1 Introduction

The increasing environmental concerns and energy problems have drawn considerable attention to Fischer–Tropsch(FT) synthesis technology which converts synthesis gas (CO + H2) from carbon feedstock such as natural gas, coal, biomass, etc. to synthetic hydrocarbon fuels and chemicals at an operation temperature from (448 to 623) K [1, 2]. The FT synthesis process generally involves complex products and reactants mixture systems of alkanes, olefins, alcohols, etc., and their mixtures with dissolved gases such as CO2, CO, H2, and N2 [2]. The thermophysical properties of reactants and products involved in FT synthesis process, particularly at elevated temperatures, are crucial for the development and industrial scaling of the reactors. These properties, including viscosity, density, surface tension, etc., have a significant impact on the hydrodynamics of the reactors. Hydrodynamics determines the phase mixing, as well as the heat and mass transfer within the reactors [3]. Nonetheless, due to the complex mixed fluids involved in FT synthesis and extreme operational conditions, the research on thermophysical properties for relevant fluids is far from sufficient.

Our group has conducted studies on the thermophysical properties of pure and mixed fluids involved in FT synthesis. Yang et al. [4,5,6] utilized pendant drop method to measure the surface tension of alkanes, alcohols, alkane mixtures, and alkanes with dissolved CO2 and N2 at temperatures up to 573 K. Cui et al. [7, 8] utilized surface light scattering method to measure the viscosity of n-heptane or n-tetradecane with dissolved CO2 at temperatures up to 473 K. Fu et al. [9, 10] and Liang et al. [11] used vibrating-wire method to measure the viscosity and density of alcohols, n-dodecane, and n-dodecane with dissolved CO2 in a temperature range of (298 to 623) K and at pressures up to 10 MPa. Yang et al. [12] measured the viscosity and density of CO2 + n-decane using a vibrating-wire viscometer coupled with a vibrating-tube densimeter at temperatures from (303 to 373) K and pressures up to 80 MPa.

Squalane plays a role in FT synthesis products and is a preferred option for modeling long-chain hydrocarbons. This particular branched alkane serves as a representative example of the paraffinic constituents in mineral oils. Despite the abundance of measured viscosity and density data of squalane in literature, experimental data for CO2 + squalane are insufficient (Table 1). Tomida et al. [13] measured density and viscosity of CO2 + squalane using a rolling ball viscometer and a glass piezometer, respectively, at four isotherms from (293.15 to 353.15) K, three pressures from (10.0 to 20) MPa, and at CO2 mole fractions varied from 0 to 0.417; they have estimated the uncertainty of viscosity to be within 3%. Ciotta et al. [14] reported both density and viscosity using a vibrating-wire instrument at temperatures from (303.15 to 448.15) K and CO2 mole fractions up to 0.788; they have performed the measurements at a very wide pressure range up to 171 MPa. Kandil et al. [15] also carried out density and viscosity measurements at temperatures between (313.15 to 363.15) K, pressures from slightly above bubble points up to about 75 MPa, and three CO2 mole fractions of 0.199, 0.299, and 0.519 with a vibrating-tube densimeter and a capillary viscometer, respectively. Zambrano et al. [16] utilized an Anton Paar DMA HPM vibrating-tube densimeter to measure the density of CO2 + squalane over a temperature range from (283.15 to 393.15) K, at pressures from (10 to 100) MPa and two CO2 mole fractions of 0.1001 and 0.2012. Although the viscosity and density of CO2 + squalane were measured up to a high pressure of 171 MPa, all the available data are for sub-saturated mixtures of CO2 in squalane and below 448 K, which is inadequate for the development of high-precision density and viscosity models for CO2 + squalane, particularly at high temperatures.

Table 1 Summary of liquid viscosity and density for CO2 + squalane in literature
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This work describes continuing work on the thermophysical properties of alkanes and alkanes with dissolved CO2 involved in FT synthesis. The viscosity and density measurements of squalane and CO2 + squalane were conducted by using a vibrating-wire apparatus. The measurements were carried out at eleven isotherms from (298.15 to 548.15) K and six pressures starting at 0.1 MPa up to 10 MPa for squalane and at eleven isotherms from (308.15.15 to 548.15) K and five pressures approximately ranging from (1 to 6) MPa for CO2 + squalane. The experimental data would serve as a foundational basis for future modeling studies on CO2 + alkane mixed fluids. Furthermore, the experimental viscosity and density data were correlated using the modified Tait-Andrade equation and Tait equation, respectively. The comparisons of correlations with the available literature data were discussed for squalene.

