characteristic equation of a passive solar still

Desalination 245 (2009) 246–265

Characteristic equation of a passive solar still

Rahul Dev*, G.N. TiwariCentre for Energy Studies, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

Tel. +91 (99) 6834 4470; email: [email protected]

Received 29 January 2008; Accepted 6 July 2008

Abstract

The characteristic equation has been used for experimental results of a passive solar still to generate linear andnon-linear characteristic equations for winter and summer conditions. The different angles of inclination ofcondensing cover (15°, 30°, 45°) have been chosen for winter and summer conditions both. It has been observed thatthe passive solar still with inclination of 45° gives better performance both in winter and summer respectively.Different water depths (0.04, 0.08, 0.12, and 0.16 m) have also been taken for solar still with 30° inclination anglefor summer weather condition. Comparisons of instantaneous gain and loss efficiencies at 0.01 and 0.04 m waterdepths for a 15° inclination angle have also been made to show the effect of water depth on the performance of solarstills. It was found that a lower water depth gives better efficiency, which is in agreement with many investigators.The instantaneous gain and loss efficiency curves have been simultaneously analyzed to give a better understandingof the performance of solar stills. The proposed method will be used to standardize the design and operationalparameter of a passive solar still, i.e. angle of inclination and water depth for highest yield for a given climaticcondition.

Keywords: Solar energy; Solar distillation; Passive solar still; Instantaneous efficiency

1. Introduction

The availability of fresh water is decreasingfrom the natural resources due to water pollutionand the receding level of underground water allover the world. Solar distillation is an economi-cal, effective and environmentally friendlymethod over all the conventional distillation

*Corresponding author.

methods (which are energy and cost intensivetechniques) for getting the pure water through theuse of solar energy.

The performance of solar distillation systemsdepends on climatic parameters such as ambienttemperature, solar radiation intensity and weathercondition etc., design parameters like inclinationangle and operational parameters like orientationof solar still and brine water depth [1]. It has been

doi:10.1016/j.desal.200 .0 .08 7 110011-9164/09/$– See front matter © 2008 Elsevier B.V. All rights reserved.

R. Dev, G.N. Tiwari / Desalination 245 (2009) 246–265 247

found that with the increment in solar radiationintensity [2] and ambient temperature, the pro-ductivity of the solar still increases [2,3]. Manyinvestigators proved that increasing the brinewater depth in solar still basis reduces daily pro-ductivity [4–6].

An extensive review of various types of solarstills was updated by Tiwari [7]. The performanceevaluation of passive solar still in terms ofyield/m2 has been studied by Tiwari and Tiwari[8]. It was concluded that the annual yield ismaximum for 30E inclination of the condensingglass cover on the basis of the monthly yield.However, it is important to mention that thesystem should be designed by using the conceptof efficiency. The first attempt to characterize theperformance of a solar still was made by Tamimi[9]. The broad domain of characteristic curves ofsolar stills between an ideal condition and theworst condition have been analyzed. The conceptof instantaneous thermal efficiency to charac-terize designs of solar still including trapezoidalcavity system was introduced by Tiwari and Noor[10]. The first attempt to plot the characteristiccurve for one-sided vertical solar stills was madeby Boukar and Harmim [11].

Tsilingiris [12] has studied the influence ofbinary mixture thermo-physical properties in theanalysis of heat and mass transfer processes insolar distillation systems. Various equations forheat transfer coefficient and other properties ofwater vapor mixture have been developed. It wasfound that variation in values of heat transfercoefficients is almost constant till 60°C; there-after at higher temperature these values changeconsiderably. A group of improved heat and masstransfer correlations in basin type solar stills hasbeen developed by Hongfei et. al. [13]. Torchia-Núñez et. al. studied the exergetic analysis of apassive solar still [14]. It was found that for thesame exergy input, a collector (basin), brine andsolar still have exergy efficiencies of 12.9%, 6%and 5% respectively. An experimental validationof thermal modeling of solar stills on the basis of

heat transfer coefficients for summer and winterconditions has been presented by Shukla andSorayan [15]. A study on water evaporation areafor increasing the yield in a solar still was carriedout by Kwatra [16]. In this study a relationshipbetween enlarged evaporation area and distillateby computer simulation was obtained. It is shownthat a gain of 19.6% in a yield when the evapo-ration area was quadrupled. For an asymptotic(infinite) area, a 30.2% increase in gain wasfound. The thermal dynamic effects of materialused for manufacturing solar stills in combinationwith weather parameter has been studied by Portaet al. [17].

In the present work, the effect of inclinationangle on instantaneous efficiency of single slopesolar still for summer and winter conditions ofNew Delhi has been analyzed. The effect of waterdepth on instantaneous efficiency has also beenstudied for summer conditions. The inclinationangles of condensing cover were taken as 15°,30° and 45° for a water depth of 0.04 m. For a30° inclination angle, water depth was taken as0.04, 0.08, 0.12, and 0.16 m alternatively. Andfor a 15° inclination angle, water depth was takenat 0.01 and 0.04 m. Experimental data collectedfor June 2004, November 2004 and April 2005are used for the analysis [18].

The aim of the present work is to get a betterunderstanding of the performance of passive solarstills through characteristic curves for summerand winter weather conditions.

2. Experimental set-up and observations

A photograph of the experimental set-up of asingle slope passive solar still with differentinclination angles of the condensing cover (15°,30°, 30°, 45°) is shown in Fig.1(a). The sche-matic diagram of the passive solar still is alsoshown in Fig. 1(b). The set-up is installed at SolarEnergy Park, Indian Institute of Technology, NewDelhi (latitude 28°35N N, longitude 77°12N E,

R. Dev, G.N. Tiwari / Desalination 245 (2009) 246–265248

Fig. 1a. Photograph of experimental set-up of solar stillsinclined at 15°, 30°, 30°, 45° from left to right.

