Measurement of gas mixing volumes by Flux Response Technology

Fluid Phase Equilibria 256 (2007) 93–98

Measurement of gas mixing volumes by Flux Response Technology N. Riesco a,∗ , G. Mason a , I.W. Cumming a , P.A. Russell b , K. Hellgardt c a

Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK b School of Science and Technology, University of Teesside, Middlesborough, Tees Valley TS1 3BA, UK c Department of Chemical Engineering and Chemical Technology, Imperial College, London SW7 2AZ, UK Received 9 August 2006; received in revised form 6 November 2006; accepted 7 November 2006 Available online 12 November 2006

Abstract Flux Response Technology is a technique based on a capillary viscometer for the measurements of flow and viscosity changes. In previous works, it has been used to measure logarithmic viscosity gradients of ideal mixtures. Here, we extend its range of application to the measurement of non-ideal mixing volumes at low temperatures. The theory describing a Flux Response apparatus has been reformulated without assuming gas ideality and taking into account the mixing volume. A systematic error in the measurement of the viscosity has been identified and estimated. We report the excess volume measurements of CO2 + CCl2 F2 mixtures obtained at 273.15 K and 1.11 bar. © 2006 Elsevier B.V. All rights reserved. Keywords: Gases; Instrumentation; Isothermal; Mixing; PVT; Viscosity; CO2 ; R12

1. Introduction

2. Theory

Flux Response viscometry using ideal mixtures has been already successfully applied at ambient [1,2] and high [3,4] temperatures, and to binary and ternary [5] mixtures. Recently, its range of application was extended to non-ideal mixtures [6]. In order to achieve this, the viscosity measurements had to be corrected, among other effects, for volume changes on mixing (i.e. non-ideality). The corrections proved to be accurate enough to provide values of the partial molar volumes. A prototype of that development was presented in ref. [7]. By design, that prototype was only able to work at ambient temperature. In the present work, the initial design has been modified to add proper temperature control and to simplify the corrections. The accuracy of the viscosity measurements has been improved and the temperature at which the mixture takes place is now controlled within ±0.02 K. We report the volume measurements of CO2 + CCl2 F2 mixtures obtained at 273.15 K and 1.11 bar using the improved apparatus. To our knowledge, the mixing volumes of this mixture have not been published before.

2.1. Capillary viscometry: differential setup Capillaries have been used for more than a century to measure changes in flow and viscosity. The impedance of a capillary can be calculated approximately with the Poiseuille equation [8] or more accurately as Berg has shown in ref. [9]. The Flux Response Technology is based on a capillary viscometer, a schematic diagram is shown in Fig. 1. Unlike traditional capillary viscometry, a Flux Response apparatus has two capillaries labelled as reference and measurement, joined together at the downstream ends. And, instead of measuring the pressure drop across a capillary, a sensitive differential pressure transducer measures the difference in the pressure drops of both capillaries. The main benefits of such a differential setup are: the use of a small range differential pressure transducer (±2 mm H2 O for the present work) that can measure small changes in pressure and the simplification of the equations, as will be shown. 2.2. Perturbation stream

∗

Corresponding author. Tel.: + 44 1509 222513; fax: + 44 1509 223923. E-mail address: [email protected] (N. Riesco).

0378-3812/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2006.11.004

The operation of a Flux Response apparatus starts with a gas mixture of known composition flowing through the reference and measurement lines. Both flows have been balanced so that

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flow plateau, the second as the mixing volume plateau and the latter as the viscosity plateau. In the following sections, we will show how to compute the partial molar volumes of a mixture and its logarithmic viscosity gradient using the transducer signal at the baseline and these three plateaus. 2.3. Measurement of flow changes

Fig. 1. Schematic diagram of a Flux Response apparatus. DPT is a differential pressure transducer. DL-1, DL-2 and DL-3 are large volumes included to delay the propagation of composition changes across the apparatus. C-1 and C2 are ideally two identical capillaries. C-1 will be referred as the measurement capillary and C-2 as the reference capillary.

