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Study on average Nusselt and Sherwood numbers in vertical plate channels with falling water film evaporation
International Journal of Heat and Mass Transfer 110 (2017) 783–788
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Study on average Nusselt and Sherwood numbers in vertical plate channels with falling water film evaporation Yangda Wan, Chengqin Ren ⇑, Yang Yang, Li Xing State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China
a r t i c l e
i n f o
Article history: Received 19 January 2017 Received in revised form 24 March 2017 Accepted 24 March 2017
Keywords: Falling water film evaporation Average Nusselt and Sherwood numbers Correlations Heat and mass transfer
a b s t r a c t In this paper, the effects of variable parameters on the average Nusselt and Sherwood numbers between moist air and liquid water film in vertical plate channels are investigated using an approach proposed by previous work. The validity of data is also investigated. Traditional correlations for the average Nusselt and Sherwood numbers are not comprehensive enough as they cannot indicate the effects of inlet parameters on heat and mass transfer process. In this research, correlations for the average Nusselt and Sherwood numbers are developed in terms of moist air inlet Reynolds number, ratio of water to moist air inlet mass flow rate, ratio of channel length to half channel width and moist air inlet dimensionless temperature. The definition of the moist air inlet dimensionless temperature takes into account the effects of moist air inlet dry-bulb and wet-bulb temperatures and water film inlet temperature. The correlations show that the moist air inlet dimensionless temperature has the greatest influence on the average Nusselt number compared with that of the other dimensionless parameters, but it has the lowest influence on the average Sherwood number. The results would be helpful to understand the heat and mass transfer behavior between moist air and liquid water film. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Falling film evaporation is widely used for effective cooling [1– 3]. The application of falling film evaporation for air conditioning can bring environmentally friendly products and reduce energy consumption. Therefore falling film evaporation devices are drawing more and more attention in recent years. Xuan et al. [4,5] presented research and application of evaporative cooling for air conditioning in China. Experimental and theoretical research works on feasibility studies, performance test and optimization as well as heat and mass transfer analysis were reviewed in detail. The energy saving potential and environmental impacts of typical evaporative cooling air conditioning systems were illustrated. Camargo et al. [6,7] developed a mathematical model for evaporative cooling air conditioning system, allowing the determination of the effectiveness of saturation. The experimental results are used to determine the heat transfer coefficient and to compare with the mathematical model. Dowdy and Karabash [8] proposed the correlation for determination of the heat transfer coefficient in a rigid cellulose evaporative medium based on the assumption that air flow is turbulent. The correlation has
⇑ Corresponding author. E-mail address: [email protected] (C. Ren). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.03.087 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
been used in literature [9–12]. Wu et al. [11] proposed a simplified cooling efficiency correlation based on material properties and configuration of wetted medium used in the cooler. The models that implement this correlation were presented in literature [13,14]. Considering that heat and mass transfer process are also affected not only by material, but also by operation conditions, such as the inlet parameters of moist air and liquid water, traditional correlations are not comprehensive enough as they cannot indicate the effects of the inlet parameters on heat and mass transfer process. Heat and mass transfer coefficients correlations developed by many studies were calculated by the logarithmic mean temperature difference or arithmetical mean temperature difference. Little study was concerned on the method of measuring coupled heat and mass transfer coefficients during falling water film evaporation process. In a previous article [15], we have developed a new approach to analyze the heat and mass transfer characteristic in vertical plate channels with falling water film evaporation. The objective of the present paper is to extend the previous work [15] to develop general correlations for heat and mass transfer coefficients. In dimensionless terms, this means developing correlations for the average Nusselt and Sherwood numbers in terms of their dependence on the influence parameters. A temperature dimensionless parameter hg;i takes into account the effects of moist