2 Experimental

2.1 Materials

Table 2 presents the information on chemical materials used in this work. CO2 was provided by Shaanxi Yulong Gas with a declared mass purity higher than 0.99999. Squalane was supplied by Acros Organics with a declared mass purity higher than 0.99.

Table 2 Specification of Chemical Materials
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2.2 Apparatus

The experimental measurements of squalane and CO2 + squalane were carried out by using a vibrating-wire apparatus described recently [11]. The apparatus employed a centerless-ground tungsten wire with a nominal radius of 0.05 mm and a nominal length of 50 mm. The wire was inserted vertically between two permanent magnets made of Sm2Co17 that can be operated at a maximum temperature of 623 K. To achieve viscosity and density measurements simultaneously, the upper end of the wire was clamped in a fixed support, while the lower end was tensioned by a sinker made of aluminum alloy 6061 and with a mass of 226.91 g. The wire, magnets, and sinker were assembled in the middle of a stainless steel high-pressure vessel by a supporting structure. During the experiment, the wire was submerged by the investigated fluid, and driven by a constant sinusoidal current to operate transverse vibration. The electromotive force in the vicinity of the fundamental transverse resonance frequency developed across the wire was detected as a function of frequency by a lock-in amplifier (model 7230, AMETEK). If all the other necessary parameters in the working equation were given, the viscosity and density of the studied fluid were obtained through a nonlinear optimization process that minimized the deviation between the experimental data and the working equation.

The temperature of the vibrating-wire apparatus was measured by a temperature measurement instrument connected with a platinum resistance thermometer which was calibrated before use. The combined expanded uncertainty of temperature measurements is estimated to be Uc(T) = 25 mK with a level of confidence of 0.95 (k = 2). For pure liquids sample and their mixtures measurements, the pressure of the experimental apparatus was regulated by a piston screw pump. For gas-saturated alkane measurements, the pressure was adjusted mainly by a high-pressure gas regulator combined with an air compressor. The pressure of the experimental apparatus was measured in external pipework using a Druck UNIK 5000 pressure transducer which has an accuracy of 0.1% for the full-scale range from (0 to 15) MPa. The output signal of the pressure transducer was read by an Agilent 34970A. The combined expanded uncertainty of the measured pressure is Uc(p) = 18 kPa (k = 2).

To ensure precise density measurement at elevated temperatures, we utilized a refined expression for f0, which is the “buoyancy-corrected” fundamental transverse resonance frequency of the wire in vacuum. This refined expression was initially presented by Ciotta and Trusler [17]. Additionally, we introduced a temperature-dependent function for the linear thermal expansion coefficient of the sinker [11]. Prior to conducting experimental measurements of the studied samples, the calibration measurements were carried out in vacuum and with toluene. In addition, validation measurements were performed with toluene at temperatures from (298 to 548) K and pressures up to 10 MPa. More details were presented in our recently published work [11]. Furthermore, we followed the same methodology as our previous work [6] to analyze the measurement uncertainty of the vibrating-wire apparatus [9]. The combined expanded uncertainties are estimated to be 2.2% for viscosity and 0.24% for density with a confidence level of 0.95 (k = 2).

3 Results

Measurements at every experimental condition were repeated more than three times and the mean of multiple repeated measurements was ascribed to be the experimental results. Table 3 displays the measured viscosities and densities of squalane. The experimental data were obtained at 11 isotherms spanning temperatures from (298.15 to 548.15) K and pressures from (0.1 to 10) MPa.

