Introduction

Study on the solubility of gases in a variety of hydrocarbons and their dissolution properties is important. Hydrogen is one of these gases which its solubility in hydrocarbons is one of the important parameters in designing, optimizing and interpreting kinetic reactions that are needed in various industrial processes and the commissioning of related equipment. Hydrogenation reactors in petrochemical complexes, are one of the equipment that hydrogen solubility in the input feed can affect the quality of products. Therefore, reliable estimation of hydrogen solubility in alkenes is needed, which results in more precise information gained for designing and optimizing hydrogenation reactors and amount of hydrogen consumed in a unit. Sagara et al. [21, 22], Vasileva et al. [32], Sokolov and Polyakov [28], and Xie et al. [33] experimentally investigated the hydrogen-olefin systems. Baird et al. [5] studied the solubility of hydrogen in shale oil experimentally. The results showed that the solubility of hydrogen is very small due to phenolic polar compounds in shale oil. Given that prediction and calculation of solubility and phase equilibrium with cubic equations of state for hydrogen-hydrocarbon systems is good, these systems are modeled more with cubic equations of state. Ferrando and Ungerer [7] modeled phase equilibria of hydrogen-1-hexene and hydrogen-1-octene systems by PR [20] equation of state using interaction parameter that presented by Moysan et al. [12]. Also Qian et al. [19] studied the phase equilibria of hydrogen-ethylene, hydrogen-propylene and hydrogen-1-hexene systems using the PR equation with binary interaction parameter obtained by group contribution method [10]. Aguilar-Cisneros et al. [1, 2] investigated the solubility of hydrogen in heavy oil cuts by combining the PR and UNIFAC model (Universal Quasi-Chemical Functional Group Activity Coefficients), with an overall error near to 15%. Torres et al. [30] presented augmented Grayson-Streed model for predicting hydrogen solubility in normal alkanes and aromatics by studying the hydrogen solubility in these hydrocarbons using the Grayson-Streed [29] model. The proposed model improves the predicted values of hydrogen solubility in normal alkanes and aromatics. Also, Nasery et al. [14] developed a model based on adaptive neuro fuzzy inference system (ANFIS) to predict hydrogen solubility in oil cut, based on available data from open literature. The average absolute deviation and coefficient of determination (R2) of model were 3.4% and 0.99, respectively, which indicates good accuracy and correlation of the model. The results showed that the model could predict experimental data with acceptable accuracy.

Solubility of gases in a solution decreases significantly by increasing or decreasing pressure that applied on the solution and according Henry’s law, the amount of gas that is dissolved at a constant temperature in a certain amount of liquid is directly proportional to the partial pressure of the gas above the solution. Due to the fact that dilute solutions follow the Henry’s law, usually the Henry’s constant of these systems is calculated. Nakahara and Hirata [13], Orentlicher and Prausnitz [15] studied Henry’s constant of hydrogen in alkenes such as ethylene and propylene, Angelo and Francesco [6] studied Henry’s constant of hydrogen in alcohols and Park et al. [17] studied Henry’s constant of hydrogen in heavy paraffins. Recently, Trinh et al. [31] have studied Henry’s constant of hydrogen in compounds containing alkanes, alcohols, aldehydes, carboxylic acids, esters and ethers.

The thermodynamic properties of dissolution, such as enthalpy, entropy, and Gibbs free energy, are important in terms of theoretically and practically. Using these quantities, one can predict and calculate behavior of the system at different conditions without laboratory data. The calculation of thermodynamic dissolution properties is carried out using the Van’t Hoff equation. Anthony et al. [4] calculated enthalpy and entropy of dissolved various gases (hydrogen, nitrogen, argon, etc.) in ionic liquid [bmim] [PF6] using the Van’t Hoff equation. In other work, they examined the enthalpy and entropy of water dissolution in a variety of ionic liquids by this equation [3]. Recently, Liu et al. [11] investigated the thermodynamic properties of the dissolution of various types of gases, including hydrogen, in the ionic liquid [emim] [Tf2N] using the Van’t Hoff equation. Also Shakeel et al. [23,24,25,26] using the Van’t Hoff equation, calculated the thermodynamic properties of dissolved solids in liquids.