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Nomenclature heat and mass transfer area [m2] specific heat capacity [kJ kg1 °C1] water to air heat capacity rate ratio half width and length of the channel respectively [m] diffusion coefficient of water vapor in air [m2 s1] dimensionless ratio of water film heat transfer coefficient to air mass transfer coefficient defined as hc;w =hD cp;dg hfg evaporation heat of water [kJ kg1] hfg;0 evaporation heat of water at reference temperature (0 °C) [kJ kg1] hc moist air side heat transfer coefficient [kW m2 °C1] hc;w water film side heat transfer coefficient [kW m2 °C1] hD moist air side mass transfer coefficient [kg m2 s1] hfg;0 normalized evaporation heat of water at reference temperature condition [°C] i specific enthalpy [kJ kg1] Lewis factor defined as hc =hD cp;dg Lef m mass flow rate [kg m1 s1] NTU m number of mass transfer units NuM , ShM average Nusselt and Sherwood numbers respectively Re Reynolds number Rwg ratio of water to moist air mass flow rate
A cp C ast w D, L Ds E
air inlet dry-bulb and wet-bulb temperatures and water film inlet temperature. The results would be helpful to understand the heat and mass transfer behavior between moist air and liquid water film. 2. Methodology 2.1. 2-D model The model investigated is a vertical plate channel with the length L and half channel width D as shown schematically in Fig. 1. The moist air flows upward and the liquid water film flows along the surface of the channel. The channel walls are thermally
T u,
v
W x, y
temperature [°C] velocity components in x, y coordinate directions, respectively [m s1] humidity ratio of moist air [kg kg1] space coordinate as shown in (Fig. 1) [m]
Greek letters aL ratio of channel length to half channel width d water film thickness [m] h dimensionless temperature k thermal conductivity [W m1 °C1] t kinematic viscosity [m2 s1] q density [kg m3] U relative humidity [%] Subscripts g; dg moist air, dry air respectively i, o inlet, outlet respectively I condition at the gas-liquid interface water film, water vapor respectively w, v wg moist air wet bulb temperature
insulated in order to investigate the evaporative cooling process associated with many engineering applications such as direct evaporative cooler, cooling tower, etc. It is assumed that the flow is incompressible and the air-vapor mixture is an ideal gas mixture. The thermo-physical properties of the dry air, water and water vapor are assumed to be constant. Dufour and Soret effects are neglected. The interface between the water film and the moist air is at thermodynamic equilibrium state [16–18]. The governing equations for the 2-D CFD model have been described in our previous study [15]. In view of the large number of parameters, a similarity analysis was performed to deduce the 2-D model equations into dimensionless forms to find the dimensionless factors affecting the average Nusselt and Sherwood num-
Fig. 1. (a) Schematic diagram of the physical. (b) Differential element of channel.
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bers in the previous study. For brevity, the correlations for the average Nusselt and Sherwood numbers are repeated as follows:
NuM ¼ f ðRwg ; Reg ; aL ; hg;i Þ
ð1Þ
ShM ¼ f ðRwg ; Reg ; aL ; hg;i Þ
ð2Þ
where Rwg ¼ mw =mg represents ratio of liquid water to moist air mass flow rate; Reg ¼ 4ðD dÞv g;i =tg represents moist air inlet Reynolds number; aL ¼ L=D represents ratio of channel length to half channel width; hg;i ¼ ðT g;i T wg;i Þ=ðT w;i T wg;i Þ represents moist air inlet dimensionless temperature. The average Nusselt and Sherwood numbers are defined as:
NuM ¼
4ðD dÞhc kg
ð3Þ
ShM ¼
4ðD dÞhD Ds qg
ð4Þ
2.2. 1-D model The 1-D model developed by our previous study [15,19] will be used in this paper for calculating the average Nusselt and Sherwood numbers, as shown in Fig. 1(b). To simplify the complexity of the heat and mass transfer analysis, the two additional assumptions for the 1-D model are made: (1) humidity ratio of air in equilibrium with the water surface is assumed to be a linear function of the water surface temperature, (2) constant heat and mass transfer coefficients along the heat exchanger surface. By principles of mass and energy conversion, a set of differential equations can be obtained: Energy balance equation for moist air
mg dig ¼ hc ðT w T g Þ þ hfg;I hD ðW I W g Þ dA
ð5Þ
Mass balance equation for moist air and liquid water
mg dW g ¼ hD ðW I W g ÞdA ¼ dmw
ð6Þ
Energy balance equation for the differential element
hc;w ðT w T I Þ ¼ hc ðT I T g Þ þ hfg;I hD ðW I W g Þ
ð7Þ ð8Þ
For specific enthalpy of moist air, the following equation applies
ig ¼ ðcp;dg þ W g cpv ÞT g þ W g hfg;0
ð10Þ
dW g ¼ ðW I W g ÞdNTU m
ð11Þ
dT w ¼
~ E B2 ðT I T w ÞdNTU m C w
ð12Þ
