ORIGINAL

Theoretical investigation on thermal performance of heat pipe flatplate solar collector with cross flow heat exchanger

Lan Xiao • Shuang-Ying Wu • Qiao-Ling Zhang •

You-Rong Li

Received: 17 November 2010 / Accepted: 4 January 2012 / Published online: 14 January 2012

� Springer-Verlag 2012

Abstract Based on the heat transfer characteristics of

absorber plate and the heat transfer effectiveness-number

of heat transfer unit method of heat exchanger, a new

theoretical method of analyzing the thermal performance of

heat pipe flat plate solar collector with cross flow heat

exchanger has been put forward and validated by com-

parisons with the experimental and numerical results in

pre-existing literature. The proposed theoretical method

can be used to analyze and discuss the influence of relevant

parameters on the thermal performance of heat pipe flat

plate solar collector.

List of symbols

Cp Specific heat capacity at constant pressure

(J kg-1 K-1)

D Heat pipe diameter (m)

G Mass flow rate of heated fluid, (kg s-1)

Io Solar intensity (W m-2)

Le Length of heat pipe evaporator section (m)

Lc Length of heat pipe condenser section (m)

N Heat pipe number

NTU Number of heat transfer unit

Nu Nusselt number

Pr Prandtl number

Re Reynolds number

S Absorbed solar intensity (W m-2)

Ta Ambient temperature (K)

Tb Wall temperature of heat pipe region (K)

Thp Working fluid temperature of heat pipe (K)

Ti Heated fluid inlet temperature (K)

UL Collector overall heat loss coefficient

(W m-2 K-1)

U Heat convection coefficient, (W m-2 K-1)

W Pitch distance between the heat pipe (m)

Greek symbols

a Absorptivity

d Absorber plate thickness (m)

k Thermal conductivity (W m-1 K-1)

s Transmissivity of glass cover, local time (h)

e Heat transfer effectiveness

g Thermal efficiency

1 Introduction

Solar thermal utilization is of great importance for envi-

ronmental protection and conventional energy saving.

A variety of flat plate solar collectors and evacuated tubular

solar collectors have been produced and applied around the

world. However, these conventional solar collectors suffer

from some drawbacks, such as reversed cycle during

cloudy periods of the day and the night, high heat capacity,

limited quantity of heat transferred by the fluid, high

pumping requirements, scale formation, freezing and cor-

rosion [1]. Heat pipes offer a promising solution to these

problems. Heat pipes are devices of very high thermal

conductance, which transfer thermal energy by two phase

circulation of fluid, and can easily be integrated into most

types of solar collector [2, 3]. The basic difference in

thermal performance between a heat pipe solar collector

L. Xiao � S.-Y. Wu (&) � Q.-L. Zhang � Y.-R. Li

Key Laboratory of Low-grade Energy Utilization Technologies

and Systems, Ministry of Education, Chongqing University,

Chongqing 400044, China

e-mail: shuangyingwu@yahoo.com.cn

L. Xiao � S.-Y. Wu � Q.-L. Zhang � Y.-R. Li

College of Power Engineering, Chongqing University,

Chongqing 400044, China

123

Heat Mass Transfer (2012) 48:1167–1176

DOI 10.1007/s00231-012-0972-3

and a conventional one lies in the heat transfer processes

from the absorber tube wall to the energy-transporting

fluid. In the case with a heat pipe, the process is evapora-

tion–condensation–convection, while for conventional

solar collectors, heat transfer occurs only in the absorber

plate. Thus, solar collectors with heat pipes have a lower

thermal mass, resulting in a reduction of start-up time.

A feature that makes heat pipes attractive for use in solar

collectors is their ability to operate like a thermal-diode,

i.e., the flow of the heat is in one direction only. This

minimizes heat loss from the transporting fluid, e.g., water,

when incident radiation is low. Another advantage is

redundancy, that is, a failure in one heat pipe would not

have a major effect on the operation of the collector. Also,

since heat pipes are sealed, by selecting suitable working

fluids, compatible with wick and pipe materials, corrosion

can be minimized. Furthermore, when the maximum

design temperature of the collector is reached, additional

heat transfer can be prevented. This would prevent over-

heating of the circulating fluid. Freezing can be eliminated

through working fluid selection, and, therefore only the

heat exchanger section must be insulated.

Numerous investigations on heat pipe solar collectors

have been conducted over years. Hussein [4, 5] investi-

gated theoretically and experimentally a thermosyphon flat

plate solar collector. The transient thermal behavior of

wickless heat pipe flat plate solar collectors was analyzed

with regard to a range of parameters. The results revealed

that the pitch distance limited the selection of an absorber

plate to one having a high value of thermal conductivity.

