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Thermal performance analysis of closed wet cooling towers under both unsaturated and supersaturated conditions
International Journal of Heat and Mass Transfer 55 (2012) 7803–7811
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Thermal performance analysis of closed wet cooling towers under both unsaturated and supersaturated conditions Wei-Ye Zheng, Dong-Sheng Zhu ⇑, Guo-Yan Zhou ⇑, Jia-Fei Wu, Yun-Yi Shi Key Laboratory of Pressure System and Safety (MOE), School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 12 April 2012 Received in revised form 3 August 2012 Accepted 3 August 2012 Available online 4 September 2012 Keywords: Poppe method Closed wet cooling tower Heat and mass transfer Thermal performance
a b s t r a c t A closed wet cooling tower (CWCT) can be regarded as a cooling tower with the packing replaced by a bank of tubes. The heat and mass transfer in CWCT has been little investigated under supersaturated condition. The governing equations for heat and mass transfer in CWCT under both unsaturated and supersaturated conditions were derived based on Poppe method. Based on a sectional method, two analytical models were proposed and a program was coded to investigate the thermal performance by automatically selecting the corresponding governing equations whether the air was supersaturated or not. The results show that the air temperature, humidity ratio and enthalpy distributions under both unsaturated and supersaturated conditions could be accurately predicted with the analytical models. Then the effect of degree of saturation of inlet air on the heat transfer rate and process-water outlet temperature was presented, and little difference was found between the unsaturated and supersaturated conditions if the outlet air was supersaturated. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Closed wet cooling towers (CWCTs) in conventional form can be regarded as a cooling tower in which the packing is replaced by a bank of tubes carrying the process fluid. CWCTs are increasingly used to reject heat from industrial processes and buildings as they have advantages over traditional cooling towers in terms of better thermal performance and lower energy consumption. There have been substantial research interests in modeling CWCTs. Numerous researchers have developed different analytical models to investigate the thermal performance of CWCTs. Zalewski and Gryglaszewski [1] presented a mathematical model of heat and mass transfer correlations in the evaporative fluid coolers. The agreement between the calculated and experimental values was improved when the correlation function of mass transfer coefficient was introduced. Correlation based models were adopted by Facão and Oliveira [2] to predict the thermal performance, the results showed that the simpler models, with a global approach, could give as good, or even better, results as models based on finite difference techniques for the small CWCT. Hasan and Sirén [3] developed a computational model to optimize flow rates and number of tubes and rows for the required cooling load and to achieve a high coefficient of performance. The ratio of computed to the ⇑ Corresponding authors. Tel./fax: +86 021 64253708. E-mail addresses: [email protected] (D.-S. Zhu), [email protected] (G.-Y. Zhou). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.08.006
experimental rejected heat was between 92 and 115%. Stabat and Marchio [4] presented a simplified model for the behavior of a CWCT. The model was based on effectiveness models by simplification of heat and mass balance and transfer equations. The error in heat transfer rate was less than 10% compared with manufacturers’ catalog data. Qureshi and Zubair [5] described a risk based approach with the fouling model to study the effect of fouling on the thermal performance parameters such as effectiveness when operating under similar operating conditions. Xia et al. [6] adopted a transient one dimensional distributed-parameter model to evaluate the CWCT performance under different operating conditions. Determination of heat and mass transfer coefficients, as well as the influence of Lewis number on the thermal performance was presented. The above analytical models have been derived with the assumption that the air is unsaturated along the CWCT. Kloppers and Kröger [7] investigated into the heat and mass transfer analysis taking into account the supersaturation for the counterflow wet cooling towers and concluded that the Poppe method was especially suited to be employed in the analysis of hybrid cooling towers as the state of the outlet air was accurately determined. While the air may be supersaturated along the CWCT has not been taken into consideration. Therefore, errors can be resulted in when evaluating the heat and mass transfer of a CWCT under supersaturated conditions. Under the supersaturated condition, the air may become saturated before it leaves the CWCT. Because the gradient for temperature and vapor concentration still exists at the air–water interface, the air becomes supersaturated and the excess
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Nomenclature A cp d G h hd hw i ima k K Lef m Pr Q Re RH
area, m2 specific heat constant pressure, J/kg K diameter, m mass velocity, kg/s m2 heat transfer coefficient, W/m2 K mass transfer coefficient, kg/m2 s convection heat transfer coefficient of deluge water, W/m2 K enthalpy, J/kg enthalpy of the air-vapor mixture per unit mass of dry air, J/kg thermal conductivity, W/m K overall heat transfer coefficient, W/m2 K Lewis number mass flow rate, kg/s Prandtl number heat transfer rate, W Reynolds number air relative humidity, %
T Ta Va w k
temperature, K air dry-bulb temperature, K air volumetric flow rate, m3/s humidity ratio, kg water vapor/kg dry air thermal conductivity of water, W/m K
Subscripts a air, or based on air-side area c convection d deluge water i inlet, or inside m mass transfer, mean o outlet, or outside s supersaturated p constant pressure or process water sw Saturated v Vapor w deluge water or wall wb wet bulb
water vapor will condense as a mist. In this paper, the thermal performance of CWCT is predicted by implementing analytical models under both unsaturated and supersaturated conditions based on Poppe method. Comparison is taken to identify the difference in the thermal performance. 2. Analytical heat and mass transfer Fig. 1 represents a schematic layout of a countercurrent CWCT. The process water, to be cooled, flows inside the tubes. Air is drawn up through the bank of tubes while deluge water is sprayed downward over the tubes. Some of the deluge water is evaporated into the air while the remainder falls back into the water sump and is recirculated. Energy is transferred from the process water through the tube wall and into the deluge water. From here, the energy is transferred into the air due to convection and evaporation. 2.1. Poppe analysis By employing these assumptions and following an approach similar to Poppe and Rögener [8] and Dreyer [9], an analytical model of the CWCTs can be derived from basic principles. An elementary control volume is shown in Fig. 2. The assumptions are as follows:
Fig. 1. Schematic layout of a CWCT.
(1) The heat and mass transfer are performed in a steady state. (2) The radiation between the CWCT and the surroundings is neglected. (3) The tube surfaces are uniformly wetted, and the air flow and deluge water are uniformly distributed throughout the CWCT. (4) The air–water interface offers a negligible resistance to heat transfer [10], hence the air–water interface air can be considered as saturated air at the deluge-water temperature. 2.1.1. Governing equations for heat and mass transfer for unsaturated air The mass balance applicable to the control volume is
Fig. 2. Control volume.
dmw ¼ ma dw mp þ ma ð1 þ wÞ þ mw ¼ mp þ ma ð1 þ w þ dwÞ þ ðmw þ dmw Þ or
ð1Þ
ð2Þ
where ma is the mass flow rate of the dry air, w is the humidity ratio.
