%Reduction des tores
\documentstyle[11pt]{article}
\pagestyle{plain} \oddsidemargin 1 cm \evensidemargin 2 cm
\textwidth 15.5 cm
 \topmargin 0 cm
  \textheight 22.5 cm
\renewcommand{\baselinestretch}{1.9}

%\theoremstyle{plain}
\newtheorem{defn}{D\'efinition}[section]

\title{THE MOISTURE FROM THE AIR AS WATER RESOURCE IN ARID REGION: HOPES, DOUBTS
AND FACTS}
\author{B. Kogan, ECOST, Israel \and
A. Trahtman \thanks{{\it e-mail}: trakht@macs.biu.ac.il } Dep. of
Math., Bar-Ilan Univ., 52900, Israel}
\date{}

\begin{document}

\maketitle
\centerline {Journal of Arid Environments vol. 53, 2(2002), 231-240}
%\medskip

\begin{abstract}
   The recovery of clean water from dew has remained a fascinating
problem in the arid regions of the globe.
  The stone heaps near the city of Feodosia in the Crimean peninsula
were considered for many years to be artificial dew-catching constructions for
obtaining drinking water.
Several attempts to reconstruct these systems have been made but they have been
considered unsuccessful because of low yield. This has caused some doubts and
negative
estimations regarding the role of the Crimean stone heaps as water collectors.
The opinion that there were no dew-catching constructions in Crimea still
dominates today.

 In this discussion we shall consider the role of the Crimean stone heaps as
water condensers
and a model of  Nikolayev, Beysens et al. (1996) of this process.
 Some conclusions will be put forward showing why this model does not correspond
with
 the system under consideration, hence concluding that the above mentioned
negative
opinion, which is based on the model, is a rather hasty conclusion.

 The traditional model of the Crimean water collector will be modified by the
consideration of the role of the draught in the process of
condensation. Qualitative and quantitative analysis of the process
and of draught outbreak will be proposed. The efficiency of the
collector will be estimated.

\end{abstract}

{\bf Keywords}: dew, condensation, water collection, arid region.

\medskip

\nopagebreak

 \section*{Introduction}

     The demand for fresh water is currently an important political, social
   and economic issue in many countries of the arid regions of the globe.
   The main sources of fresh water are rivers, lakes and artesian wells.
   But river discharge comprises
   only 7\% of the total condensation. The renewable source of fresh
 water - the moisture of atmosphere is almost not used.

  About 200 nights in Highland Negev in Israel are characterized by 100\%
humidity (Broza, 1979). Annual dewfall in coastal regions, Jerusalem and the
North
Negev is 60-120 mm (Ashbel, 1935, 1949). The number of foggy
 nights in the North Negev and Yizreel Valley is about 40 (Levi, 1967).

    We consider here the possibility of condensation of the moisture of humid
 air on some cold surface with certain swiping potential.
  Condensation of moisture remains in the shadow
 of other solutions.
Nevertheless, the experiments of obtaining water from fog or from
 moist air were conducted in not less than 22 countries (NIkolayev et al.,
1996), (Schemenauer et al., 1991). The amount of obtained water
depends upon the place, the time of year and the percentage of
moisture in the air. Most advanced systems using high elevation
 fog give 3-7 liters per day per $m^2$ of working surface
 (Schemenauer et al., 1989, 1991, 1992).
   The high quality of the atmospheric water and minimal influence on the
 environment are important positive factors considered in this approach.

The history of water collection in Crimea is one of more
fascinating and intriguing in this area. The purpose of our work
is estimation of the ability of the Crimea stone heaps to collect
dew and consideration of some negative opinions regarding this
ability, development of the traditional model of water collection
by stone heap and the role of draught for this model, qualitative
and quantitative analysis of the process and of draught outbreak,
estimation of the output.

\section{The story of the dew collection in Crimea}

This section follows mainly to  F.I. Zibold, a Crimean forester and
 engineer (Zibold, 1905). Let us consider some undoubted
facts from this work.

