Comparison heat and mass transfer coefficients in the shell side of the hollow fiber membrane module

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1 Comprison het nd mss trnsfer coefficients in the shell side of the hollow fiber membrne module Sebstin Englrt,* Wrocłw University of ience nd Technology, Fculty of Environmentl Engineering, Deprtment of Air Conditioning, Heting, Gs Engineering nd Air Protection, 4/6 Norwid st., Wrocłw, Polnd Abstrct. The rticle discusses the theory ssocited with the determintion of totl het nd mss trnsfer coefficients in membrne module. The equtions for determining Nusselt nd Sherwood numbers on the shell side of the module re presented. The design of the membrne module nd the test bench re lso discussed. The convective coefficients of het nd mss trnsfer on the shell side hve been determined on the bsis of the mesurements. Clcultion results obtined from correltion equtions were then compred with the results of experimentl investigtions of convective het nd mss trnsfer coefficients in the irwter system. The eqution is indicted pproprite for use in the cse nlysed. Introduction Membrnes re currently used in mny industries to conduct vrious types of seprtion processes. They cn lso be used in HVAC (heting, ventiltion nd ir conditioning) systems to conduct ir tretment processes for exmple drying, humidifying or evportive cooling [ 3]. The benefits of using membrnes in the ir tretment process include, for exmple, lck of direct contct between liquid nd ir, elimintion of liquid droplets in the ir strem, prevention of microorgnisms, bcteri or spores between phses nd prevention of liquid contmintion by irborne dust [4, 5]. Such benefits re not offered by trditionl ir tretment techniques. A problemtic issue my be the design of lrge scle membrne modules due to the discrepncies ssocited with the determintion of het nd mss trnsfer coefficients. An ccurte prediction of the efficiency of the hollow fibre membrne cn be limited by models of het nd mss trnsfer coefficient on the shell side. Severl correltion equtions hve been developed by vrious uthors for the different processes nd conditions of fluid flow through the membrne module, llowing the determintion of the Nu nd Sh numbers on the shell side [6, 7]. The ppliction of certin equtions my result in lrge discrepncies in the results of clcultions. * Corresponding uthor: sebstin.englrt@pwr.edu.pl The Authors, published by EDP iences. This is n open ccess rticle distributed under the terms of the Cretive Commons Attribution License 4.0 (

2 This rticle hence presents exmples of empiricl equtions for determining Nu nd Sh numbers. The results of the clcultions obtined from correltion equtions were compred with the results of experimentl investigtions of convective het nd mss trnsfer coefficients in the ir wter system. The equtions pproprite for use in the nlysed cse were then indicted. 2 Theory As membrne module, the clcultion of the het nd mss exchnger requires, mong others, determintion of hydrodynmic conditions of the flow of individul fluid strems nd determintion of het nd mss trnsfer coefficients s well s determintion of the verge temperture or concentrtion difference. The bsic equtions used in the design of exchngers re equtions tht llow us to determine the totl het nd mss flows. According to studies [8, 9] the overll mss trnsfer rte cn be determined s: Anlogously for the het trnsfer rte: N Q Kol AC () Hol AT (2) 2. Overll het nd mss trnsfer coefficients The overll het nd mss trnsfer coefficients my then be determined while ccounting for the individul resistnces. For the mss trnsfer coefficient, Mhmud et l. [0] nd Nii et l. [] indicte tht resistnce on the wter side cn be omitted. Therefore, the overll mss trnsfer coefficient from wter to ir cn be obtined from the following eqution: K ol km k (3) Similrly, one cn determine the het trnsfer coefficient [4, 2, 3]: H ol hm h (4) esistnces in equtions (3) nd (4) cn be determined using the formuls given by Zhng [2]: km D vm d d o i nd k (5) ko hm m d d o i nd h (6) ho The mount of het nd mss exchnged by the ir in the whole membrne module per unit of time is given the following equtions: 2