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The energy balance over the control volume gives
mp cpp T p þ ma ima þ mw cpw T w ¼ mp cpp ðT p þ dT p Þ þ ma ðima þ dima Þ þ ðmw þ dmw Þcpw ðT w þ dT w Þ
ð3Þ
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The convective heat transfer of sensible heat at the interface of air–water is given by
dQ c ¼ hðT w T a ÞdA
ð15Þ
where the process-water temperature Tp, and the deluge-water film temperature Tw is in °C. Neglecting the second order terms and simplify Eq. (3) to
The corresponding enthalpy transfer at the air–water interface due to the difference in the vapor concentration is then
1 dT w ¼ ðmp cpp dT p ma dima cpw T w dmw Þ mw cpw
The enthalpy of the water vapor, iv, calculated at the bulk deluge-water film temperature, is given by
ð4Þ
where ima is the enthalpy of the air-vapor mixture per unit mass of dry air, which can be expressed as
ima ¼ cpa T a þ wðifgwo þ cpv T a Þ
ð5Þ
dQ m ¼ iv dma ¼ iv hd ðwsw wÞdA
iv ¼ ifgwo þ cpv T w
ð16Þ
ð17Þ
Substituting Eqs. (15) and (16) into Eq. (14), find the total heat transfer at the air–water interface
The latent heat ifgwo is evaluated at 0 °C and the specific heats, cpa and cpv at Ta/2 °C. The enthalpy of the saturated air at the air–water interface evaluated at the bulk deluge-water film temperature is
dQ a ¼ hðT w T a ÞdA þ iv hd ðwsw wÞdA
imasw ¼ cpa T w þ wsw ðifgwo þ cpv T w Þ ¼ cpa T w þ wiv þ ðwsw wÞiv
where Lef = h/hdcpma is the Lewis factor and is an indication of the relative rates of heat and mass transfer. Bosnjakovic [11] developed an empirical relation for the Lewis factor Lef for air–water vapor systems. The Lewis factor for unsaturated air is given by
ð6Þ Subtracting Eq. (5) from Eq. (6) and ignoring the small difference in the specific heats, which are evaluated at difference temperatures. The equation can be simplified as follows:
ðimasw ima Þ ðwsw wÞiv Tw Ta ¼ cpma
ð7Þ
where cpma = cpa + wcpv. The heat transfer of process water can be written as
dQ p ¼ mp cpp dT p
dQ p ¼ KðT p T w ÞdA
KðT p T w ÞdA mp cpp
ð10Þ
ð11Þ
where di and do are the inside and outside tube diameters, kw the tube wall conductivity. The mass flow rate evaporating from the deluge water into the air vapor is expressed as
dmw ¼ hd ðwsw wÞdA
ð12Þ
where wsw is the saturated humidity ratio of the air evaluated at the bulk deluge-water film temperature. From Eqs. (2) and (12) and find
dw ¼
hd ðwsw wÞdA ma
ð13Þ
If the moist air is unsaturated, the total energy balance at the air–water interface consists of an enthalpy transfer due to the difference in temperature and the difference in vapor concentration
dQ a ¼ dQ c þ dQ m ¼ ma dima
Lef ¼ 0:8650:667
wsw þ 0:622 wsw þ 0:622 1 ln w þ 0:622 w þ 0:622
ð19Þ
ð20Þ
The enthalpy transfer to the air vapor from Eq. (14) is
hd ½Lef ðimasw ima Þ þ ð1 Lef Þðwsw wÞiv dA ma ð21Þ
If the outlet air is not saturated, the temperature and enthalpy distribution can be calculated from Eqs. (2), (4), (10), (13), (20) and (21), .
ð9Þ
where K is the overall heat transfer coefficient between the process water inside the tubes and the deluge water on the outside tubes based on the outer area of the tube, A is the outer area of the tubes. It can be written as
1 1 do do do 1 þ ¼ þ ln K hi di 2kw hw di
dQ a ¼ hd ½Lef ðimasw ima Þ þ ð1 Lef Þðwsw wÞiv dA
ð8Þ
Substituting Eq. (8) into Eq. (9) yields
dT p ¼
Substituting Eq. (7) into Eq. (18) and find
dima ¼ dQ a =ma ¼
The heat transfer from the process water to the deluge water is given by
ð18Þ
ð14Þ
where the subscripts c and m refer to the enthalpy associated with the convective heat transfer and the mass transfer.
2.1.2. Governing equations for heat and mass transfer for supersaturated air If the air is supersaturated, the excess water vapor will condense as a water mist. The control volume in Fig. 2 is also applicable if the air is supersaturated. The enthalpy of supersaturated air can be expressed as
imas ¼ cpa T a þ wsa ðifgwo þ cpv T a Þ þ ðw wsa Þcpw T a ¼ cpa T a þ wsa ½iv þ cpv ðT a T w Þ þ ðw wsa Þcpw T a
ð22Þ
where wsa is the humidity ratio of saturated air at temperature Ta. Subtracting Eq. (22) from Eq. (6), by introducing (w-wsa)cpwTw-(w-wsa)cpwTw, which adds up to zero, the temperature driving potential can be obtained
Tw Ta ¼
imasw imas ðwsw wsa Þiv þ ðw wsa Þcpw T w cpmas
ð23Þ
where cpmas is the specific heat of supersaturated air per unit mass of dry air and define as
cpmas ¼ cpa þ wsa cpv þ ðw wsa Þcpw
ð24Þ
Bourillot [12], Poppe and Rögener[8] and Kloppers and Kröger [7] proposed that the heat and mass transfer coefficients for supersaturated and unsaturated air were the same. The mass flow rate evaporating from the deluge water into the supersaturated air is expressed as
dmw ¼ hd ðwsw wsa ÞdA
ð25Þ
By applying the same analysis as in the case of unsaturated air, the governing equations can be obtained
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Table 1 Comparison between predicted and experimental outlet process-water and air temperature. Case
Process water 3
1 2 3 4 5 6 7 8
Deluge water
Air inlet
3
Tpo (°C)
3
Tao (°C)
Vp (m /h)
Tpi (°C)
Vd (m /h)
Va (m /s)
Tai (°C)
Twbi (°C)
By Exp.
By Poppe
Err (%)
By Exp.
By Poppe
Err (%)
28.47 28.47 27.24 28.28 28.31 26.91 28.11 28.22
22.91 22.54 39.22 22.93 22.56 39.27 22.99 22.61
15.07 15.32 15.25 15.09 15.24 15.19 15.08 15.22
4.00 3.87 3.79 3.17 3.05 2.97 2.35 2.22
9.9 14.6 15.0 10.1 14.7 14.9 9.8 14.7
4.1 7.1 7.6 4.0 7.1 7.6 3.9 7.1
20.03 20.13 33.27 20.43 20.51 34.02 20.91 20.91
19.94 20.07 33.15 20.40 20.46 33.99 20.95 20.94
0.25 0.25 0.48 0.29 0.15 0.32 0.34 0.24
13.78 16.38 23.58 14.5 16.82 24.23 15.14 17.26
14.37 16.51 23.32 14.86 16.91 23.96 15.29 17.36
4.28 0.79 1.10 2.48 0.54 1.11 0.99 0.58
215
190
210
180 Q , kW
Q , kW
205 200 Tpi=310K, Tai=280K Unsaturated Supersaturated
195 190
170 160 Tpi=310K, Tai=290K Unsaturated Supersaturated
150 140
185
0
20
40
60 RH, %
80
100
0
20
(a) Tpi=310K, Tai=280K
40
60 RH, %
80
100
(b) Tpi=310K, Tai=290K 160
180 160
120 Q , kW
Q , kW
140 120 Tpi=310K, Tai=300K Unsaturated Supersaturated
100
80 Tpi=310K, Tai=310K Unsaturated Supersaturated
40
80 0
20
40
60 RH, %
80
100
0
0
(c) Tpi=310K, Tai=300K
20
40
60 RH, %
80
100
(d) Tpi=310K, Tai=310K
Fig. 3. Comparison between the total heat transfer rate under unsaturated and supersaturated conditions.
dmw ¼ ma dw dT w ¼
1 ðmp cpp dT p ma dima cpw T w dmw Þ mw cpw
ð28Þ
hd ðwsw wsa ÞdA ma
ð29Þ
wsw þ 0:622 wsw þ 0:622 Lef ¼ 0:8650:667 1 ln wsa þ 0:622 wsa þ 0:622
2.1.3. Deluge-water temperature distribution Heyns [13] showed an example of the change in the delugewater temperature through the tube bundle. Since the deluge water is recirculated (Fig. 1), temperature rising due to pump work is small and can be neglected, the temperature of the deluge water at the inlet and outlet of the tube bundle is the same [3,13,14].