the difference in pressure between the points A and B is zero within the precision of the transducer. We define the transducer signal at this stage as the baseline and we will denote the molar ˙ 0. flow through the measurement capillary by N A third line, labelled as perturbation, can be added to or removed from the measurement stream at point D. The volume flow through this perturbation line is small compared to that of the measurement line, less than 3%. Once the baseline has been recorded, the perturbation flow is added to the measurement stream. At first, the gas flowing through the perturbation line is a gas mixture of the same composition as that of the measurement stream. Later, it will be replaced with one of the pure component gases. We will denote the molar flow of the pure gas in the perturbation line by n, ˙ its molar volume by V¯ p and the molar volume of the gas mixture by V¯ m . While the delay line DL-3 contains a gas mixture of the same composition as that of the measurement stream, the molar flow through the measurement capillary can be calculated using Eq. (1). Note that at this stage, the gas mixtures at point D coming from the measurement and perturbation lines have the same composition. ˙1 = N

˙ 0 V¯ m + n˙ V¯ p N V¯ m

(1)

When the pure gas in the perturbation line reaches the point D, it mixes with the measurement stream changing its molar volume, V¯ m + V¯ . In order to delay the arrival of the change in composition at the measurement capillary, the volume DL2 has been inserted. At this stage, the molar flow through the measurement capillary can be calculated using Eq. (2): ˙ ˙ V¯ m + V¯ ) ˙ 2 = (N0 + n)( N V¯ m

dp ˙ V¯ = −Kc μN (4) dl where Kc is a constant depending on the geometry of the cap˙ its molar flow and V is its illary, μ the viscosity of the fluid, N molar volume. The pressure drop across the whole length of the capillary is the given by Eq. (5),  ˙ pu − pd = Kc μN V¯ dl (5) where pu and pd stand for the upstream and downstream pressures, respectively. In a Flux Response apparatus, any change ˙ + N, ˙ will be recorded by the differential in the molar flow, N pressure transducer, DPT, as a change in the upstream pressure of the measurement capillary, pu + p:  ˙ + N) ˙ V¯ dl (6) pu + p − pd = Kc μ(N For changes in pressure p small enough the integral factor in Eqs. (5) and (6) will be approximately equal and by subtraction of both equations we can obtain:  ˙ ˙ (7) p = Kc μN V¯ dl = Kmv N where Kmv is a constant depending on the characteristics of the capillary and the gas mixture. We define, using the baseline as a reference, the following changes in the upstream pressure: p1 for the flow plateau, p2 for the mixing volume plateau and p3 for the viscosity plateau. See Fig. 2 for an illustration. In order to calculate the partial molar volume of a gas mixture, we compute the ratio of pressures for p2 and p1 . This ratio can be expressed in terms of the molar flows by using the equation above:

(2) Rmv =

Eventually, the change in composition reaches the measurement capillary, C-2, and the molar flow will be calculated using Eq. (3). ˙ ˙ V¯ m + V¯ m ) ˙ 3 = (N0 + n)( ˙ 0 + n˙ N =N ¯ Vm + V¯ m

Here we will introduce the equations that model a Flux Response apparatus operating under the condition that the composition of the gas mixture in the measurement capillary does not change. In the next section, we will remove this condition. The pressure drop of an element of length of a capillary for laminar flow is given by Eq. (4),

(3)

The differential pressure transducer will record three plateaus for each of the molar flows above. We will refer to the first as the

˙0 ˙2 −N p2 N = ˙ ˙ p1 N1 − N 0

Substitution of Eqs. (1) and (2) gives:      ˙ 0 + n˙ 1 N V¯ m + V¯ Rmv = V¯ p n˙

(8)

(9)

To manipulate this expression further, we need to make use of Eqs. (10) and (11). Eq. (10) expresses the partial molar volume in terms of the molar volume, V¯ m , and its derivative with respect

N. Riesco et al. / Fluid Phase Equilibria 256 (2007) 93–98

95

Subtraction of the equations above yields: d ˙ 0 ]V¯ m ˙ 0 + N) ˙ − μm N (p − p0 ) = −Kc [[(μm + μ)(N dl ˙ 0 + N) ˙ V¯ ] + (μm + μ)(N (15) For a Flux Response apparatus the second term is typically negligible: d ˙ 0 + N) ˙ − μm N ˙ 0 ]V¯ m (p − p0 ) ≈ −Kc [(μm + μ)(N dl (16) The error introduced by dropping that term can be evaluated using the following ratio: Fig. 2. Illustration of the upstream pressure changes using the baseline as a reference: p1 for the flow plateau, p2 for the mixing volume plateau and p3 for the viscosity plateau.

