Analytical thermal modelling of multi-basin solar still

Energy Convers. Mgmt Vol. 34, No. 12, pp. 1261-1266, 1993 Printed in Great Britain. All rights reserved

0196-8904/93 $6.00+ 0.00 Copyright © 1993 Pergamon Press Ltd

ANALYTICAL THERMAL MODELLING OF MULTI-BASIN SOLAR STILL G. N. TIWARI, t S. K. S I N G H 2 and V. P. B H A T N A G A R 3 JCentre for Energy Studies, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi-100 016, 2S.P.M. College, Biharsharif-803 I01 and 3Delhi College of Engineering, Kashmeri Gate, Delhi 6, India (Received 27 June 1992; received f o r publication 23 December 1992)

Abstract--Based on energy balances for different components of a multi-basin solar still, namely, the basin-liner, the water mass and the glass cover, analytical expressions for water and glass temperatures have been derived in terms of climatic and design parameters of the system. It is inferred that the daily yield is a maximum for the least water depth in each basin Solar distillation

Solar energy

Purification of water

NOMENCLATURE Specific heat o f water (J/kg °C) Cw C = Constants defined in text h u = Total heat transfer coefficients from water mass to glass cover for j t h effect (W/m 2 °C) h2 = Total convective and radiative heat transfer coefficient from top glass cover to ambient air (W/m E°C) h3j= Convective heat transfer coefficient either from basin liner or glass cover to water mass in j t h effect (W/m 2 °C) h b = Bottom heat transfer coefficient ('~,V/m2 °C) hewj = Convective heat transfer coefficient from water to glass cover in j t h effect (W/m 2 °C) h,wj = Evaporative heat transfer coefficient from water to glass cover in j t h effect (W/m 2 °C) l(t) = Solar intensity incident on glass cover (W/m 2) K~=Thermal conductivity of insulating material (W/m 2 °C) L = Latent heat of vaporization (J/kg) L i = Thickness of insulation (m) rhej= Rate of distillation in j t h effect (kg/s) g w j = Mass of water in j t h basin (kg) Partial saturated vapour pressure at T~(N/m 2) ewj= Partial saturated vapour pressure at Tw(N/m 2) Qew = Rate of heat flux per m 2 due to evaporation (W/m 2) l = Time (s) to= Ambient temperature (°C) L= Basin liner temperature (°C) Glass cover temperature o f j t h effect (°C) rwj= Water temperature o f j t h effect (°C) V = wind velocity (m/s) Greek letters

(ctr)b = (ctT)w = E= tr = ct, fl = r/=

Fraction of energy absorbed by basin liner Fraction o f energy absorbed by water Emissivity Stefan-Boltzmann constant (W/m: K 4) Constants defined in text Overall percentage efficiency of still INTRODUCTION

The working of solar distillation units has been broadly classified into passive and active modes of operation. In a recent review, it is reported that the overall thermal efficiency of a passive distiller is higher than that of an active one due to the lower range of operating temperature [1]. A more detailed classification is given in Table 1. In order to increase the daily output of a conventional distiller, the following effects have been studied: (i) Dye or charcoal chips [2], (ii) Back wall as reflector [3, 4], 1261

1262

TIWARI et al.:

MULTI-BASIN SOLAR STILL THERMAL MODELLING Table 1. Classification of passive solar still Passive solar still

I

I

I

I

!

I

Inclined basin solar still

I

New design solar still

I

I

Single effect distiller

With dye

I

Inclined solar still

Conventional solar still

I

Multi-wick still

1

Liferaft

Spherical Tubular

Multi-effect distiller

Without-dye

L

I

With stati nary water

i

With moving water I

I

With feeding

(regenerative still)

I

Without feeding

(iii) Back wall with cotton cloth [5], (iv) Regenerative effect in back wall and the glass cover [5], (v) Solar still with internal condenser, Ahmed [6], and the reported increase in the daily yield of the distiller is about 10-15%. In view of its increased cost of installation and maintenance, it is not appreciable. The use of latent heat of vaporization for further distillation, which is generally known as double-basin distillation [7, 8], has been shown to increase the daily yield appreciably ( - 3 5 - 4 0 % ) under a clear climatic condition. Recently, Mahdi [9] has studied the performance of a multibasin solar still and concluded that the latent heat of varporization can be used appreciably only up to the third stage. Mahdi has solved coupled differential equations based upon energy balance equations for different components of the system. In this communication, an analytical study of a multi-basin solar still (three-stage) has been done in terms of daily yield. Analytical expressions for the temperatures of different water masses and glass covers have been derived using the energy balance equations for the different components of the system. It is inferred that the daily yield is a maximum for the least water depth in each effect.

Glass cover

nl rd effect rind effect

--- DistiUote output

I st effect

k.

_:

Bos,n l,ner

Fig. 1. Cross-sectional view of three-stage distillation unit.

TIWARI et al.: MULTI-BASINSOLAR STILL THERMAL MODELLING

1263

ANALYSIS

A three-stage solar still is shown schematically in Fig. 1. The energy balance equations for the various components are as follows: For the basin liner (CtZ)bI(t) = h3j(Tb - Twj) + hb(Tb -- T,).