T wi ¼ T wo
ð32Þ
ð30Þ
dima ¼ dQ a =ma ¼
If the outlet air is supersaturated, the temperature and enthalpy distribution can be calculated from Eqs. (26)–(31).
ð27Þ
KðT p T w ÞdA mp cpp
dT p ¼
dw ¼
ð26Þ
hd ½Lef ðimasw imas þ ðw wsa Þcpw T w Þ þ ð1 Lef Þðwsw wsa Þiv dA ma ð31Þ
2.2. Solving the governing equations The governing equations for the thermal performance of the CWCT can be evaluated iteratively by the finite difference method. The air conditions in terms of temperature, humidity ratio and enthalpy for unsaturated and supersaturated air must be solved
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40
40
35
35
Tpo ,°C
Tpo , °C
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30 Tpi=310K, Tai=280K Unsaturated Supersaturated
25
20
0
20
30 Tpi=310K, Tai=290K Unsaturated Supersaturated
25
40
60
80
20
10 0
0
20
40
40
40
35
35
30 Tpi=310K, Tai=300K Unsaturated Supersaturated
20 0
20
40
100
30 Tpi=310K, Tai=310K Unsaturated Supersaturated
25
60
80
(b) Tpi=310K, Tai=290K
Tpo , °C
Tpo , °C
(a) Tpi=310K, Tai=280K
25
60
RH, %
RH, %
80
100
20 0
20
40
60
80
RH, %
RH, %
(c) Tpi=310K, Tai=300K
(d) Tpi=310K, Tai=310K
100
Fig. 4. Comparison between the process-water outlet temperatures under unsaturated and supersaturated conditions.
21.5
0.017
21.0 20.5
Tpi=310K, Tai=280K Unsaturated Supersaturated
20.0 Tao ,°C
wo , kg/kg
0.016
Tpi=310K, Tai=280K Unsaturated Supersaturated
0.015
19.5 19.0
0.014
0.013 0.000
18.5 18.0 0.001
0.002
0.003 0.004 wi , kg/kg
0.005
0.006
Fig. 5. Comparison between the outlet humidity ratio of air under unsaturated and supersaturated conditions.
by an iterative procedure. Successive iterations are needed for the values of the outlet air enthalpy imao, outlet humidity ratio wo and the inlet deluge-water temperature Tw to find the values for all the control volumes. The solution procedure is presented in Appendix A. For the unsaturated condition, governing equations for unsaturated air are solved to predict the thermal performance of the CWCT. For the supersaturated condition, a test is implemented to determine whether the air is unsaturated or supersaturated in each control volume, then the program automatically selects the corresponding governing equations. The thermo-physical properties (density, thermal conductivity, viscosity and heat capacity) of process water and saturated air at air-deluge water interface can
17.5 0.000
0.001
0.002
0.003 0.004 wi , kg/kg
0.005
0.006
Fig. 6. Comparison between the outlet air temperature under unsaturated and supersaturated conditions.
be calculated by some empirical equations, which are summarized by Kröger [14].
3. Results and discussion 3.1. Experimental validation for unsaturated air A prototype tower [15] consists of eight rows of externally galvanized steel oval tubes. The major axis of the tube is 31.8 mm, and minor axis 21.6 mm with tube wall thickness 1.5 mm. The oval
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Fig. 7. Path of air temperature and humidity indicated on a supersaturated psychrometric chart.
0.018
0.014 0.012
20 18 16
0.010
Ta ,°C
w , kg/kg
22
Tpi=310K, Tai=280K, wi=0.004kg/kg wunsaturated wsupersaturated wsaturated
0.016
0.008
Tpi=310K, Tai=280K, wi=0.004kg/kg Ta , unsaturated Ta , supersaturated
12
0.006 0.004
10
0
1
2
3
45 Tube row
6
7
8
8
Fig. 8. Distribution of humidity ratio of air under unsaturated and supersaturated conditions along the CWCT.
tubes are 1.25 m long and arranged in a stagger pattern at a transversal pitch of 53.4 mm. There are 37 oval tubes per tube row. The correlations for deluge-water film heat transfer coefficient (hw) and mass transfer coefficient (hd) [15] were expressed as 1=3 hw ¼ 350:3ð1 þ 0:0169tw ÞG0:59 Gw a
ð33Þ
hd ¼ 0:034G0:977 a
ð34Þ
for 2.57 < Ga < 4.94 kg/m2 s, 1.2 < Gw = U/d < 3.176 kg/m2 s, 11 < Tw < 28°C. The tube-side heat transfer coefficient hi in a turbulent flow was calculated as follows
hi ¼ 0:023
14
k 0:8 0:3 Re Pr di
ð35Þ
Establishing the CWCT heat and mass transfer coefficients and using the presented analytical models, it is possible to predict the thermal performance characteristics with variable outdoor air conditions and flow rates. A comparison between the predicted and experimental results for outlet process water temperature is given in Table 1. As it can be observed, the predicted results are in good agreement with the corresponding experimental data. This can verify the reliability of the prediction by Poppe analysis for unsaturated air. The predicted
1
2
3
4 5 Tube row
6
7
8
Fig. 9. Distribution of air temperature under unsaturated and supersaturated conditions along the CWCT.
outlet air temperature also shows a good agreement with experimental data. The acceptable agreement between the theoretical and experimental results for unsaturated conditions enhances our confidence on the reliability of the numerical investigation of the thermal performance for the supersaturated conditions. 3.2. Comparison and discussion Since the experimental data for the distributions of processwater temperature, dry-bulb temperature and humidity ratio of air along the CWCT are unavailable, the comparison is limited to the prediction by Poppe method under both unsaturated and supersaturated conditions. The differences between the predicted results are investigated under different operating conditions. Inlet air temperatures of 280, 290, 300 and 310 K are considered. The relative humidity of air is varied from 8 to 97%. Fig. 3 shows the comparison between the total heat transfer rate Q under unsaturated and supersaturated conditions. The predicted total heat transfer rate is the same if the air is unsaturated. It is found that the outlet air conditions are all unsaturated at inlet air temperature of 290, 300 and 310 K of RH varying from 8 to 97%. Whereas at an inlet air temperature of 280 K, the predicted results
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80
60 i , kJ/kg
show that the outlet air conditions are all supersaturated. The higher the inlet air temperature, the higher the saturation point and hence the more water vapor can be absorbed when the air gets saturated. Little difference in the total heat transfer rate is found between the unsaturated and supersaturated conditions. The difference is approximately zero. The process-water outlet temperatures under unsaturated and supersaturated conditions are shown in Fig. 4. Because the difference in the heat transfer rate is approximately zero, the corresponding process-water outlet temperatures show no difference between the results at different air conditions. The degree of saturation of the air does not have a great impact on the heat transfer rate and process-water outlet temperature between the two conditions predicted by Poppe method. Kloppers and Kröger [7] pointed out that the lines of constant air enthalpy in the supersaturated region were very close to vertical. Therefore, for a specific air enthalpy, it does not matter how much water vapor and water mist are present in the supersaturated air. Thus, the assumption that the air is unsaturated is also a very useful assumption to predict the total heat transfer rate of the CWCTs if the actual air is supersaturated. The following results are obtained for inlet air temperature of 280 K. As can be seen in Fig. 5, the difference in humidity ratio of outlet air is increasing with the increasing humidity ratio of inlet air. At 0.006 kg/kg dry air (RH = 97.7%) in a very humid condition, the difference is almost 4.53%. According to Fig. 6, the predicted outlet air temperatures are not very close to each other under unsaturated and supersaturated conditions. The maximum difference is 9.3% at humidity ratio of 0.006 kg/kg dry air (RH = 97.7%). As shown in Fig. 7, in the supersaturated region on a psychrometric chart, lines of constant air enthalpy are close to vertical, while in the unsaturated region, the lines of constant air enthalpy are far from vertical which will cause discrepancy in outlet air temperature prediction. For example, the supersaturated condition point B whose enthalpy is 60 kJ/kg dry air and humidity ratio is 0.020 kg/kg dry air, then air temperature is determined to be 18.61°C.Assuming the supersaturated point B is in the unsaturated region, if the humidity ratio and enthalpy are the same with point B, the predicted point will be A with the corresponding air temperature 11.33°C which is much lower than the actual air temperature, if the air temperature and enthalpy are the same with point B, the predicted humidity ratio is 0.0165 kg/kg dry air. Thus, the predicted air temperature will be between 11.33 and 18.61°C and the humidity ratio will be in the range of 0.0165 to 0.020 kg/kg dry air which are lower than the values of the actual supersaturated condition. Thus, the assumption that the air is unsaturated is unable to predict accurate air temperature and humidity ratio if the actual air is supersaturated. Due to the rigorous Poppe method derivation, the analytical model with Poppe method can be able to give the distributions of enthalpy, air temperature and humidity ratio along the CWCT. The air condition of temperature of 280 K and humidity ratio of 0.004 kg/kg dry air which the corresponding relative humidity is approximately 65.3% is considered. The distributions are showed as follows (i = 1 for the bottom row; i = 8 for the top row). Since the inlet air is relatively low temperature and high humid, the air immediately becomes supersaturated as it passes several tube rows, as shown in Fig. 8. The transition from unsaturation to supersaturation is decided by the humidity ratio whether is lower or higher than the saturated humidity ratio. For the unsaturated condition, the predicted humidity ratio is lower than the saturated humidity ratio, while the predicted humidity ratio is higher than the saturated humidity ratio for the supersaturated condition. As the temperature and humidity gradient still exist between the air-deluge water interface and air, it can be seen in Figs. 8 and 9, the air temperature together with the saturated humidity ratio increase. The predicted humidity ratio under unsaturated condition
Tpi=310K, Tai=280K wi=0.004kg/kg ima, unsaturated ima, supersaturated imasw, unsaturated imasw, supersaturated
40
20
0
1
2
3
4 5 Tube row
6
7
8
Fig. 10. Distribution of enthalpy under unsaturated and supersaturated conditions along the CWCT.