to the molar fraction X of the first component. It is easily derived from the definition of molar partial volume. ∂V¯ V¯ V˜ i = V¯ m + (δi1 − X) ≈ V¯ m + (δi1 − X) ∂X X

(10)

where δi1 is 1 for the first component, i.e. i = 1, and 0 otherwise. Eq. (11) expresses the molar balance at point D: ˙ 0 X + δi1 n˙ = (N ˙ 0 + n)(X N ˙ + X)

(11)

where X is the change in the molar fraction X of the mixture due to the addition of the perturbation stream. Finally, combining Eqs. (10) and (11), and substituting into Eq. (9) we obtain an expression for the partial molar volume:   1 ˜ Rmv ≈ Vi (12) V¯ p where V˜ i stands for the partial molar volume of the component i in the mixture. 2.4. Measurement of viscosity changes Consider now the case in which the composition of the gas mixture in the measurement capillary changes by a small amount. This is the case when the perturbation stream has been added to the measurement line and has reached the measurement capillary. The pressure drop before the change in composition will be described by Eq. (13) and after by Eq. (14).

˙ 0 + N) ˙ V¯ (μm + μ)(N ˙ ˙ ˙ 0 ]V¯ m [(μm + μ)(N0 + N) − μm N   ˙ N ˙ 0 )) (1 + (μ/μm ))(1 + (N/ V¯ = ˙ N ˙ 0 )) − 1 V¯ m (1 + (μ/μm ))(1 + (N/

(17)

which in the present work is on average 1% and in the most adverse case, when the measurement flow is CCl2 F2 and the perturbation is CO2 , is less than 3%. Integration of Eq. (16) from the downstream to the upstream of the measurement capillary gives an expression for the change in the upstream pressure, p:  

˙ 0 + N ˙ − μm N ˙0 p = Kvi (μm + μ) N (18) where Kvi is a constant depending on the characteristics of the capillary and the gas mixture. Note that, unlike the derivation offered in a previous paper [1], it has not been necessary to assume that the gas mixture behaves ideally. In order to calculate the logarithmic viscosity gradient of a gas mixture, we define the ratio of pressures for p3 and p1 . This ratio can be expressed in terms of molar flows and viscosities by using the equation above. ˙ 3 − μm N ˙0 p3 (μm + μ)N Rvi = = (19) ˙ ˙ p1 μ m N1 − μ m N0 Inserting the Eqs. (1) and (3) into Eq. (19), we get an expression similar to that for the mixing volume ratio, Eq. (9):      ˙ 0 + n˙ V¯ p 1 N Rvi μm + = μ (20) V¯ m μm n˙ and which can be manipulated using Eq. (11) to show the dependence on the change in the molar fraction, X: Rvi

V¯ p μ = 1 + (δi1 − X) ¯ μm X Vm

(21)

dp0 ˙ 0 V¯ m = −Kc μm N dl

(13)

Finally, the logarithmic viscosity gradient can be estimated using the following approximation:

dp ˙ 0 + N)( ˙ V¯ m + V¯ ) = −Kc (μm + μ)(N dl

(14)

∂ ln μ ln(1 + (μ/μm )) ≈ ∂X X

where p0 is the pressure along the measurement capillary before the change in composition and p after. μ is the change in viscosity due to the change in composition and V¯ the change ˙ is any possible change in molar flow. in molar volume. N

(22)

2.5. Measurement of gas mixture compositions The gas mixtures in this work were prepared, as described in ref. [1], by mixing two gas streams. Their compositions were

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Table 1 Excess volumes, mixing volume and viscosity ratios for CO2 + CCl2 F2 at 273.15 K and 1.11 bar. Xa

