HeatAndMass Transfer Relationship to Determine Shear Stress in Tubular Membrane Systems Ratkovich, Nicolas Rios; Nopens, Ingmar


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1 Aalborg Universitet HeatAnMass Transfer Relationship to Determine Shear Stress in Tubular Membrane Systems Ratkovich, Nicolas Rios; Nopens, Ingmar Publishe in: International Journal of Heat an Mass Transfer DOI (link to publication from Publisher): 0.06/j.ijheatmasstransfer Publication ate: 202 Document Version Author final version (often known as posrint) ink to publication from Aalborg University Citation for publishe version (APA): Ratkovich, N. R., & Nopens, I. (202). HeatAnMass Transfer Relationship to Determine Shear Stress in Tubular Membrane Systems. International Journal of Heat an Mass Transfer, 55(222), /j.ijheatmasstransfer General rights Copyright an moral rights for the publications mae accessible in the public portal are retaine by the authors an/or other copyright owners an it is a conition of accessing publications that users recognise an abie by the legal requirements associate with these rights.? Users may ownloa an print one copy of any publication from the public portal for the purpose of private stuy or research.? You may not further istribute the material or use it for any profitmaking activity or commercial gain? You may freely istribute the UR ientifying the publication in the public portal? Take own policy If you believe that this ocument breaches copyright please contact us at proviing etails, an we will remove access to the work immeiately an investigate your claim. Downloae from vbn.aau.k on: oktober 02, 205
2 Elsevier Eitorial System(tm) for International Journal of Heat an Mass Transfer Manuscript Draft Manuscript Number: Title: HeatAnMass Transfer Relationship to Determine Shear Stress in Tubular Membrane Systems Article Type: Full ength Article Keywors: Membrane bioreactor; heatanmass transfer analogy; shear stress; Sherwoo number; empirical moel. Corresponing Author: Mr. Nicolas Ratkovich, Ph.D Corresponing Author's Institution: First Author: Nicolas Ratkovich, Ph.D Orer of Authors: Nicolas Ratkovich, Ph.D; Pierre Berube, Professor; Ingmar Nopens, Professor Abstract: The main rawback of Membrane Bioreactors (MBRs) is the fouling of the membrane. One way to reuce this fouling is through controlling the hyroynamics of the twophase slug flow near the membrane surface. It has been proven in literature that the slug flow pattern has a higher scouring effect to remove particulates ue to the high shear rates an high mass transfer between the membrane surface an the bulk region. However, to calculate the mass transfer coefficient in an efficient an accurate way is not straightforwar. Inee, for accurate etermination, numerous complex experimental measurements are require. Therefore, this work proposes an alternative metho that uses alreay existing heat transfer relationships for two phase flow an links them through a imensionless number to the mass transfer coefficient (Sherwoo number) to obtain an empirical relationship which can be use to etermine the shear stress.
3 Manuscript HeatAnMass Transfer Relationship to Determine Shear Stress in Tubular Membrane Systems N. Ratkovich a,*, P.R. Berube b, I. Nopens a a BIOMATH  Department of Applie Mathematics, Biometrics an Process Control, Ghent University, Coupure inks 65, B9000, Ghent, Belgium. b Department of Civil Engineering, The University of British Columbia, 6250 Applie Science ane, Vancouver, BC, V6T Z4, Canaa. * Corresponing author: Tel: , Fax: , ABSTRACT The main rawback of Membrane Bioreactors (MBRs) is the fouling of the membrane. One way to reuce this fouling is through controlling the hyroynamics of the twophase slug flow near the membrane surface. It has been proven in literature that the slug flow pattern has a higher scouring effect to remove particulates ue to the high shear rates an high mass transfer between the membrane surface an the bulk region. However, to calculate the mass transfer coefficient in an efficient an accurate way is not straightforwar. Inee, for accurate etermination, numerous complex experimental measurements are require. Therefore, this work proposes an alternative metho that uses alreay existing heat transfer relationships for two phase flow an links them through a imensionless number to the mass transfer coefficient (Sherwoo number) to obtain an empirical relationship which can be use to etermine the shear stress. Keywors Membrane bioreactor, heatanmass transfer analogy, shear stress, Sherwoo number, empirical moel. INTRODUCTION Bearing in min the more stringent effluent quality stanars impose by the EU Water Framework Directive (EUWFD), wastewater treatment efficiencies nee to be improve. These improvements can be achieve both in terms of biological removal efficiency as well as in the slugewater separation step. For the last step two types of technologies exist, the Conventional Activate Sluge (CAS) systems where the separation is brought about by gravity an the Membrane Bioreactors (MBR) where the separation is achieve by filtration. The last one has proven to be a goo alternative to achieve high effluent quality compare to the CAS system. A common problem encountere with MBR systems is the fouling of the membrane resulting in a nee for its frequent cleaning an replacement []. Membrane fouling is the main bottleneck of
