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Effectiveness factors for a partially wetted catalyst based on the rivulet flow model
Journal Pre-proofs Effectiveness factors for a partially wetted catalyst based on the rivulet flow model Yanling Tang, Meichen Chen, Zhenmin Cheng, Tao Yang, Bo Chen, Hailong Ge, Xiangchen Fang PII: DOI: Reference:
S0009-2509(20)30047-6 https://doi.org/10.1016/j.ces.2020.115515 CES 115515
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Chemical Engineering Science
Received Date: Revised Date: Accepted Date:
2 December 2019 11 January 2020 25 January 2020
Please cite this article as: Y. Tang, M. Chen, Z. Cheng, T. Yang, B. Chen, H. Ge, X. Fang, Effectiveness factors for a partially wetted catalyst based on the rivulet flow model, Chemical Engineering Science (2020), doi: https:// doi.org/10.1016/j.ces.2020.115515
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Effectiveness factors for a partially wetted catalyst based on the rivulet flow model Yanling Tang, Meichen Chen, Zhenmin Cheng* State Key Laboratory of Chemical Engineering, School of Chemical Engineering, East China University of Science and Technology, Shanghai 200237, China Tao Yang, Bo Chen, Hailong Ge, Xiangchen Fang* Dalian Research Institute of Petroleum and Petrochemicals, SINOPEC, Dalian 116045, China
Abstract Partial wetting is a common phenomenon in a trickle bed reactor, which causes nonuniform mass transfer flux at the external surface of the catalyst. Previous work on effectiveness factor for a partially wetted catalyst are mainly based on the assumption that the spherical catalyst is wetted by a liquid film. In fact, it is only true when the liquid-solid contact angle is very small. In most situations, rivulet flow prevails. The purpose of the article is to calculate effectiveness factors for both gas-limited and liquid-limited reactions under the two wetting modes by using finite element method (FEM). The effects of wetting efficiency, Biot number, and Thiele modulus on effectiveness factors are investigated. On the basis of the FEM results, the intrinsic reason for the variation of effectiveness factors with respect to wetting efficiency is revealed, and good agreements are observed between this work and the conventional models. Keywords: Partially wetted catalyst; Film flow; Rivulet flow; Effectiveness factor.
*Correspondence concerning this paper should be addressed to Dr. Z. M. Cheng at [email protected] and Dr. X. C. Fang at fangxiangchen @sinopec.com. -1-
1. Introduction As one of the most common gas-liquid-solid reactors, trickle bed reactor (TBR) is often operated in the trickling flow regime, with the gas and liquid phases introduced at low to medium flow rates cocurrently from the top of the apparatus. In spite of the simple geometry, the flow dynamics around the randomly packed particles is complex. The catalyst outer surface can be partially wetted due to liquid maldistribution or to the poor wettability of the surface (Dudukovic et al., 1999; Ling et al., 2017). As an important issue to trickle-bed reactor design, this phenomenon has been taken under full consideration in numerous papers (Al-Dahhan and Duduković, 1995; Baussaron et al., 2007; Houwelingen et al., 2010). Partial wetting makes the flow field and transfer process on the surface of catalyst nonuniform. Even at the wetted external surface, static and dynamic liquid would have different contributions to mass transfer (Ravindra et al., 1997; Rajashekharam et al., 1998). The complex external environments caused by partial wetting can largely affect the yield of the desired products, which should be analyzed in detail for precise design of TBR. For liquid-limited reactions, as the liquid reactants are in contact with the catalyst only from the wetted external surface, complete wetting of the catalyst surface is essential for the full utilization of the catalyst. For gas-limited reactions, partial wetting of the catalyst surface would be favorable, as a partial of the catalyst surface can be directly exposed to the gas reactants and consequently the reactants do not have to diffuse through the liquid film, which offers an enhanced mass transfer -2-
rate. The effects of catalyst wetting efficiency on apparent reaction rate have been confirmed by a lot of experimental work (Ring and Missen, 1989; Funk et al., 1991; Horowitz et al., 1999; Massa et al., 2009; Kumar et al., 2017). In a majority of industrial operations, as the catalyst pore network is interconnected and the pore diameter in the catalyst is very small, which creates a huge capillary force and the capillary force makes the vapor much easily condensed, therefore, the catalyst pores can be believed completely filled with liquid (Mills and Duduković, 1979; Ramachandran and Smith, 1979; Herskowitz, 1981; Palmisano et al., 2003). So the problem of calculation of the effectiveness factor then reduces to a numerical solution of the governing reaction-diffusion equations with a non-uniform boundary condition, based on the local wetting situation at the surface of the catalyst. In recent years, much efforts have been made in this area, as shown in Table 1. It should be mentioned that the wetting models in Table 1 are based on the assumption that the spherical particle is wetted by liquid film. However, many researchers pointed out that on the single particle scale, the liquid flow texture consists of two features: liquid film and rivulet, over the particle surface (Hartley and Murgatroyd, 1964; Zimmerman and Ng, 1986; Maiti et al., 2006; Mederos et al., 2009; Wang et al., 2013). So it is a great pity that no work has been done to calculate effectiveness factor based on the rivulet flow model.
-3-
Table 1 Models and methods in calculating effectiveness factors: state-of-the-art status Authors Mills and Duduković (1979)
Wetting model
Method
effectiveness factors
Dual-Serie s Solution
Ramachandran and Smith (1979)
1 V
=
Weighting method
Vw
0
cdV
f w 1 f d Salb: 1 1 c c 2 dy 0 2 x x2 x x0 Sphere:
Separation of variables
Herskowitz (1981)
3 c sin d 22 0 r r 1 Cylinder:
Sakornwimon and Sylvester (1982)
Finite difference method
Palmisano et al. (2003)
Method of fundament al solution
Augier et al. (2010)
Finite element method
Bazer-Bachi et al. (2011)
Finite element method
2 2
0
c d r r 1
3 R 2 C sin d d R3Cs 0 0 2
De C d kCb n
cdV V =
V
Rx (r , )dV Rx ,maxV
In this study, a wetting experiment on a single particle was carried out to figure out the difference between film and rivulet wetting performance. On the basis of experimental verification, finite element method will be used to evaluate the pellet effectiveness factor as a function of the wetting fraction. It is the first study to -4-
calculate the effectiveness factor for a partially wetted catalyst based on both the film and the rivulet flow models. Therefore, it enables us to compare the results of the two wetting models in a wide operating condition.