Xb

0.000 0.000 0.113 0.236 0.362 0.480 0.482 0.561 0.560 0.600 0.715 0.833 1.000 1.000

0.000 0.000 0.126 0.243 0.354 0.481 0.505 0.556 0.557 0.609 0.699 0.808 1.000 1.000

CO2 perturbation

CCl2 F2 perturbation

X

Rmv,1

Rvi,1

X

Rmv,2

Rvi,2

0.026 0.026 0.027 0.017 0.020 0.018 0.018 0.011 0.012 0.013 0.010 0.005 – –

1.0103 1.0101 1.0077 1.0066 1.0043 1.0028 1.0023 1.0022 1.0023 1.0017 1.0014 1.0005 – –

1.1528 1.1496 1.1568 1.1269 1.1235 1.1015 1.0974 1.0836 1.0843 1.0758 1.0584 1.0343 – –

– – −0.004 −0.006 −0.012 −0.017 −0.017 −0.015 −0.015 −0.021 −0.025 −0.023 −0.030 −0.030

– – 1.0006 1.0009 1.0016 1.0027 1.0026 1.0032 1.0034 1.0038 1.0052 1.0061 1.0096 1.0093

– – 0.9787 0.9616 0.9349 0.9093 0.9035 0.8990 0.8982 0.8860 0.8698 0.8611 0.8511 0.8508

V E (cm3 mol−1 )c

∂ ln μ d ∂X

0.0 0.0 31.1 46.5 51.6 55.4 49.1 52.4 56.0 50.3 50.3 32.1 0.0 0.0

0.1771 0.1739 0.2016 0.1877 0.2102 0.2127 0.2142 0.2046 0.2059 0.2092 0.2073 0.1914 0.1665 0.1668

X is the molar fraction of CO2 before adding the perturbation and X + X after. a Calculated using a bubble flowmeter. b Calculated using Eq. (26). c Calculated using V ¯ m = XRmv,CO2 V¯ CO2 + (1 − X)Rmv,CCl2 F2 V¯ CCl2 F2 where V¯ CO2 = 20309 cm3 mol−1 and V¯ CCl2 F2 = 19889 cm3 mol−1 are the molar volumes of the pure components at 273.15 K and 1.11 bar and were calculated using REFPROP 7.0 [12]. d Calculated using Eqs. (21), (22) and (26).

calculated by measuring the volume flow rates of each stream with a bubble flowmeter. However, in order to make use of Eq. (21) when (δi1 − X) is close to zero, a more accurate measurement of the composition is required. To this end and following the method introduced by Heslop et al. [10], the mixture compositions have been determined using the mixing volume and viscosity ratios. By writing Eq. (21) explicitly for both perturbations, we get two expressions for the viscosity change: μ Rvi,1 V¯ 1 /V¯ m − 1 = μm X 1−X

(23)

μ Rvi,2 V¯ 2 /V¯ m − 1 = μm X −X

(24)

where Rvi,1 , Rvi,2 , V¯ 1 and V¯ 2 are the viscosity ratios and molar volumes of the CO2 and CCl2 F2 perturbations, respectively. The mixture molar volume V¯ m can be calculated using the molar partial volumes given by Eq. (12): V¯ m = V¯ 1 Rmv,1 X + V¯ 2 Rmv,2 (1 − X)

and 99%, respectively. The gas mixture was prepared by mixing two gas streams at known flow rates as described in ref. [1]. The error in the molar fraction is estimated to be ±0.03. The measurement and reference capillaries have been described in previous works, refs. [6,7]: length 1.1 m, external diameter 1/16 in. = 1.6 mm, internal diameter 0.030 in. = 0.8 mm and stainless steel. The differential pressure transducer was a Furness FCO44 with a pressure range of ±2 mmH2 O and modified as in ref. [4]. For further details, see our previous papers[1,4]. The current apparatus has been modified to improve its temperature control. Previously, the reference and measurement capillaries, C-1 and C-2, and the delay lines, DL-1 and DL2, were located in the interior of an aluminium block insulated with polyurethane and cooled directly by a Peltier module. For this work, the Peltier module was connected to a Eurotherm 825 PID controller, which kept the temperature in the aluminium block within ±0.02 K.