4 fullscale application of membrane bioreactors (MBRs) an has restricte its market breakthrough ue to the reuction of prouctivity an increase maintenance an operational cost. iterature has shown that, next to the composition of the sluge, the hyroynamics near the membrane surface play an important role. In search for better control of fouling, literature has focuse on the etermination of the fouling constituents. However, it has been shown that the hyroynamics near the membrane surface play an as important role. To reuce the fouling on the membrane air is often introuce in the sluge flow to create a gasliqui twophase crossflow, to increase the surface shear stress to remove foulants that are alreay attache an to increase the mass transfer between the cake layer an the bulk region [2]. However, the governing mechanisms are not yet completely unerstoo, which results in a trial an error approach to optimize hyroynamic control of fouling Due to the complexity involve in mass transfer measurements for twophase flows, some stuies have focuse on eveloping relationships between heat an mass transfer. This is possible because of the analogies between heat an mass transfer moels in imensionless form which are base on the transport of momentum, mass, heat an energy, an more specifically in the ewis number ( e). The latter is a imensionless number efine as the ratio of thermal to mass iffusivity [,4]: Sh Nu Sc e () Pr where Sh, Nu, Sc an Pr are the Sherwoo, Nusselt, Schmit an Prantl numbers respectively. They are efine in Tab.. In Tab., h is the convection coefficient, is the tube iameter, k is the thermal conuctivity, c c is the specific heat, p is the viscosity, k the mass m transfer coefficient, D is the iffusion coefficient an f is the ensity. These kins of analogies are commonly use in cases where it is easier to obtain heat transfer ata rather than mass transfer ata. The work presente here focuses on a better unerstaning of the mass transfer coefficient near the membrane surface using a heat transfer analogy for twophase slug flow for siestream MBR. 2. MATERIA AND METHODS 2. Description of the setup A escription of the setup that was use to collect shear stress information is given in Fig.. A plexiglas tube with a length of 2 m an an inner iameter of 9.9 mm was use. This tube is similar in geometry to the airlift tubular membranes of interest. A flow cell, locate in the mile of the plexiglas tube ( m) has two electrochemical shear probes, which are use to measure
5 surface shear stresses. A temperature controlle water bath (20 C) is use to keep the temperature of the electrolyte solution flowing through the system constant. A peristaltic pump (Masterflex S, USA) is use to recirculate the electrolytic solution from the gasliqui separator tank to the plexiglass tube at controlle liqui flow rates. Two flow meters (ColeParmer, N0820, USA) are use to monitor the liqui an gas flow rates. Five liqui flow rates (0., 0.2, 0., 0.4 an 0.5 min  ) an three gas flow rates (0., 0.2 an 0. min  ) were investigate, resulting in a total of 5 combinations. These ranges of flow rates correspon to those expecte in fullscale airlift tubular membrane systems [5]. For each experimental conition, surface shear stresses are measure for a perio of 0 secons, an recore at a frequency of 000 Hz [6]. All experimental conitions are replicate six times. The electrochemical probes are mae from two platinum wires imbee flush to the insie surface of the tube wall to avoi them having an effect on the flow fiel. A etaile escription of the irectional electrochemical probes is presente in [7]. Measurements from the irectional electrochemical probes are measure as volt an can be converte to a mass transfer coefficient ( k m ), which can be use to calculate shear stresses using the following equation [7]: τ W.56 D e k 2 m (2) f where is the iameter of the probe (m), e D is the iffusion coefficient of ferricyanie f ( m 2 s  [8]) an is the ynamic viscosity of the solution (= 0.00 Pa s). The etaile proceure to obtain Eq. (2) can be foun in [5]. Eq. (2) correlates mass transfer to shear stress, which is the objective of this work. 2.2 Slug flow To reuce the fouling on the membrane, air is introuce to create a twophase flow. In vertical tubes, there are four specific flow patterns: bubbly, slug, churn an annular flow. Their respective structure epens on the superficial velocities, surface tension an ensities of the fluis. It was foun that the setup uner stuy is operate in the slug flow region (Taylor bubbles). In the slug flow that typically buils up, three ifferent zones can be istinguishe (Fig. 2): ) the falling film zone, i.e. the zone where the bubble is passing, 2) the wake zone, i.e. the zone just behin the bubble where mixing of liqui an gas takes place an ) the liqui zone [2]. 2. Mass Transfer Coefficient Singlephase flow: During the filtration process, the separation between the sluge an the solute occurs at the membrane, giving an increase in the solute concentration near the membrane surface. This is calle concentration polarization [9] which is function of the mass
