Theoretical model of an evacuated tube heat pipe

Energy 91 (2015) 911e924

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Theoretical model of an evacuated tube heat pipe solar collector integrated with phase change material M.S. Naghavi a, *, K.S. Ong b, I.A. Badruddin a, M. Mehrali a, M. Silakhori a, H.S.C. Metselaar a, ** a b

Center for Advance Materials, Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia Department of Industrial Engineering, Faculty of Engineering and Green Technology, Universiti Tunku Abdul Rahman, Kampar, Malaysia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 January 2015 Received in revised form 26 August 2015 Accepted 28 August 2015 Available online 24 September 2015

The purpose of this paper is to model theoretically a solar hot water system consisting of an array of ETHPSC (evacuated tube heat pipe solar collectors) connected to a common manifold filled with phase change material and acting as a LHTES (latent heat thermal energy storage) tank. Solar energy incident on the ETHPSC is collected and stored in the LHTES tank. The stored heat is then transferred to the domestic hot water supply via a finned heat exchanger pipe placed inside the tank. A combination of mathematical algorithms is used to model a complete process of the heat absorption, storage and release modes of the proposed system. The results show that for a large range of flow rates, the thermal performance of the ETHPSC-LHTES system is higher than that of a similar system without latent heat storage. Furthermore, the analysis shows that the efficiency of the introduced system is less sensitive to the draw off water flowrate than a conventional system. Analysis indicates that this system could be applicable as a complementary part to conventional ETHPSC systems to be able to produce hot water at night time or at times with weak radiation. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Solar water heater Heat pipe Phase change material Latent heat system Thermal energy storage

1. Introduction Several configurations of SWH (solar water heater) systems integrated with PCM (phase change material) as LHTES (latent heat thermal energy storage) have been developed and analyzed theoretically and experimentally in recent years [1e9]. In most of the designs, the PCM was placed in the hot water storage tank as a cylindrical or spherical packed bed parts [10e13]. The reports indicated that under certain conditions such as system configuration, type of PCM, inlet water temperature and water flow rate, there are improvements in the performance of the SWH-PCM systems. Kousksou et al. [6] studied the performance enhancement of SWH system with PCM-filled cylindrical containers in the storage tank and showed that there is an improvement in performance. Esen et al. [14] optimized cylindrical PCM packages in the hot water storage tank by linking the geometrical variables and lu et al. [15] found that hot water PCM characteristics. Canbazog

* Corresponding author. Tel.: þ60 379674451. ** Corresponding author. Tel.: þ60 379674451. E-mail addresses: [email protected] (M.S. Naghavi), h.metselaar@um. edu.my (H.S.C. Metselaar). http://dx.doi.org/10.1016/j.energy.2015.08.100 0360-5442/© 2015 Elsevier Ltd. All rights reserved.

production capability increased about three fold with PCM cylinders integrated into the storage tank. Talmatsky and Kribus [16] concluded that the performance of a SWH system with PCM cylinders in the storage tank might not be substantially beneficial because the system will be very sensitive to the PCM parameters, and it could lead to system failure. System performance of a SWH depends upon various factors such as collector and storage tank design and ambient conditions. Several comparative studies have shown that the performance of ETHPSC (evacuated tube heat pipe solar collectors) is higher than flat plate collectors under different weather conditions [17e20]. From these observations, it would seem feasible to enhance the performance of an ETHPSC-PCM system. Many types of PCMs have been suggested such as paraffin waxes, fatty acids and salt hydrates. Zalba and Canbazoglu [15,21] studied a wide range of PCMs and their characteristics. Though the high thermal energy density of a LHTES system makes it an appropriate energy storage device, the main weakness is the poor thermal conductivity associated with PCMs, which strongly affects the system performance. This weakness imposes a penalty on the extraction or removal of the heat storage. The rate of the phase change process is not up to the expected level. As a result, the LHTES system remains unreliable to

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M.S. Naghavi et al. / Energy 91 (2015) 911e924

Nomenclature

Symbols Ap Ac,c Af,c cp,l cp,s cp,w di,p dw dt Ed g Gsc hf, c hf, p hpcm hst hw Ib Id Ig IT keff kf,c kf,p kl ks kT kw K la lc le lf, c lf, p lp lst lst,s L Lr m_ mw n Na Nc Nf Ng Pf,c q ro,hpa ro,hpe ro,hpc rv,hpa rv,hpe rw,hpa

pipe inner surface area cross section area of the fin on HPC face area of the fin on the HPC specific heat of liquid PCM specific heat of solid PCM specific heat of water pipe inner diameter mesh wire diameter glass tube outer diameter total delivered energy gravity extraterrestrial radiation fin height on the condenser fin height on pipe heat transfer coefficient in PCM storage tank height heat transfer coefficient in water beam solar radiation diffuse solar radiation solar radiation diffusely total incident radiation effective thermal conductivity of wick thermal conductivity of fin on condenser thermal conductivity of fin on pipe thermal conductivity of liquid PCM thermal conductivity of solid PCM hourly clearness index thermal conductivity of heat pipe wall extinction coefficient of the glass length of heat pipe adiabatic region length of heat pipe condenser region length of heat pipe evaporator region fin length on the condenser fin length on pipe length of solar absorber plate storage tank length length of a section of the storage tank latent heat of PCM usable volume loss rate percentage water mass flow rate mass of the stored hot water in water storage tank number of the day in year number of apertures per unit length of wick tenths cloud cover number of the fins reflective Index of glass perimeter of the fin on HPC heat flux heat pipe adiabatic outer radius heat pipe evaporator outer radius heat pipe evaporator outer radius vapor core radius at adiabatic section vapor core radius at evaporator section wick radius at adiabatic section

