INTRODUCTION
Almost all catalytic reactions in the petrochemical industry are reversible and therefore their conversion is limited by the thermodynamic equilibrium. This conservative limitation can be broken by using selective membranes to remove one of the products. In this section, the usage of selective membranes for the permselective removal of hydrogen in dehydrogenation reaction is introduced (She et al., 2001; Hermann et al., 1997; Abashar, 2003). These membranes, which have 100% selectivity for the removal of hydrogen, lead to MPMP. The most efficient configuration is when in the other side of the membrane is a hydrogenation reaction and the flows in the two sides of the membrane are countercurrent. Membrane catalytic dehtdrogenation reactors have been investigated and used earlier (e.g., She et al., 2001; Hermann et al., 1997; Abashar, 2003), the new contribution in this paper is having a hydrogenation reaction on the other side of the membrane. Therefore the novelty in this paper is in combining both the hydrogen removal from the dehydrogenation reaction in one side of the membrane and using it as a reactant in the other hydrogenation side. The performance of this integrated membrane reactor is estimated by the conversion of the reactant in the dehydrogenation side. This paper is concentrating on the removal of hydrogen from a dehydrogenation reaction, mainly ethylbenzene to styrene (She et al., 2001; Hermann et al., 1997; Abashar, 2003), using hydrogen permselective membranes. The yield of the reactor is measured by the Ethylbenzene conversion and yield of Styrene.
The rate of hydrogen removal from the reaction side depends upon the type of the membrane and also the hydrogen driving force between the two sides of the membranes. This driving force increases when there is a hydrogenation reaction in the other side of the membrane. In the present study a hydrogenation reaction of nitrobenzene to aniline is taking place on the other side of the membrane. A reliable mathematical model is used to investigate the characteristics of this novel membrane reactor configuration which gives a very useful step towards sustainability which is necessary but not sufficient for sustainability.
REACTION KINETICS
The reaction network for the dehydrogenation of ethylbenzene to styrene is (Abashar, 2003; Moustafa and Elnashaie, 2000; Elnashaie et al., 2000; Abdulla et al., 1993):
\(C_{6}H_{5}CH_{2}CH_{3} \Leftrightarrow C_{6}H_{5}\text{CHC}H_{2} + H_{2}\) \(\Delta H_{298} = 117.6\ \frac{\text{kJ}}{\text{mole}}\) 
(1) 
\(C_{6}H_{5}CH_{2}CH_{3} \rightarrow C_{6}H_{6} + C_{2}H_{4}\) \(\Delta H_{298} = 105.4\ \frac{\text{kJ}}{\text{mole}}\) 
(2) 
\(C_{6}H_{5}CH_{2}CH_{3} + H_{2} \rightarrow C_{6}H_{5}CH_{3} + CH_{4}\) \(\Delta H_{298} =  54.6\ \frac{\text{kJ}}{\text{mole}}\) 
(3) 
\(2H_{2}O + C_{2}H_{4} \rightarrow 2\text{CO} + 4H_{2}\) \(\Delta H_{298} = 210.2\ \frac{\text{kJ}}{\text{mole}}\) 
(4) 
\(H_{2}O + CH_{4} \rightarrow \text{CO} + 3H_{2}\) \(\Delta H_{298} = 206.1\ \frac{\text{kJ}}{\text{mole}}\) 
(5) 
\(H_{2}O + \text{CO} \rightarrow CO_{2} + H_{2}\) \(\Delta H_{298} =  41.2\ \frac{\text{kJ}}{\text{mole}}\) 
(6) 
In this network, all side reactions are irreversible with the only reversible reaction being the main reaction which produces the desired product styrene. The corresponding rate equations, expressed as functions of component partial pressure in bars, (Abashar, 2003; Moustafa and Elnashaie, 2000; Elnashaie et al., 2000; Abdulla and Elnashaie, 1993) are:
\[r_{1} = k_{1}\left( p_{\text{EB}}  p_{\text{ST}}\frac{p_{H_{2}}}{K_{A}} \right)\] 
(7) 
\[r_{2} = k_{2}p_{\text{EB}}\] 
(8) 
\[r_{3} = k_{3}p_{\text{EB}}p_{H_{2}}\] 
(9) 
\[r_{4} = k_{4}p_{H_{2}O}p_{C_{2}H_{4}}^{1/2}\] 
(10) 
\[r_{5} = k_{5}p_{H_{2}O}p_{CH_{4}}\] 
(11) 
\[r_{6} = k_{6}\left( \frac{P}{T^{3}} \right)p_{H_{2}O}p_{\text{CO}}\] 
(12) 
with rate constants defined as:
\[k_{i} = exp\left( A_{i}  \frac{E_{i}}{\text{RT}} \right)\] 
(13) 
Table 1. Arrhenius equations and equilibrium constants for ethylbenzene reactions
Reaction No.

