A SIMPLIFIED MODEL FOR ESTIMATING DISTILLATE YIELD IN LARGE SOLAR
STILLS
Arturo Palacio, Alejandro Rodríguez, Italia Millan, Jose L. Fernández
Engineering Institute, National University Autonomous of Mexico
Apartado Postal 70-472, Coyoacan, 04510, Mexico, D.F.
email: vortex@vortex.iingen.unam.mx
ABSTRACT
A simplified one-phase model based which considers an homogeneous gas mixture
with an “apparent diffusion coefficient”, has been developed to represent the flow
characteristics inside a double slope solar still. Its application for the calculation of
the heat flow and therefore of the distillate yield shows good agreement with
experimental data. Preliminary results about the asymmetry of the flow patterns
attained for symmetrical boundary conditions is presented and discussed.
INTRODUCTION
The use of solar energy in engineering applications has gained more importance
over the last decades. Several studies, theoretic as well as experimental, have been
realized on the heat transfer mechanisms that take place in triangular cavities, which
is the typical shape in the X-Y plane geometry of a solar still. Convection has been
proven to be the main heat transfer phenomenon in the water distillation process, as
concluded by several authors.
In 1979, Flack, Konopnicki and Rooke described an experimental study of the heat
transfer due to convection in a laminar flow contained in a triangular cavity
(isothermal walls, hot and cold, and adiabatic base). The results are similar to those
shown in rectangular cavities with the exception of the apex, where conduction is
proven to be the predominant heat transfer mechanism due to temperature difference
and the proximity of the walls. A second study realized by Flack et al. [1980]
concludes that under summer conditions (hot top, cold base) the air flow remains
laminar, convection near the walls is minimum and the aspect ratio at a given Gr has
very little influence over Nu. As for the winter conditions (cold top, hot base), Flack
shows the flow changes from laminar to turbulent as Gr increments and most of the
heat transfer through the base is realized via convection. The centreline temperature
profile illustrate that the temperature is almost constant in the central area of the still
but increases significantly near the base and walls. The indirect relation between the
Nu and the walls inclination angle (at a given Gr) is established too. This is confirmed
later, in a following experiment made by Poulikakos and Bejan [1982] in larger scale
triangular spaces (increased Rayleigh Nos.)
Martín del Campo et al carry out in 1988 a more extensive numerical analysis with all
possible combinations of cold, hot and adiabatic walls studied for different aspect
ratios and Gr numbers. His results embrace Flack's conclusions that bottom heated
cavities are more convection and heat transfer effective than top heated ones and
show the direct relation between Gr and the convective heat transfer as well as the
influence that horizontal gradients have over convective flow, favoring it. This
analysis also focuses on the influence that geometry has on the efficiency of the still.
Aspect ratios of 1 to 3 are proven as the most effective. Geometry impact on water
distillate becomes the principal concern that numerous authors focus on, such as
Rubio et al [2000] make a comparison between double and single slope stills finding
there is no significant difference in their productions when subjected to same water
and cover temperature and therefore demonstrating the distillate production is
independent of the cavity's geometry.
Palacio et al [6] comment on the fact that in solar stills with larger aspect ratios, the
dominating heat transfer mechanism is convection which is not as efficient as
diffusion, more proper of shallow cavity solar stills. Despite this, more recent
analytical studies done in high inclination solar stills by Palacio and Fernández
[1993], suggest that tall vertex stills are viable and their geometry can be enhanced
to facilitate the development of convection vortices (which will compensate for the
loss in heat and mass transfer by diffusion) and improve the heat transfer process as
the increased height was proven not to incur substantially in the loss of distillate
yield. This theory is reinforced by the results obtained by Porta et al [1994] on
shallow cavity stills although it is suggested, in opposition to what was concluded
before, that convection plays an important role in shallow cavity solar stills as well.
Kwantra [1996] studies the importance of the water evaporation area in a solar still.
His mathematical analysis illustrates the quantitative relationship between it and the
distillation yield. An enlarged evaporation area results in a more efficient evaporationcondensation process, increasing the yield.
Rubio et al [2002], present a procedure to estimate the glass cover production in
double slop solar stills, as a function of still temperature and area fraction, as an
extension to the model proposed by Dunkle.
The endeavor for a better understanding of the seawater distillation phenomenon and
the search for ideal geometries and conditions that will improve the distillate yield are
still an ongoing task. The intent of this paper is to report the results obtained through
the development of a mathematical model capable of reproducing actual
experimental data regarding distillate yield and its application on different cases that
will lead the path to a continuous improvement process on this field.
PROBLEM DESCRIPTION
A typical double slope solar still, consists on an insulated box with channels located
along the base of each condenser to collect the distillate yield; the glass covers are
attached to the basin at an angle  which depends on the particular design, as
shown on figure 1. The evaporation area is given by the product of the length and
the width of the basin, lw. The height h represents the distance between the water
surface and the vertex of the glass covers.
Fig. 1 Typical double slope still
In this paper, the performance of a relatively small section solar still (w=1.6 m, h=0.46
m) is compared to that of a large section still (w=5 m, h=2.5 m), both constructed and
operated in La Paz, B.C.S., Mexico, Rubio et al [2002].
Mathematical Formulation
The set of partial differential equations describing the problem examined herein, are those
of continuity, momentum and energy, together with an additional couple of turbulence
transport equations. This set of equations can be expressed in a general form as follows:

