3722
Ind. Eng. Chem. Res. 1998, 37, 3722-3730
Effect of Entrance and Exit Geometry on Pressure Drop and Flooding Limits in a Single Channel of an Internally Finned Monolith Paul J. M. Lebens,* Rolf K. Edvinsson, S. Tiong Sie, and Jacob A. Moulijn Department of Chemical Process Technology, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
Flooding limits and pressure drop of gas-liquid countercurrent flow have been determined in internally finned tubes, representing a single channel of an internally finned monolith. It was found that with the appropriate liquid outlet geometry it is possible to achieve countercurrent annular flow at velocities that are relevant for hydrotreating of petroleum fractions. Comparison with an unfinned tube showed that the longitudinal fins inside the tube stabilize the liquid layer against slug formation, especially at high liquid velocities. The pressure drop was calculated using a countercurrent film flow model. By taking the extra pressure drop caused by the end-effects into account, it was possible to predict the pressure drop of both the finned and the unfinned tubes in the wavy annular flow regime. 1. Introduction Three-phase catalytic processes, in which the reactants are in the gas phase as well as in the liquid phase, are widely applied in the chemical industry. An important example from the petroleum industry is hydrotreating of heavy oils, a process in which oil and hydrogen are passed downward over a fixed-bed of randomly packed catalyst particles. The optimal size of the particles, typically 1 to 3 mm, is determined by the balance between high-pressure drop and pore diffusion resistance. Cocurrent flow of gas and liquid is used since countercurrent flow will cause flooding at linear velocities that are of industrial interest. From a theoretical point of view, however, countercurrent operation has distinct advantages over cocurrent operation in processes where the conversion is (1) suppressed by (by)products, for example, hydrodesulfurization and hydrodenitrogenation, or (2) limited by thermodynamic equilibria, for example, hydrodearomatization. The concentration of the gaseous (inhibiting) reaction products (e.g., H2S, NH3) increases in the direction of the gas flow while it decreases in the direction of the liquid flow. Hence, in countercurrent operation the reaction conditions are most favorable at the liquid outlet (e.g., high PH2, low PH2S), where the least reactive liquid compounds must be converted (Sie and Lebens, 1997). The internally finned monolithic reactor (IFMR) is a new type of three-phase catalytic reactor that can allow countercurrent gas-liquid flow at industrial relevant flow rates (Sie et al., 1992). The catalyst, which is based on a normal monolith having longitudinal fins incorporated in the walls of the channels, makes it possible to structure the countercurrent gas-liquid flow and reduce the effect of momentum transfer. The liquid, which wets the solid surface, tends to occupy the grooves between the fins because of the capillary forces leaving space for the gas to flow upward through the central core (Figure 1). * To whom correspondence may be addressed: e-mail,
[email protected]; tel, (+31) 15 278 4396; fax, (+31) 15 278 4452.
Figure 1. Schematic representation of the concept of the internally finned monolith reactor.
Another essential function of the fins is to increase the specific surface area and at the same time increase the fractional catalyst volume. Conventional monoliths with channels (d ) 4 mm) that could allow countercurrent two-phase flow at reasonable velocities have either a too low surface-to-volume ratio (low cell density) or a too high porosity (high cell density). The internally finned monoliths are characterized by a low cell density (∼25 cpsi), a high fractional catalyst volume (∼0.57), and a high geometric surface area (∼2000 m2/m3). The latter two are comparable to a randomly packed bed of 1.5-3 mm diameter catalyst particles. The IFMR offers a high degree of freedom in design. Channel size and shape can be chosen according to the process requirements. Examples of some channel geometries and their characteristic properties are given in Figure 2 and Table 1. Gas-liquid countercurrent flow in cylindrical tubes is characterized by different flooding regimes. It is known that an increase in either the gas or the liquid flow rate leads to a change in momentum transfer that can lead to a change of flow pattern. Interfacial waves and liquid droplets make gas-liquid countercurrent flow a complex and sometimes chaotic phenomenon. Zhang and Giot (1995) have given a quantitative description of the flooding phenomenon in vertical adiabatic gas-liquid countercurrent flow. The different flow patterns were compiled in a general flow regime
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Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3723
lan and Whalley, 1985) have made detailed reviews and comparisons. Most of the models are, however, not generally applicable because of the limited experimental range and the lack of fundamental understanding of the flooding phenomena. A correlation that has been fairly successful is the correlation of Wallis (Wallis, 1969). On the basis of experiments, he suggested that flooding could be predicted by
u*GS1/2 + mWu*LS1/2 ) CW
(1)
where u*LS and u*GS are the dimensionless superficial liquid and gas velocities defined as
u*LS ) Figure 2. Examples of different cross sectional channel geometries: a, circular; b, square; c, triangular; d, hexagonal.