W I ¼ d þ eT I TI ¼
ð13Þ
w þ Lef T g þ B3 hfg;0 ðW g dÞ ½ET þ Lef B4 Þ ðE
In the above equations, B1 ¼ ð1 þ W g Rcv Þ=½1 þ
ð14Þ
Rcv Lef
ðW I W g Þ,
B3 ¼ 1 þ T I ðRcv Rcw Þ=hfg;0 and B2 ¼ 1 þ ðRcw =EÞðW I W g Þ, B4 ¼ 1 þ B3 ehfg;0 =Lef ; NTU m ¼ hD A=mg represents the number of mass transfer units, here A is the total heat and mass transfer area and will be equal to L if the channel breadth in the direction per ¼ hc;w =hD cp;dg represents dimenpendicular to x-y plane is unity; E sionless ratio of water film heat transfer coefficient to air mass transfer coefficient; C w ¼ mw cpw =mg cp;dg represents water to dry air heat capacity ratio; hfg;0 ¼ hfg;0 =cp;dg represents a normalized heat of evaporation at 0 °C; Rcv ¼ cpv =cp;dg and Rcw ¼ cpw =cp;dg represent water vapor and liquid water to dry air specific heat capacity ratios, respectively; Lef ¼ hc =hD cp;dg represents Lewis factor for airwater mixture. 3. Numerical method and model validation 3.1. 2-D model solution and model validation The 2-D model equations governing the heat and mass transfer in the channel are solved together with the boundary conditions by means of the finite volume method commercial software (Fluent). SIMPLE algorithm is used for coupling pressure and velocity. Second order upwind scheme is selected as the difference scheme. 2-D double precision style is used in the simulation for accuracy. In the setting of convergence conditions, the absolute value of energy residual is less than 1 108 , and the absolute value of
Energy balance equation for the incremental volume
dðmw cpw T w Þ mg dig ¼ 0
Lef ðT I T g ÞdNTU m B1
dT g ¼
ð9Þ
After rearrangement of (Eqs. (5)–(9)), the final forms of the developed equations are as follows:
other quality residuals are less than 1 106 . The heat and mass transfer characteristics are studied by Fluent, and it is necessary to verify the reliability of simulation method. The results derived by Yan [20] are employed to check the validation of the CFD method. Fig. 2(a) and (b) show respectively the axial distributions of local Nusselt and Sherwood numbers for: Reg ¼ 2000, T w;i ¼ 40 °C, T g;i = 20 °C, U = 50%, mw;i =0.04 kg m1 s1. It can be noted that the relative error between the present predictions and
Fig. 2. Axial distributions of (a) local Nusselt number, (b) local Sherwood number.
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Fig. 3. Calculation of the average Nusselt and Sherwood numbers.
those of Yan [20] is less than 3.0%, which is within the engineering allowable range. Therefore, it is reliable to conduct simulation on the heat and mass transfer characteristics of the 2-D CFD model. 3.2 Determination of the average Nusselt and Sherwood numbers The average Nusselt and Sherwood numbers for air flow inside vertical plate channels with falling water film evaporation are very important parameters to evaluate devices performance, which is helpful to reveal the heat and mass transfer characteristics between moist air and liquid water film. The heat and mass transfer coefficients are the primary parameters in the heat and mass transfer mathematical model, which is used for calculating all the outlet parameters and designing the devices. The average Nusselt and Sherwood numbers can be calculated by combining the 1-D model and the 2-D CFD model data. The steps for calculating the average Nusselt and Sherwood numbers are summarized in Fig. 3. (According to the output values NTU m , the average heat and mass transfer coefficients can be Lef and E, calculated, and then the average Nusselt and Sherwood numbers can be calculated according to Eqs. (3) and (4) respectively.)
40 °C and 20 °C, respectively, and the half channel width D is still fixed at 0.0015 m. As illustrated in Fig. 4, it is concluded that the method based on combining the 1-D model and the 2-D CFD model can give reasonable results of the heat and mass transfer coefficients between moist air and liquid water film, since the discrepancies of the gas-liquid interface outlet temperature T I;o between calculated results and 2-D CFD values are acceptable (within ±0.2 °C). Fig. 5 presents the effects of moist air inlet dimensionless temperature hg;i , moist air inlet Reynolds number Reg , ratio of water to
4. Results and discussion Simulation experiments are performed for a lot of cases under different operating conditions. For all these cases, the inlet water temperature and the moist air wet-bulb temperature are fixed at
Fig. 4. Comparison of calculated results with 2-D CFD model values for the gasliquid interface outlet temperature.
Y. Wan et al. / International Journal of Heat and Mass Transfer 110 (2017) 783–788
787
Fig. 5. Effects of parameters hg;i , Reg and Rwg on the average Nusselt and Sherwood numbers at various aL from 100 to 400. (a)–(b) Rwg ¼ 1:5, Reg ¼ 1100, (c)–(d) hg;i ¼ 0:6, Rwg ¼ 1:5, (e)–(f) hg;i ¼ 0:6, Reg ¼ 1100.