Frequency Factor ^{a}

Activation Energy ^{a} (kJ/mole)

12

0.85

90.9

13

14.00

208.1

14

0.56

91.5

15

0.12

104.0

16

3.21

65.7

17

21.24

73.6

Equilibrium constant \(K_{A} = exp\left(  \frac{\Delta F}{\text{RT}} \right)\)
\[\Delta F = a + \text{bbT} + cT^{2}\]
\[a = 122725.157\ kJ/kmole\]
\[\text{bb} =  126.2674\ kJ/kmole \times K\]
\[c =  2.194 \times 10^{ 3}\ kJ/kmole \times K^{2}\]

^{a} obtained from references (Moustafa and Elnashaie, 2000; Elnashaie et al., 2000; Abdulla and Elnashaie, 1993; Trefethen, 2000).

Table 2. Unit conversion constants for reaction rates of dehydrogenation network
Reaction rate equation No.

Constant to be multiplied by

7

1/100

8

1/100

9

1/10^{7}

10

1/10^{4.5}

11

1/10^{7}

12

1/10^{12}


The numerical values of A_{i} and E_{i} are given in Table 1 and are used to calculate the rate of reactions in kmol/kg cat/h. To change the units of partial pressures from bars to Pascal and the reaction rates from kmol/kg cat/h to mol/kgcat/s, the above rate equations have to be multiplied by the constants in Table 2 (Amon et al., 1999; Shu et al., 1994; Carl, 1999; Assabumrungrat et al., 2002; Bischoff, 1990; Villadsen and Michelsen, 1978; Trefethen, 2000). On the tube side, the hydrogenation reaction of nitrobenzene to aniline is given by (Amon et al., 1999):
\(C_{6}H_{5}NO_{2} + 3H_{2} \rightarrow C_{6}H_{5}NH_{2} + 2H_{2}O\) \(\Delta H_{298\ K} =  443.0\ \frac{\text{kJ}}{\text{mole}}\) 
(14) 
The rate equation of this reaction is estimated (Amon et al., 1999) by:
\[r' = \frac{k'K_{\text{NB}}K_{H_{2}}{p'}_{\text{NB}}\sqrt{{p'}_{H_{2}}}}{\left( 1 + K_{\text{NB}}{p'}_{\text{NB}} + K_{H_{2}}\sqrt{{p'}_{H_{2}}} \right)^{2}}\] 
(15) 
with reaction rate constant defined as:
\[k' = 10^{3}\exp\left( A'  \frac{E'}{\text{RT}'} \right)\] 
(16) 
where: \(A' = 0.186\), \(E' = 10.0 \times 10^{3}\ J/mole\), \(K_{H_{2}} = 4.427 \times 10^{ 3}\text{ P}a^{ 1/2}\), \(K_{\text{NB}} = 1.510 \times 10^{ 5}\text{ P}a^{ 1}\)
GOVERNING EQUATIONS FOR MEMBRANE REACTOR
To obtain the mole balance equations and the energy balance equation, a differential element inside the membrane reactor was considered. After writing the balances under the steady state assumption, both sides of the resulting equations were divided by the thickness of the differential element, which was then forced to approach zero. The resulting balances equations of the shell side can be expressed as:
Mole balances:
\[\frac{dn_{i}}{\text{dz}} = \sum_{j = 1}^{6}{\sigma_{\text{ij}}r_{j}\left( 1  \varepsilon \right)}A_{\text{cs}}\rho_{s}  2\pi\ r_{3}Na_{i}J_{i}\] 
(17) 
Energy balance:
\[\frac{\text{dT}}{\text{dz}} = \frac{\sum_{j = 1}^{6}\left\lbrack  \Delta H\left( T \right) \right\rbrack_{j}r_{j}\left( 1  \varepsilon \right)A_{\text{cs}}\rho_{s} + Q}{\sum_{i = 1}^{10}{n_{i}Cp_{i}}}\] 
(18) 
Pressure drop:
\[\frac{\text{dP}}{\text{dz}} =  \frac{G}{\rho_{g}g_{c}D_{p}}\left( \frac{1  \varepsilon}{\varepsilon^{3}} \right)\ \left\lbrack \frac{150\left( 1  \varepsilon \right)\mu_{g}}{D_{p}} + 1.75G \right\rbrack\] 
(19) 
The corresponding balance equations for the tube side can be expressed as:
Mole balance:
\[\frac{d{n'}_{i}}{\text{dz}} = \left(  1 \right)^{b}\left\lbrack \sigma_{i}r'\left( 1  \varepsilon' \right){A'}_{\text{cs}}{\rho'}_{s} + 2\pi\ r_{3}a_{i}J_{i} \right\rbrack\] 
(20) 
Energy balance:
\[\frac{dT'}{\text{dz}} = \left(  1 \right)^{b}\left\lbrack \frac{\left( 2\pi\ r_{3} \right)\sum_{i = 1}^{i}{a_{i}J_{i}\int_{T'}^{T}{Cp_{i}dT + \left\lbrack  \Delta H'\left( T' \right) \right\rbrack\ r'\left( 1  \varepsilon' \right){A'}_{\text{cs}}{\rho'}_{s}  Q}}}{\sum_{i = 1}^{4}{{n'}_{i}C{p'}_{i}}} \right\rbrack\] 
(21) 
Pressure drop:
