THIN MEMBRANE HEAT-PIPE SOLAR ABSORBER WITH
FRESNEL LENS
CONTRACT JOE3-CT98-7020
TASK 2
MATHEMATICAL MODELLING OF THE MEMBRANE HEAT-PIPE SOLAR ABSORBER
FINAL REPORT TO THE PROJECT COORDINATOR
for the period of June 2000 to November 2000
Institute of Building Technology
School of the Built Environment
The University of Nottingham,
University Park,
Nottingham
NG7 2RD
UK
November 2000
CONTENTS
-
Introduction
- Mathematical simulation of collector efficiency and radiation distribution
- Optimisation of the structure, sizes and working conditions of the membrane
heat-pipe solar collector
-
Numerical simulation of the thermal behaviour of the membrane heat-pipe solar collector
-
Conclusions
-
References
-
Nomenclature
- Appendix 1 Optimisation of the structure, sizes and working
conditions
- Appendix 2 – Numerical simulation of the thermal behaviour of the
membrane heat-pipe solar absorber
- Appendix 3 - Theoretical and experimental Investigation of the heating &
cooling process of the Membrane Solar Collector
-
Figures
1.
Introduction
A comprehensive mathematical model for a thin membrane heat-pipe solar collector has been
developed. The model investigates the optimum sizes and structure, and also assesses the steady
state and transient performance of the solar collector. The collector is designed to be a compact,
efficient and low cost heat generator achieving temperatures of up to 250oC that may be used to
generate electricity and hot water for domestic and industrial use [1].
Figures 1 and 2 show the schematic variations for a ‘normal’ and ‘thermosyphon’ type collector.
Figure 3 shows a schematic cross-section of the main items of the collector. The main body of the
collector comprises two plates separated by a thin evaporation gap. The plates are ‘spot’ welded
together creating mini-channels (or ribs) that are parallel along the width of the absorber (Figure
4). In the mathematical model each mini-channel is considered to be a single micro heat-pipe. The
micro heat-pipes connect the evaporator section to the condenser section of the collector enabling
the flow of refrigerant vapor and condensed liquid refrigerant. As shown in Figures 1, 2 and 3 two
strips of micro heat-pipes (henceforth described as super heat-pipes) are attached to the main
absorber. Fresnel lenses are used to concentrate direct radiation onto the super heat-pipes by up to
a factor of 5 to 1. In operation, it is assumed that a fraction (10%) of the direct radiation is
scattered onto the main absorber plate. A micro-pore Vacuum Super Insulation (VSI) material and
a thin aluminum sheet are fitted beneath the absorber plate to reduce heat losses. A vacuum is
maintained between the glass cover and the top-face of the absorber panel so as to reduce
radiation and convection losses.
Two variations of the thin membrane heat-pipe solar collector have been investigated. These are
classified as a ‘normal’ and a ‘thermosyphon’ collector according to the method of return of the
condensed refrigerant. In the case of the ‘normal’ collector, condensed refrigerant returns to the
evaporation section along the sides of the micro heat-pipes by the combined effect of capillary
and gravity forces. Part of the returned liquid evaporates, on absorbing solar irradiation striking
the absorber surface, and the remainder is returned to the reservoir. This collector works on the
principle of micro gravitational heat-pipes and hence are called ‘normal’ heat-pipe collectors. In
the case of the ‘thermpsyphon’ collector condensed refrigerant returns to the reservoir via a tube,
as is shown in Figure 2. Liquid refrigerant from the reservoir flows to the evaporation section by
capillary action whereby the refrigerant is vaporised, thus creating a continuous ‘thermosyphon’
effect.
2.
Mathematical
distribution
simulation
of
collector
efficiency
and
radiation
Data given in the CIBSE guide [2], and equations for determining solar radiation and collector
efficiency [3] have been used to mathematically model the solar heat-input and collector
efficiency of the membrane heat-pipe solar collector.
The basic formula for the collector efficiency is given as,

t h  t a 

Io
  fr 1 2 aa  U


The parameter U is the overall heat transfer coefficient describing radiation, convection and
conduction heat losses from the collector body to the surroundings. The mathematical model
simulating solar irradiation, collector efficiency and heat losses is described in [4] and [5]. For a
standard summer day (Figure 5) ambient temperature varies between 19oC to 26oC and the solar
radiation (Figure 6) varies from 0-1,000 W/m2. The number of daylight hours is about 10-12
hours. The theoretical efficiencies for the collector vary between 0 to 79% as shown in Figure 7.
Figure 7 also shows increase in the collector efficiency with the use of concentrating lenses. An
indication of the heat output, for a collector area of 0.25m2, is given in Figure 8.
In the mathematical simulation the solar distribution on different areas of the membrane heat-pipe
collector is determined as indicated in the example given below. The total collector area is 0.25
m2. A fraction (10%) of the concentrated direct radiation is assumed to be incident on the main
body of the absorber.
Solar irradiation: 1000 W/m2
Direct radiation: 800 W/m2
90%
Concentrated heat pipe: 720 W/m2
Diffuse radiation: 200 W/m2
10%
100%
Un-concentrated heat pipe: 280 W/m2
Solar input calculation
Area: 0.05m2; *
Concentration ratio: 5:1;
Efficiency with concentration
(150oC): 0.758
Number of micro heat pipe: 6
Solar input: 720 x 0.05 x 5 x
0.758 = 136.44 W
Solar input to each heat pipe:
136.44/4 = 34.11 W
Solar input calculation
Area: 0.2m2;
Concentration ratio: 1:1;
Efficiency without
concentration (150oC): 0.596
Number of micro heat pipe: 13
Solar input: 280 x 0.20 x 1 x
0.596 = 33.34 W
Solar input to each heat pipe:
(280x0.2x0.596)/13 = 2.57 W
EXPLANATION
Solar input = radiation x area
x concentration x efficiency
EXPLANATION
Solar input = radiation x area x
concentration x efficiency
Further simulation conditions are:
- Total area of super heat pipe = 0.07 m2, assumes 0.02 m2 not focused.
