Heat and mass transfer analysis of a clay-pot refrigerator

International Journal of Heat and Mass Transfer 55 (2012) 3977–3983

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Heat and mass transfer analysis of a clay-pot refrigerator A.W. Date Mechanical Engineering Department and Center for Technology Alternatives for Rural Areas, Indian Institute of Technology, Mumbai 400076, India

a r t i c l e

i n f o

Article history: Received 5 May 2011 Received in revised form 22 February 2012 Accepted 23 February 2012 Available online 3 May 2012 Keywords: Reynolds flow model Heat/mass transfer Clay-pot refrigerator

a b s t r a c t The simple clay-pot refrigerator is ideally suited for preserving vegetarian foods in hot and dry climates. The refrigerator works on the evaporative cooling principle. In this paper, steady-state performance of the refrigerator is analysed using Reynolds flow model of convective heat/mass transfer. For the assumed respiratory cooling load, the preservation temperature is predicted under a variety of ambient temperatures and relative humidities. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction In rural areas of India, vegetarian food is often preserved in a clay-pot refrigerator. The cooling space is a smaller clay pot inserted within a larger clay pot (see Fig. 1 - left). The annular space between the two pots is filled with sand or tiny pebbles. The voids in the annular space are occupied by water . Convective and radiative heat transfer Qin from the hot and dry surroundings evaporates this water and brings about cooling of the space in the inner pot where food is kept. Once the sensible heats are removed, the preserved foods give out respiratory cooling load qload (typically 0.05  0.2 W/kg ).1 Thus, the Steady-State Coefficient of Performance (COP) of such a refrigerator may be defined as

COP ¼

Q load qload  mfood ¼ Q in Q in

ð1Þ

This definition is of course different from that used for a conventional refrigerator in which Qin is replaced by work input Win. But, whereas one pays for Win, Qin is free. Also, Win of a conventional refrigerator is under designer’s control whereas Qin depends on ambient conditions and therefore not under designer’s control. In order to circumvent the above difficulty, here an efficiency gth is newly defined based on thermodynamic considerations as

T  T cold gth ¼ 1 T 1  T dp

ð2Þ

where the preservation temperature in the inner pot Tcold depends on the ambient conditions viz. temperature T1 and relative humidE-mail address: [email protected] In addition, there may be cooling load due to heat transfers from the top and bottom of the refrigerator. 1

0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.03.028

ity U1 (or, vapour mass fraction xv,1) and irreversible heat and mass transfer processes. On the other hand, if the moist ambient air was cooled at constant pressure then the moisture will begin to condense at what is called the dew point temperature Tdp. We may now imagine that our refrigerator is used to bring about this ideal cooling. Then, gth = 1 will represent maximum cooling performance. The main irreversible heat and mass transfers are influenced by the thermal conductivity of clay walls and sand + water as well as by surface areas of inner and outer pots which determine the internal resistance to heat transfer whereas the external resistance to heat transfer is influenced by the external (usually by natural convection) heat transfer coefficient ao. Similarly, the evaporation rate is influenced by the effective hydraulic conductivity (KH) of the outer clay wall and the external mass transfer coefficient g. Fig. 1 (right) shows the assumed model of the refrigerator. The curved surfaces are replaced by straight cylindrical surfaces of radii ri and ro with height H. The inner and outer clay wall thicknesses are bi and bo respectively. The thermal conductivity of clay is designated kcl and effective conductivity of sand+water is designated as keff. Then, dependence of gth (or COP) and Tcold will be given by

½gth ; T cold  ¼ FfðT 1 ; U1 Þ; ðr i ; r o ; H; bi ; bo Þ; ðkcl ; keff ; ai ; ao ; Þ; K H g

ð3Þ

where  is emissivity of the outer surface of the outer pot and ai,o are heat transfer coefficients associated with inner and outer pots.2 The objective of the present paper is to establish this functional dependence using the Reynolds flow model due to Spalding [1]. This 2 It must be mentioned that, as a cooling device, although the clay-pot refrigerator has been in use for very long in history (see, for example, http://en.wikipedia.org/ wiki/Pot-in-pot-refrigerator) the present author has found no heat/mass transfer related technical lierature in any professional hand-books, research journals or textbooks.

