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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 fin-and-tube heat exchangers in dehumidifying conditions Received: 19 August 2004/ Accepted: 24 November 2004/Published online: 4 March 2005 C211 Springer-Verlag 2005 Abstract This study proposed a new method, namely a tube-by-tube reduction method to analyze the perfor- mance of fin-and-tube heat exchangers having plain fin configuration under dehumidifying conditions. The mass transfer coe?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 dehumidification is comparatively independent of fin pitch. The ratio of the heat transfer characteristic to mass transfer characteristic (h c,o /h d,o C p,a ) is in the range of 0.6C241.0, and the ratio is insensitive to change of fin spacing at low Reynolds number. However, a slight drop of the ratio of (h c,o /h d,o C p,a ) is seen with the decrease of fin spacing when the Reynolds number is su?cient high. This is associated with the more pronounced influence due to condensate removal by the vapor shear. Corre- lations are proposed to describe the heat and mass performance for the present plate fin configurations. These correlations can describe 89% of the Chilton Colburn j-factor of the heat transfer (j h ) within 15% and can correlate 81% of the Chilton Colburn j-factor of the mass transfer (j m ) within 20%. Keywords Fin-and-tube heat exchanger ? Dehumidifying ? Sensible heat transfer performance ? Mass transfer performance Nomenclature A f Surface area of fin A o Total surface area A p,i Inside surface area of tubes A p,o Outside surface area of tubes b￠ p Slope of the air saturation curved between the outside and inside tube wall temperature b￠ r Slope of the air saturation curved between the mean water temperature and the inside wall temperature b￠ w,m Slope of the air saturation curved at the mean water film temperature of the fin surface b￠ w,p Slope of the air saturation curved at the mean water film temperature of the tube surface C p,a Moist air specific heat at constant pressure C p,w Water specific heat at constant pressure D c Tube outside diameter (include collar) D i Tube inside diameter f i In-tube friction factors of water F Correction factor G max Maximum mass velocity based on minimum flow area h c,o Sensible heat transfer coefficient h d,o Mass transfer coefficient h i Inside heat transfer coefficient h o,w Total heat transfer coefficient for wet external fin I o Modified Bessel function solution of the first kind, order 0 I 1 Modified Bessel function solution of the first kind, order 1 i a Air enthalpy i a,in Inlet air enthalpy i a,m Mean air enthalpy i a,out Outlet air enthalpy i g Saturated water vapor enthalpy W. Pirompugd ? S. Wongwises ( is less than 0.05, where _ Q w is the water-side heat transfer rate for _ Q w and air-side heat transfer rate _ Q a ), are considered in the final analysis. Detailed geometry used for the present plain fin-and-tube heat exchangers is tabulated in Table 1. The test fin-and-tube heat exchangers are tension wrapped having a ‘‘L’’ type fin 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 Mo?at [15], are tabulated in Table 2. 3 Data reduction 3.1 Heat transfer coe?cient (h c,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: 27±0.5C176C 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: 7±0.5C176C Water velocity inside the tube: 1.5–1.7 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 _ Q a and _ Q w ; namely, _ Q a ? _m a (i a;in C0 i a;out ), e1T _ Q w ? _m w C p;w eT w;out C0 T w;in T; e2T _ Q avg ? _ Q a t _ Q w 2 : e3T The overall heat transfer coe?cient, U o,w , is based on the enthalpy potential and is given as follows: _ Q avg ? U o;w A o Di m F; e4T where Di m is the mean enthalpy di?erence for counter flow coil, Di m ? i a;m C0 i r;m : e5T According to Bump [4] and Myers [16], for the counter flow configuration, the mean enthalpy is i a;m ? i a;in t i a;in C0i a;out ln i a;in C0 i r;out C0C1C14 i a;out C0i r;in C0C1 C0 ei a;in C0 i a;out Tei a;in C0 i r;out T ei a;in C0 i r;out TC0(i a;out C0 i r;in T ; e6T i r;m ? i r;out t i r;out C0 i r;in ln i a;in C0i r;out C0C1C14 i a;out C0 i r;in C0C1 C0 ei r;out C0i r;in )(i a;in C0i r;out ) ei a;in C0 i r;out )C0ei a;out C0i r;in T ; e7T where F in Eq. 4 is the correction factor accounting for the present cross-flow unmixed/unmixed configuration. The overall heat transfer coe?cient is related to the individual heat transfer resistance [16] as follows: 1 U o;w ? b 0 r A o h i A p;i t b 0 p A o ln D c =D i eT 2pk p L p t 1 h o;w A p;o . b 0 w;p A o C16C17 t A f g f;wet . b 0 w;m A o C16C17; e8T where h o,w ? 