Table 3 Viscosities η and densities ρ of squalane measured with a vibrating-wire apparatus at temperatures T from (298.15 to 548.15) K and pressures p up to 10 MPaa
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Table 4 illustrates the experimental viscosities and densities of CO2 + squalane. The measurements were conducted at CO2-saturated conditions across 11 isotherms ranging from (308.15 to 548.15) K and at pressures from (1 to 6) MPa. After the measurement of CO2 + squalane at 548.15 K, 2.37 MPa, the apparatus was cooled down naturally, and then the viscosity and density were remeasured at 348.15 K, 1.13 MPa. The remeasurement results are in good agreement with the initial measurements, with deviations of less than 0.6% for viscosity and 0.12% for density. In order to further verify the reliability of the experimental results, after the experiment of 548.15 K, 5.67 MPa, the apparatus was cooled down to the atmospheric temperature naturally. The sample was then discharged, and the apparatus was cleaned and vacuumed. A new squalane sample was introduced for subsequent remeasurements of viscosity and density. The remeasurements were taken at the following conditions: 498.15 K, 2.35 MPa/3.55 MPa/5.65 MPa; 523.15 K, 1.29 MPa/2.36 MPa/3.59 MPa; 548.15 K, 1.30 MPa/2.37 MPa/3.59 MPa. The remeasurement results were compared with the first measured results, showing deviations of less than 0.19% for viscosity and 0.16% for density. All the measurements indicate that the sample dose not deteriorate after the high-temperature experiment and the experimental results are stable and reliable. Our group has conducted measurements of the solubility of CO2 + squalane at temperatures ranging from (300 to 523) K and at pressures up to 10 MPa, and developed solubility predictive models by the volume-translated Peng–Robinson equation of state (VTPR) in combination with van der Waals mixing rules [6]. The mole fractions of CO2 (xCO2) calculated by solubility predictive models are also listed in Table 4. The combined expanded uncertainty of the calculated xCO2 is estimated to be 3.5% (k = 2).

Table 4 Viscosities η and densities ρ of CO2 + squalane measured with a vibrating-wire apparatus at temperatures T from (308.15 to 548.15) K and pressures p up to 6 MPaa
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Figures 1 and 2 demonstrate the viscosities and densities of squalane as a function of pressure, respectively. The trends for both viscosities and densities show a close-to-linear increase with increasing pressure, while they decrease as temperature increases. At lower temperatures, the viscosities exhibit a rapid decrease with increasing temperature, while at higher temperatures the decrease is more gradual.

Fig. 1
figure 1

Viscosities of squalane as a function of pressure: , 298.15 K; , 323.15 K; , 348.15 K; , 373.15 K; , 398.15 K; 423.15 K; , 448.15 K; , 473.15 K; , 498.15 K; , 523.15 K; , 548.15 K

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Fig. 2
figure 2

Densities of squalane as a function of pressure:, 298.15 K; , 323.15 K; , 348.15 K; , 373.15 K; , 398.15 K; , 423.15 K; , 448.15 K; , 473.15 K; , 498.15 K; , 523.15 K; , 548.15 K

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The relationship between pressure and the viscosities of CO2 + squalane is illustrated in Fig. 3. At low temperatures, the viscosities present a considerable decrease with increasing pressure as the increase of CO2 solubility in liquid squalane, while at higher temperatures, the decrease in viscosity with increasing pressure is relatively slower. Figure 4 presents the pressure-dependant densities of CO2 + squalane. Along the isotherms, for temperatures below 498.15 K, the densities of CO2 + squalane increase with increasing pressure (or xCO2), while for temperatures above 523.15 K, an opposing trend occurs where densities decrease with increasing pressure (or xCO2), but the decrease is accompanied by noticeable fluctuations and the downward trend is not very pronounced. The possible reason is that the decrease of density with the increase of pressure is smaller than the density measurement uncertainty of the vibrating-wire apparatus used at 523.15 K and 548.15 K. Along the isobars, densities decrease with increasing temperature.