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A.W. Date / International Journal of Heat and Mass Transfer 55 (2012) 3977–3983

Nomenclature A B b Bi Cp g H h k kp KH Le _ m m P Q T

surface area m 2 Spalding number clay wall thickness m Biot number specific heat J/kg  K mass transfer coefficient kg/m2  s pot height m enthalpy of the mixture J/kg thermal conductivity W/m  K permeability m2 hydraulic conductivity m/s Lewis number evaporation rate kg/s mass kg pressure N/m2 heat transfer W temperature °C

U

latent heat J/kg relative humidity

Suffixes a cl cold dp e eff i in load m M mean nc o rad ref v w 1

air clay inner pot environment dew point energy conservation principle sand + water inner pot radiation + convection cooling load mass conservation principle or mean transferred substance state mixed mean natural convection outer pot radiation reference value vapour water or w-w state ambient condition

k

Greek symbols a heat transfer coefficient W/m2  K b volumetric coefficient K1 x mass fraction gth thermodynamic efficiency  emissivity

Fig. 1. Clay pot refrigerator and assumed model.

model is presented in Section 2. The computed results for several values of parameters are presented in Section 3. Finally, conclusions are reported in Section 4. 2. Reynolds flow model 2.1. Definitions of states and phases The Reynolds flow model is an algebraic model of mass transfer _ w (kg/s) across the interface between the transferred substance m (water, in the present case) and the considered phase (stagnant surrounding air in the present case.). Fig. 2 shows the mass transfer situation. The interface in the present case is the outer surface of the outer pot and is designated by w-w. The width of the considered phase spans from w-w surface to the imaginary 1  1 surface. In the Reynolds flow model, the transferred substance is taken at uniform temperature and concentration. But, in the present case, the temperature will vary in the annular space occupied by the transferred substance. Therefore, we construct an imaginary transferred substance state M-M with uniform (or, mixed-mean) properties given by

Fig. 2. Reynolds flow model.

TM ¼

Ti þ To and xv ;M ¼ 1 2

ð4Þ

A.W. Date / International Journal of Heat and Mass Transfer 55 (2012) 3977–3983

where Ti and To are temperatures of the outer surface of the inner pot and inner surface of the outer pot, respectively as shown in Fig. 2.

The Reynolds Flow Model postulates two mass flows (see Fig 2). 1. A mass flow g Ao (kg / s) crossing the 1  1 surface flowing towards the w-w surface but carrying with it properties of the 1-state. _ w ) crossing the 1  1 surface away from 2. A mass flow (gAo þ m the w-w surface but carrying with it properties of the w-state. _ w enters the considered phase at where Ao = 2proH. Now, since m the w-w surface, mass balance over the width of the considered phase shows that hypothesising existence of fictitious mass flow (g Ao) does not create or destroy any mass in the considered phase. As such, the hypothesis claims that all phenomena of the real flow in the considered phase and their effects on the heat/mass transfer processes at the w-w surface will be unaffected by the Reynolds flow model hypothesis. With this claim, we invoke conservation of mass and energy principles. 2.2.1. Mass conservation principle Thus, invoking the mass conservation principle for water-vapour between w-w and 1  1 states, we have

_ w ¼ ðm _ w þ gAo Þxv ;w gAo xv ;1 þ m

xv ;1  xv ;w xv ;w  1

ð6Þ

ao

ð7Þ

Cpm

_ w;m is evaluated from In Eq. (6), subscript m designates that m mass-conservation principle and B is called the Spalding number. The value of xv,1 is known but that of xv,w is not known. The latter will be determined from equilibrium condition at the w-w surface when Tw is known. Further, this evaporation rate meets with resistance of the outer-pot clay wall thickness. Therefore, using Darcy’s law, the average mass flow rate over height H is

qw kp Ao pa _ w;m ¼ m lw bo

  pH=2 pw;sat  pa pa

ð8Þ

where kp (m2) is permeability. Note that pH/2/pa = 1 + 0.05 H because pa  10 m of water. Also, since saturation conditions prevail at the w-w surface, pw, sat/pa = xv,w/(0.622 + 0.378 xv,w), we have

_ w;m ¼ qw K H Ao m



0:622ð1 þ 0:05HÞ þ ð0:0189H  0:622Þ xv ;w 0:622 þ 0:378 xv ;w



ð9Þ where K H ¼ ðk pa Þ=ðlw bo Þ is hydraulic conductivity expressed in (m/s). Combining Eq. (9) with Eq. (6), we can write

