- Email: [email protected]
Modelling of heat and mass transfer in a horizontal-tube falling-film condenser for brackish water desalination in remote areas
DESALINATION ELSEVIER
Desalination 166 (2004) 17-24
www.elsevier.com/locate/desal
Modelling of heat and mass transfer in a horizontal-tube fallingfilm condenser for brackish water desalination in remote areas K . B o u r o u n i a*, M . T . C h a i b i b ~Laboratoire d'Energie Solaire, D~partement de G~nie Industriel, Ecole Nationate d'Ing~nieurs de Tunis, BP 37 Le Belv~dkre, 1012 Tunis, Tunisia Tel. +216 (1) 874700; Fax : +216 (1) 871729; email: [email protected] blnstitut National de Recherche en G~nie Rural, Eaux et Forets, PO Box 10, Ariana 2080, Tunisia
Received 5 March 2004; accepted 18 March 2004
Abstract Horizontal-tube falling film condensers are used in many fields: chemical industries, refrigeration, desalination, cooling, etc. However, their performances cannot be predicted precisely due to the complexity of the phenomena. In this paper, heat and mass transfer in a horizontal-tube falling-film condenser used in an innovative desalination plant was studied theoretically. The polypropylene exchanger was designed to work at relatively high temperatures (2535°C). In fact, the desalination unit was designed to function in remote areas where the ambient temperature frequently exceeds 30°C [1 ]. The model developed uses basic aerodynamic, hydrodynamic and heat/mass transfer information to predict the performance of the exchanger. The predicted transfer characteristics obtained from the simulations were compared to experimental data [2]. From this comparison, it was seen that the model is well able to predict the trends of heat and mass characteristics of the condenser. The influence of the different thermal, hydrodynamic and geometric parameters on the condenser performances was investigated. The variations of the distilled water amount inside the exchanger were analysed. The results were used to determine the optimum operational conditions. Finally, the model was also used to optimise the different exchanger components. Keywords: Condensation; Desalination; Horizontal tubes; Falling film; Heat transfer coefficient; Performance
1. Introduction Shell and tube condensers are widely used in refrigeration, desalination and process industries *Corresponding author.
where condensation on the outside of horizontal tubes is very common. A typical configuration is shown in Fig. 1. The condensing vapour is introduced into the top of the exchanger as saturated or superheated vapour. If the temperature of the tube bundle surface is kept below the vapour
Presented at the EuroMed 2004 conference on Desalination Strategies in South Mediterranean Countries: Cooperation between Mediterranean Countries of Europe and the Southern Rim of the Mediterranean. Sponsored by the European Desalination Society and Office National de l'Eau Potable, Marrakech, Morocco, 30 May-2 June, 2004.
0011-9164/04/$- See front matter © 2004 Elsevier B.V. All rights reserved doi; 10.1016/j.desal.2004.06.055
18
K. Bourouni, M.T. Chaibi /Desalination 166 (2004) 17-24
Condensing vapour inlet ~
O~
ing heat transfer coefficients becomes very difficult. Design and modelling of these devices rely on the accuracy of the theoretical heat transfer relationships of the condensing vapour [3]. Sarma et al. [5] studied the problem of condensation of vapours flowing with high velocity around a horizontal tube for the ease of tube wall isothermal conditions. The interfacial shear at the vapour condensate film is assessed with the help of the Colbum analogy. Extension of this result in the estimation of average condensation heat transfer coefficients was successful. Experimental investigations highlighted the difficulty of the study of this kind of exchanger. In fact, the vapour velocity distribution through tube banks is very complex [6]. On the other hand, applying models which successfully predict enhancement due to the vapour shear on single tubes to tube bank geometries is still somewhat questionable. In this paper a global model was developed to predict the condensing heat and mass transfer coefficients under tube bank conditions. The aim was to complete the model developed by Bourouni et al. [1] for the evaporator used in the desalination unit. According to this investigation, a model for the global desalination unit (evaporator and condenser) is provided.
oooooooO00 ooo
/
/ oooo
id o
o /
(b)
Fig. 1. Schematic of shell-and-tube condensing heat exchanger with vapour-downflow for (a) an in, line tube bundle configuration and (b) a staggered tube bundle configuration.