Fig. 1b. Schematic diagram of single slope passive solarstill.

altitude 216 m from mean sea level). The area ofeach solar still is 1 m2 (length = 1 m, width =1 m), facing towards the south. The height of thelower vertical sides of the solar stills was taken at0.15 m for the inclination angles (15°, 30°, 45°)and the heights of the higher sides were taken as0.42, 0.73, and 1.15 m with a water depth of0.04 m. Another solar still with an inclinationangle of 30° with a height of lower and highervertical sides — 0.25 m and 0.83 m, respectively— was taken for a comparison in performance atvarious water depths (0.08, 0.12, and 0.16 m).

The experiments were carried out on typicaldays in June 2004, November 2004 and April2005. The height of the lower side was increasedto avoid the spilling of basin water into thedistillate channel and to prevent the contact ofdistillate channel with the glass cover.

The body of all solar stills is made of glassreinforced plastic (GRP) 5 mm thick. The glass isused as a condensing cover 4 mm thick. To

increase the absorption of solar radiation, thebasin liner was blackened. The glass cover wasfixed using a rubber gasket on the top of thevertical walls of each side. A channel was madeadjacent to the smaller wall to collect the distilledwater with a plastic pipe to drain it into anexternal measuring jar [8].

The hourly variation of solar intensity [I(t)],ambient temperature (Ta), water temperature (Tw),inner–outer glass temperature (Tci, Tco) and yield(mew) are given in Table 1(a–j) [18]. To measurethe global solar intensity a calibrated solarimeter(CEL, India) of least count 2 m W/cm2 was used,given in Table 1(a–j) after converting into W/m2.A thermometer having least count 0.1°C andthermocouples, calibrated by a Zeal thermometer,were utilized for measuring the various tempera-tures. The thermometer was used to measure theambient temperature. Thermocouples were usedto measure the temperature of water surface andthe inner–outer surface of the glass cover. Ameasuring jar with least count 1 ml was utilizedfor measuring the yield.

3. Mathematical model

The following assumptions have been made towrite the energy balance equations:C The solar distillation unit is vapor-leakage

proof and is in a quasi-steady state.C The heat capacities of glass and basin material

are negligible.C Temperature-dependent heat transfer coeffi-

cients have been considered.

The following are the basic energy equationsfor each component:C Energy balance for inner surface of glass

cover, Tiwari and Tiwari [8]:

(1) 1g

g w ci ci cog

KI t h T T T T

L


C Energy balance for outer surface of glasscover:

(2) 2g

ci co co ag

KT T h T T

L

C Energy balance for water mass:

(3)

3

1d

d

w b w

ww w w ci

I t h T T

TM C h T T

t

C Energy balance for basin liner:

(4)

3b b b w b

b b a b s

I t A h T T A

h T T A A

Simplifying the above equations, we get anexpression for water temperature [8]:

(5)

1 at atw wo

f tT e T e

a

where ; ;e L a L

w w w w

I t U T Uf t a

M C M C

; ; L t bU U U 1

1

cg at

cg a

hUU

h U

3

3

bb

b

h hU

h h

and ; ;1 2e w b gH H 31

3 b

hH

h h

; ; 12

1 cg a

hH

h U

gg g

b

A

A 1 s

b bb

Ah h

A

; gcg a cg a

b

AU U

A

2

2

g g

cg a

g g

K L hU

K L h

Eq. (5) can be rearranged as follows:

(6) 1 at atew a wo

L

T I t T e T eU

After simplification of Eqs. (1) and (2), the innersurface glass temperature can be written as

(7) 1

1

w cg a a gci

cg a

h T U T I tT

h U

Assuming is negligible; Eq. (7) will be of thegfollowing form:

(8)1

1

w cg a aci

cg a

h T U TT

h U

The rate of evaporative heat transfer is given by

1

ew cg aew ew w ci w a

cg a

h Uq h T T T T

h U

(9)

From Eq. (6) one gets

(10)

1

1.

. 1 .

ew cg aew

L cg a

at ate L w a

h Uq

U h U

e I t U e T T

The instantaneous efficiency which is the ratioof the rate of evaporative heat transfer to the rateof solar radiation falling on the system will begiven by

(11)

1

1.

. 1

ew cg aewi

L cg a

w aat ate L

h Uq

I t U h U

T Te U e

I t


which can be further written as

. 1 w aat ati e

T TF e F Ue

I t

(12)

where 1

1. ew cg a

L cg a

h UF

U h U

or

(13)

w ai effeff

T TF F U

I t

where ; . 1 ateeff

e ateff LU U e

Eq. (13) is similar to the characteristic equation ofa flat-plate collector (FPC) given by Hottel–Whillier–Bliss for the forced mode of operation[19].

(14)

fi ai R Leff

T TF U

I t

It is to be noted that(1) the upward heat loss should be minimum

for FPC (the slope of curve is negative i.e.,!FRUL) where

1 exp

R f C L

C L f

F mC A U

A U F mC

FR is the collector efficiency factor[20], and(2) the upward heat loss should be maximum

for solar still for the best performance [slope ofcurve is positive, i.e., FNUeff from Eq. (13)].