1 C p;a . b 0 w;m h c;o C16C17 t y w =k w eT ; e9T y w in Eq. 9 is the thickness of the water film. A constant of 0.005 in. was proposed by Myers [16]. In practice, (y w /k w ) accounts for only 0.5–5% compared to (C p,a /b￠ w,m h c,o ), and has often been neglected by previ- ous investigators. As a result, this term is not included in the final analysis. In this study, we had proposed a row-by-row and tube-by-tube reduction method for detailed evaluation of the performance of fin-and-tube heat exchanger in- stead of conventional lump approach. Hence analysis of the fin-and-tube heat exchanger is done by dividing it into many tiny segments (number of tube row · number of tube per row · number of fin) as shown in Fig. 2.In the analysis, F is the correction factor accounting for a single-pass, cross-flow heat exchanger for one fluid mixed, other fluid unmixed that was shown by Threlkeld [20]. The tube-side heat transfer coe?cient, h i evaluated with the Gnielinski correlation [8], Fig. 2 Dividing of the fin-and-tube heat exchanger into the small pieces Table 2 Summary of estimated uncertainties Primary measurements Derived quantities Parameter Uncertainty Parameter Uncertainty Re Dc =400 Uncertainty Re Dc =5,000 _m a 0.3–1% Re Dc ±1.0% ±0.57% _m w 0.5% Re Di ±0.73% ±0.73% DP 0.5% _ Q w ±3.95% ±1.22% T w 0.05C176C _ Q a ±5.5% ±2.4% T a 0.1C176C j ±11.4% ±5.9% Table 1 Geometric dimensions of the sample plain fin-and-tube heat exchangers No. Fin thickness (mm) Sp (mm) Dc (mm) Pt (mm) Pl (mm) Row no. 1 0.115 1.08 8.51 25.4 19.05 1 2 0.120 1.63 10.34 25.4 22.00 1 3 0.115 1.93 8.51 25.4 19.05 1 4 0.115 2.12 10.23 25.4 19.05 1 5 0.120 2.38 10.34 25.4 22.00 1 6 0.115 1.12 8.51 25.4 19.05 2 7 0.120 1.58 8.62 25.4 19.05 2 8 0.115 1.95 8.51 25.4 19.05 2 9 0.120 3.01 8.62 25.4 19.05 2 10 0.130 2.11 10.23 25.4 22.00 2 11 0.115 1.12 10.23 25.4 19.05 4 12 0.115 1.44 10.23 25.4 19.05 4 13 0.115 2.20 10.23 25.4 19.05 4 14 0.130 2.10 10.23 25.4 22.00 4 15 0.130 1.72 10.23 25.4 22.00 6 16 0.130 2.08 10.23 25.4 22.00 6 17 0.130 3.03 10.23 25.4 22.00 6 759 h i ? ef i =2TeRe Di C01000TPr 1:07t12:7 ???????? f i =2 p ePr 2=3 C01T C1 k i D i ; e10T and the friction factor, f i is f i ? 1 e1:58ln Re Di C03:28T 2 : e11T The Reynolds number used in Eqs. 10 and 11 is based on the inside diameter of the tube and Re Di ? qVD i =l: In all case, the water side resistance is less than 10% of the overall resistance. In Eq. 8 there are four quantities (b￠ w,m , b￠ w,p , b￠ p and b￠ r ) involving enthalpy-temperature ratios that must be evaluated. The quantities of b￠ p and b￠ r can be calculated as b 0 r ? i s;p;i;m C0 i r;m T p;i;m C0T r;m ; e12T b 0 p ? i s;p;o;m C0 i s;p;i;m T p;o;m C0 T p;i;m : e13T The values of b￠ w,p and b￠ w,m are the slopes of satu- rated enthalpy curve evaluated at the outer mean water film temperature at the base surface and at the fin sur- face. Without loss of generality, b￠ w,p can be approxi- mated by the slope of saturated enthalpy curve evaluated at the base surface temperature [23]. The wet fin e?ciency (g f,wet ) is based on the enthalpy di?erence proposed by Threlkeld [20]. i.e., g f,wet ? i C0 i s,fm i C0i s,fb ; e14T where i s,fm is the saturated air enthalpy at the mean temperature of fin and i s,fb is the saturated air enthalpy at the fin base temperature. The use of the enthalpy potential equation, greatly simplifies the fin e?ciency calculation as illustrated by Kandlikar [10]. However, the original formulation of the wet fin e?ciency by Threlkeld [20] was for straight fin configuration (Fig. 2a). For a circular fin (Fig. 2b), the wet fin e?ciency is [23], g f;wet ? 