rw,hpe Rb S St tCHE tDHE tf, CHE tf, DHE thp twck Tamb Tf Thp Tm Tpcm Ts Tu Tw,b,i Tw,b,o Tw,i,i Tw,i,o Tw,t,i Tw,t,o Vu Vu,d Vu,r wst Xpcm Ypcm

wick radius at evaporator section beam contribution ratio absorbed solar energy Stefan number [¼clDTl/L] PCM thickness in CHE mode PCM slab thickness in DHE mode thickness of fin on heat pipe thickness of fin on water pipe thickness of the heat pipe wall thickness of the wick ambient temperature film temperature heat pipe surface temperature PCM melting temperature PCM temperature cold water temperature water operating temperature inlet water temperature in baseline system outlet water temperature in baseline system inlet water temperature in innovative system outlet water temperature in innovative system inlet water temperature to water storage tank outlet water temperature from water storage tank usable volume of hot water destroyed usable volume of hot water remained usable volume of hot water storage tank width liquid-solid location in PCM slab solid-liquid location in PCM slab

Greek

a an as bs be g d ε εg εp εs

q qz l ri rg rl rs rw mpcm mw vw (ta)

f u h

tilted incidence normal incidence of glass thermal diffusivity [¼ks/rscs] slope of solar collector expansion coefficient of the PCM surface Azimuth angle declination porosity glass emittance plate emittance sky emissivity angle of incidence zenith angle longitude diffuse reflectance of that surface diffuse reflectance of the ground liquid PCM density solid PCM density water density dynamic viscosity of the PCM dynamic viscosity of the water mean velocity of water inside the pipe appropriate transmittance-absorptance latitude hour angle efficiency


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find widespread use. Therefore, it is necessary to find a way to improve the charging/discharging rate of the PCM. Various techniques applied to the LHTES systems to increase the heat transfer rate include using extended surfaces [22]. The role of different configurations of fins in enhancing the performance of the LHTES systems have been studied extensively [23e28]. It was found that this technique had significant influence on heat transfer enhancement, although their performance depends on the configuration and system design [29e31]. An analytical study by Sharifi et al. [32] showed that heat transfer rate to PCM by heat pipes is much higher in comparison with rods and tubes. Most of the existing investigations on SWH-PCM systems have been done by integrating the PCM inside storage tanks. To the authors’ knowledge, although there are few research works on the combination of heat pipe solar collectors and PCMs in storage tanks [33e35], there is no work done on the proposed ETHPSC-PCM system where there is an intermediate heat exchanger on the manifold (condenser side) of the heat pipe. A new ETHPSC-PCM system is described in the following section. This research plans to examine, analytically, the feasibility of this system based on actual meteorological data in a tropical country (Malaysia). The charging/discharging processes during the day/night times are determined. The performance of a conventional SWH compared with this innovative design. PCM melting/solidifying processes are determined by a one-phase Stefan problem quasi-stationary approximation method, which is one of the more widely used standard analytical solutions. Calculations are done by a computational code developed in Matlab.

2. The proposed system The proposed ETHPSC-PCM system is shown in Fig. 1. It consists of an array of ETHPSCs connected to a common manifold filled with PCM. This manifold acts as an LHTES tank. The evaporator sections (HPEs) of the ETHPSC are exposed to solar radiation. The condenser sections (HPCs) are inserted into finned sockets welded into the manifold. Solar energy absorbed by the evaporator is transferred to the condenser via heat pipe action of evaporation and condensation and stored in the PCM in the LHTES tank. Domestic cold water supplied via a finned HWSHE (hot water supply heat exchanger) pipe is heated up as it flows through the LHTES tank. Single-row vertical fins are assumed welded onto the heat pipe condenser sockets and also to the HWSHE pipe line to enhance heat transfer as shown in Fig. 2. Dimensions of the LHTES tank, ETHPSC and fins are shown in Table 1.

y

LHTES tank

Heat pipe

PCM Solar absorber

HWSHE

Fin on HPC Cold-water in Fig. 1. ETHPSC-PCM system.

Three different heat transfer processes occur during charging and discharging of the heat within the PCM are illustrated in Fig. 3. Solar energy absorbed by the evaporator section of the ETHPSC is conducted to the condenser section (SEA). The heat is then transferred to the PCM and represents the CHE (charging heat exchange) process. The stored heat is discharged from the PCM to the HWSHE pipe line and represents the DHE (discharge heat exchange) process. These processes are calculated independently and in the order in which they occur. 3. Theoretical model This section presents equations for mathematical modeling of the heat transfer processes.

Hot water out

x

Fig. 2. Fin designs for HPC section (top) and water supply pipe heat exchanger (bottom).

3.1. Solar energy Solar radiation incident on and absorbed by a tilted collector surface depends upon several variables like surface material properties, convection from the surface and reflectance factors. Duffie and Beckman [36] proposed a theoretical method for a tilted collector (Appendix A):

1 þ cos bs SðtÞ ¼ Ib ðtÞ$Rb $ðtaÞb þ Id ðtÞ$ðtaÞd $ 2 1 cos bs þ IT ðtÞ$rg $ðtaÞg $ 2

(1)

The Kuala Lumpur International Airport meteorological station located as shown in Table 2.