   The population of the city of Feodosia in the coastal part of Crimea
peninsula
 in the 19th century was about 11,000. The climate was quite dry, the rains were
rare
and droughts lasting several months were normal. Zibold was unable to find
any spring or well around the city, but mentioned a large quantity of dew.
 The water supply of the city
was based on the so-called "fountains", which were big reservoirs of the water.
  The inscription in the Armenian language found on the 182 $m^3$ "Karaite
   fountain" is dated 1586. There were 8 working fountains
 in 1874 and only 5 working fountains in 1882. There were 26 fountains one
hundred years before, in 1784. In the Middle Ages, during the heyday of the
city,
 it is suspected that there were up to 100 "fountains".
The reservoirs have obtained the water from a network of
tile pipes, 5 to 7 cm in diameter, and channels filled with crushed rock.
 Nevertheless, there was no trace of springs. The pipes and channels ended in
 enormous pyramids of crushed calcareous rock of odd shapes and 5 to 10 cm in
size.

    The daily output of the "fountains" was studied twice: in October 1874 (8
"fountains", 66,000 litres), and in May 1882 (5 "fountains",
57,000 litres).
 Zibold found in 1905 10 stone heaps considered as water condensers
and printed the volumes of five of them: 2900, 1970, 1450, 1250, 1250
$m^3$. So we can consider the average stone heap volume of  1664 $m^3$.

 If we suppose that the number of active
   condensers was not less than 8 in 1874 or 5 in 1882 and not greater
  than 13 in 1874 or 10 in 1882, then we can derive that 5,100  - 11,400 litres
  were the production limits for a single stone heap per day.

A constant deterioration of considered water supply system was mentioned.
Only the remains of pipes are noticed now (Nikolayev et al., 1996).
 The pyramids of crushed rock were destroyed in the beginning of the 20th
century (Alexeyev et al., 1998).

 Nevertheless, one can find today in Crimean forests some another kind of
water condenser also called "fountains" by Crimean inhabitants.
   The size of the installation is like the size of a little house.
 The water leaks from the pipe going out from the lower part of the
 construction. This kind of installation was described in literature
(Anonymous, 1935a), (Jumikis, 1965).

\begin{figure}
\begin{picture}(100,192)
\end{picture}
\begin{picture}(225,192)

 \put(15,22.5){\line(3,-1){67.5}}
 \put(82.5,0){\line(1,0){67.5}}
 \put(150,0){\line(3,1){67.5}}

 \put(217.5,22.5){\line(0,1){33.75}}
 \put(217.5,57.75){\line(-3,1){67.5}}
 \put(150,80.25){\line(0,1){22.5}}

 \put(15,22.5){\line(0,1){33.75}}
 \put(15,57.75){\line(3,1){67.5}}
 \put(82.25,80.25){\line(0,1){22.5}}

 \put(82.25,102.75){\line(-3,1){67.5}}
 \put(15,125.75){\line(0,1){33.75}}

 \put(15,125.75){\line(1,-1){33.75}}
 \put(15,57.75){\line(1,1){33.75}}

 \put(217,125){\line(-1,-1){33.75}}
 \put(217.5,57.75){\line(-1,1){33.75}}

 \put(150,103.25){\line(3,1){67.5}}
 \put(217.5,125.75){\line(0,1){33.75}}

 \put(15,157.5){\line(3,1){67.5}}
 \put(82.5,180){\line(1,0){67.5}}
 \put(150,180){\line(3,-1){67.5}}

\put(117.5,90){\circle{30}}
 \put(91.25,64.75){$crater$}
 \put(47.75,90){\vector(-1,0){37.5}}
 \put(14,82.5){$drain$}
 \put(185.25,90){\vector(1,0){37.5}}
 \put(195,82.5){$drain$}

 \put(105,-6){\vector(-1,0){90}}
 \put(127.5,-6){\vector(1,0){90}}
\put(105,-7.5){$53 m$}

 \put(1.25,90){\vector(0,-1){90}}
 \put(1.25,112.25){\vector(0,1){67.5}}
 \put(0,97.75){$43 m$}

 \end{picture}

\begin{picture}(100,60)
\end{picture}
\begin{picture}(240,67.5)

 \put(15,22.5){\line(4,1){67.5}}
 \put(82.5,39.4){\line(6,1){15.75}}
 \put(99,42){\line(1,0){36}}
 \put(7.5,22.5){\line(1,0){221}}

 \put(217.5,22.5){\line(-4,1){67.5}}
 \put(150,39.1){\line(-6,1){15.75}}

 \put(15,22.5){\line(6,1){33.75}}
 \put(82.5,39.1){\line(-3,-1){33.75}}

 \put(217.5,22.5){\line(-6,1){33.75}}
 \put(150,39.1){\line(3,-1){33.75}}

 \put(100,42){\line(1,-1){11}}
 \put(133,42){\line(-1,-1){11}}
 \put(111,31){\line(1,0){11}}