The above justification can be observed indeveloped characteristic equations. Further, theinstantaneous loss efficiency can be obtained as

(15)

w w w woiL

M C T T

I t

With the help of Eq.(6), one gets

(16)

1 atw w

iLL

w ae L

M C e

U

T TU

I t

Eq. (15) can be further written as

(17)

w a

iL e L

T TF U

I t

where 1 at

w w

L

M C eF

U

Eq. (17) is similar to the characteristic equa-tion of FPC, i.e., Eq. (14). For solar stills, theinstantaneous loss efficiency should be minimalto make higher instantaneous gain efficiency asgiven in Eq. (13). The sum of instantaneous gainand loss efficiencies (η = ηi + ηiL) cannot be morethan the efficiency of an ideal solar still, i.e.,60%. In ideal solar stills, the instantaneous lossefficiency is minimal (for zero depth of watermass) as analyzed by Cooper [21].

4. Results and discussions

The climatic data (Table 1a–j) and designparameters (Table 2) give the data to evaluate ηi

and (Tw!Ta)/I(t) given in Table 1(a–j). The tableshows some unrealistic results of energy effi-ciencies (instantaneous gain/loss efficiencies) i.e.,either ηi, ηil >1 or ηi, ηil <0, which is not possible.Hence these results are not taken into considera-tion and are termed as ‘unrealistic results’. Thishappens because of two reasons:

1. Low solar intensity in the mornings andevenings.


Table 1aHourly observation taken for a passive solar still with an inclination angle of 15° and water depth = 0.04 m on a typicalday in June, 2004 [18]

Time(h)

I(t)(W/m2)

Ta

(EC)Tw

(EC)Tci

(EC)Tco

(EC)mew

(kg m2/h)(Tw!Ta)/I(t)(EC/W m2)

ηi Tw!Two

(EC)ηiL

7–8 AM 120 25.50 28.09 27.85 27.72 0.0115 0.022 0.064 2.59 —a

8–9 AM 260 27.10 30.74 30.40 30.32 0.0305 0.014 0.078 2.00 0.3599–10 AM 440 27.50 34.33 34.19 34.44 0.032 0.016 0.048 4.24 0.45010–11 AM 620 29.50 41.06 39.11 39.72 0.036 0.019 0.039 6.73 0.50711–12 PM 740 31.50 48.92 46.17 45.36 0.0745 0.024 0.067 7.86 0.49612–13 PM 740 32.50 56.44 52.14 51.04 0.1465 0.032 0.131 7.52 0.47413–14 PM 720 33.50 61.62 55.05 55.22 0.209 0.039 0.193 5.18 0.33614–15 PM 690 34.50 62.79 54.74 54.54 0.254 0.041 0.244 1.17 0.07915–16 PM 540 35.00 62.29 51.63 50.88 0.262 0.051 0.322 !0.50 —16–17 PM 290 34.50 61.78 50.84 48.95 0.236 0.094 0.540 !0.51 —17–18 PM 110 33.50 60.23 51.11 48.85 0.2015 0.243 — !1.55 —

aUnrealistic results.

Table 1bHourly observation taken for a passive solar still with an inclination angle of 30° and water depth = 0.04 m on a typicalday in June, 2004 [18]

Time(h)

I(t)(W/m2)

Ta

(EC)Tw

(EC)Tci

(EC)Tco

(EC)mew

(kg m2/h)ηi (To!Ta)/I(t)

(EC/W m2)Tw!Two

(EC)ηiL

7–8 AM 120 25.50 27.57 29.00 28.22 0.0145 0.080 0.017 2.07 0.8058–9 AM 260 26.50 28.88 31.73 31.03 0.0335 0.086 0.009 1.31 0.2359–10 AM 440 27.50 32.46 35.84 35.28 0.0330 0.050 0.011 3.58 0.38010–11 AM 620 29.50 38.63 41.42 41.12 0.0300 0.032 0.015 6.17 0.46411–12 PM 740 31.00 46.00 47.02 46.23 0.0585 0.052 0.020 7.37 0.46512–13 PM 740 32.50 51.90 51.20 48.91 0.1225 0.110 0.026 5.90 0.37213–14 PM 720 33.50 56.16 53.55 49.22 0.1700 0.157 0.031 4.26 0.27614–15 PM 690 34.50 58.11 53.65 47.55 0.2055 0.198 0.034 1.95 0.13215–16 PM 540 35.00 56.95 51.39 43.73 0.2055 0.253 0.041 !1.16 —a

16–17 PM 290 34.50 55.64 49.16 41.53 0.1690 0.387 0.073 !1.31 —17–18 PM 110 33.50 54.22 47.65 43.16 0.1445 0.872 0.188 !1.42 —



Table 1cHourly observation taken for a passive solar still with an inclination angle of 45° and water depth = 0.04 m on a typicalday in June, 2004 [18]

Time(h)

I(t)(W/m2)

Ta

(EC)Tw

(EC)Tci

(EC)Tco

(EC)mew

(kg m2/h)ηi (Tw!Ta)/I(t)

(EC/W m2)Tw!Two

(EC)ηiL

7–8 AM 120 25.50 26.94 27.95 28.54 0.0160 0.089 0.012 1.44 0.5608–9 AM 260 26.50 28.13 31.07 31.65 0.0420 0.107 0.006 1.19 0.2149–10 AM 440 27.50 31.54 33.75 34.28 0.0460 0.069 0.009 3.41 0.36210–11 AM 620 29.50 38.29 37.13 37.13 0.0375 0.040 0.014 6.75 0.50811–12 PM 740 31.50 45.56 41.41 40.50 0.0735 0.066 0.019 7.27 0.45812–13 PM 740 32.50 51.58 46.17 43.17 0.1650 0.148 0.026 6.02 0.38013–14 PM 720 33.50 54.98 50.05 44.59 0.2280 0.210 0.030 3.40 0.22014–15 PM 690 34.50 55.29 49.80 43.15 0.2575 0.248 0.030 0.31 0.02115–16 PM 540 35.00 53.69 48.26 43.86 0.2425 0.298 0.035 !1.60 —a