2r i M T (r 2 o C0r 2 i ) C2 K 1 (M T r i )I 1 (M T r o )C0K 1 (M T r o )I 1 (M T r i ) K 1 (M T r o )I 0 (M T r i )tK 0 (M T r i )I 1 (M T r o ) C20C21 ; e15T where M T ? ??????????? 2h o;w k f t r ; e16T The test heat exchangers are of Fig. 3c configura- tion. Hence, the corresponding fin e?ciency is calcu- lated by the equivalent circular area method as depicted in Fig. 4. Evaluation of b￠ w,m requires a trial and error proce- dure. For the trial and error procedure, i s,w,m must be calculated using the following equation: i s;w;m ? i a;m C0 C p;a h o;w g f;wet b 0 w;m h c;o C2 1C0 U o;w A o b 0 r h i A p;i t b 0 p ln D c =D i eT 2pk p L p "# ! C2ei a;m C0i r;m T: e17T An algorithm for solving the sensible heat transfer coe?cient h c,o for 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 _ Q total using Eq. (3). 2. Assume a h c,o for all elements. 3. Calculate the heat transfer performance for each segment with the following procedures. 3.1. Calculate the tube side heat transfer coe?cient of h i using Eq. 10. 3.2. Assume an outlet air enthalpy of the calculated segment. 3.3. Calculate i a,m by Eq. 6 and i r,m by Eq. 7. 3.4. Assume T p,i,m and T p,o,m . 3.5. Calculate b 0 r A o C0C1 = h i A p;i C0C1 and b 0 p A o ln D c =D i eT hi = h 2pk p L p C138. 3.6. Assume a T w,m . 3.7. Calculate the g f,wet using Eq. 15. 3.8. Calculate U o,w from Eq. 8. 3.9. Calculate i s,w,m by Eq. 17. 3.10. Calculate T w,m from i s,w,m . Fig. 3 Type of fin configuration Fig. 4 Approximation method for treating a plate fin of uniform thickness 760 3.11. If T w,m derived 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 T w,m derived in step 3.10 until T w,m is constant. 3.12. Calculate _ Q of this segment. 3.13. Calculate T p,i,m and T p,o,m from the inside convection heat transfer and the conduction heat transfer of tube and collar. 3.14. If T p,i,m and T p,o,m derived in step 3.13 are not equal that is assumed in step 3.4, the calculation step 3.5–3.13 will be repeated with T p,i,m and T p,o,m derived in step 3.13 until T p,i,m and T p,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 3.3–3.15 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 _ Q total , h c,o will be assumed a new value and the cal- culation step 3 will be repeated until the summation of _ Q for all elements is equal _ Q total . 3.2 Mass transfer coe?cient (h d,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]: di a dW a ? R ei a C0 i s;w T eW a C0 W s;w T tei g C02;501RT; e18T Where R represent the ratio of sensible heat transfer characteristics to the mass transfer performance. R ? h c;o h d;o C p;a : e19T However, for the present fin-and-tube heat ex- changer, Eq. 18 did not correctly describe the dehu- midification process on the psychrometric chart. This is because the saturated air enthalpy (i s,w ) at the mean temperature at the fin surface is di?erent from that at the fin base. In this regard, a modification of the process line on the psychrometricchart corresponding to the fin-and- tube heat exchanger is made. The derivation is as fol- lows. From the energy balance of the dehumidification one can arrive at the following expression: _m a di a ? h c;o C p;a dA p;o ei a;m C0i s;p;o;m Tt h c;o C p;a dA f ei a;m C0 i s;w;m T: e20T Note that the first 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: _m a dW a ? h d;o dA p;o eW a;m C0 W s;p;o;m T t h d;o dA f eW a;m C0 W s;w;m T: e21T Dividing Eq. 20 by Eq. 21 yields di a dW a ? R C1ei a;m C0 i s;p;o;m TtR C1ee C01TC1ei a;m C0 i s;w;m T eW a;m C0 W s;p;o;m Ttee C01TC1eW a;m C0 W s;w;m T ; e22T where e ? A o A p;o : e23T 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 coe?cient can be obtained. Analogous procedures for obtaining the mass transfer coe?cients are given as: 1. Obtain W s,p,o,m and W s,w,m from i s,p,o,m and i s,w,m from those calculation of heat transfer. 