[
a
T ) triangular; S ) square.
Figure 3. General flow regime map for countercurrent gas-liquid flow in vertical cylindrical tubes (from Zhang and Giot, 1995). Borders: s, wavy annular; - - -, slug; ‚‚‚, unstable flow.
map that is shown in Figure 3. Although it is difficult to observe each flow regime, it is believed that such a map could cover all the possible flow patterns in vertical countercurrent flow when entrance effects, multipath phenomena, and thermal effects can be neglected. In this work the 10 flow regimes are reduced to three main categories: wavy annular (SF, DW, TW), slug (L, S, B), and unstable (E, F, G, H) flow. The latter is considered to be an intermediate flow regime between wavy annular (WA) and slug flow (S) because liquid up flow will lead to accumulation of liquid at the top which eventually will result in slug flow. The transition from wavy annular flow to slug flow, either directly or indirectly via the unstable flow regime, is defined as the flooding point. Many correlations describing flooding in two-phase countercurrent flow are available in the literature. Several authors (Bankoff and Lee, 1985, 1986; McQuil-
]
1/2
gDh(FL - FG)
u*GS )
Table 1. Characteristic Properties of Some Internally Finned Monoliths basic channel cross section circular triangular square hexagonal arrangementa T T S T wall thickness 0.1p 0.2p 0.2p 0.2p number of fins 6 3 4 6 fin height 0.2p 0.1p 0.2p 0.2p fin thickness 0.1p 0.2p 0.2p 0.2p solid fraction 0.57 0.62 0.60 0.68 surface-to-volume ratio 10.0/p 8.5/p 8.1/p 8.8/p
FLuLS2
[
FGuGS2
gDh(FL - FG)
]
1/2
(2)
Although eq 1 is empirical, Turner (1966) has shown that there is also theoretical support for this type of flooding correlation. Essentially, it can be considered as an interfacial force balance between momentum flux and buoyancy. The flooding parameters mW and CW are determined from experiments. Usually mW is unity and CW ranges from 0.725 to 1 depending on the entrance and exit geometry. A few researchers have reported values for mW and CW that also depend on the fluid properties (McQuillan and Whalley, 1985; Turner, 1966; Zapke and Kro¨ger, 1996). Although a considerable amount of work has been published on flooding in large diameter cylindrical tubes, little insight is available on the hydrodynamic behavior of gas-liquid countercurrent flow in small tubes, especially when they have internal fins. With the application of an IFMR, however, pressure drop and flooding are two aspects that are of great importance for the design of such a reactor. The present study is therefore concerned with the determination of the pressure drop and the flooding limits of gas-liquid countercurrent flow in a single tube provided with internal fins. Since in the envisaged monolith the channel geometry will be nearly identical, the hydrodynamic behavior in one channel is expected to be representative for a complete monolith if gas and liquid are evenly distributed. Special attention is paid to the effect of the liquid outlet geometry, as it is known from literature that this is a critical factor (Imura et al., 1977). To investigate the effect of the fins, the hydrodynamic behavior of the finned tubes was compared with unfinned tubes. Finally, the flooding correlation of Wallis was evaluated for the measured flooding data and the pressure drop was calculated using a countercurrent laminar film flow model. 2. Theory The wavy annular film flow in the tubes can be modeled by using a straightforward film flow model. For cylindrical tubes this results in a set of equations that can be solved analytically, while for the finned tubes only a numerical solution is possible. The extra pressure drop, which is a result of entrance and exit effects, is calculated with an empirical correlation that is evaluated for single-phase flow. It is assumed that in
3724 Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998