(   ) +   (  V  ) =   (    ) + S 
 t
(1)
where t denotes time,  represents any dependent variable, V the velocity vector,  the
fluid density,  the transport coefficient of the dependent variable, S the source of  per
unit volume and  is the vector differential operator.
The dependent variables and their corresponding transport equations, are listed below in
Table 1, where u, v are the velocity components in the coordinates directions x and y
respectively, T is the temperature, which is directly calculated from the energy equation, k is
the turbulent kinetic energy and  its dissipation rate;  l is the laminar dynamic viscosity,
 t the turbulent viscosity, and  e the effective viscosity;  k and   are the standard
turbulence-model coefficients which will be described later.
The source term for the momentum equation is given by:


Sm = -  p +  g +    e (  U ) 
2

( e   U ) I 
3

(2)
TABLE 1. Transport coefficients for variable 
Continuity
Momentum


1
0
u, v
e
( l /  l + t /  T )
Temperature
T
Turbulent kinetic energy
K
( l + t /  k )
Dissipation rate of k

( l + t /   )
where p is static pressure, g gravity vector, I the unit tensor, and the superscript T denotes
the transpose of the dyadic. The Boussinesq approximation was considered to take into
account buoyancy effects.
Turbulence model
The turbulence model employed is the standard k-  model, where the kinetic energy k, and
its dissipation rate  , represent the velocity and length scale of the turbulent motion
respectively. The corresponding source terms are:
S k = ( Pk -   + G B )
S  =  C 1 P k - C 2   + C 3 G B
(3)


k
(4)
where Pk is the production rate of k, and GB is the production or destruction of k due to
buoyancy effects; these two terms may be written as:


Pk =  t  U : (  U + (  U ) )
GB = -

t
g  
t
(5)
(6)
The turbulent viscosity is calculated from the local values of k and  as follows:
t = C  k2 / 
(7)
The values assigned to the turbulent constants are the standard ones recommended by
Launder and Spalding [1974].
Boundary Conditions
Two dimensional calculations were performed neglecting the end effects of the still solar.
Assuming steady state operation conditions of the still, glass cover temperatures T g1 and
Tg2 as well as water surface temperature Tw were prescribed. The model takes into
account the friction effect at the glass covers and water surfaces through the wallfunction approach outlined by Rodi [1993]. A reference pressure value is assigned at one
cell in the middle of the domain.
The solution algorithm is based upon the well-known iterative “guess-and-correct” procedure
of Patankar and Spalding [1972], but modified according to the SIMPLEST algorithm of
Spalding [1982].
PRESENTATION AND DISCUSSION OF RESULTS
Model validation
In order to assess the accuracy of the model to represent the flow of a mass of fluid
inside a triangular cavity, the experimental results reported by Flack [1979, 1980]
were considered. In his paper dated 1979, Flack et al. [1979] measured local and
overall heat transfer data for convection air flow in triangular enclosures with two side
walls which were heated and cooled and an adiabatic bottom. The experimental data
generated for the local heat flux distributions is compared with the results obtained
from the present model in figure 2, showing an acceptable agreement.
o
6
Q2=90 y Gr2=5.60x10
1.00E+02
HOT:
Flack
laminar
Nu1
no kebuoyancy
kebuoy=0.2
1.00E+01
COLD:
Flack
laminar
no kebuoyancy
kebuoy=0.2
1.00E+00
1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
Gr1
Fig 2. Local Nusselt numbers for both hot and cold walls as a function of Grashof.
The cases where the triangular enclosures are heated or cooled from the bottom are
reported by Flack et al. [1980] and the typical local heat flux distributions for both
cases are compared with the present model in figures 3 and 4, showing again a good
agreement.
6.00E+02
HFLX
5.00E+02
Flack2
4.00E+02
q"
3.00E+02
2.00E+02
1.00E+02
0.00E+00
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 3. Heat flux distribution for hot top.
x/L
2.50E+02
HFLX1
HFLX2
2.00E+02
Flack2
1.50E+02
q"
1.00E+02
5.00E+01
0.00E+00
0
0.2
0.4
0.6
0.8
1
x/W
Figure 4. Heat flux distribution for cold top.
1.2
1
Model Application
Even though the real problem inside a solar still corresponds to that of a multiphase
flow of an air-water vapour mixture, the simplicity of the one-phase model has been
kept by considering an apparent diffusion coefficient for an homogeneous gas
mixture, as described by Palacio et al [1993]. This means that a lower Prandtl
number than the one for pure air or pure water vapour needs to be determined for the
mixture, in order to obtain an adequate calculation of the heat flow as compared with
the experimental measurements of the distillate.
Following this approach, the Prandtl number effect on the calculation of the heat flux
was investigated employing the average thermal operation conditions reported by
Rubio et al [2002] for the small section solar still: T w= 80 0C, Tg1= Tg2= 65 0C, q”=366
W/m2. Given the symmetry of the data, half of the domain was considered for the
calculation. Figure 5 shows that for Pr numbers smaller than 0.1, the heat flux
increases very rapidly, while for larger values the variation is much slower. The
required heat flux value needed to match the experimental data resulted for a Prandtl
number of 0.11.
450
Heat flux (W/m2)
400
350
300
250
small
200
large
150
100
50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Prandtl
Fig. 5. Effect of Prandtl number on heat flux calculation
The characteristic flow patterns, as well as the temperature distribution inside the
solar still are shown in figures 6 and 7 respectively. Maximum flow velocities of the
order of 0.36 m/s are present along the vertical centre line of the cavity and down the
glass cover, while a single recirculation region is formed near the centre of the
domain. Isotherms still show the characteristic shape of a convective dominating
flow.
Fig. 6 Flow patterns for the small section still
Fig. 7 Temperature distribution for the small section still
The same exercise was repeated for the large section solar still, with the following
parameters: Tw= 60 0C, Tg1= 48 0C, Tg2= 44 0C, q”=220 W/m2. The same behaviour
observed for the previous case regarding the effect of the Prandtl number on the
calculated heat flux was found, and as figure 5 reveals, the measured heat flux is
reproduced again when the Prandtl number approaches the value of 0.11, the
difference being less than 5%. It is of interest to notice on the same figure that even
though the maximum temperature difference between the water and the glass is
practically the same for both cases, the heat flux predicted for the large section still is
nearly half of that for the small section still at the low Pr number region.
Given the above operation conditions for the big section still, the flow patterns and