moist air mass flow rate Rwg on the average Nusselt and Sherwood numbers for the ratio of channel length to half channel width aL between 100 and 400. It is concluded that the average Nusselt and Sherwood numbers increase with the decrease of the ratio of channel length to half channel width. This is because a higher means a greater percentage of the fully developed region with smaller local Nusselt and Sherwood numbers than those in the entrance region. Fig. 5(a) and (b) present the effects of the moist air inlet dimensionless temperature hg;i on the average Nusselt and Sherwood numbers respectively. The average Nusselt numbers are greatly dependent on the moist air inlet dimensionless temperature. The slope becomes much steeper when hg;i > 0:5, whereas the average Sherwood numbers slightly change for hg;i from 0.2 to 1.0. From the results of Fig. 5(c) and (d), the moist air inlet Reynolds number influence on the average Nusselt and Sherwood numbers are presented. The results show that the average Nusselt and Sherwood numbers rapidly rise with the increase of the moist air inlet Reynolds number. Taking the case of aL ¼ 100 as an example, when the moist air inlet Reynolds number changes from 600 to 2100, the average Nusselt and Sherwood numbers increases from
9.4 to 11.1 and 7.5 to 9.3 respectively. Similar trends can be found in the ratio of water to moist air mass flow rate Rwg influence on the average Nusselt and Sherwood numbers, as shown in Fig. 5 (e) and (f). The heat and mass coefficients are very important for designing or simulating the thermodynamic performance of falling water film evaporation systems. Thus, the correlations of the average Nusselt and Sherwood numbers are developed as following by regression method: 0:090 NuM ¼ 8:250h0:196 Re0:101 Re0:050 g;i g wg aL
ð15Þ
0:163 Re0:213 Re0:049 ShM ¼ 3:926h0:016 g;i g wg aL
ð16Þ
The application ranges are 0:2 6 hg;i 6 1:0, 600 6 Reg 6 2100, 0:75 6 Rewg 3:5 and 100 6 aL 400 in this study. The maximum errors of above correlations are 8.0%. According to the values of exponents in the correlations, it can be concluded that hg;i has the greatest influence on NuM compared with that of the other
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dimensionless parameters, but it has the lowest influence on ShM . In addition, the signs of the exponents are also consistent with the variations of the parameters in Fig. 5. 5. Conclusion The determination of the average Nusselt and Sherwood numbers in vertical plate channels with falling water film evaporation via a new approach is reported. Many cases under different conditions are performed for providing valid and important data for falling water film evaporation. Dimensionless parameters such as moist air inlet Reynolds number, ratio of water to moist air mass flow rate, ratio of channel length to half channel width and moist air inlet dimensionless temperature have significant effect on the average Nusselt and Sherwood numbers. In particular, the moist air inlet dimensionless temperature takes into account the effects of the inlet parameters, such as moist air dry-bulb and wet-bulb temperatures and water film temperature. Finally, the correlations of the average Nusselt and Sherwood numbers are developed for designing or simulating the thermodynamic performance of falling water film evaporation systems, which will be conducted in our future research. Acknowledgment This work is supported by the 2017 Graduate Student Research Innovation Project in Hunan Province. References [1] V.E. Nakoryakov, N.I. Grigoryeva, M.V. Bartashevich, Heat and mass transfer in the entrance region of the falling film: absorption, desorption, condensation and evaporation, Int. J. Heat Mass Transf. 54 (2011) 4485–4490. [2] A. Surtaev, A. Pavlenko, Observation of boiling heat transfer and crisis phenomena in falling water film at transient heating, Int. J. Heat Mass Transf. 74 (2014) 342–352. [3] C.N. Markides, R. Mathie, A. Charogiannis, An experimental study of spatiotemporally resolved heat transfer in thin liquid-film flows falling over an inclined heated foil, Int. J. Heat Mass Transf. 93 (2016) 872–888.
[4] Y.M. Xuan, F. Xiao, X.F. Niu, X. Huang, S.W. Wang, Research and application of evaporative cooling in China: a review (I) – research, Renew. Sustain. Energy Rev. 16 (2012) 3535–3546. [5] Y.M. Xuan, F. Xiao, X.F. Niu, X. Huang, S.W. Wang, Research and application of evaporative cooling in China: a review (II) – system and equipment, Renew. Sustain. Energy Rev. 16 (2012) 3523–3534. [6] J.R. Camargo, C.D. Ebinuma, S. Cardoso, A mathematical model for direct evaporative cooling air conditioning system, Rev. Engenharia Térmic 4 (2003) 30–34. [7] J.R. Camargo, C.D. Ebinuma, J.L. Silveira, Experimental performance of a direct evaporative cooler operating during summer in a Brazilian city, Int. J. Refrig. 28 (2005) 1124–1132. [8] J.A. Dowdy, N.S. Karabash, Experimental determination of heat and mass transfer coefficients in rigid impregnated cellulose evaporative media, Ashrae Trans. 