2. Modelling Establishment 2.1 Hydrodynamic basis of the model By application of surface grafting with organic compounds, the hydrophilic alumina sphere of 5 mm in diameter was modified into different degrees of hydrophobicity with a series of liquid-solid contact angles of 14 . 48 , 88 , 119 , and 152 . The contact angle was measured using a high-speed camera with a resolution of 1280×1024@1500 fps under ambient condition. A water droplet dyed by eosin with a volume of 1 L was fed at the catalyst surface. The surface tension of the wetting liquid used in the experiments was 67 mN·m-1. The contact angle measurement on spherical surface was adopted from the method of Extrand et al. (2008), who proposed that the contact angle on the curved surface was the same as those observed on flat, horizontal surfaces. The results are illustrated in Fig. 1.
-5-
Fig. 1. Liquid flow morphology over the particle surface with different wettability It is obvious to find in most cases liquid flows over the catalyst surface in the form of rivulet, and film flow only appears when the contact angle is very small, i.e.,
14 . The most widely used catalyst support in industry is alumina, which is hydrophilic. It was verified that the contact angle between ethanol and alumina was 30 ° from the experimental work of Baussaron (2005), despite ethanol contains a hydrophilic hydroxyl group. In another work by Augier et al. (2010), they proved that it was in the rivulet flow regime for the ethanol- alumina system from CFD simulation. What is more, the liquid feedstocks used in refining and petrochemical industry are normally oils and organic compounds, which are more hydrophobic than ethanol, and would lead to a higher liquid-solid contact angle. So, the liquid-solid wetting property is always poor. Therefore, the rivulet wetting morphology is dominant in industry, and
model B (Fig .2b) is more consistent with industrial
applications than model A (Fig .2a). -6-
Fig. 2. Wetting models of a partially wetted pellet (a) Film model
(b) Rivulet model
2.2 Mathematical model For an isothermal, irreversible first order reaction, assuming that the catalyst is completely wetted internally, a steady state equation could be written as follows (Mills and Duduković, 1979): De 2C kC
(1)
Where, C is the concentration of the limiting reactant; De is the effective diffusivity for a fully internally wetted catalyst; k is the first-order kinetic rate constant. In dimensionless form, the above equation is written as
2c 2c
(2)
Where, c is the dimensionless concentration of the limiting reactant in the liquid phase, normalized by its concentration at the bulk phase; is the Thiele modulus, defined as rp k De , and rp is the radius of spherical particle. The effectiveness factor for a first order reaction is given as cdV V . The boundary condition will be different for the following two kind of reactions, since the liquid phase is assumed unvaporizing. -7-
2.2.1. Limiting reactant in the gas phase For a gas-limited reaction, the boundary condition is that of the Robin type. Over the dry portion of the catalyst, it is written as
k gs C A C A, S DeC
(3)
Where, C A is the saturated concentration of the gas reagent in the liquid phase. Generally, the transport over the dry external surface is relatively rapid, gas-solid mass transfer resistance can be negligible, hence Dirichlet boundary condition is satisfied:
C A, S C A
(4)
Over the wetted portion, where the gaseous specie has to diffuse through the gas-liquid film and the liquid- solid film and hence the boundary condition is
k gls C A C A, S DeC
(5)
1 1 1 kgls kgl H kls
(6)
Where, k gl is the gas to liquid mass transfer coefficient and kls is the liquid to solid mass transfer coefficient. The boundary condition in dimensionless form is:
c 1 , at the dry external surface
(7a)
c Biw 1 c , at the wetted external surface
(7b)
Biw is the local Biot number, Biw rp k gls De , which describes the ratio of external mass transfer rate to internal mass transfer rate. 2.2.2 Limiting reactant in the liquid phase For a catalyst pellet partially wetted by dynamic liquid over the wetted surface, -8-
the boundary condition will be written as:
kls CB ,bulk CB , S DeC
(8)
CB ,bulk is the concentration of liquid reagent in the bulk liquid phase.
A further simplification can be made on the wetted portion as
CB,S CB,bulk
(9)
If the liquid-solid mass transfer resistance is negligible compared to be reaction rate. Over the dry portion of the catalyst, as there is no mass transfer flux for the nonvolatile limiting reagent in the liquid phase, the Neuman boundary condition is applicable:
DeC 0
(10)
The boundary condition in dimensionless form is:
c 1 , at the wetted external surface
(11a)
c 0 , at the dry external surface
(11b)
3. Results and Discussion 3.1 Diffusion without reaction In order to explore the effects of difference in wetting model on diffusion, a simulation of pure diffusion without reaction was performed. The transient-state diffusion equation to be solved for a spherical particle of 5 mm in diameter is as follows
dc dt Dec With the following boundary conditions and initial conditions: -9-
(12)
c 1 , at the wetted external surface
(13-a)
c 0 , at the dry external surface
(13-b)
c 0 , at t 0
(13-c)
The diffusivity is taken as 10-9 m2/s on behalf of a wide variety of liquid systems (Satterfield and Sherwood, 1963). Fig. 3 presents the concentration distribution of two wetting models at 1000s. It is clear that model B is more favorable to diffusion than model A at low wetting fractions, as it affects a wider area. But the difference between the two models decreases as the wetting fraction increases and can be ignored when f 0.4 .
Fig. 3. Comparison between model A and model B on diffusion
3.2. Limiting reactant in the liquid phase Fig. 4 shows the variation of catalyst effectiveness factor with wetting fraction under two wetting models at different Thiele modulus. The kinetics constant has been given different values in order to cover a wide range of the Thiele modulus ( 0.01< 30 ). As model A and model B exhibit the same tendency and model A has -10-
been investigated by many scholars, the discussion in the following sections will be based on model B, however, the similarities and differences between two models are still under consideration.
Fig. 4. Catalyst effectiveness factors for liquid-limited reactions It can be obtained from Fig. 4 that the values of model B are higher than that of model A when 4 and f 0.4 . Fig .5 reveals that it is owing to the fact that the model B is more conducive for the diffusion of reactants at the same wetting fraction. Smaller wetting fraction makes the diffusion influence more obvious. But the values of model B are consistent with the values of model A when
f 0.4 or 4 , because the difference between the two models on diffusion can be ignored, as confirmed in Section 3.1.
Fig. 5. Comparison between model A and model B when f 0.1 , 1
-11-
3.2.1. Influence of the Thiele modulus It can be seen from Fig. 4 that the catalyst effectiveness factors are greatly affected by the Thiele modulus. Fig. 6 reports the variation of the internal dimensionless concentration with the Thiele modulus when the wetting fraction is 0.5. As Thiele modulus is the ratio of the intrinsic reaction rate to the internal diffusion rate, when the Thiele modulus increases, internal mass transfer resistance in the catalyst increases. As a result, the diffusion rate becomes slower with respect to the reaction rate, which leads to a decrease in the dimensionless concentration and a decrease in the catalyst effectiveness factor.