\[\frac{\text{dP}'}{\text{dz}} = \left(  1 \right)^{b}\frac{G'}{{\rho'}_{g}g_{c}{D'}_{p}}\left( \frac{1  \varepsilon'}{{\varepsilon'}^{3}} \right)\ \left\lbrack \frac{150\left( 1  \varepsilon' \right){\mu'}_{g}}{{D'}_{p}} + 1.75G' \right\rbrack\] 
(22) 
The hydrogen flux across the membrane surface obeys Sievert’s law (Shu et al., 1994), i.e.
\[J_{H_{2}} = \frac{Q_{o}\exp\left(  \frac{E_{H_{2},P}}{\text{RT}} \right)}{\delta_{H_{2}}}\left( \sqrt{P_{H_{2}}}  \sqrt{{P'}_{H_{2}}} \right)\] 
(23) 
The preexponential constant, permeation activation energy, and the thickness of the hydrogen permeation membrane are taken as 6.33×10^{7} mol/m/sec/Pa^{0.5}, 15700 J/mole, and 1~2×10^{5} m, respectively (Shu et al., 1994). Heat transfer across the membrane involves both convection from the gas mixture to the membrane, conduction across the membrane layer, and finally convection from the membrane to the second gas mixture. Radiation of heat is neglected. The membrane tube is considered to be a composite wall having a stainless steel layer coated by a thin layer of palladium. The thermal conductivities of the stainless steel layer and the palladium are taken to be 24.5 W/m×K and 93.3 W/m×K, average values over a temperature range of 2001800 K (Yaws, 1999; Assabumrungrat et al., 2002). The heat transferred per unit length from each tube is obtained from:
\[\frac{hD_{t}}{k_{g}} = 0.813\left( \frac{D_{p}G}{\mu_{g}} \right)^{0.9}\exp\left(  \frac{6D_{p}}{D_{t}} \right)\] 
(24) 
The convective heat transfer coefficients in equation (24) are calculated using Leva’s correlation (1949) (Bischoff, 1990). For the shell side in which the reacting mixture is heated up, the convective heat transfer coefficient is calculated by (Bischoff, 1990):
\[\frac{hD_{t}}{k_{g}} = 0.813\left( \frac{D_{p}G}{\mu_{g}} \right)^{0.9}\exp\left(  \frac{6D_{p}}{D_{t}} \right)\] 
(25) 
In contrast, the reacting mixture in the tube side is cooled and consequently the convective heat transfer coefficient is calculated by (Bischoff, 1990):
\[\frac{h'{D'}_{t}}{{k'}_{g}} = 3.50\left( \frac{{D'}_{p}G'}{{\mu'}_{g}} \right)^{0.7}\exp\left(  \frac{4.6{D'}_{p}}{{D'}_{t}} \right)\] 
(26) 
Physical properties such as, thermal conductivity, gas density and viscosity, and heat capacities are taken as functions of temperature from Yaws (Yaws, 1999).
BOUNDARY CONDITIONS
In the case of the cocurrent operation\(\left( b = 2 \right)\), the above system of differential equations gives an initial value problem which can be solved by a RungeKutta Verner fifth and sixth order method with an automatic step size, double precision calculation, and a relative error of 1×10^{12} to ensure high accuracy. The initial conditions are:
At: \(z = 0\)
\[\text{Shell}\ \text{Compartment} = \left\{ \begin{matrix} n_{i} = n_{\text{if}} \\ T = T_{f} \\ P = P_{f} \\ \end{matrix} \right.\ \] 
(27) 
\[\text{Tube}\ \text{Compartment} = \left\{ \begin{matrix} {n'}_{i} = {n'}_{\text{if}} \\ T' = {T'}_{f} \\ P' = {P'}_{f} \\ \end{matrix} \right.\ \] 
(28) 
For the countercurrent operation case \(\left( b = 1 \right)\), the above system of differential equations results in a split twopoint boundary value problem which can be solved by an orthogonal collocation technique (Villadsen and Michelsen, 1978; Trefethen, 2000). The boundary conditions are:
At \(z = 0\) inlet condition of dehydrogenation compartment
\[\text{Shell}\ \text{Compartment} = \left\{ \begin{matrix} n_{i} = n_{\text{if}} \\ T = T_{f} \\ P = P_{f} \\ \end{matrix} \right.\ \] 
(29) 
At \(z = L\) inlet condition of hydrogenation compartment
\[\text{Tube}\ \text{Compartment} = \left\{ \begin{matrix} {n'}_{i} = {n'}_{\text{if}} \\ T' = {T'}_{f} \\ P' = {P'}_{f} \\ \end{matrix} \right.\ \] 
(30) 
Table 3. Operating conditions for dehydrogenation (shell side) reaction of ethylbenzene to styrene
Parameter

