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Basic hydrodynamic aspects of a solar energy based desalination process
Solar Energy, Vol. 54, No. 2, pp. 125-134, 1995 Copyright0 1995Ekvier Science Ltd Printed in the USA. All rightsreserved
Pergamon
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BASIC HYDRODYNAMIC ASPECTS OF A SOLAR ENERGY BASED DESALINATION PROCESS G. A.
BEMPORAD
ISMES Spa, V.le G. Cesare 29,241OO
Bergamo,
Italy
Abstract-The theoretical feasibility of a solar energy based desalination scheme is analyzed in this study. The proposed scheme exploits the vapor pressure difference between fluids of different salinities and temperatures to produce fresh water from seawater. The scheme’s basic components are a seawater column, an injection pipe heated on top through a heat exchanger loop, a withdrawal pipe, a vacuum chamber filled with vapour, and a fresh water column cooled on top where vapour condenses into fresh water. A mathematical model was developed to simulate unsteady mass, heat and solute transfer during the desalination process. The governing equations were integrated numerically in space and time through a finite difference technique. The numerical simulations considered both steady-state and time dependent heat sources. The numerical results proved the theoretical feasibility of the proposed desalination scheme. However,‘the presence of an unsteady heat source, typical to solar energy based schemes, may lead to an unstable density profile in the water column and reduce the scheme efficiency if not properly controlled.
1. INTRODUCIlON
cuit, which also maintained the two fluids at suitably different operating temperatures. An attractive feature of the RHPDS was its design versatility which made it especially suitable for exploitation of renewable energy sources, such as solar radiation and wind energy. A similar desalination scheme was analyzed by Bemporad and Reali ( 1992). In that study, a steady heat source was provided to the desalination process by operating a solar pond. The scheme analysis showed that the mechanical energy required to produce 1 kg of fresh water from seawater was ~3 Es, where Es N 2.5 kJ is the minimum theoretical energy required to produce 1 kg of fresh water from seawater ( Spiegler, 1977 ) . Among the most efficient industrial systems for the desalination of seawater are the revem osmosis (RO) method and the mechanical vapor compression (MVC) method. The minimum energetic requirements for both the RO method and the MVC method are -lOE,. In this study we are mainly concerned with the basic hydrodynamic aspects related to the desalination process described by Reali ( 1984) and Bemporad and Reali ( 1992). These studies showed that fresh water might be produced at reasonable costs by using solar energy. However, mechanisms related with the evaporation process in the proposed scheme may induce a certain recirculation in the seawater column, therefore reducing the efficiency of the desalination process. In this framework, a mathematical model was developed to simulate the time dependent heat and salt exchanges in the seawater column.
Fresh water is an essential commodity both for developed and developing countries. Many areas depend heavily on water produced by desalination plants for their water supply. Large desalination plants are usually coupled with electric power generating plants. In remote towns, villages, and islands, potable water is sup plied from either small, local desalination plants powered by trucked-in fuel, or is transported from the nearest supply source. The utilization of solar radiation as the energy source for desalination plants may offer a valid alternative for supplying potable water. Several configurations for solar desalination plants were proposed in past studies. The numerical analyses and practical results showed the feasibility of such systems for fresh water production. For instance, experimental results and computer simulations of a solar desalination plant located in Abu Dhabi were presented by El-Nashar and El Baghdadi ( 1987) and El-Nashar ( 1990, 1992). In that plant, a solar collector field was used to intercept the sun’s radiation and convert part of it into thermal energy for subsequent use as a heat source in the distiller. Assouad and Iavan ( 1988) presented a solar desalination scheme with latent heat recovery: The scheme consisted of a humidifier, a solar still, a condenser, and a pond. Aly ( 1990) analyzed a central fuel-fired, solar-assisted dual purpose plant. In that scheme, solar energy collected at moderate temperatures was used to supplement the fuel fired in the conventional plant. A refrigerator heat pump desalination scheme (RHPDS) was proposed by Reali ( 1984). That scheme exploited the difference in vapor pressure between fluids of different salinities and temperatures. By maintaining fresh water and seawater at suitably different temperatures, seawater could evaporate into a vacuum chamber and condense in a fresh water chamber. Efficient recovery of the latent heat of condensation was performed through a refrigerator-heat-pump cir-
2. BACKGROUND In the temperature range 273-373°K the vapor pressure of seawater is N 1.84% less than that of fresh water (Spiegler, 1977). Therefore, if two vessels containing deaerated fresh water and seawater are kept at the same temperature (T), 273°K < T-c 373”K, and have their tops connected, then fresh water will distill 125
126
G. A. BEMPORAD
into the seawater vessel. Otherwise, distillation of seawater may be obtained if the distilled fresh water temperature is kept lower than the seawater temperature by a suitable amount. In the proposed desalination scheme, heat is injected into the seawater chamber in order to increase its temperature with respect to the temperature of the fluid in the fresh water chamber, as shown in Fig. 1. The seawater chamber is basically a thermohaline system subject to transfer of thermal energy, mechanical energy, salt, and water. Mathematical models for the analysis of the density stratification in such thermohaline systems were the subject of past research work (e.g., Salhotra et al., 1987; Keren et al., 1993). In these studies it was shown how the heat balance affects the water surface temperature, which affects evaporation and, hence, the water budget. The water budget in turn affects the concentration of the water column and, hence, influences the salt budget. Moreover, in a typical thermohaline system, an increase in evaporation tends to decrease the surface temperature, while an increase in surface salinity due to evaporation decreases evaporation. Mixing processes may also be present, according to the mechanical energy of the injection and withdrawal processes. The stratification patterns along the water column of temperature and salinity affect the column’s vertical stability. For instance, an increase (decrease) in the salinity of the surface layer due to net evaporation (net inflows) adds negative (positive) buoyancy to the water column, thus reducing (increasing) its stability. Evaporation from the seawater chamber to the vacuum chamber is driven by the difference between the vapor pressure of the seawater chamber and the vapor pressure of the fresh water chamber. Nearly 100 equations for calculating the rate of evaporation from both natural and thermally-loaded water bodies can be found in the literature (Sill, 1983 ) . The most common approach is to consider the evaporation to be proportional to the difference between the saturated water
vapor density at the surface temperature and the ambient water vapor density, measured at a given height. This difference must be multiplied by a function of the wind speed at a given height above the water surface. The form of this function is commonly obtained as an empirical fit to a set of site-specific field measurements. In the seawater chamber, however, evaporation occurs in the absence of wind and under vacuum conditions. Convection-like motion, which forms in completely still air, cannot be present in such conditions. The evaporative flux from a condensed phase is the difference between the flux of molecules leaving the surface and the flux of vapour molecules which strike it and condense (e.g., Kennard, 1938; Algie, 19781. Fujikawa and Akamatsu ( 1980) proposed a relationship to determine the rate of evaporation and condensation of the vapour based on the temperatures of the vapour and the liquid at the phase interface, as well as on the actual and equilibrium vapour pressures. In an analogous manner, we can assume that the evap oration per unit area qehas the following form
f(‘s)
P( Ts) (TV
+
273)l/Z
where (e.g., Jobson, 1973) p(T)
=
e(63.042-7139.6/(
T+273)-6.2558ln(
T-+273)).
L
Sea-Water Chamber
rater Chamber
e--r
Fkesh Water
I
l4 I
pa
is the fresh water vapor pressure at temperature T, T, is the temperature at the surface of the sea water column, T/is the temperature in the fresh water chamber, pf is the fresh water density, cr, = 10m7 - 10m6is an experimental coefficient (e.g., Fujikawa and Akamatsu, 1980)) f(C) is a correction factor to account for the presence of a solute concentration C. For small con-
Cooling
Fresh-
102
-
Fig. 1. Schematic of the desalination process.
GL
Sea Water
127
Basic hydrodynamic aspects centrations C the correction factor can be expressed as (Keren et al., 1993) f(C)
= 1. - cY,c;
cr, = 0.0054.