 \put(108.75,44.25){$9 m$}
 \put(133.25,30.75){\vector(0,1){11}}
 \put(133.25,42){\vector(0,-1){11}}
 \put(135,30){$2.5 m$}

 \put(7.5,22.5){\vector(0,1){15}}
 \put(7.5,37.5){\vector(0,-1){15}}
 \put(-7.5,24){$5 m$}

 \end{picture}
\end{figure}

     To verify the possibilities of dew condensation, Zibold built in 1912 in
   Crimea a  stone heap condenser model. He had kept the shape, the size and
   the structure of old prototypes. Sea-beach pebbles were used as a building
material.
  Unexpectedly, the installation yielded only 300-360 liters per day
 (Anonymous, 1935), (Nikolayev et al., 1996) and suddenly stopped functioning.
  A leakage of the bowl was suspected.
No information about functioning of the installation can be found in Zibold's
 publications. The data on daily output we know (Nikolayev et al., 1996) was
 obtained from an indirect source.
 One can suppose that Zibold considered his experiment a failure. Hence,
 this level of diurnal productivity could not be an optimal basis for the
estimation of
 the stone heap average output.

Zibold's attempt inspired some experiments with this type of water condenser
in the South of France by L. Chaptal, M. Goddard and A. Knapen.
(see (Jumikis, 1965), (Chaptal, 1932), (Knapen, 1928)).
These installations called "aerial wells" or (vapor) "captors" were analogous
to the Crimean prototypes. The size of the installations was essentially less.
Different modifications of the construction were checked.
 Some of the constructions yielded condensed water, but the amount of the water
 was less than expected. The rosy hopes of their creators failed.

From the Feodosia weather station data, the average velocity of winds is
7 m/s and winds from the sea are dominant. The average annual rainfall is
 366 mm and the average number of days with fog is 25
(Alexeyev et al., 1998), (Nikolayev et al., 1996).

\section{The discussion of the role of the Crimean stone heaps condensers}

 We can conclude that the water supply of Feodosia for many years was on the
 level of 60,000 litres per day (or greater), and we do not know today of
another source of drinking water except the stone heaps on the
mountains near this town.
  We know that the Zibold and Chaptal installations have followed the prototype
described by Zibold and have produced water, but the yield was
unexpectedly low.

    Let us consider other points of view on the Crimea stone heaps.
  There are doubts of its role as water condensers, and now it is a prevailing
point of view.
Ashbel (1949) writes: "The exact purpose of the installation in the Crimean
 peninsula described ... as "dew wells" is not known. If we must speak of "dew wells",
  we might take as an example those in the desert of Northwest Africa".
  "No large amount of water is to be expected in
  such reservoirs: but where no drinking water whatsoever is available, the few
  liters collected every night is invaluable to thirsty wayfarers".

Nikolayev, Beysens at al. (1996) have investigated the history of
moisture condensers and have refused to accept the hypothesis
concerning the Crimean installations: "The Tepe-Oba mountain is
fissured by the remains of a sophisticated system of ancient water
supply and tubes can be found within hundreds of meters of every
mound. Excavations of more than 80 mounds, however, did not reveal
any signs of a hydraulic system. On the
 contrary, tombs were in each of them. Where water still comes out of the
  broken water supply, it contains dissolved minerals and thus does not
  come out of condensers, because condensed water is almost distilled.
  Moreover, the dry remains of the ancient water tubes are covered (inside)
  by a thick layer of mineral deposits. Hence it is thought that there were
  no ancient dew condensers in the surroundings of Feodosia".
 "A model for calculating condensation rates on real dew condensers" was
proposed. This model of water condenser is considered by authors as a
 model of Zibold type condenser and a scientific base for the conclusions
 of the paper.

 We may give a reason that contradicts this assertion. First of all, the
 numerical simulation of the diurnal cycle of this model implies condensation of
 the water in the nighttime. "The condensation starts 4 h after the sunset"
 and "stops at about 09.00" (Nikolayev et al., 1996, p. 27).
  The real situation for Zibold type condensers is quite another.
  The sources concerning this kind of condenser discuss about a day condensation
 (Jumikis, 1965).
 The large stone hills are now destroyed but one can find even now  in
Crimean forests these types of such "fountains" (Anonymous, 1935a) (see the
 illustration of the installation in Jumikis (1965) as well).
   One of the authors had observed such a system some years ago. The water had
 leaked from the pipe going out from the lower part of the construction.
It was the daytime between 10.00 and 12.00 and some litres of the water were
used for dinner. There were no doubts of the role of the pipe and of the
installation.