Also, from the theoretical analysis, it was accomplished

that the condenser section aspect ratio and the heat pipe

inclination angle had a considerable effect on the con-

densation heat transfer coefficient inside the inclined

wickless heat pipes. Later, a wickless heat pipes flat plate

solar collector with a cross flow heat exchanger was

investigated theoretically and experimentally under the

meteorological conditions of Cairo, Egypt [6]. The exper-

imental and theoretical results indicated that the number of

wickless heat pipes has a significant effect on the collector

efficiency. Furthermore, Hussein et al. [1] investigated

experimentally the effect of wickless heat pipe cross sec-

tion geometry and its working fluid filling ratio on the

performance of flat plate solar collectors. The experimental

results indicate that the elliptical cross section wickless

heat pipe flat plate solar collectors have better performance

than the circular cross section ones at low water filling

ratios. The optimum water filling ratio of the elliptical

cross section wickless heat pipe solar collector is about

10%, while it is very close to 20% for the circular cross

section one.

Much effort has been made on the design of heat pipe

solar collectors in the past few years. Riffat et al. [7]

presented the results obtained from laboratory testing of

four liquid flat plate collectors, i.e., a wavy fin collector,

two flat plate heat pipe collectors, and a clip fin solar

collector. Results showed the clip fin solar collector to be

promising, with experimental efficiencies approaching

86%. An analytical model has been developed to investi-

gate the performance of a ‘mini’ gravitational heat pipe and

two ‘micro’ gravitational heat pipes of different sizes,

using water as the refrigerant. In general, the modeling

results indicate ‘micro’ gravitational heat pipes have higher

heat transport limits than ‘mini’ heat pipes of the same

cross-sectional area [8]. Riffat and Zhao [9] designed and

constructed a hybrid heat pipe solar collector/CHP system

based on the integration of a hybrid heat pipe solar col-

lector, a turbine, a boiler, condensers and pumps. The

evaluation of the system performance by a combination of

theoretical modeling and experimental testing was also

conducted. It demonstrated that using more collector units

would help to improve the system’s energy efficiency [10].

Later, the same authors designed and constructed a thin

membrane heat pipe solar collector to allow heat from solar

radiation to be collected at a relatively high efficiency

while keeping the capital cost low. The test efficiency was

found to be in the range 40–70%, which is a bit lower than

the values predicted by modeling. The factors influencing

these results were investigated [11]. A two-phase closed

thermosyphon flat plate solar collector with a shell and tube

heat exchanger was designed, constructed, and tested at

transient conditions to study its performance for different

cooling water mass flow rates at different inlet cooling

water temperatures [12]. Abu-Zour et al. [13] designed new

solar collectors integrated into louvered shading devices.

Various designs of solar louver collector were discussed

and a new type of absorber plate based on heat pipe

technology was investigated. The compatibility of louvers

with different sizes and shapes could improve solar control

and the aesthetics of building facades. Yu et al. [14]

introduced and developed the prototype of a cellular heat

pipe flat solar collector. Theoretical and experimental

research showed that the thermal performance of the new

solar heater is better than that of evacuated glass tube solar

heater or ordinary flat plate solar heater. Experimental flat

plate solar collector operating in conjunction with a closed-

end oscillating heat pipe (CEOHP) or a closed-loop oscil-

lating heat pipe with check valve (CLOHP/CV) offered

reasonably efficient and cost effective alternatives to con-

ventional solar collector system that use heat pipes

[15, 16]. Efficiencies of about 62 and 76% were attained

separately for the two systems, which are directly compa-

rable to that of the solar collector by heat pipe. CEOHP and

CLOHP/CV system offer the additional benefits of corro-

sion free operation and absence of freezing during winter

months.

1168 Heat Mass Transfer (2012) 48:1167–1176

123

On the other hand, experiments were performed to find

out how the thermal performance of a two-phase ther-

mosyphon solar collector was affected by using different

refrigerants [17]. Three refrigerants (R-134a, R407C, and

R410A) were used in identical small-scale solar water

heating systems, among which R410A was found to show

the highest solar thermal energy collection performance.

Flow and heat transfer of a plate heat pipe solar collector,

of which the condenser has the shape of a rectangular

channel, was modeled and the heat transfer coefficient

assessed, using the Fluent code. Results showed that the

Nusselt number is significantly higher than the one for

forced convection in a rectangular channel with fully

developed boundary layers. In order to enhance heat

transfer, a modification to the rectangular channel was

analyzed, using baffles to improve flow distribution and

increase velocity [18]. Azad [19] investigated theoretically

and experimentally the thermal behavior of a gravity

assisted heat pipe solar collector. A theoretical model

based on effectiveness-NTU method was developed for

evaluating the thermal efficiency of the collector, the inlet,

outlet water temperatures and heat pipe temperature.

