effect of phase change material on passive thermal heating of a greenhouse

INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. 2006; 30:221–236Published online 5 July 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/er.1132

Effect of phase change material on passive thermalheating of a greenhouse

Nisha Kumari, G. N. Tiwarin,y and M. S. Sodha

Centre for Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi, India

SUMMARY

In this study, a periodic analysis of a greenhouse with combination of phase change material (PCM) andinsulation as a north wall has been developed for thermal heating. The thermal model is based on Fourieranalysis. Effect of distribution of PCM thickness on plant and room air temperature has been studied indetail. The plant and room air temperature have been evaluated with and without north wall. Numericalcomputations have been carried out for a typical winter day of New Delhi. On the basis of numericalresults, it is inferred that (i) there is a significant effect of PCM north wall and heat capacity of planttemperature during off-sunshine hour due to storage effect and (ii) the rate of heat flux inside greenhousefrom north wall is maximum for least thickness of PCM. Copyright # 2005 John Wiley & Sons, Ltd.

KEY WORDS: greenhouse; solar energy; phase change material; thermal modelling

1. INTRODUCTION

Greenhouse is used for nursery, vegetable production and floriculture, etc. (Tiwari, 2003).Parameters to be monitored for high productivity are light, relative humidity, CO2 andtemperature inside the greenhouse. The temperature inside the greenhouse can be increased anddecreased as per heating and cooling of the greenhouse air. The heating of a greenhouse can beaccumulated either by passive or active mode. Passive mode uses the sun’s rays to heat a surfaceinside the greenhouse directly, storing the heat in a mass of concrete, rock, or water if required.At night the heated mass radiates the thermal energy to heat the air inside greenhouse fordesired air temperature. In an active mode, an additional thermal energy is fed insidegreenhouse from air heating system in addition to direct thermal heating.

The passive heating of a greenhouse can be achieved by the following methods:

* using North wall;* use of moving insulation during night;

Received 4 October 2004Revised 11 February 2005Accepted 18 March 2005Copyright # 2005 John Wiley & Sons, Ltd.

yE-mail: [email protected]

nCorrespondence to: G. N. Tiwari, Centre for Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi,India.

* rock-bed storage; and* use of isothermal mass inside greenhouse, etc.

Latent heat storage is one of the most efficient ways of storing thermal energy. Unlikethe sensible heat storage method, the latent heat storage method provides much higherstorage density, with a smaller temperature difference between storing and releasing heat(Farid et al., 2004). Phase change materials (PCMs) use chemical bonds to store and releaseheat. In a latent heat thermal storage (LHTS) system, during phase change the solid–liquidinterface moves away from the heat transfer surface. During this process, the surface heatflux decreases due to the increasing thermal resistance of the growing layer of the molten/solidified medium. In the case of solidification, conduction is the sole transport mechanism,and in the case of melting, natural convection occurs in the melt layer and this generallyincreases the heat transfer rate compared to the solidification process. This decrease of theheat transfer rate calls for the usage of proper heat transfer enhancement techniques in theLHTS systems. This is called a change in state, or ‘phase’. Initially, these solid–liquid PCMsperform like conventional storage materials; their temperature rises as they absorb solar heat.When the ambient temperature in the space around the PCM material drops, the PCMsolidifies, releasing its stored latent heat. PCMs absorb emit heat while maintaining a nearlyconstant temperature. Phase change materials (PCMs) when used for energy storage; provide ahigh-energy storage capacity, for example, 170 kJ kg�1 for calcium chloride hexahydrate(CaCl2 � 6H2O) (Sari and Kaygusuz, 2001). Greenhouse technology promises control over all theenvironmental parameters for commercial crop production and to fully exploit the yieldpotential of the crops.

In this study, the thermal performance of greenhouse with a PCM north wall (calcium chloridehexahydrate as a PCM) has been investigated theoretically in terms of room temperature, planttemperature, the various rates of heat transfer and thermal load leveling (TLL).