the wavy annular flow regime the entrance and exit effects are about the same as those in single-phase (gas) flow. Countercurrent Film Flow Model for Unfinned Cylindrical Tubes. The model for smooth countercurrent film flow used to calculate the relation between the pressure drop, liquid hold-up, and flow rates in the wavy annular flow regime is based on the following assumptions: (1) The flow is steady, laminar, isothermal, and axisymmetric. (2) The interface between the liquid film and the gas core is smooth and continuous. The small capillary ripples, which exist already at very low liquid flow rates, are assumed to have no significant effect. (3) All physical properties are considered to be constant. (4) Both the liquid and the gas phase are incompressible Newtonian fluids. (5) At the interface the gas and liquid velocity are equal. (6) Entrance and exit effects are not included in the model. For each phase, the equation of motion for the twophase flow can be derived from the Navier-Stokes equation (Bird et al. 1960). For a cylindrical tube the conservation of momentum in the radial direction can be written as:
∂ ∂ µF r uF ) -g (FF + ∆LP)r ∂r ∂r
(
)
(3)
and ∆LP ) ∆Pf/gL The boundary conditions that are needed to solve this set of partial differential equations (PDEs) include the symmetric condition at the axis, matching conditions at the interface, and the no-slip condition at the wall. With these conditions the following expressions for the liquid velocity, uL, and gas velocity, uG, are obtained:
r 2 g (F + ∆LP)R2 1 + 4µL L R g r (4) (F + FG)Rf2 ln 2µL L R
( ( ))
(( ) ( ) ) ( ( ))
Rf 2 g r 2 (FG + ∆LP)R2 + 4µG R R Rf 2 Rf g g (FL + ∆LP)R2 1 + (FL - FG)Rf2 ln 4µL R 2µL R (5)
uG )
()
Subsequently, the superficial velocities can be calculated by integration.
uGS )
( ( ) ( )) (( ) ( ( )) )
() ( )( ( )) () ()
Rf g (FG + ∆LP)R2 8µG R
Rf 4 Rf 2 g (FL + ∆LP)R2 1 + -2 + 2µL R R Rf 2 Rf g (FL - FG)Rf2 1 - ln - 1 (6) µL R R
4
+
Rf g (FL + ∆LP) R2 4µL R
2
1-
Rf R
2
Rf g (FL - FG)Rf2 2µL R
+
2
ln
Rf (7) R
The liquid hold-up, L, which is directly related to the film thickness, can be expressed as follows:
L ) 1 -
() Rf R
2
(8)
Countercurrent Film Flow Model for Finned Tubes. For the finned tubes with a cross section as depicted in Figure 4a, a few extra assumptions have been made: (1) The liquid is uniformly distributed over the subchannels so that the cross section consists of 12 equal segments (Figure 4c,d). (2) The gas-liquid interface between the fins has a constant curvature, touching the fin with the contact angle θ (Figure 4c,d). In rectangular coordinates the conservation of momentum for the computational domain as illustrated in Figure 4 takes the following form:
(
µF
with F ) G or L for gas or liquid, respectively,
uL )
uGS )
∂2uF ∂x2
+
)
∂2uF ∂y2
) -g (FF + ∆LP)
(9)
with F ) G or L for gas or liquid respectively The boundary conditions that are necessary to solve this set of PDEs include the symmetric condition on the two lines of symmetry (S1, S2), matching conditions at the interface, and the no-slip condition at the solid surface. A commercially available PDE solver, SPDE (Spde, Inc.), was used to numerically solve this boundary value problem. The program used the Galerkin finite element method (FEM) of weighted residuals with a fully automated adaptive grid refinement. The velocity profiles in the liquid and the gas phase were determined for a given liquid hold-up and pressure drop. The superficial velocities were calculated by summation. The FEM code could also be used to calculate the relation between frictional pressure drop and superficial velocity in single-phase gas flow. In this case the liquid hold-up was assumed to be zero. Entrance and Exit Effects. For single-phase flow, the frictional pressure drop was calculated with Poiseuille’s law
∆Pf ) 4fF
FFuFS2 L L ) 2B1µFuFS 2 2 Dh D
(10)
h
with fF )
B1 for laminar flow ReF
in which B1 depends on the cross sectional shape of the tube. For cylindrical tubes B1 equals 16 while for finned tubes of the used geometry B1 is equal to 14.4, according to the calculations with the FEM code. It has to be mentioned here that although B1 is smaller for the
Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3725 Table 2. Tubes Used in the Experiments tube type
material
D (mm)
Da (mm)
Dh (mm)
unfinned unfinned finned
glass perspex polycarbonate
1.75 3.5 4
1.75 3.5 3.7
1.75 3.5 1.75
Figure 4. Photo of the cross section of the finned tubes that were used in the experiments (a). Schematic representation of cross section (b) and computational domain of film flow model in an internally finned channel (c, d). Figure 6. Different outlet geometries of internally finned tubes: a, beveled; b, beveled with fins partially removed; c, funnel; d, capillary and wool. Outlets a and c were also used for the unfinned tubes.
