Zheng Bo; Tang Xiaojin; Zhang Zhanzhu; Zong Baoning

(SINOPEC Research Institute of Petroleum Processing, Beijing 100083)

Abstract: New modified combination mathematical models including the pores blocking models and the cake layer models were developed to describe the continuous cross-flow microfiltration in an airlift external loop slurry reactor. The pores blocking models were created based on the standard blocking law and the intermediate blocking law, and then the cake layer models were developed based on the hydrodynamic theory in which the calculation method of porosity of cake layer was newly corrected. The Air-Water-FCC equilibrium catalysts cold model experiment was used to verify the relevant models.Results showed that the calculated values fitted well with experimental data with a relative error of less than 10%.

Key words: airlift external-loop slurry reactor, cross-flow microfiltration, filtration resistance, mathematical model

1 Introduction

Slurry reactors have been widely used in gas-liquid-solid reactions in recent years, but the difficulties related with liquid/solid separation of the slurry have been limiting its industrial applications. Membrane filtration technology has been considered as an effective method to solve this problem[1-3], especially in an airlift external-loop slurry reactor. The circulated slurry in the reactor could produce shear flow to strengthen the cross-flow microfiltration,which can limit the cake layer growth and prolong the operating cycle for separation. However, this technology cannot thoroughly prevent small particles from blocking the membrane pores and large particles from depositing on the membrane surface. So it is necessary to deeply study the mechanism of membrane fouling and develop reasonable mathematical models to optimize and predict the separation process.

In a microfiltration process, the filtration resistance contains the membrane resistance, the pores blocking resistance, and the cake layer resistance. The membrane resistance is constant and the other two kinds of resistance will change with time, so it is essential to study the law on variation of filtration resistance. Hermia, et al.[4]built a correlation (Equation (1)) to describe the process of pores blocking and cake layer formation, and the correlation with different parameter values could be used to predict different operation period independently. But it is difficult to predict the whole operation time exactly. Altmann,et al.[5]studied the force analysis for single particle and then developed mathematical models to simulate cake layer formation, but Altmann did not consider the pores blocking. Shi, et al.[6]developed the combination mathematical models including pores blocking and cake layer formation, in which the cake porosity was supposed as a constant[5]. The pores blocking process was simulated by combination of the standard blocking law and the intermediate blocking law[4]. Li[7]simplified the pores blocking simulation and enriched the cake layer formation simulation, but he presumed transmembrane pressure as the pressure difference of cake layers to calculate the porosity of cake.

Investigation of the literature reports shows that currently there are no reasonable mathematical models to simulate the cross-flow microfiltration in an airlift external-loop slurry reactor. So in this study, new modified combination models would be developed to predict this separation process.

2 Experimental

The schematic diagram of an airlift external-loop slurry reactor with cross-flow microfiltration setup is shown in Figure 1. The experimental setup mainly consists of a slurry reactor, a filter, and a permeate tank. The diameter of the riser, the downcomer, and the gas/slurry separator is 0.28 m, 0.025 m, and 0.42 m, respectively. There are a dozen of ceramic membrane tubes with a length of 0.2 m and 19 channels with a diameter of 0.006 m in each tube.The uniform size of membrane pores is 0.2 μm and the total filtration area is equal to 0.86 m2[8].

As shown in Figure 1, there are liquid phase and solid phase in the reactor, because they are mixed so well that they could be considered as one slurry phase. The gas phase goes into the riser in the form of bubbles through a distributor and then most of them could remain in the gas/slurry separator. There are nearly no bubbles existing in the downcomer, so the slurry could circulate in the reactor because of density difference between the riser and the downcomer. When the slurry enters the downcomer, the liquid/solid mixture could be separated in the filter by transmembrane pressure. The permeate would return to the reactor and the condensed slurry could return to the riser.The permeate flux was measured by the gravimetric method and the cumulative mass distribution of solid particles was measured by a laser particle size analyzer(Mastersizer) as shown in Figure 2. Air, water and FCC equilibrium catalysts represent the gas phase, the liquid phase, and the solid phase, respectively. Materials properties are listed in Table 1.