(1)

(ocz)jl( t ) + h3j( Tb - Twj) = MwjCw(dTwfldt ) + hu( T w - Tu).

(2)

For the water mass

For the glass cover

hlj(Twj-

Tg/)= h3q+ ,)[Tw- Two+,)]

(3)

if the glass cover is in contact with water and hlj(Twj - T~) = h2(Tgi- Ta)

(4)

if the glass cover is exposed to ambient. Here, the subscripts j = 1, 2, 3 represent the first, second and third stages, respectively. Here h u = h,w;+ h~,,,j+ he,~j

(5)

h~wj= {aE [(Twj + 273) 4 - (Tv + 273)4]/(Tw - T~)}

(6)

hewj = 0.884{(Tw - Tg/) + ( P w - Pa)(Twj+ 273)/(268.9 x 103 - Pwj)}'/3

(7)

hew = 0.016 x hew x (Pw - P u ) / ( T w - Tgj),

(8)

hb = [(L,/K~) + (1/h~)]-'

(9)

h2 = 5.7 + 3.8V [10].

(10)

where

and

In order to write the above energy balances, the following assumptions have been made: (i) The glass covers and the insulating materials have very small heat capacities, (ii) The temperature remains constant throughout the thickness of the water masses and the glass covers, (iii) The whole distiller unit is vapour-tight and, (iv) The glass covers have very small inclination (10-15 °) so that their areas are nearly equal to that of the basin liner. From equation (1), Tb = [(OtZ)bI(t) + h3j Twj + hb Ta]/(h3j + hb).

(ll)

Tg/= [h u Tw + h3~+l) Twti+0]/[hu + h3q+ I)]

(12)

From equation (3)

when the glass cover is in contact with water and T~ = (h,j T~j + h 2 ra)/(h,j + h2)

(13)

when the glass cover is exposed to the ambient. From equations (2), (7), (8) and (9), for j = 1, 2, 3, one gets: IdTwlq ---~-l_j + al rwl + a2 rw2 + a3 Tw3 = f l (t)

(14)

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TIWARI

et al.: M U L T I - B A S I N S O L A R STILL T H E R M A L M O D E L L I N G

[ I

- - ~ - j + a, Tw, +a'zTw2+a'3Tw3 = f 2 ( t )

drw2]

,

(15)

dTw3-]

,,

(16)

- ~ - j + a, rw, +a'~'r,~+a';Tw~=A(t)

where a, = (U,2 + U,b)/(Mw, Cw) a2 = - Ul2/Mwl Cw,

a3 = 0

h31 (h31~)I(t)+

f~(t)=[(az)~I(t)+

UibTal/MwlC

a~ = - U12/Mw2Cw,

a~ = (Ul2 + UI3)/Mw2C~

a'3 = - U,3/Mw2Cw,

f2(t) = (aOuI(t)/Mw2Cw

al"= O,

w

a2"= - U i 3 / M w 3 C w

a~' = (U13+ U32)/Mw3Cw f3 (t) = [(az)m I ( t ) + U32Ta]/Mw3 C w UI2 =

hllh32/(hll h- h32 )

Ulb = h31 hb/(h31 + hb)

and U32 =

(17)

h13h2/(h13 -4- h2).

Multiplying equation (15) by a and equation (16) by fl and adding to equation (14)

[d(Tw, + aTw2 +//Tw3)/dtl + (al + aa'~ + fla'~')Tw, + (a2 + aa'2 + fla'2')Tw2 + (a3 + aa~ + fla'3')Tw3 =f~(t) + af2(t) + flf3(t) or

[d(Tw~ + a T w 2 + f l T w 3 ) / d t ] + C ( T w ~

+UTw2+flTw3)=f~(t)+af2(t)+flf3(t)

(18a)

where C = aj + aa'l + fla'(

(18b)

Cot = a2 + aa~ + fla'2'

(18c)

Cfl = a 3 + aa~ + fla'3'.

(18d)

From equations (18c) and (18d), one gets a = [a'2'a3 - a 2 ( a ; - C)]/{[(a'2- C)(a~' - C ) ] - a ~ a ~ }

(18e)

fl = [a2a~ - a3(a'2 - C)]/[(a'z - C)(a~' - C ) - a ~ a ~ ] .

(181)

Using equations (18e) and (18t) in equation (18b) C 3 - .41C 2 -4- A 2 C + A 3 = 0

(19a)

A1 = al +a~+a~'

(19b)

A2 = [(al a'2 - a2a~ ) + (a~a ; - a'3a~') + (a; al - a'(a3)]

(19c)

where

A3 = [al(a'z' a~ - a'2a'3') + a~ (a2a'3' - a3a'2') + a'( (a~a3 - a~a2)].

(19d)

Solutions to equation (19a) are C, = [(-D,/4) + x/(D~/4) + (E,/27)] t/3 + [ - D , / 4 ) - ~/(D~14) + (E,127)] '/3 + (A~/3) (20a)

TIWARI et al.: MULTI-BASIN SOLAR STILL THERMAL MODELLING C2, C 3 =

{--(C

I --

At)~2 +_x/(C, - A,)2/4 - [A2 + (Ct - A,)C,]}

1265 (20b)

where D, = ( - 2A ~/27) + (At A2/3) +/43

(20c)

E] = A 2 -- (A2/3).

