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Cross flow and heat transfer of hollow-fiber tube banks with complex distribution patterns and various baffle designs
International Journal of Heat and Mass Transfer 147 (2020) 118937
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Cross flow and heat transfer of hollow-fiber tube banks with complex distribution patterns and various baffle designs Kui He a, Li-Zhi Zhang a,b,⇑ a Key Laboratory of Enhanced Heat Transfer and Energy Conservation of Education Ministry, School of Chemistry and Chemical Engineering, South China University of Technology, Guangzhou 510640, China b State Key Laboratory of Sub-tropical Building Science, South China University of Technology, Guangzhou 510640, China
a r t i c l e
i n f o
Article history: Received 31 May 2019 Received in revised form 9 October 2019 Accepted 21 October 2019
Keywords: Fiber bank optimization Baffle design Cross flow Heat transfer
a b s t r a c t The pressure drop and heat transfer of hollow-fiber tube banks with various packing patterns and baffle designs are studied by combining pattern recognition technology and computational fluid dynamics. The fluid passes across the fiber bank. The inlet flow Reynolds number ranges from 20 to 180. The pressure drop and heat transfer coefficients of the contactors are highly correlated to the topological characteristics of the fringe of the fiber bank. A single parameter w, which summarizes this information, is calculated using pattern recognition technology. The pressure drop and Nusselt number of a contactor exhibit power law relations with w. Interestingly, this phenomenon is independent of the baffle design, which affects only the magnitude of the pressure drop and the heat transfer coefficient. Generally, a higher w results in contactors with worse performance. For w > 0.6, the wall effect is negligible, and all contactors show similar performance regardless of the fiber pattern. The comprehensive performance parameter £ is approximately 0.6 for all contactors with various patterns. An ‘‘insert’’ strategy that is used to reduce w is demonstrated. The results show how to optimize the structure of contactor for industrial production. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Cross flow over tube banks is a classic application of fundamental fluid dynamics. It is widely used in heat exchangers [1–4] and other engineering applications [5–8], though it differs from each other in details. Tube banks can be roughly classified based on the scale of the tubes, the flow state in the tube bank, and the spatial distribution of the tubes. There are three classes. The first class are tube banks with a uniform pattern. The tubes are usually arranged in a rectangular pattern (inline pattern) or a triangular pattern (staggered pattern). A tube bank is geometrically characterized by the tube diameter (D), stream-wise row pitch (SL ), and transverse row pitch (ST ). The tubes of the first class usually have tubes with diameters from several millimeters to several meters. Typically, the flow state for this class is fully turbulent or transitional. It is the most traditional and extensively studied class. Studies of the heat transfer due to the features of cross flow over these tube banks date back to the 1920s. Earlier experiments found that ⇑ Corresponding author at: South China University of Technology, School of Chemistry and Chemical Engineering, Key Laboratory of Enhanced Heat Transfer and Energy Conservation of Education Ministry, Wushan Road, Guangzhou 510640, China. E-mail address: [email protected] (L.-Z. Zhang). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118937 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
the coefficients for the flow are determined by the geometrical distribution of the tubes, the Reynolds number, and the Prandtl number [9–13]. The second class are tube banks with small diameters, in the sub-micrometer range. They have very fine fibers [13,14]. Unlike the first class, which has uniform fiber distribution patterns, these tube banks often have complex patterns and there may be 10,000 or more tubes. For this class, the flow state is in the creeping flow region (Re 1). Therefore, analytical methods are used to obtain the permeability [15,16]. For the third class, which are between the above two in scale, the diameters are in the range of micrometers to millimeters. They are usually used as hollow-fiber membrane contactors, which are a type of mass separation device [17–20]. The structure of a hollowfiber membrane contactor is like a tube-shell heat exchanger. In a hollow fiber contactor, typical Reynolds number ranges from decades to several hundred. There is complex flow separation among the fibers. Due to the soft and fine scale, making a uniform fiber pattern relies on specially designed frames. It can be very timeconsuming to plug numerous fibers into a frame. These designs are applicable only for lab-scale experiments since they are not suitable for mass production. In industry, the fiber tubes are first formed into banks and then made into a contactor. Fig. 1 shows the basic packing process for contactors that is currently widely
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Nomenclature a c d d D G h k m n N Nu PD pd Re S T/t u
air specific heat capacity of fluid (j kg1K1) length scale (m) mean diameter of fiber (m) diameter of contactor or width of channel (m) gravity constant (m2 s1) convective heat transfer coefficient (W m2 K1) kinetic energy, J power exponent, mass flow rate (kg s1) power exponent number Nusselt number averaged packing density local packing density Reynolds number length scale (mm) or area (m2) temperature (K) vector velocity (m s1)
Greek letters a ratio of the area of fiber bank region and the cross-section area of contactor c ratio of the area of cut-through cavity and the cross-section area of contactor D differential e dissipation rate / a comprehensive parameter, /=ðNTU rp =DP rp Þ=ðNTU inl = DPinl Þ k heat conductivity (W m1K1) l dynamic viscosity (Pa s) q air density (kg m3) r the non-dimensional open width s shear stress
n w
sensible cooling efficacy ratio of the area of cut-through cavity and the area of fiber banks
Subscripts a air av averaged c contactor or channel cut cut-through dp pressure drop f fiber fw wall of fibers i 1, 2, 3. . . is inserts lm logarithm m membrane, mean min minimum in inlet of sub-channel inl inline out outlet of sub-channel ow open width p spanwise rp random pattern rs removable inserts s solid, superficial T transverse T turbulent, total urs unremovable inserts w width, water, wall x, y, z streamwise, normalwise, spanwise
There are few studies on the cross flow over the third class of tube banks, for two reasons. First, the distribution of fibers in a bank is neither uniform nor periodic. As can be seen in cross sections, the fiber patterns vary from case to case. Second, the fiber banks are often enclosed in a shell. Therefore, there is a potential wall effect. Thus, it is not easy to derive universal laws for this class. Our recent previous study shows that there are correlations between the pressure drop, heat transfer coefficient and the topological features of hollow-fiber patterns [21]. The topological features are usually determined using pattern recognition technology. The pressure drop and heat transfer are calculated using computational fluid dynamics (CFD). However, the wall effects are still unknown. So, in this paper, this problem is addressed. This study is organized into three parts. The first section considers how the correlations are affected by the shell design and the wall effect. The second section derives universal law for these tube banks. Finally, we give our conclusions and suggestions for industrial production.
2. Methodology Fig. 1. Production of a commercial membrane contactor.
used. There are four steps: packing, gluing, drying, and cutting. In the packing process, the fibers are gathered into clusters by hand. It can be seen that the fibers are randomly distributed inside a shell.
2.1. Producing artificial fiber distribution patterns and determining their topological characteristics As described in our previous study [21], the first step is to generate an artificial fiber distribution pattern, which is imported into a CFD model. The model calculates the pressure drop and heat transfer. Fig. 2 shows the procedure for generating an artificial
K. He, L.-Z. Zhang / International Journal of Heat and Mass Transfer 147 (2020) 118937
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Fig. 2. The two steps in producing a fiber bank and some generated patterns.
fiber pattern that is like a real contactor. The small circles represent fibers. The large voids found in real contactors are generated by setting forbidden packing regions (marked with a light color) in a round region (a shell region). The diameter of the round region is 25 mm, which is the same as the inner diameter of contactors previously manufactured by us. There are two typical types of large void. The first are the cavities surrounding the fiber banks, which are called cut-through cavities. The other type of cavity are isolated cavities, which are isolated by the surrounding fibers. In places where the hollow fibers are densely populated, the void space is small and nearly uniform. These uniform voids are generated by packing circles together [22–24]. The coordinates of the center of each circle are randomly generated in the packing region. The circles are not allowed to overlap each other. The minimum space between two circles is set at 0.1 mm. Fig. 2 also shows ten generated patterns. Typical cut-through cavities are shown in patterns P6–P10 and patterns P3–P5 have isolated cavities. The generated fiber banks are imported into 2D CFD models. As depicted in Fig. 3, the patterns plus different shells form various contactors. The geometric centers of the packing region (containing a fiber bank) coincide with those of the shells. Fig. 3 shows six groups of shells with different geometries and scales. They represent two typical geometries. Groups a, b, c, and d are cylindrical shells. The open width increases from 10 mm to 24 mm. Groups e and f are box shells. The detailed parameters of these shells are
listed in Table 1. Some features of these contactors are calculated using pattern recognition technology: the average packing density (PD), the packing density of the fiber banks (pd), the ratio of the area of the cut-through cavities and the cross-sectional area of the contactor (c), the ratio of the area of the cut-through cavities and the area of the fiber banks (w), and the ratio of the area of the fiber banks and the cross-sectional area of the contactor (a). These are defined as follows:
PD ¼
npd2 4S0
ð1Þ
pd ¼
npd2 4Sfb
ð2Þ
c¼
Scut S0
ð3Þ
w¼
Scut Sfb
ð4Þ
a¼
Sfb S0
ð5Þ
Sfb ¼ S0 Scut
ð6Þ
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Fig. 3. Forming a contactor and various baffle designs.