2. Assume a value of R. 3. Calculations is performed from the first 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 (j h and j m ) The heat and mass transfer characteristics of the heat exchanger is presented by the following non-dimensional group: j h ? h c;o G max C p;a Pr 2=3 ; e24T j m ? h d;o G max Sc 2=3 : e25T 761 4 Results and discussions Heat transfer performance of the fin-and-tube heat exchangers is in terms of dimensionless parameter j h .A typical plot for examination of the influence of fin pitch is shown in Fig. 5. In this figure, 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 fin pitch. Notice that this phenomenon is quite di?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 fin 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 fin spacing. This is especially pronounced when Re Dc 5,000. For Re Dc <5,000, the heat transfer performance increases with decrease of fin 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 influence to the change of fin spacing for N=1 and 2. Apparently, the results are attributed to the presence of condensate under dehumidification. This is because the appearance of condensate plays a role to alter the airflow pattern, roughening the fin surface and providing a better mixing of the airflow. As a conse- quence, the influence of fin pitch is reduced accordingly. This phenomenon is analogous to using the enhanced fin surface in fully dry condition. For enhanced surfaces such as slit and louver fin geometry, Du and Wang [5] and Wang et al. [24, 25] reported a negligible e?ect of fin pitch even for N=1 or 2. Mass transfer performance of the present dehumidi- fying coils is termed as dimensionless j m factor. For examination of the influence 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 figure, results using the present tube-by- tube method show relatively small influence of the inlet relative humidity. This is applicable for both 1-row and 2-row configuration. 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 e?ect of inlet relative humidity is almost negligible regardless the reduction method is chosen. Hence, it is expected that the associated influence on the mass transfer performance is also small. With the ori- ginal procedures of Threlkeld method that was appli- cable to the counter-cross flow arrangement and of exclusive of the e?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 coe?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 E?ect of the fin pitch on j h between those derived by Threlkeld method and by present method Fig. 6 E?ect of the inlet relative humidity on j m between those derived by Threlkeld method and by present method for samples no. 5 and 10 762 very close fin spacing of 1.08 mm. As seen in Fig. 7 at Re Dc <1,000, the results indicate a departure of the re- duced results for more than 50% between these two methods. Moreover, there is negligible influence of inlet humidity for the present method when Re Dc 1,000 when RH=50%. This is in connection with the blow-o? of condensate at larger Re Dc which make more zoom for water vapor to condensate along the surface and even resultis in a partially dry consitions due to the rise of dew point temperature. This phenomenon becomes less pro- nounced with the rise of the number of tube row for condensate blow-o? may be blocked by the subsequent tube row. The dehumidifying process involves heat and mass transfer simultaneously, if mass transfer data are unavailable, it is convenient to employ the analogy be- tween heat and mass transfer. The existence of the heat and mass analogy is because the fact that conduction and di?usion in a liquid are governed by physical laws of identical mathematical form. Therefore, for air-water vapor mixture, the ratio of h c,o /h d,o C p,a is generally around unity, i.e., h c;o h d;o C p;a C25 1: e26T The term in Eq. 19 approximately equals to unity for dilute mixtures like water vapor in air near the atmo- spheric pressure (temperature well-below corresponding boiling point). The validity of Eq. 26 relies heavily on the mass transfer rate. The experim