saturation temperature the vapour is condensed on a bank of horizontal tubes whereupon it leaves the exchanger in a saturated or sub-cooled state [3]. Although for basic mechanisms (homogeneous, direct contact, drop and film), most condensers are designed to operate under the film condensation mode. The process is characterised by the formation of a thin film of liquid that drains under the action of gravity or surface tension or both. The presence of a film creates a barrier between the vapour and the cooled surface and thus retards the condensation process. If condensation is to be enhanced, the film thickness must be reduced. This reduction can be achieved by using, among other methods, finned surfaces instead of plan surfaces [4]. Due to the complex geometry of shell-andtube condensers, heat transfer is controlled by the combined effects of tube surface geometry, condensate inundation from the tubes above, and condensing vapour velocity (vapour shear effects). Therefore, the prediction of the condens-
2. Mathematical model
The humid air, in provenance from the evaporator, enters the condenser by the top at temperature T,.,g, humidity X,., and mass flow rate rhi,,g. The cooling water is introduced at the bottom of the condenser at temperature T~,.ctand a mass flow rate rhj,.d.i(Fig. 2). In the condenser the humid air moves down through the space between the tubes. On contact with the cold tube walls, there is film condensation coupled with latent heat restitution to the salt water circulating inside the tubes. The presence of a film creates a barrier between the
K. Bourouni, 34. 7'. Chaibi / Desalination 166 (2004) 17-24
(b)
Air inlet
19
Liquid film entrance
Cooling~l~2 water outlet Y~
I
Tin.c Cell i+l Cc Celt N (N=40)
V
Tout.g,i, Uout.g,i , Xom,i,Pout,i Fig. 2. Division of the condenser into cells and presentation of the different fluid flows.
vapour and the cooled surface and thus retards the condensation process. The humid air and liquid film flows are assumed to be approximately equal over the various channels of the heat exchanger. This allows the analysis to be limited to the consideration of one tube stage. The exchanger is divided into 40 stages (Fig. 2a). Because of the geometry of the condenser, the transfer between the fluids is a three-dimensional process (Fig. 2b). In fact, all the parameters characterising heat and mass transfer vary from cell to cell and inside each cell from one tube to the other. To simplify the modelling, we consider the mean values of the temperature and water vapour mixed fraction in each cell. The modelling problem becomes one-dimensional. The regime considered in this investigation is permanent. It is convenient to start the calculation process at the top of the exchanger (at the humid air inlet side) since the initial parameters are known there. For a typical element i, the humid air and the condensed liquid film flow between the tubes and the cooled water flows inside the tubes in direction z. The positive direction is chosen
downwards, thus the falling film has a positive velocity. The coordinates are rendered dimensionless by means of the cell height B. A fraction of the heat flux supplied by the condensation is transferred to the cooling water inside the tubes by conduction through the liquid film and the tube walls. The elaboration of the mathematical model is based on the following assumptions: • The entire surface of the tubes is covered with liquid film: there is perfect wetting. ® The film flow is steady without any intermption. ° The film thickness is small compared to the tube diameter. • The surface tension and pressure gradient effects are negligible. ° The temperature of the tube walls is uniform. • The condensation occurs on the liquid-vapour interface. ° The humid air is considered as a perfect gas. • The liquid-gas interface is at thermodynamic equilibrium, and there is no gas dissolution in the liquid.
K. Bouroung M.T. Chaibi/ Desalination 166 (2004) 17-24
20
• Heat transfer by radiation is not considered.
2.1. Energy balance In a calculation cell, the energy balance relations for the fluids are: • Cooling liquid:
rhctCPct dTcl dz
-gdint~l(Tw, int-Tcl )
(1)
The internal wall temperature Tw.i,, is deduced from the equation of the heat conduction through the tube wall. • Liquid film: The temperature gap between the film and the wall causes a heat transfer from the interface to the wall. The film thickness is neglected compared to the tube radius. The energy balance yields:
The first term on the left-hand side of Eq. (4) represents the mass flux towards the liquid film where condensation takes place and the latent heat is provided, while the second term represents the heat transferred by convection.
2.2. Boundary conditions We assume that all heat transfer crossing the film is transferred to the tube wall. The energy balance at the interface yields:
hf (Zl- Zw.ext)=gg(~- Zl)
(5)
In the case of condensation, the amount of evaporated water is given by the equation of Stefane [7] deduced from Fick's theory:
-D
<.,¢Cp¢ ez aT, = -rha C&t dz
(6)
+
(2) The coefficient D~ represents the diffusivity of vapour in the air. It is given by:
The external wall temperature Tw.e,, is deduced from the equation of the heat conduction through the liquid film. • Humid air: The heat transfer between the humid air and the liquid film can occur with or without condensation at the interface. If the vapour pressure at the interface is equal to the saturation pressure, no condensation takes place. In this case the energy balance is given by:
1.81
v
Pt
273.15
(7)
2.3. Mass balance In the case of condensation at the interface, it is convenient to add the equations showing the conservation of the mass flow between fluid flows to the energy balance equations and boundary conditions.
(3)
thi..g Cpg dz =
mc = xrh;n.g
If not, condensation takes place:
<,,-
0: 2.26(+27315)
dT,
(4)
(8)
where x represents the fraction of condensed vapour. When condensation at the air-liquid interface occurs, the humid air mass flow decreases. On the
K. Bourouni, M. 1". Chaibi / Desalination 166 (2004) 17-24 other hand, the liquid film mass flow increases. The following equations summarise this discrepancy:
rhout.g: (1 -x) mi,.g
(9)
thout,f = lhin.f + rh c
(10)
2.4. Heat transfer coefficients
In Eq. (4) h: represents the liquid film heat transfer coefficient. The results obtained from the literature show that this coefficient depends on film properties, dimension of the tube and the temperature gradient between the saturation temperature and the wall temperature. For the first cell, this coefficient is given by the Nusselt correlation [3]: .