16–17 PM 290 34.50 51.48 46.18 44.97 0.1900 0.435 0.059 !2.21 —17–18 PM 110 33.50 49.11 43.23 42.67 0.1485 0.142 0.896 !2.37 —


Table 1dHourly observation taken for a passive solar still with an inclination angle of 30° and water depth = 0.08 m on a typicalday in June, 2004 [18]

Time(h)

I(t)(W/m2)

Ta

(EC)Tw

(EC)Tci

(EC)Tco

(EC)mew


(EC/W m2)Tw!Two

(EC)ηiL

7–8 AM 160 25.50 34.63 33.91 33.37 0.019 0.077 0.057 9.13 —a

8–9 AM 310 26.50 35.09 36.88 36.63 0.035 0.075 0.028 0.46 0.1389–10 AM 490 27.50 38.50 43.99 43.92 0.023 0.031 0.022 3.41 0.65010–11 AM 620 29.50 42.57 49.14 49.15 0.010 0.010 0.021 4.07 0.61311–12 PM 700 31.00 47.78 54.70 55.00 0.008 0.008 0.024 5.21 0.69512–13 PM 730 32.50 51.03 56.95 56.95 0.031 0.028 0.025 3.25 0.41613–14 PM 710 33.50 53.30 56.08 54.74 0.062 0.058 0.028 2.27 0.29814–15 PM 660 34.50 56.96 54.81 51.97 0.087 0.087 0.034 3.66 0.51815–16 PM 530 35.00 58.86 54.14 48.17 0.117 0.146 0.045 1.90 0.33516–17 PM 330 34.50 58.61 53.07 45.77 0.141 0.284 0.073 !0.25 —17–18 PM 140 33.50 57.98 50.83 46.79 0.143 0.676 0.175 !0.63 —



Table 1eHourly observation taken for a passive solar still with an inclination angle of 30° and water depth = 0.12 m on a typicalday in June, 2004 [18]

Time(h)

I(t)(W/m2)

Ta

(EC)Tw

(EC)Tci

(EC)Tco

(EC)mew


(EC/W m2)Tw!Two

(EC)ηiL

7–8 AM 120 25.50 29.32 28.56 27.98 0.0145 0.080 0.032 3.82 —a

8–9 AM 260 26.50 29.27 30.91 30.40 0.0255 0.065 0.011 !0.05 —9–10 AM 440 27.50 30.35 35.06 34.63 0.0215 0.032 0.006 1.08 0.34410–11 AM 620 29.50 32.10 39.91 39.62 0.0165 0.018 0.004 1.75 0.39511–12 PM 740 31.50 34.99 44.46 44.60 0.0110 0.010 0.005 2.89 0.54712–13 PM 740 32.50 38.86 46.81 47.23 0.0150 0.013 0.009 3.87 0.73213–14 PM 720 33.50 42.06 47.84 46.91 0.0260 0.024 0.012 3.20 0.62214–15 PM 690 34.50 44.73 46.16 43.27 0.0415 0.040 0.015 2.67 0.54215–16 PM 540 35.00 47.06 44.94 39.59 0.0540 0.066 0.022 2.33 0.60416–17 PM 290 34.50 48.55 44.48 38.77 0.0660 0.151 0.048 1.49 0.71917–18 PM 110 33.50 49.58 42.48 39.53 0.0830 0.501 0.146 1.03 —


Table 1fHourly observation taken for a passive solar still with an inclination angle of 30° and water depth = 0.16 m on a typicalday in June, 2004 [18]

Time(h)

I(t)(W/m2)

Ta

(EC)Tw

(EC)Tci

(EC)Tco

(EC)mew


(EC/W m2)Tw!Two

(EC)ηiL

7–8 AM 170 26.50 23.96 26.81 26.18 0.0125 0.049 !0.015 !2.54 —a

8–9 AM 370 28.50 25.04 32.45 32.16 0.0200 0.036 !0.009 1.08 0.5459–10 AM 560 30.00 27.60 37.70 37.70 0.0105 0.012 !0.004 2.56 0.85310–11 AM 720 31.50 31.24 42.44 42.31 0.0060 0.006 0.000 3.64 0.94411–12 PM 830 32.50 35.11 46.02 45.56 0.0060 0.005 0.003 3.87 0.87012–13 PM 820 33.50 39.13 48.77 48.16 0.0070 0.006 0.007 4.02 0.91513–14 PM 780 34.50 42.08 50.24 49.47 0.0155 0.013 0.010 2.95 0.70614–15 PM 690 35.00 45.02 48.42 46.11 0.0285 0.027 0.015 2.94 0.79515–16 PM 410 35.00 47.59 44.44 40.08 0.0510 0.083 0.031 2.57 —16–17 PM 250 34.50 47.34 41.95 36.69 0.0700 0.186 0.051 !0.25 —17–18 PM 180 33.50 47.13 40.51 37.14 0.0730 0.269 0.076 !0.21 —



Table 1gHourly observation taken for a passive solar still with an inclination angle of 15° and water depth = 0.04 m on a typicalday in November, 2004 [18]

Time(h)

I(t)(W/m2)