Some installations analogous to the Crimean stone hills were constructed
in southern France, in 1928-1931 by Chaptal.
Let us note a Chaptal observation that "...it was quite exceptional to find dew
formed in the well during the night") (Jumikis, 1965).

    Let us go to the tubes having "tracks of mineral deposits".
    As for the purity of water from the dew condenser, note that the quality
 of condensed water was studied by Schemenauer and Cereceda (1992) and
 they establish that it could be used for drinking, but not "almost distilled".
Real condensed water contains minerals. A stream of condensed
  water will produce over centuries "a thick layer of mineral deposits".
  Therefore, the existence of a layer of mineral deposits does not
  contradict but supports the existence of a dew collector.

 The "excavations of more than 80 mounds did not reveal any signs of a
 hydraulic system". Truly, tile pipes and channels were situated outside
 the mounds.

  As for tombs found in stone heaps, we find no contradiction
 in this situation. Crimea had a large population for many centuries.
 It is difficult to imagine any place in the coastal part of Crimea without
tombs.

  The model of Nikolayev et al. (1996)
 corresponds more to the type of water condenser studied by Nilsson (1996)
(see also (Vargas et al., 1998)) where the night condensation
and cooling of the condensation surface by help of irradiation were considered
and some real results were obtained.

The non-trivial part of each study is the estimation of the losses
of the real process. For instance, the losses of unknown nature in
the  Schemenauer (1991) experiment were in the range 70\%.
 Therefore, let us evaluate the results
obtained  in the numerical experiment  (NIkolayev et al., 1996)
 on the base of  real data (Nilsson, 1996).
The values 221, 155, 61 litres per night were obtained  in the numerical
 experiment for Zibold condenser with
$S=854 m^2$.  Thus we have $0.25$, $0.18$, $0.07$ $l/m^2$, correspondingly.
 The maximum single night collection of Nilsson installation was $0.24$ $l/m^2$.
So there exists a good correspondence between two maximal values,
 but the average night dew collection of Nilsson installation
 in the dry months was only in the range 0.04 - 0.03 $l/m^2$.
 Therefore we can take into account the possible losses and estimate
the night output of Zibold condenser in the range of
 some dozens of litres. It does not correspond the reported data
 300-360 l  (NIkolayev et al., 1996) of the diurnal output of Zibold condenser.
We can suppose that the Zibold type condensers
 produced the water mainly in the daytime and the model of
 Nikolayev, Beysens et al. (1996) explains only the night
 output of the condenser, but does not explain the whole output.

The negative conclusion of the role of the Crimea stone heaps as water
 collectors seems therefore too hasty and unfounded.

\section{Traditional condensation model}
The traditional model of a functioning Zibold type condenser was proposed by
 Chaptal (1932) and Knapen (1928) (see Jumikis (1965) too). The model is based
on the results of the Zibold and Chaptal experiments.

With the decrease in air temperature at night the chilled air (now heavier than
the warm air in the inner part of the stone heap) drives out the warm air that
was
accumulated in the system during the day. The cool temperature is transmitted
by conduction and heat exchange throughout the stone heap. The wind accelerates
the chilling of the stones. As a result, the temperature of the interior of the
system reaches the temperature of the night air. The surface of the stones
becomes chilled and is thus in a condition to condense the warm, damp air
for a certain period of time.
 During the day the warm air, more or less saturated with water vapor, enters
the condensation surface and contracts. The air is gradually chilled until the
vapor
 reaches the dew point. Part of the vapor condenses on the surfaces of the
stones. After some time, the temperature  between the inside and
outside will reach equilibrium and condensation stops. The
following night, the process starts all over again.

The amount of condensation evidently depends upon the
temperature difference between the exterior and interior of the system, upon
the degree of saturation of the air and upon the properties of the condenser
 surface. The warmest day gave the largest quantity of captured water.
 There should be sufficiently effective renewal of the air in the
 interior in the night and in the day, but in the day the ventilation
should not be too high and turbulent (or else evaporation would consume
the aqueous deposit immediately as it forms). The condensation is the most
sensitive part of the process.
The permeability of the stone heap is satisfactory due to the size of
stones.

The traditional model of a functioning  Zibold type condenser and
 the model of  Nikolayev et al. (1996) differ in consideration of the working
layer
of the condenser. The yield of the condenser depends linearly on the surface
area
 of condensation (Nilsson, 1996), and the surface area depends on the depth
 of the working layer as a quadratic function.