The coefficient he; in Eq. (1) represents the convective heat transfer coefficient through a fluid flowing inside the tubes. In the turbulent regime, it is given by the correlation of Colburn
21
3
]1/4
;:,--0.72s v:(T,,~,-Tw,ex,)d=,] P: I However, for cell number i, the correlation given by Fujii et al. [3] was used:
[8]: )%1
08
hc; =0.023 din---~tRec;" Prff 3
(11)
The coefficient h, used in Eq. (4) represents the convective heat transfer coefficient in an air flow around a bundle of tubes. Based on the data available, Colburn [8] recommended the following dimensionless equation for gases flowing normally to banks of tubes for a Reynolds number from 2,000 to 32,000:
hgd~,_0.33(
Cp~g ] 1/3 dextpg Ug.max
0.¢
(12)
~ i _ 1.24[i3/4_(i_1)3/4]
(14)
To take into account the variations in fluid properties as a function of temperature, McAdams [9] proposed evaluating all the physical properties at the average temperatures of the fluids in the cell. Since the fluid properties are evaluated at average temperatures, which are unknown, and because the relations between them are not simple linear functions, the proposed equations are not linear. To express heat conservation in a given cell of the evaporator, the energy balance Eqs. (1)-(5) can be transformed into the following equation:
/~ar,.,,aT~,,,ar~,,.aV..,) -- m ~ C p A r ~ , , - ~d,.,L,g~,., A r:,~ + mc, Cpc, AT1, , - mo.,.:Cp:Ars,;- r~d~.Li~;AT3,i (15)
+ mc, Cpc, Ar,,,-mou,:Cp:AT,.~+
rCd~x,L,~,~ATa, g
mctCpdAr,,i-rhout.fCp:ATs,i-rhi,,.g.; Cpg Ara,,-mc,,L,,,~ 1= 0
where (16)
AT,,i = To~,.ct,i- Ti,.ct,i' AT2,i = Tw.im,i- ~t,i, AT3,i = T,,i- T~.~t,i, AT4,i = T,,i- ~,i, ATs,i = Ti,.f,i- To~,.f,i
22
K. Bourouni, M.T. Chaibi / Desalination 166 (2004) 17-24
The temperature gap ATs.; was deduced from the other variables. This justifies the fact that function f only depends on four independent variables. The problem is thus transformed to a non-linear multi-variable function minimisation. In each gap of the exchanger, solving heat transfer equations is equivalent to minimising the functionf. To solve this problem we used the downhill simplex method developed by Mead and Nelder [10]. This method is based on the successive evaluation of the function and not on its derivation. It consists initially in taking trial values of AT/,0,i = 1.4 and in calculating the corresponding values of the functionf. It is the initialisation of the function. This step is considered as the iteration 0. After, at each algorithm iteration j, the variable incrementation (t;. ~= ~4) giving a maximum decrease in the function is determined. The solution is obtained when the function residue is less than 10 -4. Then the quantity of condensed water in the gap, fluid temperatures, mass flow rates, and air humidity at the cell outlet are determined. The latter parameters correspond to the following cell inlet.
Uin.g
3.1. Influence o f inlet air velocity
We investigated the influence of air velocity on the mass and heat transfer characteristics of the condenser. The variation of the interfacial mass condensation with air velocity Ug is plotted in Fig. 3. An inspection of this plot reveals that m~ increases with Ug. This result can be explained by an increase in the air heat transfer coefficient with the air velocity [Eq. (12)]. This causes an increase in the heat transfer between the humid air and the cooling water. 251X)
2000
~-~ 1500 v
&
1000
&
500 ),
o 0.02
.
.
#'& ~ .
.
0.0A
.
.
.
.
0.06
4x & .
~' .
.
0.08
.
.
)
.
.
0.1
.
~
.
.
.
0.12
i
'
'
0.14
0.16
.
L
0.1~
Ug {m/s)
Fig. 3. Influence of the air velocity on the total amount of condensed water. T,,.cl = 20°C, T,,g = 70°C, Rei,.a = 14,000, Re~,f = 420, Xi,.g = 35 g/kg.
3. Numerical results
The instrumentation developed to analyse heat and mass transfer in the plant only allows the investigation of global parameters (inlet and outlet temperatures, pressures, humidity and total amount of condensed vapour). The global measured parameters were systematically compared to the data of Bourouni et al. [2] and those predicted by the model. Below, typical results obtained from simulations and experiments are presented. The basic parameter in this problem is the prediction of the amount o f condensed vapour. For a given operational condition, the variation in this parameter vs. inlet air velocity, cooling liquid temperature and mass flow rate are presented.
2500 & 2000 & ~ " 1501}
1000
& &
500
~x
i , i I , ~ ) , l
0 0
/0
. . . .
20
t
30
. . . .
i
40
. . . .
!
50
. . . .
i
60
. . . .
A
70
. . . .