- Panel sizes: 1100 mm x 250 mm (length x width);
- Length of evaporation section: 1000 mm;
- Length of condensation section: 100 mm;
- Heat inputs: 34.11 W for each concentrated super heat pipe; 2.57 W for each unconcentrated micro heat pipe in the absorber;
- Liquid fill level: 0.25m;
- Inclination angle: 60 deg.
3.
Optimisation of the structure, sizes and working conditions of the
membrane heat-pipe solar collector
‘Normal’ and ‘Thermosyphon’ micro heat-pipe
The collector is considered to consist of numerous micro heat-pipes parallel in width (see Figure
4). A micro heat-pipe was originally defined as
“a heat-pipe so small that the mean curvature of the vapor-liquid interface is necessary
comparable in magnitude to the reciprocal of the hydraulic radius of the total flow
channel” [6].
In practical terms a micro heat-pipe is a “wick-less”, non-circular channel with an approximate
equivalent diameter of 0.1mm to 1mm. In the case of the membrane heat-pipe collector the micro
heat-pipes array with a certain inclination, say 60 degrees to the horizontal surface, and hence
they are actually gravity micro heat-pipes (also called micro closed two-phase thermosyphon) in
which gravity force plays a significant role. Strictly speaking in the case of the membrane heatpipe collector, the term “gravity micro heat-pipe” is not correct since each heat-pipe cannot be
scaled down to the sub-millimeter range. Reasonable sizes are about 1mm to 3mm diameter, but
generally no more than 2mm equivalent diameter. It is therefore more precise to speak of
“miniature gravity heat-pipes”. In contrast with conventional gravity heat-pipes, some new
problems will be encountered with miniature gravity heat-pipes in the simulation, i.e., nonnegligible effect of capillary forces on fluid flow and the larger effect of viscosity on flow
resistance [7].
Due to the small radius of curvature within the ribs of the heat-pipes, the corner regions serve as
liquid arteries producing capillary pressure differences that promote the flow of the fluid from the
condenser to the evaporator. Even so, traditional steady-state modeling techniques can still be
used to formulate an initial estimate of the operational characteristics and performance limitation
of the membrane heat-pipe collector. The “heat transport capacity” which is a measure of the
performance of a micro heat-pipe has been used as the index for optimization in the mathematical
model (Appendix 1).
A computer program, based on the analysis given in Appendix 1, has been developed to
determine the optimum rib sizes of the micro heat-pipe (e.g., rib width and rib gap width) and the
working conditions (e.g., operating temperature, inclination and liquid fill level) of the membrane
solar collector [8]. The relationship between rib sizes and heat transport capacity of the collector
is simulated and the results are shown in Figure 9. The optimum rib width “A” and rib gap width
“B” is determined at the maximum heat transport capacity. Figure 9 shows the optimum rib width
and rib gap width to be 9.8mm and 0.63mm for the ‘normal’ micro heat-pipe and the
‘thermosyphon’ micro heat-pipe. However the maximum heat transport capacity for the ‘normal’
collector is greater than that for the ‘thermosyphon’ collector. This suggests that the ‘normal’
collector could be the favorable structure in the final design.
A further geometrical option for the micro heat-pipe was considered in the mathematical
modelling study. The geometrical shape of this option is as shown in Figure 11. The schematic of
the collector is shown in Figures 10 and 12. The absorber is considered to consist of 19 micro
heat-pipes parallel in width, of which 4 micro heat-pipes are focused with Fresnel lenses. This
membrane heat-pipe collector also has two variations for fluid flow and heat transfer, i.e.,
‘normal’ and ‘thermosyphon’. The results from the mathematical modeling are described below.
Optimisation simulation results
Liquid fill level
The variation of heat transport capacity with liquid fill level is simulated and the results are
shown in Figures 13 and 14. It is found that for the ‘normal’ heat pipe collector the limit of heat
transport capacity increases somewhat linearly with liquid fill level. Theoretically, the higher the
level, the larger the heat transport capacity. However, practical consideration gives the limit of
liquid fill level to be 1/4 to 1/3 of the evaporation volume. This is because liquid fill level greater
than 1/3 of the evaporation volume would cause un-evaporated liquid “blocks” being carried to
the condensation section, resulting in inefficient heat transfer [9].
For the ‘thermosyphon’ collector the heat transport capacity remains constant with variation in
liquid fill level. This is because the dominant limit for heat transport capacity is entrainment limit,
which does not vary with liquid fill level in the ‘thermosyphon’ option.
Inclination
Variation of heat transport capacity with inclination is simulated and the results are shown in
Figures 15 and 16. It is seen that the limit of heat transport capacity increases when inclination
varies from 0deg to 20deg. Further increase in inclination has no significance on the improvement
of heat transport capacity limitation in all cases.
Working temperature
Variation of heat transport capacity with working temperature is simulated and the results are
shown in Figures 17 and 18. It is found that the limit of heat transport capacity increases linearly
with working temperature. The major factor influencing heat transport capacity is entrainment
effect, which is caused by the narrow rib gap width. Variation of other geometry parameters does
not affect panel’s thermal performance significantly.