(2)
The variation of density p with respect to temperature and concentration can be expressed by the following equation of state P(T, C) = P,(
1 - PTATO
C,(C) = azc + p2
(4)
where (Ye= -30.10 J kg-’ “C-I and rB2= 4178.4 J kg-’ “C-l. The heat flux 4, associated to the evaporation qe is (5)
9, = p/M T&e where (e.g., Jobson, 1973) = 10’.[3146
- 2.36(T+
273OK)]i
is the latent heat of vaporization at temperature T. The vacuum chamber is not subject to atmospheric effects, and in the vicinity of the surface layer the vapour temperature is approximately equal to the seawater temperature. In such conditions, back radiation
Application of the heat flux balance to the interface between the seawater and the water vapour determines the evaporation rate in the desalination process. 3. THE MATHEMATICAL
MODEL
The mass, heat, and salt exchanges which characterize the proposed desalination process were analyzed by developing an appropriate mathematical model. For this purpose, we consider the seawater chamber to be formed, as shown in Figure 2, by: A salt water column, subject to a non-uniform vertical distribution of salinity and temperature. This water body is assumed to be stagnant and homogeneous in the horizontal plane. Therefore, in each elementary fluid volume of area A, and depth dz, molecular heat and salt diffusion in the vertical direction are the dominant transport mechanisms. A schematic description of the salt water column identifies: a surface layer, where water evaporates to the vacuum chamber and which is fully mixed due to the mechanical energy introduced by the fluid injection and withdrawal, a middle layer, which is not subject to fluid injection and withdrawal, and a bottom layer, which is subject to waste brine removal through a bottom wash flow. An injection anular pipe, coaxial and external to the salt water column, which carries the seawater to the surface layer. The thermal power needed for the evaporation process is delivered at the top of the injection pipe through a heat exchanger. A withdrawal pipe, coaxial and internal to the salt water column, which withdraws water from the sur-
ht y = hc
Y =
1 Water
(6)
&I = -4,.
(3)
+ PcAG)
where AT, and AC, are variations from reference property values, of density pO. Solute concentrations are expressed as a percentage of mass, & = 5 - 10e4 OC-’ is the thermal expansion coefficient, PC = 8 - 10-3%-’ is the solutal expansion coefficient. Solute concentration and temperature affects also the value of the specific heat C,; however, temperature may be neglected (e.g., Keren et al., 1993) to obtain
l,(T)
flux results negligible, and the net surface heat flux reduces to
column
: d,
Injection pipe Withdrawal -
.Bottom
.
,-‘I+-
pipe layer
Qt -?-’ I :
-5 \I
*
+-dd:W
Fig. 2. Schematic of the seawater chamber.
F
.Qb
Y =
hb
y=h
0
T
128
G.
A. BEMPORAD
face layer. Salt concentration at the surface of the water column is thereby reduced and the water body stability can be maintained. In both coaxial pipes, it is assumed that convection is the dominant mechanism for heat and salt transport. The injection and withdrawal pipes exchange thermal energy with the salt water column through their surfaces; salt is not exchanged between these pipes and the middle layer of the water column. Application of the continuity equation in the salt water column leads to the following expressions for flow rates at the surface and bottom layers, respectively,
the bottom level of the middle layer; h, is the bottom level of the water column; (J refers to the water column, (,,,) refers to the withdrawal pipe, (i) refers to the injection pipe, (s) refers to the surface layer, (b) refers to the bottom layer. With regard to salinity transfer in the water column, we obtain the following expressions for the surface, middle and bottom layers, respectively
$WMh
- WA, + -4v)
a ay
= (&‘h,sQi - (K)c,sQw - KC Qi=qe$+Qw Qb = cost
(~C)c,y=h,Ac
(7)
where Qi is the injection pipe flow rate, which provides the inflow to the surface layer of the salt water column, d, is the salt water column diameter, qe is the evaporation rate, determined by eqn ( 1) , Qwis the discharge in the withdrawal pipe and Qb is the bottom layer washllow. In the withdrawal pipe we have Qw = cost, while Qi is determined according to the evaporation rate. Application of the conservation of energy leads to the following heat balance equations for the surface, middle, and bottom layers of the salt water column, respectively
; (PC)c,b(hb -
h&L
= h’C)bQb- (P’&d?b +
KC
$ (PC)c,j=/&
(9)
where ~~ is salt diffusivity. Application of the heat conservation equation in the injection pipe leads to the following expressions for the surface and the middle parts, respectively:
= Qi((dJpT)i,y=h, - (pCpT)i+) + W,
$(PCpThAi = -Qi
g (KJ)i
- y(Ti
- Tc)rdc - y’( Ti - Ta)*di
(10)
where Ai = $ (d; - d,2)
A, =
3 (df
- d;);
A,,, = ; d2,
are the salt water column and the withdrawal pipe cross sectional areas, respectively; d, is the withdrawal pipe diameter; I is the time; KT is the heat diffisivity; y = 650 - 1200 W m-2 “C-’ is the forced convection coefficient; hl is the bottom level of the surface layer; hb is
is the injection pipe cross sectional area; di is the injection pipe external diameter; T, is the ambient temperature; y’ is the natural convection coefficient; W, is the thermal power provided to the desalination scheme by an external source. Radiation losses to the ambient should also be considered. However, appropriate insulation materials should make this additional term negligible with respect to the other terms of the heat balance equations. The equations of solute conservation, for the surface and the middle parts of the injection pipe, are respectively
129
Basic hydrodynamic aspects Analogously, the heat conservation equations for the withdrawal pipe middle and bottom sections, respectively, are ; (PCJ-LA
= ;
(~c,T)wQw - -AT, - Tc)?rdw
; (PCpThv,tv‘k(h - ho) = ((~C,T)w,y=/t, - (pc,T)w,b)Qw - ?‘(Tw,b - Tc,bbdw
(12)
The equations of solute conservation, for the surface and the middle parts of the withdrawal pipe, are ; (~ChvAw = $ (~ClwQw
$
(Pchv,ddhb
-
ho)
=
((PC)w,y=hb
-
(P%b)Qw
(13)
Equations (7- 13 ) are solved numerically for the unknown temperature T and concentration C along the salt water column, the injection pipe, and the withdrawal pipe, respectively. The density vertical distribution results from the temperature and the solute concentration vertical distributions. The governing equations of the present mathematical model are discretized in time and space assuming that in the time interval r = (t(“), t(“) + At = t(“+‘)) the time derivative is discretized by a forward Euler formula, the convective terms are discretized with an up-wind finite difference form, and the diffusive terms are discretized with a central difference form, as follows:
aPr)
A-
=
f
(n+l)
at -aft')
ay
z.z
-
f
(n)
At
(A -h-l P
or
AY
-af(r) = (.h+,-h)‘“’
ay
;
(Kg
CT)
1
=
&
AY
((K,,
+
4. NUMERICAL RESULTS
The mathematical model described in the previous sections was employed to determine the performance of the proposed desalination scheme. Both steady-state and unsteady conditions were analyzed. During all simulations, the temperature of the fresh water chamber Tf was kept at 1O’C. The diameter of the water column was d, = 0.25 m, the external diameter was di = 0.30 m and the diameter of the withdrawal pipe was d, = 0.05 m. The thermal and the solute diffisivities were assumed to be equal to KT = 1.5 - lo-’ m2/s and KC = 1.5 - 5 - 10m9m*/s. The forced convection coefficient was assumed to be y = 1000 W/m */ “C. Assuming the injection pipe to be sufficiently insulated against heat losses to the ambient, we have y’ = 0. The bottom and the surface layer of the salt water column both had a thickness equal to 0.2 m. A first set of numerical simulations was performed to simulate heat and salt transfer in the seawater column under steady-state conditions. The thermal power needed to drive the desalination process was gradually increased to reach a value of 50 W. The initial temperature and the salinity concentration for the water column were 20°C and 4%, respectively. Figure 3 shows the evaporation rate as a function of time. Approximately 15-20 days were needed in order to reach steady-state conditions. Figure 4a-b shows the temperature and the density profiles along the water column, respectively, until steady-state conditions are reached. These simulations were performed with bottom and withdrawal discharges equal to Qb = 1.O- 10m6 m3/s and Qw = 1.0. 10m6m3/s, respectively. Figure 5 shows the evaporation rate at steady state conditions as a function of the withdrawal rate Q,,,. This figure shows that minor variations in the evaporation rate are caused by a certain variation in the withdrawal rate. Smaller withdrawal rates mean smaller injection rates and, therefore, higher temperatures for the surface layer as well as in the withdrawal pipe. Although the large convection coefficient y ensures that most of this excess heat is recovered by the water column, slightly better performances are expected for smaller withdrawal rates. Unsteady conditions were simulated by performing numerical simulations with a time-dependent heat source. This heat source was assumed to be provided
K,wj+1 -Xl
- (K, + &)(A
-&))(n)
(14c)
where f ( y, t) is the generic function of time and space and Kthe generalized diffusion coefficient. In eqn ( 14b) the choice over the derivation operator is performed according to the dominant direction of the current in the injection and the withdrawal pipes. Introduction of expressions ( 14) into the governing equations for the salt water column, the injection pipe, and the withdrawal pipe leads to the numerical scheme shown in Appendix A.