3 Q mc p t t 2 (7) N m x 2 x (8) Experimentlly, tking into ccount equtions ( 2) nd (7 8), the overll het nd mss trnsfer coefficients cn be clculted s: K H ol mc p t t2 AT m x x ol 2 AC In the cse of the membrne module, when the wter circultes in closed circuit s result of het nd mss exchnge processes, the wter temperture is set (t = t w2), hence: (9) (0) t t T () t t t t Anlogously, the logrithmic men difference of concentrtions cn be determined s [4]: C C C x x C C C C x x t t t t x x (2) 2.2 Shell side het nd mss trnsfer coefficient The convective coefficients of het nd mss trnsfer on the shell side will depend on the geometry of the module, the system of membrnes in the module, or the method of fluid flow through the module. These coefficients cn be expressed respectively by Nusselt nd Sherwood number: h o d Nu o, kodo Sh (3) D Correltion equtions llowing to determine Nu nd Sh numbers on the shell side of the membrne module re summrised in Tble nd Tble 2. Nu nd Sh number cn lso be determined by wy of the nlogy to het nd mss trnsfer [5]: Nu Sh /3 / 3 (4) Pr 3

4 Tble. Correltions of Nusselt numbers for fluid flow cross tube bnk (stggered rrngement). Authors nd references Correltions Colburn [6] 0.6 / 3 Nu e Pr (5) Chen nd Wung [7] Žukusks [8] Husen [9] No Nu 0.78e Pr (6) 0.36 / 4 (Pr/ Prs ) C nd n re given in [5] Nu Ce n Pr F / b Nu 0.35F e Pr Tble 2. Shell side mss trnsfer correltions. (7) (8) Authors nd references Cote et l. [20] Yng nd Cussler [2] Kreith nd Blck [22] Costello et l. [23] Gwroński nd Wrzesińsk [24] Prsd nd Sirkr [25] Koo nd Sngni [26] Johnson et l. [27] Zhng [28] Sh Sh Correltions No Sh 0.6e (9) 0.34 Sh.38e (20) 0.59 Sh 0.39e (2) 0.53 ( ) e (22) ( ) 0.09( ) e (23) d 0.6 Sh 5.85( ) h e (24) L /3 /3 d / 3 / 3 )( ) o e (25) Sh.89( L 4 Sh 0.59 d h (26) 0.24 e L 3 Sh ( ) e 0.882d k 2 f k (27) The schemtic drwing of the module nd the wy the membrnes re rrnged s shown in Figure. For het exchngers with cross-flow, the eynolds number should be determined t the mximum velocity [29]: u d e mx o b b, umx mx u, u (28) b c where: S L, d o S b T, d o c 2 2 (b / 2) (29) 4

5 If S L = S T, dimensionless coefficient for stggered rrngement cn be determined from the following eqution [2]: (30) Experimentlly, convective het nd mss trnsfer coefficient cn be determined tking into ccount equtions (3 6) nd (9 0) s: h o m 2 hm c p t t, AT k o m 2 km x x (3) AC Fig.. hemtic of the membrne module nd the rrngement of fiber bundles. 3 Experimentl setup To determine the experimentl convective coefficients of het nd mss trnsfer on the ir side, membrne module with polypropylene cpillry membrne (PP) ws used with the following prmeters:.67 mm membrne inner dimeter, 2.7 mm membrne outer dimeter, 0.43 m verge pore size nd 0.55 m mximum pore size. The module uses 744 membrnes. The effective length of the membrnes, equl to the length of the ir inlet, ws 400 mm, which gve n effective re of 2.53 m 2. The pcking density ws 663 m 2 /m 3, pcking friction ws The membrnes were rrnged perpendiculr to the direction of the ir flow (Fig. ). The mesurements were crried out on the test bench, the digrm of which is shown in Figure 2. The test bench ws equipped with devices enbling mesurement of the following prmeters: temperture nd humidity of the outside ir; temperture nd humidity of the ir t the inlet nd outlet of the membrne module; ir flow rte in the duct; wter temperture before nd behind the module, wter flow rte. The mesurements were crried out for vrible ir nd wter flow. For ir, the flow ws chnged in the rnge of m 3 /h nd for wter: kg/h. One exmple of ir conversion in test instlltion (in membrne module) which tkes into ccount the prmeters of the outside ir during mesurements (point ) is shown in the digrm i-x (Fig. 3). 5

An exmple of ir conversion in the experimentl instlltion for m = 2 m 3 /h nd mw = 40 kg/h.