temperature distribution are very different from the ones obtained for the small
section still, the reason being the asymmetry prescribed on the glass covers
temperatures. Most of the flow on figure 8 is turning in clock-wise direction, except
for a small region located at the left end, where recirculation takes place on the
opposite direction due to the temperature distribution generated according to figure 9.
This time the maximum velocity reaches 0.465 m/s at the right side of the still, where
the largest temperature gradients takes place. For this temperature boundary
condition, nearly 54% of the heat transfer takes place on the right glass cover, where
the prescribed temperature is 4 0C lower than the other glass.
Fig. 8 Flow patterns for the large section still
Fig. 9 Temperature distribution for the large section still
Employing visualisation techniques developed by Porta et al. [1998], Poujol et al.
[1999] have detected that even for symmetrical temperature conditions on both glass
covers, the flow patterns do not attain symmetry. In order to make a preliminary
exploration of this fact, another numerical simulation was performed for the operating
conditions of the small section still, but this time considering the complete domain,
with a 50 x 30 grid shown in figure 10. Even though the general flow structure is very
similar and the maximum fluid velocity is again about 0.35 m/s, there is a clear
tendency of the flow to form two asymmetric regions, figure 11, with a stronger
recirculation present on the left side where larger temperature gradients are formed
according to figure 12. This result was obtained repeatedly after using several
different sets of relaxation parameters.
Fig 10. Computational grid employed for the complete domain
Fig 11. Flow pattern for the small section still considering the whole domain
Fig 12. Temperature distribution for the small section still considering the whole
domain
The same exercise was repeated for the large section still applying equal
temperature values at the glass covers (46 0C), and the results were qualitatively the
same as for the small one.
CONCLUSIONS AND RECOMMENDATIONS
A mathematical model was developed to evaluate the flow characteristics inside a
double slope solar still and applied to reproduce the experimental results of two
experimental solar stills located in La Paz, B.C.S., Mexico.
The results seem to indicate that if the gas mixture inside the apparatus is treated as
homogeneous, a Prandtl number value of 0.11 allows to reproduce with a very good
approximation the heat flux, and therefore the distillate yield produced, for both
experimental devices whose cavity sectional areas differ by a factor larger than 15.
Even for symmetrical temperature boundary conditions, the numerical model shows
that the temperature distribution inside the solar still, and therefore the corresponding
flow patterns are not symmetrical, which confirms the preliminary observations made
by the working group of La Paz, Mexico.
It would be recommendable to apply the model to try to represent the transient
stages of heating and cooling of the stills.
REFERENCES
Flack R. D., Konopnicki T.T. and Rooke J. H., "The Measurement of Natural
Convective Heat Transfer in Triangular Enclosure", ASME Journal of Heat Transfer
Vol. 101 No. 4, pp. 648-653, 1979.
Flack R. D., "The Experimental Measurement of Natural Convection Heat Transfer
in Triangular Enclosures Heated or Cooled from Below", ASME Journal of Heat
Transfer Vol. 102, pp.770-772, 1980.
Kwatra H., "Performance of a Solar Still: Predicted Effect of Enhanced Evaporation
Area on Yield and Evaporation Temperature", Solar Energy Vol. 56 No. 3, pp. 261266, 1996.
Launder, B E y D B, Spalding (1974), “The Numerical Computation of Turbulent
Flow”, Comp Meth Appl Mech Eng 269-289.
Martín del Campo E., Sen M. and Ramos E., "Analysis of Laminar Natural
Convection in a Triangular Enclosure", Numerical Heat Transfer Vol. 13, pp. 353-372,
1988.
Palacio A. and Fernández J. L., "Numerical Analysis of Greenhouse-Type Solar
Stills with High Inclination", Solar Energy Vol. 50 No. 6, pp. 469-476, 1993.
Porta M. A., Fernández J. L. and Chargoy N., "Influencia de la Distancia Vidrio-Agua
en Destiladores Solares de Caseta", ANES, 1994.
Porta, MA, Rubio, E, y Fernández, JL, “Visualization of natural convection inside
shallow solar stills”, Journal Experiments in Fluids, Vol. 25, pp 369-370, 1998
Poujol, FT y Fernández, JL “Evolution of flow patterns in a space with triangular top”,
Journal Experiments in Fluids. Septiembre, 1999 (review)
Poulikakos D. and Bejan A., "Natural Convection Experiments in a Triangular
Enclosure", Technical Notes Journal of Heat Transfer Vol. 105, pp. 652-655, 1982.
Rodi W (1993), “Turbulence models an their Application in Hydraulics. A state-of-theart review”, 3rd De IAHR Monograph, Balkema, Rotterdam NL.
Rubio E., Porta M. A. and Fernández J. L., "Cavity Geometry Influence on Mass Flow
Rate for Single and Double Slope Solar Stills", ”, Applied Thermal Enginnering Vol.
20, pp1105-1111, 2000.
Rubio E., Porta M., Fernández J.L., “Thermal Performance of the condensing covers
in a triangular Solar Still”, Renewable Energy, (2002), in press