6 transfer coefficient. The latter can be obtaine by electrochemical methos as presente in section 2. or by using imensionless relationships function of the Sh an epening on the flow regime (Tab. 2). However, it is important to note that the relationships are for smooth tubes only an they are not efine in the transition regime (2000<Re<4000). A weighting factor approach can be use here to etermine the Sh number in the transition regime [0]. Twophase flow: In a slug flow, each zone has its own mass transfer coefficient. Fig. illustrates the mass transfer coefficients for each zone. It is possible to observe that if the flow woul be single phase the value of the mass transfer coefficient in the liqui slug woul be lower compare to the falling film an wake zone ue to higher liqui velocities. Therefore, the mass transfer coefficient increases ue to the twophase cross flow. [2] an [] propose equations for each zone base on hyroynamics moels an mass balances of slug flow. However, these moels require extensive experimental measurements an mathematical erivation. 2.4 Heat Transfer Coefficient Singlephase flow: The heat transfer for single phase flow in a tube epens on the Nu number an the flow regime (Tab 2). The Nu number is the ratio of convective to conuctive heat transfer normal to the bounary an the Pr number is the ratio of the momentum iffusivity an the thermal iffusivity. They are the thermal counterpart for the Sh an Sc the number. Comparing the heat an mass transfer analogies for single phase flow, it is possible to observe that the structure is the same but coefficients in the equations are slightly ifferent (Tab. 2). Twophase flow: The Nu number for two phase flow ( Nu ) is efine by Nu h kc, [2]. Where h is the heat transfer coefficient for twophase flow an k c, is the thermal conuctivity coefficient for twophase flow, which is efine by k c, xkc, x k. Here, k c, G c, an k c, are G the thermal conuctivity of the liqui an gas respectively an x is the vapour quality. The heat transfer coefficient for twophase flow is efine by []: h F p 0. x F 0.55 x Fp h PrG * I Pr G p () where liqui an F is the flow pattern factor (imensionless), p h is the heat transfer coefficient for the * I is the inclination factor (imensionless). The efinition of these parameters can be foun in []. The subscripts G an are gas an liqui respectively. The heat transfer
7 coefficient for the liqui ( h ) in single phase flow are given in Tab 2 as function of the Nu number. The liqui ( Re ) an superficial gas ( Re SG ) Reynols numbers are efine by: u S Re (4) u G SG Re SG (5) G This liqui Reynols number gives a better representation for the liqui phase heat transfer ( h ) an it works well in their twophase heat transfer relationship for various gasliqui combinations an flow patterns. Eq. () is vali for Re from 750 to. 0 5 an Re from 4 to SG 2.5 HeatanMass Transfer Analogy The ewis number (Eq. ()) can be use for both laminar an turbulent regimes. Moreover, it allows to etermine either the heat or mass transfer, given one of them is known as the ewis number can be compute inepenently. The exponent n is usually. When there is filtration, it is possible to assume that the filtration has no effect on the hyroynamics ue to the fact that the permeate flow is less that % of the crossflow an it is assume that it oes not affect the slug flow. Therefore, the filtration process is not taken into account to evelop a relationship for the mass transfer coefficient.. RESUT AND DISCUSSION. Shear profiles Singlephase flow: Initial measurements were performe for singlephase flow to calibrate the wall shear stress with theoretical equations using the friction factor (Tab. 2). The friction factor is use in internal flow calculations an it expresses the linear relationship between mean flow velocity an shear stress at the wall. The mass transfer coefficient obtaine from the electrochemical setup was converte into shear stress (Fig. 4a). Also, it was foun that the flow was in laminar regime. Subsequently, the Sh number was compute an compare to the relationship for single phase flow (Tab. 2) to check the valiity of the eveque equation (Fig. 4b). Using SPSS v5 to estimate the coefficient, the propose moel becomes:.495 Re Sc Sh (6) which is 8% lower compare to the theoretical moel an can be use as a starting point for the analysis of the twophase flow.