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Table 1 Dimensions of LHTES tank, ETHPSC and fins. Manifold and evacuated tube

Heat pipe

Item

Value

Item

Value

Number of heat pipes Glass tube outer diameter (mm) Glass tube thickness (mm) Absorber plate thickness (mm) Evacuated tube length (mm) Fin thickness on the CHE (mm) Storage tank height (mm) Storage tank width (mm) Storage tank length (mm) Fin height on the CHE (mm) Fin length on the CHE (mm) Fin thickness on DHE (mm) Fin height on DHE (mm) Pipe outer diameter (mm) Length of a section (mm) Slope of solar collector (o) Glass emittance Plate emittance Reflective Index of glass

20 58 2 1 1675 2 204 204 1725 200 200 2 91 18 142 20 0.88 0.95 1.53

Evaporator length (mm) Adiabatic length (mm) Condenser length (mm) Evaporator outer radius (mm) Condenser outer radius (mm) Wall thickness (mm) Wick thickness (mm) Vapor core radius (mm) Thermal conductivity of liquid (W/mK) Thermal conductivity of wall (W/mK) Porosity Permeability Thickness of the fiber (mm) Number of meshes per unit length

1675.00 30.00 70.00 4.00 12.00 0.50 1.00 3.00 0.63 385 0.5 1.5 109 2 107 4000

3.2. Heat pipe surface temperature

3.3. Charging mode

A schematic diagram of a cylindrical heat pipe is shown in Fig. 4. Zue and Vafai [38] introduced a set of mathematical equations, which provides an analytical solution for surface temperature along the length of a low temperature heat pipe. For steady state operation, the axial temperature distribution is obtained from:

Characteristics of the paraffin wax used as the PCM and copper fin properties are shown in Table 3. During the charging mode (CHE), the heat transferred from the heat pipe will be stored in the PCM. The PCM temperature distri-

. . 1 20 3 8 ln rw;hpe rv;hpe > SðtÞ 4@ln ro;hpe rw;hpe l 1 > c > A 1þ 5; > þ þ Tpcm ðtÞ þ > > 2plc hpcm;c ro;hpe kw keff le > > > > > . . > 2 3 < ln rw;hpa rv;hpa SðtÞ 4ln ro;hpa rw;hpa 1 Thp ðx; tÞ ¼ 5; > T ðtÞ þ þ þ > > pcm 2plc hpcm;c ro;hpa kw keff > > > > > > > > SðtÞ > : Tpcm ðtÞ þ ; 2phpcm;c ro;hpc L

It is assumed that there is no radial heat transfer across the HPE and the HPC. Conduction along the length of heat pipe is negligible. The surface temperatures of solar absorber plate and HPE are assumed equal. Multiple wire mesh screen is considered as wick type of the heat pipe. The effective thermal conductivity of the liquid-saturated wick and working fluid is calculated according to Peterson [39]:

keff ¼

ke ½ke þ ks ð1 εÞðke ks Þ ke þ ks þ ð1 εÞðke ks Þ

(3)

where, ε ¼ 1 (pNadw)/4 in equation (3). Natural convection coefficient in a rectangular cavity containing paraffin wax during melting is prepared by Ref. [40]:

nh i. o1=3 hpcm;c ðtÞ ¼ 0:072 g r2l cpl k2l be ðq=2Þ m where, q ¼ (Thpc(t) Tm) in equation (4).

(4)

0 x le

le x le þ la ;

(2)

le þ la x l

bution and the interface location need to be determined. The heat equation in the PCM is a 2nd order linear partial differential equation with non-homogeneous boundary conditions. According to Fig. 3, the heat transfer from the HPC to the PCM is symmetric on both sides. Therefore, calculations are determined only for the PCM on one side of the finned PCM. One important assumption in the quasi-stationary one-phase Stefan problem is that the imposed temperature on the hot plate must be uniform. The hot plate is the finned HPC. Another assumption is that the heat transfer from the HPC fin to the PCM occurs in one dimension. The PCM is initially at ambient temperature. Heat storage in PCM is sensible storage before it reaches its melting point and after that is latent storage. The charging mode contains of both sensible and latent heating. A flowchart of the calculations for charging mode is provided in Appendix B.1. 3.3.1. PCM sensible heating One of the basic conditions in the Neumann solution for the quasi-stationary one-phase Stefan problem is that the PCM must be at its melting point. Therefore, to achieve an accurate result, transient sensible heating calculations need to be done separately. The


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Fig. 3. Heat transfer process in LHTES tank.

Table 2 Location of meteorological station [37].

Table 3 Physical properties of Paraffin wax and copper fins [41,42].

Item

Value

Property

PCM

Fin (solid)

Latitude (o) Longitude (o) Surface Azimuth angle (o) Extraterrestrial radiation (W/m2) Ground reflectance

2.44 101.42 180.00 1367.00 0.30

Melting point ( C) Density solid/liquid at 15/70 C (kg/m3) Thermal conductivity of solid/liquid (W/mK) Specific heat solid/liquid (kJ/kgK) Volume expansion at DT ¼ 20 C (%) Heat storage capacity (kJ/kg) Dynamic viscosity (kg/ms)

64 990/916 0.349/0.167 2.76/2.48 0.293 174 0.00385

e 2713 380 0.96 e e e

PCM temperature is a function of position and time, T(x,t). A slab of the PCM, 0 x l, is considered to be solid at uniform temperature. Heat transfer through the PCM is by the imposed temperature at the interface of the PCM and the HPC fin. The other three edges are assumed insulated. The mathematical model of the onedimensional transient heating process can be expressed as:

rs cs

vT v2 T ¼ ks 2 ; vt vx

(5)

The method of separation of variables is used to solve the equation with the following initial and boundary conditions [43]:

Tðx; 0Þ ¼ Tamb < Tm ;

(6a)

Tð0; tÞ ¼ Thpc ðtÞ;

(6b)

Fig. 4. Heat pipe cross section. Reprinted from Ref. [38] with permission from Elsevier.

