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ORIGINAL Worachest Pirompugd Somchai Wongwises Chi-Chuan Wang A tube-by-tube reduction method for simultaneous heat and mass transfer characteristics for plain fi n-and-tube heat exchangers in dehumidifying conditions Received: 19 August 2004/ Accepted: 24 November 2004/Published online: 4 March 2005 ? Springer-Verlag 2005 Abstract This study proposed a new method, namely a tube-by-tube reduction method to analyze the perfor- mance of fi n-and-tube heat exchangers having plain fi n confi gurationunderdehumidifyingconditions.The mass transfer coeffi cients which seldom reported in the open literature, are also presented. For fully wet con- ditions, it is found that the reduced results for both sensible heat transfer performance and the mass transfer performance by the present method are insensitive to change of inlet humidity. Unlike those tested in fully dry condition, the sensible heat transfer performance under dehumidifi cation is comparatively independent of fi n pitch. The ratio of the heat transfer characteristic to mass transfer characteristic (hc,o/hd,oCp,a) is in the range of 0.6?1.0, and the ratio is insensitive to change of fi n spacing at low Reynolds number. However, a slight drop of the ratio of (hc,o/hd,oCp,a) is seen with the decrease of fi n spacing when the Reynolds number is suffi cient high. This is associated with the more pronounced infl uence due to condensate removal by the vapor shear. Corre- lations are proposed to describe the heat and mass performance for the present plate fi n confi gurations. These correlations can describe 89% of the Chilton Colburn j-factor of the heat transfer (jh) within 15% and can correlate 81% of the Chilton Colburn j-factor of the mass transfer (jm) within 20%. Keywords Fin-and-tube heat exchanger Dehumidifying Sensible heat transfer performance Mass transfer performance Nomenclature Af Surface area of fi n AoTotal surface area Ap,iInside surface area of tubes Ap,oOutside surface area of tubes bpSlope of the air saturation curved between the outside and inside tube wall temperature brSlope of the air saturation curved between the mean water temperature and the inside wall temperature bw,mSlope of the air saturation curved at the mean water fi lm temperature of the fi n surface bw,pSlope of the air saturation curved at the mean water fi lm temperature of the tube surface Cp,a Moist air specifi c heat at constant pressure Cp,w Water specifi c heat at constant pressure DcTube outside diameter (include collar) DiTube inside diameter fiIn-tube friction factors of water FCorrection factor GmaxMaximum mass velocity based on minimum fl ow area hc,o Sensible heat transfer coeffi cient hd,o Mass transfer coeffi cient hi Inside heat transfer coeffi cient ho,w Total heat transfer coeffi cient for wet external fi n Io Modifi ed Bessel function solution of the fi rst kind, order 0 I1 Modifi ed Bessel function solution of the fi rst kind, order 1 iaAir enthalpy ia,inInlet air enthalpy ia,mMean air enthalpy ia,outOutlet air enthalpy igSaturated water vapor enthalpy W. Pirompugd S. Wongwises ( is less than 0.05, where _ Qwis the water-side heat transfer rate for _ Qwand air-side heat transfer rate _ Qa), are considered in the fi nal analysis. Detailed geometry used for the present plain fi n-and-tube heat exchangers is tabulated in Table 1. The test fi n-and-tube heat exchangers are tension wrapped having a L type fi n collar. The test conditions of the inlet air are as follows: The test conditions approximate those encountered with typical fan-coils and evaporators of air-condition- ing applications. Uncertainties reported in the present investigation, following the single-sample analysis pro- posed by Moff at 15, are tabulated in Table 2. 