Ta

(EC)Tw

(EC)mew


(EC/W m2)Tw!Two

(EC)ηiL

7–8 AM 70 16.0 8.0 0.008 0.076 !0.114 !8.0 —a

8–9 AM 230 18.0 12.0 0.019 0.055 !0.026 4.0 0.8129–10 AM 420 21.5 18.0 0.025 0.040 !0.008 6.9 0.66710–11 AM 495 24.4 27.0 0.020 0.027 0.005 9.0 0.84811–12 PM 470 27.0 37.5 0.020 0.028 0.022 10.5 —12–13 PM 405 28.6 43.0 0.050 0.082 0.036 5.5 0.63413–14 PM 330 28.5 45.5 0.100 0.201 0.052 2.5 0.35414–15 PM 165 26.9 49.0 0.140 0.563 0.134 3.5 —15–16 PM 70 25.7 48.0 0.150 — 0.319 !1.0 —16–17 PM 30 23.7 46.0 0.145 — 0.743 !2.0 —17–18 PM 0 20.5 43.5 0.135 — — !2.5 —


Table 1hHourly observation taken for a passive solar still with an inclination angle of 30° and water depth = 0.04 m on a typicalday in November, 2004 [18]

Time(h)

I(t)(W/m2)

Ta

(EC)Tw

(EC)mew


(EC/W m2)Tw!Two

(EC)ηiL

7–8 AM 70 16.0 10.0 0.009 0.085 !0.086 !6.0 —8–9 AM 230 18.0 12.5 0.025 0.072 !0.024 2.5 0.5079–10 AM 420 21.5 18.0 0.025 0.040 !0.008 5.5 0.61110–11 AM 495 24.4 25.0 0.010 0.013 0.001 7.0 0.66011–12 PM 470 27.0 32.5 0.008 0.011 0.012 7.5 0.74512–13 PM 405 28.6 38.0 0.030 0.049 0.023 5.5 0.63413–14 PM 330 28.5 43.0 0.065 0.131 0.044 5.0 0.70714–15 PM 165 26.9 44.0 0.100 0.402 0.104 1.0 0.28315–16 PM 70 25.7 43.0 0.112 —a 0.247 –1.0 —16–17 PM 30 23.7 41.0 0.110 — 0.577 –2.0 —17–18 PM 0 20.5 37.5 0.100 — — –3.5 —



Table 1iHourly observation taken for a passive solar still with an inclination angle of 45° and water depth = 0.04 m on a typicalday in November, 2004 [18]

Time(h)

I(t)(W/m2)

Ta

(EC)Tw

(EC)mew


(EC/W m2)Tw!Two

(EC)ηiL

7–8 AM 70 16 10 0.009 0.085 !0.086 !6.0 —8–9 AM 230 18 12.5 0.031 0.089 !0.024 2.5 0.5079–10 AM 420 21.5 18 0.04 0.063 !0.008 5.5 0.61110–11 AM 495 24.4 25 0.024 0.032 0.001 7.0 0.66011–12 PM 470 27 35 0.012 0.017 0.017 10.0 0.99312–13 PM 405 28.6 43 0.05 0.082 0.036 8.0 0.92213–14 PM 330 28.5 47 0.112 0.225 0.056 4.0 0.56614–15 PM 165 26.9 47 0.151 0.608 0.122 0.0 —15–16 PM 70 25.7 44 0.151 —a 0.261 !3.0 —16–17 PM 30 23.7 41 0.131 — 0.577 !3.0 —17–18 PM 0 20.5 37 0.113 — — !4.0 —


Table 1jHourly observation taken for a passive solar still with an inclination angle of 15° and water depth = 0.01 m on a typicalday in April, 2005

Time(h)

Ta

(EC)Tco

(EC)Tci

(EC)Tw

(EC)I (t)(W/m2)

mw

(kg m2/h)(Tw!Ta)/I(t) ηi Tw!Two ηiL

7 AM 20 22.1 22.3 21.5 0 0.00 —a — 1.50 —8 AM 24 24.4 25.3 24.4 120 0.01 0.003 0.055 2.90 0.2829 AM 29 31.4 26.7 34.0 440 0.01 0.011 0.015 9.60 0.25510 AM 32 45.2 30.5 48.1 600 0.07 0.027 0.077 14.10 0.27411 AM 34 52.7 35.8 56.0 700 0.17 0.031 0.161 7.90 0.13212 PM 37 59.6 41.7 63.0 780 0.28 0.033 0.238 7.00 0.10513 PM 39 62.5 48.7 64.8 760 0.34 0.049 0.297 1.80 0.02814 PM 39 57.5 47.1 68.2 600 0.36 0.057 0.398 3.40 0.06615 PM 38 58.3 46.1 68.8 540 0.27 0.0570 0.332 0.60 0.01316 PM 38 53.9 43.5 62.4 300 0.24 0.081 0.531 !6.40 —17 PM 36 48.9 41.1 56.1 140 0.17 0.144 0.806 !6.30 —


Table 2Design parameters of the solar still

Sr. no. Parameters Values

1. Ag 1.04 m2, 1.16 m2, 1.42 m2 for 15°, 30°, 45° respectively

2. As 0.16 m2, 0.32 m2, 0.48 m2,0.64 m2, 0.72 m2 for 0.04, 0.08, 0.12, 0.16and 0.18 m respectively

3. Ab 1 m2

4. αb 0.30–0.80 (depending uponcondition of basin)

5. αg 0.056. Rg 0.057. Rw 0.058. α 5.67×10!8 W/m2 K9. εw 0.9510. L 2390×103 J/kg11. εg 0.9412. Lg 4×10!3 m13. Kg 0.78 W/m K14. Lb 5×10!3

15. Kg 0.0351 W/m K16. ρw 0.3517. Cw 4200 J/kg K18. t 3600 s19. h2 5.7+3.8 W/m2K20. Mw 40, 80, 120, 160 and 180 kg

for 0.04 m, 0.08 m, 0.012 m,0.16 m and 0.18 m respectively

21. hi 2.8 W/m2 K

2. Cooling of water takes place when no solarradiation from evening till the next morning.

The obtained result of ηi and x = (Tw!Ta)/I(t)has been plotted both linearly and non-linearlyfor all types of solar stills studied, similarly asshown in Fig. 2a and 2b for a 15° inclinationangle with 0.04 m water depth for summer andwinter conditions. The time period, i.e. 10 AM–4 PM, was chosen to plot the characteristic curvesbecause it is a high solar intensity period. Thenon-linear characteristic curves are analyzed as

1

1. ew cg a

L cg a

h UF

U h U

Fig. 2a. Linear characteristic curve for a single slope solarstill at 15E inclination and at a depth of 0.04 m for themonth of June 2004.