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3.4. Discharging mode

vT ¼0 vt x¼tCHE

(6c)

The solution of this heat conduction problem is as below:

Tðx; tÞ ¼

Thpc ðtÞ Tamb x þ Tamb L

þ

∞ X

2 2 a2 sn p t L2

Cn e

n¼1

npx ; sin L (7)

where, x denotes the position of the PCM in distance from the hot Z lpcm Thpc ðtÞ Tamb npx Sin dx; plate (HPC) and Cn ¼ l 2 pcm lpcm lpcm 0 n ¼ 1; 2; 3; …. 3.3.2. PCM latent heating During melting, one more boundary condition at the interface location, X(t), will be added to the problem. Therefore, the equation (5) must be solved by the following initial and boundary conditions for the range of 0 < x < X(t):

Tpcm ð0; tÞ ¼ Thp ðtÞ;

(8a)

Tpcm Xpcm ðtÞ; t ¼ Tm ;

(8b)

0

rl LXpcm ðtÞ ¼ kl

vTpcm ; vt x¼Xpcm ðtÞ

(8c)

Xpcm ð0Þ ¼ 0

(8d)

The values of the PCM properties are in the liquid phase. Existence of boundary condition (8c) makes the heat equation very complicated. According to the Neumann solution for the one-phase Stefan problem [44], the quasi-stationary solution for expression of the interface location and temperature distribution in the PCM in terms of the original physical variables for charging process takes the form:

Xpcm ðtÞ ¼ hf ;c

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2kl Stl tÞ=ðrcl Þ

Tpcm ðx; tÞ ¼ Thpc ðtÞ DTl

(9)

x ; 0 x Xpcm ðtÞ; Xpcm ðtÞ

(10)

where, DTl ¼ Thpc(t) Tm. Existence of fins on the HPC increases the rate of the heat transfer rate from the object to surrounding material. It was assumed that the temperature distribution in the fin surface is uniform. To include the influence of this assumption in the calculation process, the efficiency of the fin is determined from the following equations [45]:

hf $c ðtÞ ¼

qf ðtÞ ; qmax ðtÞ

The PCM inside the LHTES tank is divided into a finite number of slabs (j), as shown in Fig. 5. Energy balance is applied to each slab. Outlet water temperature of each slab is equal to the inlet water temperature of the next slab. The solidification rate of the PCM in each slab depends on the inlet water temperature. The finned HWSHE pipe is placed at the middle of the tank, so the PCM in each slab is divided to two equal parts on the upper side and lower side of the finned HWSHE pipe. Heat transfer occurs from both sides of the fin. The flowchart of the discharging mode is shown in Appendix B.2. The PCM solidifies due to the cold supply water entering at velocity v and temperature Tw,i,i into the storage tank. The unknown variables are the solideliquid interface progression rate in each slab and the outlet water temperature after each slab. The determination of these variables must be done simultaneously. It is assumed that in a physical process of discharging - when Stefan number is small (St < 2) e the PCM temperature remains constantly in the range of melting point, while the solideliquid interface is progressing [44]. Thus, the supply water temperature is rising while the slab j is solidifying. After the slab j fully solidified, it has effectively lost its thermal inertia, and its temperature will rapidly fall to the inlet water temperature at its location, and the water will quickly respond to one less slab. It means that supply water initially was warming up by N slabs, but after the earliest PCM slabs solidified, the supply water will be warmed up by (N-1) slabs. For this method of solution, few assumptions must be considered. As indicated in Fig. 3, it is assumed that in discharging mode the thermal energy transmission from the PCM to the finned HWSHE pipe occurs only in one direction (on the y axis). In addition, heat transmission in the direction of x can be ignored, when the axial temperature drop is not too substantial. It as assumed that heat conduction is negligible inside the water, which is thermally at the steady state, so that its temperature depends merely on the pipe direction, which is reasonable for fairly rapid flow, after an initial transient, if the pipe is not too narrow and not as well long. Another assumption is that the latent heat of the PCM is so large that the surface temperature of the PCM at fins, and the pipe walls can be taken as essentially equal to Tm during the entire solidification process. This assumption implies a uniform temperature of Tm in the PCM throughout the solidification. Large latent heat (small Stefan number) implies that the front will not penetrate very far into the PCM, and a large conductivity will tend to level up the temperature, so the assumption is plausible at least. If in charge

(11)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh mhf ; c 1 qf ðtÞ ¼ hpcm;c ðtÞPf ;c kf ;c Ac;c qðtÞ sinh mhf ; c

(12)

and

qmax ðtÞ ¼ hpcm ðtÞAf ;c qðtÞ where, q(t) ¼ (Thpc(t)Tpcm(t)), mðtÞ ¼

(13) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hpcm;c ðtÞPf ;c =kf ;c Af ;c . The

value of the efficiency of the fin will be multiplied to the equation (9), which causes to slow down the interface growth rate.