In this work, the solubility of hydrogen in olefins that exist in a variety of petrochemical streams is considered and will be investigated. The studied olefins in this work are ethylene, propylene, 1-butene, 1-hexene, 1-heptene and 1-octene. A thermodynamic model was developed for investigating the solubility of hydrogen in these olefins. In this model, the PR equation of state is used, and the interaction parameter between hydrogen and olefins is obtained by fitting the experimental data of hydrogen solubility in olefins. Interaction parameters are also calculated using methods such as Moysan and GCM and are compared with the interaction parameter obtained by fitting. Using methods such as Moysan and GCM to calculate the interaction parameter in equation of state, it is possible to predict the hydrogen solubility in olefins. Henry’s constant of hydrogen is also rarely investigated in the previous work. In this work, Henry’s constant of hydrogen is calculated and evaluated in all systems studied with the PR equation of state. Finally, the thermodynamic properties of dissolution such as enthalpy (heat of solution), entropy and Gibbs free energy of dissolution are calculated for these systems because these properties are necessary to describe these solution and design of processes. These properties are also obtained for the first time and are not observed in previous work.

Thermodynamic modeling

The Peng–Robinson equation of state is written as follows [20]:

$$P = \frac{RT}{{\upsilon - b}} - \frac{a}{{\upsilon \left( {\upsilon + b} \right) + b\left( {\upsilon - b} \right)}}.$$
(1)

With

$$a = a_{c} \,\alpha \left( T \right),$$
(2)
$$b = 0.0777960739\frac{{RT_{c} }}{{P_{c} }},$$
(3)
$$a_{c} = 0.457235529\frac{{R^{2} T_{c}^{2} }}{{P_{c} }},$$
(4)
$$\alpha \left( T \right) = \left[ {1 + m\left( {1 - \sqrt {\frac{T}{{T_{c} }}} } \right)} \right]^{2} ,$$
(5)

where P, R, T, v, \({T}_{c}\), \({P}_{c}\) and \(\omega\) are pressure, universal gas constant, temperature, molar volume, critical temperature, critical pressure and acentric factor, respectively. The parameter m is calculated according the value of the acentric factor:

$$\omega < 0.49:m = 0.37464 + 1.54226\omega - 0.426992\omega {}^{2},$$
(6)
$$\omega > 0.49:m = 0.379642 + 1.48503\omega - 0.164423\omega {}^{2} + 0.016666\omega^{3} .$$
(7)

a and b for the mixtures are:

$$a = \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{N} {z_{i} z_{j} \sqrt {a_{i} a_{j} } \left[ {1 - k_{ij} \left( T \right)} \right]} } ,$$
(8)
$$b = \sum\limits_{i = 1}^{N} {z_{i} b_{i} } ,$$
(9)

where in Eqs. (8) and (9), \({z}_{i}\), \({z}_{j}\) and \({k}_{ij}\) are mole fraction of component i and j in mixture and binary interaction parameter, respectively. Binary interaction parameters were calculated using three methods:

  • Fitting

  • Group contribution method

  • Moysan correlation

In the first method, binary interaction parameter was obtained by fitting experimental hydrogen solubility by equation of state. The binary interaction parameter between hydrogen and the alkenes has been evaluated by minimizing the following objective function:

$${\text{O}}.{\text{F}}. = \left( \frac{1}{N} \right)\sum\limits_{i = 1}^{N} {{\text{Abs}}\left( {\frac{{P_{{\exp_{i} }} - P_{{{\text{cal}}_{i} }} }}{{P_{{\exp_{i} }} }}} \right)} ,$$
(10)

where N is the number of points used, Pexp is the experimental pressure from literatures and Pcal is the calculated pressure with the PR78 equation of state that Binary interaction parameters were calculated using fitting the equation of state on experimental data, group contribution method and Moysan correlation. Bubble point calculations have been used to calculate the total pressure in these systems.