Value and dimension

Length of the reactor

3.0 m

Crosssectional area of the shell side

3.0 m^{2}

Feed molar flow rates of:

Ethylbenzene

30.0^{ }mol/s

Styrene

0.1861 mol/s

Hydrogen

0.0 mol/s

Benzene

0.03056 mol/s

Ethylene

0.0 mol/s

Toluene

0.2444 mol/s

Methane

0.0 mol/s

Carbon monoxide

0.0 mol/s

Carbon dioxide

0.0 mol/s

Steam

140.0 mol/s

Inlet temperature

850 K

Inlet pressure

4.5×10^{5 }Pa

Catalyst density ^{b}

1500 kg/m^{3}

Diameter of catalyst particle

4.7×10^{3 }m

Void fraction

0.48


OPERATING CONDITIONS ON BOTH SIDES OF THE REACTOR
The operating conditions for both sides of the reactor are given in Tables 3 and 4. For the dehydrogenation reaction of ethylbenzene to styrene, the feed molar flow rates are the same as those presented by (Moustafa and Elnashaie, 2000; Elnashaie et al., 2000; Abdulla and Elnashaie, 1993), whereas the molar flow rate of nitrobenzene is based on stoichiometry. Excess steam is provided in the feed line to prevent the formation of coke on the catalyst.
Table 4. Operating conditions
Parameter

Value and dimension

No.of hydrogenation tubes

1270

Outer radius of a hydrogenation tube

0.0318 m

Thickness of the stainless hydrogenation tube

0.0030 m

Total Crosssectional area of the tube side available for flow

3.310 m^{2}

Feed molar flow rates of:

Nitrobenzene

10.0 mol/s

Hydrogen

0.0 mole/s

Aniline

0.0 mole/s

Steam

100.0 mole/s

Inlet temperature

860 K

inlet pressure

1.1×10^{5 }Pa

Catalyst density

1400 Kg/m^{3}

Diameter of catalyst particle

4.7×10^{3 }m

Void fraction

0.46


RESULTS AND DISCUSSION
The simulation results for the coupled membrane reactor system can confirm the potential of coupling the two reactions. In general, the achievable performance as far as conversion of ethylbenzene and yield of styrene are concerned is much better than that of the corresponding uncoupled industrial fixed bed reactors operated at the same conditions. The coupled membrane reactor also has the potential to give better conversion of ethylbenzene and yield of styrene by increasing the length of the reactor since the two profiles continue to evolve with distance:
Hydrogen Molar Flow Rate
Hydrogen molar flow rates for an uncoupled fixed bed reactor and for coupled cocurrent and countercurrent cases membrane reactors are shown in Figure 2. Hydrogen is generated in the uncoupled fixed bed reactor case while producing styrene, which is the key component, is low. For the coupled cocurrent case, on the other hand, hydrogen produced on the dehydrogenation side diffuses immediately through the membrane walls of the hydrogenation tubes where it meets nitrobenzene to react and produce aniline. The comparable rates of the net production of hydrogen on the dehydrogenation side and the rate of diffusion of hydrogen through the palladium membrane prevent the hydrogen from accumulating on the dehydrogenation side where it reduces the net reaction rate. The coupled countercurrent case is of special interest due to the presence of a maximum where the flow rate of hydrogen reaches a maximum and then decreases. At the feed end of the dehydrogenation side (\(z = 0\)), the rate of production of hydrogen by reactions (1), (4), (5), and (6) surpasses both the rate of consumption of hydrogen by reaction (3) and the rate of diffusion of hydrogen through the palladium membrane. This leads to an increase in the number of moles of hydrogen until a point is reached where the hydrogen flow rate has reached its maximum value, i.e. when the rate of change of molar flow rate of hydrogen at that point with respect to the length of the reactor is zero. At this point, the rate of the production of hydrogen is balanced by both the rate of consumption of hydrogen by reaction (3) and the rate of diffusion of hydrogen through the membrane. Beyond that point, the process is dominated by consumption of hydrogen by reaction (3) and diffusion of hydrogen through the membrane. Consequently, the molar flow rate of hydrogen decreases along the reactor. This trend induces similar behavior in the hydrogenation tubes. As nitrobenzene moves from its feed point, located at \(z = 3.0\ m\), it reacts with the permeate hydrogen. Another maximum is reached where the rate of diffusion of hydrogen through the membrane is balanced by its rate of consumption by the hydrogenation reaction. This point is located to the left (on Figure 2) of that on the dehydrogenation side. After passing this maximum, the rate of consumption of hydrogen dominates causing the amount of hydrogen in the hydrogenation tubes to drop quickly.
Conversion of Ethylbenzene:
The predicted conversion of ethylbenzene on the dehydrogenation side is shown in Figure 3 for the same three cases. For the operating conditions chosen, the conversion reaches 23.4% for the uncoupled adiabatic fixed bed case, 54.6% for the membrane reactor with cocurrent flow, and 61.7% for the membrane reactor in the countercurrent flow configuration. Again, coupling of the two reactions has the potential to provide significant improvement in reactor performance. Moreover, the heat generated by the exothermic hydrogenation reaction is put to good use, rather than being simply rejected to cooling water.
Yield of Styrene
Three cases are investigated: uncoupled fixed bed case corresponding to the classical industrial operation, cocurrent coupled case, and countercurrent coupled case. Results for the three cases are shown in Figure 4. The lowest yield by a considerable margin is predicted for the uncoupled adiabatic fixed bed where the maximum yield is 18.9%. For the membrane reactor, the predicted yield increases to 52.5% for the cocurrent flow configuration and 57.7% for the countercurrent case.
Conversion of Nitrobenzene
Figure 5 plots the predicted conversion of nitrobenzene as a function of distance along the reactor for the hydrogenation reaction to aniline. The countercurrent membrane reactor is seen to give higher conversion than the cocurrent case because of the large driving forces. The conversion for the cocurrent case is predicted to reach 51.1%, while 57.9% is calculated for the countercurrent case.
