The Analysis of TwoPhase Condensation Heat Transfer Models Based on the Comparison of the Boundary Condition


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1 Acta Polytechnica Hungarica Vol. 9, No. 6, 2012 The Analysis o TwoPhase Condensation Heat Transer Models Based on the Comparison o the Boundary Condition Róbert Sánta College o Applied Sciences Subotica Tech 16 Marka Oreskovica, Subotica, Serbia Abstract: The article aims to present, analyze and select twophase condensation heat transer coeicients or the rerigerantside in the condenser o the heating and cooling system o a heat pump. The heat transer models are analyzed or condensation in horizontal smooth tubes. The mathematical models published by various authors reer to the range o heterogeneous condensation. The inal aim o the analysis is to select the optimal model rom the examined twophase condensation heat transer models. The selection o the condensing rerigerant twophase heat transer model is based on the boundary condition. The applied method o analysis is numericalgraphical. Keywords: heat pump; condenser; R134a; heat transer; boundary condition Nomenclature D inside diameter o tube [m] Greek Symbols x vapor quality [] α heat transer coeicient[w/m 2 K] G mass velocity [kg/m 2 s] λ thermal conductivity [W/mK] T temperature [K] ρ density [Ns/m 2 ] p pressure [Pa] ε void raction [] p* reduced pressure [] η dynamic viscosity [Ns/m 2 ] Re Reynold`s number [] Pr Prandt number [] Subscript F two phase multiplier g vapor Fr Froude number [] liquid Z Shah`s correlating parameter [] k two phase X tt Martinelli`s correlating parameter [] C Constant [] 167
2 R. Sánta The Analysis o Twophase Condensation Heat Transer Models Based on the Comparison o the Boundary Condition 1 Introduction The condenser heat exchanger plays a signiicant role in the structure and operation o the heat pump as it aects the system s coeicient o perormance (COP). The motivation or the composition o this article was the selection o the optimal twophase heat transer mathematical model o rerigerant among the analyzed mathematical models in the horizontal tube o the condenser. The structure and dimensions o the condenser have a signiicant impact on the heat transer intensity. The parameters that inluence heat transer are the heat transer coeicients o the rerigerant and o the heated water. The earliest determination o the condensation heat transer coeicient o the twophase rerigerant lowing in horizontal smooth tubes was carried out by Boyko and Kruzhilin [1] and Akers et al. [2]. For the stationery condensation, there are a large number o mathematical models or dierent conditions and rerigerants, authored by Cavallini and Zecchin [3, 4], Shah [5], M. K. Dobson, J. C. Chato [6], and J. R. Thome et al. [7]. The common characteristics o these heat transer models are that under the same conditions they give a high dierence in values o heat transer coeicient. M. M. Awad et al. [8] in their mathematical model used the heat transer correlations. A number o researchers have dealt with this ield o science and implemented heat transer coeicients in numerous mathematical models o heat pump; see J. Nyers et al. [912]. 2 The Physical Model o Condenser with Horizontal Smooth Tubes The condenser is a heat exchanger where the higher temperature rerigerant gives heat to the lower temperature heating medium. The analyzed condenser has parallel, straight and smooth tube bundles with bales. The rerigerant lows inside tubes, while the heated water lows in the shell tube. The process in the condenser on the rerigerant side is made up o three sections: A the Superheated vapor section B the Condensation section C the Subcooled liquid section Ater compression, the vapor is singlephase and superheated. In the superheated section, the value o vapor quality equals
3 Acta Polytechnica Hungarica Vol. 9, No. 6, 2012 The vapor is in contact with the tube wall, whose temperature is lower than the saturation temperature, and thereore the heat transers to the wall and vapor condenses. Heat is then transerred rom the tube wall to the water by conduction. In the condensation section, the condensate and the vapor have a heterogeneous low, which is characterized by intensive turbulence. At the beginning o the condensation section, the low pattern is annular, because in the core the velocity o the vapor is much higher than the velocity o the liquid. In the annular low regime at the liquidvapor interace, the dominant orce is tangential stress, with the gravitational orce playing a less important role. As condensation continues, the velocity o the vapor phase reduces and the dominant orce shits rom tangential orce to gravitational orce. The liquid phase accumulates at the bottom o the tube. Condensation takes place mainly at the top o the tube because the liquid layer is thin, and thereore heat resistance is smaller. Figure 1 Condensation in ully annular low in horizontal tube In the continuation o the condensation, the vapor continually condenses, and in the cross section area the surace o liquid increases, and the low pattern changes to slug and plug pattern. The size o the vapor slug reduces urther and a bubbly low pattern develops. At the end o condensation section, the vapor quality reduces to zero and the low in the tube becomes a singlephase low (liquid low). Figure 2 Flow pattern map or condensation in a horizontal tube 169