Fig. 6. Variation of the internal dimensionless concentration with the Thiele modulus in the case at f =0.5 (a) Symmetry axis view
(b) Horizontal symmetry plane view -12-
3.2.2. Influence of the wetting efficiency The variation of effectiveness factor with wetting fraction exhibits three different tendencies based on the Thiele modulus. At low values of the Thiele modulus
0.1 , compared with the reaction rate, the diffusion rate is fast enough, which means the internal mass transfer resistance can be ignored in the catalyst. Accordingly, the reactants entering from the wetted zone can quickly diffuse throughout the catalyst, even a tiny wetting fraction can make the effectiveness factor equal to 1. Fig. 7 presents the variation of the internal dimensionless concentration with the wetting fraction in the case when the Thiele modulus is 0.1. It proves that even when the wetting fraction is only 0.05, internal dimensionless concentration is equal to 1 owing to the relatively rapid diffusion.
Fig. 7. Variation of the internal dimensionless concentration with the wetting fraction in the case when =0.1 The relation between catalyst effectiveness factor and wetting fraction is nonlinear at intermediate values of the Thiele modulus 0.1< <4 . Variation of internal dimensionless concentration with wetting faction in the case when the Thiele modulus is 1 is given in Fig. 8. It can be seen that the reactants entering from the wetted zone can diffuse into the dry zone, but the concentration gradually decreases -13-
due to diffusion resistance. So, the effectiveness factor varies largely at low wetting fractions, but when the wetting fraction reaches a certain extent, increasing the wetting fraction does not significantly increase the catalyst efficiency.
Fig. 8. Variation of the internal dimensionless concentration with the wetting fraction in the case when =1 (a) Symmetry axis view
(b) Horizontal symmetry plane view
The performance becomes different at high values of the Thiele modulus 4 . Fig. 9 shows the variation of internal dimensionless concentration with wetting fraction when the Thiele modulus is equal to 15. As is observed, the reactants only exist within a short distance apart from the wetted external surface. This is because the diffusion rate is very slow compared with the reaction rate, mass transfer therefore -14-
becomes the controlling step. Thus, the reactants are completely consumed just after entering the catalyst, and will not spread to dry regions due to internal diffusion resistance. Therefore, catalyst effectiveness factor is proportional to the wetting fraction.
Fig. 9. Variation of the internal dimensionless concentration with the wetting fraction in the case when =15 3.2.3. Modification of the Thiele modulus under partial wetting condition To obtain an understanding of the behavior of partially wetted catalyst, it is necessary to start from the effectiveness factor for a catalyst with homogeneous boundary condition. When the external mass-transfer can be ignored, we have: For a semiinfinite slab:
=
tanh
,
where
V S
k De
(14) And for a sphere:
3
2
coth 1 , where rp
3V k De S
k De
(15)
When the limiting reactant is in the liquid phase, based on the work of Aris (1957), Duduković (1977) considered that only the wetted external surface can supply
-15-
the reagent, so that Thiele modulus can be modified as
V f S
k De
f for a
semiinfinite slab to account for the effects of partial wetting. Thus, an expression for a partially wetted semiinfinite slab was proposed
tanh f f
(16)
It is also possible to apply in the case of a sphere with a radius of rp 3V S , so that Eq. (15) becomes
3 f 2 2 coth 1 f f
(17)
In order to explore the condition under which the Thiele modulus can be modified as f , effectiveness factors from FEM simulation are reported as a function of
f in Fig. 10 . It can be concluded that the Thiele modulus can be modified as f only when f 0.4 or 4 . The reason is that the conventional model does not take the interaction between the dry zone and the wetted zone into account. However, as discussed in above sections, the reactants entering from the wetted zone will diffuse into the dry zone when 4 . Hence the values of FEM will be greater than the predicted values of the conventional model. The smaller the Thiele modulus, the faster the diffusion rate inside the catalyst pellet is. As a consequence, the dry and the wetted regions of the catalyst will interact strongly with each other, consequently the difference between the FEM and the conventional model becomes larger.
-16-
Fig. 10. Effectiveness factor as a function of f
3.3. Limiting reactant in the gas phase Opposed to the liquid-limited reactions where the reactants enter only from the wetted regions, the problem becomes more complicated for the gas-limited reactions by the fact that the reactants enter via both dry and wet regions of the catalyst at different mass transfer rates. So the catalyst effectiveness factors are the results of the interaction of Thiele modulus, wetting fraction and Biot number . 3.3.1. Influence of the Thiele modulus The effects of Thiele modulus on effectiveness factor are given in Fig. 11 for wetting fraction of 0.5 and Biot number of 1, the increase of Thiele modulus is made by increasing the kinetics constant. When the Thiele modulus is 0.1, the internal mass diffusion resistance is negligible. So, diffusion is fast enough to make internal dimensionless concentration to be 1. When the Thiele modulus increases gradually, on one hand, the internal diffusion resistance increases; on the other hand, as the Biot number is fixed, the external diffusion resistance also increases, making the reactant -17-
concentration at the wetted external surface decreases, all these factors lead to the decrease of catalyst effectiveness factor.
Fig. 11. Variation of dimensionless concentration with Thiele modulus for gas-limited reactions in the case when f 0.5 , Biw 1 (a) Symmetry axis view (b) Horizontal symmetry plane view
3.3.2. Influence of the Biot number According to Fig. 12, it is clear that the catalyst effectiveness factor increases with the increase of Biot number. This is because when the Thiele modulus is fixed, the larger the Biot number, the smaller the external diffusion resistance is. Thus the mass transfer flux from the wetted zone into the catalyst will be greater, and the -18-
concentration of the wetted surface will increase.
Fig. 12. Variation of the dimensionless concentration with Biot number for gas-limited reactions in the case when f 0.5 and =1 . (a) Symmetry axis view
(b) Horizontal symmetry plane view
Fig. 13. Effects of Biot number on effectiveness factor at different Thiele modulus Fig. 13 illustrates the change of catalyst effectiveness factor by increasing Biot -19-
number at different Thiele modulus. It can be found that at the low Thiele modulus, the internal mass diffusion resistance is negligible, the reactants entering from the dry zone can diffuse all over the catalyst, the effectiveness factor achieves a high value, so it is meaningless to increase the Biot number at the wetted zone. While at low wetting fraction, increasing the Biot number does not significantly improve the effectiveness factor because the range of impact is narrow. However, when the wetting fraction is gradually increased and internal mass transfer resistance cannot be ignored, reactants coming from the dry zone cannot diffuse sufficiently to the wetted zone, external mass transfer on the wetted zone greatly affects the observed reaction rate. Hence the reduction of the external mass transfer resistance for the rate-limiting gaseous reactant on partially wetted pellets leads to higher observed reaction rates. 3.3.3. Influence of wetting efficiency The effects of wetting fraction on catalyst effectiveness factors are discussed based on Fig. 14. Obviously, when the Biot number is small and the external mass diffusion resistance at the wetted external surface is severe, the gas-limited reactions present the same tendency as the liquid-limited reactions for the same reason. In the case when 0.1 , the effectiveness factor is independent of the wetting fraction. While at the intermediate values of the Thiele modulus are taken 0.1< <4 , the catalyst effectiveness factor decreases with the increase of the wetting fraction and decreases more rapidly in the high wetting fraction. When the Thiele modulus is large enough 4 , the catalyst effectiveness factor is inversely proportional to the wetting fraction. However, once the Biot number is large, the difference in mass -20-
transfer flux between the dry and wetted zones is not significant, it means that the interaction between the dry and wet zones is not obvious, leading to the fact that the dry and wetted zones can be treated separately, which can be seen in Fig. 15. Thus, the catalyst effectiveness factor can be considered as weighted average of dry and wet zones, and it is inversely proportional to the wetting fraction even at intermediate values of the Thiele modulus.