Table 1 Geometric parameters used in the study. Parameter
ST (mm)
Sp (mm)
Dc (mm)
df (mm)
D (mm)
Group Group Group Group Group Group
10 15 20 24 30 36
23 20 15 7 15 15
36 36 36 36 36 36
1.45 1.45 1.45 1.45 1.45 1.45
25 25 25 25 25 25
a b c d e f
where n, d, S0 , Scut , and Sfb are the number of packed circles, the mean diameter of the circles (fibers), the cross-sectional area of the total packing region, the area of the cut-through cavities, and the area of the fiber bank region, respectively. Of these, n, d, and S0 are already known and are constant for all contactors. Scut (or w) is calculated for both real and artificial contactors as following: (i) The pictures are transformed into binary ones. The fibers are identified by a pattern recognition algorithm and forms points set of the centers of the fibers. The fibers recognition algorithm is mainly based on Hough transform [25]; (ii) the Delaunay subdivision [26] is conducted on the centers of the fibers and boundary fibers. The boundary fibers are packed on the wall of a shell. In this way, the complex porous structure is reflected as a net-like topology. It is shown in Fig. 4 that the Scut is the total area of the marked triangles. The triangles making up Scut contain boundary fibers. The area of the other triangles is included in Sfb . A MATLAB code is developed to conduct these calculations. In summary, the code firstly reads the pattern figures, recognizes the fibers, conducts the Delaunay subdivision, and then calculates the sum of the areas of the marked triangles. The code will be integrated with hardware (camera) in future. Fig. 5 shows the parameters defined for different groups of contactors. The parameter a ranges from 0.69 (P5) down to 0.54 (P6). Correspondingly, P5 has the lowest packing density and the highest mean area of voids among the fibers. P6 is the opposite one. The pressure drop and heat transfer coefficient are determined by the ratio of fluid flowing through the cut-through area and the fluid flowing through the fiber bank (which is a porous medium). It is hard to estimate them precisely. Here, we consider that
Fig. 4. Delaunay subdivision of the void spaces in a contactor (CP6): (a) An artificially created contactor, (b) the Delaunay subdivision of the contactors. 1-a wall boundary circle; 2-the inner shell wall of a contactor.
they depend solely on the ratio for the area of the cut-through cavities c and the ratio for the packing area a. c is largest for P6 and smallest for P1. If c is larger, more fluid bypasses the fiber banks and the contactor has a lower heat transfer coefficient. The parameter w is the ratio of the area of the cut-through cavities and the area of the fiber banks. It is a measure of the bypass effect. Other parameters may also be used to characterize the complex porous structure among the fiber banks, but c and w are the easiest to obtain with pattern recognition technology because the identified triangles contain special vertexes (boundary points).
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Fig. 5. The parameters used to characterize contactors: (a) a and c, (b) w.
2.2. The CFD modelling The CFD models are based on the following simplifications: (i) the 3D physical geometry is simplified as a 2D problem due to the cross flow, (ii) the heat loss from the contactor to the environment is negligible, and (iii) all the walls are smooth. In a real application, the contactors are arranged in a rectangular channel. The experimental setup is described in our previous work [21,27]. The domain of a CFD model consists of two parts: the channel and the contactor. It is shown in Fig. 6(a). The lengths of the inlet and outlet regions are set to 25D, where D is the width of the wind channel, about 36 mm. The mean diameter of the fibers is about 1.5 mm. The corresponding number of packed fibers is 100. The average packing density is 0.35. In the fiber bank region, the coordinates of the fiber (circle) centers are abstracted from the artificially generated patterns. A non-uniform mesh is generated for the fiber bank region. There is a more refined mesh between two fibers. The enhanced wall treatment is used for all cases. Fig. 6(b) is a magnified view of the mesh in the fiber bank. There is a uniform rectangular mesh in the inlet and outlet regions. In the normal direction to the wall, there is a compressed mesh (Fig. 6(c)). The commercial CFD code Fluent 14.5 was used to simulate the fluid flow and heat transfer for these artificially generated
contactors. The conservation equations of mass, momentum, and energy are as follows: Mass conservation:
! ! r qu ¼ 0
ð7Þ
Momentum conservation:
! !! ! ! r q u u ¼ r P þ r s þ q! g
ð8Þ
Energy conservation:
!
! !
!
! ! qcp ! u rT þ u rP ¼ r krT þ s u
ð9Þ h!
!
i
! u þ r u T . The s ¼ l r! above equations are closed by the RNG k e turbulence model. The RNG k e turbulence model is suitable for a flow with recirculation where, for incompressible viscous flow,
and the separation of the flow over a tube bank. The latter cannot be captured by the standard k e turbulence model [28]. They are captured with the additional strain rate and the turbulent viscosity term in the RNG k e model. We used the RNG k e turbulence model based on our previous work. The general form of the RNG k e turbulence model in Cartesian coordinates is
@ qkuj @ ¼ @xj @xj @ qeuj @ ¼ @xj @xj
lþ
lt rk
lþ
lt re
@k @xj
@e @xj
þ Pk þ Pkb qe
e
ð10Þ
e2
þ C 1e ðPk þ C 3e P eb Þ C 2e q k k
ð11Þ P k ¼ l t S2
lt ¼ C l q
ð12Þ 2
k
ð13Þ
e
In the current study, the buoyancy effect is neglected, so Pkb ¼ 0 and P eb ¼ 0. The other constants are as follows:
1=2 C l g3 ð1 g=g0 Þ C 2e ¼ C 2e þ ; g ¼ Sk=e; S ¼ 2Sij Sij ; 1 þ bg3 1 @uj @ui ; C l ¼ 0:0845; C 1e ¼ 1:42; þ Sij ¼ 2 @xi @xj Fig. 6. The computational domain and mesh: (a) The computational domain of the channel containing a contactor, (b) the zoomed-in mesh at region 2 of fiber bank, (c) the zoomed-in mesh at region 1 of channel inlet.
C 2e ¼ 1:68;
g0 ¼ 4:38:
rk ¼ 0:7194; re ¼ 0:7194; b ¼ 0:012; and
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2.3. Boundary conditions and solution controls
3. Results and discussion
The boundary conditions are set as follows. The incompressible air is considered as a working fluid with constant thermo-physical properties. This is due to the small temperature changes in real contactors. The range of Re selected for the numerical study is between 20 and 180, which is commonly used in real situations. A larger inlet velocity generates a higher pressure drop. The
3.1. Pressure drop and Nu for various contactors
Reynolds number (Re ¼ qdV c =l) is based on the channel air velocity and the average outer diameter of the fibers. The corresponding velocity of the air at the inlet of the contactor is between 0.2 and 1.8 m s1. On the outer wall of the fibers and on the wall of the channel, the non-slip boundary condition is imposed. The temperature of the air at the inlet is 280 K. The outer wall of the fibers is kept at a constant wall temperature of 305 K. The heat loss via the wall of the wind channel and through the wall of the contactor shell is ignored. To facilitate comparisons, thermal isolation is assumed. A rubber gasket is used to seal the gap between the channel wall and the contactors, and an isothermal and non-slip boundary condition is used. The main fluid properties are the conductivity of air ka = 0.026 W m1 K1, the kinematic viscosity of air ma = 1.5 10-5 m2 s1, and the density of air, 1.22 kg m3. These are consistent with those in our previous study [21]. The governing equations listed in the last section are solved on a multi-block mesh. A second-order upwind scheme is implemented for the convection terms. The semi-implicit method for a pressure-linked equation (SIMPLE) algorithm [29] is used for the pressure and velocity coupling. The residuals for the continuity and energy equations are of the order of 10-6 and 10-10. The simulations are accelerated by the multi-grid method [30].
2.4. Validation of the numerical model The above numerical model was previously validated by simulating both standard inline tube banks and randomly picked real contactors [21]. The simulation results were compared to the corresponding experimental data. A mesh-independent test was also carried out. Two parameters of a contactor are used for this: the pressure drop and the average Nusselt number. The average Nu is
Nu ¼
dh ka
ð14Þ
where ka ; h;and d are the thermal conductivity of air, the convective heat transfer coefficient, and the averaged diameter of a fiber, respectively. We use Newton’s law of cooling:
Sm hDtm ¼ ma C p ðt out tin Þ
Dt m ¼
ðtout tfw Þ ðtin t fw Þ t fw Þ ln ððttoutt Þ in
ð15Þ
ð16Þ
fw
where t out ; t in , and t fw are the average temperatures at the inlet and outlet of the test region and the outer wall of the fibers. Sm ; C p and ma are the total membrane area, the heat capacity of air and the mass flow rate of air, respectively. A detailed comparison shows that the predictions of the CFD models agree well with the experimental results [21]. The maximum difference for the pressure drop is 17% and for Nu, 19%. A similar mesh density, turbulent model and boundary conditions were used in this study and these models were not specifically validated.