93 (1987) 382–395. [9] A. Franco, D.L. Valera, A. Peña, Energy efficiency in greenhouse evaporative cooling techniques: cooling boxes versus cellulose pads, Energies 7 (2014) 1427–1447. [10] G. Heidarinejad, V. Khalajzadeh, S. Delfani, Performance analysis of a groundassisted direct evaporative cooling air conditioner, Build. Environ. 45 (2010) 2421–2429. [11] J.M. Wu, X. Huang, H. Zhang, Theoretical analysis on heat and mass transfer in a direct evaporative cooler, Appl. Therm. Eng. 29 (2009) 980–984. [12] J. Lin, K. Thu, T.D. Bui, R.Z. Wang, K.C. Ng, K.J. Chua, Study on dew point evaporative cooling system with counter-flow configuration, Energy Convers. Manage. 109 (2016) 153–165. [13] J.M. Wu, X. Huang, H. Zhang, Numerical investigation on the heat and mass transfer in a direct evaporative cooler, Appl. Therm. Eng. 29 (2009) 195–201. [14] A. Fouda, Z. Melikyan, A simplified model for analysis of heat and mass transfer in a direct evaporative cooler, Appl. Therm. Eng. 31 (2011) 932–936. [15] C.Q. Ren, Y.D. Wan, A new approach to the analysis of heat and mass transfer characteristics for laminar air flow inside vertical plate channels with falling water film evaporation, Int. J. Heat Mass Transf. 103 (2016) 1017–1028. [16] H.C. Chang, E.A. Demekhin, Complex Wave Dynamics on Thin Films, Elsevier Press, Amsterdam, 2002. [17] S.M. Huang, Z.R. Zhong, M.L. Yang, Conjugate heat and mass transfer in an internally-cooled membrane-based liquid desiccant dehumidifier (IMLDD), J. Membr. Sci. 508 (2016) 73–83. [18] Y.D. Wan, C.Q. Ren, L. Xing, Y. Yang, Analysis of heat and mass transfer characteristics in vertical plate channels with falling film evaporation under uniform heat flux/uniform wall temperature boundary conditions, Int. J. Heat Mass Transf. 108 (2017) 1279–1284. [19] C.Q. Ren, An analytical approach to the heat and mass transfer processes in counterflow cooling towers, J. Heat Transf. 128 (11) (2006) 1142–1148. [20] W.M. Yan, Convective heat and mass transfer from a falling film to a laminar gas stream, Heat Mass Transf. 29 (2) (1993) 79–87.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Study on average Nusselt and Sherwood numbers in vertical plate channels with falling water film evaporation Yangda Wan, Chengqin Ren ⇑, Yang Yang, Li Xing State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China
a r t i c l e
i n f o
Article history: Received 19 January 2017 Received in revised form 24 March 2017 Accepted 24 March 2017
Keywords: Falling water film evaporation Average Nusselt and Sherwood numbers Correlations Heat and mass transfer
a b s t r a c t In this paper, the effects of variable parameters on the average Nusselt and Sherwood numbers between moist air and liquid water film in vertical plate channels are investigated using an approach proposed by previous work. The validity of data is also investigated. Traditional correlations for the average Nusselt and Sherwood numbers are not comprehensive enough as they cannot indicate the effects of inlet parameters on heat and mass transfer process. In this research, correlations for the average Nusselt and Sherwood numbers are developed in terms of moist air inlet Reynolds number, ratio of water to moist air inlet mass flow rate, ratio of channel length to half channel width and moist air inlet dimensionless temperature. The definition of the moist air inlet dimensionless temperature takes into account the effects of moist air inlet dry-bulb and wet-bulb temperatures and water film inlet temperature. The correlations show that the moist air inlet dimensionless temperature has the greatest influence on the average Nusselt number compared with that of the other dimensionless parameters, but it has the lowest influence on the average Sherwood number. The results would be helpful to understand the heat and mass transfer behavior between moist air and liquid water film. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Falling film evaporation is widely used for effective cooling [1– 3]. The application of falling film evaporation for air conditioning can bring environmentally friendly products and reduce energy consumption. Therefore falling film evaporation devices are drawing more and more attention in recent years. Xuan et al. [4,5] presented research and application of evaporative cooling for air conditioning in China. Experimental and theoretical research works on feasibility studies, performance test and optimization as well as heat and mass transfer analysis were reviewed in detail. The energy saving potential and environmental impacts of typical evaporative cooling air conditioning systems were illustrated. Camargo et al. [6,7] developed a mathematical model for evaporative cooling air conditioning system, allowing the determination of the effectiveness of saturation. The experimental results are used to determine the heat transfer coefficient and to compare with the mathematical model. Dowdy and Karabash [8] proposed the correlation for determination of the heat transfer coefficient in a rigid cellulose evaporative medium based on the assumption that air flow is turbulent. The correlation has