Figure 1 Schematic diagram of the experimental setup

Figure 2 Size distribution of particles

Table 1 Materials properties (T=293 K, P=0.1 MPa)

3 Results and Discussion

In this study, new modified combination models were developed to simulate the whole filtration process.The first process was simulated by the modified pores blocking models based on the research of Hermia,[4]et al. and Shi,[6]et al. before cake layer formation, and the second process was described by the new modified cake layer models based on the study of Altmann,[5]et al. and Li[7].

3.1 Force analysis for single particle

Starting point of the models study is the consideration of the forces acting on a single particle. By means of this idea we can know what size of particles can reach the membrane to block the pores and then form a cake layer. Altmann[5]found that when the small particles deposited on the surface of membrane or cake layer, there would be six forces acting on it as shown in Figure 3. The adhesive force (Fa)and friction force (Ff) were larger than the hydrodynamic force including the drag force of the permeate (Fy), the lift force of shear flow (FL), and the drag force of cross-flow(FD). So once a small particle is deposited on the surface,it would never have a chance to return to the slurry. Only larger particles or particle agglomerates could be removed from cake layer by the shear flow. Altmann gave a critical diameter ds,critbased on the balance between Fyand FLacting on a streaming particle. The proportion of Fyand FLcould determine if the particle could be transported to the membrane to form deposits. Only particles, the diameter of which was smaller than ds,crit, could deposit on the layer under certain conditions.

Figure 3 Force analysis for single streaming particle and deposited particle

The modified Fycan be calculated by Equation (2).

Hence λ can be obtained by Equations (3—5), which should be used when the particles concentration of slurry was high.

Then FLcan be obtained by Equation (6).

The τ value in Equation (6) can be calculated by Equations (7—9).

The density of slurry ρs,eand viscosity of slurry μs,ecan be calculated by Equations (10—11)[9].

Based on the balance between Fyand FL, the ds,critvalue is calculated as shown in Equation (12).

3.2 Pores blocking

Before cake layer formation caused by particles deposition, small particles which could reach the surface of membrane would block the membrane pores. Some particles could enter the interior of pores and then deposit on the walls of pores, resulting in membrane pores constriction. Some other larger particles may block pores or settle down on other deposited particles, which would reduce the active membrane surface. Shi considered that there are two pores blocking mechanism before cake layer formation. The first pores blocking process may use the modified standard blocking models to simulate and the other case may use the modified intermediate blocking models to simulate[4,6].

3.2.1 Standard blocking models

The permeate volumetric rate Qpcan be calculated by Equations (13—14).

The αsvalue, which changes with time, can be corrected by Equations (15—17).

The P1(t) value can be obtained by Equations (18—20).When ds,critis greater than the diameter of membrane pores with a diameter of 0.2 μm, the Q`(ds,crit) value is equal to 1.

Base on Equations (13—20) and the Darcy’s law, the filtration resistance Rsta(t) resulted from standard blocking mechanism can be calculated by Equation (21).

3.2.2 Intermediate blocking models

The permeate volumetric rate Qpis calculated by Equations (22—23).

The S value, which changes with time, can be corrected by Equations (24—26).

The P2(t) value can be obtained by Equations (27—28).

Based on Equations (22—28) and the Darcy’s law, the filtration resistance Rint(t) resulted from intermediate blocking mechanism can be calculated by Equation (29).

Before cake layer formation, the filtration resistance consists of the standard blocking resistance Rsta(t) and the intermediate blocking resistance Rint(t), while the weight of each resistance is P1(t) and P2(t). The total blocking resistance Rmi(t) is calculated by Equation (30), and the permeate flux variation with time J(t) can be obtained by Equation (31).

3.3 Cake layer formation

When the pores blocking process finished, the particles began to deposit on the surface of membrane or to form the cake layer. As more particles deposited on the layer,the cake resistance increased dramatically, which was determined by the structure of cake and the thickness of layer. So after a transition time t0, the permeate flux decreased which mainly depended on the cake layer formation.

Base on the research of Altmann[5], the particle mass transported to the membrane surface at a certain permeate flux m(t) is calculated by Equation (12) and Equation (32).