Figure 18 also shows that the heat transport capacity does not change beyond a working
temperature of 175oC, for the ‘thermosyphon’ collector (for the geometry given in Figure 11).
However, this could not be concluded as the case, since the simulation was not carried out beyond
200oC.
It is interesting to note that the geometry given in Figure 4 result in higher values of heat transport
capacity for the ‘normal’ collector whereas the geometry given in Figure 11 result in higher limits
of heat transport capacity for the ‘thermosyphon’ option.
4.
Numerical simulation of the thermal behaviour of the membrane heat pipe solar
collector
Mathematical theory
The finite element method is used to analyse the heat and mass transfer in each micro heat-pipe.
The grid division used for the simulation is as shown in Figure 19. The length step is taken as
1mm and each length step is taken as the unit of an element where differential equations given in
Appendix 2 are applied for simulation. The thermal performance of the collector in both steady
state and transient state operations are analysed in detail.
The high thermal conductivity of heat-pipes is the result of evaporation and condensation
processes occurring within the heat-pipe. Determination of the evaporation and condensation rate
plays a key role in evaluating the thermal characteristics and heat transport limitations of heatpipes. In the numerical model, an expression for the free molecular flow mass flux of evaporation
j, presented by Collier [10] and later used by Colwell and Chang [11] is employed, as described in
Appendix 2.
Numerical simulation conditions
A numerical simulation program has been developed to describe the thermal behavior of the
micro heat-pipe and the solar collector with part-surface concentration. The basic conditions for
simulation are summarised in Tables 1 and 2.
Table 1. Summary of the simulation conditions of the panel collector
(For the geometrical shape as shown in Figure 4)
Length of
evaporation
section, mm
1000
Solar
irradiation,
W/m2
1000
Length of
condensation
section, mm
Width of the
panel, mm
Concentrated
area,
mm x mm
100
250
1000 x 50
Unconcentrated
area, mm x
mm
1000 x 180
Rib width a, Rib gap width
Working
mm
b,
temperature,
oC
mm
9.8
Efficiency of Solar input to Efficiency of
Ratio of
Solar input to
the panel area
the area
the panel area concentration the area with
without
without
with
concentration,
concentration concentration, concentration
W
W
59.6%
33.34/2.57
75.8%
5
136.45/34.11
each
each
0.63
150
Liquid fill
level, mm
Inclination of
the panel,
deg
250
60
Table 2. Summary of the simulation conditions of the panel collector
(For the geometrical shape as shown in Figure 11)
Length of
evaporation
section, mm
Length of
condensation
section, mm
Width of the
panel, mm
Concentrated
area,
mm x mm
1000
100
250
1000 x 50
Solar
irradiation,
W/m2
1000
Unconcentrated
area, mm x
mm
1000 x 180
Rib width a, Rib gap width
Working
mm
b,
temperature,
oC
mm
5.00
Efficiency of Solar input to Efficiency of
Ratio of
Solar input to
the panel area
the area
the panel area concentration the area with
without
without
with
concentration,
concentration concentration, concentration
W
W
59.6%
33.34/2.57
75.8%
5
136.45/34.11
each
each
1.00
150
Liquid fill
level, mm
Inclination of
the panel,
deg
250
60
Numerical simulation results [12], [13], [14], [15]
Liquid and vapour cross section areas
Figures 20 and 21 show the variation of liquid cross-sectional areas with height position above
liquid fill level. Figures 22 and 23 show the variation of vapour cross sectional areas with height
position above liquid fill level.
For the ‘normal’ type collector it is seen that liquid cross section area increases along the height
position in the evaporation section and decreases along the height position in the condensation
section. The vapour cross section area varies in the opposite trend.
For the ‘thermosyphon’ type collector it is seen that liquid cross sectional area decreases along the
height position in the evaporation section and increases along the height position in the
condensation section. The vapor cross sectional area varies in the opposite trend. The trend for the
‘thermosyphon’ type collector is different to that for the ‘normal’ micro heat pipes, and this is
caused by capillary effect in the pipes. In the evaporation section, the higher the position the less
the capillary force, and consequently the less the liquid cross section area. In condensation
section, condensation results in an increase of liquid cross sectional area and decrease of vapor
cross sectional area along the height.
In all cases, the vapour cross sectional area is seen to be greater for the ‘normal’ collector than
that for the ‘thermosyphon’ option. This suggest the ‘normal’ collector to be more effective as a
heat-pipe absorber.
Liquid and Vapour pressures
Figures 24 and 25 show the variation of vapour and liquid pressures for the ‘normal’ collector. It
is seen that vapor-liquid pressure difference in the collector with concentration is less than that in
the collector without concentration, although the mass flow rate in the collector with
concentration is greater than that in the collector without concentration (see Figures 32 and 33).
This behaviour demonstrates that concentration can promote reduced flow resistance and
increased heat transfer.
Figures 26 and 27 show the variation of vapour and liquid pressures for the ‘thermosyphon’
collector. It is seen that there is a significant pressure drop in the liquid phase along the height
both in the concentrated and un-concentrated situations for the geometry of Figure 4. However,
there is a significant pressure drop in the liquid phase along the height only in the concentrated
situation for the geometry of Figure 11. The change in the vapour pressure is, however,
negligible. This is because the geometry structure of Figure 4 has much more narrow corner area
than that of the geometry structure of Figure 11, which would cause larger flow resistance in the
fluid flow.