⇑ Corresponding author. E-mail address: [email protected] (C. Ren). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.03.087 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
been used in literature [9–12]. Wu et al. [11] proposed a simplified cooling efficiency correlation based on material properties and configuration of wetted medium used in the cooler. The models that implement this correlation were presented in literature [13,14]. Considering that heat and mass transfer process are also affected not only by material, but also by operation conditions, such as the inlet parameters of moist air and liquid water, traditional correlations are not comprehensive enough as they cannot indicate the effects of the inlet parameters on heat and mass transfer process. Heat and mass transfer coefficients correlations developed by many studies were calculated by the logarithmic mean temperature difference or arithmetical mean temperature difference. Little study was concerned on the method of measuring coupled heat and mass transfer coefficients during falling water film evaporation process. In a previous article [15], we have developed a new approach to analyze the heat and mass transfer characteristic in vertical plate channels with falling water film evaporation. The objective of the present paper is to extend the previous work [15] to develop general correlations for heat and mass transfer coefficients. In dimensionless terms, this means developing correlations for the average Nusselt and Sherwood numbers in terms of their dependence on the influence parameters. A temperature dimensionless parameter hg;i takes into account the effects of moist
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Y. Wan et al. / International Journal of Heat and Mass Transfer 110 (2017) 783–788
Nomenclature heat and mass transfer area [m2] specific heat capacity [kJ kg1 °C1] water to air heat capacity rate ratio half width and length of the channel respectively [m] diffusion coefficient of water vapor in air [m2 s1] dimensionless ratio of water film heat transfer coefficient to air mass transfer coefficient defined as hc;w =hD cp;dg hfg evaporation heat of water [kJ kg1] hfg;0 evaporation heat of water at reference temperature (0 °C) [kJ kg1] hc moist air side heat transfer coefficient [kW m2 °C1] hc;w water film side heat transfer coefficient [kW m2 °C1] hD moist air side mass transfer coefficient [kg m2 s1] hfg;0 normalized evaporation heat of water at reference temperature condition [°C] i specific enthalpy [kJ kg1] Lewis factor defined as hc =hD cp;dg Lef m mass flow rate [kg m1 s1] NTU m number of mass transfer units NuM , ShM average Nusselt and Sherwood numbers respectively Re Reynolds number Rwg ratio of water to moist air mass flow rate
A cp C ast w D, L Ds E
air inlet dry-bulb and wet-bulb temperatures and water film inlet temperature. The results would be helpful to understand the heat and mass transfer behavior between moist air and liquid water film. 2. Methodology 2.1. 2-D model The model investigated is a vertical plate channel with the length L and half channel width D as shown schematically in Fig. 1. The moist air flows upward and the liquid water film flows along the surface of the channel. The channel walls are thermally
T u,
v
W x, y
temperature [°C] velocity components in x, y coordinate directions, respectively [m s1] humidity ratio of moist air [kg kg1] space coordinate as shown in (Fig. 1) [m]
Greek letters aL ratio of channel length to half channel width d water film thickness [m] h dimensionless temperature k thermal conductivity [W m1 °C1] t kinematic viscosity [m2 s1] q density [kg m3] U relative humidity [%] Subscripts g; dg moist air, dry air respectively i, o inlet, outlet respectively I condition at the gas-liquid interface water film, water vapor respectively w, v wg moist air wet bulb temperature
insulated in order to investigate the evaporative cooling process associated with many engineering applications such as direct evaporative cooler, cooling tower, etc. It is assumed that the flow is incompressible and the air-vapor mixture is an ideal gas mixture. The thermo-physical properties of the dry air, water and water vapor are assumed to be constant. Dufour and Soret effects are neglected. The interface between the water film and the moist air is at thermodynamic equilibrium state [16–18]. The governing equations for the 2-D CFD model have been described in our previous study [15]. In view of the large number of parameters, a similarity analysis was performed to deduce the 2-D model equations into dimensionless forms to find the dimensionless factors affecting the average Nusselt and Sherwood num-
Fig. 1. (a) Schematic diagram of the physical. (b) Differential element of channel.
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Y. Wan et al. / International Journal of Heat and Mass Transfer 110 (2017) 783–788
bers in the previous study. For brevity, the correlations for the average Nusselt and Sherwood numbers are repeated as follows:
NuM ¼ f ðRwg ; Reg ; aL ; hg;i Þ
ð1Þ
ShM ¼ f ðRwg ; Reg ; aL ; hg;i Þ
ð2Þ