In the second method, the GCM [10] was used to calculate the binary interaction parameter:

$$k_{ij} \left( T \right) = \frac{{ - \frac{1}{2}\sum\limits_{k = 1}^{{N_{g} }} {\sum\limits_{l = 1}^{{N_{g} }} {\left( {\alpha_{ik} - \alpha_{jk} } \right)\left( {\alpha_{il} - \alpha_{jl} } \right)A_{kl} \left( {\frac{298.15}{T}} \right)^{{\left( {\frac{{B_{kl} }}{{A_{kl} }} - 1} \right)}} - \left( {\frac{{\sqrt {a_{i} \left( T \right)} }}{{b_{i} }} - \frac{{\sqrt {a_{j} \left( T \right)} }}{{b_{j} }}} \right)} } }}{{2\frac{{\sqrt {a_{i} \left( T \right)a_{j} \left( T \right)} }}{{b_{i} b_{j} }}}},$$
(11)

where in Eq. (11), Ng is the number of different groups defined by the method and \({a}_{ik}\) is the fraction of molecule i occupied by group k. \({A}_{kl}={A}_{lk}\) and \({B}_{kl}={B}_{lk}\) (where k and l are two different groups) are constant parameters that were determined by Qian et al. [19].

The third method for calculating the binary interaction parameter between hydrogen and alkenes for this equation of state is the proposed equation by Moysan et al. [12]:

$$k_{{H_{2} ,j}} = 1 + \frac{{0.0417\left( {T_{{r,H_{2} }} - 1} \right) - 1}}{{\sqrt {\alpha_{{H_{2} }} } }}.$$
(12)

In Eq. (12), \({T}_{r,H2}\) is reduced temperature of hydrogen and \({\alpha }_{H2}\) is defined by Eq. (5).

Thermodynamic parameters for dissolution of hydrogen

The thermodynamic parameters for dissolution of hydrogen in various alkenes were expressed (or measured) in terms of dissolution enthalpy \(\left( {\Delta H^{ \circ } } \right)\), Gibbs free energy \(\left( {\Delta G^{ \circ } } \right)\), and dissolution entropy \(\left( {\Delta S^{ \circ } } \right)\). The \(\Delta H^{ \circ }\) values for dissolution behavior of hydrogen in each alkene were determined using Van’t Hoff equation [23,24,25,26]. The \(\Delta H^{ \circ }\) was calculated using Eq. (13):

$$\Delta H^{ \circ } = - R\frac{\partial \ln \left( x \right)}{{\partial \left( {\frac{1}{T} - \frac{1}{{T_{{{\text{mean}}}} }}} \right)}}.$$
(13)

where R is the universal gas constant (8.314 J/mol K), and Tmean, is the mean of the experimental temperatures. The \(\Delta H^{ \circ }\) values were obtained from the slope of the plot of ln(x) versus \(\left( {\frac{1}{T} - \frac{1}{{T_{{{\text{mean}}}} }}} \right)\). The \(\Delta G^{ \circ }\) for the dissolution of hydrogen can be calculated as:

$$\Delta G^{ \circ } = - RT*{\text{intercept}}{.}$$
(14)

In which the intercept was obtained from the plot of ln(x) versus \(\left( {\frac{1}{T} - \frac{1}{{T_{{{\text{mean}}}} }}} \right)\). Finally, from these evaluated \(\Delta H^{ \circ }\) and \(\Delta G^{ \circ }\) values, the \(\Delta S^{ \circ }\) of dissolution were obtained using Eq. (15):

$$\Delta S^{ \circ } = \frac{{\Delta H^{ \circ } - \Delta G^{ \circ } }}{{T_{{{\text{mean}}}} }}.$$
(15)

Results and discussion

Correlation and prediction of hydrogen solubility

Table 1 presents the critical properties of the compounds used. The Binary interaction parameter is obtained at each temperature by fitting the equation of state on experimental data, group contribution method (GCM) and Moysan correlation. The binary interaction parameters are given in Table 2.

Table 1 Critical properties and acentric factors
Full size table
Table 2 Binary interaction parameter and calculated Henry’s constant values
Full size table

The average absolute deviation (AAD) between experimental data and correlation/prediction of H2 solubility in the different alkenes is given in Table 3. The results show that using the PR equation with kij obtained from the fitting of data, has lower average error in comparison with the kij obtained from the Moysan and GCM method, which was predictable.