Fig. 7 shows the pressure drop of 60 different contactors. The contactors are classified into six groups. For instance, Ema1 is based on shell group a and pattern P1. In each group, there are ten studied cases and 9 flow Reynolds numbers. Therefore, 540 simulations are used to generate the data. Each simulation last about 1 h. These constants are obtained by B-spline regression of these data. The lines in the figures are the pressure drop of contactors with fibers uniformly arranged in a rectangular pattern, which are used as a reference. In a rectangular pattern, the pitch of the fibers is equal in the horizontal and vertical directions, being about 1.5d. Therefore, the local and average packing densities of the reference contactor are 0.35. As mentioned before, for groups a to d, the inner size of the shell is constant. However, the open width of the shell (Sw ) increases. This means that the effect of the contraction of the inlet flow decreases. As the open width is 10 mm, the pressure drop for different contactors is high (compared to the pressure head of 600–700 Pa for a normal 500 W wind blower), ranging from 200 Pa to 350 Pa at Re = 200. The pressure drop is always lower than that for contactors with an inline pattern. As the open width increases, the amplitudes of the pressure drop reduce but there is more variability in the data. In summary, although the pressure drop for contactors with the same pattern varies from case to case, there are some common features for different groups. For instance, contactors with P1 have a pressure drop that is higher or slightly lower than contactors with an inline pattern. Moreover, contactors with P10 have the lowest pressure drop when there is no strong flow contraction effect (Sw = 15 mm, 20 mm, or 24 mm). Groups e and f have a box-shaped shell and the two walls are further apart. The effect of the wall reduces as the inner volume of the shell increases. Unlike groups a to d, there is almost no flow contraction effect and the inlet flow is almost uniform. Further, the pressure drop can be ranked for the models in descending order as Eme1 > Eme4 > Eme3 > Eme5 > Eme2 > Eme7 > Eme8 > Eme9 > Eme6 > Eme10. This consistent descending order may be because there is no flow contraction effect and the effect of the wall can be ignored. Thus, the pressure drop mainly depends on the fringe shape and local packing density of a fiber bank, which will be discussed further in Section 3.2. The Nusselt numbers are shown in Fig. 8. There are more common features among the different groups. For groups a, b, and c, the descending order of Nu is almost the same: P1 > P5 > P4 > P3 > P8 > P2 > P7 > P9 > P10 P6. For groups d, e, and f, the descending order for Nu is P1 > P5 > P4 > P3 > P2 > P7 > P8 > P9 > P6 > P10. The top four high Nusselt numbers are for contactors with patterns P1, P5, P4, and P3. As the wall effect decreases, the order changes for contactors with P2, P7, P8, and P9. The contactors with P1 have the highest Nu. The contactors with P6 and P10 have the lowest Nu. To give an intuitive understanding, the velocity and temperature contours of typical contactors with shell c and shell e are exhibited in Fig. 9. For the contactors with P1, the hollow fibers are distributed in a ribbon in the vertical direction. The distribution of the flow inside the contactors is uneven. The maldistribution of the flow occurs in two classes. The first class, referred to as the inner flow maldistribution, is due to the uneven distribution of fibers. This maldistribution inside the fiber bank is similar in contactors with P1 and P10. The second class of flow maldistribution originates from the complex flow channel formed by the outer fringe of a fiber bank and the shell wall. For different designs of
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Fig. 7. Pressure drops of different contactors: (a)-(f) pressure drops of contactors for group A-F.
the shell, the velocity contours show distinctive features with the second class of maldistribution. For contactors with P1, the velocity contours are almost symmetrical, despite the change of shell. By comparison, the contactors with P10 have an unsymmetrical velocity contour. The high-velocity region is shown for the cut-through cavity region. The temperature contours show similar features. Thus, fiber distribution patterns like P6 and P10 should be avoided since the fibers are concentrated in a certain region and there are large cut-through cavities. These features can easily be identified by pattern recognition technology.
3.2. The effects of shell design The general effect of the shell wall (baffle design) is shown in Fig. 10(a) and Fig. 10(b), which shows the correlation of the pressure drop and Nu with Sw . As Sw increases, the pressure drop for contactors with various patterns reduces monotonically. The rate of decrease varies from case to case. The rates for contactors with P1 decrease more slowly than for other patterns. By contrast, the rate of decrease for contactors with P10 is the largest. The pressure drop of contactors with various patterns decreases almost linearly.
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Fig. 8. The Nusselt numbers of different contactors: (a)-(f) Nusselt numbers of contactors for group A-F.
This feature is not seen in the correlation of Nu with Sw . As Sw increases from 10 mm to 24 mm, Nu reduces slightly or is mainly unchanged. It suddenly falls when Sw is increased to 30 mm. The differences among the contactors with various patterns are more distinctive, compared to the pressure drop. The ranking of Nu is very clear and consistent, in descending order: P1 > P5 > P4 > P3. The Nu for contactors with P2 and P8 are very close. The other similar pairs are P7 and P9, and P6 and P10. Fig. 10(a) and Fig. 10(b) may be useful for manufacturers. Increasing the open width of shells leads to a decrease in the pressure drop while the heat transfer coefficient is almost unchanged for Sw 24 mm. We suggest using an open width of between 20 mm and 24 mm. If the open width is increased, the poor distribution patterns leads to a reduction in heat and mass transfer.
As mentioned before, these poor patterns show common features that can easily be identified by pattern recognition technology. Here the pressure drops and Nusselt numbers of various contactors are correlated to a single parameter w. These parameters quantitatively characterize the ratio of fluid flow via the cutthrough cavity and the fiber bank. Fig. 10(c) and Fig. 10(d) show the corresponding correlations. rw is the ratio of Sw and the inner diameter of a contactor (25 mm). Interestingly, although rw ð¼ SDw Þ changes from 0.4 to 1.4, the contactors with various fiber distribution patterns show similar correlations. The pressure drop and Nusselt number decrease with w in power laws:
DP ¼ C1 Rem1 wn1 m2
Nu ¼ C2 Re
n2
w
ð17Þ ð18Þ
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to the accuracy reduction in the predictive model. Taking more details of the pattern into the correlations by advanced technology, such as machine learning, may increase the accuracy of the predictions. However, the simplicity of the correlations is preferred, so it is not considered in this study. 3.3. A universal law for contactors with different patterns In the last section, it was shown that changing Sw has a significant effect on the pressure drop and heat transfer. However, the similarity found in correlations (especially in Nu) indicates that there may be a universal law for contactors with various patterns. To abstract the universal law, it is necessary to introduce a comprehensive parameter to evaluate the performance of a contactor:
£¼
NTU rp =DPrp NTU inl =DPinl
ð19Þ
where NTU rp and DPrp are the Number of Heat Transfer (NTU) and pressure drop of contactors with an arbitrary pattern. NTU inl and DP inl are the NTU and pressure drop of the contactors with an inline pattern. The NTU is defined as follows.
NTU ¼
hSm cp ma
ð20Þ
Fig. 9. Contours of velocity and temperature for contactors with a good pattern and a poor pattern (P1 & P10): Velocity contours of contactors for P1 & P10, where the velocity is scaled by Vmax of (a) 6.3 m s1, (c) 2.7 m s1, (e) 10 m s1, (g) 6.3 m s1. (b), (d), (f), (h) The corresponding temperature contours of the contactors (legend unit: K).
The constants in the correlations are listed in Table 2. The exponents m1 ; m2 and n2 are almost unchanged, at about 1.71. 0.95 and 1.45, respectively. The exponents n1 converges to around 1.2 at large rw . The convergence of the exponents indicates that the effect of wall reduces as the open width of the shell increases. The consistent correlations among different groups illustrate that the heat and mass transfer of the contactors are governed by the bypass effect. w reflects this effect. In order to test the reliability of the correlations. Three contactors with patterns PT1-PT3 and shell c are generated. The pressure drop and Nusselt number predicted by the correlation are compared to the corresponding CFD data. They are shown in Fig. 10(e) and Fig. 10(f). The w is 0.302, 0.211 and 0.173 for the PT1-PT3, respectively. The difference between the predicted result and CFD data is in the range of 3.3%-31%. The accuracy is suitable for engineering application. It must be pointed out that only considering w is to obtain simple correlations. The reason for such consideration has been given in previous study [21]. The loss of other topological information leads to accuracy reduction in the correlations. Some contactors with equal w may have different Nu and pressure drops. For example, the AP7 and AP8 have similar w approximating to 0.56. The data in Fig. 7 and 8 shows the differences. The pressure drop of AP8 overshoots that of AP7 by 8.3%-18%. The Nusselt number of AP8 overshoots that of AP7 by 11%-35%. Fig. 2 shows that the topology details of P7 and P8 are different. Apparently, the x-axial symmetry of the outline of P8 is lower than that of P7. The pressure drop of AP8 becomes slightly higher with the increase in Reynolds numbers. Stronger flow separation and flow wake are introduced by P8 (not shown). For another instance, the w of PT3 and CP4 approximate 0.18. The pressure drop of PT3 overshoots that of CP4 by 12%-25%. The Nusselt number of PT3 overshoots that of CP4 by 12%-35%. It is seen in Fig. 10 that there is large cavities among the fibers in PT3. Such cavity leads to higher packing densities at the other regions. The loss of information reflecting such large cavity features and outline features in the correlations leads
Fig. 11 shows the correlations of £ with Re ¼ q d V c =l for various contactors. For Sw ¼10 mm, 15 mm, 20 mm, and 24 mm, £ increases with Re and converges to a certain value at high Reynolds number. For Sw ¼30 mm and 36 mm, £ decreases dramatically and then converges to a constant value. Interestingly, £ is approximately 0.6 ± 0.1 at high Reynolds number, regardless of the fiber distribution pattern and shell inner volume. This is attributed to the diminishing effect of the wall. The comprehensive performance of contactors with large Sw and various patterns is actually similar. The comprehensive performance of contactors with unevenly distributed fibers reaches around 60% of that for a contactor with a regular inline pattern. For contactors with smaller Sw and various patterns, the enhancement of the comprehensive performance depends on Reynolds number and patterns. The £ of most contactors is between 0.2 and 0.6 of that for a contactor with a regular inline pattern. The contactors with P1 & P3 & P4 & P5 shows better comprehensive performance. The improvement is mainly because of the Reynolds number effect. A smaller Sw leads to a stronger flow contraction and a higher pressure inside the contactor, which makes the flow contact with the fiber more uniform. 3.4. Optimization of fiber banks of contactors The research above indicates that the performance of a contactor is mainly determined by the single parameter w. This parameter is mainly dependent on the fringe shape of the packed fiber bank and the corresponding shell design. These contactors show similar performance as a contactor with a uniform inline pattern. Considering the cost of packing, precisely controlling the distribution of fibers is not necessary. Based on the present study, an easy method of packing is suggested to avoid creating a poor contactor. Fig. 12 shows the insertion of a solid cylinder into the fiber bank of a contactor. The cylinder serves as a support and reduces the large voids in the boundary region. The cylinder could be left in the contactor or it could easily be removed from the contactor after sealing. If it is to be left in the contactor, we suggest that it is made from a cheap and light material, such as polystyrene (PS). In this case, the solid cylinder occupies some of the space in the contactor and squeezes the fibers to the rest region of the shell. If it is to be removed, then it can be made of wax. In this case, a hollow space will remain after sealing. Where should the solid cylinders
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Fig. 10. Correlations of pressure drop and Nusselt number with Sw and w: (a) Correlations of DP with Sw atRe = 140, (b) correlations of Nuwith Sw atRe = 140, (c) correlations of DP with w at Re = 100, (d) correlations of Nu with w at Re = 100. 1–6 are the regression curves for contactors with row = 0.4–1.4; (e) lines with solid box, circle and triangle are predicted pressure drop of PT1, PT2 and PT3, respectively; (f) lines with solid box, circle and triangle are predicted Nu number of PT1, PT2 and PT3, respectively. The hollow symbols are corresponding CFD data.