80
Ti~cl (°C)
Fig. 4. Influence of the ~,.c~ on the total amount of condensed water. ~,.~ = 70°C, Re~,.,.l= 14,000, Rei,f--- 420, Ui,.g = 1 re~s, Xi,.g = 35 g/kg.
K. BourounL M.T. Chaibi / Desalination 166 (2004) 17-24
3.2. I n f l u e n c e o f c o o l i n g w a t e r inlet t e m p e r a t u r e
m
i n . c l
In order to analyse the influence of the cooling liquid inlet temperature on the condensation, the variations in the total quantity o f condensed water vs. the T~,,clt are presented in Fig. 4. According to this figure, it can be seen that the amount of condensed water decreases significantly when T~,.cI increases. At 35°C, which is the mean cooling liquid temperature, the amount of condensed water cannot exceed t m3/d.
4.
g h h
C o n c l u s i o n s
An analytical study to predict the heat and mass transfer characteristics of the plastic horizontal-tube falling-film condenser was developed. The variations in fluid properties during their progression in the exchanger due to temperature evolution were considered. The elaborated model is based on the resolution of heat and mass transfer equations in each cell of the exchanger. This resolution was effected by using fluid properties calculated at the average temperature in the cell. The influence of inlet air velocity and cooling liquid inlet temperature on heat and mass transfer in the exchanger was investigated. Numerical results show an increase in condensing exchanger performance when the inlet cooling water decreases and air velocity increases. For the conditions corresponding to the real environment of southern Tunisia, the model predicted that the amount of condensed water cannot exceed 2 m3/d.
Lv ni M P Pr
Ro Rect
---
Acceleration of gravity, kg/m2s Heat transfer coefficient, w/m2°C - Average heat transfer coefficient, w/m2°C - Latent heat of condensation, kJ/kg - Mass flux, kg/s - Molecular weight, kg/(kg mole) - Pressure, Pa - - D i m e n s i o n l e s s Prandtl number = - -
--
Rein.g
--
T T U
-- -
--
X
- -
X
--
Z
- -
Z
- -
V
---
Thermal conductivity, W/m°C Density, kg/m3 - Kinematics viscosity, m2/s - - Dynamic viscosity, kg/m s
Subscripts a c
el ev
ext 5.
S y m b o l s
B
Cp
--
d Ov
f
--
Cell height, m Specific heat, J/Kg °C Tube diameter, m Diffusivity o f v a p o u r in air, mZ/s Function
Perfect gas constant, J/(mole °C) Dimensionless Reynolds number for cooling liquid = (di,, Ucl)/(Vc/) Dimensionless Reynolds number for air = (dext Ug)/(Vg) Local temperature, °C ' Average temp., °C = (r; ~ + Tout)~2 Velocity, m/s Evaporated water fraction Absolute humidity, kg/kg Coordinates Dimensionless coordinates = (z/B)
Greek
P
--
f
--
g
--
i !
---
in int
---
j
23
--
Dry air Condensed Cooling liquid Evaporated External Liquid film Gas Cell step Interface Inlet Internal Iteration step
24
K. BourounL M.T. Chaibi / Desalination 166 (2004) 17-24
out
--
Outlet
sat
--
Saturated
v
--
Vapour
w
--
Tube
vs
--
Saturant
wall vapour
References [ 1] K. Bourouni, R. Martin, L. Tadrist and H. Tadrist, Proc. Heat transfer and evaporation in cross flow plastic heat exchanger for water desalination. IDA Congress on Desalination and Water Reuse, Madrid, 127 (1997) 145. [2] K. Bourouni, M.T. Chaibi, R. Martin and L. Tadrist, Heat transfer and evaporation in geothermal desalination units, Appl. Energy, 64 (1999) 129-147. [3] M.W. Browne and P.K. Bansal, An overview of condensation heat transfer on horizontal tube bundles, Appl. Thermal Engn., 19 (1999) 565-594. [4] J.V. Lunardini and A. Aziz, Effect of condensation on performance and design of extended surfaces, Cerel Report 95-20, 1995.
[5] P.K. Sarma, B. Vijayalakshrni, F. Mayinger and S. Kakak, Turbulent film condensation on horizontal tube with external flow of pure vapors, Int. J. Heat and Mass Transfer, 41 (1988) 537-545. [6] H. Honda and S. Nozu, Effect of drainage strips on the condensation heat transfer performance of horizontal finned-tubes, in: Heat Transfer Science and Technology, Hemisphere, New York, 1987, pp. 455-462. • [7] A. Agunaoun and A. Daif, Evaporation en convection forc6e d'un film mince s'ecoulant en r6gime permanent, laminaire et sans onde sur une surface plane inclin6e, Int. J. Heat Mass Trans., 18 (1994) 2947-2956. [8] A.P. Colbum, A method of correlating forced convection heat transfer data and comparison with fluid friction, Trans. AIChE, 29 (1933) 174-179. [9] W.H. McAdams, Heat Transmission, McGraw-Hill, New York, 1954. [10] H.W. Press, P.F. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes, in: The Art of Scientitle Computing, Cambridge University Press, London, 1986.