Fig. 3
figure 3

Viscosities of CO2 + squalane as a function of pressure: , 308.15 K; , 323.15 K; , 348.15 K; , 373.15 K; , 398.15 K; , 423.15 K; , 448.15 K; , 473.15 K; , 498.15 K; , 523.15 K; , 548.15 K

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Fig. 4
figure 4

Densities of CO2 + squalane as a function of pressure: , 308.15 K; , 323.15 K; , 348.15 K; , 373.15 K; , 398.15 K; , 423.15 K; , 448.15 K; , 473.15 K; , 498.15 K; , 523.15 K; , 548.15 K

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Figure 5 depicts a comparison of viscosities between squalane and CO2 + squalane at temperatures from (308.15 to 548.15) K. At the same temperature and pressure of CO2 + squalane, the viscosities of squalane were calculated from Eq. 1 described in Sect. 4. It is apparent that the viscosities of CO2 + squalane are always lower than that of squalane at the same temperature and pressure. In addition, the viscosities of squalane increase slightly with increasing pressures, while viscosities of CO2 + squalane decrease significantly with increasing pressures. For example, at a temperature of 308.15 K and a pressure of 1.10 MPa, the viscosity of pure squalane is about 1.4 times as much as CO2 + squalane. However, at a higher pressure of 5.07 MPa and the same temperature of 308.15 K, the viscosity difference is significantly greater, the viscosity of pure squalane is about 5.1 times as much as CO2 + squalane. This substantial variation is mainly due to the increased concentration of CO2 in liquid squalane, rather than the rise in pressure.

Fig. 5
figure 5

Comparison of the viscosities between squalane and CO2 + squalane at temperatures from (308.15 to 548.15) K. Hollow dots (squalane): , 308.15 K; , 348.15 K; , 398.15 K; , 448.15 K; , 498.15 K; , 548.15 K. Solid dots (CO2 + squalane): , 308.15 K;, 348.15 K;, 398.15 K; , 448.15 K; , 498.15 K;, 548.15 K

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Figure 6 shows the density comparisons between squalane and CO2 + squalane over a temperature range of (308.15 to 548.15) K. Densities of squalane were calculated from Eq. 5 described in Sect. 4 at the same condition of CO2 + squalane. It is evident that at low temperatures (e.g., 308.15 K, 348.15 K), the densities of CO2 + squalane are greater than that of squalane due to the influence of CO2. However, at a temperature of 398.15 K, the densities of CO2 + squalane are nearly identical to that of pure squalane. As the temperature rises further (e.g., 448.15 K, 498.15 K, 548.15 K), CO2 dissolves into pure squalane resulting in a decrease in density. Therefore, the dissolution of CO2 widens the density range of CO2 + squalane when compared to pure squalane.

Fig. 6
figure 6

Comparison of the densities between squalane and CO2 + squalane at temperatures from (308.15 to 548.15) K. Hollow dots (squalane): , 308.15 K; , 348.15 K; , 398.15 K; , 448.15 K; , 498.15 K; , 548.15 K. Solid dots (CO2 + squalane): , 308.15 K; , 348.15 K; , 398.15 K; , 448.15 K; , 498.15 K; , 548.15 K

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4 Correlations

4.1 Correlations for Squalane

To correlate the viscosity data of squalane, the modified Tait-Andrade equation [11] was used:

$$\eta = A_{\eta } \exp \left( {\frac{{B_{\eta } }}{{T + C_{\eta } }}} \right)\left( {\frac{p + E}{{p_{0} + E}}} \right)^{D}$$
(1)

where, p0 = 0.1 MPa; Aη and Cη are constants; Bη is a cubic function of reduced temperature; D and E are quadratic functions of reduced temperature.

$$B_{\eta } = \sum\limits_{i = 0}^{3} {b_{\eta ,i} } \left( {T/T_{0} } \right)^{i}$$
(2)
$$D = \sum\limits_{i = 0}^{2} {d_{i} } \left( {T/T_{0} } \right)^{ - i}$$
(3)
$$E = \sum\limits_{i = 0}^{2} {e_{i} } \left( {T/T_{0} } \right)^{i}$$
(4)

where, T0 = 298.15 K; bη,i, di, and ei are adjusted coefficients by fitting to the measured viscosity data.

A modified Tait equation [11] was used to correlate the density data of squalane

$$\rho = \rho_{0} \left [ {1 - C\ln \left( {\frac{p + B}{{p_{0} + B}}} \right)} \right]^{ - 1}$$
(5)

where, p0 = 0.1 MPa; ρ0 is the density at p = p0; C is a constant. ρ0 and B are cubic functions of reduced temperature as follows:

$$\rho_{0} = \sum\limits_{i = 0}^{3} {a_{i} } \left( {T/T_{0} } \right)^{i}$$
(6)
$$B = \sum\limits_{i = 0}^{3} {b_{i} } \left( {T/T_{0} } \right)^{i}$$
(7)

in which, ai and bi are determined by fitting to the experimental densities.