lie between the saturated and supersaturated humidity ratio predicted under supersaturated condition. Fig. 10 shows the distribution of enthalpy under unsaturated and supersaturated conditions along the CWCT. No significant difference is found between the two conditions. That the enthalpy of water mist in the air is small compared to the total air enthalpy may be an explanation for it. 4. Conclusion The governing equations based on Poppe method are derived to investigate the thermal performance of CWCT under both unsaturated and supersaturated conditions. A detailed program with finite difference method is presented to solve the governing equations. Little difference in heat transfer rate and process-water temperature is found between the prediction under the unsaturated and supersaturated conditions. The results show that it is very useful to predict the total heat transfer rate and outlet process-water temperature of the CWCTs that the air is assumed to be unsaturated if the actual air is supersaturated. It indicates that supersaturation is likely to occur at low inlet air temperature. It is clear from the discussion, the predicted air temperatures and humidity ratio are not very close to the two conditions if the outlet air is supersaturated. The distribution of air temperature, enthalpy and humidity ratio along the CWCT would provide a better understanding of the heat and mass transfer of CWCT. The present method can be employed to predict the air condition in the supersaturated state along the CWCT. Acknowledgements The work was financially supported by Cultivation Fund for Interdisciplinary and Major Project (MOE) (2010JB0621) and Natural Science Foundation of China under Grant No. 50905060. Appendix A. Solving procedure The finite difference method is employed to solve the governing equations for unsaturated and supersaturated conditions based on Poppe method. Eqs. (2), (4), (10), (13), (20), and (21) represent the governing equations for unsaturated air and Eqs. (26)–(31) for supersaturated air. The CWCT is divided into a number of control volumes, as shown in Fig. A.1. The governing equations for heat and mass transfer are calculated for each control volume. The solution starts by taking the process-water inlet control volume and proceeds in the direction of process water flow. The thermo-physical
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Deluge water
(1,1)
(1,2)
(1,n-1)
(1,n)
Process water
(2i-1,1)
(2i-1,2)
(2i-1,n-1)
(2i-1,n)
(2i,1)
(2i,2)
(2i,n-1)
(2i,n)
(N,1)
(N,2)
dL
(N,n-1)
(N,n)
Air
(a) tube coil with N tube rows and n control volumes per tube row Fig. A.1. Flowchart of the calculating process.
Fig. A.2. Flowchart diagram for Poppe-type method.
(b) control volume
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properties (density, thermal conductivity, viscosity and heat capacity) of process water and saturated air at the air-deluge water interface in current control volume are defined based on the outlet temperature of the preceding control volume. The equations must be solved by an iterative procedure. Successive iterations are needed for the values of the outlet air enthalpy imao, outlet humidity ratio wo and the inlet deluge-water temperature Twi to find the values for all the control volumes. The flowchart diagrams for the thermal performance prediction under both unsaturated and supersaturated conditions by Poppe method are showed in Fig. A.2. First, iterations are carried out to check the average of computed inlet air enthalpy with the inlet air state imai. Second, iterations are executed to compare the average computed inlet air humidity ratio with the inlet air state wi. Finally, iterations are implemented to approach equality of Twi and Two. Under the supersaturated condition, test is implemented to determine whether the air is unsaturated or supersaturated in each control volume. For the control volume (2i-1,j), the air is assumed to be unsaturated, the air temperature Ta(2i-1,j) is determined from Eq. (5) by iterative with w(2i-1,j) and ima(2i-1,j) known. Then assume the air is saturated and determine the wetbulb temperature Twb(2i-1,j) from the following Eq. (A.1). If Ta(2i-1,j) < Twb(2i-1,j), which is impossible, the air is supersaturated, otherwise the air is unsaturated and Ta(2i-1,j) is the dry-bulb temperature of the air. The dry-bulb temperature of supersaturated air Ta(2i-1,j) is determined from Eq. (22) by iterative with w(2i-1,j) and ima(2i-1,j) known. w¼
2501:6 2:3263ðT wb 273:15Þ 0:62509pv wb 2501:6 þ 1:8577ðT 273:15Þ 4:184ðT wb 273:15Þ P a 1:005pv wb 1:00416ðT T wb Þ 2501:6 þ 1:8577ðT 273:15Þ 4:184ðT wb 273:15Þ
ðA:1Þ
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References [1] W. Zalewski, P.A. Gryglaszewski, Mathematical model of heat and mass transfer processes in evaporative fluid coolers, Chem. Eng. Process. 36 (1997) 271–280. [2] J. Facão, A.C. Oliveira, Thermal behaviour of closed wet cooling towers for use with chilled ceilings, Appl. Therm. Eng. 20 (2000) 1225–1236. [3] A. Hasan, K. Sirén, Theoretical and computational analysis of closed wet cooling towers and its applications in cooling of buildings, Energ. Build. 34 (2002) 477–486. [4] P. Stabat, D. Marchio, Simplified model for indirect-contact evaporative cooling-tower behaviour, Appl. Energ. 78 (2004) 433–451. [5] B.A. Qureshi, S.M. Zubair, A comprehensive design and rating study of evaporative coolers and condensers. Part I. Performance evaluation, Int. J. Refrig. 29 (2006) 645–658. [6] Z.Z. Xia, C.J. Chen, R.Z. Wang, Numerical simulation of a closed wet cooling tower with novel design, Int. J. Heat Mass Transfer 54 (2011) 2367–2374. [7] J.C. Kloppers, D.G. Kröger, A critical investigation into the heat and mass transfer analysis of counterflow wet-cooling towers, Int. J. Heat Mass Transfer 48 (2005) 765–777. [8] M. Poppe, H. Rögener, Evaporative Cooling Systems. VDI-Wärmeatlas, Section Mh, Berlin, 1984. [9] A.A. Dreyer, Analysis of Evaporative Coolers and Condensers, M.Sc thesis, University of Stellenbosch, Republic of South Africa, 1988. [10] ASHRAE, ASHRAE Systems and Equipment, in: American Society of Heating Refrigeration and Air Conditioning Engineers, USA, 1992. [11] F. Bosnjakovic, Technische Thermodinamik, Theodor Steinkopf, Dresden, 1965. [12] C. Bourillot, Numerical model for calculating the performance of an evaporative cooling tower, EPRI, Report CS-3212-SR, 1983. [13] J.A. Heyns, D.G. Kröger, Experimental investigation into the thermal-flow performance characteristics of an evaporative cooler, Appl. Therm. Eng. 30 (2010) 492–498. [14] D.G. Kröger, Air-cooled heat exchangers and cooling towers: thermal-flow performance evaluation and design, Pennwell Corp., Tulsa. Oklahoma, USA, 2004. [15] W.-Y. Zheng, D.-S. Zhu, J. Song, L.-D. Zeng, H.-j. Zhou, Experimental and computational analysis of thermal performance of the oval tube closed wet cooling tower, Appl. Therm. Eng. 35 (2012) 233–239.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Thermal performance analysis of closed wet cooling towers under both unsaturated and supersaturated conditions Wei-Ye Zheng, Dong-Sheng Zhu ⇑, Guo-Yan Zhou ⇑, Jia-Fei Wu, Yun-Yi Shi Key Laboratory of Pressure System and Safety (MOE), School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 12 April 2012 Received in revised form 3 August 2012 Accepted 3 August 2012 Available online 4 September 2012 Keywords: Poppe method Closed wet cooling tower Heat and mass transfer Thermal performance