3 Data reduction 3.1 Heat transfer coeffi cient (hc,o) Basically, the present reduction method is based on the Threlkeld 20 method. Some important reduction pro- Fig. 1 Schematic of experimental setup Dry-bulb temperatures of the air:270.5?C Inlet relative humidity for the incoming air: 50% and 90% Inlet air velocity:From 0.3 m/s to 4.5 m/s Inlet water temperature:70.5?C Water velocity inside the tube:1.51.7 m/s 758 cedures for the original Threlkeld method is described as follows. The total heat transfer rate used in the calculation is the mathematical average of _ Qaand _ Qw; namely, _ Qa _ ma(ia;in? ia;out),1 _ Qw _ mwCp;wTw;out? Tw;in;2 _ Qavg _ Qa _ Qw 2 :3 The overall heat transfer coeffi cient, Uo,w, is based on the enthalpy potential and is given as follows: _ Qavg Uo;wAoDimF;4 where Dim is the mean enthalpy diff erence for counter fl ow coil, Dim ia;m? ir;m:5 According to Bump 4 and Myers 16, for the counter fl ow confi guration, the mean enthalpy is ia;m ia;in ia;in? ia;out lnia;in? ir;out ? ia;out? ir;in ? ? ia;in? ia;outia;in? ir;out ia;in? ir;out ? (ia;out? ir;in ;6 ir;m ir;out ir;out? ir;in lnia;in? ir;out ? ia;out? ir;in ? ? ir;out? ir;in)(ia;in? ir;out) ia;in? ir;out) ? ia;out? ir;in ;7 where F in Eq. 4 is the correction factor accounting for the present cross-fl ow unmixed/unmixed confi guration. The overall heat transfer coeffi cient is related to the individual heat transfer resistance 16 as follows: 1 Uo;w b0rAo hiAp;i b0pAoln Dc=Di 2pkpLp 1 ho;wAp;o . b0w;pAo ? Afgf;wet . b0w;mAo ? ?; 8 where ho,w 1 Cp;a . b0w;mhc;o ? yw=kw ;9 yw in Eq. 9 is the thickness of the water fi lm. A constant of 0.005 in. was proposed by Myers 16. In practice, (yw/kw) accounts for only 0.55% compared to (Cp,a/bw,mhc,o), and has often been neglected by previ- ous investigators. As a result, this term is not included in the fi nal analysis. In this study, we had proposed a row-by-row and tube-by-tube reduction method for detailed evaluation of the performance of fi n-and-tube heat exchanger in- stead of conventional lump approach. Hence analysis of the fi n-and-tube heat exchanger is done by dividing it into many tiny segments (number of tube row number of tube per row number of fi n) as shown in Fig. 2. In the analysis, F is the correction factor accounting for a single-pass, cross-fl ow heat exchanger for one fl uid mixed, other fl uid unmixed that was shown by Threlkeld 20. The tube-side heat transfer coeffi cient, hievaluated with the Gnielinski correlation 8, Fig. 2 Dividing of the fi n-and-tube heat exchanger into the small pieces Table 2 Summary of estimated uncertainties Primary measurementsDerived quantities ParameterUncertaintyParameterUncertainty ReDc=400 Uncertainty ReDc=5,000 _ ma0.31% ReDc1.0%0.57% _ mw0.5% ReDi0.73%0.73% DP0.5% _ Qw3.95%1.22% Tw0.05?C _ Qa5.5%2.4% Ta0.1?Cj11.4%5.9% Table 1 Geometric dimensions of the sample plain fi n-and-tube heat exchangers No.Fin thickness (mm) Sp (mm) Dc (mm) Pt (mm) Pl (mm) Row no. 10.1151.088.5125.419.051 20.1201.6310.3425.422.001 30.1151.938.5125.419.051 40.1152.1210.2325.419.051 50.1202.3810.3425.422.001 60.1151.128.5125.419.052 70.1201.588.6225.419.052 80.1151.958.5125.419.052 90.1203.018.6225.419.052 100.1302.1110.2325.422.002 110.1151.1210.2325.419.054 120.1151.4410.2325.419.054 130.1152.2010.2325.419.054 140.1302.1010.2325.422.004 150.1301.7210.2325.422.006 160.1302.0810.2325.422.006 170.1303.0310.2325.422.006 759 hi fi=2ReDi? 1000Pr 1:07 12:7 ffiffi ffiffi ffiffi ffiffi fi=2 p Pr2=3? 1 ? ki Di ;10 and the friction factor, fiis fi 1 1:58ln ReDi? 3:282 :11 The Reynolds number used in Eqs. 10 and 11 is based on the inside diameter of the tube and ReDi qVDi=l: In all case, the water side resistance is less than 10% of the overall resistance. In Eq. 8 there are four quantities (bw,m, bw,p, bpand br) involving enthalpy-temperature ratios that must be evaluated. The quantities of bpand brcan be calculated as b0r is;p;i;m? ir;m Tp;i;m? Tr;m ;12 b0p is;p;o;m? is;p;i;m Tp;o;m? Tp;i;m :13 The values of bw,pand bw,mare the slopes of satu- rated enthalpy curve evaluated at the outer mean water fi lm temperature at the base surface and at the fi n sur- face. Without loss of generality, bw,pcan be approxi- matedbytheslopeofsaturatedenthalpycurve evaluated at the base surface temperature 23. The wet fi n effi ciency (gf,wet ) is based on the enthalpy diff erence proposed by Threlkeld 20. i.e., gf,wet i ? is,fm i ? is,fb ;14 where is,fmis the saturated air enthalpy at the mean temperature of fi n and is,fbis the saturated air enthalpy at the fi n base temperature. The use of the enthalpy potential equation, greatly simplifi es the fi n effi ciency calculation as illustrated by Kandlikar 10. However, the original formulation of the wet fi n effi ciency by Threlkeld20wasforstraight fi n confi guration (Fig. 2a). For a circular fi n (Fig. 2b), the wet fi n effi ciency is 23, gf;wet 2ri MT(r2 o? r2i) ? K1(MTri)I1(MTro) ? K1(MTro)I1(MTri) K1(MTro)I0(MTri) K0(MTri)I1(MTro) ? ;15 where MT ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 2ho;w kft r ;16 The test heat exchangers are of Fig. 3c confi gura- tion. Hence, the corresponding fi n effi ciency is calcu- latedbytheequivalentcircularareamethodas depicted in Fig. 4. Evaluation of bw,mrequires a trial and error proce- dure. For the trial and error procedure, is,w,mmust be calculated using the following equation: is;w;m ia;m? Cp;aho;wgf;wet b0w;mhc;o ?1 ? Uo;wAo b0r hiAp;i b0pln Dc=Di 2pkpLp # ! ? ia;m? ir;m:17 An algorithm for solving the sensible heat transfer coeffi cient hc,ofor the present row-by-row and tube-by- tube approach is given as follows: 1. Based on the measurement information, calculate the total heat transfer rate _ Qtotalusing Eq. (3). 