where Rwg ¼ mw =mg represents ratio of liquid water to moist air mass flow rate; Reg ¼ 4ðD dÞv g;i =tg represents moist air inlet Reynolds number; aL ¼ L=D represents ratio of channel length to half channel width; hg;i ¼ ðT g;i T wg;i Þ=ðT w;i T wg;i Þ represents moist air inlet dimensionless temperature. The average Nusselt and Sherwood numbers are defined as:
NuM ¼
4ðD dÞhc kg
ð3Þ
ShM ¼
4ðD dÞhD Ds qg
ð4Þ
2.2. 1-D model The 1-D model developed by our previous study [15,19] will be used in this paper for calculating the average Nusselt and Sherwood numbers, as shown in Fig. 1(b). To simplify the complexity of the heat and mass transfer analysis, the two additional assumptions for the 1-D model are made: (1) humidity ratio of air in equilibrium with the water surface is assumed to be a linear function of the water surface temperature, (2) constant heat and mass transfer coefficients along the heat exchanger surface. By principles of mass and energy conversion, a set of differential equations can be obtained: Energy balance equation for moist air
mg dig ¼ hc ðT w T g Þ þ hfg;I hD ðW I W g Þ dA
ð5Þ
Mass balance equation for moist air and liquid water
mg dW g ¼ hD ðW I W g ÞdA ¼ dmw
ð6Þ
Energy balance equation for the differential element
hc;w ðT w T I Þ ¼ hc ðT I T g Þ þ hfg;I hD ðW I W g Þ
ð7Þ ð8Þ
For specific enthalpy of moist air, the following equation applies
ig ¼ ðcp;dg þ W g cpv ÞT g þ W g hfg;0
ð10Þ
dW g ¼ ðW I W g ÞdNTU m
ð11Þ
dT w ¼
~ E B2 ðT I T w ÞdNTU m C w
ð12Þ
W I ¼ d þ eT I TI ¼
ð13Þ
w þ Lef T g þ B3 hfg;0 ðW g dÞ ½ET þ Lef B4 Þ ðE
In the above equations, B1 ¼ ð1 þ W g Rcv Þ=½1 þ
ð14Þ
Rcv Lef
ðW I W g Þ,
B3 ¼ 1 þ T I ðRcv Rcw Þ=hfg;0 and B2 ¼ 1 þ ðRcw =EÞðW I W g Þ, B4 ¼ 1 þ B3 ehfg;0 =Lef ; NTU m ¼ hD A=mg represents the number of mass transfer units, here A is the total heat and mass transfer area and will be equal to L if the channel breadth in the direction per ¼ hc;w =hD cp;dg represents dimenpendicular to x-y plane is unity; E sionless ratio of water film heat transfer coefficient to air mass transfer coefficient; C w ¼ mw cpw =mg cp;dg represents water to dry air heat capacity ratio; hfg;0 ¼ hfg;0 =cp;dg represents a normalized heat of evaporation at 0 °C; Rcv ¼ cpv =cp;dg and Rcw ¼ cpw =cp;dg represent water vapor and liquid water to dry air specific heat capacity ratios, respectively; Lef ¼ hc =hD cp;dg represents Lewis factor for airwater mixture. 3. Numerical method and model validation 3.1. 2-D model solution and model validation The 2-D model equations governing the heat and mass transfer in the channel are solved together with the boundary conditions by means of the finite volume method commercial software (Fluent). SIMPLE algorithm is used for coupling pressure and velocity. Second order upwind scheme is selected as the difference scheme. 2-D double precision style is used in the simulation for accuracy. In the setting of convergence conditions, the absolute value of energy residual is less than 1 108 , and the absolute value of
Energy balance equation for the incremental volume
dðmw cpw T w Þ mg dig ¼ 0
Lef ðT I T g ÞdNTU m B1
dT g ¼
ð9Þ
After rearrangement of (Eqs. (5)–(9)), the final forms of the developed equations are as follows:
other quality residuals are less than 1 106 . The heat and mass transfer characteristics are studied by Fluent, and it is necessary to verify the reliability of simulation method. The results derived by Yan [20] are employed to check the validation of the CFD method. Fig. 2(a) and (b) show respectively the axial distributions of local Nusselt and Sherwood numbers for: Reg ¼ 2000, T w;i ¼ 40 °C, T g;i = 20 °C, U = 50%, mw;i =0.04 kg m1 s1. It can be noted that the relative error between the present predictions and
Fig. 2. Axial distributions of (a) local Nusselt number, (b) local Sherwood number.
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Fig. 3. Calculation of the average Nusselt and Sherwood numbers.
those of Yan [20] is less than 3.0%, which is within the engineering allowable range. Therefore, it is reliable to conduct simulation on the heat and mass transfer characteristics of the 2-D CFD model. 3.2 Determination of the average Nusselt and Sherwood numbers The average Nusselt and Sherwood numbers for air flow inside vertical plate channels with falling water film evaporation are very important parameters to evaluate devices performance, which is helpful to reveal the heat and mass transfer characteristics between moist air and liquid water film. The heat and mass transfer coefficients are the primary parameters in the heat and mass transfer mathematical model, which is used for calculating all the outlet parameters and designing the devices. The average Nusselt and Sherwood numbers can be calculated by combining the 1-D model and the 2-D CFD model data. The steps for calculating the average Nusselt and Sherwood numbers are summarized in Fig. 3. (According to the output values NTU m , the average heat and mass transfer coefficients can be Lef and E, calculated, and then the average Nusselt and Sherwood numbers can be calculated according to Eqs. (3) and (4) respectively.)