4 R. Sánta The Analysis o Twophase Condensation Heat Transer Models Based on the Comparison o the Boundary Condition A subcooling section is only ormed in exceptional circum stances, maybe in the crosslow condensers. In the case o shelltube condensers, the subcooling is realized in a separate heat exchanger (subcooler). In the subcooler, the rerigerant is in the singlephase (liquid phase). 3 Annular Flow Condensation One o the most important low regimes is annular low, which is characterized by a phase interace separating a thin liquid ilm rom the gas low in the core region. This low regime is the most investigated one both analytically and experimentally because o its practical signiicance. Much o the condensation process occurs in the annular low regime. Thereore, many o the existing intube condensation correlations are based on the annular low regime. These correlations are classiied into three categories, that is, shearbased correlations, boundary layerbased correlations, and twophase multiplierbased correlations. The majority o smoothtube heat transer correlations are o the twophase multiplierbased variety. 4 The TwoPhase Multiplierbased Heat Transer Correlations in Condensation Section The simplest method o heat transer prediction in the annular low regime is the twophase multiplier approach. Twophase multiplierbased correlations were pioneered or predicting convective evaporation data (Dengler and Addoms 1956) and were adapted or condensation by Shah (1979). The twophase multiplier based heat transer correlations typically result in the ollowing orm: Where: α = α k = α F ρ n m ( C, Re, Pr ) and η F = k x,,, Fr ρ g η g The twophase multiplier can depend on more dimensionless groups than those indicated in Eq. 1; the shown groups are the most prevalent. Several examples o twophase multiplierbased condensing correlations are available, including those o Akers et al. (1959), Boyko and Kruzhilin (1967), Cavallini and Zecchin (1974), Tang (1998) and Dobson and Chato (1998). k (1) 170
5 Acta Polytechnica Hungarica Vol. 9, No. 6, The Cavallini and Zecchin Correlation Cavallini and Zecchin developed a semiempirical equation that has a simple orm. The mathematical model or heat transer by Cavallini and Zecchin correlation is: α k = λ 0.05 Re Pr (2) D The equivalent Reynolds number is: 0.5 ρ ηg Re = Re g + Re ρg η where Re v and Re l are the Reynolds number o the liquid and vapor phase, respectively, which can be calculated by Eq. 4. G ( 1 x) d i Re l =, μl G x d i Re g = (4) μl In Eq. 2 the application range o Cavallini and Zecchin`s correlation was summarized as ollows: 10 d i = 8mm, 30 < < 50 C, < l v 3 T sat ρ ρ < 2 10, 10 μ μ < 2 10, < l v < 3 l < Rel 0.8 Pr < 20, Flow regime: Annular low Rerigerants tested: R11, R12, R21, R22, R113, R114, R134a, R410A, R 407C 3 (3) 4.2 Shah Correlation The Shah correlation takes into account the pressure o the rerigerant, in addition to the quality o the mixture. This can also be used to ind the local condensation heat transer coeicient. The heat transer coeicient is a product o liquid heat transer coeicient given by the DittusBoelter equation and an additional term. In 1979, Shah presented the ollowing correlation: α α k = 1+ Z (5) 171
6 R. Sánta The Analysis o Twophase Condensation Heat Transer Models Based on the Comparison o the Boundary Condition The liquid heat transer coeicient is calculated by using the DittusBoelter equation: λ α = Re Pr (6) D The mathematical model or heat transer by the Shah correlation is: ( x) x 1 α k = α ( 1 x) + *0.38 (7) p The application range o Shah s correlation was summarized as ollows: 7 < d i < 40, 10 < P sat < 9.87MPa, 21 < T sat < 310 C, 10.8 < G kg cr < 2, m s 0 l.5 < Pr, 350 < Rel, Flow regime: Annular low Rerigerants tested: R718, R11, R12, R22, R113, methanol, ethanol, benzene, toluene, and ethylene 4.3 The Boyko and Kruzhilin Correlation The Boyko and Kruzhilin correlation is an adaptation o the Mikheev correlation. The correlation is simple to use, generally conservative, and suiciently accurate. This correlation takes into account the heat transer coeicient in singlephase low, the density o the twophase low and vapor quality. The mathematical model or heat transer by the Boyko and Kruzhilin correlation is: ρ l α 1 k = α + x (8) ρ g The liquid heat transer coeicient is obtained as ollows: λ α = Re Pr (9) D The application range o Boyko and Kruzhilin s correlation was summarized as ollows: 1500 < Re < Flow regime: Annular low Rerigerants tested: steam, R22 172