Fig. 2b. Polynomial characteristic curve for a single slopesolar still at 15E inclination and at a depth 0.04 m for themonth of June 2004.

[Eq. (13)] is a function of temperature-dependentheat transfer coefficients which gives non-linea-rity in a characteristic curve with (Tw!Ta)/I(t). IfFN [Eq. (13)] becomes constant, then one can getthe linear characteristic curve. The parameter FNcan be taken as a constant till 60EC temperatureof water because of a slight variation in values ofheat transfer coefficients. The variation in valuesof heat transfer coefficients after water tempera-ture of 60EC is considerably more [12]. Hence,


Fig. 3a. Linear characteristic curve for a single slope solarstill at 30E inclination and at a depth of 0.16 m for themonth of June 2004.

Fig. 3b. Polynomial characteristic curve for a single slopesolar still at 30E inclination and at a depth of 0.16 m forthe month of June 2004.

non-linear behavior of characteristic equations isdue to temperature dependence of the internalheat transfer coefficient, which has a significanteffect on Ut and UL [Eq. (5), unlike the flat platecollector]. The linear and polynomial charac-teristic curves using Microsoft Excel 2003 havebeen obtained. The results with constants withcoefficient of correlations have been shown in the

same figures. Figs. 3a and 3b show the typicalresults of solar stills inclined at 30° with a waterdepth of 0.16 m for summer conditions.

For comparison of ηi at different inclinationangles, Figs. 4a and 4b show the variation of ηi

with (Tw!Ta)/I(t). This indicates that the incli-nation with 45° gives the best performance forsummer conditions. An unusual result is foundthat the maximum instantaneous efficiency forthe solar still inclined at 15° is higher than that of45°. This may be because of low solar intensityalthough it is higher at all other values of x =(Tw!Ta)/I(t). Similar linear and non-linearequations have been developed for differentinclinations for winter conditions in Figs. 5a and5b, showing the best solar still with a 45° incli-nation. Figs. 5a and 5b show that the instan-taneous efficiency is increased for 30° and 45°inclined solar stills but decreased for 15° inclinedsolar still. This is due to the low solar inclinationangle in winter conditions in comparison tosummer conditions. The data of x = (Tw!Ta)/I(t)and ηi taken to plot the instantaneous efficiencycurves have been reduced due to short daylengthin winter conditions. The results of both summerand winter conditions have been tabulated inTable 3a and 3b with threshold intensity, coeffi-cient of correlation and root mean square of %deviation. The solar still inclined at 15° and0.04 m water depth in November shows a non-linear characteristic equation with unrealisticresults i.e. imaginary roots of the equation. Hencefor calculation of values of x, the differentiationmethod of maxima and minima is used. Thisgives a minimum value of y = 0.0146 at x =0.0115. This result indicates that at x <0.0115,instantaneous efficiency again starts increasing.This happens because of low values of solarintensity, ambient and water temperature in themorning time. Hence the value of x for minimumvalue of y can be taken for getting the thresholdsolar intensity. Short daylength in winter con-ditions reduces the number of data which are usedto plot the characteristic curves (Table 3a and b).


Fig. 4a. Comparison of linear characteristic curves for asingle slope solar still at different angles of inclination atdepth 0.04 m for the month of June 2004.

Fig. 4b. Comparison of polynomial characteristic curvesfor a single slope solar still at different angles of incli-nation at depth 0.04 m for the month of June 2004.

Fig. 5a. Comparison of linear characteristic curves for asingle slope solar still at different angles of inclination atdepth 0.04 m for the month of November 2004.

Fig. 5b. Comparison of polynomial characteristic curvesfor a single slope solar still at different angles of incli-nation at depth 0.04 m for the month of November 2004.


Table 3aLinear characteristic equations for a single slope passive solar still at different angles of inclination of condensing coverat brine water depth = 0.04 m

Sr.no.

Angle of inclinationof condensing cover

Linearcharacteristicequations

Value of x forthreshold intensity(Ithreshold )

Coefficient ofcorrelation (R2)

Root meansquare of %deviation

For June 2004 (summer)1. 15° y = 9.7073x!0.1706 x = 0.0175 0.9803 9.812. 30° y = 9.9438x!0.1500 x = 0.0151 0.9959 2.613. 45° y = 15.186x!0.2293 x = 0.0151 0.975 7.94For November 2004 (winter)4. 15° y = 5.9801x!0.1143 x = 0.0191 0.9743 28.545. 30° y = 4.4265x!0.0579 x = 0.0131 0.9992 6.116. 45° y = 6.0246x!0.1237 x = 0.0205 0.9986 6.25

Table 3bNon-linear characteristic equations for a single slope passive solar still at different angles of inclination of condensingcover at brine water depth = 0.04 m

Sr.no.