Optimum value of evaporator length to condenser length

ratio was also determined. Also, the performance of a

wick-assisted heat pipe solar collector was investigated

theoretically and experimentally. The simulated useful heat

flux, heat pipe temperature, water outlet temperature and

efficiency were in good agreement with experimental

results [20].

As seen from these extensive literature reviews, most

investigations on the performance analysis of heat pipe

solar collector have been accomplished by experimental

and numerical methods. The well known Hottel-Willer

equation [21], which was originally applicable to perfor-

mance evaluation of conventional flat plate solar collectors,

has been widely adopted for theoretical analysis on thermal

performance of heat pipe solar collectors. In fact, the flow

and heat transfer characteristic of heat pipe flat plate solar

collector is quite different from that of conventional flat

plate solar collector due to their disparate configurations.

Typically, conventional solar collectors use pipes attached

to the collecting plate where working fluid (water or air)

circulates either naturally or forcibly and transfers heat,

hence, the wall temperature of pipes gradually increases

along the working fluid flow direction. However, for a heat

pipe flat plate solar collector, the axial temperature dif-

ference of each heat pipe is approximately zero because of

heat pipe’s principle of vapor–liquid phase transition,

whereas the water flowing through a rectangular cross-flow

channel absorbs the heat from heat pipes, and the tem-

perature of water increases. There is no doubt that the

thermal performance of heat pipe solar collector should

have its own rule which is unlike that of conventional

flat plate solar collectors. Therefore, in this paper, the

derivation of new analytical solutions for thermal evalua-

tion of the heat pipe flat plate solar collector with cross

flow heat exchanger will be proceeded base on basic heat

transfer process of collector and effectiveness-number of

heat transfer unit (e-NTU) method of heat exchanger

design.

2 Computational model and theoretical analysis

2.1 Computational model

Figure 1 shows a heat pipe flat plate solar collector, which

consists of N wickless heat pipes. The evaporator sections

with length Le of the wickless heat pipes are welded to

absorber plate made of copper sheets. The condenser sec-

tions with length Lc of the wickless heat pipes are

immersed in a cooling manifold. The pitch distance

between the heat pipe is W, the heat pipe diameter is D and

the absorber plate thickness is d. The absorber plate and

heat pipe are made of copper, and the thermal conductivity

is k. The left and right sides of absorber plate which

contain point B0 and A0, respectively are adiabatic walls.

The ambient temperature is Ta. Solar intensity is Io, while

the solar energy intensity absorbed by absorber plate

is S (without glass cover: S = Ioa; with glass cover:

S = Io(as), where a is absorptivity of absorber plate and

heat pipe wall, s is transmissivity of glass cover. This study

will discuss the case with glass cover). The wall temper-

ature of each heat pipe region A1–B1, A2–B2, A3–B3,…, AN–

BN is assumed to be uniform, which are Tb1, Tb2, Tb3, …,

TbN, respectively. All these temperatures are unknown and

require to be solved.

As displayed in Fig. 2, for a collector with N heat pipes,

the heated fluid flows orderly through the condenser sec-

tions from the 1st heat pipe to the Nth one in a crosswise

manner. The heated fluid outlet temperature of the 1st

condenser section of heat pipe equals to the heated fluid

inlet temperature of the 2nd one, and so on. The final

temperature of heated fluid through N heat pipes is T0N,

which needs to be solved, and the heated fluid inlet tem-

perature Ti is a known quantity.

2.2 Theoretical analysis

For the energy balance equation and its solution for the

heat pipe collector as shown in Fig. 1, some hypotheses

have been made:

• The system is in quasi-steady state.

• The phase change of the working fluid inside the

wickless heat pipe occurs at approximately constant

Heat Mass Transfer (2012) 48:1167–1176 1169

123

temperature and the temperature difference between the

evaporator and condenser sections of the heat pipe is

less than 2�C.

• The thickness of absorber plate with high heat

conduction performance is much less than the width,

so the temperature variation along the thickness of

absorber plate is negligible.

• The overall heat loss coefficient between the absorber

plate of collector and the external environment is

treated as a constant.

• The heat pipe collector is well insulated and the heat

loss through the insulation is neglected. Meanwhile,

both end sides of the absorber plate are adiabatic.

Since the absorber plate is made of copper which is good

conductor of heat, the temperature gradient of the absorber

plate along the axial direction of heat pipe is negligible,

therefore the heat transfer process in the regions B0–A1,

B1–A2, B2–A3, …, BN–A0 of absorber plate is similar to the

heat conduction problem of a fin.

2.2.1 The region B0–A1

One-dimensional steady-state heat conduction of a fin with

width of (W - D)/2 is assumed for the heat transfer pro-

cess in the region B0–A1 of absorber plate. A differential

element with width of dx, as shown in Fig. 3, is taken to

analyze the energy balance. The assumption that the length

of collector along the axial direction of heat pipe is set to

be 1 are made. The net input heat rate from the solar

irradiation Qnet is

Qnet ¼ qnetdx ¼ ½S� ULðT � TaÞ�dx ð1Þ

where qnet is the net input heat flux; UL is the collector

overall heat loss coefficient.