be placed in a contactor? Fig. 12 shows cylinders with a diameter of 10 mm inserted into different locations of a shell. The solid black circles (IS1, IS3, and IS5) and dotted circles (IS2, IS4 and IS6) represent unremovable and removable cylinders, respectively. Three typical positions are studied. The polar coordinates of the center of the circles are (0°, 0), (90°, 7.4 mm), and (180°, 7.4 mm).
In the simulations, 100 fibers were packed into the rest of the space in the contactor. The average local packing density (pd) was about 0.43. Fig. 13 shows the pressure drop, heat transfer coefficient, sensible cooling efficacy (n) and Number of Heat Transfer (NTU) of various artificially controlled fiber patterns. The sensible cooling efficacy is defined as follows.
11
K. He, L.-Z. Zhang / International Journal of Heat and Mass Transfer 147 (2020) 118937 Table 2 Exponent constants in the correlations. Constants
C1
m1
n1
C2
m2
n2
R2dp
R2Nu
row = 0.4 row = 0.6 row = 0.8 row = 0.96 row = 1.2 row = 1.44
0.0239 0.0139 0.00652 0.00235 0.00143 0.000685
1.764 1.718 1.696 1.723 1.715 1.771
0.312 0.647 0.818 0.966 1.343 1.179
0.0418 0.0275 0.011 0.00429 0.00545 0.006
0.896 0.897 0.923 0.975 1.122 1.066
1.419 1.439 1.481 1.459 1.486 1.408
0.97 0.98 0.95 0.94 0.96 0.94
0.90 0.90 0.91 0.88 0.96 0.95
Fig. 11. / for contactors with various patterns and baffle designs. (a)–(f) / for contactors with various baffle designs.
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K. He, L.-Z. Zhang / International Journal of Heat and Mass Transfer 147 (2020) 118937
n¼
Fig. 12. Contactors with various inserts. The dotted circles and solid circles represent removable inserts and unremovable inserts, respectively, where the diameter (Dis) of the cylindrical inserts are 10 mm. The center polar coordinates for inserts IS1-IS6 are (0°, 0 mm), (0°, 0 mm), (90°, 7.4 mm), (90°, 7.4 mm), (180°, 7.4 mm), (180°, 7.4 mm), respectively.
t out tin tfw t in
ð21Þ
An unremovable cylinder results in a much higher pressure drop that is more than 74–170% (¼ ðDPurs DPrs Þ=DPrs ) of that for contactors with a removable cylinder. Thus, we recommend that the insert is removed after sealing. For the contactors with removable cylinders, the different positions of the insert result in slightly different pressure drops. The insert at (180°, 7.4 mm) has the lowest pressure drop. Next are the inserts at (0°, 0 mm) and (90°, 7.4 mm). The relative difference is 7–32%. The characteristics of the cylinder and its position have only a slight effect on the Nusselt number. For inserts at (0°, 0 mm) and (90°, 7.4 mm), the unremovable cylinder results in a slightly higher heat transfer coefficient (within 25%). For a cylinder at (180°, 7.4 mm), the unremovable insert results in a slightly lower heat transfer coefficient. Fig. 13 (c) and Fig. 13 (d) show that the contactor with unremovable inserts at (90°, 7.4 mm) is of the highest sensible cooling efficacy and NTU as well as the highest pressure drop. It is considered that the presence of an unremovable insert forces more fluid to bypass the fiber bank via the boundary region, where fewer fibers are distributed. However, considering the significantly increased pressure drop, the contactor with IS5 is suggested. Fig. 14 shows contactors with artificially controlled patterns and the corresponding velocity and temperature contours. It shows that cylinders and a hollow space among the fibers induce different velocity fields. Generally, the presence of an unremovable cylinder (Fig. 14(c), (g), and (k)) results in a higher average packing density. The flow is forced to contact with the fibers, producing a high
Fig. 13. Pressure drop, Nusselt number, sensible cooling efficacy (n) and Number of Heat transfer Unit (NTU) of contactors with unremovable inserts and removable inserts: (a) DP of contactors with inserts IS1-IS6, (b) Nu of contactors with inserts IS1-IS6, (c) sensible cooling efficacy, (d) Number of Heat transfer Unit (NTU).
K. He, L.-Z. Zhang / International Journal of Heat and Mass Transfer 147 (2020) 118937
13
(2) Different baffle designs lead to quantitative changes in the pressure drop and heat transfer coefficient. However, the power law correlations are mainly unchanged and have very similar features. There are slight differences in the power exponents. (3) The parameter £ was defined to compare the comprehensive performance of contactors. When w exceeds 0.6, £ approaches 0.6, regardless of the fiber pattern and baffle design. In other words, the effects of the wall and the fiber distribution can be ignored beyond this critical value. (4) The fiber bank patterns can be improved by inserting a solid cylinder into the shell of the contactor. A removable cylinder effectively prevents the deterioration of the heat transfer coefficient and results in a slight increase in the pressure drop. The effects of the position of the insert were also discussed.
Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgments The project is supported by the Natural Science Foundation of China (NSFC), (grant No. 51706073; No.51936005) and the National Key R&D Program of China (grant No. 2017YFE0116100). It is also supported by the Key Project of Science Research Plan of Guangzhou City (grant No. 201904020027). References
Fig. 14. Contours of velocity and temperature for contactors with different inserts. Velocity contours (legend unit: m s1) for Vmax = (a) 8.7 m s1, (c) 12.1 m s1, (e) 10.2 m s1, (g) 17.1 m s1, (i) 8.5 m s1, and (k) 12.3 m s1; (b), (d), (f), (h), (j), (l) The corresponding temperature contours (legend unit: K).
velocity region. Compared to removable inserts that leave a hollow space, it has a higher pressure drop and a higher heat transfer coefficient. 4. Conclusions and suggestions The cross flow and heat transfer in hollow-fiber banks with random distribution patterns and various baffle designs have been studied. The inlet flow Reynolds number ranges from 20 to 180. A single topological parameter, which mainly depends on the fringe shape of a fiber bank, was calculated using pattern recognition technology. Good correlations between the pressure drop and heat transfer with the defined topological parameter are found for different baffle designs. A universal law is abstracted for this class of tube banks. A method to optimize fiber banks based on this novel method are also demonstrated. Following conclusions and suggestions are summarized: (1) The pressure drop and averaged Nusselt number of a contactor are highly correlated with a single topological parameter w in a power law, which is mainly determined by the shape of the fringe of the fiber bank and can easily be calculated by pattern recognition technology.
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[18] M.C. Yang, E.L. Cussler, Designing hollow-fiber contactors, AIChE J. 32 (11) (1986) 1910–1916. [19] R. Prasad, K.K. Sirkar, Dispersion-free solvent extraction with microporous hollow-fiber modules, AIChE J. 34 (2) (1988) 177–188. [20] C.Y. Feng, K.C. Khulbe, T. Matsuura, A.F. Ismail, Recent progresses in polymeric hollow fiber membrane preparation, characterization and applications, Sep. Purif. Technol. 111 (25) (2013) 43–71. [21] K. He, L.Z. Zhang, Fluid flow and heat transfer of cross flow hollow fiber membrane contactors with randomly distributed fibers: A topological study, Int. J. Heat Mass Transf. 135 (2019) 186–198. [22] B. Widom, Random sequential addition of hard spheres to a volume, J. Chem. Phys. 44 (1966) 3888. [23] B.D. Lubachesky, F.H. Stillinger, Geometric properties of random disk packings, J. Stat. Phys. 60 (5–6) (1990) 561–583. [24] G. Catalanotti, On the generation of RVE-based models of composites reinforced with long fibres or spherical particles, Compos. Struct. 138 (2016) 84–95.
