main gas flow at point A are joined in front of the third delay line. As mentioned already, this arrangement maintains the constant flowrate and composition requirement in BPR-2(M). Special fittings are used with valve 3PSV-2 which bring the main gas flow into the valve and then out again through a concentric tube. The arrangement minimises the dead volume in the pipework that occurs with the valve. The fittings are more fully described by Addison et al. (1994) who used them to produce dispersion-free fronts. Because the flowrate of the perturbation gas is very small, it takes days to purge the gas in the mass flow regulators. It even takes hours to displace gas in the tubes after changing from one perturbation gas to another. On the other hand, keeping the perturbation flowrate constant is very important and adjustments to mass flow controllers MFC-1, MFC-2 or MFC-3 must be minimised because they take many hours to settle down again. Ideally they should be left alone. This is why the SSI 02-0182 three-way valves are combined with lengths of high resistance capillary tube (about the same as the upstream capillary chokes) and connected to valves which by-pass the mass flow regulators. Opening a valve flushes the pipework with new perturbation gas. This operation is done with the perturbation gas going into the purge stream and so the measurement stream is not disturbed. These modifications introduce no dead volume and enable the perturbation gases to be changed in a few minutes. This speeds up the experimental process greatly. Before adding the perturbation gas the pressure at both sides of the perturbation stream addition valve have to be exactly equal, otherwise a sharp deviation in pressure occurs when the perturbation valve is turned. Because they are all connected together just before the third delay line, both main gas flows and the purge flow plus the perturbation gas flow all have
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the same outlet pressure. Slight adjustment of the purge flow using MFC-4 changes the pressure drop through the third downstream capillary choke and alters the pressure at the perturbation stream addition valve. Adjustment is carried out until turning the perturbation stream addition valve 3PSV-2 gives an even step change in the response of the pressure transducer with no overshoot and no undershoot. (vi) Gas mixtures The gas blender has two main roles. First, it enables the main-flow composition to be easily and quickly set to approximately to the desired value. It maintains this composition constant even when the pressure downstream changes. Second, it enables the approximately set main-flow composition to be measured. Because of the blender’s design, it is possible to shut off the flow of either of the components and for the blender to continue to deliver the other component at the same flowrate as it was previously contributed to the mixture. This feature enables the composition of the mixture delivered by the blender to be precisely determined by measuring the individual flowrates of the gases making up the mixture with a soap-film meter. The flowrates were measured by timing the passage of the soap film between particular graduation marks. Mole fractions thus determined do not depend on the volumetric calibration of the meter tube. Determining the gas mixture composition has been a problem in work on the viscosity of gas mixtures and in some work has been the greatest source of uncertainty (see, for example, Kestin and Leidenfrost, 1959; and Iwasaki and Kestin, 1963). Kestin’s team overcame the problem by weighing the gases into a bottle (Kestin et al., 1966). In more recent work, Matthews et al. (1982) used a gas-density balance. Our use of the blender has the advantage that the composition may be changed quickly. In other work we had come to distrust the analysis of the nominal 50% argon—50% nitrogen mixture we used as one of the perturbation gases. A null experiment comparing the ‘50—50’ mixture with gas from the blender showed that the mixture composition was 47 mole% argon. In the null experiment the ‘50—50’ mixture was used as the perturbation gas and the main gas stream was provided by the gas blender. The composition of the main gas stream was adjusted until adding perturbation gas produced no discernible viscosity step. The ‘47% argon’ main gas stream did exactly this and this composition has been used in the analysis of our results. (vii) ¹emperature control The entire apparatus is not in a temperature-controlled bath. This may seem surprising but it was done for convenience because it eased making changes and developments to the pneumatic system. The gradient of the relative viscosity with composition curve depends much less strongly on temperature than the viscosity—composition function. However, it did prove necessary to enclose the apparatus in a closed
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metal cabinet so that it was kept at constant, but not controlled, temperature. The box isolated the apparatus from rapid changes in the environment. We believe that radiation heat transfer causes most thermal noise in the apparatus, and this is why the simple metal cabinet worked so well. EXPERIMENTAL RESULTS
(i) Data acquisition All the development work was carried out using a chart recorder to monitor the output from the differential pressure transducer. However, actual data acquisition of the pressure changes was carried out using a 16 bit analogue-to-digital converter and ¼orkBench PC software to digitise the signal from the transducer 25 times a second. When the system is properly balanced, carrying out a run is fairly quick. We found it useful to record a complete cycle of the addition and removal of perturbation gas. The cycle links the addition of the perturbation gas, and the subsequent viscosity step, followed by the removal of the perturbation gas and its viscosity step all in one continuous data file. A run took about 10 min. 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 typical data record is shown in Fig. 4. This is a direct plot of the raw digitised data file. The noise is lost in the thickness of the plotline. In Fig. 4 the variation of the differential pressure transducer signal for both the addition and removal of the perturbation gas are shown. The line AE is the baseline of the signal. After the addition of the perturbation stream, the voltage signal increases rapidly indicating the additional flow of the perturbation gas. The record now becomes horizontal, B, whilst the composition front caused by adding the perturbation gas is moving through the delay line. When the front arrives at the downstream capillary the resulting change of viscosity changes the signal from B to C and the system reaches a new steady state. The change between B and C is not as steep as that between A and B because there is dispersion in the delay line (Addison et al., 1994). The perturbation gas is removed after the vertical broken line and the response is repeated in the opposite direction. The character of the noise is shown in Fig. 5. Plotted in Fig. 5 relative to arbitrary datum points are 10 s stretches of plateaus A, B and C in Fig. 4. The stretches are 10—20 s, 60—70 s and 110—120 s and each stretch contains 250 points. The resolution of the digitisation is just sufficient to reveal the short-term noise. (ii) Data analysis The plateaus of both flowrate and viscosity are relatively free of noise. The data records were analysed using a separate computer program. The program calculates the average of several hundred data points in two parts of the plateau region of each step
Fig. 4. A typical experimental result. The mole fraction of argon in the main gas flow is 0.686 and nitrogen is used as the perturbation gas. This is a computer-generated plot of the original raw digitised data file. The noise is lost in the thickness of the plotline. The noise is displayed in Fig. 5 where the boxed stretches are plotted much enlarged. See also Table 1.