a b s t r a c t A closed wet cooling tower (CWCT) can be regarded as a cooling tower with the packing replaced by a bank of tubes. The heat and mass transfer in CWCT has been little investigated under supersaturated condition. The governing equations for heat and mass transfer in CWCT under both unsaturated and supersaturated conditions were derived based on Poppe method. Based on a sectional method, two analytical models were proposed and a program was coded to investigate the thermal performance by automatically selecting the corresponding governing equations whether the air was supersaturated or not. The results show that the air temperature, humidity ratio and enthalpy distributions under both unsaturated and supersaturated conditions could be accurately predicted with the analytical models. Then the effect of degree of saturation of inlet air on the heat transfer rate and process-water outlet temperature was presented, and little difference was found between the unsaturated and supersaturated conditions if the outlet air was supersaturated. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Closed wet cooling towers (CWCTs) in conventional form can be regarded as a cooling tower in which the packing is replaced by a bank of tubes carrying the process fluid. CWCTs are increasingly used to reject heat from industrial processes and buildings as they have advantages over traditional cooling towers in terms of better thermal performance and lower energy consumption. There have been substantial research interests in modeling CWCTs. Numerous researchers have developed different analytical models to investigate the thermal performance of CWCTs. Zalewski and Gryglaszewski [1] presented a mathematical model of heat and mass transfer correlations in the evaporative fluid coolers. The agreement between the calculated and experimental values was improved when the correlation function of mass transfer coefficient was introduced. Correlation based models were adopted by Facão and Oliveira [2] to predict the thermal performance, the results showed that the simpler models, with a global approach, could give as good, or even better, results as models based on finite difference techniques for the small CWCT. Hasan and Sirén [3] developed a computational model to optimize flow rates and number of tubes and rows for the required cooling load and to achieve a high coefficient of performance. The ratio of computed to the ⇑ Corresponding authors. Tel./fax: +86 021 64253708. E-mail addresses: [email protected] (D.-S. Zhu), [email protected] (G.-Y. Zhou). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.08.006
experimental rejected heat was between 92 and 115%. Stabat and Marchio [4] presented a simplified model for the behavior of a CWCT. The model was based on effectiveness models by simplification of heat and mass balance and transfer equations. The error in heat transfer rate was less than 10% compared with manufacturers’ catalog data. Qureshi and Zubair [5] described a risk based approach with the fouling model to study the effect of fouling on the thermal performance parameters such as effectiveness when operating under similar operating conditions. Xia et al. [6] adopted a transient one dimensional distributed-parameter model to evaluate the CWCT performance under different operating conditions. Determination of heat and mass transfer coefficients, as well as the influence of Lewis number on the thermal performance was presented. The above analytical models have been derived with the assumption that the air is unsaturated along the CWCT. Kloppers and Kröger [7] investigated into the heat and mass transfer analysis taking into account the supersaturation for the counterflow wet cooling towers and concluded that the Poppe method was especially suited to be employed in the analysis of hybrid cooling towers as the state of the outlet air was accurately determined. While the air may be supersaturated along the CWCT has not been taken into consideration. Therefore, errors can be resulted in when evaluating the heat and mass transfer of a CWCT under supersaturated conditions. Under the supersaturated condition, the air may become saturated before it leaves the CWCT. Because the gradient for temperature and vapor concentration still exists at the air–water interface, the air becomes supersaturated and the excess
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Nomenclature A cp d G h hd hw i ima k K Lef m Pr Q Re RH
area, m2 specific heat constant pressure, J/kg K diameter, m mass velocity, kg/s m2 heat transfer coefficient, W/m2 K mass transfer coefficient, kg/m2 s convection heat transfer coefficient of deluge water, W/m2 K enthalpy, J/kg enthalpy of the air-vapor mixture per unit mass of dry air, J/kg thermal conductivity, W/m K overall heat transfer coefficient, W/m2 K Lewis number mass flow rate, kg/s Prandtl number heat transfer rate, W Reynolds number air relative humidity, %
T Ta Va w k
temperature, K air dry-bulb temperature, K air volumetric flow rate, m3/s humidity ratio, kg water vapor/kg dry air thermal conductivity of water, W/m K
Subscripts a air, or based on air-side area c convection d deluge water i inlet, or inside m mass transfer, mean o outlet, or outside s supersaturated p constant pressure or process water sw Saturated v Vapor w deluge water or wall wb wet bulb
water vapor will condense as a mist. In this paper, the thermal performance of CWCT is predicted by implementing analytical models under both unsaturated and supersaturated conditions based on Poppe method. Comparison is taken to identify the difference in the thermal performance. 2. Analytical heat and mass transfer Fig. 1 represents a schematic layout of a countercurrent CWCT. The process water, to be cooled, flows inside the tubes. Air is drawn up through the bank of tubes while deluge water is sprayed downward over the tubes. Some of the deluge water is evaporated into the air while the remainder falls back into the water sump and is recirculated. Energy is transferred from the process water through the tube wall and into the deluge water. From here, the energy is transferred into the air due to convection and evaporation. 2.1. Poppe analysis By employing these assumptions and following an approach similar to Poppe and Rögener [8] and Dreyer [9], an analytical model of the CWCTs can be derived from basic principles. An elementary control volume is shown in Fig. 2. The assumptions are as follows:
Fig. 1. Schematic layout of a CWCT.
(1) The heat and mass transfer are performed in a steady state. (2) The radiation between the CWCT and the surroundings is neglected. (3) The tube surfaces are uniformly wetted, and the air flow and deluge water are uniformly distributed throughout the CWCT. (4) The air–water interface offers a negligible resistance to heat transfer [10], hence the air–water interface air can be considered as saturated air at the deluge-water temperature. 2.1.1. Governing equations for heat and mass transfer for unsaturated air The mass balance applicable to the control volume is
Fig. 2. Control volume.
dmw ¼ ma dw mp þ ma ð1 þ wÞ þ mw ¼ mp þ ma ð1 þ w þ dwÞ þ ðmw þ dmw Þ or
ð1Þ
ð2Þ
where ma is the mass flow rate of the dry air, w is the humidity ratio.