7 Acta Polytechnica Hungarica Vol. 9, No. 6, Akers Correlation Akers et al. (1959) developed a twophase multiplierbased correlation that became known as the equivalent Reynolds number model. This model deines the allliquid mass low rate that provides the same heat transer coeicient as an annular condensing low. The mathematics model or heat transer by the Akers correlation is: α k = λ 0.05 Re Pr (10) D where: the equivalent mass velocity is: G = G ( x) ekv. 1 ρ + x ρ g The multiplier actors unction o the Reynolds number is: C = and n = 0. 8 i Re (11) 1 C = 5.03 and n = i Re Flow regime: Annular low Rerigerants tested: R12, Propane, Methanol 4.5 Dobson et al. Dobson et al. (1998) proposed separate correlations or the wavy and annular low regimes. For the annular low regime, they correlated condensation data by assuming that the ratio o the twophase Nusselt number to the Nusselt number predicted by a singlephase correlation is exclusively a unction o the Martinelliparameter. Utilizing the DittusBoelter correlation to predict the singlephase Nusselt number, regression analysis o annular low data yielded the ollowing orm: 0.8 l 0.4 l Nu = Re Pr g( X ) (12) Where: g ( X tt ) = 1+ X tt tt The unction g at the end o equation 12 represents the twophase multiplier, and the values o the constant and the exponent are similar to the values rom convective evaporation correlations [Wattelet et al., 1992]. 173
8 R. Sánta The Analysis o Twophase Condensation Heat Transer Models Based on the Comparison o the Boundary Condition The mathematical model or heat transer by the Dobson et al. correlation or annular low is: λ = α k Rel Prl (13) D X tt where the application range o Dobson et al. s correlation was summarized as ollows: d i = 4.57mm, 35 < T sat < 60 C, 75 < G 500 kg cr < 2, 0.1 < x < 0.9, m s Flow regime: Annular low Rerigerants tested: R22, R134a, R410A 5 Comparison o Heat Transer Correlations Many correlations that are available come with no explicit range o parameters over which they can be expected to give accurate results. Thereore, a design engineer requiring a suitable heat transer correlation typically encounters a series o seemingly contradictory reports about which correlation is best. The heat transer process in annular twophase low is similar to that in singlephase low o the liquid, and thus their ratio may be characterized by a twophase multiplier. This concept uses the same rationale as the LockhartMartinelli (1949) twophase multiplier, developed or the prediction o twophase rictional pressure drop. This work investigated the comparison o the twophase correlations on the basis o the abovementioned similarities. The deviation o values o the two phase heat transer correlation was examined in the values o boundary condition, i.e. singlephase heat transer coeicient values o liquid. The liquid phase heat transer coeicient was examined the reerence value. The singlephase heat transer coeicients are typically predicted by the Dittus and Boelter (1930) [13] correlation, which results in the ollowing orm: α k = λ Re Pr (14) D Shah [14] also compared the condensation heat transer correlations. Shah s basis o comparison was his own model, namely the twophase heat transer correlation. 174
9 Acta Polytechnica Hungarica Vol. 9, No. 6, Initial Condition and Values The mathematical models are simulated by the use o the sotware tool MathCAD. The initial conditions and values or the simulation o the stationary condensation are: Rerigerant: R134a Mass velocity: G = kg 2 m s Vapor quality ranged: x = 10 [] Pressure in the inlet o condenser: p k = 15 [bar] Inner diameter o tube: d= 6 [mm] Below, the reader can ind graphs drawn or the considered condensation heat transer correlations, showing the deviation o the condensation heat transer coeicients rom the reerence value and average o local condensation heat transer in unction o vapor quality and dierent mass velocity. 7 Results and Discussions Figure 3 kg Condensation heat transer coeicients low in tube with reerence value shown, G = 100 m 2 s 175
10 R. Sánta The Analysis o Twophase Condensation Heat Transer Models Based on the Comparison o the Boundary Condition Figure 4 Condensation heat transer coeicients low in tube with reerence value shown, G = 200 kg m 2 s Figure 5 Condensation heat transer coeicients low in tube with reerence value shown, G = 300 kg 2 m s 176
11 Acta Polytechnica Hungarica Vol. 9, No. 6, 2012 Figure 6 Condensation heat transer coeicients low in tube with reerence value shown, G = 400 kg 2 m s Figure 7 kg Condensation heat transer coeicients low in tube with reerence value shown, G = m s 177
