SOCRATES PROGRAMME
A EUROPEAN SUMMER SCHOOL
AN INTENSIVE COURSE ON
‘ICT Tools on PV - Systems Engineering: Teaching & Learning’
3 – 18 July, 2006
Technological Educational Institute of Patra
Design, Simulation and Performance of Reflecting Parabolic Solar
Collector
GEORGE BARAKOS
Department of Mechanical Engineering
Technological Educational Institute of Patra
GR26334 GREECE
barakos@teipat.gr
Introduction:
In this lecture a study of mini parabolic collectors systems is presented. The mini
Parabolic Collector Systems (P.C.S.) are designed as to be integrated on the roofs of a
residential buildings or in a larger scale on the roofs of bigger buildings, (offices,
store houses, green farm houses). The common approach to locate collector’s arrays
on roofs is to slope them at approximately the optimum solar collector angle.
Then a detailed parametric analysis of the constructed parabolic trough-concentrating
collectors (P.T.C.) by using a simulation method developed is made. The simulation
procedure reveals the individual contributions of both direct and diffuse components
of the solar radiation to the flux reaching the cylindrical absorber of the P.T.C.
In addition, one may study the effects of various optical and thermal parameters of
the collector’s performance.
Collector’s Geometric Elements:
In this section an analysis of the geometrical characteristics of the mini parabolic solar
collectors is made.
The data of the mini parabolic system (fig.1) used for our study are provided. Let us
suppose that the length, l=1250mm, is big enough to avoid end effects, as the image
from the end of the trough is formed beyond the end of the receiver [1].
The length or aperture 2 y s , and the height h of the parabola are given as data input
according to the designer.
From the equation of the parabola one may obtain the focus length f.
f
y2
4 z
1
Let the cylindrical absorber tube have diameter d. Here some definitions about linear
concentrating systems or parabolic trough concentrating systems may be given [2, 3,
4].
Fig.1: Characteristics of the parabola

Aperture is the opening through which the solar radiation enters. AA’=2ys

Concentrator or optical system is the part of the collector that directs
radiation onto the receiver.

Area concentration ratio C or geometric concentration ratio C is the ratio
of the area of aperture to the area of the receiver.

Flux concentration ratio is defined as the ratio of the average energy flux on
the receiver to that on the aperture.

Acceptance angle 2θα is the angular range over which all or almost all rays
are accepted without moving all or part of the collector. The half acceptance
angle for the parabolic trough collector is given at the position y by the
formulae:
d
(1)
sin  y 
  y 2 
2f 1    
  2f  
This means that a ray hits the absorber if the angle of incidence is    y .
2
   y Is the angle at which the rays reach the receiver tube tangentially.
 y : Increases with y.
For y   y S the above formulae gives
  , the largest angle for which all
incident rays are accepted. This acceptance, half – angle of the parabolic trough, can
be written in terms of rim angle  R and concentration ratio C.
sin   
sin  R
C
(2)
Rim angle φR. This angle is related to aperture and focal length f by
y
   2y
tan R   s  s or
2f
 2  4f
 8f y s  
tan  R  

2
 16f y s   1 
(3)
The linear concentrating systems to be used in this study will be fixed, integrated
on the roof, oriented to “Non Track” the sun, the concentration ratios must be
intermediate for the linear imaging collectors like trough parabolic concentrating
system.
For these systems of relatively low concentration ratio, part of the diffuse radiation
is reflected to the receiver, with the amount depending on the acceptance angle of the
concentrator. In general, concentrators with receivers much smaller than the aperture
(big concentration ratio) are effective only on beam radiation.
Construction, material details and geometry of the concentrating collector
system:
In R.E.S. Laboratory in order to study the behavior of mini parabolic collector
systems and compare theoretical and experimental results, a construction of two
panels of mini parabolic collectors has been made.
For the construction of the reflecting parabola aluminum foils type MIRO 27Hochglanz with a selective surface from ALANOD company were used. The foil
thickness is Sf = 0,5 mm, the absorber tube is copper painted black, the absorber
tube thickness is St = 1 mm. As absorber, copper tubes with various external
diameters (e.g. 10mm, 15mm, 18mm, 22mm), were used.
Let a reflecting parabola with an aperture 2ys , height h, which implies a focal
distance f = ys2 / 4h
The aluminum foil takes the form of the trough parabola via two base parabolas and
two head parabolas made by C.N.C. machine (see figures 2a ,2b,2c and 2d).
3
Fig.2a, b: Head and base parabola.
Fig. 2c, d: CNC machine where our base and head parabola constructed
Those parabolas (base and head) are constructed from common aluminum foil of
2mm thickness.
The base parabola using cutting tool with radius R must have:
Aperture
:
bb = 2ys + 2Sf / sinψ – 2R/sinψ
(4)
Height
:
hb = h + Sf - R
(5)
Focal distance :
fb = (bb/2)2/ 4hb
(6)
The head parabola must have:
Aperture
:
bh = 2ys + 2Sf / sinψ
(7)
Height
:
hh = h + R
(8)
4
Focal distance
:
fh = (bh/2)2/ 4hh
(9)
The sinus ψ is due to the fact that the tangent to the parabola on the upper extreme A
of the aperture, see fig. 8 comes from:
ys
 dz 
 2y 
(10)
tan   
 dy 
   4f   2f