Table 3 Average absolute deviation (AAD) for solubility of hydrogen in alkenes
Full size table

Comparison of the modeling results using the kij obtained from the Moysan and GCM method shows that the average error associated with the Moysan method is less than the GCM. Of course, this result was unexpected because the GCM method is a newer method than Moysan and is expected to have better results. The reason for the weak results of the GCM method may be the optimal parameters of the model (Akl and Bkl) and the components used to optimize the parameters of the GCM model. For example, optimization of the parameters of the GCM model was done with cyclic and high-carbon number hydrocarbons. But in the present work, hydrocarbons with a low carbon number are modeled. So that average error between experimental hydrogen solubility and modeling results by GCM method for systems hydrogen + ethylene or propylene was high.

Another reason that could lead to errors in the results of the prediction of general models such as GCM and Moysan is the temperature range in which the parameters of these models is fitted in this range. If the fitting of the parameters was performed at high temperature and these parameters used at low temperatures, this can be caused the high error and the weakness of the model results (for example, the prediction results of the hydrogen + 1-butene system at 253.2 K using the Moysan kij).

Figure 1 shows solubility of hydrogen in alkenes using PR EoS with kij by Fitting, GCM and Moysan correlations.

Fig. 1
figure 1

Correlation and prediction of solubility of hydrogen in alkenes using PR EoS with kij by fitting, GCM and Moysan correlations. Solid points (●): experimental data. Solid line (——): solubility correlation using PR EoS with kij by fitting, dashed line (---): solubility prediction using PR EoS with kij by GCM, dotted line (…..): solubility prediction using PR EoS with kij by Moysan correlation. Systems: a hydrogen + ethylene, b hydrogen + propylene ,c hydrogen + 1-butene, d hydrogen + 1-hexene, e hydrogen + 1-heptene, f hydrogen + 1-octene

Full size image

Henry’s law

Henry’s law is used to describe the low solubility of hydrogen in a variety of alkene solvents. Based on definition, the Henry’s constant of a solute in a solvent, H1,2 is given as:

$${H}_{\mathrm{1,2}}=\underset{{x}_{1}\to 0}{\mathrm{lim}}\left(\frac{{f}_{1}^{l}}{{x}_{1}}\right)=\underset{{x}_{1}\to 0}{\mathrm{lim}}\left({\varphi }_{1}P\right),$$
(16)

where \({f}_{1}^{l}\), \({x}_{1}\), and \({\varphi }_{1}\) are the fugacity, mole fraction and fugacity coefficient of hydrogen in the liquid phase, respectively. The Henry’s constant was calculated using the PR equation and binary interaction parameter derived from the fitting of experimental solubility data. Accordingly, at each temperature, pressure and mole fraction of hydrogen in liquid phase, fugacity coefficient of hydrogen in liquid phase was calculated. The calculated fugacity coefficient was multiplied by system pressure. The obtained values were plotted in terms of hydrogen mole fraction in the liquid phase. Draw the best passing line through the points, and the intersection of this line with the vertical axis gives the Henry’s constant at that temperature. Figure 2 shows how to use this method to calculate the Henry’s constant of hydrogen in ethylene at 123.15 K. The Henry’s constant of hydrogen in some alkenes are given in Table 2. Also, Fig. 3 presents the temperature dependency of these Henry’s constants.

Fig. 2
figure 2

Evaluation of Henry’s constant of hydrogen in ethylene at T = 123.15 K

Full size image
Fig. 3
figure 3

Experimental and calculated Henry’s constant of hydrogen in some alkenes studied in this work vs. temperature

Full size image

According to Fig. 3, Henry’s constant of hydrogen decrease with increasing temperature, which show that with increasing temperature, the solubility of hydrogen in alkenes increases (see Fig. 1; Tables 2, 3, columns 2, 3). Temperature dependency of Henry’s constant of hydrogen in ethylene and propylene is higher than that of other alkenes, which indicates that the temperature has a greater effect on the solubility of hydrogen in ethylene and propylene.

The calculated Henry’s constant values of hydrogen in ethylene in this work and the values calculated by Nakahara and Hirata (BWR equation of state, [13], were compared to experimental data [13], and Orentlicher and Prausnitz [15] at temperature 173 and 199 K. also, that comparison was made for calculated Henry’s constant of hydrogen in propylene at temperature 200 K. The Comparison showed that average absolute deviation (AAD%) between experimental data and calculated Henry’s constant in this work was 9.80% and AAD% between experimental data and calculated Henry’s constant by BWR equation of state [13], was 8.85%. These comparisons show that the equation of state and method used in this work to obtain the Henry’s constant of hydrogen in the alkenes is reliable and the errors associated to this model are in the level and size of previous work. These results were shown in Table 4.