Fig. 14. Catalyst effectiveness factors for gas-limited reactions
Fig. 15. Variation of the internal dimensionless concentration in the catalyst with the wetting fraction in the case when =2.5 , Bi w 10 3.3.4. Comparison with the conventional models The mathematical expression of effectiveness factor for a partially wetted catalyst is complicated to derive due to the fact that different boundary conditions must be satisfied with the wetted and the dry surface, even for the simplest geometry like 1-D slab (Valerius et al., 1996; Beaudry et al., 2010). -21-
For a catalyst with homogeneous boundary condition, when taking external mass-transfer limitations into account, we have For a semiinfinite slab:
=
tanh tanh 1 Bi
(18)
And for a sphere:
3 coth 1
2 1 coth 1 Bi
(19)
For the gas-limited reactions in a partially wetted catalyst, the most widely accepted model is “weighting method”, which was proposed by Ramachandran and Smith (1979). The model is derived from a partially wetted infinite slab and is based on the assumption that the reagent entering through the dry and the wetted parts of the slab can be treated separately and do not interact throughout the slab volume:
f w 1 f d
1 f tanh f tanh 1 tanh Biw 1 tanh Bid
(20)
The equivalent form of Eq. (20) for a sphere is
3 f coth 1
3 1 f coth 1
1 coth 1 Biw 1 coth 1 Bid 2
2
(21)
When the gas-solid mass transfer resistance can be ignored, Eq. (19) becomes
3 f coth 1
1 coth 1 Biw 2
3 1 f coth 1
2
(22)
The Biot number at the wetted external surface is generally more than 10 in a trickle bed (Iliuta et al., 1999), so the conventional models and FEM are compared only when Biw 10 in this paper. -22-
Fig. 16. Comparison between conventional model and FEM when Biw 10 Fig. 16 reveals that the model value is in good agreement with FEM, which means Eq. (22) can be used to predict the effectiveness factors. This is because when the difference of mass transfer flux between dry and wetted external surface is small, unsymmetrical boundary conditions do not have a major influence on the concentration profiles in the catalyst, making “weighting method” accurate (Ramachandran and Smith, 1979). It is noteworthy that =0.2 at f 0 when =15 , which is equal to 3 . This is because when 15 , ( coth 1) 1 , consequently, in the case when f 0
3( coth 1)
2
3
(23)
And in another case, when f = 1,
3( coth 1) 3 1 coth 1 Biw 1 Biw 2
(24)
This is verified by FEM through =0.08 at f 1 . As a result, the simplified form of effectiveness factors for gas-limited reactions can be written as
-23-
3 1 f 3f 1 Biw
(25)
4. Conclusion By the wetting experiment, it is verified that film flow only appears when the contact angle is very small, in most cases liquid flows over the catalyst surface in the form of rivulet. The simulation of diffusion shows that rivulet wetting model is more conducive for the diffusion than film wetting model but the difference between them can be ignored when f 0.4 . Using FEM simulation, it is verified that Thiele modulus can be modified as f when f 0.4 or 4 to account for the effects of partial wetting for liquid-limited reactions, and the “weighting method” can be applied for gas-limited reactions.
Acknowledgment The authors are grateful to the financial supports from the National Natural Science Foundation of China (21676085) and the Key Research Project “Packaged Technology
Development
in
Residue
Oil
Hydroprocessing
Applications” supported by the SINOPEC.
Notation Bid Biot number at the dry external surface, dimensionless Biw Biot number at the wetted external surface, dimensionless
C concentration of Limiting reactant, mol m3 -24-
and
Industrial
C A saturation concentration of gaseous reagent in the liquid phase, mol m3 C A, S surface concentration of gaseous reactant, mol m3 CB ,bulk concentration of liquid reagent in the bulk liquid phase, mol m3
CB.S surface concentration of liquid reactant, mol m3
c dimensionless concentration of limiting reactant, dimensionless De effective diffusivity for fully internal wetted catalyst, m 2 s f the fraction of solid surface wetted by the liquid, dimensionless H Henry’s law constant, Pa m3 mol
k first-order kinetic rate constant. s 1
k gs gas-solid mass transfer coefficient, m s
kls liquid-solid mass transfer coefficient, m s k gls gas-liquid-solid mass transfer coefficient, m s rp radius of spherical particle, m S total pellet external area, m
V total pellet volume, m
2
3
Greek Letters
efficiency factor, dimensionless Thiele modulus, dimensionless
contact angle,
-25-
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dioxide in a trickle-bed reactor. Industrial & Engineering Chemistry Research 36, 5125-5132. Ring, Z.E., Missen, R.W., 1989. Trickle ‐ bed reactors: An experimental study of partial wetting effect. AIChE Journal 35, 1821-1828. Sakornwimon, W., Sylvester, N.D., 1982. Effectiveness factors for partially wetted catalysts in trickle-bed reactors. Industrial & Engineering Chemistry Process Design and Development 21, 16-25. Satterfield. C.N., Sherwood. T.K., The role of diffusion in catalysis. Addison-Wesley Publishing Company, Inc., 1963. Valerius, G., Zhu, X., Hofmann, H., 1996. Modelling of a trickle-bed reactor I. Extended definitions and new approximations. Chemical Engineering & Processing 35, 1-9. Wang, Y., Chen, J., Larachi, F., 2013. Modelling and simulation of trickle-bed reactors using computational fluid dynamics: A state-of-the-art review. Canadian Journal of Chemical Engineering 91, 136-180. Zimmerman, S.P., Ng, K.M., 1986. Liquid distribution in trickling flow trickle-bed reactors. Chemical Engineering Science 41, 861-866.
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Author contributions Yanling Tang: Conceptualization, Methodology, Software, Investigation,WritingOriginal Draft. Meichen Chen: Validation, Formal analysis, Visualization,Software. Zhenmin Cheng*: Conceptualization, Resources, Writing-Review & Editing, Supervision, Project administration, Funding acquisition . Tao Yang：Resources, Writing-Review & Editing, Supervision,Data Curation. Bo Chen: Resources, Writing-Review & Editing, Supervision,Data Curation. Hailong Ge: Resources, Writing-Review & Editing, Supervision,Data Curation. Xiangchen Fang*: Conceptualization, Resources, Writing-Review & Editing, Supervision, Project administration, Funding acquisition .