Vapour,liquid and wall temperatures
Figures 28 and 29 show the variation of vapor, liquid & wall temperatures with height position
above the liquid level for the ‘normal’ collector. It is seen that there is little difference between
vapor and liquid temperatures in both the un-concentrated and concentrated cases. However, a
larger temperature difference exists between the pipe inner wall and the vapor area as shown in
Table 3.
Table 3. Temperature difference between pipe inner wall and vapour area
Un-concentrated case
Concentrated case
Evaporation
Condensation
Evaporation
Condensation
section
section
section
section
Geometry (Fig. 4)
0.1
-0.1
1.6
-1.6
Geometry (Fig. 11)
0.42
-0.42
3.8
-3.8
The temperature distribution in the ‘thermosyphon’ collector is similar to that for the ‘normal’
option, which is shown in Figures 30 and 31.
Mass flow rates
For the ‘normal’ collector the variation in mass flow rates for the concentrated/un-concentrated
case with height position above the liquid fill level is shown in Figures 32 and 33. It is seen that
mass flow rate increases with height position in the evaporation section and decreases with height
position in the condensation section. The mass flow rate in the concentrated case is larger than
that in the un-concentrated case.
For the ‘thermosyphon’ collector, the variation in mass flow rates for both liquid and vapour are
shown in Figures 34 and 35.
The results show that with concentration a higher mass flow rate can be achieved which enhances
the effectiveness of the collector.
Start-up conditions
The start-up process, for both geometries investigated (Figures 4 and 11), is simulated and the
results are shown in Figures 36 and 37. The results show that for the concentrated case, start up
conditions can be achieved much quicker than for the un-concentrated case. This is due to the
concentrated heat being focused on a much smaller area. Heat transfer is therefore accelerated
giving a rapid increase in the working fluid temperature.
The geometries investigated show similar trends. However the start up for the geometry shown in
Figure 11 is much slower than that for the geometry shown in Figure 4.
5.
Conclusions
A comprehensive and detailed mathematical model of the thin membrane heat-pipe solar absorber
has been developed. The model has been used to investigate the sizes, structure and thermal
performance of two geometries (Figures 4 and 11). A ‘normal’ and ‘thermosyphon’ variation of
the membrane collector was simulated for the two geometries considered.
In the simulation, the solar absorber plate is considered to be a number of micro heat pipes
parallel along its width. The cross section of each micro heat pipe is taken as a rib. In the
simulation the heat transport capacity, which is a measure of the capacity of the heat-pipe to carry
heat, was taken as the index of performance. The simulation results show the optimised sizes of
the rib to be 9.8mm rib width and 0.63mm gap width. Due to manufacturing and economic
considerations a further geometrical option was simulated with rib sizes of 5mm rib width and
1mm rib gap width.
Further simulation was carried out to investigate the variation of heat transport capacity with
liquid fill level, inclination and working temperature of the collector. In all cases, the results
suggest higher limits of heat transport capacity are obtained for the ‘normal’ type collector with
the geometry given in Figure 4. However, the geometry given in Figure 11 results in higher limits
of heat transport capacity for the ‘thermosyphon’ option.
-
Although the heat transport capacity increases somewhat linearly (for the ‘normal’ type
collector) with liquid fill level, practical limitations suggest the liquid fill level to be 1/4 to
1/3 of the evaporation volume since un-evaporated liquid “blocks” being carried to the
condensation section of the absorber would result in inefficient heat transfer [9].
-
For both ‘normal’ and ‘thermosyphon’ heat-pipe solar absorbers, the limit of heat
transport capacity increases when the inclination varies from 0 to 20deg. Further increase
in inclination has no significance on the improvement of heat transport limit.
-
For both ‘normal’ and ‘thermosyphon’ heat-pipe solar absorbers, the limit of heat
transport capacity increases with working temperature. Therefore higher working
temperature, greater than 200oC, would result in higher thermal performance of the
collector.
Numerical simulation was carried out to investigate liquid/vapour cross sectional areas, pressures,
temperature and mass flow rate. Also, start-up condition for the absorber was simulated.
-
For ‘normal’ heat pipe solar absorber, liquid cross sectional area increases along the
height position in the evaporation section and decreases along the height position in the
condensation section, whilst the vapor cross sectional area varies in the opposite trend. For
‘thermosyphon’ heat pipe solar absorber, variation trend of both liquid and vapor cross
sectional areas is opposite to that of ‘normal’ heat pipe solar absorber.
-
For ‘normal’ heat pipe solar absorber, vapor-liquid pressure difference in a concentrated
pipe is less than that in an un-concentrated pipe. For ‘thermosyphon’ heat pipe solar
absorber, vapor pressure remains nearly constant along the height for both concentrated
and un-concentrated pipes. However, larger liquid pressure drop exists in concentrated
pipes compared to un-concentrated pipes.
-
For both ‘normal’ and ‘thermosyphon’ heat pipe solar absorber, there is little difference
between vapor and liquid temperatures in both un-concentrated and concentrated areas.
However, a smaller temperature difference exists between the pipe inner wall and vapor
area, +0.42oC (evaporation section)/-0.42oC (condensation section) for the un-
concentrated pipe as compared with +3.8oC (evaporation section)/-3.8oC (condensation
section) for the concentrated pipe.
-
For ‘normal’ heat pipe solar absorber, the mass flow rates (vapor and liquid) increase with
height position in the evaporation section and decrease with height position in the
condensation section. For ‘thermosyphon’ heat pipe solar absorber, vapor mass flow rates
increase with height position in the evaporation section and decrease with height position
in the condensation section. Whilst the liquid mass flow rate varies with the opposite
trend. In both cases, the mass flow rates in concentrated pipes are higher than that in unconcentrated pipes.