A narrow working layer of Zibold installation of depth 0.3 m is considered
 in the last model. The working layer in the traditional model is essentially
greater.
 If we suppose that  effective ventilation can be expected up to
 3 m,  then it gives us 100 times greater upper bound on the yield
 of the Zibold type condenser.
 Hence, the surface of condensation does not influence the upper bound
 for the yield of the Zibold system in the traditional model. We must consider
 more essential restrictions on the productivity  such as heat
 capacity of the active layer. The diurnal output of the Zibold type condenser
in the traditional model therefore is measured as l/(grade $m^3$) (or kg/(grade
$m^3$)).

Let us use the following notation:
 specific heat capacity of the stones $c_s= 1090$ $J kg^{-!} K$,
   latent heat of condensation $L = $ 2 260 000 $J kg ^{-1}$  (Nikolayev et
al., 1996),
  density of the stones $d_s$ is 2500 kg $m^{-3}$ - 2700 kg $m^{-3}$
 (Nikolayev et al., 1996), (Alexeyev et al., 1998)l.

 For specific volume $m_{w}$ of the condensed water during one day
per one grade we have

\begin{equation}
 m_{w} ={K_s c_s d_s \over  L} \label{m}
\end{equation}

where $K_s$ denotes the ratio of volumes of the stones of the heap
  and the heap.
The heating of the air is not considered here for the sake of
simplicity. $K_s=0.5$ (Nikolayev et al., 1996) or $0.7$ (Alexeyev
et al. 1998). Because the difference is essential, let us carry
the necessary calculations.

We begin from the case of low density for to find a lower bound on
the parameter. Let us consider every stone as ellipsoid with axes
$a,b,c$ that is inscribed in regular parallelepiped of size 2a X
2b X 2c. Suppose that the hill consists of such parallelepipeds
and distinct parallelepipeds have no intersection.

 The volume of ellipsoid is equal to
${4\over 3} \pi abc$ (Beyer, 1991),
 the volume of the
parallelepiped is $8abc$. So for the desired ratio
in the case of low density we have

\centerline{$K_s = {4\over 3}{ \pi abc \over 8abc} \simeq 0.5236$}

For the case of high density and for the upper bound
on $K_s$ let us consider the stones as spheres
with radius $R$. Suppose that the centers of three neighbor
spheres form a
 regular triangle with side $2R$ and the centers of
 four neighbor spheres form a regular triangular pyramid
with the same edge $2R$.
The attitude of the regular triangle is $R\sqrt{3}$,
The attitude of the regular triangular pyramid is

\centerline{$\sqrt( (R\sqrt 3 )^2 - ({R\sqrt 3 \over 3})^2) = 2R\sqrt{2 \over
3}$}

So for the desired ratio
in the case of high density we have

\centerline{$K_s = {4\over 3}{\pi R^3 \over 8R^3} {2R \over R\sqrt{3}} {2R \over
2R\sqrt{2 \over 3}} \simeq 0.7405$}

The real experiment with beach pebbles gives us some values from
0.56 to 0.62. So let us accept the average value of these
theoretical and practical results: $K_s=0.6$.

From ($\ref{m}$) for specific diurnal output $m_w$ we have

\begin{equation}
m_w =  0.72 {kg \over m^3} - 0.78 {kg \over m^3}           \label{mw}
\end{equation}

 per grade without consideration of losses.

(Let us remember that the real values obtained in Zibold and Chaptal
installations lay between 0.31  and 0.05 kg/$m^3$.
 In the fog collector installations (Schemenauer et.al,1989),
 the real deposit was 2.9 times less than the theoretical but here the gap
is greater).

Let $T_s$ be temperature of stones in hill, $T_a$ is the temperature of
air, $T_d$ denotes the temperature of dew point. Suppose $T_a>T_s$, $T_d>T_s$.
 Let $V$ be volume of active layer of stone hill. The diurnal mass of
 condensed water may be presented in the following form:

\begin{equation}
 m_d ={K_dV c_s d_s \over L}( min(T_a,T_d) -T_s)=  m_wV \Delta T
\end{equation}

The average volume $V$ of the mound was about 1664 $m^3$ (Zibold, 1905)).
 So, the stone heap can yield a maximum of 1200 - 1300 litres per grade every
day.
 The difference between the air temperature and the temperature of stones during
 the day is not greater than the difference between the day and night
temperatures of the air.
If we suppose that the last is about 5-10 grades and the air is saturated
 ($5 < \Delta T <10$) then
 we obtain 6,000-13,000 litres as an upper bound of diurnal output for the
stone heap. It is possible in an ideal situation when the day air temperature is
 equal to that of the dew point, the permeability of the stone heap is optimal,
the
 evaporation is minimal and the drain inside the stone heap is satisfactory.
The real yield was essentially less because of losses. It corresponds with the
real data from Feodosia (Zibold, 1905) and the estimation found above on the
base of the data (5,100  - 11,400 litres).