0
200
400
600
800
lr,OO
ttme(hours)
Fig. 3. Evaporation rate in the transient period and at steadystate.
G.
130
40
50
temperature
(C)
A.
BEMPORAD
salt water column, respectively. The inversion in the density profile near the surface shown in this figure may lead to a certain mixing in the surface. Figure 8ab shows the temperature and the salinity profiles in the withdrawal pipe. Due to the large heat exchange between the water column and the withdrawal pipe, the temperatures in the withdrawal pipe rapidly approach the temperatures of the water column. The salinity concentration, instead, is significantly different since no salt transfer is allowed between the two water bodies.
(a) 5. DISCUSSION
a
E g * 5 m
6 4 2 0 1000
1005
1010 density
1015
1020
1025
(kg/m3)
(b) Fig. 4. Seawater column in the transient period and at steadystate; (a) temperature profiles, and (b) density profiles.
by a set of solar panels. The available radiant solar energy on a solar panel (in W/m *) was determined at ENEL-DSR-CRIS-Milan0 (Reali, 1993) for a typical sunny day of May in Milan. Previous studies (e.g., Duff, 1988 ) recorded instantaneous collection efficiencies up to 60% for temperature differences up to almost 80°C during most daylight hours. Monthly collection efficiencies above 50% were attained in space heating applications in Northern Italy and Germany. In this study the efficiency of the solar panel was assumed as 50% for a panel surface of 0.25 m2. The available radiant energy used to drive the desalination process is shown in Fig. 6, together with the resulting evaporation rate. In this figure, t = 0 refers to 5 A.M. Figure 7a-b shows the development during the day of the temperature and density profiles in the
wthdrawal
rate
(m3is)
Fig. 5. Evaporation rate at steady state versus the withdrawal rate for various heat sources.
The numerical simulations presented in the previous section showed that most of the heat derived from the heat source could be maintained in the upper parts of the water column and employed for the evaporation process. At steady-state, the thermal power supplied to the seawater chamber was equal to 50 W, while the evaporation rate was 2 - lo-* m3 s-l. The thermal power needed to raise the temperature of that water from 20°C to 67°C was about 4 W, and the thermal power needed to evaporate it at 67°C was about 45 W (eqn [ 5 I). Thermal losses of about 1 W are caused by the bottom layer wash-flow and the withdrawal pipe salt removal. No other losses are present, since the seawater chamber was assumed to be completely insulated. Therefore, about 98% of the power supplied to the seawater chamber was employed to evaporate the fresh water. Under unsteady conditions, the total energy supplied to the seawater chamber during the typical sunny day was equal to 2.8 MJ. The total amount of fresh water evaporated during that day was equal to 1.08 . 10T3 m3. To raise this amount of seawater to the temperature of the surface layer and to provide the latent heat of evaporation about 2.7 MJ had to be supplied. According to these simulations, about 96% of the energy supplied during the day was employed to make the water evaporate. However, this last energetic balance is valid provided that the mixing caused by the density inversion is limited to the upper zone of the seawater column. The stability of the seawater column is of primary importance for the desalination process. Seawater is heated and injected on top of the seawater column through the injection pipe. From the surface layer some
time (s)
Fig. 6. Evaporation rate and radiant energy as a function of time during a sunny day.
131
Basic hydrodynamic aspects
---
51840 s trne 34560 s
.--
time 17280 s
---- me
40
50
temperature
(C)
60
70
(a)
density (kgim3) 08
Fig. 7. Transient conditions in the water column; (a) temperature profiles, and (b) density profiles.
water evaporates, therefore increasing the surface layer salinity. If not properly controlled, the surface layer becomes heavier than the lower layers, therefore causing a vertical recirculation in the water column. For that reason, a certain quantity of salty water must be withdrawn from the internal pipe as a wash flow. If the stable configuration cannot be maintained, the efficiency of the desalination scheme may be significantly reduced, since the water mixing will cause the warmer water to reach the lowest parts of the water column. Instead, higher efficiencies are obtained when most of the available heat remain in the surface to drive the evaporation process. The performance of the desalination process is influenced by several factors, such as withdrawal rate, thermal power input, temperature of the fresh water chamber, and others. For instance, the numerical simulations performed in this study have shown that higher efficiencies may be obtained by recirculating water through the withdrawal pipe at very low rates. Moreover, an unsteady heat source is most likely to cause a surface density inversion. In fact, heat is more rapidly lost from the surface layer of the water column than salt, and, therefore, the density of such a layer slowly increases until it reaches an unstable condition. A possible way to avoid density inversion is to use steady sources of thermal power. As shown by the numerical simulations, the direct application of a solar panel to the seawater chamber does not seem to offer the conditions required for maximum efficiency unless mechanisms to store solar energy on a daily or better weekly basis are introduced. The solar pond, for instance, could offer such benefits, since its built-in storage ca-
pability make it the natural choice for such applications. In these conditions, the proposed desalination scheme may offer a simple alternative to the existing desalination schemes which exploit solar energy to drive the desalination process. The expression for the evaporation rate was determined on the basis of basic theories on evaporation. However, in the desalination scheme under investigation the evaporation rate is also a function of the geometry of the vacuum chamber as well as of the quantity of gases dissolved in the seawater [e.g., Kennard, 19381. These factors, not considered in the present analysis, may significantly affect the evaporation rate. The order of magnitude of the experimental evaporation constant (Y,,,used in this study was derived from previous investigations as well as from preliminary experiments performed at ENELDSR-CRIS (Reali, 1993). Since the evaporation process is influenced by several factors, many of which were not considered in the present study, extensive experimental work should be performed in order to determine the value of this constant under the various conditions. However, through the present theoretical study it was possible to determine the major issues which characterize the proposed desalination scheme, namely the possibility of density inversion in the saltwater column and the difficulties in operating the scheme under an unsteady heat source. 6. CONCLUSIONS
A mathematical model was developed to simulate the complex hydrodynamic phenomena occurring in
10
/““i
8
0
40
20
60
temperature
il YO
(C)
(a)
1
A’ ,,’
2 ,
0 38
/(/
I],. 39
I
/,
111, f
40
,*’
/’ 4’
42
(W Fig. 8. Transient conditions in the withdrawal pipe; (a) temperature profiles, and (b) salinity profiles.
132
GA.
BEMPORAD
a water column during a desalination process. The basic equations which govern heat and mass transfer in the desalination process were integrated through a finite difference numerical scheme. The numerical results theoretically proved the feasibility of the proposed desalination process. Maintenance of a stable water column made possible the utilization of most of the heat absorbed by the solar panel for the evaporation process. These preliminary results proved the feasibility of a continuous production of fresh water through the proposed desalination scheme. Acknowledgments-This study was funded by the Hydraulic and Structural Research Center of ENEL Spa. The author is grateful to Dr. M. Reah of ENEL Spa for his comments and guidance during the preparation of this manuscript.