8 The mass transfer coefficient for the probe was efine in Eq. (2), form which is necessary to extrapolate to the mass transfer coefficient for the tube as follows: k m e w D 2 f e (7) Now the objective is to etermine the wall shear stress from the heat transfer point of view (using the Nu number). It is possible to write the wall shear stress as function of the Sh number as follows:.56 D f w Sh (8) 2 e The ewis number can be written as: Sc Pr Sh Nu (9) Combining Eq. (8) an (9) yiels:.56 D Sc f w Nu 2 (0) Pr e This is a general equation vali for single phase flow. Nevertheless, the behaviour of twophase flow is ifferent an, hence, some corrections are neee. Twophase flow: Typical voltage results obtaine using the electrochemical shear probes, an the corresponing shear stresses, are presente in [5] an will not be shown here. It is important to highlight, nevertheless, that gas slugs rising in vertical tubes were observe to perioically coalesce when trailing slugs reache the wake of the leaing slugs, accelerating the tailing slugs to finally coalesce with the leaing slug. For this reason, the shear stress profiles inuce by successive slugs were not exactly the same. As a result, the profile of shear stresses in successive shear events, inuce by rising gas slugs, varie consierably over time. Shear Stress Histograms (SSH) were use to explore the effect of the ifferent experimental conitions investigate (Fig. 5) on the resulting shear stresses [5].
9 From Fig. 6, it is possible to istinguish two peaks in the SSH: one peak occurs at positive shear value an is cause by the liqui slugs an a secon peak occurs at a negative shear value an is cause by the gas slugs. The magnitue of the frequency for both peaks is, however, ifferent for the ifferent gasliqui flow rate combinations. Therefore, Eq. (0) can be written for two zones, instea of zones for simplicity (Fig. 2 an ): One zone for the liqui slug ( ls ) an one zone for the gas slug ( gs ) (this zone will inclue the falling film zone an the wake zone, because in the SSH, the wake zone cannot be istinguishe):.56 D Sc Pr f w, ls 2 Nu () e w, gs.56 D 2 e f Sc Pr Nu (2) From the SSH, it is possible to get the average shear stress for the liqui an the gas slug peaks to fee Eq. () an (2) respectively. Due to the fact that the length of the bubbles is ifferent, cause by coalescence, the values obtaine from the SSH istribution are just averages. Therefore, a correction factor nees to be ae. This correction factor shoul consier the fact that the hyraulic iameter changes in the falling film zone. The correction factor is function of the Reynols number as follows: a2 i t a Re () This empirical factor consiers several characteristics of the flow, such as: coalescence of bubbles, bubble length, hyraulic iameter an transition regime, as the transition regime is not efine. Eq. () an (2) become:.56 D Sc 2 Pr e w, ls tls Nu (4).56 D Sc w, gs t 2 gs Nu (5) Pr e It is important to highlight that the power in the t coefficients of Eq. (4) an (5) is just to maintain the same exponent of the Nu number. For simplicity, the first term of the equation is groupe in a constant.