[25] D.H. Ballard, Generalizing the Hough transform to detect arbitrary shapes, Pattern Recogn. 13 (2) (1981) 111–122. [26] M.V. Kreveld, M. Löffler, R.I. Silveira, Optimization for first order Delaunay triangulations, Comp. Geom. 43 (4) (2010) 377–394. [27] K. He, Q.Q. Chen, C.C. Huang, L.Z. Zhang, Fluid flow and mass transfer in an industrial-scale hollow fiber membrane contactor scaled up with small elements, Int. J. Heat Mass Transf. 127 (2018) 289–301. [28] S.M. Huang, D. Qiu, Z.H. Wu, M.L. Yang, S.H. Zeng, Y.H. Chen, Fluid flow and heat transfer across a curved hollow fiber membrane tube bank (CHFMTB): Effects of the tube deformations, Int. J. Heat Mass Transf. 116 (2018) 471–483. [29] S.V. Patankar, D.B. Spalding, A calculation procedure for heat, mass and momentum transfer in three dimensional parabolic flows, Int. J. Heat Mass Transf. 15 (10) (1972) 1787–1806. [30] P. Wesseling, An introduction to multigrid method, John Wiley and Sons, Chichester, UK, 1992.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Cross flow and heat transfer of hollow-fiber tube banks with complex distribution patterns and various baffle designs Kui He a, Li-Zhi Zhang a,b,⇑ a Key Laboratory of Enhanced Heat Transfer and Energy Conservation of Education Ministry, School of Chemistry and Chemical Engineering, South China University of Technology, Guangzhou 510640, China b State Key Laboratory of Sub-tropical Building Science, South China University of Technology, Guangzhou 510640, China
a r t i c l e
i n f o
Article history: Received 31 May 2019 Received in revised form 9 October 2019 Accepted 21 October 2019
Keywords: Fiber bank optimization Baffle design Cross flow Heat transfer
a b s t r a c t The pressure drop and heat transfer of hollow-fiber tube banks with various packing patterns and baffle designs are studied by combining pattern recognition technology and computational fluid dynamics. The fluid passes across the fiber bank. The inlet flow Reynolds number ranges from 20 to 180. The pressure drop and heat transfer coefficients of the contactors are highly correlated to the topological characteristics of the fringe of the fiber bank. A single parameter w, which summarizes this information, is calculated using pattern recognition technology. The pressure drop and Nusselt number of a contactor exhibit power law relations with w. Interestingly, this phenomenon is independent of the baffle design, which affects only the magnitude of the pressure drop and the heat transfer coefficient. Generally, a higher w results in contactors with worse performance. For w > 0.6, the wall effect is negligible, and all contactors show similar performance regardless of the fiber pattern. The comprehensive performance parameter £ is approximately 0.6 for all contactors with various patterns. An ‘‘insert’’ strategy that is used to reduce w is demonstrated. The results show how to optimize the structure of contactor for industrial production. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Cross flow over tube banks is a classic application of fundamental fluid dynamics. It is widely used in heat exchangers [1–4] and other engineering applications [5–8], though it differs from each other in details. Tube banks can be roughly classified based on the scale of the tubes, the flow state in the tube bank, and the spatial distribution of the tubes. There are three classes. The first class are tube banks with a uniform pattern. The tubes are usually arranged in a rectangular pattern (inline pattern) or a triangular pattern (staggered pattern). A tube bank is geometrically characterized by the tube diameter (D), stream-wise row pitch (SL ), and transverse row pitch (ST ). The tubes of the first class usually have tubes with diameters from several millimeters to several meters. Typically, the flow state for this class is fully turbulent or transitional. It is the most traditional and extensively studied class. Studies of the heat transfer due to the features of cross flow over these tube banks date back to the 1920s. Earlier experiments found that ⇑ Corresponding author at: South China University of Technology, School of Chemistry and Chemical Engineering, Key Laboratory of Enhanced Heat Transfer and Energy Conservation of Education Ministry, Wushan Road, Guangzhou 510640, China. E-mail address: [email protected] (L.-Z. Zhang). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118937 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
the coefficients for the flow are determined by the geometrical distribution of the tubes, the Reynolds number, and the Prandtl number [9–13]. The second class are tube banks with small diameters, in the sub-micrometer range. They have very fine fibers [13,14]. Unlike the first class, which has uniform fiber distribution patterns, these tube banks often have complex patterns and there may be 10,000 or more tubes. For this class, the flow state is in the creeping flow region (Re 1). Therefore, analytical methods are used to obtain the permeability [15,16]. For the third class, which are between the above two in scale, the diameters are in the range of micrometers to millimeters. They are usually used as hollow-fiber membrane contactors, which are a type of mass separation device [17–20]. The structure of a hollowfiber membrane contactor is like a tube-shell heat exchanger. In a hollow fiber contactor, typical Reynolds number ranges from decades to several hundred. There is complex flow separation among the fibers. Due to the soft and fine scale, making a uniform fiber pattern relies on specially designed frames. It can be very timeconsuming to plug numerous fibers into a frame. These designs are applicable only for lab-scale experiments since they are not suitable for mass production. In industry, the fiber tubes are first formed into banks and then made into a contactor. Fig. 1 shows the basic packing process for contactors that is currently widely
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Nomenclature a c d d D G h k m n N Nu PD pd Re S T/t u
air specific heat capacity of fluid (j kg1K1) length scale (m) mean diameter of fiber (m) diameter of contactor or width of channel (m) gravity constant (m2 s1) convective heat transfer coefficient (W m2 K1) kinetic energy, J power exponent, mass flow rate (kg s1) power exponent number Nusselt number averaged packing density local packing density Reynolds number length scale (mm) or area (m2) temperature (K) vector velocity (m s1)
Greek letters a ratio of the area of fiber bank region and the cross-section area of contactor c ratio of the area of cut-through cavity and the cross-section area of contactor D differential e dissipation rate / a comprehensive parameter, /=ðNTU rp =DP rp Þ=ðNTU inl = DPinl Þ k heat conductivity (W m1K1) l dynamic viscosity (Pa s) q air density (kg m3) r the non-dimensional open width s shear stress
n w
sensible cooling efficacy ratio of the area of cut-through cavity and the area of fiber banks
Subscripts a air av averaged c contactor or channel cut cut-through dp pressure drop f fiber fw wall of fibers i 1, 2, 3. . . is inserts lm logarithm m membrane, mean min minimum in inlet of sub-channel inl inline out outlet of sub-channel ow open width p spanwise rp random pattern rs removable inserts s solid, superficial T transverse T turbulent, total urs unremovable inserts w width, water, wall x, y, z streamwise, normalwise, spanwise
There are few studies on the cross flow over the third class of tube banks, for two reasons. First, the distribution of fibers in a bank is neither uniform nor periodic. As can be seen in cross sections, the fiber patterns vary from case to case. Second, the fiber banks are often enclosed in a shell. Therefore, there is a potential wall effect. Thus, it is not easy to derive universal laws for this class. Our recent previous study shows that there are correlations between the pressure drop, heat transfer coefficient and the topological features of hollow-fiber patterns [21]. The topological features are usually determined using pattern recognition technology. The pressure drop and heat transfer are calculated using computational fluid dynamics (CFD). However, the wall effects are still unknown. So, in this paper, this problem is addressed. This study is organized into three parts. The first section considers how the correlations are affected by the shell design and the wall effect. The second section derives universal law for these tube banks. Finally, we give our conclusions and suggestions for industrial production.
2. Methodology Fig. 1. Production of a commercial membrane contactor.
used. There are four steps: packing, gluing, drying, and cutting. In the packing process, the fibers are gathered into clusters by hand. It can be seen that the fibers are randomly distributed inside a shell.
2.1. Producing artificial fiber distribution patterns and determining their topological characteristics As described in our previous study [21], the first step is to generate an artificial fiber distribution pattern, which is imported into a CFD model. The model calculates the pressure drop and heat transfer. Fig. 2 shows the procedure for generating an artificial
K. He, L.-Z. Zhang / International Journal of Heat and Mass Transfer 147 (2020) 118937
3
Fig. 2. The two steps in producing a fiber bank and some generated patterns.
fiber pattern that is like a real contactor. The small circles represent fibers. The large voids found in real contactors are generated by setting forbidden packing regions (marked with a light color) in a round region (a shell region). The diameter of the round region is 25 mm, which is the same as the inner diameter of contactors previously manufactured by us. There are two typical types of large void. The first are the cavities surrounding the fiber banks, which are called cut-through cavities. The other type of cavity are isolated cavities, which are isolated by the surrounding fibers. In places where the hollow fibers are densely populated, the void space is small and nearly uniform. These uniform voids are generated by packing circles together [22–24]. The coordinates of the center of each circle are randomly generated in the packing region. The circles are not allowed to overlap each other. The minimum space between two circles is set at 0.1 mm. Fig. 2 also shows ten generated patterns. Typical cut-through cavities are shown in patterns P6–P10 and patterns P3–P5 have isolated cavities. The generated fiber banks are imported into 2D CFD models. As depicted in Fig. 3, the patterns plus different shells form various contactors. The geometric centers of the packing region (containing a fiber bank) coincide with those of the shells. Fig. 3 shows six groups of shells with different geometries and scales. They represent two typical geometries. Groups a, b, c, and d are cylindrical shells. The open width increases from 10 mm to 24 mm. Groups e and f are box shells. The detailed parameters of these shells are
listed in Table 1. Some features of these contactors are calculated using pattern recognition technology: the average packing density (PD), the packing density of the fiber banks (pd), the ratio of the area of the cut-through cavities and the cross-sectional area of the contactor (c), the ratio of the area of the cut-through cavities and the area of the fiber banks (w), and the ratio of the area of the fiber banks and the cross-sectional area of the contactor (a). These are defined as follows:
PD ¼
npd2 4S0
ð1Þ
pd ¼
npd2 4Sfb
ð2Þ
c¼
Scut S0
ð3Þ
w¼
Scut Sfb
ð4Þ
a¼
Sfb S0
ð5Þ
Sfb ¼ S0 Scut
ð6Þ
4
K. He, L.-Z. Zhang / International Journal of Heat and Mass Transfer 147 (2020) 118937
Fig. 3. Forming a contactor and various baffle designs.