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The energy balance over the control volume gives
mp cpp T p þ ma ima þ mw cpw T w ¼ mp cpp ðT p þ dT p Þ þ ma ðima þ dima Þ þ ðmw þ dmw Þcpw ðT w þ dT w Þ
ð3Þ
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The convective heat transfer of sensible heat at the interface of air–water is given by
dQ c ¼ hðT w T a ÞdA
ð15Þ
where the process-water temperature Tp, and the deluge-water film temperature Tw is in °C. Neglecting the second order terms and simplify Eq. (3) to
The corresponding enthalpy transfer at the air–water interface due to the difference in the vapor concentration is then
1 dT w ¼ ðmp cpp dT p ma dima cpw T w dmw Þ mw cpw
The enthalpy of the water vapor, iv, calculated at the bulk deluge-water film temperature, is given by
ð4Þ
where ima is the enthalpy of the air-vapor mixture per unit mass of dry air, which can be expressed as
ima ¼ cpa T a þ wðifgwo þ cpv T a Þ
ð5Þ
dQ m ¼ iv dma ¼ iv hd ðwsw wÞdA
iv ¼ ifgwo þ cpv T w
ð16Þ
ð17Þ
Substituting Eqs. (15) and (16) into Eq. (14), find the total heat transfer at the air–water interface
The latent heat ifgwo is evaluated at 0 °C and the specific heats, cpa and cpv at Ta/2 °C. The enthalpy of the saturated air at the air–water interface evaluated at the bulk deluge-water film temperature is
dQ a ¼ hðT w T a ÞdA þ iv hd ðwsw wÞdA
imasw ¼ cpa T w þ wsw ðifgwo þ cpv T w Þ ¼ cpa T w þ wiv þ ðwsw wÞiv
where Lef = h/hdcpma is the Lewis factor and is an indication of the relative rates of heat and mass transfer. Bosnjakovic [11] developed an empirical relation for the Lewis factor Lef for air–water vapor systems. The Lewis factor for unsaturated air is given by
ð6Þ Subtracting Eq. (5) from Eq. (6) and ignoring the small difference in the specific heats, which are evaluated at difference temperatures. The equation can be simplified as follows:
ðimasw ima Þ ðwsw wÞiv Tw Ta ¼ cpma
ð7Þ
where cpma = cpa + wcpv. The heat transfer of process water can be written as
dQ p ¼ mp cpp dT p
dQ p ¼ KðT p T w ÞdA
KðT p T w ÞdA mp cpp
ð10Þ
ð11Þ
where di and do are the inside and outside tube diameters, kw the tube wall conductivity. The mass flow rate evaporating from the deluge water into the air vapor is expressed as
dmw ¼ hd ðwsw wÞdA
ð12Þ
where wsw is the saturated humidity ratio of the air evaluated at the bulk deluge-water film temperature. From Eqs. (2) and (12) and find
dw ¼
hd ðwsw wÞdA ma
ð13Þ
If the moist air is unsaturated, the total energy balance at the air–water interface consists of an enthalpy transfer due to the difference in temperature and the difference in vapor concentration
dQ a ¼ dQ c þ dQ m ¼ ma dima
Lef ¼ 0:8650:667
wsw þ 0:622 wsw þ 0:622 1 ln w þ 0:622 w þ 0:622
ð19Þ
ð20Þ
The enthalpy transfer to the air vapor from Eq. (14) is
hd ½Lef ðimasw ima Þ þ ð1 Lef Þðwsw wÞiv dA ma ð21Þ
If the outlet air is not saturated, the temperature and enthalpy distribution can be calculated from Eqs. (2), (4), (10), (13), (20) and (21), .
ð9Þ
where K is the overall heat transfer coefficient between the process water inside the tubes and the deluge water on the outside tubes based on the outer area of the tube, A is the outer area of the tubes. It can be written as
1 1 do do do 1 þ ¼ þ ln K hi di 2kw hw di
dQ a ¼ hd ½Lef ðimasw ima Þ þ ð1 Lef Þðwsw wÞiv dA
ð8Þ
Substituting Eq. (8) into Eq. (9) yields
dT p ¼
Substituting Eq. (7) into Eq. (18) and find
dima ¼ dQ a =ma ¼
The heat transfer from the process water to the deluge water is given by
ð18Þ
ð14Þ
where the subscripts c and m refer to the enthalpy associated with the convective heat transfer and the mass transfer.
2.1.2. Governing equations for heat and mass transfer for supersaturated air If the air is supersaturated, the excess water vapor will condense as a water mist. The control volume in Fig. 2 is also applicable if the air is supersaturated. The enthalpy of supersaturated air can be expressed as
imas ¼ cpa T a þ wsa ðifgwo þ cpv T a Þ þ ðw wsa Þcpw T a ¼ cpa T a þ wsa ½iv þ cpv ðT a T w Þ þ ðw wsa Þcpw T a
ð22Þ
where wsa is the humidity ratio of saturated air at temperature Ta. Subtracting Eq. (22) from Eq. (6), by introducing (w-wsa)cpwTw-(w-wsa)cpwTw, which adds up to zero, the temperature driving potential can be obtained
Tw Ta ¼
imasw imas ðwsw wsa Þiv þ ðw wsa Þcpw T w cpmas
ð23Þ
where cpmas is the specific heat of supersaturated air per unit mass of dry air and define as
cpmas ¼ cpa þ wsa cpv þ ðw wsa Þcpw
ð24Þ
Bourillot [12], Poppe and Rögener[8] and Kloppers and Kröger [7] proposed that the heat and mass transfer coefficients for supersaturated and unsaturated air were the same. The mass flow rate evaporating from the deluge water into the supersaturated air is expressed as
dmw ¼ hd ðwsw wsa ÞdA
ð25Þ
By applying the same analysis as in the case of unsaturated air, the governing equations can be obtained
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Table 1 Comparison between predicted and experimental outlet process-water and air temperature. Case
Process water 3
1 2 3 4 5 6 7 8
Deluge water
Air inlet
3
Tpo (°C)
3
Tao (°C)
Vp (m /h)
Tpi (°C)
Vd (m /h)
Va (m /s)
Tai (°C)
Twbi (°C)
By Exp.
By Poppe
Err (%)
By Exp.
By Poppe
Err (%)
28.47 28.47 27.24 28.28 28.31 26.91 28.11 28.22
22.91 22.54 39.22 22.93 22.56 39.27 22.99 22.61
15.07 15.32 15.25 15.09 15.24 15.19 15.08 15.22
4.00 3.87 3.79 3.17 3.05 2.97 2.35 2.22
9.9 14.6 15.0 10.1 14.7 14.9 9.8 14.7
4.1 7.1 7.6 4.0 7.1 7.6 3.9 7.1
20.03 20.13 33.27 20.43 20.51 34.02 20.91 20.91
19.94 20.07 33.15 20.40 20.46 33.99 20.95 20.94
0.25 0.25 0.48 0.29 0.15 0.32 0.34 0.24
13.78 16.38 23.58 14.5 16.82 24.23 15.14 17.26
14.37 16.51 23.32 14.86 16.91 23.96 15.29 17.36
4.28 0.79 1.10 2.48 0.54 1.11 0.99 0.58
215
190
210
180 Q , kW
Q , kW
205 200 Tpi=310K, Tai=280K Unsaturated Supersaturated
195 190
170 160 Tpi=310K, Tai=290K Unsaturated Supersaturated
150 140
185
0
20
40
60 RH, %
80
100
0
20
(a) Tpi=310K, Tai=280K
40
60 RH, %
80
100
(b) Tpi=310K, Tai=290K 160
180 160
120 Q , kW
Q , kW
140 120 Tpi=310K, Tai=300K Unsaturated Supersaturated
100
80 Tpi=310K, Tai=310K Unsaturated Supersaturated
40
80 0
20
40
60 RH, %
80
100
0
0
(c) Tpi=310K, Tai=300K
20
40
60 RH, %
80
100
(d) Tpi=310K, Tai=310K
Fig. 3. Comparison between the total heat transfer rate under unsaturated and supersaturated conditions.