2. WORKING PRINCIPLE OF A GREENHOUSE

Green house is usually a covered structure of plastic film, which is transparent toshort wavelength radiation and opaque to long wavelength radiation. The solar radiation,S(t), is incident on the canopy of greenhouse as shown in Figure 1(a). A fraction of solarenergy fgSðtÞg; is reflected back from canopy and a part of the rest radiation, fð1� gÞSðtÞg;is transmitted inside the greenhouse. Out of this transmitted radiation, fð1� gÞtSðtÞg; afraction of this fFnð1� gÞtSðtÞg; falls on the north wall. After reflection from the surface,part of incident solar radiation on north wall, anFnð1� gnÞtSðtÞ; is absorbed by itself andrest is conducted. There are convective and radiative losses from the wall to room air.fagð1� FnÞð1� gÞtSðtÞg is reflected from the floor and fagð1� ggÞð1� FpÞð1� FnÞð1� gÞtSðtÞgis absorbed by the floor and rest is conducted inside the ground. After absorption bythe floor, there are convective and radiative losses from the floor to room air. The convectedand radiated energy from the floor raises the temperature of air inside greenhouse. Thetransmitted solar radiation through north canopy cover Fnð1� gÞ2t2SðtÞ is generally signi-ficant during the winter month. This can be retained inside the greenhouse by providingbrick north wall and more energy can be stored using phase change material with this wall(Tiwari, 2002).

Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Energy Res. 2006; 30:221–236

N. KUMARI, G. N. TIWARI AND M. S. SODHA222

3. ANALYSIS OF PHASE CHANGE MATERIAL

The energy balance at different boundaries of PCM as shown in Figure 1(b), areAt y ¼ �l1:

atIðtÞ ¼ �KldTl

dy

��y¼�l1

þUtðTljy¼�l1 � TrÞ ð1Þ

and(a)

Liquid Phase Solid Phase

I(t) Kl Ks

Cl Cs

Tl Ts

y=-l1 y= 0 y=l2

L (b)

Insulation PCM

Canopy Cover

Room air

Plant

Keff

Ceff

Ki

Ci

ρρ

x=0 y=L1y=L

Li=L1-L

y=0

Sun

Figure 1. (a) Greenhouse with insulated PCM north wall; and (b) PCM wall.


EFFECT OF PHASE CHANGE MATERIAL ON THERMAL HEATING OF A GREENHOUSE 223

At y ¼ 0:

Tlðy ¼ 0Þ ¼ Tsðy ¼ 0Þ ð2Þ

�KldTl

dy

��y¼0¼ �Ks

dTs

dy

��y¼0þ ’q ð3Þ

At y ¼ l2:

�KsdTs

dy

��y¼l2

¼ h2ðTs jy¼l2 � TaÞ ð4Þ

Here

Tl ¼ A1yþ B1 and Ts ¼ A2yþ B2 ð5Þ

where A1, A2, B1 and B2 are constants.Substituting the values of Tl and Ts from Equation (5) in Equations (1)–(4), one gets

A1ðKl þUtl1Þ �UtB1 ¼ �atIðtÞ �UtTa ð6Þ

B1 ¼ B2 ¼ T0 ð7Þ

�KlA1 � ’mcB1 þ KsA2 ¼ � ’mcTi ð8Þ

A2ðKs þ h2l2Þ þ h2B2 ¼ h2Ta ð9Þ

Since the entire mass of the PCM does not attain the same temperature, an average temperatureover the entire thickness has been taken. The steady-state temperature averaged over thethickness of the PCM can be obtained from.

%T ¼1

L

Z l2

l1

TðyÞ dy

¼1

L

Z 0

�l1TlðyÞ dyþ

Z l2

0

TsðyÞ dy� �

¼1

L�A1

l212þ T0l1 þ A2

l222� T0l2

� �ð10Þ

The effective thermal conductivity is obtained by taking the average of the thermal conductivityof the solid and the liquid portions i.e.

Keff ¼ Ks 1�l1

L

� �þ Kl

l1

L

� �ð11aÞ

By energy conservation,

MCeff ð %T � TaÞ ¼MCsð %T � TaÞ þMH0l1

L

� �ð11bÞ



By using average temperature from Equation (10) in the above equation, one can get theexpression for effective specific heat of PCM as

Ceff ¼ Cs þH0½l1=L�ð %T � TaÞ

ð12Þ

Here l1=L is the liquid fraction and M is the total mass of the PCM. The numerical values ofCs;Ks;Kl and Ho (Ismail and Castro, 1997; Ismail and Henriquez, 1997) have been taken as

Cs ¼ 1:64 kJ kg�1 8C�1

Ks ¼ 0:40Wm�1 8C�1

Kl ¼ 0:66Wm�1 8C�1

Ho ¼ 150 kJ kg�1

4. THERMAL ANALYSIS

In order to write the energy balance equations for different component of the system, thefollowing assumptions have been made:

(1) The orientation of the greenhouse is east west.(2) Absorptivity and heat capacity of the enclosed air is neglected.(3) Thermal properties of material/air are temperature independent.