Temperature Profiles
Temperature profiles for the adiabatic fixed bed reactor and both the cocurrent and countercurrent membrane reactors are plotted in Figures 6 and 7 for different number of membrane tubes. Heat is continuously supplied from the exothermic hydrogenation nitrobenzenetoaniline reaction on the tube side to the endothermic dehydrogenation reaction of ethylbenzene on the shell side. The temperature variation is larger at the inlet of the membrane reactor in the coupled countercurrent case than for coupled cocurrent flow. The fall in temperature for both cases of the coupled membrane reactor is much less than that for the uncoupled fixed bed reactor. In the countercurrent case of the coupled reactor, the temperature on the dehydrogenation side drops because the heat transferred from the nitrobenzene side is decreased due to the low driving force, i.e. temperatures on both sides becomes comparable.
Predicted temperature profiles on the hydrogenation side where the nitrobenzene to aniline reaction occurs are also plotted Figures 6 and 7 for different number of membrane tubes. The temperature rises from 860 K at the feed point to slightly more than 920 K for the coupled cocurrent flow case and slightly more than 945 K for the coupled countercurrent flow configuration. The large temperature rise for the coupled countercurrent coupled case is due to the large driving force causing higher diffusion rate of hydrogen and consequently increasing the nitrobenzene conversion. However, the temperature reaches a peak where both heat generated on the nitrobenzene side due to the reaction and energy carried by the permeating hydrogen is balanced by heat transfer through the membrane to the ethylbenzene side. Later, it decreases because of the dominance of the heat transferred through the membrane.
Practical Considerations
Coupling the exothermic hydrogenation reaction with the endothermic dehydrogenation reaction is predicted to be capable of providing a significant improvement in reactor performance and energy integration. However, it must be noted that palladium membranes are currently limited to temperatures of ~ 900 K. The maximum temperature can be reduced by increasing the number of the membranes, giving more surface area permitting more heat transfer rate across the membrane. This is illustrated in Figures 6 and 7 where the axial temperature profiles in both compartments of the coupled membrane reactor are plotted with flow rate kept unaltered. Note that the maximum temperature can be maintained below 880 K, with countercurrent operation giving higher maximum temperature than the cocurrent case. We also note that the countercurrent configuration is likely to be very difficult to start up in practice. Hence, cocurrent operation Process control and safety aspects are also expected to be easier to provide for the cocurrent configuration despite the advantages of countercurrent operation, which is more likely to be practical for the foreseeable future. Finally, while homogeneous onedimensional models, like that used here, provide a good initial sense of what could be achieved in coupled fixed bed reactors, more comprehensive heterogeneous models with fewer simplifying assumptions are needed prior to proofofconcept experiments on such reactors.
CONCLUSIONS
The performance of a newly configured membrane reactor with two reactions, an exothermic hydrogenation and an endothermic dehydrogenation reaction in parallel, has been modeled and evaluated for: cocurrent and countercurrent operation. Pseudo homogeneous models have been used to describe the behavior of the system. In the range of the parameters used to investigate the performance of the membrane reactor, the simulation predicts a considerable increase in the conversion of ethylbenzene and yield of styrene when the dehydrogenation reaction of ethylbenzene to styrene is coupled with the hydrogenation reaction of nitrobenzene. The results from the countercurrent mode of operation were in all cases higher than those from the cocurrent mode of operation due to the large driving forces. The simulation results suggest that coupling may be feasible in this case with promising performance. The results show that the counter current configuration with efficient removal of hydrogen will be the best to achieve MPMP which is the best for sustainable development. However, the performance of the reactor needs to be proven experimentally and tested over a range of parameters under practical operating conditions. The novelty of this reactor achieving MPMP is in combining both the hydrogen removal from the dehydrogenation reaction in one side of the membrane and using it as a reactant in the other hydrogenation side. The performance of the reactor is measured by the Ethylbenzene conversion and yield of Styrene.