dmw ¼ ma dw dT w ¼
1 ðmp cpp dT p ma dima cpw T w dmw Þ mw cpw
ð28Þ
hd ðwsw wsa ÞdA ma
ð29Þ
wsw þ 0:622 wsw þ 0:622 Lef ¼ 0:8650:667 1 ln wsa þ 0:622 wsa þ 0:622
2.1.3. Deluge-water temperature distribution Heyns [13] showed an example of the change in the delugewater temperature through the tube bundle. Since the deluge water is recirculated (Fig. 1), temperature rising due to pump work is small and can be neglected, the temperature of the deluge water at the inlet and outlet of the tube bundle is the same [3,13,14].
T wi ¼ T wo
ð32Þ
ð30Þ
dima ¼ dQ a =ma ¼
If the outlet air is supersaturated, the temperature and enthalpy distribution can be calculated from Eqs. (26)–(31).
ð27Þ
KðT p T w ÞdA mp cpp
dT p ¼
dw ¼
ð26Þ
hd ½Lef ðimasw imas þ ðw wsa Þcpw T w Þ þ ð1 Lef Þðwsw wsa Þiv dA ma ð31Þ
2.2. Solving the governing equations The governing equations for the thermal performance of the CWCT can be evaluated iteratively by the finite difference method. The air conditions in terms of temperature, humidity ratio and enthalpy for unsaturated and supersaturated air must be solved
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40
40
35
35
Tpo ,°C
Tpo , °C
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30 Tpi=310K, Tai=280K Unsaturated Supersaturated
25
20
0
20
30 Tpi=310K, Tai=290K Unsaturated Supersaturated
25
40
60
80
20
10 0
0
20
40
40
40
35
35
30 Tpi=310K, Tai=300K Unsaturated Supersaturated
20 0
20
40
100
30 Tpi=310K, Tai=310K Unsaturated Supersaturated
25
60
80
(b) Tpi=310K, Tai=290K
Tpo , °C
Tpo , °C
(a) Tpi=310K, Tai=280K
25
60
RH, %
RH, %
80
100
20 0
20
40
60
80
RH, %
RH, %
(c) Tpi=310K, Tai=300K
(d) Tpi=310K, Tai=310K
100
Fig. 4. Comparison between the process-water outlet temperatures under unsaturated and supersaturated conditions.
21.5
0.017
21.0 20.5
Tpi=310K, Tai=280K Unsaturated Supersaturated
20.0 Tao ,°C
wo , kg/kg
0.016
Tpi=310K, Tai=280K Unsaturated Supersaturated
0.015
19.5 19.0
0.014
0.013 0.000
18.5 18.0 0.001
0.002
0.003 0.004 wi , kg/kg
0.005
0.006
Fig. 5. Comparison between the outlet humidity ratio of air under unsaturated and supersaturated conditions.
by an iterative procedure. Successive iterations are needed for the values of the outlet air enthalpy imao, outlet humidity ratio wo and the inlet deluge-water temperature Tw to find the values for all the control volumes. The solution procedure is presented in Appendix A. For the unsaturated condition, governing equations for unsaturated air are solved to predict the thermal performance of the CWCT. For the supersaturated condition, a test is implemented to determine whether the air is unsaturated or supersaturated in each control volume, then the program automatically selects the corresponding governing equations. The thermo-physical properties (density, thermal conductivity, viscosity and heat capacity) of process water and saturated air at air-deluge water interface can
17.5 0.000
0.001
0.002
0.003 0.004 wi , kg/kg
0.005
0.006
Fig. 6. Comparison between the outlet air temperature under unsaturated and supersaturated conditions.
be calculated by some empirical equations, which are summarized by Kröger [14].
3. Results and discussion 3.1. Experimental validation for unsaturated air A prototype tower [15] consists of eight rows of externally galvanized steel oval tubes. The major axis of the tube is 31.8 mm, and minor axis 21.6 mm with tube wall thickness 1.5 mm. The oval
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Fig. 7. Path of air temperature and humidity indicated on a supersaturated psychrometric chart.
0.018
0.014 0.012
20 18 16
0.010
Ta ,°C
w , kg/kg
22
Tpi=310K, Tai=280K, wi=0.004kg/kg wunsaturated wsupersaturated wsaturated
0.016
0.008
Tpi=310K, Tai=280K, wi=0.004kg/kg Ta , unsaturated Ta , supersaturated
12
0.006 0.004
10
0
1
2
3
45 Tube row
6
7
8
8
Fig. 8. Distribution of humidity ratio of air under unsaturated and supersaturated conditions along the CWCT.
tubes are 1.25 m long and arranged in a stagger pattern at a transversal pitch of 53.4 mm. There are 37 oval tubes per tube row. The correlations for deluge-water film heat transfer coefficient (hw) and mass transfer coefficient (hd) [15] were expressed as 1=3 hw ¼ 350:3ð1 þ 0:0169tw ÞG0:59 Gw a
ð33Þ
hd ¼ 0:034G0:977 a
ð34Þ
for 2.57 < Ga < 4.94 kg/m2 s, 1.2 < Gw = U/d < 3.176 kg/m2 s, 11 < Tw < 28°C. The tube-side heat transfer coefficient hi in a turbulent flow was calculated as follows
hi ¼ 0:023
14
k 0:8 0:3 Re Pr di
ð35Þ
Establishing the CWCT heat and mass transfer coefficients and using the presented analytical models, it is possible to predict the thermal performance characteristics with variable outdoor air conditions and flow rates. A comparison between the predicted and experimental results for outlet process water temperature is given in Table 1. As it can be observed, the predicted results are in good agreement with the corresponding experimental data. This can verify the reliability of the prediction by Poppe analysis for unsaturated air. The predicted
1
2
3
4 5 Tube row
6
7
8
Fig. 9. Distribution of air temperature under unsaturated and supersaturated conditions along the CWCT.