Energy balance equations for different components of the greenhouse system, as shown inFigure 1(a), has been written as below:North wall

aNFNð1� gNÞtSðtÞ ¼ ’qcw þ ’qrw½ �AN þ �ANKeff@TE

@y

��y¼0

" #

or

aNFNð1� gNÞtSðtÞ ¼ hNAnðTE jy¼0 � TrÞ þ �ANKeff@TE

@y

��y¼0

!ð13Þ

At y ¼ L

Continuity of temperature

TEjy¼L ¼ TIjy¼L ð14Þ

Continuity of flux

�Keff@TE

@y

��y¼L¼ �Ki

@TI

@y

��y¼L

ð15Þ

At interface y ¼ L1

�Ki@TI

@y

��y¼L1

¼ hiðTIjy¼L1� TaÞ ð16Þ



Plant mass

apð1� gpÞFpð1� FNÞð1� gÞtSðtÞ ¼ ½’qep þ ’qcp þ ’qrp�AP þMpCpdTp

dt

or

apð1� gpÞFpð1� FNÞð1� gÞtSðtÞ ¼ AphpðTp � TrÞ þMpCpdTp

dtð17Þ

Floor

agð1� ggÞð1� FpÞð1� FNÞð1� gÞtSðtÞ ¼ AghgðT jx¼0 � TrÞ � KgAg@T

@x

��x¼0

ð18aÞ

As x!1; T is finite ð18bÞ

Room air

½’qcw þ ’qrw�AN þ ’qep þ ’qcp þ ’qrp� �

AP þ ’qcgAg ¼X

hðtÞðTr � TaÞ þ ’maCaðTr � TaÞ

orANhNðT jx¼0 � TrÞ þ AphpðTp � TrÞ þ AghgðT jx¼0 � TrÞ

¼ hðtÞX

AiðTr � TaÞ þ ’maCaðTr � TaÞ þ hdAdðTr � TaÞ ð19Þ

where SðtÞ ¼ SIiAi; Ii is the sum of beam and diffuse radiation given in Figure 2 on ith walls androofs of a greenhouse shown in Figure 1(a) (Liu and Jordan, 1962). In the above equation,

’ma ¼ Ca ¼VrtCa ¼

V n 1:2

ð3600=NÞn 1000 ¼

1

3NV ¼ 0:33NV

Since solar radiation and ambient air temperature are periodic in nature and hence parametersdepending on these can be represented mathematically in the form of Fourier series (Threlkeld,1970; Baldasano et al., 1998), as

SðtÞ ¼ Sto þReX6n¼1

Stneinot ð20Þ

Ta ¼ Tao þReX6n¼1

Taneinot ð21Þ

where

Tan ¼ Tane�icn ; Stn ¼ Stne

�ifn ; cn ¼ tan�1Bn

An

� �and fn ¼ tan�1

Bn

An

� �

Fourier coefficients namely Sto; Stn; cn; Tao; Tan and fn have been evaluated for a typical day ofwinter month as shown in Figures 2 and presented in Table I.

Due to periodic nature of S(t) and Ta, TE, TI, T, Tp and Tr can be expressed as follows:

TE ¼ Ayþ BþXðCne

bEy þDne�bEyÞeinot ð22Þ

TI ¼ A0yþ B0 þXðC0ne

bIy þD0ne�bIyÞeinot ð23Þ



0

20

40

60

80

100

120

140

160

180

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (Hours)

Diff

use

Rad

iatio

n (W

/m2 )

SW

SR

NR

NW

EW

WW

SW,NW,EW,WW

SR,NR

(i)

0

100

200

300

400

500

600

700

800

6 8 10 12 14 16 18 20Time (Hours)

Bea

m R

adia

tion

(W

/m2 )

SW

SR

NR

NW

EW

WW

SW

SR

NR

EW

WW

(ii)

0

2000

4000

6000

8000

10000

12000

14000

6 9 12 15 18

Time (Hours)

Tot

al R

adia

tion

(Wat

ts)

SR

NR

NW

EW

WW

SW

(a)

(b)

Figure 2. (a) Hourly variation of (i) diffuse radiation and (ii) beam radiation ondifferent walls of greenhouse for a typical winter day in New Delhi; and(b) hourly variation of total solar radiation incident on greenhouse for a typical winter

day in New Delhi.