NOMENCLATURE
AC water column cross sectional area ( m2) 4 injection pipe cross sectional area ( m2) A, withdrawal pipe cross sectional area (m*)
c
solute concentration (%) bottom layer solute concentration (% ) CC water column solute concentration (96) injection pipe solute concentration (% ) Ci G surface layer solute concentration (% ) CIV withdrawal layer solute concentration (5%) CP specific heat (Jkg-’ “C-i] dc diameter of the salt water column (m) 4 diameter of the injection pipe (m) dw diameter of the withdrawal pipe (m) -5 fresh water/seawater separation energy (J) gravitational acceleration ( mse2) f elevation of the bottom of the water column (m) ht elevation of the bottom of the surface layer (m) h elevation of the bottom of the middle layer (m) h, elevation of the water column (m) 1. latent heat of vaporization (Jkg-‘) P vapour pressure (Pa) injection pipe discharge ( m3 s-‘) 2 flow in the bottom layer of the’water column ( m3 s-i) QW withdrawal pipe discharge ( m3 s-’) qe evaporation discharge (ms-’ ) T temperature (“C) Tb bottom layer temperature (“C) TC water column temperature (“C) Ti injection pipe temperature (“C) T, surface layer temperature (“C) TIV withdrawal layer temperature (“C) W thermal power ( W) Y vertical coordinate (m) fft empirical coefficient in eqn (2) (dimensionless) empirical coefficient in eqn (4) (J kg-’ “C-l) a2 empirical coefficient in eqn (4) (J kg-’ “C-‘) & emoirical coefficient (ka m-*Pa-‘s-’ OK”‘) am sol&al expansion coeffi>ent (Q -’) ’ Bc thermal expansion coefficient (“C-‘) t% forced convection coefficient (W m-* “C-l) Y Y’ natural convection coefficient ( W mm2“C-l) cb
xc or Y p &
solute molecular diffusivity (m2s-‘) heat molecular diffusivity (m*s-I) kinematic viscosity (m*s-‘) density (kg m-“) evaporation thermal energy flux ( W m-*) 4. net thermal energy flux (W m-*) Subscripts and superscripts 6 of the bottom layer c of the water column f of the fresh water i of the injection pipe 0 of a reference state s of the surface layer w of the withdrawal pipe
REFERENCES
S. H. Algie, Kinetic theories of evaporation, J. Chem. Phys. 69(2), (1978). S. E. Aly, Analysis of a fuel-solar assisted central dual purpose plant, Desalination 78, 363-379 ( 1990). Y. Assouad and Z. Lavan, Solar desalination with latent heat recovery, ASME J. Solar Energy Engrg. 110, 14-16 (1988). G. A. Bemporad and M. Reali, A solar pond desalination scheme for fresh water and salt production, 2nd International Congress on Energy, Environment and Technological Innovation (October 1992), Rome, Italy, Universim La Sapienza. W. S. Duff, Experimental results from twelve evacuated collector installations, Proceedings of the I988 ASES Annual Meeting (June 1988), Cambridge, MA, ISES. A. M. El-Nashar and A. M. El Baghdadi, Seawater distillation by solar energy, Desalination 61,49-66 ( 1987). A. M. El-Nashar, Computer simulation of the performance of a solar desalination plant, Solar Energy 44, 193-205 (1990). A. M. El-Nashar, Optimizing the operating parameters of a solar desalination plant, Solar Energy 48,207-2 13 ( 1992). S. Fujikawa and T. Akamatsu, Effects of the non-equilibrium condensation of vapour on the pressure wave produced by the collapse of a bubble in a liquid, J. of Fluid Mech. 97(3), 481-512 (1980). E. H. Kennard, Kinetic theories of gases, McGraw-Hill, New York (1938). Y. Keren, H. Rubin, J. Atkinson, M. Priven, and G. A. Bemporad, Theoretical and experimental comparison of conventional and advanced solar pond performance, Solar Energy 51(4), 255-270 (1993). H. H. Jobson, The dissipation of excess heat from water systems, ASCE-J. Power Engrg. 99(l), ( 1973). M. Reali, A refrigerator-heat-pump desalination scheme for fresh water and salt recovery, Energy 9,583-588 ( 1984). A. M. Salhotra, E. E. Adams, and D. R. F. Harleman, Vertical mixing in thermohaline system, 3rd Internntional Symposium on StratiJied Flows, Pasadena, CA ( 1987). B. L. Sill, Free and forced convection effects on evaporation, ASCE, J. OfHydrauIic Engineering 109(9), 1216-1229 (1983). K. S. Spiegler, Saltwater Purification, 2nd Ed, PZenum Press, NY(l977).
APPENDIX: THE NUMERICAL MODEL
Introducing the expressions in eqn ( 14) into the equations of heat balance (8) we obtain, for the surface, middle, and bottom layers of the seawater column, respectively
133
Basic hydrodynamic aspects
where E = pC,,T is a compounded variable. Analogously, from eqn (9) we obtain for the surface, middle, and bottom layers, respectively, the following discretired equations of salt balance
where n = pC is another compounded variable. By solving eqns (A2) and (Al) it is possible to determine ,$s”, n$“, for j = 1,2, . . . , N - 1, N, where j = 1 refers to the bottom layer, j = N to the surface layer, and (,) refers to the seawater column. Discretiration in time and space of the heat balance equations for the injection pipe eqn ( 10) leads to, for the surface and the middle parts, respectively:
g
- [i$
gy’ - gj A,
At
=
’
(Eij -
_Q!“’
’
&i-d’” -
AY
h,)A, = Qj”)(&_,,r_, - &N) + WI(“)
(h, -
At
y( Ti,
-
TcJ)(“)adc - y’( TiJ -
T,J’“‘rd,
- eu(( T,,
+
273)4 -
(T, +
273)“)%d,
(A3)
Analogously, discretization in time and space of the salt balance equations for the injection pipe, eqn ( 1 1 ), leads to, for the surface and the middle parts, respectively:
qy;” - $J, At $+I) -
At
(h - hMi = Q;“‘(m,,-,
““’
Ai
-Qi
=
(II)
(t)ij -
- KV)
mJ-1
P’
(A4b)
AY
Discretization of the heat conservation equations ( 12) for the withdrawal pipe middle and bottom sections, respectively, leads to p;”
- p/J .
At
A
=
(‘%+I- hd(“’_
Q
w w
y(
E!P - GyI At
T,,,
Tcj)Gend
_
Iv
AY
(hb - ~o)Aw = Qw(S,2 - .!&I) (“I - y( Tw,, -
Tc,,)‘“‘sd,(hb
- h,)
(As)
Analogously, discretization of the salt conservation equations ( 13) for the withdrawal pipe middle and bottom sections, respectively, leads to @+I’ -
VW.1 At
p;
,
A
=
*’
Q
('lw,j+l-
v
%,j)(")
AY
(A6) The four physical variables, Ti(“+‘), $+I), py’), (CJj”+‘) for the water column, the injection pipe and the withdrawal pipe, respectively, are obtained by solving for each time step, at each computational node, the additional system of four coupled equations
G.A.BEMPORAD
134
[pj(c&Jjq(n+‘) (“+I)
I&C,1 [(qj]‘“+”
p:n+‘ =) /I,( 1 -
= =
=
$+‘)
$+l)
&+‘)
&(Tj (“+I)- 7-J + &(C:n+‘) - CO))
(‘47)
Initial conditions on q, Cj, p,, (C,)j for j = 1, 2, . . , N - 1, N, for the seawater column, the injection pipe and the withdrawal pipe, complete the numerical formulation. Given the different mathematical characteristics of the advection and of the diffusion equations, respectively, the solution of the numerical problems is obtained by adopting a sub-cycling process, where for the convective terms a time step n times smaller is utilized.