10 a.56 D 2 e f Sc Pr.56 D 2 e f Sc Pr (6) Rewriting Eq. (4) an (5) an combining with () yiels: w, ls a 2, a, ls Rei Nu a 2, a Re Nu ls a (7) gs a (8) w, gs, gs i TP The correction factor can now be etermine from fitting Eq. (7) an (8) to experimentally gathere ata. This was one through a powerlaw regression with the software SPSS v5 using: a w ls 2, ls, a, ls Rei t a Nu a 2, gs w gs a,, gs Rei t a Nu l g (9) (20) The problem now arises as to which Reynols number to use (liqui or superficial gas Reynols number) in the correction factor. For this purpose, the R 2 can be use as gooness of fit criteria. Results are summarize in Tab.. From Tab., the values that are in bol provie the best fit to the experimental ata. Both liqui an gas slugs were foun to be more epenent of the Re rather than the Re. The SG Re consiers the mixture velocity an the voi fraction of the gas slug which is clearly important to account for the liqui an gas slugs. On the other han, the Re was expecte to yiel a ba SG correlation for the liqui slug (i.e. no liqui velocity is inclue in the Reynols number). However, it was expecte that it woul provie a goo correlation for the gas slug, which is clearly not the case. The reason for that coul be that it shoul inclue the combine liqui an gas velocities to account for the increase in gas velocity ue to buoyancy effects. Therefore, it was chosen to use liqui an gas slug. Re in Eq. (9) an (20). Fig. 6 shows the powerlaw relationships for the From Fig. 6, it is possible to observe that the liqui Reynols number is aequate to fit the empirical Eq. (9) an (20) to experimental ata. The recovere parameters for Eq. (7) an (8) are shown in Tab. 4. Therefore the final expressions of Eq. (7) an (8) have the form:
11 Nu Nu w, ls Re (2) (22) w, gs 8 Re From Fig. 6, it is possible to observe that the results of the heatanmass transfer relationship are aequate to preict the shear stress for the liqui an gas slug. The above analysis inicates that relatively simple imensionless moels can be use to escribe the shear stress in the slug flow. Note that since the relationships presente in Eq. (2) an (22) are empirical, care must be taken when using them for esign purposes. It is worth mentioning that this kin of analogies assume Newtonian behaviour. Given that sluge only exhibits slight nonnewtonian behaviour (flow behaviour inex close to unity it is assume). CONCUSIONS To etermine the mass transfer coefficient experimentally is an aruous task an requires a lot of time an experimental work. Besies, it can only be one for solutions where the mass iffusion coefficient an the chemical reactions are well known. Therefore, to apply it in an activate sluge, which is a heterogeneous mixture causes severe ifficulties. To overcome this, a setup with shear probes an an electrolytic solution was use to measure the shear stress an the mass transfer coefficient. Base on that, a heat transfer relationship, which is well stuie in the literature, is suggeste, to etermine the shear stress using the Sherwoo number. A valiation with experimental measurements was mae an prove that this type of analogy is vali. The outcome of the mass transfer coefficient was valiate an an empirical expression was evelope in function of the Nusselt number. ACKNOWEDMENT This research project has been supporte by a Marie Curie Early Stage Research Training Fellowship of the European Community s Sixth Framework Programme uner contract number MESTCT Funing for the infrastructure use to measure surface shear stress was provie by the Natural Science an Engineering Research Council of Canaa (NSERC). NOMENCATURE C Concentration, g  c p Specific heat, kj kg  K  e Tube iameter, m Probe iameter, m D f Diffusion coefficient, m 2 s 
12 F p Flow pattern factor, imensionless h heat transfer coefficient, W m 2 K  * I Inclination factor, imensionless J Flux, m 2 s  k c Thermal conuctivity, W m  K  k m Mass transfer coefficient, m s  Nu Pr Re Sc Sh t x Nusselt number, imensionless Prantl number, imensionless Reynols number, imensionless Schmit number, imensionless Sherwoo number, imensionless Correction factor, imensionless Vapour quality, imensionless Greek symbols Voi fraction, imensionless Density, kg m  Viscosity, Pa s Wall shear stress, Pa Subscript B G gs Bulk Gas Gas slug ls M SG W iqui iqui slug Membrane Superficial gas Twophase Wall