For further analysis, the statistical values of fitting, the average absolute relative deviations (AAD), the maximum absolute relative deviations (MAD), and the average relative bias (Bias) were defined as follows:

$${\text{AAD}}/\% = \frac{100}{N}\sum\limits_{i = 1}^{i = N} {\left| {\frac{{X_{i,\exp } - X_{{i,{\text{cal}}}} }}{{X_{{i,{\text{cal}}}} }}} \right|}$$
(8)
$${\text{MAD}}/\% = 100 \cdot \max \left( {\left| {\frac{{X_{i,\exp } - X_{{i,{\text{cal}}}} }}{{X_{{i,{\text{cal}}}} }}} \right|} \right)$$
(9)
$${\text{Bias / \% = }}\frac{100}{N}\sum\limits_{i = 1}^{i = N} {\left( {\frac{{X_{i,\exp } - X_{{i,{\text{cal}}}} }}{{X_{{i,{\text{cal}}}} }}} \right)}$$
(10)

Table 5 presents the coefficients obtained from viscosity and density correlations for squalane, respectively. The AADs of viscosity and density are 0.37% and 0.02%, respectively, indicating that the correlations are accurate in describing the experimental data in this work.

Table 5 Coefficients of modified Tait-Andrade equation and Tait equation and for squalane
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The viscosity and density comparisons between experimental data and the correlations were made at experimental conditions of this work. The viscosity deviations are demonstrated in Fig. 7. Except for several points where the deviation is large, the viscosity deviations are generally within ± 3%. The largest deviation is the viscosity at 353.15 K, 0.1 MPa measured by Krahn and Luft [18] using a rolling ball viscometer, with a deviation of 7.33%. In addition, the viscosity at 313.15 K, 10 MPa measured by Comunas et al. [19] shows a large negative deviation of -6.86%. The density deviations are almost within ± 0.3% as shown in Fig. 8. The data point with the maximum deviation was measured by Rowane et al. [20] using a rolling ball densitometer/viscometer at 372.6 K, 4 MPa, with a deviation of − 1.11%. Additionally, the density data reported by Schmidt et al. [21] utilizing the vibrating-wire method demonstrate an overall negative deviation ranging from (− 0.42 to − 0.26) %.

Fig. 7
figure 7

Squalane viscosity deviations as a function of temperature between experiment and correlation. , This work; , Krahn and Luft [18]; , Kumagai et al. [22]; , Pensado et al. [23]; , Dubey and Sharma [24]; , Ciotta et al. [14]; , Dubey et al. [25]; , Dubey et al. [26]; , Harris [27]; , Comuñas et al. [28]; , Mylona et al. [29]; , Comuñas et al. [19]; , Gaciño et al. [30]; , Schmidt et al. [21]; , Rowane et al. [20]; , Klein et al. [31]; , Bürk et al. [32]

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Fig. 8
figure 8

Squalane density deviations as a function of temperature between experiment and correlation., This work; , Korosi and Kováts [33];, Fandiño et al., [34]; , Kumagai et al. [22];, Dubey and Sharma [24]; , Ciotta et al., [14]; , Dubey et al. [25]; , Dubey et al. [26];, Harris [27];, Korotkovskii et al. [35]; , Schmidt et al. [21]; , Rowane et al. [20]; , Bürk et al. [32]

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4.2 Correlations for CO2 + Squalane

Utilizing the viscosity data of binary mixtures composed of methane and carbon dioxide with hydrocarbons up to n-hexadecane, Thol and Richter [36] conducted an analysis of the extended corresponding states method, two different entropy scaling approaches, and the friction theory regarding their efficacy in predicting viscosity with increasing asymmetry of the binary mixtures. According to their analysis, the precise prediction of viscosity for asymmetry mixtures remains a challenge primarily due to limited experimental data and the significant influence of high asymmetry. Similarly, the accurate density modeling of asymmetry mixtures also presents significant difficulty. Tait-like equations were most commonly used to correlate the viscosities and densities of CO2 + alkane mixtures at every single composition [14, 16, 37]. While the correlations may not accurately extrapolate to other state points, they closely match the experimental data when used for interpolation.