Figure 5. Schematic representation of experimental setup: (a) tubing pump; (b) rotameter; (c) storage tank; (d) bottom vessel; (e) tube; (f) top vessel; (g) differential pressure transmitter; (h) mass flow controller; (i) gas supply; (j) detail of liquid inlet; (k) computer PC-386; (l) i/o unit; (v) video recordings.
finned tubes, the pressure drop is much higher since the hydraulic diameter of the finned tube is two times smaller. The entrance and exit effects, which have been neglected in eq 10 as well as in the basic equation of the countercurrent film flow model, can be taken into account by empirically adding an extra contribution to the pressure drop:
∆Pe ≡
(
( ))
FGuGS2 L B2 ReGB3 2 Dh
B4
(11)
For laminar flow eq 10 can be modified to:
(
( ))
FGuGS2 B1 L L ∆Pt ) 4 + B2 ReGB3 2 ReG Dh Dh
B4
(12)
where B2 is considered to take into account the extra energy dissipation caused by the gas inlet while B3 and B4 take into account the development of the velocity profile inside the tube. Consequently, B2 is assumed to be dependent on the entrance and exit geometry of the tube, while B3 and B4 are assumed to be constant for each cross sectional shape. Since all inlet effects are related to each other, it is assumed that they can be combined in one empirical term. 3. Experimental Section The hydrodynamic behavior of the gas-liquid countercurrent flow was studied in the setup shown in
Figure 5. The gas (air) flows upward due to a pressure gradient while the liquid flows downward due to gravity. The gas flow is controlled by a digital mass flow controller (Brooks Instruments model 5860S, (1% full scale) and the liquid is circulated with a tubing pump (Masterflex model 7550-62). Both the mass flow controller and the liquid pump are controlled by a computer (PC-386 25 MHz, RS-232). The liquid flow rate is measured with a rotameter ((2% full scale). The pressure drop between the upper and lower vessel is measured with a differential pressure transmitter (Cole and Parmer Ashcroft model XLdp, (0.5% full scale). The analogue signal (4-20 mA) of the differential pressure transmitter is converted to a digital signal with an Opto 22 model 200 i/o unit. A cross sectional view of the internally finned test tube that was used is shown in Figure 4a. The inner diameter of the tube was 4 mm while the apparent diameter (Da) and the hydraulic diameter (Dh) of the tube were about 3.7 and 1.75 mm, respectively. Two unfinned tubes were used with inner diameter of 3.5 and 1.75 mm, comparable to the Da and Dh of the finned tube. The tubes were typically 1 m long and were made of transparent polycarbonate, Perspex, or glass (Table 2). The finned tubes were manufactured by assembling 5 cm pieces. Wetting was complete for n-decane, which was used as the model liquid because its properties are representative for heavy oil under hydrotreating conditions (T ) 573-672 K, P ) 5-15 MPa). The flooding limits, which are characterized by a transition of the flow pattern from wavy annular flow to slug flow, were determined both indirectly by measuring the pressure drop and directly by visual inspection including video recordings (VHS camera). In the experiments, which were all performed at room temperature and atmospheric pressure, liquid always entered the tube by means of overflow while it was drained off with varying outlet geometries. An overview of these liquid outlet geometries is given in Figure 6. 4. Results and Discussion In initial experiments it was found that countercurrent annular flow in both finned and unfinned tubes is strongly limited by the behavior of the liquid exiting at the bottom end of the tube. Consider the case when the
3726 Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998
Figure 7. Visualization of outlet effects in a tubes with (a) a flat and (b) a beveled outlet.
Figure 9. Transition from wavy annular flow to slug flow in an internally finned tube for different outlet angles: R ) 60, 65, 70°; Vis. ) visually observed; ∆P ) determined by pressure drop.