-
For both ‘normal’ and ‘thermosyphon’ heat pipe solar absorber, the temperature of the
working fluid in un-concentrated pipes increases slowly and the start up time is about
1350 seconds (22.5minutes). Whilst the temperature of the working fluid in concentrated
pipes increases rapidly and the start-up time is about 140 seconds (2.3minutes).
6. References
1. F G Best and R Lloyd, Thin Membrane Heatpipe Solar Absorber with Fresnel Lens, Industrial
Design Consultants Ltd, EC Contract JOE3-CT98-7020, June 1999, pp. 1-30.
2. CIBSE Guide, Volume A, Section A2: Weather and Solar Data, The chartered Institution of
Building Service Engineers, London, UK, 1988, A2-21-A2-30.
3. Duffie and John A, Solar Engineering of Thermal Process, C1980.
4. Walled Yagoub, Membrane Heatpipes Solar Collector, M.Sc Thesis of School of the Built
Envirinment, The University of Nottingham, October 1998, 48.
5. Riffat, S. B., Doherty, P. S., Zhao, X., Simulation of Annual Climate, Solar Irradiation and
Collector Efficiency, Working paper No 1 on the Mathematical Modelling of the Membrane
Heat Pipe Solar Collector. July 1999, 1-16.
6. Cotter. T. P., Principles and Prospects of Micro Heat Pipes, Proceedings of the 5 th
International Heat Pipe Conference, JaTech, Tokyo, 1984, pp. 328-335.
7. M. Groll, S. Rosler, Operation Principles and Performance of Heat Pipes and Closed TwoPhase Thermosyphons. Journal of Non-Equilibrium Thermodynamics, Vol. 17(2)91-192, pp.
112-114, 1992.
8. Riffat, S. B., Doherty, P. S., Zhao, X., Simulation of Optimum Structure, Sizes and Working
Conditions of the Panel Collector, Working Paper No 2 on the Mathematical Modelling of the
Membrane Solar Panel. August 1999, 1-24.
9. Harada, K., Inoue, S., Fujita, J, Suematsu, H., Wakiyama, Y., Heat transfer characteristics of
large heat pipe, Hitachi Zosen Tech. Rev. 41, pp. 167, 1980.
10. Collier, J. C., Convective Boiling and Condensation, McGraw-Hill, New York, 1981.
11. Colwell, G. T., and Chang, W. S., Measurement of the Transient Behaviour of a Capillary
Structure Under Heavy Thermal Loading, International Journal of Heat and Mass Transfer,
Vol. 27, No. 4, 1984, 541-551.
12. Riffat, S. B., Doherty, P. S., Zhao, X., Numerical Simulation of the Thermal Behavior of the
Membrane Heat Pipes Solar Collector, Working Paper No 3 on the Mathematical Modelling
of the Membrane Solar Panel. September 1999, 1-16.
13. Riffat, S. B., Doherty, P. S., Zhao, X., Simulation of Optimum Structure, Sizes and Working
Conditions of the Panel Collector-Thermosyphon Option, Working Paper No 7 on the
Mathematical Modelling of the Membrane Solar Panel. January 2000, 1-24.
14. Riffat, S. B., Doherty, P. S., Zhao, X., Numerical Simulation of the Thermal Behavior of the
Membrane Heat Pipes Solar Collector-Thermosyphon Option, Working Paper No 8 on the
Mathematical Modelling of the Membrane Solar Panel. January 2000, 1-30.
15. Riffat, S. B., Doherty, P. S., Zhao, X., Mathematical Simulation of the Membrane Heat Pipes
Solar Collector- New Structure Design, Working Paper No 9 on the Mathematical Modelling
of the Membrane Solar Panel. April 2000, 1-16.
16. Babin, B. R., Peterson, G. P., and Wu, D., Steady-State Modelling and Testing of a Micro
Heat Pipe, Journal of Heat Transfer, Vol. 112, 1990, 596-601.
7. Nomenclature
Solar radiation:
 ---------- efficiency of solar collector
fr ---------- correction factor
1 ----------- transmittance of cover plate 1
2 ------------ transmittance of cover plate 2
aa ------------ absorptivity of absorber plate
U ------------ overall heat transfer coefficient
th ----------- hot side temperature of the collector
ta ----------- ambient temperature
I0 ----------- solar radiation striking on the collector
%
W/m2K
o
C
o
C
W/m2
Optimization (appendix 1)
Geometry parameters:
Av----- vapour area in the cross section of one rib
Al------ liquid area in the cross section of one rib
ri------ equivalent radius of the cross section
rv----equivalent radius of the vapor area
m
rh,w--- hydraulic radius of the triangle grooved wicks with liquid.