Fig. 5. The character of the short-term noise is shown here by plotting the 10—20 second (A), 60—70 s (B) and 110—120 s (C) stretches of plataus A, B and C in Fig. 4 relative to arbitrary datum points. Each of these stretches contains 250 points. The resolution of the digitisation is just sufficient to reveal the short-term noise.
and records the maximum and minimum values in the ranges examined. The maximum and minimum values give an indication of the extremes of noise in the signal. Values of each step for the data in Fig. 4 are listed in Table 1. Close examination shows that the pressure step sizes obtained when the perturbation stream is added (p !p "43.278 and p !p "!6.411) are not 1 0 2 1 exactly the same as those observed (p !p " 1 0 !43.255 and p !p "6.364) when the perturbation 2 1 stream is removed. The reason is that the viscosity of the gas in the downstream capillary tubes is different
Viscometry by flow and composition
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Table 1. Typical transducer voltage values (in mV) for each plateau in Fig. 4 Plateau
Average
Minimum
Maximum
A B C D E
11.559 54.837 48.426 5.171 11.535
11.545 54.813 48.413 5.154 11.508
11.572 54.858 48.441 5.200 11.554
for the two steps: it is k for adding the perturbation stream and k#dk when removing the perturbation stream. The analysis of this systematic difference will be given in a subsequent paper. We made measurements across the composition range for the nitrogen—argon system at 24°C (the laboratory temperature). Three perturbation gases were used, pure argon, pure nitrogen, and a commercial bottled mixture of approximately 50% argon and 50% nitrogen. The results are shown in Fig. 6. (iii) Comparison with published results We would like to compare our results for the nitrogen—argon system with published results. There are no published direct measurements of the gradient of viscosity with composition, but there are well-established values for the viscosities of the single components, nitrogen and argon. For us to find the viscosity of argon relative to nitrogen we need to integrate eq. (15)
P
P
1 p !p 1 1 1 dk 2 1 dX " dX A3 A3 p !p (XT !X0 ) k dX 0 1 0 A3 A3 0 A3
P
"
1 d ln k k dX "ln A3 A3 dX k 0 A3 N2
(16)
using the experimental values for d ln k/dX shown A3 in Fig. 6. Figure 6 shows that d ln k/dX varies alA3 most linearly with argon mole fraction, so the missing infinite-dilution (‘pure-component’) values of d ln k/ dX were estimated by linear extrapolation from the A3 last or first pair of points, as appropriate. Integration was performed numerically using the trapezium rule for unequal intervals
Fig. 6. Experimental values of the logarithmic viscosity derivative, d ln k/dX , for nitrogen—argon mixtures at 24°C. A3 These results were obtained (.) by adding argon perturbation gas, (m) by adding nitrogen perturbation gas and (r) by adding mixed perturbation gas.