Table I. Fourier coefficient of: (a) total solar intensity falling on greenhouse; and (b) ambient air temperature.

n 0 1 2 3 4 5 6

(a)Stn 9308.55 �14024 5179 176.9971 �937 521.99 �289.99fn 3.1417 6.2835 0.0057 3.1417 6.2838 3.1424

(b)Tan 13.4250 �4.0893 0.4800 0.0112 0.0190 �0.0209 �0.0135cn 4.0606 1.1727 0.0000 5.9497 3.3124 3.3124



T ¼ Aoxþ Bo þXðCnoe

bnox þDnoe�bnoxÞeinwt ð24Þ

Tp ¼ Tpo þReX6n¼1

Tpneinot ð25Þ

Tr ¼ Tro þReX6n¼1

Trneinot ð26Þ

where

bE ¼noreffCeff

2Keff

� �1=2ð1þ iÞ and

bI ¼norICI

2KI

� �1=2ð1þ iÞ

Using the condition given in Equations (18b) and (24), it gives

Ao ¼ 0; Cno ¼ 0 and hence

T ¼ Bo þX

Dnoe�bnoxeinot ð27Þ

With the help of Equations (20)–(27), time independent and dependent parts of Equations(13)–(19) form 7� 7 matrices. The matrix for time independent and time dependent part aregiven in Appendix. Thermal load levelling (TLL) for thermal heating of greenhouse have beencalculated by using the following expression:

TLL ¼ ðTr;max � Tr;minÞ=ðTr;max þ Tr;minÞ ð28Þ

For thermal heating of greenhouse TLL should be minimum for minimum fluctuation i.e.minimum ðTr;max � Tr;minÞ; the value of numerator and maximum value of denominator inEquation (28).

5. NUMERICAL RESULTS AND DISCUSSION

Equations (A8) and (A9) of Appendix have been computed by using Matlab for matrixinversion for a given design parameters of Table II and climatic parameters as shown inFigure 2. After knowing the unknown constants of Equations (A8) and (A9), varioustemperatures namely plant (Tp), room (Tr), PCM (TE), insulation (TI) and floor (T) given byEquations (22)–(26) have been evaluated and the variation of these temperatures has beenshown in Figures 3–5. One can observe that there is a significant rise in the plant temperaturedue to PCM north wall as shown in Figure 3(a). It is due to trapping of solar radiation falling onnorth wall inside greenhouse. The hourly variation of plant (Tp), room (Tr) and ambient air (Ta)temperature with PCM north wall have been shown in Figure 3(b). It is clear that the planttemperature (Tp) is higher than room temperature as expected. It is due to the fact that the plantreceives direct as well indirect thermal energy. The result is reverse in the night.



Table II. Input parameters used for computation.

Parameters Values Parameters Values Parameters Values Parameters Values

An 12m2 Fn 0.40 an 0.6 V 60

Ap 72m2 Fp 0.80 ap 0.4 N 0

Ag 24m2 gg 0.10 ag 0.4 Cp 4190

Ai 70.88m2 gn 0.05 hn 2.8Wm�2 8C�1 o 7.2� 10�5

Ad 2m2 gp 0.10 hp 15Wm�2 8C�1 t 0.65

Ki 0.057Wm�1 8C�1 Ci 1000 J kg�1 8C�1 hi 5.7Wm�2 8C�1

Keff 0.53Wm�1 8C�1 Ceff 6670 J kg�1 8C�1 hg 5.7Wm�2 8C�1

Kg 0.52Wm�1 8C�1 Cg 1880 J kg�1 8C�1 hd 1Wm�2 8C�1

Figure 3. Hourly variation of: (a) plant temperature without north wall and with north wall; and (b) plant,room and ambient air temperatures.

Figure 4. Effect of heat capacity by changing the plant mass: (a) on hourly variation of plant temperature;and (b) on hourly variation of room temperature.



Effect of isothermal mass on hourly variation of plant and room temperatures with PCMnorth wall has been shown in Figures 4(a) and (b), respectively. It is seen that the maxima of theplant and room temperature decreases with increases of mass of the plant due to storage effect.The minima of these temperatures increase during off-sunshine hour.

Figure 5(a) shows the hourly variation of PCM surface temperature (TE), insulationtemperature (TI) and floor temperature (T ) and indicates that the PCM surface temperature ishigher than other temperature due to direct exposure of north PCM wall. The correspondingrate of heat flux available at each surface has also been shown in Figure 5(b). Further it is to be

Figure 5. Effect of: (a) hourly variation of PCM surface temperature (TE), insulation temperature (TI) andfloor temperature (T ); and (b) hourly variation of heat flux from north wall to inside greenhouse, outside

greenhouse and at floor of greenhouse.