6 Fig. 2. hemtic digrm nd view of the test bench. F fn, MO mesuring orifice, o rotmeter, recorder, WT wter reservoir, M membrne module, P pump, eg thyristor controller, sensors of: t temperture, humidity, Δp differentil pressure. Fig. 3. An exmple of ir conversion in the experimentl instlltion for m = 2 m 3 /h nd mw = 40 kg/h. 4 esults nd discussion The experimentl vlues of the Nusselt nd Sherwood numbers were obtined by verging results from experimentl dt determined t different wter nd ir flows. The vlues obtined in this wy were compred with the vlues obtined from existing correltions describing convective het nd mss trnsfer coefficients on the externl side of the membrne module (ccording to Tbles nd 2). The results of the nlysis re presented in digrms in the form of the dependence of Nu = f (e) nd Sh = f (e) (Fig. 4 nd Fig. 4b). The nlysis shows tht Nu nd Sh numbers re functions incresing with n increse in e number nd virtully independent of the strem of wter flowing through the membrnes. The vrition of Nu number rnged from 0.92 to.46 nd in the cse of Sh number it ws from 0.78 to.59. Figure 4 shows tht good fit between experimentl nd clculted dt cn be obtined fter pplying eqution (4) for correltion (24) ccording to Prsd nd Sirkr [25] nd worse from eqution (25) by Koo nd Sngni [26]. The remining equtions significntly overstte the obtined results in reltion to the experimentl dt. The membrne module is similr to shell nd tube het exchnger. However, there re severl differences between 6

7 these systems, for exmple: membrnes re flexible, fibers hve smll dimeter. As result, the flow in the shell side hs much more complicted profile thn tht in het exchngers. This my be the reson overstte Nu numbers obtined from equtions (5) (8). In the cse of mss trnsfer, the experimentl dt could lso be found between the results obtined from equtions (24) nd (25) (Fig. 4 b). Hence, not tking into ccount effect of module geometry, such s membrne pcking frction rtio, the equtions typicl for cross flow hve significntly overestimted the obtined vlues in reltion to the experimentl dt obtined. 00,00 Nuexp, mw=00 kg/h Nuexp, mw=50 kg/h 00,00 Shexp, mw=00 kg/h Shexp, mw=50 kg/h Nuexp, mw=40 kg/h Shexp, mw=40 kg/h 0,00 Colburn Chen nd Wung 0,00 Cote et l. Yng nd Cussler Zukusks Costello et l. Nu,00 Husen Prsd nd Sirkr Koo et l. Nuexp Sh,00 Gwroński nd Wrzesińsk Kreith nd Blck Prsd nd Sirkr Koo et l. Johnson et l. 0, e 0, b e Zhng Shexp Fig. 4. Comprison of shell side het () nd mss (b) trnsfer between model predictions nd observtion in module. 5 Conclusions Among the equtions compred here describing the mss trnsfer on the shell side of the module, the best fit of experimentl dt ws obtined from the eqution (24) developed by Prsd nd Sirkr [25]. The verge difference between the experimentl vlues of Sh numbers nd the clculted from this eqution ws 7%. After pplying the eqution describing the nlogy between het nd mss trnsfer, eqution (24) by Prsd nd Sirkr cn be used to determine the Nusselt number. In this cse, the verge difference between the experimentl Nu numbers nd the clculted did not exceed 8%. Nomenclture A re (m 2 ) dimensionless longitudinl pitch b dimensionless trnsverse pitch C concentrtion (kg/m 3 ) c dimensionless digonl pitch c p specific het (J/(kgK)) d dimeter (m) D diffusivity (m 2 s) D vm effective moisture diffusivity (m 2 s) H ol overll het trnsfer (W/(m 2 K)) h o convective het trnsfer (W/(m 2 K)) K ol overll mss trnsfer (m/s) k o convective het trnsfer (m/s) L length of hollow fiber (m) m mss flow rte (kg/s) N mss trnsfer rte (kg/m 3 ) Q het trnsfer rte (W) t temperture (K ) u pproch velocity (m/s) u mx mximum velocity (m/s) x moisture content (kg/kg) pcking frction het conductivity (W/(mK)) kinemtic viscosity (m 2 /s) membrne thickness (m) 7