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Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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Highlights 1 A wetting experiment on a single particle was carried out to figure out the difference between film and rivulet wetting model. 2 The effectiveness factors under the two wetting modes was calculated by using finite element method. 3 Rivulet wetting model is more conducive for the diffusion than film wetting model but the difference between them can be ignored when f 0.4 . 4 Thiele modulus can be modified as f when f 0.4 or 4 to account for the effects of partial wetting for liquid-limited reactions.
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S0009-2509(20)30047-6 https://doi.org/10.1016/j.ces.2020.115515 CES 115515
To appear in:
Chemical Engineering Science
Received Date: Revised Date: Accepted Date:
2 December 2019 11 January 2020 25 January 2020
Please cite this article as: Y. Tang, M. Chen, Z. Cheng, T. Yang, B. Chen, H. Ge, X. Fang, Effectiveness factors for a partially wetted catalyst based on the rivulet flow model, Chemical Engineering Science (2020), doi: https:// doi.org/10.1016/j.ces.2020.115515
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© 2020 Published by Elsevier Ltd.
Effectiveness factors for a partially wetted catalyst based on the rivulet flow model Yanling Tang, Meichen Chen, Zhenmin Cheng* State Key Laboratory of Chemical Engineering, School of Chemical Engineering, East China University of Science and Technology, Shanghai 200237, China Tao Yang, Bo Chen, Hailong Ge, Xiangchen Fang* Dalian Research Institute of Petroleum and Petrochemicals, SINOPEC, Dalian 116045, China
Abstract Partial wetting is a common phenomenon in a trickle bed reactor, which causes nonuniform mass transfer flux at the external surface of the catalyst. Previous work on effectiveness factor for a partially wetted catalyst are mainly based on the assumption that the spherical catalyst is wetted by a liquid film. In fact, it is only true when the liquid-solid contact angle is very small. In most situations, rivulet flow prevails. The purpose of the article is to calculate effectiveness factors for both gas-limited and liquid-limited reactions under the two wetting modes by using finite element method (FEM). The effects of wetting efficiency, Biot number, and Thiele modulus on effectiveness factors are investigated. On the basis of the FEM results, the intrinsic reason for the variation of effectiveness factors with respect to wetting efficiency is revealed, and good agreements are observed between this work and the conventional models. Keywords: Partially wetted catalyst; Film flow; Rivulet flow; Effectiveness factor.
*Correspondence concerning this paper should be addressed to Dr. Z. M. Cheng at [email protected] and Dr. X. C. Fang at fangxiangchen @sinopec.com. -1-
1. Introduction As one of the most common gas-liquid-solid reactors, trickle bed reactor (TBR) is often operated in the trickling flow regime, with the gas and liquid phases introduced at low to medium flow rates cocurrently from the top of the apparatus. In spite of the simple geometry, the flow dynamics around the randomly packed particles is complex. The catalyst outer surface can be partially wetted due to liquid maldistribution or to the poor wettability of the surface (Dudukovic et al., 1999; Ling et al., 2017). As an important issue to trickle-bed reactor design, this phenomenon has been taken under full consideration in numerous papers (Al-Dahhan and Duduković, 1995; Baussaron et al., 2007; Houwelingen et al., 2010). Partial wetting makes the flow field and transfer process on the surface of catalyst nonuniform. Even at the wetted external surface, static and dynamic liquid would have different contributions to mass transfer (Ravindra et al., 1997; Rajashekharam et al., 1998). The complex external environments caused by partial wetting can largely affect the yield of the desired products, which should be analyzed in detail for precise design of TBR. For liquid-limited reactions, as the liquid reactants are in contact with the catalyst only from the wetted external surface, complete wetting of the catalyst surface is essential for the full utilization of the catalyst. For gas-limited reactions, partial wetting of the catalyst surface would be favorable, as a partial of the catalyst surface can be directly exposed to the gas reactants and consequently the reactants do not have to diffuse through the liquid film, which offers an enhanced mass transfer -2-
rate. The effects of catalyst wetting efficiency on apparent reaction rate have been confirmed by a lot of experimental work (Ring and Missen, 1989; Funk et al., 1991; Horowitz et al., 1999; Massa et al., 2009; Kumar et al., 2017). In a majority of industrial operations, as the catalyst pore network is interconnected and the pore diameter in the catalyst is very small, which creates a huge capillary force and the capillary force makes the vapor much easily condensed, therefore, the catalyst pores can be believed completely filled with liquid (Mills and Duduković, 1979; Ramachandran and Smith, 1979; Herskowitz, 1981; Palmisano et al., 2003). So the problem of calculation of the effectiveness factor then reduces to a numerical solution of the governing reaction-diffusion equations with a non-uniform boundary condition, based on the local wetting situation at the surface of the catalyst. In recent years, much efforts have been made in this area, as shown in Table 1. It should be mentioned that the wetting models in Table 1 are based on the assumption that the spherical particle is wetted by liquid film. However, many researchers pointed out that on the single particle scale, the liquid flow texture consists of two features: liquid film and rivulet, over the particle surface (Hartley and Murgatroyd, 1964; Zimmerman and Ng, 1986; Maiti et al., 2006; Mederos et al., 2009; Wang et al., 2013). So it is a great pity that no work has been done to calculate effectiveness factor based on the rivulet flow model.
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Table 1 Models and methods in calculating effectiveness factors: state-of-the-art status Authors Mills and Duduković (1979)
Wetting model
Method
effectiveness factors
Dual-Serie s Solution
Ramachandran and Smith (1979)
1 V
=
Weighting method
Vw
0
cdV
f w 1 f d Salb: 1 1 c c 2 dy 0 2 x x2 x x0 Sphere:
Separation of variables
Herskowitz (1981)
3 c sin d 22 0 r r 1 Cylinder:
Sakornwimon and Sylvester (1982)
Finite difference method
Palmisano et al. (2003)
Method of fundament al solution
Augier et al. (2010)
Finite element method
Bazer-Bachi et al. (2011)
Finite element method
2 2
0
c d r r 1
3 R 2 C sin d d R3Cs 0 0 2
De C d kCb n
cdV V =
V
Rx (r , )dV Rx ,maxV
In this study, a wetting experiment on a single particle was carried out to figure out the difference between film and rivulet wetting performance. On the basis of experimental verification, finite element method will be used to evaluate the pellet effectiveness factor as a function of the wetting fraction. It is the first study to -4-
calculate the effectiveness factor for a partially wetted catalyst based on both the film and the rivulet flow models. Therefore, it enables us to compare the results of the two wetting models in a wide operating condition.