(25)

Substracting Eq. (23) from Eq. (24) and replacing Eq. (25) gives us the mixture molar fraction: X=

V¯ 2 (Rvi,2 − Rmv,2 ) ¯ V2 (Rvi,2 − Rmv,2 ) − V¯ 1 (Rvi,1 − Rmv,1 )

(26)

The mixture molar fractions obtained using Eq. (26) and those obtained using a bubble flowmeter are in good agreement within the accuracy of the bubble flowmeter (see Table 1). 3. Experimental All the measurements were carried out using mixtures of CO2 and CCl2 F2 at 273.15 K and 1.11 bar. The temperature was measured using a platinum resistance thermometer calibrated against the ice point. The CO2 gas was supplied by BOC and the CCl2 F2 by CK Gas. The suppliers claimed a purity better than 99.99%

Fig. 3. Excess volumes V E at 273.15 K and 1.11 bar for CO2 + CCl2 F2 mixtures vs. X, the molar fraction of CO2 . Symbols: experimental results (䊉). Solid curve: Redlich-Kister fitting.

N. Riesco et al. / Fluid Phase Equilibria 256 (2007) 93–98

In the previous design, the delay line DL-3 was kept outside the aluminium block at lab temperature. This resulted in a molar flow during the flow plateau given by Eq. (27): ˙ 1 = N

˙ 0 V¯ m (T ) + n˙ V¯ p (T  ) N V¯ m (T )

(27)

˙  is the actual molar flow during the flow plateau when where N 1 the delay line DL-3 is kept at lab temperature, T  the temperature in the lab and T is the working temperature. In order to avoid this correction, in the present work, the delay line DL-3 was moved into the aluminium block.

given by Eq. (29): 1  E E )2 (Vcalx − Vexp s(V E ) = N −k

97

(29)

was 4 cm3 mol−1 . The largest excess volume was 56.0 cm3 mol−1 , which yields a 7% error. The results of the smoothing have been shown in Fig. 3. 4.2. Viscosity logarithmic gradient

4. Results

The logarithmic viscosity gradient measurements are shown in Table 1. These measurements have been fitted to a Sutherland equation:

4.1. Excess volumes

μ=

The excess volume measurements are collected in Table 1, along with the ratio measurements for the CO2 and CCl2 F2 perturbations. In order to determine the excess volumes, the molar volumes of CO2 and CCl2 F2 at 273.15 K and 1.11 bar were calculated using REFPROP 7.0 [12]: 20309 cm3 mol−1 for CO2 and 19889 cm3 mol−1 for CCl2 F2 . The excess volume V E was fitted by unweighted least-squares polynomial regression to a Redlich–Kister expansion: V E = X(1 − X)

k 

Ai (2X − 1) = 222X(1 − X) cm3 mol−1

i=0

(28) where the number of coefficients k was determined by applying an F-test [11] at 95% confidence level. The standard deviation of the fitting, for N experimental data points and k coefficients,

μ2 (1 − X) μ1 X + X + Φ12 (1 − X) (1 − X) + Φ21 X

(30)

where μ1 and μ2 are the viscosities of CO2 and CCl2 F2 respectively, and Φ12 and Φ21 are two constants known as Sutherland coefficients. The fitting proceduce is a similar to that introduced in ref. [3] by Buffham et al., who wrote the Sutherland equation for the logarithmic viscosity gradient as a function of the viscosity ratio μ2 /μ1 and the Sutherland coefficients Φ12 and Φ21 : Φ12 /[X + Φ12 (1 − X)]2 − (μ2 /μ1 )Φ21 /[(1 − X) + Φ21 X]2 ∂ ln μ = ∂X X/[X + Φ12 (1 − X)] +(μ2 /μ1 )(1 − X)/[(1 − X) + Φ21 X]

(31)

The results of the fit are shown in Fig. 4. The fitted Sutherland coeffients are Φ12 = 1.696 and Φ21 = 0.536 and the fitted viscosity ratio is μ2 /μ1 = 0.8207, which differs in less than 3% from that calculated using refs. [13,14].