Pergamon
0038-092X/95$9.50 + 00
0038-092X( 94)00110-3
BASIC HYDRODYNAMIC ASPECTS OF A SOLAR ENERGY BASED DESALINATION PROCESS G. A.
BEMPORAD
ISMES Spa, V.le G. Cesare 29,241OO
Bergamo,
Italy
Abstract-The theoretical feasibility of a solar energy based desalination scheme is analyzed in this study. The proposed scheme exploits the vapor pressure difference between fluids of different salinities and temperatures to produce fresh water from seawater. The scheme’s basic components are a seawater column, an injection pipe heated on top through a heat exchanger loop, a withdrawal pipe, a vacuum chamber filled with vapour, and a fresh water column cooled on top where vapour condenses into fresh water. A mathematical model was developed to simulate unsteady mass, heat and solute transfer during the desalination process. The governing equations were integrated numerically in space and time through a finite difference technique. The numerical simulations considered both steady-state and time dependent heat sources. The numerical results proved the theoretical feasibility of the proposed desalination scheme. However,‘the presence of an unsteady heat source, typical to solar energy based schemes, may lead to an unstable density profile in the water column and reduce the scheme efficiency if not properly controlled.
1. INTRODUCIlON
cuit, which also maintained the two fluids at suitably different operating temperatures. An attractive feature of the RHPDS was its design versatility which made it especially suitable for exploitation of renewable energy sources, such as solar radiation and wind energy. A similar desalination scheme was analyzed by Bemporad and Reali ( 1992). In that study, a steady heat source was provided to the desalination process by operating a solar pond. The scheme analysis showed that the mechanical energy required to produce 1 kg of fresh water from seawater was ~3 Es, where Es N 2.5 kJ is the minimum theoretical energy required to produce 1 kg of fresh water from seawater ( Spiegler, 1977 ) . Among the most efficient industrial systems for the desalination of seawater are the revem osmosis (RO) method and the mechanical vapor compression (MVC) method. The minimum energetic requirements for both the RO method and the MVC method are -lOE,. In this study we are mainly concerned with the basic hydrodynamic aspects related to the desalination process described by Reali ( 1984) and Bemporad and Reali ( 1992). These studies showed that fresh water might be produced at reasonable costs by using solar energy. However, mechanisms related with the evaporation process in the proposed scheme may induce a certain recirculation in the seawater column, therefore reducing the efficiency of the desalination process. In this framework, a mathematical model was developed to simulate the time dependent heat and salt exchanges in the seawater column.
Fresh water is an essential commodity both for developed and developing countries. Many areas depend heavily on water produced by desalination plants for their water supply. Large desalination plants are usually coupled with electric power generating plants. In remote towns, villages, and islands, potable water is sup plied from either small, local desalination plants powered by trucked-in fuel, or is transported from the nearest supply source. The utilization of solar radiation as the energy source for desalination plants may offer a valid alternative for supplying potable water. Several configurations for solar desalination plants were proposed in past studies. The numerical analyses and practical results showed the feasibility of such systems for fresh water production. For instance, experimental results and computer simulations of a solar desalination plant located in Abu Dhabi were presented by El-Nashar and El Baghdadi ( 1987) and El-Nashar ( 1990, 1992). In that plant, a solar collector field was used to intercept the sun’s radiation and convert part of it into thermal energy for subsequent use as a heat source in the distiller. Assouad and Iavan ( 1988) presented a solar desalination scheme with latent heat recovery: The scheme consisted of a humidifier, a solar still, a condenser, and a pond. Aly ( 1990) analyzed a central fuel-fired, solar-assisted dual purpose plant. In that scheme, solar energy collected at moderate temperatures was used to supplement the fuel fired in the conventional plant. A refrigerator heat pump desalination scheme (RHPDS) was proposed by Reali ( 1984). That scheme exploited the difference in vapor pressure between fluids of different salinities and temperatures. By maintaining fresh water and seawater at suitably different temperatures, seawater could evaporate into a vacuum chamber and condense in a fresh water chamber. Efficient recovery of the latent heat of condensation was performed through a refrigerator-heat-pump cir-
2. BACKGROUND In the temperature range 273-373°K the vapor pressure of seawater is N 1.84% less than that of fresh water (Spiegler, 1977). Therefore, if two vessels containing deaerated fresh water and seawater are kept at the same temperature (T), 273°K < T-c 373”K, and have their tops connected, then fresh water will distill 125
126
G. A. BEMPORAD
into the seawater vessel. Otherwise, distillation of seawater may be obtained if the distilled fresh water temperature is kept lower than the seawater temperature by a suitable amount. In the proposed desalination scheme, heat is injected into the seawater chamber in order to increase its temperature with respect to the temperature of the fluid in the fresh water chamber, as shown in Fig. 1. The seawater chamber is basically a thermohaline system subject to transfer of thermal energy, mechanical energy, salt, and water. Mathematical models for the analysis of the density stratification in such thermohaline systems were the subject of past research work (e.g., Salhotra et al., 1987; Keren et al., 1993). In these studies it was shown how the heat balance affects the water surface temperature, which affects evaporation and, hence, the water budget. The water budget in turn affects the concentration of the water column and, hence, influences the salt budget. Moreover, in a typical thermohaline system, an increase in evaporation tends to decrease the surface temperature, while an increase in surface salinity due to evaporation decreases evaporation. Mixing processes may also be present, according to the mechanical energy of the injection and withdrawal processes. The stratification patterns along the water column of temperature and salinity affect the column’s vertical stability. For instance, an increase (decrease) in the salinity of the surface layer due to net evaporation (net inflows) adds negative (positive) buoyancy to the water column, thus reducing (increasing) its stability. Evaporation from the seawater chamber to the vacuum chamber is driven by the difference between the vapor pressure of the seawater chamber and the vapor pressure of the fresh water chamber. Nearly 100 equations for calculating the rate of evaporation from both natural and thermally-loaded water bodies can be found in the literature (Sill, 1983 ) . The most common approach is to consider the evaporation to be proportional to the difference between the saturated water
vapor density at the surface temperature and the ambient water vapor density, measured at a given height. This difference must be multiplied by a function of the wind speed at a given height above the water surface. The form of this function is commonly obtained as an empirical fit to a set of site-specific field measurements. In the seawater chamber, however, evaporation occurs in the absence of wind and under vacuum conditions. Convection-like motion, which forms in completely still air, cannot be present in such conditions. The evaporative flux from a condensed phase is the difference between the flux of molecules leaving the surface and the flux of vapour molecules which strike it and condense (e.g., Kennard, 1938; Algie, 19781. Fujikawa and Akamatsu ( 1980) proposed a relationship to determine the rate of evaporation and condensation of the vapour based on the temperatures of the vapour and the liquid at the phase interface, as well as on the actual and equilibrium vapour pressures. In an analogous manner, we can assume that the evap oration per unit area qehas the following form
f(‘s)
P( Ts) (TV
+
273)l/Z
where (e.g., Jobson, 1973) p(T)
=
e(63.042-7139.6/(
T+273)-6.2558ln(
T-+273)).
L
Sea-Water Chamber
rater Chamber
e--r
Fkesh Water
I
l4 I
pa
is the fresh water vapor pressure at temperature T, T, is the temperature at the surface of the sea water column, T/is the temperature in the fresh water chamber, pf is the fresh water density, cr, = 10m7 - 10m6is an experimental coefficient (e.g., Fujikawa and Akamatsu, 1980)) f(C) is a correction factor to account for the presence of a solute concentration C. For small con-
Cooling
Fresh-
102
-
Fig. 1. Schematic of the desalination process.
GL
Sea Water
127
Basic hydrodynamic aspects centrations C the correction factor can be expressed as (Keren et al., 1993) f(C)
= 1. - cY,c;
cr, = 0.0054.