In terms of the heat transfer analysis of B0–A1 region,

the heat transfer direction is parallel to ?x axes, thus the

energy balance of the differential element control volume

in Fig. 3 as a whole gives

Fig. 1 The heat pipe flat plate

solar collector. a The physical

model; b section A–A

Fig. 2 Temperature variation of heated fluid in the condenser section

of heat pipe

Fig. 3 The heat transfer process in B0–A1 region of absorber plate

1170 Heat Mass Transfer (2012) 48:1167–1176

123

Qx þ Qnet ¼ Qxþdx ð2Þ

In accordance with the Fourier’s law of heat conduction

and in combination with Eqs. 1, 2 would be

d2T

dx2þ S� ULðT � TaÞ

kd¼ 0 ð3Þ

The general solution of Eq. 3 is

T ¼ C0;1 expðmxÞ þ C0;2 expð�mxÞ þ ðS=UL þ TaÞ ð4Þ

where C0,1 and C0,2 are integration constants, m ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

UL=ðkdÞp

.

The corresponding boundary conditions (the point B0 is

coordinate origin) are

dT

dx

�

�

�

�

x¼0

¼ 0; T jx¼ðW�DÞ=2¼ Tb1 ð5Þ

Thus

C0;1 ¼ C0;2 ¼Tb1 � ðS=UL þ TaÞ

2chðmH=2Þ ð6Þ

where H = W - D.

Substituting Eq. 6 into Eq. 4, the final solution for the

temperature distribution is then

T ¼ chðmxÞchðmH=2Þ Tb1 � ðS=UL þ TaÞ½ � þ S=UL þ Ta ð7Þ

The heat flow rate which enters into the vertical plane

that contains point A1 by conduction can be expressed as

QA1 ¼ �kdLe

dT

dx

�

�

�

�

x¼H=2

¼ 1

2HLeF S� ULðTb1 � TaÞ½ � ð8Þ

where F is the standard fin efficiency for straight fins with

rectangular profile, and obtained from F ¼ ½thðmH=2Þ�=ðmH=2Þ.

2.2.2 The region B1–A2

When the heated fluid flows through the condenser of the

1st heat pipe to the Nth one successively, the temperature

of heated fluid gradually goes up. Accordingly, the working

fluid temperature of heat pipe rises up, from Thp1 to Thpn.

Heat flow directions of the regions B1–A2, B2–A3, B3–A4,

…, BN-1–AN are reverse to ?x axes. In a similar manner,

the heat transfer process in region B1–A2 of absorber plate

can be considered as one dimension steady-state heat

conduction problem of a fin with width of (W - D).

Combining the thermal conduction equation Eq. 3 with

boundary conditions x = 0, T = Tb1; x = W - D,

T = Tb2, the integration constants of Eq. 4 are obtained

C1;2 ¼½Tb1 � ðS=UL þ TaÞ� expðmHÞ

2shðmHÞ� ½Tb2 � ðS=UL þ TaÞ�

2shðmHÞ ;

C1;1 ¼ Tb1 � C1;2 � ðS=UL þ TaÞ

ð9Þ

The heat flow rate into the vertical plane that contains

point B1 is calculated by

QB1 ¼ �kdLe

dT

dx

�

�

�

�

x¼0

¼ �kdmLeðTb1 � 2C1;2 � S=UL � TaÞ

ð10Þ

On the basis of the temperature distribution of region

B1–A2 in absorber plate, the heat flow rate out of the

vertical plane that contains point A2 is written as

QA2 ¼ �kdLe

dT

dx

�

�

�

�

x¼W�D

¼ �kdmLe C1;1 expðmHÞ � C1;2 expð�mHÞ� �

ð11Þ

2.2.3 The region A1–B1 of No.1 heat pipe

The gained solar energy in the region A1–B1 of No.1 heat

pipe is

QA1B1 ¼ ðp=2ÞDLe S� ULðTb1 � TaÞ½ � ð12Þ

The useful energy obtained by the No.1 heat pipe is

Qhp1 ¼ GCpðT01 � TiÞ ð13Þ

where T01 is heated fluid outlet temperature of No.1 heat

pipe.

The energy balance of the region A1–B1 of No.1 heat

pipe as a whole gives

QA1 þ QA1B1 þ QB1j j ¼ Qhp1 ð14Þ

Then the following equation can be obtained,

ð1=2ÞHLeF S�ULðTb1�TaÞ½ �þðp=2ÞDLe½S�ULðTb1�TaÞ�þkdmLeðTb1�2C1;2�S=UL�TaÞ¼GCpðT01�TiÞ

ð15Þ

Equation 15 contains unknown temperature Tb1, Tb2,

T01.