_ w;m ¼ gAo  B where m ð0:622  0:0189HÞxv ;1  0:622ð1 þ 0:05HÞ B ¼ ð:622  0:0189HÞðxv ;w  1Þ  Bim ð0:622 þ 0:378xv ;w Þ

ð11Þ

where hMw = Cpw (Tw  Tref) is the enthalpy of the transferred substance (water) in the w-state and Qrad and Qnc are heat transfers due to radiation and natural convection respectively. Similarly, between 1  1 and M-M surfaces

_ w hM þ Q load ¼ Q L þ ðm _ w þ gAo Þhw gAo h1 þ m

ð12Þ

where hM = Cpw (TM  Tref). QL is the heat conduction in the sand+ water layer. Combining Eqs. (11) and (12) gives

_ w ðhMw  hM Þ  Q rad  Q nc Q load  Q L ¼ m   T þ To _ w Cpw T w  i ¼m  Q rad  Q nc 2

ð13Þ ð14Þ

where the left hand side of this equation can be evaluated from heat conduction considerations as



 keff Ai ðT i  T o Þ ri ln ðr o =r i Þ   kcl Ao ðT o  T w Þ ¼ bo

Q load  Q L ¼

ð15Þ ð16Þ

Combining the last two equations, it can be shown that

Ti  Tw ¼



 bo ri lnðr o =r i Þ ðQ load  Q L Þ þ keff Ai kcl Ao

ð17Þ

Similarly, heat transfer across S-S surface gives

and, since the Lewis number Le ’ 1 (for our air-water vapour system) the mass transfer coefficient g (kg/m2-s) can be estimated [1,2] from natural convection heat transfer coefficient ao as



_ w hMw ¼ Q rad þ Q nc þ ðm _ w þ gAo Þhw gAo h1 þ m

ð5Þ

where xv is vapour mass-fraction. Of course, mass fraction of air will be xa = 1  xv. Upon rearrangement,

g’

and Bim  g/(qw KH) may be viewed as mass transfer Biot number. 2.2.2. Energy conservation principle Writing the energy conservation principle for the air-water vapour mixture between 1  1 and w-w surfaces, we have

2.2. Reynolds flow hypothesis

_ w;m ¼ gAo  B where B ¼ m

3979

ð10Þ

T cold  T i ¼

  Q load 1 bi þ Ai ai kcl

ð18Þ

Adding the last two equations,

T cold  T w ¼

    Q load 1 bi bo r i ln ðr o =r i Þ þ ðQ load  Q L Þ þ þ keff Ai Ai ai kcl kcl Ao ð19Þ

Now, using Eqs. (16) and (17), it can be shown that

Tw 

    Ti þ To 2bo ri ln ðr o =r i Þ ðQ load  Q L Þ þ ¼ keff Ai 2 kcl Ao

ð20Þ

Substituting this equation in Eq. (14), we have

  1 _ w Cpw 2bo ri ln ðr o =r i Þ m Q load  Q L ¼ ðQ rad þ Q nc Þ 1 þ þ keff Ai 2 kcl Ao ð21Þ Hence, substitution in Eq. (19) gives   bo r i ln ðr o =r i Þ  ðQ rad þ Q nc Þ T cold  T w ¼  þ kcl Ao keff Ai   1   _ w Cpw 2bo m r i ln ðr o =r i Þ Q 1 bi þ þ load þ  1þ 2 kcl Ao Ai ai kcl keff Ai

ð22Þ _ w from the energy conserFinally, using Eq. (11), we evaluate m vation principle as