2. Modelling Establishment 2.1 Hydrodynamic basis of the model By application of surface grafting with organic compounds, the hydrophilic alumina sphere of 5 mm in diameter was modified into different degrees of hydrophobicity with a series of liquid-solid contact angles of 14 . 48 , 88 , 119 , and 152 . The contact angle was measured using a high-speed camera with a resolution of 1280×1024@1500 fps under ambient condition. A water droplet dyed by eosin with a volume of 1 L was fed at the catalyst surface. The surface tension of the wetting liquid used in the experiments was 67 mN·m-1. The contact angle measurement on spherical surface was adopted from the method of Extrand et al. (2008), who proposed that the contact angle on the curved surface was the same as those observed on flat, horizontal surfaces. The results are illustrated in Fig. 1.
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Fig. 1. Liquid flow morphology over the particle surface with different wettability It is obvious to find in most cases liquid flows over the catalyst surface in the form of rivulet, and film flow only appears when the contact angle is very small, i.e.,
14 . The most widely used catalyst support in industry is alumina, which is hydrophilic. It was verified that the contact angle between ethanol and alumina was 30 ° from the experimental work of Baussaron (2005), despite ethanol contains a hydrophilic hydroxyl group. In another work by Augier et al. (2010), they proved that it was in the rivulet flow regime for the ethanol- alumina system from CFD simulation. What is more, the liquid feedstocks used in refining and petrochemical industry are normally oils and organic compounds, which are more hydrophobic than ethanol, and would lead to a higher liquid-solid contact angle. So, the liquid-solid wetting property is always poor. Therefore, the rivulet wetting morphology is dominant in industry, and
model B (Fig .2b) is more consistent with industrial
applications than model A (Fig .2a). -6-
Fig. 2. Wetting models of a partially wetted pellet (a) Film model
(b) Rivulet model
2.2 Mathematical model For an isothermal, irreversible first order reaction, assuming that the catalyst is completely wetted internally, a steady state equation could be written as follows (Mills and Duduković, 1979): De 2C kC
(1)
Where, C is the concentration of the limiting reactant; De is the effective diffusivity for a fully internally wetted catalyst; k is the first-order kinetic rate constant. In dimensionless form, the above equation is written as
2c 2c
(2)
Where, c is the dimensionless concentration of the limiting reactant in the liquid phase, normalized by its concentration at the bulk phase; is the Thiele modulus, defined as rp k De , and rp is the radius of spherical particle. The effectiveness factor for a first order reaction is given as cdV V . The boundary condition will be different for the following two kind of reactions, since the liquid phase is assumed unvaporizing. -7-
2.2.1. Limiting reactant in the gas phase For a gas-limited reaction, the boundary condition is that of the Robin type. Over the dry portion of the catalyst, it is written as
k gs C A C A, S DeC
(3)
Where, C A is the saturated concentration of the gas reagent in the liquid phase. Generally, the transport over the dry external surface is relatively rapid, gas-solid mass transfer resistance can be negligible, hence Dirichlet boundary condition is satisfied:
C A, S C A
(4)
Over the wetted portion, where the gaseous specie has to diffuse through the gas-liquid film and the liquid- solid film and hence the boundary condition is
k gls C A C A, S DeC
(5)
1 1 1 kgls kgl H kls
(6)
Where, k gl is the gas to liquid mass transfer coefficient and kls is the liquid to solid mass transfer coefficient. The boundary condition in dimensionless form is:
c 1 , at the dry external surface
(7a)
c Biw 1 c , at the wetted external surface
(7b)
Biw is the local Biot number, Biw rp k gls De , which describes the ratio of external mass transfer rate to internal mass transfer rate. 2.2.2 Limiting reactant in the liquid phase For a catalyst pellet partially wetted by dynamic liquid over the wetted surface, -8-
the boundary condition will be written as:
kls CB ,bulk CB , S DeC
(8)
CB ,bulk is the concentration of liquid reagent in the bulk liquid phase.
A further simplification can be made on the wetted portion as
CB,S CB,bulk
(9)
If the liquid-solid mass transfer resistance is negligible compared to be reaction rate. Over the dry portion of the catalyst, as there is no mass transfer flux for the nonvolatile limiting reagent in the liquid phase, the Neuman boundary condition is applicable:
DeC 0
(10)
The boundary condition in dimensionless form is:
c 1 , at the wetted external surface
(11a)
c 0 , at the dry external surface
(11b)
3. Results and Discussion 3.1 Diffusion without reaction In order to explore the effects of difference in wetting model on diffusion, a simulation of pure diffusion without reaction was performed. The transient-state diffusion equation to be solved for a spherical particle of 5 mm in diameter is as follows
dc dt Dec With the following boundary conditions and initial conditions: -9-
(12)
c 1 , at the wetted external surface
(13-a)
c 0 , at the dry external surface
(13-b)
c 0 , at t 0
(13-c)
The diffusivity is taken as 10-9 m2/s on behalf of a wide variety of liquid systems (Satterfield and Sherwood, 1963). Fig. 3 presents the concentration distribution of two wetting models at 1000s. It is clear that model B is more favorable to diffusion than model A at low wetting fractions, as it affects a wider area. But the difference between the two models decreases as the wetting fraction increases and can be ignored when f 0.4 .
Fig. 3. Comparison between model A and model B on diffusion
3.2. Limiting reactant in the liquid phase Fig. 4 shows the variation of catalyst effectiveness factor with wetting fraction under two wetting models at different Thiele modulus. The kinetics constant has been given different values in order to cover a wide range of the Thiele modulus ( 0.01< 30 ). As model A and model B exhibit the same tendency and model A has -10-
been investigated by many scholars, the discussion in the following sections will be based on model B, however, the similarities and differences between two models are still under consideration.
Fig. 4. Catalyst effectiveness factors for liquid-limited reactions It can be obtained from Fig. 4 that the values of model B are higher than that of model A when 4 and f 0.4 . Fig .5 reveals that it is owing to the fact that the model B is more conducive for the diffusion of reactants at the same wetting fraction. Smaller wetting fraction makes the diffusion influence more obvious. But the values of model B are consistent with the values of model A when
f 0.4 or 4 , because the difference between the two models on diffusion can be ignored, as confirmed in Section 3.1.
Fig. 5. Comparison between model A and model B when f 0.1 , 1
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3.2.1. Influence of the Thiele modulus It can be seen from Fig. 4 that the catalyst effectiveness factors are greatly affected by the Thiele modulus. Fig. 6 reports the variation of the internal dimensionless concentration with the Thiele modulus when the wetting fraction is 0.5. As Thiele modulus is the ratio of the intrinsic reaction rate to the internal diffusion rate, when the Thiele modulus increases, internal mass transfer resistance in the catalyst increases. As a result, the diffusion rate becomes slower with respect to the reaction rate, which leads to a decrease in the dimensionless concentration and a decrease in the catalyst effectiveness factor.