2.2.4 The region An–Bn of No.n (n = 2, 3, 4,…, N - 1)

heat pipe

For the region An–Bn of No.n (n = 2, 3, 4,…, N - 1) heat

pipe, the heat transfer process of each region is similar.

Applying the analysis method of the region A1–B1, we also

have

Heat Mass Transfer (2012) 48:1167–1176 1171

123

kdmLeðTbn � 2Cn;2 � S=UL � TaÞþ ðp=2ÞDLe½S� ULðTbn � TaÞ�¼ GCPðT0n � T0ðn�1ÞÞ þ kdmLe½Cn�1;1 expðmHÞ� Cn�1;2 expð�mHÞ� ð16Þ

where

Cn�1;2 ¼½Tbðn�1Þ � ðS=UL þ TaÞ� expðmHÞ

2shðmHÞ� ½Tbn � ðS=UL þ TaÞ�

2shðmHÞ ð17aÞ

Cn�1;1 ¼ Tbðn�1Þ � Cðn�1Þ;2 � ðS=UL þ TaÞ ð17bÞ

Cn;2 ¼½Tbn � ðS=UL þ TaÞ� expðmHÞ

2shðmHÞ

�½Tbðnþ1Þ � ðS=UL þ TaÞ�

2shðmHÞn 2 ð2; 3; 4; 5; . . .. . .;N � 1Þ

ð17cÞ

In which, Tb(n-1) is the region An-1–Bn-1 temperature of

No.(n - 1) heat pipe; Tbn is the region An–Bn temperature

of No.n heat pipe; Tb(n?1) is the region An?1–Bn?1

temperature of No.(n ? 1) heat pipe; T0(n-1) is heated

fluid inlet temperature of No.n heat pipe. T0n is heated fluid

outlet temperature of No.n heat pipe. Among these, Tb(n-1),

Tbn, T0(n-1) and T0n are unknown.

2.2.5 The region AN–BN of No.N heat pipe

Since the analysis method of the region AN–BN of

No.N heat pipe is almost the same as that of the region A1–

B1 of No.1 heat pipe except for different boundary condi-

tions, thus neglected for brevity’s sake. The right side

surface of absorber plate is adiabatic. The appropriate

boundary conditions (the point BN is coordinate origin)

would be

T jx¼0¼ TbN ;dT

dx

�

�

�

�

x¼ðW�DÞ=2

¼ 0 ð18Þ

Therefore, application of the above boundary conditions

yields the integration constants of general solution Eq. 4 as

CN;1 ¼TbN � ðS=UL þ TaÞ

1þ expðmHÞ ;

CN;2 ¼ TbN � CN;1 � ðS=UL þ TaÞð19Þ

The heat flow rate into the vertical plane that contains

point BN is calculated by

QBN ¼ �kdLe

dT

dx

�

�

�

�

x¼0

¼ �kdmLeðCN;1 � CN;2Þ ð20Þ

The energy balance equation for the region AN–BN of

No.N heat pipe is written as

QBNj j þ QANBN ¼ QANj j þ QhpN ð21Þ

where QANBN is the gained solar energy in the region AN–

BN of No.N heat pipe; QAN is the heat flow rate out of the

vertical plane that contains point AN.

The useful energy gained by the No.N heat pipe is given

by

QhpN ¼ GCp T0N � T0ðN�1Þ� �

ð22Þ

Combining Eqs. 20, 21 and 22 leads to:

kdmLeð2CN;1 � TbN þ S=UL þ TaÞþ ðp=2ÞDLe½S� ULðTbN � TaÞ�¼ GCpðT0N � T0ðN�1ÞÞ þ kdmLe½CN�1;1 expðmHÞ� CN�1;2 expð�mHÞ� ð23Þ

It is clear that Eq. 23 contains unknown temperature

Tb(N-1), TbN, T0(N-1) and T0N.

2.2.6 The relation T0n with Tbn

For No.1 heat pipe, the heat transfer effectiveness e1 and

the number of heat transfer unit NTUc1 are defined as

e1 ¼ 1� expð�NTUc1Þ; NTUc1 ¼ ðAc1Uc;01Þ=ðGCPÞð24Þ

where Uc,01 is heat convection coefficient of heated fluid

and the condenser section, it can be obtained from

Nu = 0.26Re0.6Pr1/3 [22]; Ac1 is the heat exchange area of

heated fluid and the condenser section of heat pipe.

From e1 ¼ ðT01 � TiÞ=ðTc;01 � TiÞ, we have

T01 ¼ Ti þ ½1� expð�NTUc1Þ�ðTc;01 � TiÞ ð25Þ

where Tc,01 is the working fluid temperature of No.1 heat

pipe condenser section and equal to the working fluid

temperature of No.1 heat pipe Thp1. Considering that phase

transition heat transfer coefficient of heat pipe is rather high,

and the temperature difference of phase transition heat

transfer is much smaller than that of single phase

convection heat transfer, namely, Tc,01 = Thp1 & Tb1.