(2)
The variation of density p with respect to temperature and concentration can be expressed by the following equation of state P(T, C) = P,(
1 - PTATO
C,(C) = azc + p2
(4)
where (Ye= -30.10 J kg-’ “C-I and rB2= 4178.4 J kg-’ “C-l. The heat flux 4, associated to the evaporation qe is (5)
9, = p/M T&e where (e.g., Jobson, 1973) = 10’.[3146
- 2.36(T+
273OK)]i
is the latent heat of vaporization at temperature T. The vacuum chamber is not subject to atmospheric effects, and in the vicinity of the surface layer the vapour temperature is approximately equal to the seawater temperature. In such conditions, back radiation
Application of the heat flux balance to the interface between the seawater and the water vapour determines the evaporation rate in the desalination process. 3. THE MATHEMATICAL
MODEL
The mass, heat, and salt exchanges which characterize the proposed desalination process were analyzed by developing an appropriate mathematical model. For this purpose, we consider the seawater chamber to be formed, as shown in Figure 2, by: A salt water column, subject to a non-uniform vertical distribution of salinity and temperature. This water body is assumed to be stagnant and homogeneous in the horizontal plane. Therefore, in each elementary fluid volume of area A, and depth dz, molecular heat and salt diffusion in the vertical direction are the dominant transport mechanisms. A schematic description of the salt water column identifies: a surface layer, where water evaporates to the vacuum chamber and which is fully mixed due to the mechanical energy introduced by the fluid injection and withdrawal, a middle layer, which is not subject to fluid injection and withdrawal, and a bottom layer, which is subject to waste brine removal through a bottom wash flow. An injection anular pipe, coaxial and external to the salt water column, which carries the seawater to the surface layer. The thermal power needed for the evaporation process is delivered at the top of the injection pipe through a heat exchanger. A withdrawal pipe, coaxial and internal to the salt water column, which withdraws water from the sur-
ht y = hc
Y =
1 Water
(6)
&I = -4,.
(3)
+ PcAG)
where AT, and AC, are variations from reference property values, of density pO. Solute concentrations are expressed as a percentage of mass, & = 5 - 10e4 OC-’ is the thermal expansion coefficient, PC = 8 - 10-3%-’ is the solutal expansion coefficient. Solute concentration and temperature affects also the value of the specific heat C,; however, temperature may be neglected (e.g., Keren et al., 1993) to obtain
l,(T)
flux results negligible, and the net surface heat flux reduces to
column
: d,
Injection pipe Withdrawal -
.Bottom
.
,-‘I+-
pipe layer
Qt -?-’ I :
-5 \I
*
+-dd:W
Fig. 2. Schematic of the seawater chamber.
F
.Qb
Y =
hb
y=h
0
T
128
G.
A. BEMPORAD
face layer. Salt concentration at the surface of the water column is thereby reduced and the water body stability can be maintained. In both coaxial pipes, it is assumed that convection is the dominant mechanism for heat and salt transport. The injection and withdrawal pipes exchange thermal energy with the salt water column through their surfaces; salt is not exchanged between these pipes and the middle layer of the water column. Application of the continuity equation in the salt water column leads to the following expressions for flow rates at the surface and bottom layers, respectively,
the bottom level of the middle layer; h, is the bottom level of the water column; (J refers to the water column, (,,,) refers to the withdrawal pipe, (i) refers to the injection pipe, (s) refers to the surface layer, (b) refers to the bottom layer. With regard to salinity transfer in the water column, we obtain the following expressions for the surface, middle and bottom layers, respectively
$WMh
- WA, + -4v)
a ay
= (&‘h,sQi - (K)c,sQw - KC Qi=qe$+Qw Qb = cost
(~C)c,y=h,Ac
(7)
where Qi is the injection pipe flow rate, which provides the inflow to the surface layer of the salt water column, d, is the salt water column diameter, qe is the evaporation rate, determined by eqn ( 1) , Qwis the discharge in the withdrawal pipe and Qb is the bottom layer washllow. In the withdrawal pipe we have Qw = cost, while Qi is determined according to the evaporation rate. Application of the conservation of energy leads to the following heat balance equations for the surface, middle, and bottom layers of the salt water column, respectively
; (PC)c,b(hb -
h&L
= h’C)bQb- (P’&d?b +
KC
$ (PC)c,j=/&
(9)
where ~~ is salt diffusivity. Application of the heat conservation equation in the injection pipe leads to the following expressions for the surface and the middle parts, respectively:
= Qi((dJpT)i,y=h, - (pCpT)i+) + W,
$(PCpThAi = -Qi
g (KJ)i
- y(Ti
- Tc)rdc - y’( Ti - Ta)*di
(10)
where Ai = $ (d; - d,2)
A, =
3 (df
- d;);
A,,, = ; d2,
are the salt water column and the withdrawal pipe cross sectional areas, respectively; d, is the withdrawal pipe diameter; I is the time; KT is the heat diffisivity; y = 650 - 1200 W m-2 “C-’ is the forced convection coefficient; hl is the bottom level of the surface layer; hb is
is the injection pipe cross sectional area; di is the injection pipe external diameter; T, is the ambient temperature; y’ is the natural convection coefficient; W, is the thermal power provided to the desalination scheme by an external source. Radiation losses to the ambient should also be considered. However, appropriate insulation materials should make this additional term negligible with respect to the other terms of the heat balance equations. The equations of solute conservation, for the surface and the middle parts of the injection pipe, are respectively
129
Basic hydrodynamic aspects Analogously, the heat conservation equations for the withdrawal pipe middle and bottom sections, respectively, are ; (PCJ-LA
= ;
(~c,T)wQw - -AT, - Tc)?rdw
; (PCpThv,tv‘k(h - ho) = ((~C,T)w,y=/t, - (pc,T)w,b)Qw - ?‘(Tw,b - Tc,bbdw
(12)
The equations of solute conservation, for the surface and the middle parts of the withdrawal pipe, are ; (~ChvAw = $ (~ClwQw
$
(Pchv,ddhb
-
ho)
=
((PC)w,y=hb
-
(P%b)Qw
(13)
Equations (7- 13 ) are solved numerically for the unknown temperature T and concentration C along the salt water column, the injection pipe, and the withdrawal pipe, respectively. The density vertical distribution results from the temperature and the solute concentration vertical distributions. The governing equations of the present mathematical model are discretized in time and space assuming that in the time interval r = (t(“), t(“) + At = t(“+‘)) the time derivative is discretized by a forward Euler formula, the convective terms are discretized with an up-wind finite difference form, and the diffusive terms are discretized with a central difference form, as follows:
aPr)
A-
=
f
(n+l)
at -aft')
ay
z.z
-
f
(n)
At
(A -h-l P
or
AY
-af(r) = (.h+,-h)‘“’
ay
;
(Kg
CT)
1
=
&
AY
((K,,
+
4. NUMERICAL RESULTS
The mathematical model described in the previous sections was employed to determine the performance of the proposed desalination scheme. Both steady-state and unsteady conditions were analyzed. During all simulations, the temperature of the fresh water chamber Tf was kept at 1O’C. The diameter of the water column was d, = 0.25 m, the external diameter was di = 0.30 m and the diameter of the withdrawal pipe was d, = 0.05 m. The thermal and the solute diffisivities were assumed to be equal to KT = 1.5 - lo-’ m2/s and KC = 1.5 - 5 - 10m9m*/s. The forced convection coefficient was assumed to be y = 1000 W/m */ “C. Assuming the injection pipe to be sufficiently insulated against heat losses to the ambient, we have y’ = 0. The bottom and the surface layer of the salt water column both had a thickness equal to 0.2 m. A first set of numerical simulations was performed to simulate heat and salt transfer in the seawater column under steady-state conditions. The thermal power needed to drive the desalination process was gradually increased to reach a value of 50 W. The initial temperature and the salinity concentration for the water column were 20°C and 4%, respectively. Figure 3 shows the evaporation rate as a function of time. Approximately 15-20 days were needed in order to reach steady-state conditions. Figure 4a-b shows the temperature and the density profiles along the water column, respectively, until steady-state conditions are reached. These simulations were performed with bottom and withdrawal discharges equal to Qb = 1.O- 10m6 m3/s and Qw = 1.0. 10m6m3/s, respectively. Figure 5 shows the evaporation rate at steady state conditions as a function of the withdrawal rate Q,,,. This figure shows that minor variations in the evaporation rate are caused by a certain variation in the withdrawal rate. Smaller withdrawal rates mean smaller injection rates and, therefore, higher temperatures for the surface layer as well as in the withdrawal pipe. Although the large convection coefficient y ensures that most of this excess heat is recovered by the water column, slightly better performances are expected for smaller withdrawal rates. Unsteady conditions were simulated by performing numerical simulations with a time-dependent heat source. This heat source was assumed to be provided
K,wj+1 -Xl
- (K, + &)(A
-&))(n)
(14c)
where f ( y, t) is the generic function of time and space and Kthe generalized diffusion coefficient. In eqn ( 14b) the choice over the derivation operator is performed according to the dominant direction of the current in the injection and the withdrawal pipes. Introduction of expressions ( 14) into the governing equations for the salt water column, the injection pipe, and the withdrawal pipe leads to the numerical scheme shown in Appendix A.