Table 4 Experimental Henry’s constant data and calculated by equation of states
Full size table

Thermodynamic parameters for dissolution of hydrogen in Alkenes

The values of thermodynamic parameters for dissolution of hydrogen in Alkenes are listed in Table 4. The \({\Delta H}^{^\circ }\) values for the dissolution of hydrogen were positive for all solvents studied. The positive values of \({\Delta H}^{^\circ }\) indicated endothermic dissolution of hydrogen. The \({\Delta G}^{^\circ }\) values for the dissolution of hydrogen were also observed to be positive for all the solvents studied. Positive values of \({\Delta G}^{^\circ }\) indicating that the dissolution of hydrogen was spontaneous and the positive values of \({\Delta G}^{^\circ }\) in Alkene solvents were possible due to strong molecular interactions between hydrogen-solvent molecules in comparison with weak molecular interactions between hydrogen–hydrogen and solvent–solvent molecules. As can be seen from Table 5, all dissolution entropy of hydrogen in ethylene, propylene, 1-butene, and 1-hexene at 100 bar, and 1-octene at 45.5 and 101.3 bar are negative that indicate the dissolution of hydrogen are not an entropy-driven process. Other value of ΔS° of hydrogen in alkene (1-hexene at P = 150 bar, P = 200 bar, P = 250 bar and 1-octene at P = 202.7 bar, P = 304.0 bar) are positive, that indicate the dissolution of hydrogen are an entropy-driven process.

Table 5 Thermodynamic parameters of hydrogen dissolution at different pressures and the mean of the experimental temperatures (Tmean)
Full size table

Also according to Table 5 (4) the values of \({\Delta H}^{^\circ }\) and \({\Delta G}^{^\circ }\) for the dissolution of hydrogen in alkenes were very low, which indicated that relatively low energy is required for the solubilization of hydrogen in alkenes.

In this work, the thermodynamic properties for dissolution of nitrogen, helium, argon and methane in alkenes have been studied, too (Table 6). Comparison of the values of the thermodynamic properties of hydrogen, nitrogen, helium, argon and methane shows that the enthalpy of light gases in alkenes is very low. Also, the positive values of enthalpy of hydrogen and helium in alkenes indicates an increase solubility by increasing the temperature and negative values of enthalpy of nitrogen, argon and methane in alkenes indicates a decrease in solubility with increasing temperature.

Table 6 Thermodynamic dissolution parameters of nitrogen, helium, argon and methane in some alkenes at different pressures, and the mean of the experimental temperatures (Tmean)
Full size table

Conclusion

The solubility of hydrogen in ethylene, propylene, 1-butene, 1-hexene, 1-heptene and 1-octene was investigated. The PR equation was used to correlate and predict the hydrogen solubility in alkenes. Binary interaction parameter was obtained by fitting the equation of state on experimental data, group contribution method and Moysan correlations. The results showed PR EoS with fitting binary interaction parameter had lowest absolute average error than Moysan and group contribution method, respectively. Because most dilute solutions follow Henry’s law, Henry’s constant was calculated for all studying systems using PR equations of state with fitting binary interaction parameter. With increasing temperature, Henry’s constant of hydrogen in alkenes decreased and hydrogen solubility increased in these systems. Also Henry’s constants of hydrogen in ethylene and propylene were more than in other alkenes. These binary interaction parameters and Henry’s constant can be used in simulation software to simulate industrial processes containing hydrogen and alkenes (petrochemical industries). In addition, Thermodynamic analysis indicated endothermic and spontaneous dissolution behavior of hydrogen in the studied alkene solvents. But results show the dissolution of hydrogen in some of alkenes is an entropy-driven process, such as 1-hexene and 1-octene. Also some recommendations for future work are: (a) using statistical associating fluid theory (SAFT) family equation of state to model system containing hydrogen, (b) ternary systems and mixtures containing hydrogen will be investigated, (c) another olefin such as branched and cyclo-olefin will be investigated.