Fig. 5. Arrangement of PCM slabs.


process the PCM is not fully melted, then, in discharging process, only molten mass fraction of the PCM is considered as an active mass of the PCM slab. 3.4.1. Heat transfer to finned HWSHE pipe The rate of change of the stored thermal energy in the PCM slab obtains by:

_ qðtÞ ¼ q_ p ðtÞ þ q_ f ;p ðtÞ

(14)

where q_ p ðtÞ is the heat flux to the pipe and q_ f ;p ðtÞ is the heat flux through the fins. Total heat transfer to the supply water via the pipe wall and fins can be calculated as:

_ qðtÞ ¼ Ap hw ðtÞqðtÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coshmlf ; p 1 þ Nf hf $p hpcm;d ðtÞPf ;p kf ;p Ac;f qðtÞ sinhmlf ;p

where, m ¼

(15) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hpcm;d ðtÞPf ;p =kf ;p Af ;f . hpcm,d is calculated from equa-

tion (4) while q ¼ (Tm Tw,i,i(t)). Efficiency of the fin on the pipe, hf.p, is calculated by placing related variables to the equation (11). Then, by taking into account all parameters, outlet water temperature from each slab of the PCM will be obtained by:

_ Tw;i;o ðx; tÞ ¼ Tw;i;i ðx; tÞ þ qðtÞ cp;w m_

(16)

The convective heat transfer coefficient of the water could be estimated based on the thermal behavior of the flow in an isothermal pipe when the thermal entrance region is not fully developed and ReD < 2500 [46]:

hw ¼

kw dp;i

!

0

1 1=3

0:0018Gz B C @3:657 þ A; 2 2 0:04 þ Gz 3

(17)

where, Gz ¼ (ReDPrdi,p)/x, ReD ¼ (rwvwdp,i)/mw and Pr ¼ (cp,imw)/kw. 3.4.2. PCM solidifying To find out the solidification time of the PCM slabs, the equation (5) in the PCM must be solved again with different initial and boundary conditions. In this case, the heat equation must be solved based on the imposed heat flux (q_ < 0):

Tpcm ðx; 0Þ ¼ Tm ; 0

rs LYpcm ðtÞ ¼ ks

ks

(18a) vTpcm ; vt y¼Ypcm ðtÞ

vTpcm _ ¼ qðtÞ; vt y¼0

Ypcm ð0Þ ¼ 0

(18b)

(18c) (18d)

Values of variable are in liquid phase. In this method, the case of a melted slab at its melt temperature Tm is being solidified by an _ imposed flux qðtÞ at the fin faces. According to the Neumann solution method for a one-phase quasi-stationary Stefan problem, by including their initial and boundary conditions, the equation for the solideliquid interface location takes the form:

_ Ypcm ðtÞ ¼ qðtÞt rs L

(19)

If the melted PCM slabs are at a temperature higher than the melting temperature, to be able to include the effect of stored

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sensible heat, the latent heat magnitude can be modified by following equation [44]:

1 L ¼ L þ cl ðTL Tm Þ 2

(20)

where, TL ¼ meanðTpcm ðx ¼ 0 : lpcm ; tend ÞÞ is the mean temperature of the PCM at the end of the melting time. 3.5. System performance and efficiency This section compares the thermal performance of the proposed system with a conventional system. In the latter one, the hot water is stored in the storage tank without water draw-off. This system is considered as a baseline system. By considering the effects of stratification, efficiencies and potential of delivering hot water could be computed. 3.5.1. Thermal analysis of the ETHPSC baseline system In the baseline SWH system, cold water passes directly over the HPC in the manifold and the heated water stored in an insulated storage tank. By considering the same equations for heat pipe surface temperature mentioned in Section 3.2, the heat transfers from the HPC to the supply water are determined. The flowchart of the baseline ETHPSC system is provided in Appendix B.3. Each unit of solar panel consists of an array of ETHPSCs. Therefore, the thermal performance of the array of the HPCs can be analyzed based on external forced convection over a tube. Churchill and Bernstein [46] presented correlations for heat transfer from a cylinder to the environment:

dp;i hw ¼ kw

" 5=8 #4=5 ! 1=2 0:62ReD Pr 1=3 ReD 0:3 þ

1=4 1 þ 282000 1 þ 0:4 Pr 2=3 (21)

Heat transfer rate is determined from:

. _ p;w Tw;b;o ¼ Tw;b;i þ hw Ap Thpc Tw;b;i mc All properties are determined temperature Tf ¼ (Thpc þ Tw,b,i)/2.

(22) at

a

film

3.5.2. Usable hot water and efficiency During the day, solar heat is collected and stored in the LHTES tank. No water is drawing off during the day. The primary objective of the storage tank is to maintain the hot water temperature. The extent to which heat is degraded over time needs to be quantified. Effective operation of hot water stored in a water tank relies on natural thermal stratification [47]. Destratification arises due to both heat convection from the hot water layer to lower temperature layers and vertical conduction in storage tank wall, even when there is no water draw-off [48]. A recent research by Armstrong et al. [49], determined the loss rate of usable hot water in a standby hot water tank due to the effect of the destratification. They found that for a tank with a 74 L capacity and 0.7 mm thickness made of copper, which is cladded with an external 50 mm thick layer of rigid polyurethane insulating foam (0.028 W/mK) [50], hot water usable volume loss rate over 12 h is 2.1 L/hour. In other words, by dividing the usable volume loss rate to the tank capacity, 3% of the usable hot water will be destroyed in the tank. It is also assumed that the heat losses from the tank wall to the environment are negligible compared to the conjugate exchange with the water. Armstrong [47] defined usable hot water volume as equal to the volume of fluid from a tank that can be mixed to a useful final

918


operating temperature, Tu. The usable hot water volume could be computed by:

Vu ðtÞ ¼ mw

12 X Tw;t;o ðtÞ Tu Tu Tw;t;i ðtÞ

(23)

hr¼1

Since the tank is considered to be in unused mode, based on the usable volume loss rate percentage (Lr), the usable volume destruction per hour could be computed as follows:

Vu;d ðtÞ ¼ Vu ðtÞ Lr ð12 tÞ

(24)

The term of (12 t) presents the time that the produced hot water in the solar collector will be on standby in the tank. Therefore, the total remaining usable hot water after 12 h can be determined by:

Vu;r ¼

12 X

Vu ðtÞ Vu;d ðtÞ

(25)

hr¼1

Destratification does not occur in the PCM tank, so there is no destructive energy in the PCM tank (Vu,d(t) ¼ 0). According to equations (23) and (25) the usable hot water and total usable hot water volume produced by the PCM tank could be computed. The total delivered energy by usable hot water from the water tank and from the PCM tank can be calculated by:

Ed ¼ Vu;r cp;w

Tu Tw;t;i ðtÞ

(26)

The energy efficiency of the systems is determined by:

htot ¼

Ed S

(27)

4. Results and discussion This research studies the feasibility of the introduced system as a complementary system to current ETHPSC systems to be able to produce hot water at night time or at times with weak radiation. This system could be applicable for the purpose of instant and sustainable production of hot water in industries or households, by using a combined configuration of standard and “PCM integrated” ETHPSC systems. The main goal that was pursued in this analysis is the theoretical evaluation of storage capacity of the proposed system of merged ETHPSC and LHTES systems. In the analysis arrangement, during the daytime water will not pass through the manifold, and the system is only in charging mode and after sunset, the system controller lets water flow through the pipe built into the tank. Accuracy of the theoretical modeling of a system depends on the validity of the equations employed all assumptions mode. Duffie and Beckman [37] provided a set of mathematical equations to determine absorbable incident solar energy. This equation is validated by Soto et al. [51]. Equation (4) determines the natural convection coefficient of melted PCM in a rectangular cavity was experimentally obtained by Marshal [40]. Zue and Vafai developed equation (2) for the heat pipe surface temperature and showed that they were in very close agreement with experimental measurement [38]. Equations (17) and (21) are used to obtain the heat transfer coefficient of supply water for flow inside a pipe and for flow across a circular cylinder. The highest error for the former equation is proven experimentally to be less than 7% [52] which for the latter equation is completely coincident with experimental data [53]. In mathematical textbooks, the quasi-stationary approximation method is a well-known modeling method to determine reasonable tradeoffs between charge and discharge parameters to store all available heat and totally discharge the heat during a

discharge process. Alexiads and Solomon [44] showed that for a Stefan number of StL < 2, the error of this method is less than 10%. Prior analysis shows that the highest Stefan numbers in charging and discharging modes are 1.19 and 0.46, respectively. The result by phase-change interface equation introduced in Section 3 is compared with the numerical method introduced by Cheng [54]. In Fig. 6 the interface progression of a material with latent heat of 100 MJ/kg, density of 1 kg/m3 and melting point of 0 C with a constant imposed temperature 2 C inside a rectangular chamber in duration of the two million seconds. The developments of the interface from both methods are very close. 4.1. Solar energy Actual meteorological data for Kuala Lumpur, Malaysia on 22nd February 2010 is chosen for this simulation. Calculations show that absorbable solar energy to the collector plate is approximately 75% of recorded solar radiation (Fig. 7). The percentage of absorbed energy depends on several factors such as longitude and latitude of the place, the angle of the solar system, glass and plate emittance and irradiation, cloudiness and ambient temperature. 4.2. Charging process The PCM is initially in solid state. During the initial charging process, sensible heat storage occurs. After reaching the PCM's melting temperature latent heat storage takes over. The temperature of the melted fraction rises up continuously as long as the heat source is active. Fig. 8 shows the temperature gradient history in the PCM. As explained in Section 2 (Fig. 3), regarding symmetry matter, the determination is done on only one side of the PCM volume. The vertical axis in Fig. 8 represents the thickness of the PCM at each side of the finned HPC, while the horizontal axis is the duration of the day time. The temperature gradient fluctuation depends on heat flux variations. The fin efficiency of HPC is about 90% for copper material. As mentioned earlier, the system consists of 20 heat pipes. During the charging mode is defined in a way that the storage tank is in standby mode (supply water does not pass through the pipe). The thermoclines and the interface growth pattern of the PCM in all 20 slabs are similar. The progression of the liquidesolid interface is shown in Fig. 9. Around 07:00, the layer of the PCM adjacent to the HPC immediately starts to melt. At the end of the day, around 65 mm of the thickness of PCM at each side of the finned HPC has melted. The rate at which the interface progresses mainly depend on the availability of the solar radiation, and the PCM properties. It is observed that the interface progression rate is faster in the afternoon. This is attributed to two reasons. One is due to the higher temperature

Fig. 6. Comparison of the interface location.


1000 800 600 400

Total Absorbable Ambient T

200 0 6

8

10

12 Time

14

16

140 Temperature (oC)

Radiation (W/m2)

1200

180

Temperature (oC)

30 29 28 27 26 25 24 23 22

1400

919

100 HPE HPC PCM 1cm PCM mean

60

18

Fig. 7. Solar radiation and ambient temperature for Kuala Lumpur, Malaysia on 22.2.2010.