Table 1 Geometric parameters used in the study. Parameter
ST (mm)
Sp (mm)
Dc (mm)
df (mm)
D (mm)
Group Group Group Group Group Group
10 15 20 24 30 36
23 20 15 7 15 15
36 36 36 36 36 36
1.45 1.45 1.45 1.45 1.45 1.45
25 25 25 25 25 25
a b c d e f
where n, d, S0 , Scut , and Sfb are the number of packed circles, the mean diameter of the circles (fibers), the cross-sectional area of the total packing region, the area of the cut-through cavities, and the area of the fiber bank region, respectively. Of these, n, d, and S0 are already known and are constant for all contactors. Scut (or w) is calculated for both real and artificial contactors as following: (i) The pictures are transformed into binary ones. The fibers are identified by a pattern recognition algorithm and forms points set of the centers of the fibers. The fibers recognition algorithm is mainly based on Hough transform [25]; (ii) the Delaunay subdivision [26] is conducted on the centers of the fibers and boundary fibers. The boundary fibers are packed on the wall of a shell. In this way, the complex porous structure is reflected as a net-like topology. It is shown in Fig. 4 that the Scut is the total area of the marked triangles. The triangles making up Scut contain boundary fibers. The area of the other triangles is included in Sfb . A MATLAB code is developed to conduct these calculations. In summary, the code firstly reads the pattern figures, recognizes the fibers, conducts the Delaunay subdivision, and then calculates the sum of the areas of the marked triangles. The code will be integrated with hardware (camera) in future. Fig. 5 shows the parameters defined for different groups of contactors. The parameter a ranges from 0.69 (P5) down to 0.54 (P6). Correspondingly, P5 has the lowest packing density and the highest mean area of voids among the fibers. P6 is the opposite one. The pressure drop and heat transfer coefficient are determined by the ratio of fluid flowing through the cut-through area and the fluid flowing through the fiber bank (which is a porous medium). It is hard to estimate them precisely. Here, we consider that
Fig. 4. Delaunay subdivision of the void spaces in a contactor (CP6): (a) An artificially created contactor, (b) the Delaunay subdivision of the contactors. 1-a wall boundary circle; 2-the inner shell wall of a contactor.
they depend solely on the ratio for the area of the cut-through cavities c and the ratio for the packing area a. c is largest for P6 and smallest for P1. If c is larger, more fluid bypasses the fiber banks and the contactor has a lower heat transfer coefficient. The parameter w is the ratio of the area of the cut-through cavities and the area of the fiber banks. It is a measure of the bypass effect. Other parameters may also be used to characterize the complex porous structure among the fiber banks, but c and w are the easiest to obtain with pattern recognition technology because the identified triangles contain special vertexes (boundary points).
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Fig. 5. The parameters used to characterize contactors: (a) a and c, (b) w.
2.2. The CFD modelling The CFD models are based on the following simplifications: (i) the 3D physical geometry is simplified as a 2D problem due to the cross flow, (ii) the heat loss from the contactor to the environment is negligible, and (iii) all the walls are smooth. In a real application, the contactors are arranged in a rectangular channel. The experimental setup is described in our previous work [21,27]. The domain of a CFD model consists of two parts: the channel and the contactor. It is shown in Fig. 6(a). The lengths of the inlet and outlet regions are set to 25D, where D is the width of the wind channel, about 36 mm. The mean diameter of the fibers is about 1.5 mm. The corresponding number of packed fibers is 100. The average packing density is 0.35. In the fiber bank region, the coordinates of the fiber (circle) centers are abstracted from the artificially generated patterns. A non-uniform mesh is generated for the fiber bank region. There is a more refined mesh between two fibers. The enhanced wall treatment is used for all cases. Fig. 6(b) is a magnified view of the mesh in the fiber bank. There is a uniform rectangular mesh in the inlet and outlet regions. In the normal direction to the wall, there is a compressed mesh (Fig. 6(c)). The commercial CFD code Fluent 14.5 was used to simulate the fluid flow and heat transfer for these artificially generated
contactors. The conservation equations of mass, momentum, and energy are as follows: Mass conservation:
! ! r qu ¼ 0
ð7Þ
Momentum conservation:
! !! ! ! r q u u ¼ r P þ r s þ q! g
ð8Þ
Energy conservation:
!
! !
!
! ! qcp ! u rT þ u rP ¼ r krT þ s u
ð9Þ h!
!
i
! u þ r u T . The s ¼ l r! above equations are closed by the RNG k e turbulence model. The RNG k e turbulence model is suitable for a flow with recirculation where, for incompressible viscous flow,
and the separation of the flow over a tube bank. The latter cannot be captured by the standard k e turbulence model [28]. They are captured with the additional strain rate and the turbulent viscosity term in the RNG k e model. We used the RNG k e turbulence model based on our previous work. The general form of the RNG k e turbulence model in Cartesian coordinates is
@ qkuj @ ¼ @xj @xj @ qeuj @ ¼ @xj @xj
lþ
lt rk
lþ
lt re
@k @xj
@e @xj
þ Pk þ Pkb qe
e
ð10Þ
e2
þ C 1e ðPk þ C 3e P eb Þ C 2e q k k
ð11Þ P k ¼ l t S2
lt ¼ C l q
ð12Þ 2
k
ð13Þ
e
In the current study, the buoyancy effect is neglected, so Pkb ¼ 0 and P eb ¼ 0. The other constants are as follows:
1=2 C l g3 ð1 g=g0 Þ C 2e ¼ C 2e þ ; g ¼ Sk=e; S ¼ 2Sij Sij ; 1 þ bg3 1 @uj @ui ; C l ¼ 0:0845; C 1e ¼ 1:42; þ Sij ¼ 2 @xi @xj Fig. 6. The computational domain and mesh: (a) The computational domain of the channel containing a contactor, (b) the zoomed-in mesh at region 2 of fiber bank, (c) the zoomed-in mesh at region 1 of channel inlet.
C 2e ¼ 1:68;
g0 ¼ 4:38:
rk ¼ 0:7194; re ¼ 0:7194; b ¼ 0:012; and
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2.3. Boundary conditions and solution controls
3. Results and discussion
The boundary conditions are set as follows. The incompressible air is considered as a working fluid with constant thermo-physical properties. This is due to the small temperature changes in real contactors. The range of Re selected for the numerical study is between 20 and 180, which is commonly used in real situations. A larger inlet velocity generates a higher pressure drop. The
3.1. Pressure drop and Nu for various contactors
Reynolds number (Re ¼ qdV c =l) is based on the channel air velocity and the average outer diameter of the fibers. The corresponding velocity of the air at the inlet of the contactor is between 0.2 and 1.8 m s1. On the outer wall of the fibers and on the wall of the channel, the non-slip boundary condition is imposed. The temperature of the air at the inlet is 280 K. The outer wall of the fibers is kept at a constant wall temperature of 305 K. The heat loss via the wall of the wind channel and through the wall of the contactor shell is ignored. To facilitate comparisons, thermal isolation is assumed. A rubber gasket is used to seal the gap between the channel wall and the contactors, and an isothermal and non-slip boundary condition is used. The main fluid properties are the conductivity of air ka = 0.026 W m1 K1, the kinematic viscosity of air ma = 1.5 10-5 m2 s1, and the density of air, 1.22 kg m3. These are consistent with those in our previous study [21]. The governing equations listed in the last section are solved on a multi-block mesh. A second-order upwind scheme is implemented for the convection terms. The semi-implicit method for a pressure-linked equation (SIMPLE) algorithm [29] is used for the pressure and velocity coupling. The residuals for the continuity and energy equations are of the order of 10-6 and 10-10. The simulations are accelerated by the multi-grid method [30].
2.4. Validation of the numerical model The above numerical model was previously validated by simulating both standard inline tube banks and randomly picked real contactors [21]. The simulation results were compared to the corresponding experimental data. A mesh-independent test was also carried out. Two parameters of a contactor are used for this: the pressure drop and the average Nusselt number. The average Nu is
Nu ¼
dh ka
ð14Þ
where ka ; h;and d are the thermal conductivity of air, the convective heat transfer coefficient, and the averaged diameter of a fiber, respectively. We use Newton’s law of cooling:
Sm hDtm ¼ ma C p ðt out tin Þ
Dt m ¼
ðtout tfw Þ ðtin t fw Þ t fw Þ ln ððttoutt Þ in
ð15Þ
ð16Þ
fw
where t out ; t in , and t fw are the average temperatures at the inlet and outlet of the test region and the outer wall of the fibers. Sm ; C p and ma are the total membrane area, the heat capacity of air and the mass flow rate of air, respectively. A detailed comparison shows that the predictions of the CFD models agree well with the experimental results [21]. The maximum difference for the pressure drop is 17% and for Nu, 19%. A similar mesh density, turbulent model and boundary conditions were used in this study and these models were not specifically validated.