Figure 8. Pressure gradient versus superficial gas velocity for various superficial liquid velocities in a 1.0 m long beveled finned tube (R ) 70°).
liquid outlet is cut perpendicular to the axis of the tube (R ) 0°). A droplet will be formed, temporarily closing off the entrance for the gas. Subsequently, the pressure in the lower vessel increases until it is sufficiently high to force the liquid droplet back into the tube. At very low superficial liquid velocities (uLS) this results in a fluctuating flow at the bottom which propagates up along the tube when uLS is increased. Even in a wider tube, where such a total closure does not occur, a fluctuating behavior can still be observed. In this case the acceleration of the gas leads, in accordance to the law of Bernoulli, to a lower local pressure. The pendent liquid is then sucked inward and forms a liquid plug (Figure 7a). Beveling the tube at an oblique angle at the bottom can alleviate this problem. English et al. (1963) have shown that tubes (d ) 1.9 cm) with such a beveled end have a lower flooding tendency. In agreement with their finding, it was found in this study that beveling small diameter tubes significantly shifts the onset of slugging to higher velocities. This was found for both finned and unfinned tubes. Flow Regime Transition in Tubes With Beveled Liquid Outlet. Figure 8 shows results of pressure drop measurements in a 1 m long finned tube with a 70° beveled liquid outlet. In the annular flow regime the pressure drop is low and essentially proportional to the superficial gas velocity (uGS). In this regime the liquid is falling down as a film with ripples or small waves on the surface. It can be seen in Figure 8 that, in this regime, uLS has only a small effect on the pressure drop. However, above a certain critical uGS the pressure drop increases suddenly. This is a result of a flow transition
from annular to slug flow initiated at the liquid outlet of the tube. This onset of slugging depends on uLS and shifts toward lower uGS when uLS is increased. In agreement with the results of Jeong and No (1996), this outlet disturbance is restricted to the bottom region of the tube for low uLS, while it leads to complete flooding for high uLS. A collection of the transition points in a plot of uLS against uGS for three internally finned tubes (IFTs) with different outlet angles (R) is shown in Figure 9. The cut off angle (R) appears to affect the transition in two ways. For low uLS the onset of slugging shifts toward higher uGS with increasing R while for low uGS it will shift toward higher uLS. The first shift can be explained by the enlarged inlet gap at higher R resulting in a lower local uGS at the entrance. As a consequence both the pressure and the shear rate on the liquid decrease with increasing R. As a result the transition from annular to slug flow occurs at higher uGS. An even greater effect of R can be observed at high uLS. Here, increased R enhances the liquid drain along the edge of the outlet angle so that the transition takes place at higher velocities (Figure 7b). A similar experiment has been performed with four 1 m long unfinned tubes having an internal diameter of 3.5 mm and R ranging from 50 to 80°. The results in Figure 10 show that the onset of slugging significantly shifts toward higher uGS and uLS when R is raised from 50 to 60°. The onset of slugging can be moved to even higher velocities when R is increased up to 70°. This is close to the optimum, and any further increase of R up to 80° leads to a smaller wavy annular flow window. A qualitative explanation for this is given later. In agreement with the results of Jeong and No (1996), two different transitions can be observed, exit and entrance flooding initiated at the liquid outlet and inlet, respectively. At both transitions, circumferencial waves bridge together and close off the open space for the gas completely. In the case of exit flooding the transition is determined by the propulsive forces of the gas phase while in the case of entrance flooding a nonuniform liquid flow causes the instabilities (Jeong and No, 1996). Consequently, the exit flooding primarily depends on uGS while the entrance flooding primarily depends on uLS.
Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3727 Table 3. Fitted Parameters and Validity Range of the Flooding Correlation of Wallis outlet unfinned tubes R ) 60° R ) 70° R ) 80° finned tubes R ) 70° R ) 70° (2 cm) R ) 70° (7 cm) funnel capillary
Figure 10. Transition from wavy annular flow (WA) to slug flow (S) in an unfinned tube (d ) 3.5 mm) for different outlet angles (R).
Figure 11. (i) Transition from wavy annular flow to slug flow in an internally finned capillary for different outlets (symbols). (ii) Equation of Wallis fitted to the experimental data (dashed lines).