Experimental viscosity and density data of CO2 + squalane were fitted to the modified Tait-Andrade equation and Tait equation mentioned above, respectively. According to Binti Mohd Taib [38], parameters in the equations could be expressed as functions of the mole fraction of CO2, x:

$$A_{\eta } = A_{\eta ,0} + A_{\eta ,1} x$$
(11)
$$B_{\eta } = B_{\eta ,0} + B_{\eta ,1} x$$
(12)
$$C_{\eta } = C_{\eta ,0} + C_{\eta ,1} x$$
(13)
$$D = \sum\limits_{i = 0}^{2} {\left( {d_{i,0} + d_{i,1} x} \right)\left( {T/T_{0} } \right)^{ - i} }$$
(14)
$$E = \sum\limits_{i = 0}^{2} {\left( {e_{i,0} + e_{i,1} x} \right)\left( {T/T_{0} } \right)^{i} }$$
(15)
$$\rho_{0} = \sum\limits_{i = 0}^{2} {\left( {a_{i,0} + a_{i,1} x} \right)\left( {T/T_{0} } \right)^{i} }$$
(16)
$$B = \sum\limits_{i = 0}^{2} {\left( {b_{i,0} + b_{i,1} x} \right)\left( {T/T_{0} } \right)^{i} }$$
(17)
$$C = c_{0} + {\text{c}}_{1} x$$
(18)

Coefficients of viscosity and density correlations for CO2 + squalane are presented in Table 6. The AADs for viscosity and density are 0.65% and 0.04%, respectively, indicating that the correlations are able to accurately reproduce our measured data.

Table 6 Coefficients of modified Tait-Andrade equation and Tait equation for CO2 + squalane
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The deviations between the experimental data and calculated values from correlations for viscosity and density are shown in Figs. 9 and  10, respectively. The viscosity deviations are generally within 0.1%, and the density deviations are almost within 0.1%. These deviations are evenly distributed on both sides of the zero line, except for the viscosity deviations at the maximum temperature of 548.15 K. As can be seen from Fig. 9, the viscosity deviation absolute values increase with the increasing pressure reaching 4.67% at 5.67 MPa. The possible reason for the large deviations is that the viscosity value changes sharply from the experimental temperature of 308.15 K to 548.15 K, from 12.946 mPa·s to 0.393 mPa·s. The modified Tait-Andrade equation adopted in this section may not be able to describe such a drastic change in viscosity very accurately.

Fig. 9
figure 9

CO2 + squalane viscosity deviations between experiment and correlation. , 308.15 K; , 323.15 K; , 348.15 K; , 373.15 K; , 398.15 K; , 423.15 K; , 448.15 K; , 473.15 K; , 498.15 K; , 523.15 K; , 548.15 K

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Fig. 10
figure 10

CO2 + squalane density deviations between experiment and correlation. , 308.15 K; , 323.15 K; , 348.15 K; , 373.15 K; , 398.15 K; , 423.15 K; , 448.15 K; , 473.15 K; , 498.15 K; , 523.15 K; , 548.15 K

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5 Conclusions

In this work, viscosity and density measurements of squalane and CO2 + squalane were conducted using a vibrating-wire apparatus up to high temperatures of 548.15 K. The experimental data were fitted by the modified Tait equation for density data and the modified Tait-Andrade equation for viscosity data, respectively. The developed empirical correlations of pure squalane and CO2 + squalane binary mixtures are able to represent the measured data of this work almost within experimental uncertainties. By comparing the squalane viscosity and density correlations with existing literature data, it was found that the viscosity and density deviations were largely within 0.3% and 3%, respectively. The viscosity and density of CO2-saturated squalane were first measured. The experimental data obtained in this work can fill in the gaps in experimental viscosity and density data of CO2 + squalane systems under CO2-saturated conditions and at high-temperature regions. Furthermore, these data can serve as a fundamental basis for theoretical and empirical modeling, as well as molecular simulations.