(20d)

Multiplying equation (18a) e c' and integrating between 0 and t, one gets its solution as:

Twt +OtTw2+flTw3 =fl(t)+°tf2(t)+flf3(t) (1 -e-°)+(Tw~o+~Tw2o+flTw~o)e -c' C

(21)

wheref~ (t) etc. and Two0etc., are the averaged values between the interval 0 and t and the values at t = 0, respectively. The average water temperature can be obtained as

Tw' + ctj T~2 + flj Tw3 = t x

( rw' + ~j Tw2 + g Tw3) dt =

Cj

"

('--e-Cjt) (--e-Cj') 1 -C-~ / +(Tw~°+~jTw2°+fljTw3°) 1 C--~jt j = F j ( t ) ( s u p p o s e d ) .

(22)

Here, j = 1, 2, 3 correspond to the three values of C, c( and fl as given in equations (20a), (20b), (18e) and (180. Solutions to equations (22), for j = 1, 2, 3, are Tw. = Fl (t) -- ~l Tw2- fl. Tw3 Tw2 =

(fl, - fl2)[F2(t) - F3 (/)] -- (f12- f13)[Fl (t) -- F2 (t)]

(23a) (23b)

( ~ l - - f12) (0~2 - - 0~3) - - fl2]~3(~1 - - ~ 2 )

Tw3 - -

(Gt2

-

~3) IF1 (t)

-

(~2 -

-

-

~3)(/~,

F2 (t)] - (~, -- ~2) [F2(t) - F3 (t)] -

~2) - (~, -

(23c)

~2)(/~2 -/~3)

Using equations (23a), (23b) and (23c) the average temperature of the glass covers can be obtained from equations (12) and (13). Knowing the temperatures of the glass covers and the water masses, the rate of heat flux per m 2 due to evaporation can be obtained from the following equation

Qewj= hew(Tw - T~)

(24)

rh~j- Q~wj

(25)

where h~wj is given by equation (8). The rate of distillation per m 2 is L

The overall thermal efficiency is

Z Q~wj × 100. = ~ l(t)At RESULTS

(26)

AND DISCUSSION

For the efficiency as well as the daily yield of the still to be high, according to equations (26) and (24), the water temperatures Twj should be high. This requires that the 1.h.s. of equation (22), which gives the linear sum of the average water temperatures, should be high. Considering the following cases: Case (i) Cjt ~ 1: Under this condition [(1 -e-Cj')/Cjt]--*l, and the r.h.s, of equation (22) becomes independent of the system and the climatic parameters, and hence, there is no yield.

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TIWARI et al.:

MULTI-BASIN SOLAR STILL THERMAL MODELLING

Case (ii) Cj t >> 1: Under this condition [(1 -e-Cjt)/Cjt]~O, and the r.h.s, of equation (22) is a maximum and strongly dependent upon the system and the climatic parameters. Thus, for the daily yield to be high, Cj should be large, for a given t (1 h). This requires, according to equation (18b) that the values of the a's should be high, and hence, the water mass Mw, i.e. the water depth in each basin should be small. If Mw3, the water mass in the third effect, is large, it is obvious from equations (16) and (17), that Tw3 becomes constant and nearly equal to the ambient temperature. This is the case of fl = 0, when the third stage gives no yield and the unit becomes just a two-effect one. And if Mw2 is also very large, the second effect also becomes inoperative as Tw2, equal to the ambient temperature, and the unit becomes equivalent to a single-basin system. Acknowledgement--The authors are grateful to Professor P. B. Sharma, Principal, Delhi College of Engineering, Delhi for his continuous support during the research work reported in the paper.

REFERENCES 1. G. N. Tiwari, Recent advances in solar distillation. In Solar Energy and Energy Conservation (Edited by Raj Kamal et al.), Chap. II, pp. 32-149. Wiley Eastern, New Delhi (1992). 2. M. S. Sodha, A. Kumar, G. N. Tiwari and G. C. Pandey, Appl. Energy 7, 147 (1980). 3. P. Bivulswas and S. Tadtiam, Int. Symposium Workshop on Renewable Energy Sources--Lahore. Elsevier, Amsterdam (1984). 4. A. Tamini, Int. J. Sol. Wind Technol. 4, 443 (1987). 5. P. Wibulswas, Renewable Energy Rev. J. 6, I01 (1984). 6. S. T. Ahmed, Sol. Wind Technol. 5, 637 (1988). 7. M. S. Sodha, J. K. Nayak, G. N. Tiwari and A. Kumar, Energy Convers. Mgmt 20, 23 (1980). 8. O. C. lloeje, Instersol 85, Vol. 2, p. 1339. Pergamon Press, Oxford (1985). 9. N. A. Mahdi, Energy 17, 87 (1992). 10. J. A. Duffle and W. A. Beckman, Solar Engineering of Thermal Processes. Wiley, New York (1980).