Flow Regime Transition in IFTs with Specially Adapted Liquid Outlets. Since the flow transition in an IFT with beveled liquid outlet (Figure 9) is still determined by exit effects, special geometries were tested. As can be seen in Figure 11, removing the fins 2 cm from a beveled liquid outlet results in a large improvement. The fins evidently obstruct the liquid flow from the edge of the outlet to the tip (Figure 7b), initiating slug flow at relatively low uGS. Removing the fins, however, enhances the liquid drain along the edge of the outlet that results in a shift of the flow transition toward higher uGS. It can also be seen that removing the fins for longer distance, here 7 cm, has no further effect on the flow transition. From visual observation it is clear that slugging is no longer initiated at the liquid outlet but at the point where the fins start. In another experiment a funnel was connected to the bottom end. The function of the funnel is to drain the liquid from the tube and to let the velocity profile in the gas develop gradually. Figure 11 shows that this geometry leads to an enlargement of the wavy annular flow region. From visual observation it was clear that
CW ( 5%
uGS interval
0.781 ( 0.005 0.858 ( 0.007 0.779 ( 0.007 0.679 ( 0.006 0.824 ( 0.011 0.888 ( 0.01 poor fit 1.176 ( 0.015
1.1 < uGS < 2.0 1.4 < uGS < 2.8 0.8 < uGS < 2.6 uGS < 0.8 0.6 < uGS < 1.2 0.4 < uGS < 2.0 1.8 < uGS < 2.6
the transition was still initiated at the bottom of the tube. A special outlet was then made in which gas was introduced through a capillary inserted into the bottom end while the liquid was drained through a wad of mineral wool. The results in Figure 11 show that the transition from annular to slug flow shifted dramatically to higher uGS. Moreover, it was observed that the transition was no longer initiated at the bottom end of the tube but that bridging of the waves inside the tube caused slugs to be formed. The transition can thus be interpreted as the maximum attainable limit for countercurrent annular flow in an IFT. Evaluation of Wallis Correlation. The flooding correlation of Wallis describes the trend of the exit flooding for both the finned and unfinned tubes well. Dependent on the outlet geometry, a flooding parameter CW can be determined to describe the data of the experiments with the beveled outlets within 10% accuracy. Especially at higher uLS the correlation describes the flooding data very well. It can be seen in Figure 11 that the trend of the results with the funnel is different from the trend predicted by the Wallis equation. The modified outlet effect causes this tube to behave slightly different from the beveled ones. The fitted flooding parameters together with the validity range are summarized in Table 3. The value of CW ranges from 0.679 to 1.176, for the least and most ideal outlet geometry, respectively. This is close to the range of 0.725-1.0 that has been reported in the literature (McQuillan and Whalley, 1985). From the results in Table 3, it can be deduced that for a preliminary design of an IFM reactor with a beveled liquid outlet, a value of CW ranging from 0.7 to 0.85 can be used as a good estimation of the maximum operation limit for organic liquids. It is expected that the surface tension of the liquid will influence the flooding limits in these small capillary tubes. Liquids with high surface tension, like water, tend to bridge together and initiate flooding at lower velocities. In these cases a modified Wallis correlation, among others proposed by Zapke and Kro¨ger (1996), or a Kutateladze-type correlation (McQuillan and Whalley, 1985) would be more appropriate. Effect of Gas Inlet on Pressure Drop in Single Phase Flow. Single-phase experiments in which the pressure drop was measured as a function of uGS were performed to investigate the effect of the outlet geometry (gas inlet) on the momentum transfer. The influence of R was studied in 1, 0.5, and 0.25 m long tubes having an internal diameter of 3.5 mm. It can be seen in Figure 12a that pressure drop caused by the in- and outlet (∆Pe) increases with increasing R. The contribution to the total pressure drop, which is about 150 Pa at uGS ) 3.0 m/s, is significant. The influence of R on the pressure drop can also be expressed qualitatively in terms of B2. Equation 11 was therefore fitted to all experimental
3728 Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998
Figure 12. Contribution of the entrance and exit effects to the pressure gradient (∆Pe/L) for an unfinned tube (d ) 3.5 mm, L ) 1 m) with different outlet angles (R). Dashed lines represent eq 11 fitted to the data (a). Outlet characteristics constant versus outlet angle (b).