0
200
400
600
800
lr,OO
ttme(hours)
Fig. 3. Evaporation rate in the transient period and at steadystate.
G.
130
40
50
temperature
(C)
A.
BEMPORAD
salt water column, respectively. The inversion in the density profile near the surface shown in this figure may lead to a certain mixing in the surface. Figure 8ab shows the temperature and the salinity profiles in the withdrawal pipe. Due to the large heat exchange between the water column and the withdrawal pipe, the temperatures in the withdrawal pipe rapidly approach the temperatures of the water column. The salinity concentration, instead, is significantly different since no salt transfer is allowed between the two water bodies.
(a) 5. DISCUSSION
a
E g * 5 m
6 4 2 0 1000
1005
1010 density
1015
1020
1025
(kg/m3)
(b) Fig. 4. Seawater column in the transient period and at steadystate; (a) temperature profiles, and (b) density profiles.
by a set of solar panels. The available radiant solar energy on a solar panel (in W/m *) was determined at ENEL-DSR-CRIS-Milan0 (Reali, 1993) for a typical sunny day of May in Milan. Previous studies (e.g., Duff, 1988 ) recorded instantaneous collection efficiencies up to 60% for temperature differences up to almost 80°C during most daylight hours. Monthly collection efficiencies above 50% were attained in space heating applications in Northern Italy and Germany. In this study the efficiency of the solar panel was assumed as 50% for a panel surface of 0.25 m2. The available radiant energy used to drive the desalination process is shown in Fig. 6, together with the resulting evaporation rate. In this figure, t = 0 refers to 5 A.M. Figure 7a-b shows the development during the day of the temperature and density profiles in the
wthdrawal
rate
(m3is)
Fig. 5. Evaporation rate at steady state versus the withdrawal rate for various heat sources.
The numerical simulations presented in the previous section showed that most of the heat derived from the heat source could be maintained in the upper parts of the water column and employed for the evaporation process. At steady-state, the thermal power supplied to the seawater chamber was equal to 50 W, while the evaporation rate was 2 - lo-* m3 s-l. The thermal power needed to raise the temperature of that water from 20°C to 67°C was about 4 W, and the thermal power needed to evaporate it at 67°C was about 45 W (eqn [ 5 I). Thermal losses of about 1 W are caused by the bottom layer wash-flow and the withdrawal pipe salt removal. No other losses are present, since the seawater chamber was assumed to be completely insulated. Therefore, about 98% of the power supplied to the seawater chamber was employed to evaporate the fresh water. Under unsteady conditions, the total energy supplied to the seawater chamber during the typical sunny day was equal to 2.8 MJ. The total amount of fresh water evaporated during that day was equal to 1.08 . 10T3 m3. To raise this amount of seawater to the temperature of the surface layer and to provide the latent heat of evaporation about 2.7 MJ had to be supplied. According to these simulations, about 96% of the energy supplied during the day was employed to make the water evaporate. However, this last energetic balance is valid provided that the mixing caused by the density inversion is limited to the upper zone of the seawater column. The stability of the seawater column is of primary importance for the desalination process. Seawater is heated and injected on top of the seawater column through the injection pipe. From the surface layer some
time (s)
Fig. 6. Evaporation rate and radiant energy as a function of time during a sunny day.
131
Basic hydrodynamic aspects
---
51840 s trne 34560 s
.--
time 17280 s
---- me
40
50
temperature
(C)
60
70
(a)
density (kgim3) 08
Fig. 7. Transient conditions in the water column; (a) temperature profiles, and (b) density profiles.
water evaporates, therefore increasing the surface layer salinity. If not properly controlled, the surface layer becomes heavier than the lower layers, therefore causing a vertical recirculation in the water column. For that reason, a certain quantity of salty water must be withdrawn from the internal pipe as a wash flow. If the stable configuration cannot be maintained, the efficiency of the desalination scheme may be significantly reduced, since the water mixing will cause the warmer water to reach the lowest parts of the water column. Instead, higher efficiencies are obtained when most of the available heat remain in the surface to drive the evaporation process. The performance of the desalination process is influenced by several factors, such as withdrawal rate, thermal power input, temperature of the fresh water chamber, and others. For instance, the numerical simulations performed in this study have shown that higher efficiencies may be obtained by recirculating water through the withdrawal pipe at very low rates. Moreover, an unsteady heat source is most likely to cause a surface density inversion. In fact, heat is more rapidly lost from the surface layer of the water column than salt, and, therefore, the density of such a layer slowly increases until it reaches an unstable condition. A possible way to avoid density inversion is to use steady sources of thermal power. As shown by the numerical simulations, the direct application of a solar panel to the seawater chamber does not seem to offer the conditions required for maximum efficiency unless mechanisms to store solar energy on a daily or better weekly basis are introduced. The solar pond, for instance, could offer such benefits, since its built-in storage ca-
pability make it the natural choice for such applications. In these conditions, the proposed desalination scheme may offer a simple alternative to the existing desalination schemes which exploit solar energy to drive the desalination process. The expression for the evaporation rate was determined on the basis of basic theories on evaporation. However, in the desalination scheme under investigation the evaporation rate is also a function of the geometry of the vacuum chamber as well as of the quantity of gases dissolved in the seawater [e.g., Kennard, 19381. These factors, not considered in the present analysis, may significantly affect the evaporation rate. The order of magnitude of the experimental evaporation constant (Y,,,used in this study was derived from previous investigations as well as from preliminary experiments performed at ENELDSR-CRIS (Reali, 1993). Since the evaporation process is influenced by several factors, many of which were not considered in the present study, extensive experimental work should be performed in order to determine the value of this constant under the various conditions. However, through the present theoretical study it was possible to determine the major issues which characterize the proposed desalination scheme, namely the possibility of density inversion in the saltwater column and the difficulties in operating the scheme under an unsteady heat source. 6. CONCLUSIONS
A mathematical model was developed to simulate the complex hydrodynamic phenomena occurring in
10
/““i
8
0
40
20
60
temperature
il YO
(C)
(a)
1
A’ ,,’
2 ,
0 38
/(/
I],. 39
I
/,
111, f
40
,*’
/’ 4’
42
(W Fig. 8. Transient conditions in the withdrawal pipe; (a) temperature profiles, and (b) salinity profiles.
132
GA.
BEMPORAD
a water column during a desalination process. The basic equations which govern heat and mass transfer in the desalination process were integrated through a finite difference numerical scheme. The numerical results theoretically proved the feasibility of the proposed desalination process. Maintenance of a stable water column made possible the utilization of most of the heat absorbed by the solar panel for the evaporation process. These preliminary results proved the feasibility of a continuous production of fresh water through the proposed desalination scheme. Acknowledgments-This study was funded by the Hydraulic and Structural Research Center of ENEL Spa. The author is grateful to Dr. M. Reah of ENEL Spa for his comments and guidance during the preparation of this manuscript.