_ w;e ¼ m

Q rad þ Q nc þ gAo ðh1  hw Þ hMw  hw

ð23Þ

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A.W. Date / International Journal of Heat and Mass Transfer 55 (2012) 3977–3983

where

h1 ¼ Cpa ðT 1  T ref Þ þ ½ðCpv  Cpa ÞðT 1  T ref Þ þ kref xv ;1 hw ¼ Cpa ðT w  T ref Þ þ ½ðCpv  Cpa ÞðT w  T ref Þ þ kref xv ;w hMw ¼ Cpw ðT w  T ref Þ and kref is latent heat of water at Tref. 2.3. Solution procedure Eqs. (10)–(23) are solved iteratively since the temperature Tw at the outer surface of the outer pot is unknown. The main steps are: 2.3.1. Preliminary steps 1. Specify geometry parameters: ri, ro, bi, bo and H. 2. Specify ambient parameters T1 and U1 (or xv,1, see Appendix) and respiratory load Qload. 3. Specify properties: Cpw, Cpa, Cpv, KH, ka, kcl and keff. 2.3.2. Begin iterations 4. Assume Tw and evaluate xv,w from correlation given in Appendix. 5. Evaluate xv,mean = 0.5(xv,1 + xv,w). Hence, evaluate Cpm = Cpa(1  xv,mean) + Cpvxv,mean. 6. Evaluate ao from McAdams [3] correlation for natural convection from a vertical surface NuH = 0.59(GrH Pr)0.25 where

NuH ¼

ao H ka

and GrH ¼

9:81 bðT 1  T w ÞH3

m2a

where b = 1./Tmean and Tmean = 0.5(Tw + T1) (K) and ka and ma are evaluated at Tmean. 7. Knowing ao, evaluate g from Eq. (7) and hence, Bim, B⁄ and _ w;m from Eq. (10). m 8. Now, evaluate (Qrad + Qnc) = (arad + ao)Ao(T1  Tw) where





arad ¼ r T 21 þ T 2w ðT 1 þ T w Þ

1. Hydraulic conductivity: Abu-Zeig and Atoun [4] and Gil et al. [5], among others, report considerable variation 2  109 < KH(m/s) < 3  108. 2. Thermal conductivity of clay: The values vary 1 < kcl (W/m-K) < 2.5; the higher value is typically taken for fired brick. 3. Thermal conductivity of sand + water: The value of keff depends on the material of solid particles used and the void space occupied by water. The plausible range of values is 1 < keff (W/m-K) < 3.5. 4. Respiratory load: For different food materials the values of respiratory load vary 0.2 < qload(W/kg of food) < 2 [6]. In view of the above variations, we set-up a reference case with values shown in Table 1. The total cooling heat load comprising respiratory (3 kg food) + leakage from top and bottom of the regfrigerator (0.4 W) is taken as Qload = 3  0.2 + 0.4 = 1 W. Calculations are first presented for the reference case. Then, effects of parameters are individually assessed keeping all other values corresponding to the reference case. In all calculations, the inner heat transfer coefficient is taken as ai = ao, Pr = 0.7 and emissivity  = 1. 3.2. Reference case Table 2 shows typical computed results for one case of T1 = 40 °C and U1 = 10% for Qload = 1 W by way of an example. The results are obtained by iteratively determining Tw. The results show that Ti < To < Tw but Tcold > Ti as expected. Also, the dew point corresponding to ambient consitions Tdp < Tcold as expected giving gth = 0.4748. The conduction heat transfer QL > Qin but (Qin + Qload) > QL as expected giving COP = 0.02727. Also, anc < arad indicating importance of accounting for radiation heat transfer. Corresponding to Tw = 27.536 °C, xv,w = 0.02317 giving Spalding number B = 0.0189. The mass transfer driving force B is further attenuated to B⁄ = 0.00207 due to Darcy resistance. Finally, the mass transfer Biot number Bim > > 1. 3.3. Parametric variations

_ w;e from Eq. (23). 9. Now, evaluate m 10. Calculate percentage difference



m _ w;m

_ w;e  m

 100 F ¼



_ w;m m If F > 0.01, revise Tw and go to step 4. 11. Continue till convergnce and evaluate COP = Qload/(Qrad + Qnc) _w¼m _ w;e ¼ m _ w;m . Also evaluate and Tcold from Eq. (22) with m gth from Eq. (2). Solutions Tcold, gth and COP for given T1, U1 and Qload are of interest.

3.3.1. Effect of T1 and U1 Table 3 shows the effect of relative humidity U1 at 3 values of T1. It is seen that at each value of T1, compared to dry ambient, the _ w decreases whereas the outer pot surface temevaporation rate m perature Tw increases with increase in U1. Since the temperature difference T1  Tw deceases with increase in U1, value of anc decreases. Due to higher absolute temperatures, however, arad increases. The value of Spalding number B, as expected, decreases with increase in U1 but that of B⁄, though smaller than B due to Darcy resistance, increases somewhat. The value of COP increases

Table 2 Results for the reference case: T1 = 40 °C, U1 = 10% (Tdp = 2.57 °C) , Qload = 1 W.