data (L ) 0.25, 0.5, 1 m) for every R using the method of least squares. The values of the fit parameters B3 and B4 are -0.0336 and 0.156, respectively. An example of the fit is shown in Figure 12a. The dependency of B2 on R is shown in Figure 12b. As expected, B2 increases with increasing R. For an R of 80°, however, the increase of B2 is larger than for lower R. Apparently, more energy is dissipated at higher R. Although it is expected that with increasing R (larger inlet gap) the pressure at the gas inlet decreases because of the Bernoulli effect, it seems that the contraction losses and disturbances of the velocity profile overall lead to a higher pressure drop. In the case of two-phase flow, where momentum is transferred from the gas phase to the liquid film, this leads to an increased interfacial shear rate that can promote slug formation. A balance between enhanced liquid drainage and the increased shear rate at higher R apparently determines the optimum at which countercurrent annular flow can occur. From Figures 10 and 12b it can be concluded that this is close to R ) 70°. The single phase frictional pressure drop in an IFT was calculated numerically with the film flow model by assuming zero liquid hold-up. The geometrical constant B1 was determined by matching eq 10 with the numerical results. The value of B1 was found to be 14.4, which is lower than the value of B1 for the circular tubes. This might seem a little bit strange since the pressure drop in the IFT is much higher. However, to compare the
Figure 13. Lines of constant pressure drop (∆P/Pa) calculated with the countercurrent film flow model and measured experimentally in a 1.0 m long (a) finned and (b) unfinned cylindrical tube (d ) 3.5 mm). In addition the lines of constant liquid holdup (L) calculated with the model, the measured flooding points, and the flooding curve fitted with the correlation of Wallis are shown.
pressure drop, B1 should be combined with the hydraulic diameter (B1/Dh2 in eq 10). For the finned tubes B1/ Dh2 ) 4.7 mm-2 while for circular tubes this parameter is 1.3 mm-2. The pressure drop will therefore be higher in the finned tubes. In addition to B1, the fit parameters that determine the entrance effect, B2 and B3, were determined from single phase experiments in a 1 m long IFT with R ) 70°. The values of B2 and B3 were found to be 6.2 and 0.071, respectively. Evaluation of Countercurrent Film Flow Model for Two-Phase Flow. Assuming similar pressure drop contribution of the entrance in two-phase flow as in single-phase flow, lines of constant pressure drop have been calculated with the countercurrent film flow model. In Figure 13 the model results are shown together with the experimental results for a finned (R ) 70°) and an unfinned tube (R ) 70°). Since the model is only valid in the wavy annular flow region, comparison can only be made on the left-hand side of the flooding curve. It can been seen in both figures that the calculated
Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3729
Figure 14. (i) Comparison of flooding curves of a few finned tubes with different liquid outlet geometries (closed symbols) and two unfinned tube (open symbols). (ii) Typical operation conditions of industrial (HDS) trickle bed reactor (TBR).
pressure drop is in good agreement with the experimental data. Figure 13 also shows lines of constant liquid hold-up (L) calculated with the countercurrent film flow model. Since in the wavy annular flow regime the pressure drop calculations are in good agreement with the experiments, it is reasonable to assume that the values of the calculated L approach the actual L. It can be seen in Figure 13 that the gas flow rate has almost no effect on the L. This indicates that momentum transfer from the gas to the liquid inside the tube is almost negligible; thus flooding caused by partial flow reversal can be excluded. From Figure 13b and Figure 10, it can be deduced that entrance flooding takes place at a certain L. This supports the idea that entrance flooding is a liquid-controlled transition. On comparing of the L lines in the Figure 13, it is evident that for a certain uLS the L in finned tubes is, as expected, higher than that in unfinned tubes. Comparison Between Finned and Unfinned Tubes. Figure 14 shows the flooding curves of the 3.5 and 1.75 mm diameter unfinned tubes and some finned tubes. On comparison of the beveled finned tubes with the unfinned tubes, two distinct differences can be observed. At low uLS, where the transition is primarily determined by the outlet (exit flooding), the wavy annular region is larger for the unfinned tubes while at higher uLS, where inlet effects determine the flooding mechanism for unfinned tubes (entrance flooding), the finned tubes are more appropriate for obtaining wavy annular flow. Visual observation demonstrated that for the finned tubes the interfacial waves at the liquid inlet flow down irregularly separated by the fins. Since expansion of the waves in an azimuthal direction is not possible, the waves are restricted to a single subchannel so that a total closure caused by coalescence of the waves will happen less frequently. Hence, for the finned tubes entrance flooding was of no importance within the measured flow range. The fins obviously stabilize the two-phase flow at higher uLS. Implication for an IFMR. With respect to the application of an IFMR, the stabilization of the flow as discussed in the previous paragraph can be of major importance. In an industrial scale reactor, in which the