outlet air temperature also shows a good agreement with experimental data. The acceptable agreement between the theoretical and experimental results for unsaturated conditions enhances our confidence on the reliability of the numerical investigation of the thermal performance for the supersaturated conditions. 3.2. Comparison and discussion Since the experimental data for the distributions of processwater temperature, dry-bulb temperature and humidity ratio of air along the CWCT are unavailable, the comparison is limited to the prediction by Poppe method under both unsaturated and supersaturated conditions. The differences between the predicted results are investigated under different operating conditions. Inlet air temperatures of 280, 290, 300 and 310 K are considered. The relative humidity of air is varied from 8 to 97%. Fig. 3 shows the comparison between the total heat transfer rate Q under unsaturated and supersaturated conditions. The predicted total heat transfer rate is the same if the air is unsaturated. It is found that the outlet air conditions are all unsaturated at inlet air temperature of 290, 300 and 310 K of RH varying from 8 to 97%. Whereas at an inlet air temperature of 280 K, the predicted results
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80
60 i , kJ/kg
show that the outlet air conditions are all supersaturated. The higher the inlet air temperature, the higher the saturation point and hence the more water vapor can be absorbed when the air gets saturated. Little difference in the total heat transfer rate is found between the unsaturated and supersaturated conditions. The difference is approximately zero. The process-water outlet temperatures under unsaturated and supersaturated conditions are shown in Fig. 4. Because the difference in the heat transfer rate is approximately zero, the corresponding process-water outlet temperatures show no difference between the results at different air conditions. The degree of saturation of the air does not have a great impact on the heat transfer rate and process-water outlet temperature between the two conditions predicted by Poppe method. Kloppers and Kröger [7] pointed out that the lines of constant air enthalpy in the supersaturated region were very close to vertical. Therefore, for a specific air enthalpy, it does not matter how much water vapor and water mist are present in the supersaturated air. Thus, the assumption that the air is unsaturated is also a very useful assumption to predict the total heat transfer rate of the CWCTs if the actual air is supersaturated. The following results are obtained for inlet air temperature of 280 K. As can be seen in Fig. 5, the difference in humidity ratio of outlet air is increasing with the increasing humidity ratio of inlet air. At 0.006 kg/kg dry air (RH = 97.7%) in a very humid condition, the difference is almost 4.53%. According to Fig. 6, the predicted outlet air temperatures are not very close to each other under unsaturated and supersaturated conditions. The maximum difference is 9.3% at humidity ratio of 0.006 kg/kg dry air (RH = 97.7%). As shown in Fig. 7, in the supersaturated region on a psychrometric chart, lines of constant air enthalpy are close to vertical, while in the unsaturated region, the lines of constant air enthalpy are far from vertical which will cause discrepancy in outlet air temperature prediction. For example, the supersaturated condition point B whose enthalpy is 60 kJ/kg dry air and humidity ratio is 0.020 kg/kg dry air, then air temperature is determined to be 18.61°C.Assuming the supersaturated point B is in the unsaturated region, if the humidity ratio and enthalpy are the same with point B, the predicted point will be A with the corresponding air temperature 11.33°C which is much lower than the actual air temperature, if the air temperature and enthalpy are the same with point B, the predicted humidity ratio is 0.0165 kg/kg dry air. Thus, the predicted air temperature will be between 11.33 and 18.61°C and the humidity ratio will be in the range of 0.0165 to 0.020 kg/kg dry air which are lower than the values of the actual supersaturated condition. Thus, the assumption that the air is unsaturated is unable to predict accurate air temperature and humidity ratio if the actual air is supersaturated. Due to the rigorous Poppe method derivation, the analytical model with Poppe method can be able to give the distributions of enthalpy, air temperature and humidity ratio along the CWCT. The air condition of temperature of 280 K and humidity ratio of 0.004 kg/kg dry air which the corresponding relative humidity is approximately 65.3% is considered. The distributions are showed as follows (i = 1 for the bottom row; i = 8 for the top row). Since the inlet air is relatively low temperature and high humid, the air immediately becomes supersaturated as it passes several tube rows, as shown in Fig. 8. The transition from unsaturation to supersaturation is decided by the humidity ratio whether is lower or higher than the saturated humidity ratio. For the unsaturated condition, the predicted humidity ratio is lower than the saturated humidity ratio, while the predicted humidity ratio is higher than the saturated humidity ratio for the supersaturated condition. As the temperature and humidity gradient still exist between the air-deluge water interface and air, it can be seen in Figs. 8 and 9, the air temperature together with the saturated humidity ratio increase. The predicted humidity ratio under unsaturated condition
Tpi=310K, Tai=280K wi=0.004kg/kg ima, unsaturated ima, supersaturated imasw, unsaturated imasw, supersaturated
40
20
0
1
2
3
4 5 Tube row
6
7
8
Fig. 10. Distribution of enthalpy under unsaturated and supersaturated conditions along the CWCT.
lie between the saturated and supersaturated humidity ratio predicted under supersaturated condition. Fig. 10 shows the distribution of enthalpy under unsaturated and supersaturated conditions along the CWCT. No significant difference is found between the two conditions. That the enthalpy of water mist in the air is small compared to the total air enthalpy may be an explanation for it. 4. Conclusion The governing equations based on Poppe method are derived to investigate the thermal performance of CWCT under both unsaturated and supersaturated conditions. A detailed program with finite difference method is presented to solve the governing equations. Little difference in heat transfer rate and process-water temperature is found between the prediction under the unsaturated and supersaturated conditions. The results show that it is very useful to predict the total heat transfer rate and outlet process-water temperature of the CWCTs that the air is assumed to be unsaturated if the actual air is supersaturated. It indicates that supersaturation is likely to occur at low inlet air temperature. It is clear from the discussion, the predicted air temperatures and humidity ratio are not very close to the two conditions if the outlet air is supersaturated. The distribution of air temperature, enthalpy and humidity ratio along the CWCT would provide a better understanding of the heat and mass transfer of CWCT. The present method can be employed to predict the air condition in the supersaturated state along the CWCT. Acknowledgements The work was financially supported by Cultivation Fund for Interdisciplinary and Major Project (MOE) (2010JB0621) and Natural Science Foundation of China under Grant No. 50905060. Appendix A. Solving procedure The finite difference method is employed to solve the governing equations for unsaturated and supersaturated conditions based on Poppe method. Eqs. (2), (4), (10), (13), (20), and (21) represent the governing equations for unsaturated air and Eqs. (26)–(31) for supersaturated air. The CWCT is divided into a number of control volumes, as shown in Fig. A.1. The governing equations for heat and mass transfer are calculated for each control volume. The solution starts by taking the process-water inlet control volume and proceeds in the direction of process water flow. The thermo-physical
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Deluge water
(1,1)
(1,2)
(1,n-1)
(1,n)
Process water
(2i-1,1)
(2i-1,2)
(2i-1,n-1)
(2i-1,n)
(2i,1)
(2i,2)
(2i,n-1)
(2i,n)
(N,1)
(N,2)
dL
(N,n-1)
(N,n)
Air
(a) tube coil with N tube rows and n control volumes per tube row Fig. A.1. Flowchart of the calculating process.
Fig. A.2. Flowchart diagram for Poppe-type method.
(b) control volume
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properties (density, thermal conductivity, viscosity and heat capacity) of process water and saturated air at the air-deluge water interface in current control volume are defined based on the outlet temperature of the preceding control volume. The equations must be solved by an iterative procedure. Successive iterations are needed for the values of the outlet air enthalpy imao, outlet humidity ratio wo and the inlet deluge-water temperature Twi to find the values for all the control volumes. The flowchart diagrams for the thermal performance prediction under both unsaturated and supersaturated conditions by Poppe method are showed in Fig. A.2. First, iterations are carried out to check the average of computed inlet air enthalpy with the inlet air state imai. Second, iterations are executed to compare the average computed inlet air humidity ratio with the inlet air state wi. Finally, iterations are implemented to approach equality of Twi and Two. Under the supersaturated condition, test is implemented to determine whether the air is unsaturated or supersaturated in each control volume. For the control volume (2i-1,j), the air is assumed to be unsaturated, the air temperature Ta(2i-1,j) is determined from Eq. (5) by iterative with w(2i-1,j) and ima(2i-1,j) known. Then assume the air is saturated and determine the wetbulb temperature Twb(2i-1,j) from the following Eq. (A.1). If Ta(2i-1,j) < Twb(2i-1,j), which is impossible, the air is supersaturated, otherwise the air is unsaturated and Ta(2i-1,j) is the dry-bulb temperature of the air. The dry-bulb temperature of supersaturated air Ta(2i-1,j) is determined from Eq. (22) by iterative with w(2i-1,j) and ima(2i-1,j) known. w¼
2501:6 2:3263ðT wb 273:15Þ 0:62509pv wb 2501:6 þ 1:8577ðT 273:15Þ 4:184ðT wb 273:15Þ P a 1:005pv wb 1:00416ðT T wb Þ 2501:6 þ 1:8577ðT 273:15Þ 4:184ðT wb 273:15Þ
ðA:1Þ
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