Fig. 7 shows the pressure drop of 60 different contactors. The contactors are classified into six groups. For instance, Ema1 is based on shell group a and pattern P1. In each group, there are ten studied cases and 9 flow Reynolds numbers. Therefore, 540 simulations are used to generate the data. Each simulation last about 1 h. These constants are obtained by B-spline regression of these data. The lines in the figures are the pressure drop of contactors with fibers uniformly arranged in a rectangular pattern, which are used as a reference. In a rectangular pattern, the pitch of the fibers is equal in the horizontal and vertical directions, being about 1.5d. Therefore, the local and average packing densities of the reference contactor are 0.35. As mentioned before, for groups a to d, the inner size of the shell is constant. However, the open width of the shell (Sw ) increases. This means that the effect of the contraction of the inlet flow decreases. As the open width is 10 mm, the pressure drop for different contactors is high (compared to the pressure head of 600–700 Pa for a normal 500 W wind blower), ranging from 200 Pa to 350 Pa at Re = 200. The pressure drop is always lower than that for contactors with an inline pattern. As the open width increases, the amplitudes of the pressure drop reduce but there is more variability in the data. In summary, although the pressure drop for contactors with the same pattern varies from case to case, there are some common features for different groups. For instance, contactors with P1 have a pressure drop that is higher or slightly lower than contactors with an inline pattern. Moreover, contactors with P10 have the lowest pressure drop when there is no strong flow contraction effect (Sw = 15 mm, 20 mm, or 24 mm). Groups e and f have a box-shaped shell and the two walls are further apart. The effect of the wall reduces as the inner volume of the shell increases. Unlike groups a to d, there is almost no flow contraction effect and the inlet flow is almost uniform. Further, the pressure drop can be ranked for the models in descending order as Eme1 > Eme4 > Eme3 > Eme5 > Eme2 > Eme7 > Eme8 > Eme9 > Eme6 > Eme10. This consistent descending order may be because there is no flow contraction effect and the effect of the wall can be ignored. Thus, the pressure drop mainly depends on the fringe shape and local packing density of a fiber bank, which will be discussed further in Section 3.2. The Nusselt numbers are shown in Fig. 8. There are more common features among the different groups. For groups a, b, and c, the descending order of Nu is almost the same: P1 > P5 > P4 > P3 > P8 > P2 > P7 > P9 > P10 P6. For groups d, e, and f, the descending order for Nu is P1 > P5 > P4 > P3 > P2 > P7 > P8 > P9 > P6 > P10. The top four high Nusselt numbers are for contactors with patterns P1, P5, P4, and P3. As the wall effect decreases, the order changes for contactors with P2, P7, P8, and P9. The contactors with P1 have the highest Nu. The contactors with P6 and P10 have the lowest Nu. To give an intuitive understanding, the velocity and temperature contours of typical contactors with shell c and shell e are exhibited in Fig. 9. For the contactors with P1, the hollow fibers are distributed in a ribbon in the vertical direction. The distribution of the flow inside the contactors is uneven. The maldistribution of the flow occurs in two classes. The first class, referred to as the inner flow maldistribution, is due to the uneven distribution of fibers. This maldistribution inside the fiber bank is similar in contactors with P1 and P10. The second class of flow maldistribution originates from the complex flow channel formed by the outer fringe of a fiber bank and the shell wall. For different designs of
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Fig. 7. Pressure drops of different contactors: (a)-(f) pressure drops of contactors for group A-F.
the shell, the velocity contours show distinctive features with the second class of maldistribution. For contactors with P1, the velocity contours are almost symmetrical, despite the change of shell. By comparison, the contactors with P10 have an unsymmetrical velocity contour. The high-velocity region is shown for the cut-through cavity region. The temperature contours show similar features. Thus, fiber distribution patterns like P6 and P10 should be avoided since the fibers are concentrated in a certain region and there are large cut-through cavities. These features can easily be identified by pattern recognition technology.
3.2. The effects of shell design The general effect of the shell wall (baffle design) is shown in Fig. 10(a) and Fig. 10(b), which shows the correlation of the pressure drop and Nu with Sw . As Sw increases, the pressure drop for contactors with various patterns reduces monotonically. The rate of decrease varies from case to case. The rates for contactors with P1 decrease more slowly than for other patterns. By contrast, the rate of decrease for contactors with P10 is the largest. The pressure drop of contactors with various patterns decreases almost linearly.
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Fig. 8. The Nusselt numbers of different contactors: (a)-(f) Nusselt numbers of contactors for group A-F.
This feature is not seen in the correlation of Nu with Sw . As Sw increases from 10 mm to 24 mm, Nu reduces slightly or is mainly unchanged. It suddenly falls when Sw is increased to 30 mm. The differences among the contactors with various patterns are more distinctive, compared to the pressure drop. The ranking of Nu is very clear and consistent, in descending order: P1 > P5 > P4 > P3. The Nu for contactors with P2 and P8 are very close. The other similar pairs are P7 and P9, and P6 and P10. Fig. 10(a) and Fig. 10(b) may be useful for manufacturers. Increasing the open width of shells leads to a decrease in the pressure drop while the heat transfer coefficient is almost unchanged for Sw 24 mm. We suggest using an open width of between 20 mm and 24 mm. If the open width is increased, the poor distribution patterns leads to a reduction in heat and mass transfer.
As mentioned before, these poor patterns show common features that can easily be identified by pattern recognition technology. Here the pressure drops and Nusselt numbers of various contactors are correlated to a single parameter w. These parameters quantitatively characterize the ratio of fluid flow via the cutthrough cavity and the fiber bank. Fig. 10(c) and Fig. 10(d) show the corresponding correlations. rw is the ratio of Sw and the inner diameter of a contactor (25 mm). Interestingly, although rw ð¼ SDw Þ changes from 0.4 to 1.4, the contactors with various fiber distribution patterns show similar correlations. The pressure drop and Nusselt number decrease with w in power laws:
DP ¼ C1 Rem1 wn1 m2
Nu ¼ C2 Re
n2
w
ð17Þ ð18Þ
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to the accuracy reduction in the predictive model. Taking more details of the pattern into the correlations by advanced technology, such as machine learning, may increase the accuracy of the predictions. However, the simplicity of the correlations is preferred, so it is not considered in this study. 3.3. A universal law for contactors with different patterns In the last section, it was shown that changing Sw has a significant effect on the pressure drop and heat transfer. However, the similarity found in correlations (especially in Nu) indicates that there may be a universal law for contactors with various patterns. To abstract the universal law, it is necessary to introduce a comprehensive parameter to evaluate the performance of a contactor:
£¼
NTU rp =DPrp NTU inl =DPinl
ð19Þ
where NTU rp and DPrp are the Number of Heat Transfer (NTU) and pressure drop of contactors with an arbitrary pattern. NTU inl and DP inl are the NTU and pressure drop of the contactors with an inline pattern. The NTU is defined as follows.
NTU ¼
hSm cp ma
ð20Þ
Fig. 9. Contours of velocity and temperature for contactors with a good pattern and a poor pattern (P1 & P10): Velocity contours of contactors for P1 & P10, where the velocity is scaled by Vmax of (a) 6.3 m s1, (c) 2.7 m s1, (e) 10 m s1, (g) 6.3 m s1. (b), (d), (f), (h) The corresponding temperature contours of the contactors (legend unit: K).