rce--- capillary radius in the beginning of the evaporation section
------- wetting angle of liquid-vapor surface
leff---- effective length of the heat pipe
rn------ critical radius of bubble generation
di ---- equivalent diameter of the cross section
le ----- length of evaporation section
lc ----- length of condensation section
la ----- length of adiabatic section
rhl ----- hydraulic radius of liquid cross section
rhv----- hydraulic radius of vapour cross section
Fv----- frictional resistance coefficient
Properties of working fluid-water:
Tv-----absolute temperature of vapor
hfg-----latent heat of vaporisation
------- surface tension
------- specific heat ratio
Rv----- water vapor constant
K ------ thermal conductivity of liquid
Mv----- Mach number of vapour flow
Rev --- Reynolds number of vapour flow
g ------- gravitational acceleration
pc,m—maximum capillary force
keff--- effective thermal conductivity of the wicks
C,C1,C2 --- coefficient of vapour phase resistance calculation
K ------ coefficient for liquid phase resistance calculation
------- angle of inclination relative to horizontal surface (variable)
qc------ heat input
qs,m---- sonic limit for heat transport
qe,m---- entrainment limit for heat transport
qb,m---- boiling limit for heat transport
qv,m---- viscous limit for heat transport
qmax-----maximum heat transport capacity
in
the
m2
m2
m
cross section
m
deg
m
m
m
m
m
m
m
m
K
J/kg
N/m
J/kg.K
W/m.deg
N/m2
m/s2
Pa
W/m.deg
deg
W
W
W
W
W
W
G
-----minimum
filled
kg
pc------- net capillary pressure difference
p1------- radial hydrostatic pressure drop
p2------- axial hydrostatic pressure drop
p3----- viscous pressure drop occurring in the liquid phase
p4----- viscous pressure drop occurring in the vapour phase
liquid
mass
Pa
Pa
Pa
Pa
Pa
Numerical simulation (appendix 2)
o
t------- working temperature
C
T------ time interval used in the iteration
second
o
t ------ temperature increase during the time interval
C
p ------ pressure
Pa
 ------ density
kg/m3
hfgv0---- latent heat of vaporisation in the first section of the element
J/kg
hfgv1---- latent heat of vaporisation in the second section of the element
J/kg
hfgl----- latent heat of vaporisation in liquid-vapor interface of the element
J/kg
Cp ----- specific heat of liquid
kJ/kg.K
 ------ viscosity
N.s/m2
j-------- free molecular flow mass flux of evaporation
kg/m2.s
M ------ molecular weight, 18
R ------- universal gas constant, 8317
J/kg.K
qec----- heat transfer rate on the vapour-liquid interface in the element for the
steady state process
W
q ----- heat increase in an element unit at T time interval in transient state
W
mec--- evaporation-condensation rate in the liquid-vapour interface of the element
kg/s
Pcl------ capillary pressure
Pa
pcl---- capillary pressure difference between the two sections of the element
Pa
prg---- radial hydrostatic pressure difference
Pa
pag---- axial hydrostatic pressure difference
Pa
pvl----- difference of the vapor and liquid pressure in the element
Pa
pfv----- friction pressure loss occurring in the vapour phase
Pa
pmv--- pressure loss due to momentum change occurring in the vapor phase
Pa
pfl----- friction pressure loss occurring in the liquid phase
Pa
pml---- pressure loss due to momentum change occurring in the liquid phase
Pa
u ------- velocity
m/s
u ----- velocity difference between the two sections of the element
m/s
ms0---- mass flow rate in the first cross section of the element
kg/s
ms1---- mass flow rate in the second cross section of the element
kg/s
ms--- difference of mass flow rates between the two sections of the element
kg/s
ms----- average mass flow rate in the element
kg/s
Subscripts
v------ vapor
l------- liquid
s------- solid
APPENDIX 1
OPTIMIZATION OF THE STRUCTURE, SIZES AND WORKING CONDITIONS
The heat transport capacity of a micro heat pipe is usually subject to following limits






Sonic limit
Viscous limit
Entrainment limit
Boiling limit
Capillary limit
Filled liquid volume limit.
Formula of the limits are shown as follows:
Sonic limit
qs ,m  Av  v h fg (RvTv /( 2(  1))1/ 2
(1)
Entrainment limit:
qe,m  Av h fg (v /( 2rh,w ))1/ 2
(2)
Boiling limit:
qb,m 
Viscous limit:
qv,m 
Filled liquid mass:
31 1 2 d i 2 1 / 3 1 / 3
G  (0.8lc  0.8le  la )(
) qc
h fg g
Capillary limit:
2Leff keff Tv
2
 pc,m )
h fg v ln( ri / rv ) rn
(
rv h fg v pv Av
16v Leff
pcp1+p2+p3+p4
(3)
(4)
(5)
(6)
The operation of the heat pipe requires that the capillary pumping pressure be greater than the
sum of all pressure drops in the flow path. This relationship can be expressed as equation 6.
Whereby, pc is net capillary pressure difference; p1is radial hydrostatic pressure drop; p2 is
axial hydrostatic pressure drop; p3 is viscous pressure drop occurring in the liquid phase; p4 is
viscous pressure drop occurring in the vapor phase; the formula for these items are as follows;
pc=2cos/rce
(7)
p1= -lgdvcos
(8)
p2= -lgLpsin
(9)
pl  (
l
KAl h fg  l
) Leff qc
K=rhl2/8
p 4 
Mv 
(11)
C ( Fv Re v )  v
2(rhv ) 2 Av  v h fg
leff qc
2rhv q c
Av  v h fg
Re v 
Av  v h fg ( RvTv ) 0.5
Whereby; for laminar flow and incompressible flow
Rev2300, Mv0.2
FvRev=16, C=1.0
For laminar flow and compressible flow
Rev2300, Mv0.2, FvRev=16
C  C1  (1  (
 1
(12)
(13)
qc
) M v 2 ) 0.5
2
For turbulent flow and incompressible flow
Rev2300, Mv0.2, FvRev=0.038
C  C2  (
(10)
2 hv qc 0.75
)
Av h fg  v
For turbulent flow and compressible flow
Rev2300, Mv0.2
FvRev=0.038
C=C1C2
(14)
APPENDIX 2
NUMERICAL SIMULATION OF THE THERMAL BEHAVIOUR OF THE MEMBRANE
HEAT PIPE SOLAR COLLECTOR
Mathematical theory and numerical analytical method
The finite element method is employed to analyze the heat and mass transfer in each micro heat
pipe. The grid division is shown in Figure 19. The length step is taken as 1mm and each length
step is taken as the unit of an element where differential equations below are applied for
simulation. The thermal performances in both steady state and transient state operation are
analyzed in detail. Since the high thermal conductivity of heat pipes is the result of evaporation
and condensation process occurring within the heat pipe. Determination of the evaporation and
condensation rate plays a key role in evaluating the thermal characteristics and heat transport
limitations of these heat pipes. In this model, an expression for the free molecular flow mass flux
of evaporation j, presented by Collier[10] and later used by Colwell and Chang [11] is employed.