40 °C and 20 °C, respectively, and the half channel width D is still fixed at 0.0015 m. As illustrated in Fig. 4, it is concluded that the method based on combining the 1-D model and the 2-D CFD model can give reasonable results of the heat and mass transfer coefficients between moist air and liquid water film, since the discrepancies of the gas-liquid interface outlet temperature T I;o between calculated results and 2-D CFD values are acceptable (within ±0.2 °C). Fig. 5 presents the effects of moist air inlet dimensionless temperature hg;i , moist air inlet Reynolds number Reg , ratio of water to
4. Results and discussion Simulation experiments are performed for a lot of cases under different operating conditions. For all these cases, the inlet water temperature and the moist air wet-bulb temperature are fixed at
Fig. 4. Comparison of calculated results with 2-D CFD model values for the gasliquid interface outlet temperature.
Y. Wan et al. / International Journal of Heat and Mass Transfer 110 (2017) 783–788
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Fig. 5. Effects of parameters hg;i , Reg and Rwg on the average Nusselt and Sherwood numbers at various aL from 100 to 400. (a)–(b) Rwg ¼ 1:5, Reg ¼ 1100, (c)–(d) hg;i ¼ 0:6, Rwg ¼ 1:5, (e)–(f) hg;i ¼ 0:6, Reg ¼ 1100.
moist air mass flow rate Rwg on the average Nusselt and Sherwood numbers for the ratio of channel length to half channel width aL between 100 and 400. It is concluded that the average Nusselt and Sherwood numbers increase with the decrease of the ratio of channel length to half channel width. This is because a higher means a greater percentage of the fully developed region with smaller local Nusselt and Sherwood numbers than those in the entrance region. Fig. 5(a) and (b) present the effects of the moist air inlet dimensionless temperature hg;i on the average Nusselt and Sherwood numbers respectively. The average Nusselt numbers are greatly dependent on the moist air inlet dimensionless temperature. The slope becomes much steeper when hg;i > 0:5, whereas the average Sherwood numbers slightly change for hg;i from 0.2 to 1.0. From the results of Fig. 5(c) and (d), the moist air inlet Reynolds number influence on the average Nusselt and Sherwood numbers are presented. The results show that the average Nusselt and Sherwood numbers rapidly rise with the increase of the moist air inlet Reynolds number. Taking the case of aL ¼ 100 as an example, when the moist air inlet Reynolds number changes from 600 to 2100, the average Nusselt and Sherwood numbers increases from
9.4 to 11.1 and 7.5 to 9.3 respectively. Similar trends can be found in the ratio of water to moist air mass flow rate Rwg influence on the average Nusselt and Sherwood numbers, as shown in Fig. 5 (e) and (f). The heat and mass coefficients are very important for designing or simulating the thermodynamic performance of falling water film evaporation systems. Thus, the correlations of the average Nusselt and Sherwood numbers are developed as following by regression method: 0:090 NuM ¼ 8:250h0:196 Re0:101 Re0:050 g;i g wg aL
ð15Þ
0:163 Re0:213 Re0:049 ShM ¼ 3:926h0:016 g;i g wg aL
ð16Þ
The application ranges are 0:2 6 hg;i 6 1:0, 600 6 Reg 6 2100, 0:75 6 Rewg 3:5 and 100 6 aL 400 in this study. The maximum errors of above correlations are 8.0%. According to the values of exponents in the correlations, it can be concluded that hg;i has the greatest influence on NuM compared with that of the other
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dimensionless parameters, but it has the lowest influence on ShM . In addition, the signs of the exponents are also consistent with the variations of the parameters in Fig. 5. 5. Conclusion The determination of the average Nusselt and Sherwood numbers in vertical plate channels with falling water film evaporation via a new approach is reported. Many cases under different conditions are performed for providing valid and important data for falling water film evaporation. Dimensionless parameters such as moist air inlet Reynolds number, ratio of water to moist air mass flow rate, ratio of channel length to half channel width and moist air inlet dimensionless temperature have significant effect on the average Nusselt and Sherwood numbers. In particular, the moist air inlet dimensionless temperature takes into account the effects of the inlet parameters, such as moist air dry-bulb and wet-bulb temperatures and water film temperature. Finally, the correlations of the average Nusselt and Sherwood numbers are developed for designing or simulating the thermodynamic performance of falling water film evaporation systems, which will be conducted in our future research. Acknowledgment This work is supported by the 2017 Graduate Student Research Innovation Project in Hunan Province. References [1] V.E. Nakoryakov, N.I. Grigoryeva, M.V. Bartashevich, Heat and mass transfer in the entrance region of the falling film: absorption, desorption, condensation and evaporation, Int. J. Heat Mass Transf. 54 (2011) 4485–4490. [2] A. Surtaev, A. Pavlenko, Observation of boiling heat transfer and crisis phenomena in falling water film at transient heating, Int. J. Heat Mass Transf. 74 (2014) 342–352. [3] C.N. Markides, R. Mathie, A. Charogiannis, An experimental study of spatiotemporally resolved heat transfer in thin liquid-film flows falling over an inclined heated foil, Int. J. Heat Mass Transf. 93 (2016) 872–888.
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