Angle ofinclination ofcondensingcover

Polynomial characteristicequations

Value of x fory = 0 or ymin

threshold intensity(Ithreshold )



For June 2004 (summer)1. 15° y = 56.723x2 + 5.5246x!0.00981 x = 0.0153 0.9831 6.082. 30° y = 2.5438x2 + 9.7893x!0.1478 x = 0.0150 0.9959 2.573. 45° y = 169.28x2 + 6.2209x!0.1155 x = 0.0136 0.978 5.80For November 2004 (winter)4. 15° y = 116.73x2 – 2.6938x+ 0.0302a x = 0.0115b 1 0.065. 30° y = 7.6294x2 + 3.4236x!0.0344 x = 0.0098 1 0.016. 45° y = !13.662x2 + 8.243x!0.1939 x = 0.5788 1 0.04

aUnrealistic results; bCalculated by differentiation method of maxima and minima.

Table 3cLinear characteristic equations for a single slope passive solar still at 30° angle of inclination of condensing cover atdifferent brine water depths for the month of June, 2004

Sr.no.

Brine waterdepth (m)

Linear characteristicequations

Value of x forthreshold intensity(Ithreshold )



1. 0.04 y = 9.9438x!0.15 x = 0.0151 0.9959 2.612. 0.08 y =5.69911x!0.1085 x = 0.0191 0.9832 13.343. 0.12 y = 3.9159x!0.0205 x = 0.0052 0.9935 5.244. 0.16 y =3.2624 x!0.0182 x = 0.0056 0.9984 17.78


Table 3dNon-linear characteristic equations for a single slope passive solar still at 30° angle of inclination of condensing cover atdifferent brine water depths for the month of June, 2004

Sr.no.

Brine waterdepth (m)

Polynomial characteristicequations

Value of x for y = 0 orymin threshold intensity(Ithreshold)



1. 0.04 y = 2.5438x2 + 9.7893x!0.1478 x = 0.0150 0.9959 2.572. 0.08 y =-83.786x2 + 11.638x!0.2087 x = 0.1177 0.9888 10.153. 0.12 y = !23.097x2 +4.6453x!0.0256 x = 0.19545 0.9940 6.064. 0.16 y = 23.918x2 + 2.3314 x!0.0116 x = 0.00474 1 1.30

Fig. 6a. Comparison of linear characteristic curves for asingle slope solar still at a 30E angle of inclination atdifferent depths for the month of June 2004.

It is important to mention that the intercept(constant c in linear equation y = mx + c) inEq. (13) is c = FN(ατ)eff, which is positive. How-

Fig. 6b. Comparison of polynomial characteristic curvesfor a single slope solar still at a 30Eangle of inclination atdifferent depths for the month of June 2004.

ever, in the developed characteristic equation,(Table 3a–d), it is negative. This indicates thatthere is a threshold intensity, Ithreshold, which canbe obtained for ηi = 0, i.e.,

threshold

w aT Tx

I t


This can be easily understood by the followingcase:

For the solar still at a 15° inclination angle andwater depth of 0.04 m for summer conditions,

y = 9.7073x!0.1706;

which gives after solving the equation

threshold

0.0175w aT Tx

I t

2

threshold

28.09 25.5150 W/m

0.0175I t

This indicates that the solar still will operateefficiently after 150 W/m2, i.e. after 8:00 AM inthe summer (Table 1a).

The value of x for threshold intensity for eachcase has also been given in the same table for thelinear and non-linear characteristic equations.This indicates the level of solar intensity requiredfor the functioning of solar still in the terms ofourly yield, i.e. water temperature higher thaninner temperature of glass cover. For polynomialcharacteristic equation at higher depth (Fig. 3b),this values is not required due to the positivevalue of ηi for all values of (Tw!Ta)I(t). This maybe due to a heat storage effect. Linear charac-teristic equations shown in Table 3a indicate thatfor a solar still at a 45° inclination angle, thecoefficient of x i.e., FNUe!at is higher than otherinclination angle solar stills, which is the result ofmore heat input in the solar still due to which heatloss also increases.

Figs. 6a and b show the comparison of ηi ofsolar stills by linear and non-linear behavior atdifferent water depths for summer conditions.The results are given in Tables 3c and 3d. A solarstill with a 30° inclination angle with lower andhigher vertical sides (0.25 m and 0.83 m, respec-tively) is utilized for the corresponding experi-

Fig. 7a. Linear efficiency gain and loss curve of the solarstill with a 15E inclination angle and water depth =0.04 m in June 2004.

Fig. 7b. Non-linear efficiency gain and loss curve of thesolar still with a 15E inclination angle and water depth at0.04m in June 2004.

ment. One can conclude that irrespective of linearand non-linear behavior, solar stills give maxi-mum instantaneous efficiency at the least water


Fig. 8a. Linear efficiency gain and loss curve of the solarstill with a 15E inclination angle and water depth =0.01 m in April 2005.

Fig. 8b. Non-linear efficiency gain and loss curve of thesolar still witha 15E inclination angle and water depth =0.01 m in April 2005.

depth (0.04 m) under study. These results are inaccordance with the results reported by variousscientists [4–6]. Linear characteristic equationsshown in Table 3c indicate that for solar stillsoperating at higher water depths, the coefficientof x, i.e., FNUe!at is lowest because of the heatstorage effect.