3. Results and discussion

Tw = 27.536

3.1. Manner of presentation As noted under Preliminary Steps, the model requires several input parameters. Among these, the most uncertain parameters are:

To = 27.104 Ti = 20.371 Tcold = 22.23

_ w ¼ 2:224  106 m QL = 37.635 Qin = 36.67 COP = 0.02727

anc = 3.851 arad = 6.55 Bim = 473.79 gth = 0.4748

g = 0.00379 B = 0.0189 B⁄ = 0.00207 xv,w = 0.02317

Table 1 Parametric values-reference case Geometry (cm)

Sp Heat J/kg-K

Th conductivity W/m-K

Hy conductivity m/s

Others

ri = 7.5 ro = 15 bi = bo = 0.5 H = 30

Cpa = 1005 Cpv = 1880 Cpw = 4186

ka = 0.027 kcl = 1.5 keff = 2.0

KH = 8  109

Qload = 1 W ma = 16.5  106 m2/s Tref = 0 °C kref = 2503 kJ/kg

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A.W. Date / International Journal of Heat and Mass Transfer 55 (2012) 3977–3983 Table 3 Effect of T1 and U1 – Qload = 1 W.

U1

_ w 106 m

Tw

anc

arad

B

B⁄

COP

Tcold

Tdp

gth

T1 = 35 °C 0 10 20 30 40 50

2.228 2.216 2.21 2.2 2.193 2.184

22.852 24.38 25.78 27.15 28.47 29.87

3.84 3.71 3.58 3.44 3.28 3.09

6.25 6.29 6.34 6.38 6.419 6.46

0.0179 0.016 0.0142 0.0124 0.0106 0.0087

0.00208 0.00214 0.00222 0.0023 0.00241 0.00255

0.0288 0.0333 0.0387 0.0458 0.0558 0.072

17.94 20.44 22.74 24.97 27.15 29.47

1.176 8.66 14.8 19.35 23.0

0.4025 0.4655 0.4965 0.5016 0.4608

T1 = 40 °C 0 10 20 30 40 50

2.24 2.233 2.19 2.233 2.163 2.16

25.667 27.536 29.317 30.947 32.518 33.967

3.99 3.85 3.70 3.55 3.38 3.204

6.49 6.55 6.61 6.66 6.71 6.75

0.0124 0.0189 0.0167 0.0145 0.0124 0.0104

0.0020 0.00206 0.00214 0.00222 0.00233 0.00245

0.0235 0.0273 0.0321 0.0382 0.0468 0.0588

19.16 22.23 25.16 27.85 30.45 32.87

2.57 12.74 19.09 23.79 27.56

0.4747 0.5444 0.5811 0.5891 0.5732

T1 = 45 °C 0 10 20 30 40 50

2.226 2.208 2.19 2.172 2.153 2.134

28.323 30.556 32.62 34.55 36.345 38.14

4.13 3.98 3.82 3.67 3.50 3.30

6.74 6.81 6.87 6.94 7.0 7.06

0.0248 0.0221 0.0195 0.017 0.0146 0.01207

0.00193 0.00199 0.00206 0.00214 0.00224 0.00236

0.0195 0.0227 0.0267 0.0319 0.0389 0.0497

20.03 23.73 27.18 30.38 33.37 36.376

6.3 16.79 23.54 28.22 32.12

0.5500 0.6317 0.6813 0.6930 0.6700

irrespective of T1 and U1, the water evaporation rate is relatively unaffected. In order to further appreciate the influence of T1 and U1, values of g th are plotted in Fig. 3. It is seen that for a fixed ambient temperature, gth is maximum at U1 ’ 40% whereas, gth increases with T1.

1 0.9 0.8 = 45 °C

T

= 40 °C

T

= 35 °C

3.3.2. Effect of Qload Table 4 shows the effect of assumed Qload on COP, gth and Tcold at T1 = 40 °C by way of an example. Compared to the reference case of Qload = 1 W, both COP and Tcold increase but gth decreases with increase in Qload as expected.