 The Zibold hopes were more optimistic - 55,000 liters per day from one stone
heap (Jumikis, 1965). Therefore, his real results imply deep disappointment.
The value obtained in Zibold installations is about 0.3 l/$m^3$
and in Chaptal installations - 0.05- 0.2 l/$m^3$  (Jumikis, 1965).
The daily output per grade was essentially less.
So, these installations could not be considered as completely perfect.

Some modification of traditional model will be considered below.

\section{The role of air draught in day condensation}

   One of the most important features of the Crimean dew collector is a crater
 in the central part of the stone heap
(Alexeyev et al., 1998), (Nikolayev et al., 1996), (Zibold, 1905). In the day
time the sun heats the stones and they heat the air in the crater.
 For to estimate the nascent temperature gradient
let us consider the axial section of the inner surface
of the crater as a non-singular curve of a second
degree (hyperbola, parabola or ellipse).
The horizontal section of the inner surface
of the crater will be considered as a circle.
Therefore the inner surface of the crater
may be considered as a surface of rotation
with focus on the vertical axis of symmetry
of the surface. Suppose also that the surface
is specular and gray.

\begin{picture}(100,135)
\end{picture}
\begin{picture}(180,135)

\put(100,70){\oval(80,80)[b]} \put(100,70){\oval(70,70)[b]}
\put(100,70){\oval(60,60)[b]} \put(100,70){\oval(50,50)[b]}
\put(100,70){\oval(40,40)[b]} \put(100,70){\oval(30,30)[b]}
\put(100,70){\oval(20,20)[b]} \put(100,70){\circle{8}}
\put(106,74){$focus$}

 \put(60,70){\vector(1,0){27}}
 \put(60,60){\vector(4,1){28}}
 \put(60,50){\vector(2,1){29}}
 \put(61,40){\vector(3,2){31}}
 \put(66,34){\vector(1,1){27}}
 \put(73,30){\vector(2,3){20}}
 \put(80,30){\vector(1,2){13.8}}
 \put(91,30){\vector(1,4){6.9}}
 \put(100,30){\vector(0,1){27}}
 \put(109,30){\vector(-1,4){6.9}}
 \put(120,30){\vector(-1,2){13.8}}
\put(127,30){\vector(-2,3){20}}
 \put(135,35){\vector(-1,1){27}}


 \put(80,123){\line(4,-1){64}}
 \put(81,121){\vector(-1,-4){3}}
 \put(94,117){\vector(-1,-4){3}}
 \put(110,113){\vector(-1,-4){3}}
 \put(126,109){\vector(-1,-4){3}}
 \put(140,105){\vector(-1,-4){3}}
 \put(114,117){$beam$ $radiation$}

   \end{picture}

 The distance $d$ between the focus and the
vertex of the surface (the bottom point of the
crater) can be calculated in the standard way
from the equation of the axial section of the
surface

\centerline{$Ax^2 +2Bxy +Cy^2 +Dx +Ey+G=0$}

One of the coefficients is here free and
five others can be calculated by considering
five distinct points on the section. Moreover,
due to symmetry,  we need only coordinates of
two points regarding to the bottom vertex.
The equation may be transformed to the canonical
form  (Berger, 1987)  or some other manual in analytical
geometry), then we calculate the desired focal
distance $d$ of the curve. From scanty date
we have in the case of Zibold installation,
the focal distance can be estimated very roughly:
2 - 4 $m$.

The solar radiation received at the inner surface
of the crater is partially absorbed by the surface,
 partially reflected and emitted.

 \centerline{$ e=e_{absorb} + e_{refl}$}

Relationships among absorption capacity, emittance and reflectance
depend upon the material of the surface. For our case we have the
following estimation (Duffy et al., 1974)
\begin{equation}  \label{er}
  0.1e< e_{refl} < 0.5e
\end{equation}
The energy absorbed by unit of surface is equal to

\centerline{$e_{absorb}\cos(\alpha)$}

where the angle $\alpha$ is the angle between the normal vector of
the surface and angle of incidence of ray. Therefore the maximum
of absorbed energy received at the point where both directions
coincide. This point has changed his position during the day in
neighborhood of the vertex of the surface.