liquid is probably not distributed evenly all the time, slightly nonuniform distribution can lead to a situation in which the channels are not operated in one point but on an isobaric line of different uLS and uGS (shown in Figure 13). For an unfinned monolith this can easily lead to entrance flooding since this transition is very sensitive for fluctuations in uLS. In an internally finned monolith, entrance effects play no significant part so that flooding will not occur. To facilitate a comparison with a trickle bed reactor (TBR), Figure 14 also indicates the range of uLS and uGS corresponding with the operation limits of an industrial TBR for HDS applications at reaction conditions. It follows that an IFT allows countercurrent operation in the annular flow regime at industrially relevant velocities. Although the outlet geometries with the funnel and the capillary are not (easily) applicable in industry, they have proven that the intrinsic flow transition of an IFT takes place at relatively high velocities so that an industrial IFMR can be operated comfortable within the stable annular flow region. The beveled outlet with the fins removed seems the easiest solution. In practice, this configuration can be realized by aligning a finned monolith block with a conventional unfinned monolith having the same cell density and V-grooves milled at the bottom end. An IFMR with just a beveled outlet could be a good alternative for a TBR with a low throughput. 5. Conclusion An initial study in small tubes, corresponding to the channels of an internally finned monolith, has shown that countercurrent gas-liquid flow is feasible at industrially relevant flow rates. The liquid and gas velocities that are typical in commonly applied industrial reactors are smaller than the velocities at which flooding occurs in these tubes. Careful design of the bottom end of the tube is however necessary to cut back the effect of the liquid outlet on the flooding behavior. A comparison with a circular unfinned tube has shown that the fins stabilize the flow, especially at higher liquid velocities. It appears that the structured catalyst creates a structured two-phase flow that lessens the tendency to form slugs. The flooding results with decane and air showed that the correlation of Wallis is a suitable criterion to estimate exit flooding in both finned and unfinned small diameter tubes. It is however expected that for high surface tension liquids, like water, a modified correlation has to be used. In the wavy annular flow regime, the pressure drop of the two phase flow can be calculated with a falling film model assuming a laminar falling film and a countercurrent laminar gas flow. The extra pressure drop caused by the gas inlet can be estimated from single-phase experiments. Finally, it can be concluded that these promising results show that it is interesting to further investigate the possibilities of applying monolithic structures for countercurrent gas-liquid catalytic processes. Nomenclature A ) cross section available for gas and liquid flow (m2) B1 ) geometric frictional constant B2 ) empirical constant dependent on liquid outlet geometry
3730 Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 B3 ) geometric empirical constant B4 ) geometric empirical constant CW ) flooding parameter of Wallis D ) tube diameter (m) Da ) apparent tube diameter ()[4*A/π]1/2) (m) Dh ) hydraulic tube diameter (m) ∆P ) pressure drop ∆LP ) modified pressure drop (eq 3) (kg/m3) p ) pitch between channels of a monolith (m) f ) Fanning friction factor g ) gravitational acceleration (m/s2) K1 ) constant (m/s) K2 ) constant (m/s) L ) length (m) mW ) flooding parameter of Wallis N ) number of cells r ) radial coordinate (m) R ) tube radius or position of the film (m) Re ) Reynolds number u ) velocity [m/s] u* ) dimensionless velocity (eq 2) x ) Cartesian coordinate (m) y ) Cartesian coordinate (m) Greek Symbols R ) angle of liquid outlet L ) liquid hold-up µ ) viscosity (Pa s) F ) density (kg/m3) Subscripts e ) exit/entrance effect f ) frictional or film F ) fluid G ) gas L ) liquid S ) superficial s1 ) symmetry line 1 s2 ) symmetry line 2 t ) total Abbreviations in Figure 3 B ) bridging and disintegration DW ) disturbance wave annular film flow E ) entrainment and wave breakup F ) film flow reversal
G ) gas upflow H ) hanging film L ) liquid downflow S ) slug flow SF ) smooth film TW ) turbulent wave annular film flow
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Received for review November 25, 1997 Revised manuscript received April 30, 1998 Accepted May 5, 1998 IE9708586