Therefore, Eq. 25 may be rearranged in the following form:

T01 ¼ Ti þ ½1� expð�NTUc1Þ�ðTb1 � TiÞ ð26Þ

Proceeding in a similar fashion, for No.n heat pipe

(n = 2, 3, 4,…, N), it is readily shown that

T0n ¼ T0ðn�1Þ þ 1� expð�NTUcnÞ½ � Tbn � T0ðn�1Þ� �

ð27Þ

Again,

en ¼ 1� expð�NTUcnÞ; NTUcn ¼ ðAcnUc;0nÞ=ðGCPÞð28Þ

It is assumed that each heat pipe is the same, namely

Acn = Ac1, Uc,01 = Uc,0n, we have

1172 Heat Mass Transfer (2012) 48:1167–1176

123

e1 ¼ e2 ¼ e3 ¼ e4 ¼ � � � ¼ en ¼ e ð29Þ

From Eqs. 26 to 29, the following equations are

obtained:

T01 ¼ Ti þ eðTb1 � TiÞ ¼ eTb1 þ ð1� eÞTi ð30Þ

T02 ¼ eTb2 þ ð1� eÞeTb1 þ ð1� eÞ2Ti ð31Þ

T0n ¼ Tið1� eÞn þ ð1� eÞn�1eTb1 þ ð1� eÞn�2eTb2

þ � � � þ ð1� eÞeTbðn�1Þ þ eTbn ð32Þ

where n 2 ð2; 3; 4; . . .;NÞ.From the relation T0n with Tbn, it is shown that Eqs. 15,

16 and 23 consist of N equations which contain N unknown

quantities. Obviously, closed form solutions can be

obtained from these equations. Namely, we can gain the

temperature Tb1, Tb2, Tb3,…, TbN of region A1–B1, A2–B2,

A3–B3,…, AN–BN of the heat pipes. Meanwhile, the heated

fluid outlet temperature of each heat pipe T01, T02, T03,…,

T0N can be found out. Further, the thermal efficiency of the

heat pipe flat plate solar collector g can be calculated as [6]

g ¼ GCpðT0N � TiÞI0Acoll

ð33Þ

where Acoll is area of collector, Acoll = NWLe.

3 Results validation

Take the collector parameters in Ref. [6], the heat pipe was

made of copper with length of 0.92 m and outer diameter

D of 0.0127 m. The lengths of evaporator and condenser

section, i.e., Le and Lc are 0.75 and 0.1 m, respectively. The

glass cover plate was sized in 0.76 m 9 1.9 m 9 0.004 m,

of which the transmissivity s is 0.9. The absorber plate was

made of copper with size of 1.89 m 9 0.75 m and it was

coated with anodic alumina spectral selective absorption

material of which the absorptivity a is 0.94. The condenser

section of heat pipe is placed in the heated fluid channels of

1.9 m 9 0.1 m 9 0.03 m. The heated fluid, namely water

flows in succession over the condenser section of the heat

pipe in a crosswise manner. The heat loss coefficient

between the absorber plate and environment UL is 8.6

W/m2 K [23]. Other parameters, such as solar intensity Io

and ambient temperature Ta are from Ref. [6]. Initial wall

temperature distributions were assumed for each heat pipe,

then the Gauss–Seidel iteration method was employed to

solve the above mentioned N equations, and the calculation

results are discussed in the following. For the convenience

of illustration and comparison, ‘‘th’’ means the theoretical

calculations in this paper; ‘‘exp’’ and ‘‘nu’’, respectively

denote the experimental and numerical results in Ref. [6].

The theoretical water outlet temperature in this paper is

compared with the experimental and numerical values in

Ref. [6] when the water flow rate G = 0.0458 kg/s, as

presented in Fig. 4a. Very small relative deviation (only

-0.1 to 0.4%) has been detected. Figure 4b demonstrates

comparisons of the collector thermal efficiency for the

same case. The relative deviation between theoretical and

experimental values is only -1.7 to 7.5%, while the rela-

tive deviation between the numerical and experimental

values in Ref. [6] was found to be -4.5 to 6.6%.

Similarly, Fig. 5a gives the comparisons of the theo-

retical water outlet temperature in this paper with the

experimental and numerical values in Ref. [6] when the

water flow rate G = 0.0125 kg/s. Again, minor relative

deviation exists between the two, which is only -1 to

0.8%. Figure 5b presents the collector thermal efficiency

comparisons. The theoretical values obtained in this paper

Fig. 4 The water outlet temperature and collector thermal efficiency

at G = 0.0458 kg/s in comparison with the experimental and

numerical results in Ref. [6]

Heat Mass Transfer (2012) 48:1167–1176 1173

123

agree well with the experimental values with relative

deviation of -5.2 to 5.8%. Whereas, the relative deviation

of the numerical and experimental values in Ref. [6] was

-4.9 to 29.2%.