NOMENCLATURE
AC water column cross sectional area ( m2) 4 injection pipe cross sectional area ( m2) A, withdrawal pipe cross sectional area (m*)
c
solute concentration (%) bottom layer solute concentration (% ) CC water column solute concentration (96) injection pipe solute concentration (% ) Ci G surface layer solute concentration (% ) CIV withdrawal layer solute concentration (5%) CP specific heat (Jkg-’ “C-i] dc diameter of the salt water column (m) 4 diameter of the injection pipe (m) dw diameter of the withdrawal pipe (m) -5 fresh water/seawater separation energy (J) gravitational acceleration ( mse2) f elevation of the bottom of the water column (m) ht elevation of the bottom of the surface layer (m) h elevation of the bottom of the middle layer (m) h, elevation of the water column (m) 1. latent heat of vaporization (Jkg-‘) P vapour pressure (Pa) injection pipe discharge ( m3 s-‘) 2 flow in the bottom layer of the’water column ( m3 s-i) QW withdrawal pipe discharge ( m3 s-’) qe evaporation discharge (ms-’ ) T temperature (“C) Tb bottom layer temperature (“C) TC water column temperature (“C) Ti injection pipe temperature (“C) T, surface layer temperature (“C) TIV withdrawal layer temperature (“C) W thermal power ( W) Y vertical coordinate (m) fft empirical coefficient in eqn (2) (dimensionless) empirical coefficient in eqn (4) (J kg-’ “C-l) a2 empirical coefficient in eqn (4) (J kg-’ “C-‘) & emoirical coefficient (ka m-*Pa-‘s-’ OK”‘) am sol&al expansion coeffi>ent (Q -’) ’ Bc thermal expansion coefficient (“C-‘) t% forced convection coefficient (W m-* “C-l) Y Y’ natural convection coefficient ( W mm2“C-l) cb
xc or Y p &
solute molecular diffusivity (m2s-‘) heat molecular diffusivity (m*s-I) kinematic viscosity (m*s-‘) density (kg m-“) evaporation thermal energy flux ( W m-*) 4. net thermal energy flux (W m-*) Subscripts and superscripts 6 of the bottom layer c of the water column f of the fresh water i of the injection pipe 0 of a reference state s of the surface layer w of the withdrawal pipe
REFERENCES
S. H. Algie, Kinetic theories of evaporation, J. Chem. Phys. 69(2), (1978). S. E. Aly, Analysis of a fuel-solar assisted central dual purpose plant, Desalination 78, 363-379 ( 1990). Y. Assouad and Z. Lavan, Solar desalination with latent heat recovery, ASME J. Solar Energy Engrg. 110, 14-16 (1988). G. A. Bemporad and M. Reali, A solar pond desalination scheme for fresh water and salt production, 2nd International Congress on Energy, Environment and Technological Innovation (October 1992), Rome, Italy, Universim La Sapienza. W. S. Duff, Experimental results from twelve evacuated collector installations, Proceedings of the I988 ASES Annual Meeting (June 1988), Cambridge, MA, ISES. A. M. El-Nashar and A. M. El Baghdadi, Seawater distillation by solar energy, Desalination 61,49-66 ( 1987). A. M. El-Nashar, Computer simulation of the performance of a solar desalination plant, Solar Energy 44, 193-205 (1990). A. M. El-Nashar, Optimizing the operating parameters of a solar desalination plant, Solar Energy 48,207-2 13 ( 1992). S. Fujikawa and T. Akamatsu, Effects of the non-equilibrium condensation of vapour on the pressure wave produced by the collapse of a bubble in a liquid, J. of Fluid Mech. 97(3), 481-512 (1980). E. H. Kennard, Kinetic theories of gases, McGraw-Hill, New York (1938). Y. Keren, H. Rubin, J. Atkinson, M. Priven, and G. A. Bemporad, Theoretical and experimental comparison of conventional and advanced solar pond performance, Solar Energy 51(4), 255-270 (1993). H. H. Jobson, The dissipation of excess heat from water systems, ASCE-J. Power Engrg. 99(l), ( 1973). M. Reali, A refrigerator-heat-pump desalination scheme for fresh water and salt recovery, Energy 9,583-588 ( 1984). A. M. Salhotra, E. E. Adams, and D. R. F. Harleman, Vertical mixing in thermohaline system, 3rd Internntional Symposium on StratiJied Flows, Pasadena, CA ( 1987). B. L. Sill, Free and forced convection effects on evaporation, ASCE, J. OfHydrauIic Engineering 109(9), 1216-1229 (1983). K. S. Spiegler, Saltwater Purification, 2nd Ed, PZenum Press, NY(l977).
APPENDIX: THE NUMERICAL MODEL
Introducing the expressions in eqn ( 14) into the equations of heat balance (8) we obtain, for the surface, middle, and bottom layers of the seawater column, respectively
133
Basic hydrodynamic aspects
where E = pC,,T is a compounded variable. Analogously, from eqn (9) we obtain for the surface, middle, and bottom layers, respectively, the following discretired equations of salt balance
where n = pC is another compounded variable. By solving eqns (A2) and (Al) it is possible to determine ,$s”, n$“, for j = 1,2, . . . , N - 1, N, where j = 1 refers to the bottom layer, j = N to the surface layer, and (,) refers to the seawater column. Discretiration in time and space of the heat balance equations for the injection pipe eqn ( 10) leads to, for the surface and the middle parts, respectively:
g
- [i$
gy’ - gj A,
At
=
’
(Eij -
_Q!“’
’
&i-d’” -
AY
h,)A, = Qj”)(&_,,r_, - &N) + WI(“)
(h, -
At
y( Ti,
-
TcJ)(“)adc - y’( TiJ -
T,J’“‘rd,
- eu(( T,,
+
273)4 -
(T, +
273)“)%d,
(A3)
Analogously, discretization in time and space of the salt balance equations for the injection pipe, eqn ( 1 1 ), leads to, for the surface and the middle parts, respectively:
qy;” - $J, At $+I) -
At
(h - hMi = Q;“‘(m,,-,
““’
Ai
-Qi
=
(II)
(t)ij -
- KV)
mJ-1
P’
(A4b)
AY
Discretization of the heat conservation equations ( 12) for the withdrawal pipe middle and bottom sections, respectively, leads to p;”
- p/J .
At
A
=
(‘%+I- hd(“’_
Q
w w
y(
E!P - GyI At
T,,,
Tcj)Gend
_
Iv
AY
(hb - ~o)Aw = Qw(S,2 - .!&I) (“I - y( Tw,, -
Tc,,)‘“‘sd,(hb
- h,)
(As)
Analogously, discretization of the salt conservation equations ( 13) for the withdrawal pipe middle and bottom sections, respectively, leads to @+I’ -
VW.1 At
p;
,
A
=
*’
Q
('lw,j+l-
v
%,j)(")
AY
(A6) The four physical variables, Ti(“+‘), $+I), py’), (CJj”+‘) for the water column, the injection pipe and the withdrawal pipe, respectively, are obtained by solving for each time step, at each computational node, the additional system of four coupled equations
G.A.BEMPORAD
134
[pj(c&Jjq(n+‘) (“+I)
I&C,1 [(qj]‘“+”
p:n+‘ =) /I,( 1 -
= =
=
$+‘)
$+l)
&+‘)
&(Tj (“+I)- 7-J + &(C:n+‘) - CO))
(‘47)
Initial conditions on q, Cj, p,, (C,)j for j = 1, 2, . . , N - 1, N, for the seawater column, the injection pipe and the withdrawal pipe, complete the numerical formulation. Given the different mathematical characteristics of the advection and of the diffusion equations, respectively, the solution of the numerical problems is obtained by adopting a sub-cycling process, where for the convective terms a time step n times smaller is utilized.