Desalination 166 (2004) 17-24
www.elsevier.com/locate/desal
Modelling of heat and mass transfer in a horizontal-tube fallingfilm condenser for brackish water desalination in remote areas K . B o u r o u n i a*, M . T . C h a i b i b ~Laboratoire d'Energie Solaire, D~partement de G~nie Industriel, Ecole Nationate d'Ing~nieurs de Tunis, BP 37 Le Belv~dkre, 1012 Tunis, Tunisia Tel. +216 (1) 874700; Fax : +216 (1) 871729; email: [email protected] blnstitut National de Recherche en G~nie Rural, Eaux et Forets, PO Box 10, Ariana 2080, Tunisia
Received 5 March 2004; accepted 18 March 2004
Abstract Horizontal-tube falling film condensers are used in many fields: chemical industries, refrigeration, desalination, cooling, etc. However, their performances cannot be predicted precisely due to the complexity of the phenomena. In this paper, heat and mass transfer in a horizontal-tube falling-film condenser used in an innovative desalination plant was studied theoretically. The polypropylene exchanger was designed to work at relatively high temperatures (2535°C). In fact, the desalination unit was designed to function in remote areas where the ambient temperature frequently exceeds 30°C [1 ]. The model developed uses basic aerodynamic, hydrodynamic and heat/mass transfer information to predict the performance of the exchanger. The predicted transfer characteristics obtained from the simulations were compared to experimental data [2]. From this comparison, it was seen that the model is well able to predict the trends of heat and mass characteristics of the condenser. The influence of the different thermal, hydrodynamic and geometric parameters on the condenser performances was investigated. The variations of the distilled water amount inside the exchanger were analysed. The results were used to determine the optimum operational conditions. Finally, the model was also used to optimise the different exchanger components. Keywords: Condensation; Desalination; Horizontal tubes; Falling film; Heat transfer coefficient; Performance
1. Introduction Shell and tube condensers are widely used in refrigeration, desalination and process industries *Corresponding author.
where condensation on the outside of horizontal tubes is very common. A typical configuration is shown in Fig. 1. The condensing vapour is introduced into the top of the exchanger as saturated or superheated vapour. If the temperature of the tube bundle surface is kept below the vapour
Presented at the EuroMed 2004 conference on Desalination Strategies in South Mediterranean Countries: Cooperation between Mediterranean Countries of Europe and the Southern Rim of the Mediterranean. Sponsored by the European Desalination Society and Office National de l'Eau Potable, Marrakech, Morocco, 30 May-2 June, 2004.
0011-9164/04/$- See front matter © 2004 Elsevier B.V. All rights reserved doi; 10.1016/j.desal.2004.06.055
18
K. Bourouni, M.T. Chaibi /Desalination 166 (2004) 17-24
Condensing vapour inlet ~
O~
ing heat transfer coefficients becomes very difficult. Design and modelling of these devices rely on the accuracy of the theoretical heat transfer relationships of the condensing vapour [3]. Sarma et al. [5] studied the problem of condensation of vapours flowing with high velocity around a horizontal tube for the ease of tube wall isothermal conditions. The interfacial shear at the vapour condensate film is assessed with the help of the Colbum analogy. Extension of this result in the estimation of average condensation heat transfer coefficients was successful. Experimental investigations highlighted the difficulty of the study of this kind of exchanger. In fact, the vapour velocity distribution through tube banks is very complex [6]. On the other hand, applying models which successfully predict enhancement due to the vapour shear on single tubes to tube bank geometries is still somewhat questionable. In this paper a global model was developed to predict the condensing heat and mass transfer coefficients under tube bank conditions. The aim was to complete the model developed by Bourouni et al. [1] for the evaporator used in the desalination unit. According to this investigation, a model for the global desalination unit (evaporator and condenser) is provided.
oooooooO00 ooo
/
/ oooo
id o
o /
(b)
Fig. 1. Schematic of shell-and-tube condensing heat exchanger with vapour-downflow for (a) an in, line tube bundle configuration and (b) a staggered tube bundle configuration.
saturation temperature the vapour is condensed on a bank of horizontal tubes whereupon it leaves the exchanger in a saturated or sub-cooled state [3]. Although for basic mechanisms (homogeneous, direct contact, drop and film), most condensers are designed to operate under the film condensation mode. The process is characterised by the formation of a thin film of liquid that drains under the action of gravity or surface tension or both. The presence of a film creates a barrier between the vapour and the cooled surface and thus retards the condensation process. If condensation is to be enhanced, the film thickness must be reduced. This reduction can be achieved by using, among other methods, finned surfaces instead of plan surfaces [4]. Due to the complex geometry of shell-andtube condensers, heat transfer is controlled by the combined effects of tube surface geometry, condensate inundation from the tubes above, and condensing vapour velocity (vapour shear effects). Therefore, the prediction of the condens-
2. Mathematical model
The humid air, in provenance from the evaporator, enters the condenser by the top at temperature T,.,g, humidity X,., and mass flow rate rhi,,g. The cooling water is introduced at the bottom of the condenser at temperature T~,.ctand a mass flow rate rhj,.d.i(Fig. 2). In the condenser the humid air moves down through the space between the tubes. On contact with the cold tube walls, there is film condensation coupled with latent heat restitution to the salt water circulating inside the tubes. The presence of a film creates a barrier between the
K. Bourouni, 34. 7'. Chaibi / Desalination 166 (2004) 17-24
(b)
Air inlet
19
Liquid film entrance
Cooling~l~2 water outlet Y~
I
Tin.c Cell i+l Cc Celt N (N=40)
V
Tout.g,i, Uout.g,i , Xom,i,Pout,i Fig. 2. Division of the condenser into cells and presentation of the different fluid flows.