Let us now consider the distribution of energy in the
crater and higher. This distribution depends upon
direction of reflection of sun rays and of absorbing
capacity of air, dust particles in the air and water
vapor.
Due to geometrical properties of the inner
surface of the crater,
the reflected rays are directed to the focus of the
surface.

 Let us consider a sequence of surfaces $S_i$, where
$S_0$ is the inner surface of crater, every $S_i$ has
the same form, the same vertical axis and the same low
focus. Suppose that for focal
distance $d_i$ of $S_i$

\begin{equation}
d_{i+1} = d_i - \Delta d                \label {dl}
\end{equation}


Every layer between two neighboring surfaces absorbs the
same amount of energy, whence for every two equal volumes
of air $V_1$ and $V_2$ with absorbed energy $e(V_1)$,
$e(V_2)$ and layer focal distances $d(V_1)$,
$d(V_2)$ we have

\begin{equation}
 {e(V_1) \over e(V_2)} = {d^2(V_2) \over d^2(V_1)}                         \label {ed}
\end{equation}

Therefore the maximum of absorbed energy is concentrated in
little lens containing the point of focus and is growing
with approaching to this point. According the Stefan-Boltzmann
formula (Duffy et al., 1974)

\begin{equation}
e = \sigma T^4                   \label {t4}
\end{equation}

where the Stefan-Boltzmann constant $\sigma$ is equal to
 $5.6697 \circ  10^{-8}W/m^2 K$,
T is the absolute temperature.

Let us consider two equal volumes of air $V_1$ and $V_2$ in
above-mentioned  neighboring layers of air with absolute
temperatures $T$ and $T+\Delta T$, absorbed energy $e$ and
$e+\Delta e$,  layer focal distances  $d$ and $d-\Delta d$,
correspondingly. ($\ref {t4}$), ($\ref {dl}$)  and ($\ref {ed}$)
imply in an area where the beam radiation is essentially less than
the reflected radiation

\centerline{${(T+\Delta T)^4 \over T^4} = {e +\Delta e \over e} = { (d-\Delta d)^2  \over d^2}$}

Therefore $(1 + {2 \Delta T \over T} + ({ \Delta T \over T})^2)=( 1 - { \Delta d \over d})$,
whence

\centerline{${2\Delta T \over T} \approx {-\Delta d \over d}$}

The integration  of corresponding differential equation gives us the connection
between the absolute temperature $T$ and  focal distance of the layer $d$

\centerline{$T \sqrt d  =C$}

where $0 < < d < < d_0$ ,  the constant $C$ depends on absorptive
properties of the air. In view of ($\ref {er}$) and ($\ref {ed}$),
the equation predominates for $d$ less than $0.3d_0 - 0.7d_0$,
scattering implies $0 < < d$.

This establishes a temperature gradient in the vertical direction.
More precisely, it is the direction from the point of maximal
radiation to the point of focus. Let us notice that the wind can
dissipate the considered lens of hot air, but only the strong wind
can destroy it completely and dispose the temperature gradient.

The temperature gradient creates draught in vertical direction
stimulating the heat exchange in the stone hill.
Thus we have continuous, moderate and stable
 draught some hours after sunrise. The air draught involves the surface of the
 stones in the inner part of the mound in the process of condensation.
 Two processes - condensation and heat exchange - are working together and the
 role of each process depends upon the humidity of the air. Every point between
 the  outer surface of the heap and the inner surface of the crater participate
in both processes. If the moist air from the sea reaches the stone hill with the
 beginning of draught (it depends on wind and distance from the sea)
 then the condensation predominates. In the case of low
 humidity the heat exchange prevails and the condensation surface remains dry.
 Therefore, the condensation is the most sensitive and unstable part of the
 process. Maximal values of the daily output of the process are given by
the formula ($\ref {mw}$): about 0.75  kg $m^{-3}$ per grade.