12 R. Sánta The Analysis o Twophase Condensation Heat Transer Models Based on the Comparison o the Boundary Condition Figure 8 The deviation o the condensation heat transer coeicients rom the reerence value, when the vapor quality x=0 Conclusions Figure 9 The average condensation heat transer coeicient, G = 400 kg m 2 s The condensation heat transer coeicients linearly increases in the unction o the vapor quality o the rerigerant. For vapor quality x=0.01 (x approximately 0), the values o the condensation heat transer coeicient obtained rom the analyzed models are approximately equal to the values o the models which have been used exclusively or the liquid phase x=0, published by several authors. The Cavalinni et al. condensation heat transer coeicient is an exception to this rule. 178
13 Acta Polytechnica Hungarica Vol. 9, No. 6, 2012 In the case o pure liquid phase, i.e. x=0, the examined condensation heat transer correlation provide real values, whereas the Dobson et al. correlation is an exception. The Dobson et al. condensation heat transer correlation is not suitable or determining the single phase heat transer coeicient. The heat transer coeicient o the Shah condensation correlation diers in the smallest amount rom the reerence value out o the examined correlations. This deviation is only 8.36%, while the heat transer coeicient o the Cavalinni et al. correlation provides the highest level o deviation. The author examined the heat transer coeicients deviation rom the reerence value at x=0 vapor quality and dierent mass velocity o rerigerant. The Dobson correlation values are an exception to this, where the vapor quality was x=0.01. In the case o the pure vapor phase, i.e. x=1 vapor quality, the examined condensation heat transer correlation presents real values o heat transer coeicient; however, the Shah and Dobson correlation values are exceptions to this rule. The majority o the condensation heat transer correlations provide the highest condensation heat transer coeicient near x = 0.99, Shah s correlation being an exception. For vapor quality x=0.99, (x approximately 1),the calculated values o the condensation heat transer coeicients by using the investigated models are much higher than by the value o single heat transer coeicients suggested by other authors. The deviation between reerence value and calculated heat transer coeicients is constant with an increase in the mass velocity o the rerigerant. The average values o the local heat transer coeicients also provide signiicant dispersion. Based on the above analysis and acts, Shah s model is considered as an optimum. Reerences [1] Boyko, Kruzhilin: Heat Transer and Hydraulic Resistance during Condensation o Steam in a Horizontal Tube and in a Bundle o Tubes, International Journal Heat and Mass Transer 10, 1967, [2] Akers, Deans, Crosser: Condensing Heat Transer within Horizontal Tubes, Chem. Eng. Prog. Symp. Series 55, 1959, [3] Cavallini, Zecchin: High Velocity Condensation o Organic Rerigerants Inside Tubes, 8 th International Congress o Rerigeration, Vol. 2, Washington DC, 1971, pp [4] Cavallini, Zecchin: A Dimensionless Correlation or Heat Transer in Forced Convection Condensation. 6 th Int. Heat Transer Con. Tokyo, 1974, pp
14 R. Sánta The Analysis o Twophase Condensation Heat Transer Models Based on the Comparison o the Boundary Condition [5] Shah: A General Correlation or Heat Transer During Flow Condensation Inside Pipes, Journal o Heat and Mass Transer 22, 1979, [6] M. K. Dobson, J. C. Chato: Condensation in Smooth Horizontal Tubes, Journal o Heat Transer 120, 1998, [7] J. R. Thome, J. El Hajal, A. Cavallini: Condensation in Horizontal Tubes, Part 2: New Heat Transer Model Based on Flow Regimes, International Journal o Heat and Mass Transer 46, 2003, [8] M. M. Awad, H. M. Mostaa, G. I. Sultan, A. Elbooz, A. M. K. Elghonemy: Perormance Enhancement o Aircooled Condensers, Acta Polytechnica Hungarica, Volume 4, No. 2, pp , 2007 [9] J. Nyers, A. Nyers: COP o HeatingCooling System with Heat Pump, 3 th International Symposium Express 2011, Proceedings 1721, Subotica, Serbia [10] J. Nyers et al.: Energy Optimum o Heating System with Heat Pump, 6 th International Multidisciplinary Conerence, Proceedings , Baia MareNagy Bánya, Romania 2005 [11] J. Nyers, G. Stoyan: A Dynamical Model Adequate or Controlling the Evaporator o a Heat Pump, International Journal o Rerigeration, 1994, Vol. 17, No. 2, pp [12] J. Nyers et al.: Csőköteges elpárologtató hőátadási tényezőjének matematikai modelljei kétázisú hűtőközegre, Magyar Épületgépészet, LIX. évolyam, 2010/6. szám, Budapest, Hungary [13] Dittus, Boelter: Heat Transer in Automobile Radiators o the Tubular Type. Publications in Engineering, Vol. 2, University o Caliornia, Berkeley, p. 443 (1930) [14] Shah: An Improved and Extended General Correlation or Heat Transer during Condensation in Plain Tubes, HVAC&R Research, pp
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