vapour and the cooled surface and thus retards the condensation process. The humid air and liquid film flows are assumed to be approximately equal over the various channels of the heat exchanger. This allows the analysis to be limited to the consideration of one tube stage. The exchanger is divided into 40 stages (Fig. 2a). Because of the geometry of the condenser, the transfer between the fluids is a three-dimensional process (Fig. 2b). In fact, all the parameters characterising heat and mass transfer vary from cell to cell and inside each cell from one tube to the other. To simplify the modelling, we consider the mean values of the temperature and water vapour mixed fraction in each cell. The modelling problem becomes one-dimensional. The regime considered in this investigation is permanent. It is convenient to start the calculation process at the top of the exchanger (at the humid air inlet side) since the initial parameters are known there. For a typical element i, the humid air and the condensed liquid film flow between the tubes and the cooled water flows inside the tubes in direction z. The positive direction is chosen
downwards, thus the falling film has a positive velocity. The coordinates are rendered dimensionless by means of the cell height B. A fraction of the heat flux supplied by the condensation is transferred to the cooling water inside the tubes by conduction through the liquid film and the tube walls. The elaboration of the mathematical model is based on the following assumptions: • The entire surface of the tubes is covered with liquid film: there is perfect wetting. ® The film flow is steady without any intermption. ° The film thickness is small compared to the tube diameter. • The surface tension and pressure gradient effects are negligible. ° The temperature of the tube walls is uniform. • The condensation occurs on the liquid-vapour interface. ° The humid air is considered as a perfect gas. • The liquid-gas interface is at thermodynamic equilibrium, and there is no gas dissolution in the liquid.
K. Bouroung M.T. Chaibi/ Desalination 166 (2004) 17-24
20
• Heat transfer by radiation is not considered.
2.1. Energy balance In a calculation cell, the energy balance relations for the fluids are: • Cooling liquid:
rhctCPct dTcl dz
-gdint~l(Tw, int-Tcl )
(1)
The internal wall temperature Tw.i,, is deduced from the equation of the heat conduction through the tube wall. • Liquid film: The temperature gap between the film and the wall causes a heat transfer from the interface to the wall. The film thickness is neglected compared to the tube radius. The energy balance yields:
The first term on the left-hand side of Eq. (4) represents the mass flux towards the liquid film where condensation takes place and the latent heat is provided, while the second term represents the heat transferred by convection.
2.2. Boundary conditions We assume that all heat transfer crossing the film is transferred to the tube wall. The energy balance at the interface yields:
hf (Zl- Zw.ext)=gg(~- Zl)
(5)
In the case of condensation, the amount of evaporated water is given by the equation of Stefane [7] deduced from Fick's theory:
-D
<.,¢Cp¢ ez aT, = -rha C&t dz
(6)
+
(2) The coefficient D~ represents the diffusivity of vapour in the air. It is given by:
The external wall temperature Tw.e,, is deduced from the equation of the heat conduction through the liquid film. • Humid air: The heat transfer between the humid air and the liquid film can occur with or without condensation at the interface. If the vapour pressure at the interface is equal to the saturation pressure, no condensation takes place. In this case the energy balance is given by:
1.81
v
Pt
273.15
(7)
2.3. Mass balance In the case of condensation at the interface, it is convenient to add the equations showing the conservation of the mass flow between fluid flows to the energy balance equations and boundary conditions.
(3)
thi..g Cpg dz =
mc = xrh;n.g
If not, condensation takes place:
<,,-
0: 2.26(+27315)
dT,
(4)
(8)
where x represents the fraction of condensed vapour. When condensation at the air-liquid interface occurs, the humid air mass flow decreases. On the
K. Bourouni, M. 1". Chaibi / Desalination 166 (2004) 17-24 other hand, the liquid film mass flow increases. The following equations summarise this discrepancy:
rhout.g: (1 -x) mi,.g
(9)
thout,f = lhin.f + rh c
(10)
2.4. Heat transfer coefficients
In Eq. (4) h: represents the liquid film heat transfer coefficient. The results obtained from the literature show that this coefficient depends on film properties, dimension of the tube and the temperature gradient between the saturation temperature and the wall temperature. For the first cell, this coefficient is given by the Nusselt correlation [3]: .