There is another situation at night. The stone heap may be considered as
 a hot island in the cold night air because the cooling of the air is more
 rapid than that of the stones. This creates an uprising
 stream of air in the heap. The process of ventilation
ends when the temperature of the stones achieves that of the air.
   Both processes of day condensation and night cooling are not independent.
The day condensation stimulates heating of the inner part of the stone
 hill and as a result the night draught. The night cooling of the surface
 of the stones promotes the day condensation on the surface.

\begin{thebibliography}{99}

\bibitem{Al} Alexeyev V.V., Berezkin M. Ju., (1998). Fresh water from air,
{\it Priroda}, {\bf 6}: 90-96.
 \bibitem{AC} Anonymous, (1935a). Aerial wells and soil moisture.
{\it  Trop. Agr}., 13(2):34.
 \bibitem{An} Anonymous, (1935). Stenograph of the proceedings of the 1-st conf.
 on the condens. of atm. vapor. Moscow,Leningrad, Manuscript kept in
 Feodosian museum (Russian). (See too Rapport CEA-Saclay, DIST,
No. 95002495, 1995, France)
  \bibitem{Ah} Ashbel D., (1935). On the importance of dew in Palestine, {\it  J.
of Palestine Oriental Soc}., {\bf 16}: 316-321.
\bibitem{As} Ashbel D., (1949). Frequency and distribution of dew in Palestine,
{\it  Geographical Review}, {\bf 39}: 291-297.
\bibitem{Bg} Berger M., (1987). {\it Geometry}, II, Berlin,Springer, 402 pp.
\bibitem{By} Beyer W. H., (1991). {\it Standard mathematical tables and formulas},
West Palm Beach, Fla, CRC Press, 982  pp.
 \bibitem{Be} Beysens D., (1991). Steuer A., Guenoun P., Fritter F., Knobler
C.M., How does dew form?, {\it Phase transition}, {\bf31}: 219-246.
\bibitem{Br} Broza M., (1979). Dew, fog and hydroscopic food as a source
of water for desert arthropods, {\it  J. Arid Env.},2(1), 43-49.
 \bibitem{Ch} Chaptal L, (1932). La capitation de la vapeur d'eau atmospherique,
{\it  La Nature}, {\bf 60}, 2893: 449-454.
\bibitem{DB} Duffy J. A., Beckman W.A., (1974). {\it Solar energy thermal processes}.
NY, Wiley and Sons, 386 pp.
\bibitem{Ju} Jumikis A.R., (1965). Aerial wells: secondary sources of water, {\it
Soil SCi}., {\bf 100}: 83-95.
\bibitem{Le} Levi M., (1967). Fog in Israel, {\it  Israel J. of Earth Sci.}, {\bf
16}: 7-21.
\bibitem{Kn} Knapen A., (1928). Memoires sur le puits aerien. {\it  Bull. Soc.
Ing. Civils France}, 139-140.
\bibitem{KT} Kogan B., Trahtman A., (1999). Atmospheric precipitation as water
resource, {\it Env. challenge for the next millennium}, Jerusalem, 141.
\bibitem{NB} Nikolayev V.S., Beysens D., Gioda A., Milmouk I., Katiushin E.,
 Morel J.P., (1996). Water recovery from dew, {\it  J. of Hydrology}, {\bf 182}:
19-35.
\bibitem{Ni} Nilsson T., (1996). Initial experiments on dew collection in
 Sweden and Tanzania, {\it  Sol. Energy Materials and Sol. Cells}, {\bf 40}: 23-32.
 \bibitem{SA} Schemenauer R.S., Cereceda P., (1991). Fog water collection
in arid coastal location, {\it Ambio}, {\bf 20}: 303-308.
 \bibitem{SC}R.S. Schemenauer R.S., Cereceda P., (1992). The quality of fog water
 collected for domestic and agricultural use in Chile, {\it J. of Appl.
Meteorology}, {\bf 31}:, 275-290.
 \bibitem{SJ} Schemenauer R.S., Joe P.L., (1989). The collection efficiency of a
 massive fog collector, {\it Atm. Res.}, {\bf 24}: 53-69.
  \bibitem{Va} Vargas W.E., Lushuku E.M., G.A. Niklasson G.A., Nilsson T.M.J.,
 (1998). Light scattering coatings: theory and solar applications, {\it Sol. Energy
       Math. and Sol.  Cells}, {\bf 54}: 343-350.
  \bibitem{Zi} Zibold F.I., (1905). The role of the underground dew in water
 supply of Feodosia, {\it Trudy opytnyh lesnichestv, No. III, Manuscript kept in
 Feodosian museum} (Russian). (See too Rapport CEA-Saclay, DIST,
No. 95002495, 1995, France)
  \end{thebibliography}

 \end{document}