This is shown below.
j [
M
]0.5 ( p s  pv )
2R(t  273)
(1)
When ps is greater than pv, j is positive and the liquid evaporates. When ps is less than pv, j is
negative and the vapor condenses. In the development of the numerical model presented here, the
evaporation-condensation rate mec is assumed to be proportional to the liquid-vapor interface
area. In each grid section of the heat pipe, mec is expressed as,
mec  jWec x
(2)
For a given latent heat hfg, the rate of heat released or absorbed in any section can be determined
from
qec  h fg mec
(3)
For a heat pipe to operate properly, the capillary pressure difference, plus gravity difference, must
be sufficient to overcome the liquid and vapor pressure losses.
Capillary pressure is expressed as
pcl  2 cos / rce
(4)
Since both pcl and rce are functions of the length position x, therefore
pcl  2 cos rce

x
x
rce 2
Expressing this over a finite interval yields
pcl  (2 cos / rce 2 )rce
(5)
(6)
Gravity pressure differences include axial and radial pressure differences. They are expressed
respectively as follows [4]:
p ag    l g sin x
(7)
p rg  2  l g cos rv
(8)
The pressure loss of liquid and vapor are expressed as
pvl  p fv  p mv  p fl  p ml
(9)
The pressure losses result from liquid and vapor friction, and the momentum changes occurring in
any one section along the heat pipe.
The pressure losses due to friction for laminar flow can be expressed as
p fv  (2 v xuv) / rhv 2
(10)
p fl  (2  l xul ) / rhl 2
(11)
The total pressure drop due to the momentum change can be expressed as
pml  (ul msl  msl ul  msl ul ) / Al
(12)
pmv  (uv msv  msv uv  msv uv ) / Av
(13)
For steady state model the energy equation can be written as
qin  qout
(14)
qin  ms0 h fgv 0  msh fgl
(15)
qout  ms1h fgv1
(16)
For transient state model the energy equation can be written as
qin  qout  qr
(17)
qr    l Al C pl tl dx    v Av C pv t v dx    s As C ps t s dx
(18)
The continuity equation can be written as
  v Av dx   l Al dx  cons tan t
(19)
The total volume and area inside the heat pipe can be defined and expressed, respectively, as
vol  voll  volv
A  Al  Av
(20)
(21)
This kind of micro heat pipe consists of two basic sections, the evaporation section and the
condensation section. Each section has a different set of boundary conditions, and as a result have
been treated independently.
Evaporation Region: The single boundary condition used in the evaporator section is the time
dependent heat flux. For a specific input heat flux, the saturation pressure at a given location can
be obtained by using a combination of the mass flux expression given in Eq. 1 and the energy
conservation equation given in Eq. 14.
For steady state model
ps 
qec
( M / 2R(273  t l ))1/ 2 xWec h fg
 pv
(22)
qin  qec
(23)
qin  qec  ql  q s  qv
(24)
q s  (C ps  s As xt s ) / t
(25)
ql  (C pl  l Al xt l ) / t
(26)
qv  (C pv  v Av xtv ) / t
(27)
For transient state model
Since tl is a function of ps, Eqs. (24-27) are coupled and can be solved using an iterative method
with relaxation to obtain values for tl, qec, ql and qs. Because the difference between the
boundary and liquid temperatures tbl is proportional to the input heat flux, the boundary
temperature can be obtained by adding tbl to tl.
Condenser Region: In the condensation section, the boundary temperature of the heat pipe was
assumed to be constant, resulting in governing equations similar to those used in evaporation
section.
qec  h fg Wec x(
M
)1/ 2 ( p s  pv )
2R(t l  273)
(28)
Rearranging eq. 22, we get
For steady state model
qec  qout
(29)
For transient state model
qout  qec  ql  q s  qv
(30)
APPENDIX 3
THEORETICAL AND EXPERIMENTAL INVESTIGATION OF THE HEATING &
COOLING PROCESS OF THE MEMBRANE SOLAR COLLECTOR
1. Theoretical simulation
The theoretical simulation is based on the conditions below:_








Collector structure: shown in Figure 3-1.
Working fluid: water
Initial working fluid temperature: 25oC, Maximum working fluid temperature: 200oC.
Ambient temperature: 25oC.
Solar irradiation: 800 W/m2.
Coefficients used for efficiency calculation:
o 1=0.96
o 2=0.9
o aa=1
o =0.1
o U=1.68
Coolant-oil in stagnant state, but there was convection and conduction loss to ambient in
the condensation section area.
Irradiation strikes the plate continuously during the period considered.