20

6

8

10

12 Time

14

16

18

Fig. 10. HPE, HPC and PCM temperatures.

and of the PCM 1 cm away from the finned HPC and mean PCM temperature. The mean PCM temperature is the average temperature of the PCM slab at the time. In the afternoon, at low solar radiation, the HPE and the HPC temperatures are almost equal due to the very low temperature difference between the HPC and the layer of the PCM in contact with it. The PCM mean temperature remains steady in the afternoon. A higher temperature of the front layer of the PCM causes the internal heat flux to the rear layer of the PCM; and with the lapse of time, there will be a smaller temperature gradient in the PCM. Due to the fact that the heat pipe heat transmission occurs only in one direction from the HPE to the HPC (acts like a one way valve [56]), the stored energy would not be wasted out of the storage tank.

4.3. Discharging process Fig. 8. PCM temperature history. Vertical axis is thickness of PCM slab (tCHE) in CHE mode.

gradient between the condenser and the PCM during the morning heating up time and the other is the presence of liquid PCM around the fin with its inherent low convective heat transfer coefficient. Fig. 10 shows the temperatures at the HPE and the HPC sections,

0.07 0.06

Thickness(m)

0.05 0.04 0.03 0.02 0.01 0

6

8

10

12 Time

14

Fig. 9. Liquid-solid interface in charging mode.

16

18

The direction of the heat transfer in the CHE mode is parallel to the x axis, while in the DHE mode it is parallel to the y axis (Fig. 3). The discharge process is assumed to start after sunset when the controller allows the supply water to pass through the finned HWSHE pipe. By paying attention to the fact that in the DHE, convective heat transfer inside the pipe is coupled to the heat removal from the PCM, the most valuable parameters to observe in the storage tank are outlet water temperature and solideliquid interface progression in different slabs of the storage tank. The simulated results of outlet supply water temperature are shown in Fig. 11 after 18:00 h at four different flow rates of 50, 60, 70 and 80 L per hour (lph). The inlet cold water inlet temperature is kept at 26 C. By increasing the flow rate, operating time of the system will decrease. The recommended operating temperature for domestic use is in the range of 38e41 C [55]. It is assumed at 39 C in this research. Hence, the operating time of the system with outlet water temperature over 39 C degrades from over 4 h to almost 3 h. The highest outlet supply water temperature is 53 C for a flow rate of 50 lph, while this is about 47 C for 80 lph. Refer to the explanation in Section 3.3, in discharging mode, due to the increment of water temperature during passing through the finned HWSHE pipe, the phase change behavior of the PCM varies over the length of the storage tank. According to the Fig. 3, the heat transfer from the melted PCM to the finned pipe is symmetric in both rectangular slabs. Fig. 12 shows the progression of the solideliquid interface in the LHTES tank during discharging mode for five slabs of the LHTES tank at four different flow rates. The results indicate that

920


Fig. 13. Outlet water temperature (Tw,b,o) for conventional ETHPSC system at different flow rates.

Fig. 11. Outlet supply water temperature (Tw,i,o) for different flow rates.

water starts at the same time as the PCM tank. While the outlet water temperature from the water tank is at a higher temperature than the operating temperature (39 C), it must be mixed with cold water (26 C) to reach to balance the temperature. Fig. 13 shows the outlet water temperature (Tw,b,o) simulated for a baseline ETHPSC system with 20 heat pipes at different flow rates. The magnitude of the produced hot water at operating temperature and the total efficiency of the systems can be estimated by equations introduced in Section 3.5. Fig. 14 compares usable volume (Vu) and efficiency (Eff) between the innovative and baseline systems. Generally, the thermal performance of the baseline system decreases when the draw-off flow rate increases. As a result, usable

earlier PCM slabs solidify sooner than later ones, because of the larger temperature difference of the inlet supply water and PCM. Due to the high temperature of the melted PCM at the end of the charging mode, refer to equation (20), a modified latent heat value of 229 kJ/kg was used for the interface progression determination.

4.4. Comparison with baseline system As described in Section 3.5, to be able to compare baseline and innovative systems, it is assumed that the produced hot water is stored in the water tank until afternoon and then release of the hot 0.1

0.07

0.08 0.07

0.06 0.05 0.04 0.03

0.01

50 lph 18

0.1

19

20 Time

21

22

0.07

60 lph 18

0.08 0.07

0.06 0.05 0.04 0.03

19

20 Time

21

22

Slab 1 Slab 5 Slab 10 Slab 15 Slab 20

0.09

Thickness (m)

0.08

0 0.1


0.09

Thickness (m)

0.04 0.02

0.01

0.06

0.05 0.04 0.03

0.02

0.02

0.01 0

0.06 0.05 0.03

0.02 0


0.09

Thickness (m)

0.08 Thickness (m)

0.1


0.09

0.01

70 lph 18

19

20 Time

21

22

0

80 lph 18

19

20 Time

21

Fig. 12. Progression of solideliquid interface in slabs 1, 5, 10, 15 and 20 in for different flowrates.