8

T

8

ηth

0.7

8

0.6 0.5 0.4

10

20

Φ(%)

30

40

50

Fig. 3. Typical variation of gth with U1-data of Table 3.

with U1 because of reduced Qin. Finally, the temperature inside the inner pot Tcold increases with U1, as expected. Most importantly, the value of Tcold remains greater than dew-point temperature Tdp (see Appendix) corresponding to T1 and U1 with the difference (Tcold  Tdp) decreasing with increase in U1. Finally, note that

3.3.3. Effect of keff Table 5 shows the effect of thermal conductivity of sand + water keff. The table shows that compared to reference case of keff = 2 W/ m-K, the values of Tcold are reduced for keff = 1 at each U1. At keff = 3, the corresponding values of Tcold are higher. The table confirms our expectation that value of keff should be as low as possible so as to enhance COP and gth. 3.3.4. Effect of kcl Table 6 shows the effect of thermal conductivity of clay kcl. It is seen that compared to the reference value of kcl = 1.5 W/m-K, Tcold reduces for kcl = 1 and increases for kcl = 2, though these effects are very marginal. 3.3.5. Effect of KH Finally, Table 7 shows effect of hydraulic conductivity KH to be very marginal irrespective of the value of U1.

Table 4 Effect of Qload at T1 = 40 °C.

U1 (%)

Qload (Ref) = 1 W

Qload = 2 W

COP

Tcold

0 10 20 30 40 50

0.0235 0.0273 0.0321 0.0382 0.0468 0.0588

19.16 22.23 25.16 27.85 30.45 32.87

Qload = 3 W

gth

COP

Tcold

0.4747 0.5444 0.5811 0.5891 0.5732

0.047 0.0545 0.0642 0.0765 0.0936 0.117

20.95 24.09 27.08 29.87 32.57 35.106

gth

COP

Tcold

gth

0.425 0.474 0.4844 0.4584 0.3934

0.0706 0.0818 0.0962 0.1147 0.1404 0.1765

22.756 25.95 29.01 31.88 34.69 37.33

0.3753 0.4031 0.3883 0.3275 0.2146

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A.W. Date / International Journal of Heat and Mass Transfer 55 (2012) 3977–3983

Table 5 Effect of keff (W/m-K) at T1 = 40 °C and Qload = 1 W.

U (%)

keff = 1

keff (Ref) = 2

COP

Tcold

0 10 20 30 40 50

0.02353 0.02727 0.03207 0.03825 0.0468 0.0588

11.37 15.51 19.43 23.06 26.54 29.76

keff = 3

gth

COP

Tcold

0.654 0.7545 0.8101 0.8304 0.8230

0.0235 0.0273 0.0321 0.0382 0.0468 0.0588

19.16 22.23 25.16 27.85 30.45 32.87

gth

COP

Tcold

gth

0.4747 0.5444 0.5811 0.5891 0.5732

0.02353 0.0272 0.03207 0.03825 0.04682 0.05883

21.76 24.47 27.05 29.45 31.76 33.91

0.4149 0.475 0.5045 0.5083 0.4895

Table 6 Effect of kcl (W/m-K) at T1 = 40 °C and Qload = 1 W.

U (%)

kcl = 1

kcl (Ref) = 1.5

COP

Tcold

0 10 20 30 40 50

0.02353 0.02727 0.03207 0.03825 0.0468 0.0588

18.922 22.027 24.97 27.71 30.34 32.788

kcl = 2

gth

COP

Tcold

0.4802 0.5513 0.5877 0.5959 0.5797

0.0235 0.0273 0.0321 0.0382 0.0468 0.0588

19.16 22.23 25.16 27.85 30.45 32.87

gth

COP

Tcold

gth

0.4748 0.5444 0.5810 0.5891 0.5731

0.02353 0.0272 0.03207 0.03825 0.04682 0.05883

19.279 22.33 25.24 27.925 30.515 32.92

0.4721 0.5414 0.5775 0.5851 0.5691

Table 7 Effect of KH (m/s) at T1 = 40 °C and Qload = 1 W.