Fig. 6. Variation of the internal dimensionless concentration with the Thiele modulus in the case at f =0.5 (a) Symmetry axis view
(b) Horizontal symmetry plane view -12-
3.2.2. Influence of the wetting efficiency The variation of effectiveness factor with wetting fraction exhibits three different tendencies based on the Thiele modulus. At low values of the Thiele modulus
0.1 , compared with the reaction rate, the diffusion rate is fast enough, which means the internal mass transfer resistance can be ignored in the catalyst. Accordingly, the reactants entering from the wetted zone can quickly diffuse throughout the catalyst, even a tiny wetting fraction can make the effectiveness factor equal to 1. Fig. 7 presents the variation of the internal dimensionless concentration with the wetting fraction in the case when the Thiele modulus is 0.1. It proves that even when the wetting fraction is only 0.05, internal dimensionless concentration is equal to 1 owing to the relatively rapid diffusion.
Fig. 7. Variation of the internal dimensionless concentration with the wetting fraction in the case when =0.1 The relation between catalyst effectiveness factor and wetting fraction is nonlinear at intermediate values of the Thiele modulus 0.1< <4 . Variation of internal dimensionless concentration with wetting faction in the case when the Thiele modulus is 1 is given in Fig. 8. It can be seen that the reactants entering from the wetted zone can diffuse into the dry zone, but the concentration gradually decreases -13-
due to diffusion resistance. So, the effectiveness factor varies largely at low wetting fractions, but when the wetting fraction reaches a certain extent, increasing the wetting fraction does not significantly increase the catalyst efficiency.
Fig. 8. Variation of the internal dimensionless concentration with the wetting fraction in the case when =1 (a) Symmetry axis view
(b) Horizontal symmetry plane view
The performance becomes different at high values of the Thiele modulus 4 . Fig. 9 shows the variation of internal dimensionless concentration with wetting fraction when the Thiele modulus is equal to 15. As is observed, the reactants only exist within a short distance apart from the wetted external surface. This is because the diffusion rate is very slow compared with the reaction rate, mass transfer therefore -14-
becomes the controlling step. Thus, the reactants are completely consumed just after entering the catalyst, and will not spread to dry regions due to internal diffusion resistance. Therefore, catalyst effectiveness factor is proportional to the wetting fraction.
Fig. 9. Variation of the internal dimensionless concentration with the wetting fraction in the case when =15 3.2.3. Modification of the Thiele modulus under partial wetting condition To obtain an understanding of the behavior of partially wetted catalyst, it is necessary to start from the effectiveness factor for a catalyst with homogeneous boundary condition. When the external mass-transfer can be ignored, we have: For a semiinfinite slab:
=
tanh
,
where
V S
k De
(14) And for a sphere:
3
2
coth 1 , where rp
3V k De S
k De
(15)
When the limiting reactant is in the liquid phase, based on the work of Aris (1957), Duduković (1977) considered that only the wetted external surface can supply
-15-
the reagent, so that Thiele modulus can be modified as
V f S
k De
f for a
semiinfinite slab to account for the effects of partial wetting. Thus, an expression for a partially wetted semiinfinite slab was proposed
tanh f f
(16)
It is also possible to apply in the case of a sphere with a radius of rp 3V S , so that Eq. (15) becomes
3 f 2 2 coth 1 f f
(17)
In order to explore the condition under which the Thiele modulus can be modified as f , effectiveness factors from FEM simulation are reported as a function of
f in Fig. 10 . It can be concluded that the Thiele modulus can be modified as f only when f 0.4 or 4 . The reason is that the conventional model does not take the interaction between the dry zone and the wetted zone into account. However, as discussed in above sections, the reactants entering from the wetted zone will diffuse into the dry zone when 4 . Hence the values of FEM will be greater than the predicted values of the conventional model. The smaller the Thiele modulus, the faster the diffusion rate inside the catalyst pellet is. As a consequence, the dry and the wetted regions of the catalyst will interact strongly with each other, consequently the difference between the FEM and the conventional model becomes larger.
-16-
Fig. 10. Effectiveness factor as a function of f
3.3. Limiting reactant in the gas phase Opposed to the liquid-limited reactions where the reactants enter only from the wetted regions, the problem becomes more complicated for the gas-limited reactions by the fact that the reactants enter via both dry and wet regions of the catalyst at different mass transfer rates. So the catalyst effectiveness factors are the results of the interaction of Thiele modulus, wetting fraction and Biot number . 3.3.1. Influence of the Thiele modulus The effects of Thiele modulus on effectiveness factor are given in Fig. 11 for wetting fraction of 0.5 and Biot number of 1, the increase of Thiele modulus is made by increasing the kinetics constant. When the Thiele modulus is 0.1, the internal mass diffusion resistance is negligible. So, diffusion is fast enough to make internal dimensionless concentration to be 1. When the Thiele modulus increases gradually, on one hand, the internal diffusion resistance increases; on the other hand, as the Biot number is fixed, the external diffusion resistance also increases, making the reactant -17-
concentration at the wetted external surface decreases, all these factors lead to the decrease of catalyst effectiveness factor.
Fig. 11. Variation of dimensionless concentration with Thiele modulus for gas-limited reactions in the case when f 0.5 , Biw 1 (a) Symmetry axis view (b) Horizontal symmetry plane view
3.3.2. Influence of the Biot number According to Fig. 12, it is clear that the catalyst effectiveness factor increases with the increase of Biot number. This is because when the Thiele modulus is fixed, the larger the Biot number, the smaller the external diffusion resistance is. Thus the mass transfer flux from the wetted zone into the catalyst will be greater, and the -18-
concentration of the wetted surface will increase.