Fig. 4. Logarithmic viscosity gradient and viscosity at 273.15 K and 1.11 bar for CO2 + CCl2 F2 mixtures vs. X, the molar fraction of CO2 . Symbols: experimental results (䊉). Solid curve: Sutherland equation.

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5. Conclusions The main purpose of this work was to apply the Flux Response viscometry to non-ideal gas mixtures. To achieve this end, we have made two contributions. On the experimental hand, we have improved the Flux Response apparatus by adding a temperature control better than ±0.02 K. And, in order to avoid any temperature corrections in the perturbation flow, we have moved the delay line DL-3 into the temperature-controlled aluminium block. On the theoretical hand, we have formulated a theory that takes into account the mixing volume of gas mixtures and does not require the assumption of ideal gas. As well, we have identified a systematic error made when measuring viscosity changes and have provided an estimation for it. New volumetric and viscometric data have been reported for CO2 + CCl2 F2 mixtures at 273.15 K and 1.11 bar. List of symbols k number of Redlich–Kister coefficients Kc capillary constant Kmv Flux Response constant for measurements of flow changes Kvi Flux Response constant for measurements of viscosity changes N number of measurements ˙ n˙ N, molar flow p pressure p change in the upstream pressure of the measurement capillary s estimated standard deviation T working temperature T lab temperature V¯ molar volume VE excess volume V¯ change in the molar volume X molar fraction of CO2 X change in the molar fraction for CO2 Greek letters Kronecker delta (1 if i = 1, 0 otherwise) δi1 μ viscosity μ change in viscosity Subscripts c capillary d downstream

i m mv p u vi 0 1 2 3

i = 1 stands for CO2 and i = 2 stands for CCl2 F2 mixture mixing volume perturbation upstream viscosity baseline flow plateau mixing volume plateau viscosity plateau

Acknowledgement This work has been funded by the Engineering and Physical Sciences Research Council (EPSRC), grant reference GR/ S76236/01. References [1] G. Mason, B.A. Buffham, M.J. Heslop, B. Zhang, Chem. Eng. Sci. 53 (1998) 2665–2674. [2] G. Mason, B.A. Buffham, M.J. Heslop, P.A. Russell, B. Zhang, Chem. Eng. Sci. 55 (2000) 5747–5754. [3] B.A. Buffham, G. Mason, M.J. Heslop, P.A. Russell, Chem. Eng. Sci. 57 (2002) 4493–4504. [4] P.A. Russell, B.A. Buffham, G. Mason, M.J. Heslop, AIChE J. 49 (2003) 1986–1994. [5] P.A. Russell, B.A. Buffham, G. Mason, D.J. Richardson, M.J. Heslop, Fluid Phase Equilib. 215 (2004) 195–205. [6] P.A. Russell, B.A. Buffham, G. Mason, K. Hellgardt, Measurement of the gradient of viscosity with composition of gas mixtures, 15th Symposium on Thermophysical Properties, Boulder, June 22–27, 2003. [7] P.A. Russell, B.A. Buffham, G. Mason, K. Hellgardt, A new method to measure partial molar volumes of binary gas mixtures, 15th Symposium on Thermophysical Properties, Boulder, June 22–27, 2003. [8] M. Kawata, K. Kurase, A. Nagashima, K. Yoshida, Capillary viscometers, in: W.A. Wakeham, et al. (Eds.) Measurement of the Transport Properties of Fluids, Blackwell Oxford, 1991. [9] R.F. Berg, Metrologia 42 (2005) 11–23. [10] M.J. Heslop, G. Mason, B.A. Buffham, Chem. Eng. Res. Des. 78 (2000) 1061–1065. [11] P.R. Bevington, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York, 1969. [12] E.W. Lemmon, M.O. McLinden, M.L. Huber, Reference Fluid Thermodynamic and Transport Properties (REFPROP) Version 7. 0, National Institute of Standards and Technology (NIST), Boulder, CO, 2002. [13] A. Fenghour, W.A. Wakeham, V. Vesovic, J. Phys. Chem. Ref. Data 27 (1998) 31–44. [14] B. Latto, A.J. Al-Saloum, J. Mech. Eng. Sci. 12 (1970) 135–142.