U1 (%)

KH = 2  109

KH (Ref) = 8  109

COP

Tcold

0 10 20 30 40 50

0.02354 0.02729 0.0321 0.03828 0.04686 0.0589

19.165 22.236 25.157 27.86 30.464 32.88

KH = 3  108

gth

COP

Tcold

0.4746 0.5445 0.5806 0.5883 0.5723

0.0235 0.0273 0.0321 0.0382 0.0468 0.0588

19.16 22.23 25.16 27.85 30.45 32.87

gth

COP

Tcold

gth

0.4746 0.5445 0.5806 0.5883 0.5723

0.02348 0.02721 0.032 0.03816 0.04668 0.05863

19.14 22.208 25.126 27.83 30.434 32.85

0.4745 0.5444 0.5806 0.5882 0.5722

4. Conclusions Correlation is valid for 20 < Tw (C) < 100. In this paper, the steady state performance of a clay-pot refrigerator is analysed using the Reynolds flow model of mass transfer due to Spalding [1]. The main conclusions are 1. For a given geometry, thermal and hydraulic conductivities and cooling load, all parameters including COP, gth and inner pot temperature Tcold show expected magnitudes irrespective of ambient T1 and U1. 2. For a given T1, COP and Tcold increase with increase in U1 but, (Tcold  Tdp) decreases with increase in U1. Consequently, gth increases with increase in T1 for a given U1. 3. The effect of increasing Qload is to increase Tcold as expected. 4. The effect of thermal conductivity of sand + water keff is most pronounced. Lower value of keff is to be preferred to reduce Tcold and increase gth. 5. The effects of thermal conductivity kcl and hydraulic conductivity KH on Tcold are found to be marginal.

2. Knowing relative humidity U1,xv,1 is evaluated from [7]

xv ;1 ¼

pv ;1 W1 where W 1 ¼ 0:622  1 þ W1 ptot  pv ;1

! and

    pv ;1 U1 p F T1 where s ¼ 1  and sat ¼ exp ¼ 1s psat 100 pcr T cr F ¼ a1  s þ a2  s1:5 þ a3  s3 þ a4  s3:5 þ a5  s4 þ a6  s7:5 a1 ¼ 7:85951783; a2 ¼ 1:84408295; a3 ¼ 11:7866497; a4 ¼ 22:6807411; a5 ¼ 15:9618719; a6 ¼ 1:80122502 Here, Tcr = 647.096 K, pcr = 220.64 bar and ptot = 1.01324 bar 3. Dew point temperature Tdp may be evaluated from [7]

T dp ¼ Appendix A

  237:7  c 17:271  T 1 U1 where c ¼ þ ln 17:271  c 237:7 þ T 1 100

1. Knowing Tw. xv,w is evaluated from [2]

xv ;w ’ 3:416  103 þ ð2:7308  104 ÞT w þ ð1:372  105 ÞT 2w þ ð8:2516  10 ÞT 3w  ð6:9092  109 ÞT 4w þ ð3:5313  1010 ÞT 5w  ð3:7037  1012 ÞT 6w þ ð6:1923  1015 ÞT 7w þ ð9:9349  1017 ÞT 8w

References

8

[1] D.B. Spalding, Convective Mass Transfer, Edward Arnold (publishers) Ltd, London, 1963. [2] M. Crawford, W.M. Kays, Convective Heat and Mass Transfer, McGraw-Hill Int Edition, New York, 1993. [3] W.H. McAdams, Heat Transmission, 3rd ed., McGraw-Hill, New York, 1954.

A.W. Date / International Journal of Heat and Mass Transfer 55 (2012) 3977–3983 [4] M.M. Abu-Zreig, M.F. Atoun, Hydraulic chrarcteristics and seepage modelling of clay pitchers produced in jordan, Canadian Biosystems Engineering 46 (2004) 1.15–1.20. [5] M. Gil, L.R. Sinobas, L. Juana, Evolution of spherical cavity radius generated around a subsurface drip emitter, Biosciences Discussion 7 (2010) 1935–1958.

3983

[6] Y.A. Cengel, A.J. Ghajjar, Heat and Mass Transfer: Fundamentals and Applications, 4th ed., McGraw-Hill, Ryerson, 2006. [7] W. Wagner, A. Pruss, International equations for the saturation properties of ordinary water substance. revised according to the international temperature scale of 1990, J. Phy. Chem. Ref. Data 22 (3) (1993) 783–787.