According to the above analysis, fairly good agreements

have been observed in the theoretical and experimental

results, especially the water outlet temperature. This con-

firms that the theoretical method raised in this paper is

feasible and effective, and can be used with confidence for

the thermal performance analysis of heat pipe flat plate

solar collector.

In addition, seen from the theoretical values of water

outlet temperature and the collector thermal efficiency in

this paper and the experimental values in Ref. [6], the

relative deviation of collector thermal efficiency is always

greater than that of water outlet temperature. For example,

at 16:00 in Fig. 4, the water outlet temperature calculated

in this paper is only 0.3% higher than the experiment value

in Ref. [6], while the relative deviation of thermal effi-

ciency is enlarged to 7.5%. This is due to that, for the

calculation of relative deviation of water outlet tempera-

ture, it is defined as the calculated and experimental water

outlet temperature difference divided by the experimental

value. The relative deviation of collector thermal efficiency

is defined as the calculated and experimental water outlet

temperature difference divided by the actual water tem-

perature rise. Because the water temperature rise in actual

process is not too much (in Ref. [6], it is only several

Celsius degree), a small deviation of water outlet temper-

ature will result in great thermal efficiency deviation,

especially for the cases of weak solar radiation or increased

water mass flow rate. Compared Fig. 4 with Fig. 5, it is

noticed that as the water mass flow rate increases, the

thermal efficiency deviation of the theoretical and experi-

mental value increases, which is owning to that the

increase of water mass flow rate leads to the decrease of

water temperature rising. This also confirms the above

analysis.

To further validate the theoretical method proposed in

this paper, comparisons of water outlet temperature and

collector thermal efficiency between the theoretical and

experimental results in Ref. [6] for the collector with dif-

ferent heat pipe number and the water mass flow rate

G = 0.0292 kg/s have been presented in Fig. 6. It suggests

that as the heat pipe number increases, the theoretical

results obtained by the present method become closer to the

experimental results in Ref. [6]. With N = 10, 12 and 14,

the relative deviations of water outlet temperature are 1.56

to 2.4%, -0.83 to -1.97%, -0.91 to 1.4%, and the relative

deviations for collector thermal efficiency are 16.7 to 38%,

-8 to -14%, -6.8 to 13.9%. It should be noted that, the

collector thermal efficiency showed the highest when

N = 12 in Ref. [6]. However, the theoretical results in this

paper reveal that, the collector thermal efficiency increase

slightly as the heat pipe number decreases, namely, the

collector thermal efficiency does not change significantly if

the heat pipe number changes a little. Take the moment of

12:30 as an example, for the collector with N = 10, 12 and

14, their thermal efficiencies are 64, 63, 61%, respectively;

however, the results in Ref. [6] stated that the collector

thermal efficiencies respectively are 54, 70, and 64% with

N = 10, 12 and 14, which shows much more significant

change than that of this paper.

4 Conclusions

The equations describing the temperature distributions of

the absorber plate and the relationships between each heat

Fig. 5 The water outlet temperature and collector thermal efficiency

at G = 0.0125 kg/s in comparison with the experimental and

numerical results in Ref. [6]

1174 Heat Mass Transfer (2012) 48:1167–1176

123

pipe wall temperature and the water temperature rise in the

process of flowing over the condenser section of each heat

pipe were derived. The theoretical values of water outlet

temperature and the collector thermal efficiency have been

calculated when the relevant parameters, such as the solar

intensity, water inlet temperature and environmental tem-

perature are known. On the basis of present analysis it is

concluded that:

1. Using the present theoretical method, the values of water

outlet temperature and the collector thermal efficiency

agree well with those in Ref. [6], especially the water

outlet temperature values. Because of the collector

thermal efficiency definition, the relative deviation of

collector thermal efficiency is generally greater than that

of water outlet temperature.

2. As an increased the heated fluid temperature rise can

reduce the relative deviation of water outlet temper-

ature and collector thermal efficiency, this theoretical

method is more suitable for the collector under certain

conditions, i.e., intense solar radiation, small heated

fluid mass flow rate, and a collector with more heat

pipes.

3. The theoretical method proposed in this paper can be

used for heat pipe flat plate solar collector performance

analysis and assessment. It is an efficient approach to

analyze and discuss the influences of solar intensity,

water inlet temperature, environmental temperature,

heated fluid mass flow rate and collector’s structure

parameters on the heat pipe flat plate solar collector

performance.