2. Assume a hc,ofor all elements. 3. Calculate the heat transfer performance for each segment with the following procedures. 3.1. Calculate the tube side heat transfer coeffi cient of hiusing Eq. 10. 3.2. Assume an outlet air enthalpy of the calculated segment. 3.3. Calculate ia,mby Eq. 6 and ir,mby Eq. 7. 3.4. Assume Tp,i,mand Tp,o,m. 3.5. Calculate b0rAo ?= h iAp;i ? andb0pAoln Dc=Di hi = h 2pkpLp?. 3.6. Assume a Tw,m. 3.7. Calculate the gf,wetusing Eq. 15. 3.8. Calculate Uo,wfrom Eq. 8. 3.9. Calculate is,w,mby Eq. 17. 3.10. Calculate Tw,mfrom is,w,m. Fig. 3 Type of fi n confi guration Fig. 4 Approximation method for treating a plate fi n of uniform thickness 760 3.11. If Tw,mderived in step 3.10 is not equal that is assumed in step 3.6, the calculation step 3.7 3.10 will be repeated with Tw,mderived in step 3.10 until Tw,mis constant. 3.12. Calculate _ Q of this segment. 3.13. Calculate Tp,i,mand Tp,o,mfrom the inside convection heat transfer and the conduction heat transfer of tube and collar. 3.14. If Tp,i,mand Tp,o,mderived in step 3.13 are not equal that is assumed in step 3.4, the calculation step 3.53.13 will be repeated with Tp,i,mand Tp,o,mderived in step 3.13 until Tp,i,mand Tp,o,m are constant. 3.15. Calculate the outlet air enthalpy by Eq. 1 and the outlet water temperature by Eq. 2. 3.16. If the outlet air enthalpy derived in step 3.15 is not equal that is assumed in step 3.2, the cal- culation step 3.33.15 will be repeated with the outlet air enthalpy derived in step 3.15 until the outlet air enthalpy is constant. 4. If the summation of _ Q for all elements is not equal _ Qtotal, hc,owill be assumed a new value and the cal- culation step 3 will be repeated until the summation of _ Q for all elements is equal _ Qtotal. 3.2 Mass transfer coeffi cient (hd,o) For the cooling and dehumidifying of moist air by a cold surface involves simultaneously heat and mass transfer, and can be described by the process line equation from Threlkeld 20: dia dWa R ia? is;w Wa? Ws;w ig? 2;501R;18 Where R represent the ratio of sensible heat transfer characteristics to the mass transfer performance. R hc;o hd;oCp;a :19 However, for the present fi n-and-tube heat ex- changer, Eq. 18 did not correctly describe the dehu- midifi cation process on the psychrometric chart. This is because the saturated air enthalpy (is,w) at the mean temperature at the fi n surface is diff erent from that at the fi n base. In this regard, a modifi cation of the process line on the psychrometric chart corresponding to the fi n-and- tube heat exchanger is made. The derivation is as fol- lows. From the energy balance of the dehumidifi cation one can arrive at the following expression: _ madia hc;o Cp;a dAp;oia;m? is;p;o;m hc;o Cp;a dAfia;m? is;w;m: 20 Note that the fi rst term on the right-hand side de- notes the sensible heat transfer whereas the second term is the latent heat transfer. Conservation of the water condensate gives: _ madWa hd;odAp;oWa;m? Ws;p;o;m hd;odAfWa;m? Ws;w;m:21 Dividing Eq. 20 by Eq. 21 yields dia dWa R ? ia;m? is;p;o;m R ? e ? 1 ? ia;m? is;w;m Wa;m? Ws;p;o;m e ? 1 ? Wa;m? Ws;w;m ; 22 where e Ao Ap;o :23 By assuming a value of the ratio of heat transfer to mass transfer, R and by integrating Eq. 22 with an iterative algorithm, the mass transfer coeffi cient can be obtained. Analogous procedures for obtaining the mass transfer coeffi cients are given as: 1. Obtain Ws,p,o,mand Ws,w,mfrom is,p,o,mand is,w,m from those calculation of heat transfer. 2. Assume a value of R. 3. Calculations is performed from the fi rst element to the last element, employing the following procedures: 3.1. Assume an outlet air humidity ratio. 3.2. Calculate the outlet air humidity ratio of each element by Eq. 22. 