Figure 6. Hourly variation of heat flux from north wall: (a) inside; and (b) outside greenhouse for differentthickness of PCM (L) and insulation (Li) for a given total thickness of north wall (0.20m).



noted that the rate of heat flux from surface of PCM north wall to inside room is highestas expected.

Figures 6 and 7 show the hourly variation of the rate of heat flux coming insidethe greenhouse and going out of the greenhouse through PCM north wall for total thicknessof 0.20 and 0.10m, respectively. It is clear that the rate of heat flux inside greenhouseis maximum for least thickness of PCM i.e. 0.033 and 0.016m and correspondingly the

Figure 7. Hourly variation of heat flux from north wall: (a) inside; and (b) outside greenhouse for differentthickness of PCM (L) and insulation (Li) for a given total thickness of north wall (0.10m).

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.1 0.2 0.3 0.4 0.5 0.6

Fraction of PCM thickness over total insulated PCM wall {L/(L+Li)}

The

rmal

Loa

d Le

velin

g (T

LL)

L+Li=.10 m

L+Li=.20 m

Figure 8. The variation of thermal load leveling (TLL) with fraction of PCMthickness for constant PCM with insulation wall.



rate of heat flux going out of north wall is minimum as shown in Figures 6(b) and 7(b),respectively.

The thermal load levelling (TLL) for both cases with L=ðLþ LiÞ has been shownin Figure 8. It is clear that the minimum thermal load levelling is achieved at L=ðLþ LiÞ ¼0:30 for both thickness of PCM wall. This indicates the less fluctuation in room air tempe-rature.

6. CONCLUSIONS

Following conclusions have been drawn:

(i) The room temperature is higher during night than ambient air temperature due to PCMnorth wall, Figure 3(b).

(ii) The fluctuation in the rate of heat transfer from PCM north wall to room air reduces withincrease of PCM thickness, Figures 6(a) and 7(a).

(iii) The thermal load leveling (TLL) is minimum for L=ðLþ LiÞ ¼ 0:30 to total (10, 20 cm)thickness of insulated PCM north wall, Figure 8.

NOMENCLATURE

Ad = area of door (m2)Ag = area of greenhouse floor (m2)Ai = area of walls and roofs of greenhouseAn = area of north wall (m2)Ap = total surface area of plants (m2)C = specific heat (J kg�1 8C�1)Fn = fraction of solar energy falling at north wallFp = fraction of solar energy falling on the plantshd = overall heat transfer coefficient from door to ambient air (Wm�2 8C�1)hi = convective and radiative heat transfer coefficient between PCM north wall to

ambient (Wm�2 8C�1)hn = convective heat transfer coefficient from north wall to room air (Wm�2 8C�1)h(t) = overall heat transfer coefficient from room air to ambient air through canopy

(Wm�2 8C�1)h2 = heat transfer coefficient from PCM to ambient (Wm�2 8C�1)Ho = latent heat (J kg�1)Ii = total radiation on different walls of greenhouseI(t) = intensity of solar radiationKeff = effective thermal conductivity of PCM (Wm�1K�1)Ki = thermal conductivity of insulation (Wm�1K�1)l1, l2 = length (m)L = thickness of PCM wall

’m = flow rate (kg s�1)Mp = mass of the plant (kg)



n = nth harmonicN = number of air changesStn = time-dependent Fourier coefficient of total solar radiation on greenhouseSto = time-independent Fourier coefficient of total solar radiation on greenhouseS(t) = total solar radiation available on greenhouse canopy cover (W)t = time (s)T = temperature (8C)Ta = ambient air temperature (8C)Tan = time-dependent Fourier coefficient of ambient temperature (8C)Tao = time-independent Fourier coefficient of ambient temperature (8C)TE = effective temperature of PCM wall (8C)Ti = interface temperature inside PCM wall (8C)TI = insulation temperature (8C)Tp = plant temperature (8C)Tr = room air temperature (8C)%T = hourly average temperature (8C)T |x=0 = floor surface temperature of greenhouse (8C)T |y=0 = surface temperature of north wall (8C)Ut = overall heat loss coefficient of collector (Wm�2 8C�1)V = volume of greenhouse (m3)y = coordinate system along thickness of PCM wall