13 REFERENCES [] S. Ju, The MBR book, Elsevier [2] R. Ghosh, Z.F. Cui, Mass transfer in gassparge ultrafiltration: Upwar slug flow in tubular membranes, J.Membr.Sci. 62 (999) [] S.A. Shirazi, E. AlAsani, J.R. Shaley, E.F. Rybicki, A mechanistic moel for preicting heat an mass transfer in vertical twophase flow, Proceeings of the ASME Heat Transfer/Fluis Engineering Summer Conference 2004, HT/FED 2004, Jul (2004) [4] E. Asani, S.A. Shirazi, J.R. Shaley, E.F. Rybicki, Valiation of mass transfer coefficient moels use in preicting CO 2 corrosion in vertical twophase flow in the oil an gas prouction, Corrosion 2006, September 0, September 4, (2006) [5] N. Ratkovich, C.C.V. Chan, P.R. Berube, I. Nopens, Experimental stuy an CFD moelling of a twophase slug flow for an airlift tubular membrane, Chemical Engineering Science. 64 (2009) [6] C.C.V. Chan, P.R. Berube, E.R. Hall, Shear profiles insie gas sparge submerge hollow fiber membrane moules, J.Membr.Sci. 297 (2007) [7].P. Reiss, T.J. Hanratty, An Experimental Stuy of the Unsteay Nature of the Viscous Sublayer, AICHE J. 9 (96) [8] J.M. Rosant, iquiwall shear stress in stratifie liqui/gas flow, J.Appl.Electrochem. 24 (994) [9] M. Muler, Basic Principles of Membrane Technology, Springer 998. [0] N.S. Cheng, Formulas for friction factor in transitional regimes, J.Hyraul.Eng. 4 (2008) [] D. Zheng, D. Che, Experimental stuy on hyroynamic characteristics of upwar gasliqui slug flow, Int.J.Multiphase Flow. 2 (2006) [2] D. Kim, A.J. Ghajar, R.. Dougherty, Robust heat transfer correlation for turbulent gasliqui flow in vertical pipes, J.Thermophys.Heat Transfer. 4 (2000) [] A.J. Ghajar, C.C. Tang, Importance of NonBoiling TwoPhase Flow Heat Transfer in Pipes for Inustrial Applications, Heat Transfer Eng. (200) [4] T. Taha, Z.F. Cui, CFD moelling of slug flow in vertical tubes, Chemical Engineering Science. 6 (2006)
14 ist of figures Figure. Description of the electrochemical shear measurement setup [5]. Figure 2. Zones in the slug flow [2] (left) an numerical simulation of a Taylor bubble rising through stagnant glycerine in a vertical tube [4] (right) Figure. Mass transfer coefficients in ifferent zones of the slug flow [2]. Figure 4. Reynols number vs a) shear stress an b) Sh number comparison for both the experimental ata an theoretical equations. Figure 5. Relative frequency vs shear stress for a liqui combination of 0. min  an three gas flow rates (0., 0.2 an 0. min  ) [5]. Figure 6. iqui Reynols number vs the correction factor of Eq. (22) an (2) for the liqui an gas slug respectively.
15 Cover etter HeatAnMass Transfer Relationship to Determine Shear Stress in Tubular Membrane Systems Corresponing Author: Nicolas Ratkovich Aress: Department of Civil Engineering, Aalborg University, Sohngaarsholmsvej 57, DK9000 Aalborg, Denmark Telephone: Fax: aress: Dear eitor The breakthrough of membrane bioreactors (MBR) in wastewater treatment is still hampere by poor unerstaning of the fouling phenomena an of the air scouring use to cure it. Mathematical moels combine with experimental ata have proven to be a goo combination to buil up process knowlege an eventually optimize the system in terms of esign an operation. The objective of this stuy was to calculate the mass transfer coefficient in an efficient an accurate way. Inee, for accurate etermination, numerous complex experimental measurements are require. Therefore, this work proposes an alternative metho that uses alreay existing heat transfer relationships for two phase flow an links them through a imensionless number to the mass transfer coefficient (Sherwoo number) to obtain an empirical relationship which can be use to etermine the shear stress. Sincerely, Nicolas Ratkovich
16 Figure Click here to ownloa high resolution image
17 Figure 2 Click here to ownloa high resolution image
18 Figure Click here to ownloa high resolution image
19 Figure 4a Click here to ownloa high resolution image
20 Figure 4b Click here to ownloa high resolution image
21 Figure 5 Click here to ownloa high resolution image
22 Figure 6 Click here to ownloa high resolution image
23 *Conflict of Interest Statement There are not any actual or potential conflict of interest of this work
24 Table Table. Dimensionless heat an mass transfer numbers Heat transfer Mass Transfer h Nu k c km Sh D f c p Pr k c Sc D f Table 2. Relationship among wall shear stress, mass an heat transfer Dimensionless numbers Sh Mass transfer Wall friction Heat transfer function, Re, Sc f function,re Nu function,re, Pr km Sh D f Sc D f u 8 w Re f 2 u h Nu k c cp Pr k c aminar (Re < 2000) Turbulent Sh.62 Re Sc 0.8 (Re < 2000) Sh 0.04Re Sc Analogy f 0.25log f 64Re Sh Nu * The subscripts B an W are for the bulk an wall respectively Re Sc Pr e 2 Nu.86 Re Pr Nu Re 0.8 Pr B W B W
25 Table. R 2 of the ifferent Reynols number. t l t g Re Re SG Table 4. Parameters of Eq. (7) an (8) iqui slug Gas slug a a a
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