Acknowledgments This work is supported by National Natural

Science Foundation of China (Project No.51076171). The authors

also wish to thank the support from Natural Science Foundation

Project of CQ CSTC (CSTC, 2010BB6062) and Project No. CDJXS

10141147 supported by Fundamental Research Funds for the Central

Universities.

References

1. Hussein HMS, El-Ghetany HH, Nada SA (2006) Performance of

wickless heat pipe flat plate solar collectors having different pipes

cross sections geometries and filling ratios. Energy Convers

Manag 47(11–12):1539–1549

2. Hussein HMS, Mohamad MA, El-Asfouri AS (1999) Transient

investigation of a thermosyphon flat plate solar collector. Appl

Therm Eng 19(7):789–800

3. Hussein HMS, Mohamad MA, El-Asfouri AS (1999) Optimiza-

tion of a wickless heat pipe flat plate collector. Energy Convers

Manag 40(18):1949–1961

4. Hussein HMS (2002) Transient investigation of a two phase

closed thermosyphon flat-plate solar water heater. Energy Con-

vers Manag 43(18):2479–2492

5. Hussein HMS (2003) Optimization of a natural circulation two

phase closed thermosyphon flat plate solar water heater. Energy

Convers Manag 44(14):2341–2352

6. Hussein HMS (2007) Theoretical and experimental investigation

of wickless heat pipes flat plate solar collector with cross flow

heat exchanger. Energy Convers Manag 48(4):1266–1272

7. Riffat SB, Doherty PS, Abdel Aziz EI (2000) Performance testing

of different types of liquid flat plate collectors. Int J Energy Res

24(13):1203–1215

8. Riffat SB, Zhao X, Doherty PS (2002) Analytical and numerical

simulation of the thermal performance of ‘mini’ gravitational and

‘micro’ gravitational heat pipes. Appl Therm Eng 22(9):1047–

1068

9. Riffat SB, Zhao X (2004) A novel hybrid heat pipe solar col-

lector/CHP system—part I: system design and construction.

Renew Energy 29(15):2217–2233

10. Riffat SB, Zhao X (2004) A novel hybrid heat-pipe solar col-

lector/CHP system—part II: theoretical and experimental inves-

tigations. Renew Energy 29(12):1965–1990

Fig. 6 The water outlet temperature and collector thermal efficiency

at N = 10, 12, 14 in comparison with the experimental and numerical

results in Ref. [6]

Heat Mass Transfer (2012) 48:1167–1176 1175

123

11. Riffat SB, Zhao X, Doherty PS (2005) Developing a theoretical

model to investigate thermal performance of a thin membrane

heat-pipe solar collector. Appl Therm Eng 25(5–6):899–915

12. Nada SA, El-Ghetany HH, Hussein HMS (2004) Performance of

a two-phase closed thermosyphon solar collector with a shell and

tube heat exchanger. Appl Therm Eng 24(13):1959–1968

13. Abu-Zour AM, Riffat SB, Gillott M (2006) New design of solar

collector integrated into solar louvers for efficient heat transfer.

Appl Therm Eng 26(16):1876–1882

14. Yu ZT, Hu YC, Hong RH, Cen KF (2005) Investigation and

analysis on a cellular heat pipe flat solar heater. Heat Mass Transf

42(2):122–128

15. Rittidech S, Wannapakne S (2007) Experimental study of the

performance of a solar collector by closed-end oscillating heat

pipe (CEOHP). Appl Therm Eng 27(11–12):1978–1985

16. Rittidech S, Donmaung A, Kumsombut K (2009) Experimental

study of the performance of a circular tube solar collector with

closed-loop oscillating heat-pipe with check valve (CLOHP/CV).

Renew Energy 34(10):2234–2238

17. Esen M, Esen H (2005) Experimental investigation of a

two-phase closed thermosyphon solar water heater. Sol Energy

79(5):459–468

18. Facao J, Oliveira AC (2005) The effect of condenser heat transfer

on the energy performance of a plate heat pipe solar collector. Int

J Energy Res 29(10):903–912

19. Azad E (2008) Theoretical and experimental investigation of heat

pipe solar collector. Exp Thermal Fluid Sci 32(8):1666–1672

20. Azad E (2009) Performance analysis of wick-assisted heat pipe

solar collector and comparison with experimental results. Heat

Mass Transf 45(5):645–649

21. Hottel HC, Willier A (1955) Evaluation of flat plate solar col-

lector performance. In: Transactions of conference on the use

of solar energy, vol II, thermal processes. University of Arizona,

Arizona, pp 74–104

22. Chi SW (1976) Heat pipe theory and practice. Hemisphere

Publishing Corp, Washington

23. Sodha MS (2006) Performance evaluation of solar PV/T system:

an experimental validation. Sol Energy 80:751–759

1176 Heat Mass Transfer (2012) 48:1167–1176

123