The constants in the correlations are listed in Table 2. The exponents m1 ; m2 and n2 are almost unchanged, at about 1.71. 0.95 and 1.45, respectively. The exponents n1 converges to around 1.2 at large rw . The convergence of the exponents indicates that the effect of wall reduces as the open width of the shell increases. The consistent correlations among different groups illustrate that the heat and mass transfer of the contactors are governed by the bypass effect. w reflects this effect. In order to test the reliability of the correlations. Three contactors with patterns PT1-PT3 and shell c are generated. The pressure drop and Nusselt number predicted by the correlation are compared to the corresponding CFD data. They are shown in Fig. 10(e) and Fig. 10(f). The w is 0.302, 0.211 and 0.173 for the PT1-PT3, respectively. The difference between the predicted result and CFD data is in the range of 3.3%-31%. The accuracy is suitable for engineering application. It must be pointed out that only considering w is to obtain simple correlations. The reason for such consideration has been given in previous study [21]. The loss of other topological information leads to accuracy reduction in the correlations. Some contactors with equal w may have different Nu and pressure drops. For example, the AP7 and AP8 have similar w approximating to 0.56. The data in Fig. 7 and 8 shows the differences. The pressure drop of AP8 overshoots that of AP7 by 8.3%-18%. The Nusselt number of AP8 overshoots that of AP7 by 11%-35%. Fig. 2 shows that the topology details of P7 and P8 are different. Apparently, the x-axial symmetry of the outline of P8 is lower than that of P7. The pressure drop of AP8 becomes slightly higher with the increase in Reynolds numbers. Stronger flow separation and flow wake are introduced by P8 (not shown). For another instance, the w of PT3 and CP4 approximate 0.18. The pressure drop of PT3 overshoots that of CP4 by 12%-25%. The Nusselt number of PT3 overshoots that of CP4 by 12%-35%. It is seen in Fig. 10 that there is large cavities among the fibers in PT3. Such cavity leads to higher packing densities at the other regions. The loss of information reflecting such large cavity features and outline features in the correlations leads
Fig. 11 shows the correlations of £ with Re ¼ q d V c =l for various contactors. For Sw ¼10 mm, 15 mm, 20 mm, and 24 mm, £ increases with Re and converges to a certain value at high Reynolds number. For Sw ¼30 mm and 36 mm, £ decreases dramatically and then converges to a constant value. Interestingly, £ is approximately 0.6 ± 0.1 at high Reynolds number, regardless of the fiber distribution pattern and shell inner volume. This is attributed to the diminishing effect of the wall. The comprehensive performance of contactors with large Sw and various patterns is actually similar. The comprehensive performance of contactors with unevenly distributed fibers reaches around 60% of that for a contactor with a regular inline pattern. For contactors with smaller Sw and various patterns, the enhancement of the comprehensive performance depends on Reynolds number and patterns. The £ of most contactors is between 0.2 and 0.6 of that for a contactor with a regular inline pattern. The contactors with P1 & P3 & P4 & P5 shows better comprehensive performance. The improvement is mainly because of the Reynolds number effect. A smaller Sw leads to a stronger flow contraction and a higher pressure inside the contactor, which makes the flow contact with the fiber more uniform. 3.4. Optimization of fiber banks of contactors The research above indicates that the performance of a contactor is mainly determined by the single parameter w. This parameter is mainly dependent on the fringe shape of the packed fiber bank and the corresponding shell design. These contactors show similar performance as a contactor with a uniform inline pattern. Considering the cost of packing, precisely controlling the distribution of fibers is not necessary. Based on the present study, an easy method of packing is suggested to avoid creating a poor contactor. Fig. 12 shows the insertion of a solid cylinder into the fiber bank of a contactor. The cylinder serves as a support and reduces the large voids in the boundary region. The cylinder could be left in the contactor or it could easily be removed from the contactor after sealing. If it is to be left in the contactor, we suggest that it is made from a cheap and light material, such as polystyrene (PS). In this case, the solid cylinder occupies some of the space in the contactor and squeezes the fibers to the rest region of the shell. If it is to be removed, then it can be made of wax. In this case, a hollow space will remain after sealing. Where should the solid cylinders
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Fig. 10. Correlations of pressure drop and Nusselt number with Sw and w: (a) Correlations of DP with Sw atRe = 140, (b) correlations of Nuwith Sw atRe = 140, (c) correlations of DP with w at Re = 100, (d) correlations of Nu with w at Re = 100. 1–6 are the regression curves for contactors with row = 0.4–1.4; (e) lines with solid box, circle and triangle are predicted pressure drop of PT1, PT2 and PT3, respectively; (f) lines with solid box, circle and triangle are predicted Nu number of PT1, PT2 and PT3, respectively. The hollow symbols are corresponding CFD data.
be placed in a contactor? Fig. 12 shows cylinders with a diameter of 10 mm inserted into different locations of a shell. The solid black circles (IS1, IS3, and IS5) and dotted circles (IS2, IS4 and IS6) represent unremovable and removable cylinders, respectively. Three typical positions are studied. The polar coordinates of the center of the circles are (0°, 0), (90°, 7.4 mm), and (180°, 7.4 mm).
In the simulations, 100 fibers were packed into the rest of the space in the contactor. The average local packing density (pd) was about 0.43. Fig. 13 shows the pressure drop, heat transfer coefficient, sensible cooling efficacy (n) and Number of Heat Transfer (NTU) of various artificially controlled fiber patterns. The sensible cooling efficacy is defined as follows.
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K. He, L.-Z. Zhang / International Journal of Heat and Mass Transfer 147 (2020) 118937 Table 2 Exponent constants in the correlations. Constants
C1
m1
n1
C2
m2
n2
R2dp
R2Nu
row = 0.4 row = 0.6 row = 0.8 row = 0.96 row = 1.2 row = 1.44
0.0239 0.0139 0.00652 0.00235 0.00143 0.000685
1.764 1.718 1.696 1.723 1.715 1.771
0.312 0.647 0.818 0.966 1.343 1.179
0.0418 0.0275 0.011 0.00429 0.00545 0.006
0.896 0.897 0.923 0.975 1.122 1.066
1.419 1.439 1.481 1.459 1.486 1.408
0.97 0.98 0.95 0.94 0.96 0.94
0.90 0.90 0.91 0.88 0.96 0.95
Fig. 11. / for contactors with various patterns and baffle designs. (a)–(f) / for contactors with various baffle designs.
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n¼
Fig. 12. Contactors with various inserts. The dotted circles and solid circles represent removable inserts and unremovable inserts, respectively, where the diameter (Dis) of the cylindrical inserts are 10 mm. The center polar coordinates for inserts IS1-IS6 are (0°, 0 mm), (0°, 0 mm), (90°, 7.4 mm), (90°, 7.4 mm), (180°, 7.4 mm), (180°, 7.4 mm), respectively.
t out tin tfw t in
ð21Þ
An unremovable cylinder results in a much higher pressure drop that is more than 74–170% (¼ ðDPurs DPrs Þ=DPrs ) of that for contactors with a removable cylinder. Thus, we recommend that the insert is removed after sealing. For the contactors with removable cylinders, the different positions of the insert result in slightly different pressure drops. The insert at (180°, 7.4 mm) has the lowest pressure drop. Next are the inserts at (0°, 0 mm) and (90°, 7.4 mm). The relative difference is 7–32%. The characteristics of the cylinder and its position have only a slight effect on the Nusselt number. For inserts at (0°, 0 mm) and (90°, 7.4 mm), the unremovable cylinder results in a slightly higher heat transfer coefficient (within 25%). For a cylinder at (180°, 7.4 mm), the unremovable insert results in a slightly lower heat transfer coefficient. Fig. 13 (c) and Fig. 13 (d) show that the contactor with unremovable inserts at (90°, 7.4 mm) is of the highest sensible cooling efficacy and NTU as well as the highest pressure drop. It is considered that the presence of an unremovable insert forces more fluid to bypass the fiber bank via the boundary region, where fewer fibers are distributed. However, considering the significantly increased pressure drop, the contactor with IS5 is suggested. Fig. 14 shows contactors with artificially controlled patterns and the corresponding velocity and temperature contours. It shows that cylinders and a hollow space among the fibers induce different velocity fields. Generally, the presence of an unremovable cylinder (Fig. 14(c), (g), and (k)) results in a higher average packing density. The flow is forced to contact with the fibers, producing a high
Fig. 13. Pressure drop, Nusselt number, sensible cooling efficacy (n) and Number of Heat transfer Unit (NTU) of contactors with unremovable inserts and removable inserts: (a) DP of contactors with inserts IS1-IS6, (b) Nu of contactors with inserts IS1-IS6, (c) sensible cooling efficacy, (d) Number of Heat transfer Unit (NTU).
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(2) Different baffle designs lead to quantitative changes in the pressure drop and heat transfer coefficient. However, the power law correlations are mainly unchanged and have very similar features. There are slight differences in the power exponents. (3) The parameter £ was defined to compare the comprehensive performance of contactors. When w exceeds 0.6, £ approaches 0.6, regardless of the fiber pattern and baffle design. In other words, the effects of the wall and the fiber distribution can be ignored beyond this critical value. (4) The fiber bank patterns can be improved by inserting a solid cylinder into the shell of the contactor. A removable cylinder effectively prevents the deterioration of the heat transfer coefficient and results in a slight increase in the pressure drop. The effects of the position of the insert were also discussed.
Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgments The project is supported by the Natural Science Foundation of China (NSFC), (grant No. 51706073; No.51936005) and the National Key R&D Program of China (grant No. 2017YFE0116100). It is also supported by the Key Project of Science Research Plan of Guangzhou City (grant No. 201904020027). References
Fig. 14. Contours of velocity and temperature for contactors with different inserts. Velocity contours (legend unit: m s1) for Vmax = (a) 8.7 m s1, (c) 12.1 m s1, (e) 10.2 m s1, (g) 17.1 m s1, (i) 8.5 m s1, and (k) 12.3 m s1; (b), (d), (f), (h), (j), (l) The corresponding temperature contours (legend unit: K).
velocity region. Compared to removable inserts that leave a hollow space, it has a higher pressure drop and a higher heat transfer coefficient. 4. Conclusions and suggestions The cross flow and heat transfer in hollow-fiber banks with random distribution patterns and various baffle designs have been studied. The inlet flow Reynolds number ranges from 20 to 180. A single topological parameter, which mainly depends on the fringe shape of a fiber bank, was calculated using pattern recognition technology. Good correlations between the pressure drop and heat transfer with the defined topological parameter are found for different baffle designs. A universal law is abstracted for this class of tube banks. A method to optimize fiber banks based on this novel method are also demonstrated. Following conclusions and suggestions are summarized: (1) The pressure drop and averaged Nusselt number of a contactor are highly correlated with a single topological parameter w in a power law, which is mainly determined by the shape of the fringe of the fiber bank and can easily be calculated by pattern recognition technology.
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