Greek letters

a = absorptivityg = reflectivityr = densityt = transmissivity of canopy cover

Subscripts

a = ambientcg = convection through groundcp = convection by plantscw = convection through north walleff = effective physical properties of PCMep = transpiration by plantsg = ground/floorI = insulationl = liquid phase of PCMN = north wallp = plantr = roomrp = radiation by plantsrw = radiation through north walls = solid phase of PCM



APPENDIX

A.1. Time-independent part

The time-independent part of Equations (13)–(19) can be written in the following form:

anFnð1� gnÞtSto ¼ �AnKeffAþ hnAnB� hnAnTro ðA1Þ

ALþ B ¼ A0Lþ B0 ðA2Þ

�Keff :A ¼ �KiA0 ðA3Þ

�KiA0 ¼ hiL1A

0 þ hiB0 � hiTao ðA4Þ

apð1� gpFpÞð1� FNÞð1� gÞtSto ¼ AphpTpo � AphpTro ðA5Þ

agð1� ggÞð1� FpÞð1� FNÞð1� gÞtSto ¼ AghgBo � AghgTro ðA6Þ

ðAghg þ ANhNÞBo � ðANhN þ Aphp þ Aghg þ hiAi þ 0:33NV þ hdAdÞTro þ AphpTpo

¼ ðhiAi þ 0:33NV þ hdAdÞTao ðA7Þ

The above equations have been arranged into 7� 7 matrix as

�ANKeff hNAN 0 0 0 0 �hNAN

L 1 �L �1 0 0 0

�Keff 0 Ki 0 0 0 0

0 0 �Ki þ hiL1 �hi 0 0 0

0 0 0 0 Aphp 0 �Aphp

0 0 0 0 0 Aghg �Aghg

0 0 0 0 Aphp ðAghg þ ANhNÞ X

2666666666666664

3777777777777775

A

B

A0

B0

Tpo

Bo

Tro

2666666666666664

3777777777777775

¼

aNFNð1� gNÞtSto

0

0

�hiTao

apð1� gpÞFpð1� FNÞð1� gÞtSto

agð1� ggÞð1� FpÞð1� FNÞð1� gÞtSto

ð�hiAi þ 0:33NV þ hdAdÞTao

2666666666666664

3777777777777775

ðA8Þ

where X ¼ �ðANhN þ Aphp þ Aghg þ hiAi þ 0:33NV þ hdAdÞ:



A.2. Time-dependent part

Similarly the coefficients of the time-dependent part can be obtained by solving the followingmatrix:

X1 X2 0 0 0 0 ð�hNANÞ

�expðbeLÞ X3 expðbiLÞ expð�biLÞ 0 0 0

X4 X5 ð�KibiÞexpðbiLÞ ðKibiÞexpð�biLÞ 0 0 0

0 0 X6 ð�ðKibiÞ þ hiÞexpð�biL1Þ 0 0 0

0 0 0 0 X7 0 ð�AphpÞ

0 0 0 0 0 X8 ð�AghgÞ

0 0 0 0 �ðAphpÞ �ðAghg þ ANhNÞ X9

2666666666666664

3777777777777775

�

Cn

Dn

C0n

D0n

Tpn

Dno

Trn

2666666666666664

3777777777777775

¼

aNFNð1� gNÞtStn

0

0

�hiTan

apð1� gpÞFpð1� FNÞð1� gÞtStn

agð1� ggÞð1� FpÞð1� FNÞð1� gÞtStn

ð�hiAi þ 0:33NV þ hdAdÞTan

2666666666666664

3777777777777775

ðA9Þ

where

X1 ¼ �AnKeffbe þ hNAN

X2 ¼ AnKeffbe þ hNAN

X3 ¼ �expð�beLÞ

X4 ¼ ðKeffbeÞexpðbeLÞ

X5 ¼ ð�KeffbeÞexpð�beLÞ

X6 ¼ ððKibiÞ þ hiÞexpðbiL1Þ

X7 ¼MpCpðinoÞ þ Aphp

X8 ¼ Aghg þ KgAgbn

X9 ¼ ANhN þ Aphp þ hiAi þ hdAd þ Aghg þ 0:33NV

The matrices given in Appendix have been solved by using Matlab 6.1 for constants A, B, A0; B0;Tpo; Bo; Tro and Cn; Dn; C0n; D

0n; Tpn; Dno; Trn for a given design in Table I and climatical

parameters in Figure 2. After knowing these constants, the hourly variation of TE, TI, T, Tp andTr can be evaluated from Equations (22)–(27).



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effect of phase change material on passive thermal heating of a greenhouse

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