The comparisons of linear and non-linearinstantaneous gain efficiency and instantaneousloss efficiency have been given in Figs. 7a and 7band Figs. 8a and 8b. Figs. 7a and b show theresults of a passive solar still with a 15E incli-nation angle at a water depth of 0.04 m for thetesting period 10 AM–2 PM in June 2004. Itshows that as the gain efficiency increases, theloss efficiency decreases with the sum of instan-taneous gain efficiency and loss efficiencyremaining almost constant, lower than the idealefficiency, i.e. 60% [21]. Non-linear curves wereanalyzed because of the terms FO and (Tw!Ta)I(t)in Eq. (17), which gives nonlinearity in instan-taneous loss efficiency. Figs. 8a and 8b show theresults of a passive solar still with a 15E inclina-tion angle at a water depth of 0.01 m for thetesting time period of 10 AM–2 PM in April2005. One can observe that the maximum instan-taneous gain efficiency is 19% at x = 0.039 at awater depth of 0.04 m in June 2004. On the otherhand, it is 39.8% at x = 0.0487 at a water depth of0.01 m in April 2005 in a similar testing timeperiod in the summer. Similarly, the minimumloss efficiency is 34% at x = 0.039 at a waterdepth of 0.04 m in June 2004 and 2.8% atx = 0.339 at a water depth of 0.01 m in April2005.

One can observe that the instantaneous gainefficiency at lower water depths is found to behigher than that of higher water depths because ofthe storage effect of water. Similarly, the instan-taneous loss efficiency at lower depth is foundlower than that of higher water depths because ofthe water storage effect.


5. Conclusions

On the basis of the present study, the follow-ing conclusions have been drawn:

1. The optimum inclination angle for the bestperformance of single slope solar stills is 45° onthe basis of instantaneous efficiency. Hence, theinclination angle should be 45° for single slopepassive solar stills from the design point of view.

2. The minimum basin water depth (0.04 m) isthe best for higher yields among different waterdepths of 0.04, 0.08, 0.12, and 0.16 m. However,it should be 0.01 m for maximum yield at a 15°inclination with minimum heat storage from theoperational point of view.

3. The developed characteristic equations inTable 3a–d should be used to design the solar stillfor given parameters like water temperature,ambient temperature and solar radiation. If oneknows these parameters, then the values of x =(Tw!Ta)/I(t) would be known. Then one canestablish a comparison of efficiencies by usingthese characteristic equations for respective solarstills. Hence, finally one can select the best solarstill by using these characteristic equations beforemanufacturing.

4. The linear and non-linear equations shouldbe used with minimum root mean square percen-tage error and coefficient of correlation amongthe equations for removing the chances of moreerrors to select the design of solar stills.

6. Recommendations

The characteristic equation obtained for apassive solar still with a 15° inclination angle ata water depth 0.01 m (Table 1j) considering onlythe testing time period from 10 AM–2 PM isgiven by:

1. The linear characteristic equation: y =28.291x!0.6944 (with coefficient of correlationR2 = 0.9136, root mean square of %deviation =14.45, for threshold intensity x = 0.0245).

2. The non-linear characteristic equation: y =4919.7x2!269.42x + 3.7651 (with coefficient ofcorrelation R2 = 0.9911, root mean square of%deviation = 3.19, for threshold intensity x=0.0273, ymin = 0.076 calculated by the differen-tiation method of maxima and minima).

These characteristic equations should be usedto design solar stills for an inclination angle of15° at a water depth of 0.01 m. In the samemanner the characteristic equations can beobtained for other solar stills with differentinclination angles during time period of 10 AM–2 PM, which should be recommended for the bestperformance of solar stills. The various passivesolar stills made of any material should be testedat 0.01 m water depth in the basin during 10 AMto 2 PM.

8. Symbols

Ab — Area of basin liner, m2

AC — Area of solar water collector, m2

As — Area of sidewalls losing heat, m2

Ag — Area of glass cover, m2

Cf — Specific heat of fluid, J/kg KCw — Specific heat of water, J/kg KFR — Flow rate factor for collector

— Average value of f(t) for time inter- f tval 0 to t

h1 — Total heat transfer coefficient forwater to glass cover, W/m2 K

h2 — Convective heat transfer coefficientfrom glass to ambient, W/m2 K

h3 — Convective heat transfer coefficientfrom basin liner to water, W/m2 K

hb — Overall heat transfer coefficientfrom basin to ambient through bot-tom and side insulation, W/m2 K

hew — Evaporative heat transfer coefficientfrom water to glass, W/m2 K

I(t) — Solar radiation on horizontal sur-face, W/m2


Kg — Thermal conductivity of glass,W/mK

Lg — Thickness of glass, mMw — Mass of water, kg

— Mass flow rate of water in solarmcollector, kg/s

— Hourly yield through solar still, kg/hewm— Rate of evaporative heat transferewq

within solar still from water to glass,W/m2 K

t — Time, sdt — Small time interval, sTa — Ambient temperature, °CTb — Basin liner temperature, °CTci — Temperature of inner surface of

glass, °CTco — Temperature of outer surface of

glass, °CTw, Two — Water temperature, ECUb — Overall bottom loss coefficient,

W/m2 KUcg!a — Overall heat transfer coefficient

from inner glass surface to ambient,W/m2 K

Ueff — Effective overall heat transfer co-efficient, W/m2 K

UL — Overall heat loss coefficient throughsolar still, W/m2 K

UNL — Collector heat loss factor, W/m2 KUt — Overall bottom loss coefficient,

W/m2 KGreek

αNb — Solar flux absorbed by basin linerαNg — Solar flux absorbed by glassαNw — Solar flux absorbed by waterτ — Transmissivity of glassηi — Instantaneous efficiencyηiL — Instantaneous loss efficiency

Acknowledgement

We are thankful to Mr. V.K. Dwivedi,research scholar, Centre for Energy Studies,Indian Institute of Technology Delhi, New Delhi,India, for providing the data for a passive solarstill with a 15° inclination angle at a water depthof 0.01 m for the month of April, 2005, for theanalysis.

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characteristic equation of a passive solar still

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