The total particle mass mt(t) is obtained by Equation (33).

The pressure difference of cake layer ΔPccan be calculated by Equation (34).

The ΔPcvalue also can be obtained by the Ergun Equation as shown in Equation (35). The thickness of cake layer h(t) can be calculated by Equation (36). Based on Equations (35—37), the porosity of cake layer εc, which changes with time, can be calculated.

The local cake resistance rccan be calculated by Equation(38), and then the resistance of cake layer Rccan be obtained by Equation (39).

During the cake layer formation process, the permeate flux variation with time J(t) can be obtained by Equation(40).

3.4 Models validation

The scheme for calculation of the models is shown in Figure 4 and the numerical method was adopted to solve it.To verify the mathematical models, a cold model experiment was developed. When the experimental conditions covered:T=293 K, P=0.1 MPa, αs=5%, ΔP=50 kPa, um=0.633 m/s,αsm=20%, εm=0.4, and t0=1 800 s, the initial permeate flux J0was equal to 3.62×10-5m/s[8].

Figure 4 Calculation of the cross-flow filtration models

Figure 5 shows the comparison between the theoretical and experimental results, and the relative error of calculation is shown in Figure 6. It is found that there was a good match of the calculated numerical values with the experiments having a relative error of less than 10%. It can illustrate that the new modified combination models can be applied for the prediction of cross-flow microfiltration process in the airlift external loop slurry reactor.

4 Conclusions

1) New modified combination mathematical models including the blocking filtration law and the cake layer filtration law were developed to describe the continuous cross-flow microfiltration in an airlift external loop slurry reactor.

Figure 5 Comparison of theoretical and experimental results■—Exp;—Cal

Figure 6 Relative error analysis■—Cal

2) The combination of standard blocking models and intermediate blocking models were used to simulate pores blocking. The new modified cake filtration models were used to simulate the cake layer formation, in which the method for calculation of cake layer porosity was created.3) The models were verified by the Air-Water-FCC equilibrium catalysts cold model experiment, and the calculated values of the models fitted well with the experimental results along with a relative error of less than 10%.

Acknowledgement:This work was financially supported by the National Key Research & Development Program of China(2016YFB0301600).

Nomenclature

Parameter

A——Filtration area of membrane, m2;

Cs——Solid mass concentration in slurry, kg/m3;

ds——Diameter of particle, m;

Jp——Permeate flux/(m/s);

K——Constant in pores blocking models, m-3;

l——Height of membrane tube, m;

L——Length of membrane pore, m;

m(t)——Particle mass transported to the membrane surface at some point, kg/(m2·s);

p——Reactor pressure, kPa;

ΔP——Transmembrane pressure, Pa;

ΔPm——Fluid resistance loss in membrane tube, Pa;

q(x)——Mass density distribution of particles in slurry,m-1;

Q(x)——Cumulative mass distribution of particles in slurry;

Q’(x)——Cumulative distribution of membrane pores;

Q——Volume flow rate, m3/s;

rc——Local cake resistance, m-2;

R——Filtration resistance, m-1;

Re——Reynolds number;

S——Solid mass concentration in slurry, kg/m3;

t——Time, s;

t0——Transition time, s;

T——Temperature, K;

u——Flow velocity, m/s;

V——Permeate volume, m3;

Xs——Average diameter of deposited particles, m;

τ——Flow shear stress, Pa;

ρ——Density, kg/m3;

μ——Viscosity, Pa·s;

α——Phase holdup;

ζ——Friction coefficient;

φ——Shape factor of particles, φ=1;

ε ——Porosity;

αsm——Particles volume fraction near membrane;

Subscript

c——Cake layer;

crit——Critical ;

e——Equivalent value;

in——Entrance of membrane tube;

int——Intermediate blocking;

m——Membrane or membrane tube;

mix——Membrane pores blocked by Standard blocking

and Intermediate blocking;

out——Exit of membrane tube;

p——Permeate;

s——Solid phase;

s,e——Equivalent value of slurry phase character;

sta——Standard blocking;

0——Initial state.