Fig. 12. Variation of the dimensionless concentration with Biot number for gas-limited reactions in the case when f 0.5 and =1 . (a) Symmetry axis view
(b) Horizontal symmetry plane view
Fig. 13. Effects of Biot number on effectiveness factor at different Thiele modulus Fig. 13 illustrates the change of catalyst effectiveness factor by increasing Biot -19-
number at different Thiele modulus. It can be found that at the low Thiele modulus, the internal mass diffusion resistance is negligible, the reactants entering from the dry zone can diffuse all over the catalyst, the effectiveness factor achieves a high value, so it is meaningless to increase the Biot number at the wetted zone. While at low wetting fraction, increasing the Biot number does not significantly improve the effectiveness factor because the range of impact is narrow. However, when the wetting fraction is gradually increased and internal mass transfer resistance cannot be ignored, reactants coming from the dry zone cannot diffuse sufficiently to the wetted zone, external mass transfer on the wetted zone greatly affects the observed reaction rate. Hence the reduction of the external mass transfer resistance for the rate-limiting gaseous reactant on partially wetted pellets leads to higher observed reaction rates. 3.3.3. Influence of wetting efficiency The effects of wetting fraction on catalyst effectiveness factors are discussed based on Fig. 14. Obviously, when the Biot number is small and the external mass diffusion resistance at the wetted external surface is severe, the gas-limited reactions present the same tendency as the liquid-limited reactions for the same reason. In the case when 0.1 , the effectiveness factor is independent of the wetting fraction. While at the intermediate values of the Thiele modulus are taken 0.1< <4 , the catalyst effectiveness factor decreases with the increase of the wetting fraction and decreases more rapidly in the high wetting fraction. When the Thiele modulus is large enough 4 , the catalyst effectiveness factor is inversely proportional to the wetting fraction. However, once the Biot number is large, the difference in mass -20-
transfer flux between the dry and wetted zones is not significant, it means that the interaction between the dry and wet zones is not obvious, leading to the fact that the dry and wetted zones can be treated separately, which can be seen in Fig. 15. Thus, the catalyst effectiveness factor can be considered as weighted average of dry and wet zones, and it is inversely proportional to the wetting fraction even at intermediate values of the Thiele modulus.
Fig. 14. Catalyst effectiveness factors for gas-limited reactions
Fig. 15. Variation of the internal dimensionless concentration in the catalyst with the wetting fraction in the case when =2.5 , Bi w 10 3.3.4. Comparison with the conventional models The mathematical expression of effectiveness factor for a partially wetted catalyst is complicated to derive due to the fact that different boundary conditions must be satisfied with the wetted and the dry surface, even for the simplest geometry like 1-D slab (Valerius et al., 1996; Beaudry et al., 2010). -21-
For a catalyst with homogeneous boundary condition, when taking external mass-transfer limitations into account, we have For a semiinfinite slab:
=
tanh tanh 1 Bi
(18)
And for a sphere:
3 coth 1
2 1 coth 1 Bi
(19)
For the gas-limited reactions in a partially wetted catalyst, the most widely accepted model is “weighting method”, which was proposed by Ramachandran and Smith (1979). The model is derived from a partially wetted infinite slab and is based on the assumption that the reagent entering through the dry and the wetted parts of the slab can be treated separately and do not interact throughout the slab volume:
f w 1 f d
1 f tanh f tanh 1 tanh Biw 1 tanh Bid
(20)
The equivalent form of Eq. (20) for a sphere is
3 f coth 1
3 1 f coth 1
1 coth 1 Biw 1 coth 1 Bid 2
2
(21)
When the gas-solid mass transfer resistance can be ignored, Eq. (19) becomes
3 f coth 1
1 coth 1 Biw 2
3 1 f coth 1
2
(22)
The Biot number at the wetted external surface is generally more than 10 in a trickle bed (Iliuta et al., 1999), so the conventional models and FEM are compared only when Biw 10 in this paper. -22-
Fig. 16. Comparison between conventional model and FEM when Biw 10 Fig. 16 reveals that the model value is in good agreement with FEM, which means Eq. (22) can be used to predict the effectiveness factors. This is because when the difference of mass transfer flux between dry and wetted external surface is small, unsymmetrical boundary conditions do not have a major influence on the concentration profiles in the catalyst, making “weighting method” accurate (Ramachandran and Smith, 1979). It is noteworthy that =0.2 at f 0 when =15 , which is equal to 3 . This is because when 15 , ( coth 1) 1 , consequently, in the case when f 0
3( coth 1)
2
3
(23)
And in another case, when f = 1,
3( coth 1) 3 1 coth 1 Biw 1 Biw 2
(24)
This is verified by FEM through =0.08 at f 1 . As a result, the simplified form of effectiveness factors for gas-limited reactions can be written as
-23-
3 1 f 3f 1 Biw
(25)
4. Conclusion By the wetting experiment, it is verified that film flow only appears when the contact angle is very small, in most cases liquid flows over the catalyst surface in the form of rivulet. The simulation of diffusion shows that rivulet wetting model is more conducive for the diffusion than film wetting model but the difference between them can be ignored when f 0.4 . Using FEM simulation, it is verified that Thiele modulus can be modified as f when f 0.4 or 4 to account for the effects of partial wetting for liquid-limited reactions, and the “weighting method” can be applied for gas-limited reactions.
Acknowledgment The authors are grateful to the financial supports from the National Natural Science Foundation of China (21676085) and the Key Research Project “Packaged Technology
Development
in
Residue
Oil
Hydroprocessing
Applications” supported by the SINOPEC.
Notation Bid Biot number at the dry external surface, dimensionless Biw Biot number at the wetted external surface, dimensionless
C concentration of Limiting reactant, mol m3 -24-
and
Industrial
C A saturation concentration of gaseous reagent in the liquid phase, mol m3 C A, S surface concentration of gaseous reactant, mol m3 CB ,bulk concentration of liquid reagent in the bulk liquid phase, mol m3
CB.S surface concentration of liquid reactant, mol m3
c dimensionless concentration of limiting reactant, dimensionless De effective diffusivity for fully internal wetted catalyst, m 2 s f the fraction of solid surface wetted by the liquid, dimensionless H Henry’s law constant, Pa m3 mol
k first-order kinetic rate constant. s 1
k gs gas-solid mass transfer coefficient, m s
kls liquid-solid mass transfer coefficient, m s k gls gas-liquid-solid mass transfer coefficient, m s rp radius of spherical particle, m S total pellet external area, m
V total pellet volume, m
2
3
Greek Letters
efficiency factor, dimensionless Thiele modulus, dimensionless
contact angle,
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Author contributions Yanling Tang: Conceptualization, Methodology, Software, Investigation,WritingOriginal Draft. Meichen Chen: Validation, Formal analysis, Visualization,Software. Zhenmin Cheng*: Conceptualization, Resources, Writing-Review & Editing, Supervision, Project administration, Funding acquisition . Tao Yang：Resources, Writing-Review & Editing, Supervision,Data Curation. Bo Chen: Resources, Writing-Review & Editing, Supervision,Data Curation. Hailong Ge: Resources, Writing-Review & Editing, Supervision,Data Curation. Xiangchen Fang*: Conceptualization, Resources, Writing-Review & Editing, Supervision, Project administration, Funding acquisition .
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Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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Highlights 1 A wetting experiment on a single particle was carried out to figure out the difference between film and rivulet wetting model. 2 The effectiveness factors under the two wetting modes was calculated by using finite element method. 3 Rivulet wetting model is more conducive for the diffusion than film wetting model but the difference between them can be ignored when f 0.4 . 4 Thiele modulus can be modified as f when f 0.4 or 4 to account for the effects of partial wetting for liquid-limited reactions.
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