The coefficient he; in Eq. (1) represents the convective heat transfer coefficient through a fluid flowing inside the tubes. In the turbulent regime, it is given by the correlation of Colburn
21
3
]1/4
;:,--0.72s v:(T,,~,-Tw,ex,)d=,] P: I However, for cell number i, the correlation given by Fujii et al. [3] was used:
[8]: )%1
08
hc; =0.023 din---~tRec;" Prff 3
(11)
The coefficient h, used in Eq. (4) represents the convective heat transfer coefficient in an air flow around a bundle of tubes. Based on the data available, Colburn [8] recommended the following dimensionless equation for gases flowing normally to banks of tubes for a Reynolds number from 2,000 to 32,000:
hgd~,_0.33(
Cp~g ] 1/3 dextpg Ug.max
0.¢
(12)
~ i _ 1.24[i3/4_(i_1)3/4]
(14)
To take into account the variations in fluid properties as a function of temperature, McAdams [9] proposed evaluating all the physical properties at the average temperatures of the fluids in the cell. Since the fluid properties are evaluated at average temperatures, which are unknown, and because the relations between them are not simple linear functions, the proposed equations are not linear. To express heat conservation in a given cell of the evaporator, the energy balance Eqs. (1)-(5) can be transformed into the following equation:
/~ar,.,,aT~,,,ar~,,.aV..,) -- m ~ C p A r ~ , , - ~d,.,L,g~,., A r:,~ + mc, Cpc, AT1, , - mo.,.:Cp:Ars,;- r~d~.Li~;AT3,i (15)
+ mc, Cpc, Ar,,,-mou,:Cp:AT,.~+
rCd~x,L,~,~ATa, g
mctCpdAr,,i-rhout.fCp:ATs,i-rhi,,.g.; Cpg Ara,,-mc,,L,,,~ 1= 0
where (16)
AT,,i = To~,.ct,i- Ti,.ct,i' AT2,i = Tw.im,i- ~t,i, AT3,i = T,,i- T~.~t,i, AT4,i = T,,i- ~,i, ATs,i = Ti,.f,i- To~,.f,i
22
K. Bourouni, M.T. Chaibi / Desalination 166 (2004) 17-24
The temperature gap ATs.; was deduced from the other variables. This justifies the fact that function f only depends on four independent variables. The problem is thus transformed to a non-linear multi-variable function minimisation. In each gap of the exchanger, solving heat transfer equations is equivalent to minimising the functionf. To solve this problem we used the downhill simplex method developed by Mead and Nelder [10]. This method is based on the successive evaluation of the function and not on its derivation. It consists initially in taking trial values of AT/,0,i = 1.4 and in calculating the corresponding values of the functionf. It is the initialisation of the function. This step is considered as the iteration 0. After, at each algorithm iteration j, the variable incrementation (t;. ~= ~4) giving a maximum decrease in the function is determined. The solution is obtained when the function residue is less than 10 -4. Then the quantity of condensed water in the gap, fluid temperatures, mass flow rates, and air humidity at the cell outlet are determined. The latter parameters correspond to the following cell inlet.
Uin.g
3.1. Influence o f inlet air velocity
We investigated the influence of air velocity on the mass and heat transfer characteristics of the condenser. The variation of the interfacial mass condensation with air velocity Ug is plotted in Fig. 3. An inspection of this plot reveals that m~ increases with Ug. This result can be explained by an increase in the air heat transfer coefficient with the air velocity [Eq. (12)]. This causes an increase in the heat transfer between the humid air and the cooling water. 251X)
2000
~-~ 1500 v
&
1000
&
500 ),
o 0.02
.
.
#'& ~ .
.
0.0A
.
.
.
.
0.06
4x & .
~' .
.
0.08
.
.
)
.
.
0.1
.
~
.
.
.
0.12
i
'
'
0.14
0.16
.
L
0.1~
Ug {m/s)
Fig. 3. Influence of the air velocity on the total amount of condensed water. T,,.cl = 20°C, T,,g = 70°C, Rei,.a = 14,000, Re~,f = 420, Xi,.g = 35 g/kg.
3. Numerical results
The instrumentation developed to analyse heat and mass transfer in the plant only allows the investigation of global parameters (inlet and outlet temperatures, pressures, humidity and total amount of condensed vapour). The global measured parameters were systematically compared to the data of Bourouni et al. [2] and those predicted by the model. Below, typical results obtained from simulations and experiments are presented. The basic parameter in this problem is the prediction of the amount o f condensed vapour. For a given operational condition, the variation in this parameter vs. inlet air velocity, cooling liquid temperature and mass flow rate are presented.
2500 & 2000 & ~ " 1501}
1000
& &
500
~x
i , i I , ~ ) , l
0 0
/0
. . . .
20
t
30
. . . .
i
40
. . . .
!
50
. . . .
i
60
. . . .
A
70
. . . .