3.3. If the outlet air humidity ratio obtained from step 3.2 is not equal to the assumed value of step 3.1, the calculation steps 3.1 and 3.2 will be re- peated. 4. If the summation of the outlet air humidity ratio for each element of the last row is not equal to the measured outlet air humidity ratio, assuming a new R value and the calculation step 3 will be repeated until the summation of the outlet air humidity ratio of the last row is equal to the measured outlet air humidity ratio. 3.3 Chilton-Colburn j-factor for heat and mass transfer (jhand jm) The heat and mass transfer characteristics of the heat exchanger is presented by the following non-dimensional group: jh hc;o GmaxCp;a Pr2=3;24 jm hd;o Gmax Sc2=3:25 761 4 Results and discussions Heat transfer performance of the fi n-and-tube heat exchangers is in terms of dimensionless parameter jh. A typical plot for examination of the infl uence of fi n pitch is shown in Fig. 5. In this fi gure, the reduced results by the present tube-by-tube method and those by the ori- ginal Threlkeld method having N=2 is shown. For heat transfer performance, reduced results from both meth- ods are nearly the same. This is somehow expected be- cause the present tube-by-tube approach is originated from the Threlkeld method. From the results, one can see that the heat transfer performance is relatively insensitive to the fi n pitch. Notice that this phenomenon is quite diff erent from that tested in fully dry conditions. As reported by Wang et al. 22 and Rich 17, the heat transfer performance is independent of fi n pitch when N 4 operated at fully dry conditions. However, for N=1 or 2, Wang and Chi 21 reported that the heat transfer performance drops with the increase of fi nspacing.Thisisespeciallypronouncedwhen ReDc5,000. For ReDc5,000, the heat transfer performance increases with decrease of fi n pitch. This phenomenon is seen for N 2, and is espe- cially pronounced for N=1. By contrast, the present sensible heat transfer performance exhibits a compara- tively insensitive infl uence to the change of fi n spacing for N=1 and 2. Apparently, the results are attributed to the presence of condensate under dehumidifi cation. This is because the appearance of condensate plays a role to alter the airfl ow pattern, roughening the fi n surface and providing a better mixing of the airfl ow. As a conse- quence, the infl uence of fi n pitch is reduced accordingly. This phenomenon is analogous to using the enhanced fi n surface in fully dry condition. For enhanced surfaces such as slit and louver fi n geometry, Du and Wang 5 and Wang et al. 24, 25 reported a negligible eff ect of fi n pitch even for N=1 or 2. Mass transfer performance of the present dehumidi- fying coils is termed as dimensionless jmfactor. For examination of the infl uence of inlet humidity on the mass transfer characteristics between the present method and that of original Threlkeld method, a typical com- parison for sample no. 5 and 10 is illustrated in Fig. 6. As seen in the fi gure, results using the present tube-by- tube method show relatively small infl uence of the inlet relative humidity. This is applicable for both 1-row and 2-row confi guration. By contrast, for the reduced results by the original Threlkeld method, one can see about 20 40% increase of mass transfer performance when the inlet relative humidity is increased from 50% to 90%. For the heat transfer performance, as aforementioned previously, the eff ect of inlet relative humidity is almost negligible regardless the reduction method is chosen. Hence, it is expected that the associated infl uence on the mass transfer performance is also small. With the ori- ginal procedures of Threlkeld method that was appli- cable to the counter-cross fl ow arrangement and of exclusive of the eff ect of primary surface, the reduced results are somewhat misleading. Hence the present tube-by-tube method is more appropriate than the ori- ginal procedures of Threlkeld method in reducing the mass transfer coeffi cient under fully wet conditions. The departure of the reduced results between Threlkeld method and the present method increases with the mass transfer rate. This can be made clear from Fig. 7 with a Fig. 5 Eff ect of the fi n pitch on jhbetween those derived by Threlkeld method and by present method Fig. 6 Eff ect of the inlet relative humidity on jmbetween those derived by Threlkeld method and by present method for samples no. 5 and 10 762 very close fi n spacing o