8 Subscripts: inlet, 2 outlet, ir, d dry bulb, f frctl, h hydrulic, i inner, m membrne, o outer, w wter The work ws relized within the lloction No. 040/0007/7 wrded for Fculty of Environmentl Engineering Wroclw University of ience nd Technology by Ministry of ience nd Higher Eduction in yers eferences. S. Englrt, E3S Web Conf. 7, 0002 (207) 2. S. Englrt, E3S Web Conf. 22, (207) 3. J. Woods, enew. Sust. Energ. ev. 33, (204) 4. S. Bergero, A. Chiri, Appl. Therm. Eng. 2, 9 35 (200) 5. D.W. Johnson, C. Yvuzturk, J. Pruis, J. Membr. i. 227, 59 7 (2003) 6. S. Shen, S.E. Kentish, G.W. Stevens, Solvent Extr. Ion Exc. 28, (200) 7. S.M. Hung, L.Z. Zhng, enew. Sust. Energ. ev. 28, (203) 8. K.. Kistler, E.L. Cussler, Trns. IChemE, Prt A 80, (2002) 9. V.Y. Dinore, G.F Versteeg, Int. J. Het nd mss 48, (2005) 0. H. Mhmud, A. Kumr,.M. Nrbitz, T. Mtsuur, J. Membr. i. 79, 29 4 (2000). S. Nii,.S. Jebson, E.L. Cussler, J. Membr. i. 20, (2002) 2. L.Z. Zhng, Int. J. Het. Mss. Trn. 55, (202) 3. N.T. Chrles, D.W. Johnson, J. Membr. i. 39, (2008) 4. A. Kmieć, S. Englrt, A. Ludwińsk, Theory nd technique of fluidiztion (Publishing House of Wroclw University of Technology, Wroclw, 2007) 5. F.P. Incroper, D.P. DeWitt, T.L. Bergmn, A.S. Lvine, Fundmentls of Het nd Mss Trnsfer (John Wiley & Sons publishers Inc, New York, 2007) 6. A.P. Colburn, Trns. Am. Inst. Chem. Eng. 29, (933) 7. C.J. Chen, T.S. Wung, ASME J. Het Trnsfer, (989) 8. A. Žukusks, Advnces in Het Trnsfer 8, (972) 9. H. Husen, Het Trnsfer in Counterflow, Prllel Flow nd Cross Flow (McGrw- Hill, New York, 983) 20. P. Cote, J. Bersillon, A. Huyrd, J. Membr. i. 47, 9 06 (989) 2. M. Yng, E.L Cussler, AIChE J. 32,, (986) 22. F. Kreith, W.Z. Blck, Bsic Het Trnsfer (Hrper & ow, New York, 980) 23. M.J. Costello, A.G. Fne, P.A.Hogn,.W. hofield, J. Membr. i. 80, (993) 24.. Gwroński, B. Wrzesińsk, J. Membr. i. 68, (2000) 25.. Prsd, K.K. Sirkr, AIChE J. 34, 2, (988) 26. S. Koo, A.S. Sngni, J. Fluid. Mech. 484, (2003) 27. D.W. Johnson, M.J.Semmens, J.S. Gulliver, J. Membr. i. 28, 67 8 (997) 28. L.Z. Zhng, Int. J. Het. Mss. Trn. 54, (20) 29. W.A. Khn, J.. Culhm, M.M. Yovnovich, J. Thermophys Het Trnsfer 20, (2006) 8

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