The selectivity model which is the primary key in industries that are significant for optimization of process and reactors via statistical method was studied. The RSM was employed to improve the selectivity model of products. The results have a good agreement between experimental data and predicted model. The results are presented in Table 3. Finally, the minimum amount of undesirable and maximum amount of desirable products are concluded from optimization process which are reported in Table 4. All analysis and calculations were performed with Minitab software.
Table 3 Selectivity model of products as well as predicted value by RSM
Table 4 Optimization results
Selectivity model of methane
Figure 2 illustrates the surface and contour plot of methane selectivity that affects the operating conditions (e.g. temperature, pressure and H2/CO ratio). As can be seen, the methane selectivity decrease by increasing the temperature at constant pressure and H2/CO ratio but this increase is smooth at constant H2/CO ratio. The methane selectivity increases by increasing the H2/CO ratio. As it is clear for CH4 product, there is a minimum selectivity in the low H2/CO ratio. As can be seen, at high temperature and low H2/CO ratio, the amount of methane selectivity is reduced. CH4 selectivity varies parabolically with pressure. By increasing the pressure, the methane selectivity decreases and then increases. Therefore, the minimum amount of methane selectivity increases. The proposed selectivity model is as follows:
$${\text{S}}\;\left( {\text{methane}} \right) = 2.441 - 0.002281{{T}} - 0.1505{{P}} - 0.347{{N}} + 0.006733{{P}}^{2} + 0.0654{{N}}^{2} + 0.000103{{T}}{{P}} + 0.000400{{T}}{{N}}$$
(7)
According to Eq. (7), H2/CO ratio has the most effect on the selectivity of methane. Also, the interaction between temperature and H2/CO ratio has an important effect. The effect of temperature can be ignored in methane selectivity because of low coefficient value.
Selectivity model for ethane
Figure 3 illustrates the surface and contour plot of ethane selectivity. As can be seen, the effect of temperature, pressure and H2/CO ratio are shown in Fig. 3. The selectivity of ethane increases with increase in pressure. By increasing the H2/CO ratio, the selectivity of ethane increases first and then decrease. As can be seen, at constant temperature and high pressure, the maximum amount of ethane was achieved. By increasing temperature, selectivity of ethane decreases and then increases. The ethane selectivity varies parabolically with temperature and H2/CO ratio. The selectivity model of ethane can be proposed as follows:
$${\text{S}}\left( {\text{ethane}} \right) = 4.540 \, - 0.01742{{T}} + 0.1058{{P}} + 0.2375{{N}} + 0.000016{{T}}^{2} {-}0.0650{{N}}^{2} {-}0.000160{{T}}{{P}}{-}0.00496{{P}}{{N}}$$
(8)
According to Eq. (8), H2/CO ratio has the most effect on the model because its coefficient is the most and has positive effect. Also, the important interaction between parameter belongs to pressure and H2/CO ratio.
Selectivity model for ethylene
Figure 4 illustrates the contour and surface plot of operating conditions effect on ethylene selectivity. As can be seen, the ethylene selectivity increased with increase in temperature but at high pressure, temperature has no effect on the ethylene selectivity therefore, the effect of temperature at high pressure can be neglected. By increasing H2/CO ratio, the selectivity increases, so the maximum amount of ethylene was achieved. The ethylene selectivity varies parabolically with pressure. The maximum selectivity of ethylene occurs at low pressure, but by increasing the pressure, ethylene selectivity decreases and then increases. The selectivity model of ethylene is given as follows:
$${\text{S}}\left( {\text{ethylene}} \right) = - \,0.511 \, + \, 0.000858{{T}} + 0.0090{{P }} + 0.2708{{N }} + 0.002068{{P}}^{2} {-}0.0699{{N}}^{2} {-}0.000071{{T}}{{P}}{-}0.000103{{T}}{{N}}$$
(9)
As can be seen in Eq. (9), H2/CO ratio is the important parameter in the model because it has the most effect on the model. Also, an important interaction is T*H2/CO ratio and because of its negative coefficient, it has negative effect on the selectivity of ethylene.
The model for CO conversion
Figure 5 shows the effect of operating conditions such as temperature, pressure and H2/CO ratio on CO conversion model. The figures are in form of surface and contours. As can be seen in Fig. 5, at constant pressure, temperature has no effect on conversion but at constant H2/CO ratio, by increasing the temperature, the conversion decreases. This reduction at low pressure has great intensity, while at high pressure, increase in temperature has no effect on conversion. At high temperature and low H2/CO ratio, by increasing the pressure, CO conversion increases. By increasing the H2/CO ratio, the CO conversion decreases and then increases.
The model for CO conversion can be presented as follows:
$${\text{X}}\left( {\text{CO}} \right) = 262.3{-}0.2938{{T}}{-}13.46{{P}}{-}118.8{{N}} + 40.68{{N}}^{2} + 0.03326{{T}}{{P}}{-}3.823{{P}}{{N}}$$
(10)
In equation of CO conversion model, H2/CO ratio is the most effective parameter in the model. It has negative effect on the model. Also, the interaction between pressure and H2/CO ratio has the most effect on the model.
Optimization
To increase the desired product and decrease the undesired products, optimization is necessary. The RSM methodology is an appropriate method for creating the connection between inputs and outputs of process. Using RSM to optimize outputs of process, the operating cost can be reduced and plant profits can be increased. In this study, the optimization of products are done in two ways. As can be seen in Table 4, in three-objectives part, to increase the selectivity of ethylene (olefin) and decrease the selectivity of methane and ethane (paraffin), T = 600.49 K, P = 3.07 bar and H2/CO = 1 should be adjusted. In three single-objective part, to maximize the selectivity of ethylene as an olefin, the temperature, pressure and H2/CO ratio have been set at 618 K, 3 bar and 1.4848, respectively. Also, to decrease the selectivity of methane, the temperature, pressure and H2/CO ratio have been set at 618 K, 6.46 bar and 1, respectively. Figure 6 indicates the optimization plot in three objectives.