Figure 1. Collector structure adopted for the theoretical simulation
Outer cover
Lens
Lens
String vest
Fibre insulation
plate
VSI materials
Aluminiu
m
The simulation results are shown in Figure 3-2. It is found that:
o Evaporation section (absorber) temperature increased linearly from 25oC to 175oC in the 60
minutes until the irradiation is removed.
o Condensation section temperature increased rapidly from 25oC to 135oC in the first 10
minutes, and then increased slowly until the irradiation is removed.
o Outer cover temperature increased from 25oC to 45oC in the 60 minutes until the irradiation is
removed.
o Collector efficiency decreased from 0.78 to 0.5 with the increase of temperature.
2. Panel 5 test
In our understanding, panel 5 test was carried out on the condition below:







Collector structure: similar to Figure 1 but without Fresnel Lens, normal type.
Working fluid: water
Initial working fluid temperature: 26.5oC.
Ambient temperature: 27.5-30oC.
Solar irradiation: 805 W/m2.
Coolant-oil had enough volume to absorb the heat discharge from condenser, Pump off.
Test time duration: 198 minutes.
The test results are outlined in Figure 3-3. It is assumed that:
o Evaporation surface temperature = (ch 3+ch 4 +ch 5+ch 7+ch 8)/5;
o Condensation surface temperature =(ch 1 + ch 2)/2;
o Outside temperature of outer cover = channel 6.
3. Panel 7 test
In our understanding, panel 7 test was carried out on the condition below:







Collector structure: similar to Figure 1 but without Fresnel Lens, normal type.
Working fluid: water
Initial working fluid temperature: 25.5oC.
Ambient temperature: 24.44oC-25.04oC.
Solar irradiation: 1021 W/m2.
41.8gms oil in heat exchanger, pump on
Irradiation: 1021W/m2.
The test results are outlined in Figure 3-4. It is assumed that:
o Evaporation section temperature = (ch 3+ch 4 +ch 5)/3;
o Condensation section temperature = (ch 1+ch 2)/2;
o Outer cover temperature;
o Temperature of bottom of tray;
o Ambient temperature.
4. Panel 11 test
In our understanding, panel 11 test was carried out on the condition below:







Collector structure: similar to Figure 1 but with Fresnel Lens, normal type.
Working fluid: water
Initial working fluid temperature: 20oC.
Ambient temperature: 19.81-22.00oC.
Solar irradiation: 1021 W/m2.
39.6gms Coolant-oil in the heat exchanger, pump on.
Irradiation: 1021 W/m2.
The test results are outlined in Figure 3-5. It is assumed that:
o Evaporation section temperature = (ch 3+ch 4 +ch 5)/2;
o Condensation section temperature = (ch 1+ch 2)/2;
o Outer-cover temperature = channel 6;
o Tray bottom temperature = channel 7;
o Ambient temperature =channel 8.
Therotical simulation of the heating process of the heat pipe solar
collector
200
180
160
140
Temperature, oC
120
100
Evaporator surface
Condenser surface
80
Outer cover
Ambient
60
40
20
0
0
10
20
30
40
50
60
Time, minutes
Figure 3-2. Heating & cooling process of the solar collector- theoretical simulation
70
180.00
160.00
140.00
Temperature, oC
120.00
100.00
80.00
60.00
Condenser outer surface
Outer cover
Evaporator outer surface
40.00
20.00
0.00
0
50
100
150
200
250
Time, minutes
Figure 3-3. Test result of panel 5
Test results of Panel 7
180.00
160.00
Condenser outer surface
Evaporator outer surface
Outer cover
Bottom of tray
Ambient
Temperature,oC
140.00
120.00
100.00
80.00
60.00
40.00
20.00
0.00
0
20
40
60
Time, minutes
Figure 3-4. Test result of panel 7
80
100
120
Test result of panel 11
180.00
Condenser outer surface
160.00
Evaporator outer surface
Temperature, oC
140.00
Outer cover
Bottom of tray
120.00
Ambient
100.00
80.00
60.00
40.00
20.00
0.00
0
20
40
60
80
100
120
Time, minutes
Figure 3-5. Test result of panel 11
5. Comparison, conclusions and suggestions
The variation of temperatures with time for the cases of theoretical simulation & experimental test
are outlined in Table 1.
Table 1. Variation of temperatures with time for the cases of theoretical simulation and
experimental test
Case
Time
Temperatu Temperatu
Temperature variationduration
re
re
outide of outcover
(test 2 and 3)
(minutes) variation- variationreservoir liquid
Evaporatio condensati
(test 4 and 5)
n area
on area
Theoretica
60
25~175
25~145
25~45
l
simulation
Panel 5
180
25~160
25~155
25~90
Panel 7
110
25~170
25~155
25~90
Panel 11
110
25~170
25~160
25~80
It is concluded:



The test temperature increases in condensation area are higher than that from theoretical
simulation. This shows that the collector has lower heat loss to ambient than anticipated
situation.
The test temperature increases of outer cover surfaces are higher than that from
theoretical simulation. This shows that the collector has higher heat loss, and hence poorer
efficiency.
Theoretical efficiency of the collector is about 0.5~0.78, and the test efficiency is about
0.4~0.7 within the temperature range above.
It is suggested:

Lower efficiency is possibly caused by the structure of condenser since there are a number
of restrictions along the flow channel of each micro heat pipe, which results in higher flow
resistance and reduced heat transport capacity. Original condenser structure is shown in
Figure 3-6.
Figure 3-6 Schematic showing of the condensation section configuration

The condenser structure may be modified as shown in Figure 3-7 Thus both vapor and
condensed liquid flow more smoothly without dramatic resistance loss.
Figure 3-7 Schematic showing of the condensation section configuration (modified)