different. However, it should be realised that the error in d ln k/dX is not constant across the composition A3 range. The best results are obtained, and the need to extrapolate avoided, by using nitrogen perturbations into argon-rich mixtures and argon perturbations into nitrogen-rich mixtures. If these values are used then the ratio becomes 1.2670. Matthews et al. (1982) give the ratio of the viscosities as 1.2718 at 298.18 K which is close to our temperature. Their ratio and ours differ by about 0.5%. Clarke and Smith (1968) give a smoothed value of 1.2699 at 297 K, which is in between. CONCLUSIONS
CA B A BD
k 1 N~1 ln A3" + k 2 N2 n/1 #
d ln k dX A3 n`1
d ln k [(X ) !(X ) ], (17) A3 n`1 A3 n dX A3 n
where n indicates the nth of a set of N experimental values. No attempt was made to smooth the data before integration. The viscosity of argon relative to nitrogen was found to be 1.2630 using the argon perturbation, 1.2700 using the nitrogen perturbation and 1.2734 using the nitrogen—argon mixture as the perturbation gas. These values are all slightly
A capillary viscometer has been developed that can measure the gradient of the viscosity—composition curve of a gas with a precision of about 1%. The theory of the method using differential-sized perturbations is confirmed by the close agreement between the viscosity of argon relative to nitrogen obtained by integrating the experimental results and previously published experimental values. The method is relatively fast and only involves measuring pressure differences. It seems likely that the technology can be developed to enable measurements to be made at more extreme conditions than we have used. The easiest further development might be to
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measure gas relative-viscosities at high temperature: the viscometric tubes can be at elevated temperature and the pressure transducer at ambient conditions. High-pressure measurements pose no difficulty in principle, but the transducer must be capable of sensing small pressure differences while being capable of withstanding high pressure. It must be mounted in a pressurised case or must itself be sufficiently rugged. Corrosive gases could be measured by purging or flexibly sealing the transducer connections or by making the transducer of sufficiently resistant materials. Acknowledgements This work was supported by the Engineering and Physical Sciences Research Council and by the Department of Chemical Engineering, Loughborough University. NOTATION
K Kc ¸ M p
pz P BPR Q R ¹ Xi z
Kc ¸ viscometer constant length of capillary tube molar flowrate in viscometric capillary tube pressure at viscometric capillary tube entrance; p at the start of an experiment, 0 p1 after the perturbation has caused the flowrate to change, p2 when the composition in the viscometric capillary has changed pressure at a distance z from capillary tube entrance system exhaust pressure set with a backpressure regulator volumetric flowrate the gas constant temperature mole fraction of ith species; X0i in main stream, XT in perturbation stream i distance from capillary tube entrance
Greek letter k viscosity REFERENCES
Addison, P. A., Buffham, B. A., Mason, G. and Yadav, G. D. (1994) Gas-phase dispersion assessed from tracer holdup measurements on a packed bed: theory, apparatus and experimental test (or How to measure dispersion with a pressure gauge). Chem. Engng Sci. 49, 561—572.
Bird, G. A. (1994) Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford. Buffham, B. A., Mason, G. and Meacham, R. I. (1993) Absolute gas chromatography. Proc. Roy Soc. ¸ond. A 440, 291—301. Chapman, S. and Cowling, T. G. (1970) ¹he Mathematical ¹heory of Non-ºniform Gases, Cambridge University Press, Cambridge. Clarke, A. G. and Smith, E. B. (1968) Low temperature viscosities of argon, krypton and xenon. J. Chem. Phys. 48, 3988—3991. Heslop, M. J., Buffham, B. A., Mason, G. and Ireland, N. (1996) New method to estimate binary gasmixture viscosities. Institution of Chemical Engineers Research Event, Leeds. Vol. 2, pp. 934—936. Heslop, M. J., Buffham, B. A. and Mason, G. (1997) Simultaneous ternary adsorption isotherms from micro-plant flowrate measurements. Institution of Chemical Engineers Jubilee Research Event, Nottingham. Vol. 2, pp. 1129—1132. Iwasaki, H. and Kestin, J. (1963) The viscosity of argon—helium mixtures. Physica 29, 1345—1372. Kestin, J., Kobayashi, Y. and Wood, R. T. (1966) The viscosity of four binary, gas mixtures at 20°C and 30°C. Physica 32, 1065—1089. Kestin, J. and Leidenfrost, W. (1959) The effect of pressure on the viscosity of N —CO mixtures. 2 2 Physica 25, 525—535. Kestin, J. and Wakeham, W. A. (1988) ¹ransport Properties of Fluids: ¹hermal Conductivity, »iscosity and Diffusion Coefficient, pp. 73—148. Hemisphere, New York. Mason, G., Buffham, B. A. (1996) Adsorption isotherms from composition and flow-rate transient times in chromatographic columns. II. Effect of pressure changes. Proc. Roy. Soc. ¸ond. A 452, 1287—1300. Mason, G., Buffham, B. A. and Heslop, M. J. (1997) Gas adsorption isotherms from composition and flow-rate transient times in chromatographic columns. III. Effect of gas viscosity changes. Proc. Roy. Soc. ¸ond. A, 453, 1569—1592. Matthews, G. P., Schofield, H., Smith, E. B. and Tindell, A. R. (1982) Viscosities of gaseous argon—nitrogen mixtures. J. Chem. Soc. Faraday ¹rans. I, 78, 2529—2534. Touloukian, Y. S., Saxena, S. C. and Hestermans, P. (1975) ¹hermophysical Properties of Matter: ¹he ¹PRC Data Series, Vol. 11, »iscosity. IFI/Plenum Press, New York.