22


921

thermal conductivity is effectively reduced to the heat conduction properties of its solid metal casing alone. As most heat pipes are constructed of copper, an overheated heat pipe will generally continue to conduct heat at around 1/80 of the original flux via conduction only rather than conduction and evaporation [56]. This fact explains why the temperature of the HPC and the HPC meet each other at afternoon. Moreover, consideration of this point plays an important role in the selection of the proper PCM for such LHTES systems. 5. Conclusions

Fig. 14. Comparison of efficiency and usable hot water volume for different flow rates.

hot water volume decreases. The thermal efficiency of the innovative system does not drop significantly for different flow rates and remains in the range 55e60%. The evaporator and condenser surface temperatures of the first and last heat pipes of the present system are compared with those from a baseline system in Fig. 15 for two supply water flow rates at 50 and 80 lph. The HPC temperature fluctuation in both cases follows different patterns. The main reason is that in the baseline system, cold water passing through the manifold cools it down, while at the innovative system the surrounding material is stationary and it causes to low temperature difference between the finned HPC and the PCM. It may cause less performance by the heat pipe and some heat loss in the system (through heat conduction into the heat pipe wall). Physically, when the HPE is heated above a certain temperature, all the working fluid in the heat pipe vaporizes and the condensing process ceases; in such conditions, the heat pipe's 200

HPEi H PEi HPCi H PCi HPEb1 H PEb1 b1 50 50 HPCb1 H PCb1 5500 HPEb20 H PEb2 b20 50 50 HPCb20 H PCb20 550 0

180

Temperature (oC)

160 140

50 lph

A solar water heater system incorporating evacuated tube heat pipe solar collector and PCM heat storage was investigated theoretically. The analysis was conducted with actual meteorological data of a typical day in Malaysia. Analysis of the ETHPSC and PCM charging and discharging processes were presented. The one-phase Stefan problem quasi-stationary approximation solution was used for the PCM charging/discharging processes. The results showed that the thermal performance of the proposed system was higher than a baseline system for supply water flow rate higher than 55 lph. The analysis also shows that the sensitivity of the efficiency of the proposed system to the supply water flow rate is less than the baseline system. Further comparative and optimization studies are required to understand and to optimize factors such as collector area, fin design and draw-off schedule on the performance of the system. Experimental study of this research is under development and will be reported separately. Acknowledgments The authors would like to acknowledge the University of Malaya for financial support from the High Impact Research Grant (HIRG) scheme (project No: UM.C/HIR/MoHE/ENG/21) to carry out this research. Appendix A The solar radiation on a tilted surface includes three components: direct, isotropic diffuse and ground reflected diffuse. It is not possible to calculate the reflected energy term in detail. Standard practice is to assume that there is one horizontal surface with a large diffusely reflecting ground. The total incident radiation on the absorber surface can be written as [37]:

120 100 80 60 40

IT ¼ Ib Rb þ Id

20 6 200

10

12 Time

14

16

HPEi H PEi HPCi H PCi HPEb1 H PEb b1 1 80 80 HPCb1 H PCb1 8800 HPEb20 H PEb b2 20 8800 HPCb20 H PCb20 880 0

180 160 Temperature (oC)

8

140

18

1 þ cos bs 1 cos bs þ Ig rg 2 2

(A1)

where (1 þ cosbs/2) and (1 cosbs/2) are the view factors from the collector to the sky and from the collector to the ground, respectively. The absorbable solar radiation for a tilted collector (MJ/m2) is computed by:

80 lph

SðtÞ ¼ Ib Rb ðtaÞb þ Id ðtaÞd

120

1 þ cos bs 2

þ Irg ðtaÞg

1 cos bs 2

(A2)

100

Suitable transmittance (t) and absorptance (a) values are taken from Duffie and Beckman [37]. The ratio of total radiation on the tilted surface to that on the horizontal surface is determined by:

80 60 40 20

6

8

10

12 Time

14

16

18

Fig. 15. Comparison of HPE and the HPC temperature at two supply water flow rates.

Rb ¼

cos q cos qz

(A3)

where, q the angle between the beam radiation and the normal to the surface is determined from:

922


Io ¼

cos q ¼ sin d sin 4 cos bs sin d cos 4 sin bs cos g þ cos d cos 4 cos bs cos u þ cos d sin 4 sin bs cos g cos u þ cos d sin bs sin g sin u (A4)

qz zenith angle is determined according to Gupta et al. [57]: i0:5 o. n h cos1 ðf =gÞ cos qz ¼ f $cos1 ðf =gÞ þ g 1 ðf =gÞ2 (A5)

where f ¼ sind$sin4 and g ¼ cosd$cos4. The declination d is given by Iqbal [58]:

12 3600 360N Gsc 1 þ 0:033 cos p 365

cos 4 cos dðsin u2 sin u1 Þ p ðu2 u1 Þ sin 4 sin d þ 180

The ratio of diffuse and beam radiations are calculated by:

8 > > 1:0 0:09kT ; ðkT 0:22Þ Id < 0:9511 0:1604kT þ 4:388k2T 16:638k3T ¼ 4 > IT > : þ12:336kT ; ð0:22 < kT 0:80Þ 0:165; ðkT > 0:8Þ

Ib ¼

0:006758 cos 2 B þ 0:000907 sin 2 B 0:002679 cos 3 B

(A9)

and

d ¼ 0:006918 0:399912 cos B þ 0:070257 sin B

(A8)

1

Id IT

IT

(A10)

þ 0:00148 sin 3 B (A6) where B ¼ (n 1)360/365. An hourly clearness index kT is defined as:

kT ¼

IT Io

The value of Io can be calculated by:

Appendix B Flowchart diagrams of solution processes are prepared in this section: B.1. ETHPSC-PCM system e charging mode

(A7) t, x, represent time and position at thickness of the PCM, respectively.


923

B.2. ETHPSC-PCM system e discharging mode

B.3. Baseline ETHPSC system

j represents number of the PCM slab that are completely solidified. k is number of the HP. Y is solideliquid interface location. Properties of the water are calculated at Tf ¼ (Tw,i þ Tm)/2.

Properties of the water are calculated at Tf ¼ (Tw,in þ Thpc)/2.Tf needs to be find by trial and error method.

924


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