80
Ti~cl (°C)
Fig. 4. Influence of the ~,.c~ on the total amount of condensed water. ~,.~ = 70°C, Re~,.,.l= 14,000, Rei,f--- 420, Ui,.g = 1 re~s, Xi,.g = 35 g/kg.
K. BourounL M.T. Chaibi / Desalination 166 (2004) 17-24
3.2. I n f l u e n c e o f c o o l i n g w a t e r inlet t e m p e r a t u r e
m
i n . c l
In order to analyse the influence of the cooling liquid inlet temperature on the condensation, the variations in the total quantity o f condensed water vs. the T~,,clt are presented in Fig. 4. According to this figure, it can be seen that the amount of condensed water decreases significantly when T~,.cI increases. At 35°C, which is the mean cooling liquid temperature, the amount of condensed water cannot exceed t m3/d.
4.
g h h
C o n c l u s i o n s
An analytical study to predict the heat and mass transfer characteristics of the plastic horizontal-tube falling-film condenser was developed. The variations in fluid properties during their progression in the exchanger due to temperature evolution were considered. The elaborated model is based on the resolution of heat and mass transfer equations in each cell of the exchanger. This resolution was effected by using fluid properties calculated at the average temperature in the cell. The influence of inlet air velocity and cooling liquid inlet temperature on heat and mass transfer in the exchanger was investigated. Numerical results show an increase in condensing exchanger performance when the inlet cooling water decreases and air velocity increases. For the conditions corresponding to the real environment of southern Tunisia, the model predicted that the amount of condensed water cannot exceed 2 m3/d.
Lv ni M P Pr
Ro Rect
---
Acceleration of gravity, kg/m2s Heat transfer coefficient, w/m2°C - Average heat transfer coefficient, w/m2°C - Latent heat of condensation, kJ/kg - Mass flux, kg/s - Molecular weight, kg/(kg mole) - Pressure, Pa - - D i m e n s i o n l e s s Prandtl number = - -
--
Rein.g
--
T T U
-- -
--
X
- -
X
--
Z
- -
Z
- -
V
---
Thermal conductivity, W/m°C Density, kg/m3 - Kinematics viscosity, m2/s - - Dynamic viscosity, kg/m s
Subscripts a c
el ev
ext 5.
S y m b o l s
B
Cp
--
d Ov
f
--
Cell height, m Specific heat, J/Kg °C Tube diameter, m Diffusivity o f v a p o u r in air, mZ/s Function
Perfect gas constant, J/(mole °C) Dimensionless Reynolds number for cooling liquid = (di,, Ucl)/(Vc/) Dimensionless Reynolds number for air = (dext Ug)/(Vg) Local temperature, °C ' Average temp., °C = (r; ~ + Tout)~2 Velocity, m/s Evaporated water fraction Absolute humidity, kg/kg Coordinates Dimensionless coordinates = (z/B)
Greek
P
--
f
--
g
--
i !
---
in int
---
j
23
--
Dry air Condensed Cooling liquid Evaporated External Liquid film Gas Cell step Interface Inlet Internal Iteration step
24
K. BourounL M.T. Chaibi / Desalination 166 (2004) 17-24
out
--
Outlet
sat
--
Saturated
v
--
Vapour
w
--
Tube
vs
--
Saturant
wall vapour
References [ 1] K. Bourouni, R. Martin, L. Tadrist and H. Tadrist, Proc. Heat transfer and evaporation in cross flow plastic heat exchanger for water desalination. IDA Congress on Desalination and Water Reuse, Madrid, 127 (1997) 145. [2] K. Bourouni, M.T. Chaibi, R. Martin and L. Tadrist, Heat transfer and evaporation in geothermal desalination units, Appl. Energy, 64 (1999) 129-147. [3] M.W. Browne and P.K. Bansal, An overview of condensation heat transfer on horizontal tube bundles, Appl. Thermal Engn., 19 (1999) 565-594. [4] J.V. Lunardini and A. Aziz, Effect of condensation on performance and design of extended surfaces, Cerel Report 95-20, 1995.
[5] P.K. Sarma, B. Vijayalakshrni, F. Mayinger and S. Kakak, Turbulent film condensation on horizontal tube with external flow of pure vapors, Int. J. Heat and Mass Transfer, 41 (1988) 537-545. [6] H. Honda and S. Nozu, Effect of drainage strips on the condensation heat transfer performance of horizontal finned-tubes, in: Heat Transfer Science and Technology, Hemisphere, New York, 1987, pp. 455-462. • [7] A. Agunaoun and A. Daif, Evaporation en convection forc6e d'un film mince s'ecoulant en r6gime permanent, laminaire et sans onde sur une surface plane inclin6e, Int. J. Heat Mass Trans., 18 (1994) 2947-2956. [8] A.P. Colbum, A method of correlating forced convection heat transfer data and comparison with fluid friction, Trans. AIChE, 29 (1933) 174-179. [9] W.H. McAdams, Heat Transmission, McGraw-Hill, New York, 1954. [10] H.W. Press, P.F. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes, in: The Art of Scientitle Computing, Cambridge University Press, London, 1986.