A

A
The equation of the parabola in a (O, y, z) system is:
y2 = 4 f z
(11)
The aluminum reflecting parabola foil for every channel of the P.C.S. will have a
mean length equal to:
 4h  b 2  16h 2
1 2
b m2
m
m
2
Lm 
b m  16h m 
ln m
2
8h m 
bm





(11)
where,
b m  2y s 
Sf
sin 
(12)
1
hm  h  Sf
2
For the overall dimensions of the parabolic trough collector see figure 3a and 3b.
Fig.3a: The parabolic trough collector
of single aperture 100mm.
Fig.3b: The parabolic trough collector of
single aperture 160mm.
5
Determination of Solar Radiation Parameters concerning Parabolic Collectors:
The determination of the solar radiation intense (direct, diffuse and global), can be
done experimentally, using pyranometers, or theoretically, using prediction methods.
Solar vectors:
In a specific instant time, fig. 4 shows the position of the sun in the O(x,y,z)
orthogonal system, where (x, y) plane is the horizontal plane.
Fig.4: Sun’s position on O(x,y,z) coordinate system
The sun’s unit vector in this system determined by:



n s  cos  cos  s i  cos  sin  s j  sin k
(13)
  
Where i , j, k are the unit vectors of the axis x, y, z.
As the real situation is a parabolic concentrating system on the building’s roof with
inclination to horizontal βo and orientation from south eo, the sun’s unit vector with
reference to an O(x1, y1, z1) system with an axis along the absorber tube central line
x1, the y1 normal to x1 and z1, where z1 axis is normal to the roofs plane, can be
determined by an orthogonal transformation, see fig.5
Fig’s 4 and 5 shows the roof’s inclination angle βο, the roof’s orientation angle eo, the
solar altitude α = 90 – θz, the solar azimuth angle γs and how one with two rotations,
first around the y axis and then around the z axis, goes from O (x, y, z) system to the
O (x1, y1, z1) orthogonal system.
Fig.5: Sun’s and collector’s position on O(x1 ,y1 ,z1 ) coordinate system
The sun’s unit vector as it is given by (13) in the O (x, y, z) system with
transformation matrix using the Eulerian angles [5,6]:
l 1  cos  cos e ,
m 1  sin e ,
n 1   sin  cos e
l 2   sin ecosβ,
m 2  cos e ,
n 2  sinesinβ
l 3  sin ,
m 3  0,
n 3  cos 
(14)
provides the sun’s unit vector in the O(x1,y1,z1) system







n s  (n s ) x1 i1  (n s ) y1 j1  (n s ) z1 k 1
(15)
The components of the solar unit vector in the coordinate system Ο (x1 ,y1 ,z1) are:
 
(n s ) x1  (n s  i1 )   cos  cos e cos  cos  s  cos  sin  s sin e  sin  sin  cos e
 
(n s ) y1  (n s  j1 )   cos  sin  s cos e  cos  cos  s sin e cos   sin e sin  sin 
(16)
 
(n s ) z1  (n s  k 1 )   cos  cos  s sin   sin  cos 

 
Where i1 , j1 and k 1 are the unit vectors of the axis x1, y1, z1 respectively:
7







i1  l 1 i  m 1 j  n 1k  cos  cos e i  sin e j  sin  cos ek







j1  l 2 i  m 2 j  n 2 k   cos  sin e i  cos e j  sin e sin  k






k 1  l 3 i  m 3 j  n 3 k  sin  i  cos  k
The parameters
(17)
l ,m ,n  , l ,m ,n  , l ,m ,n 
(14) are the direction
2
2
2
3
3
3
  
cosines of the three unit vectors i1 , j1 , k 1 of the axis x1 ,y1 ,z1 respectively.[6]
1
1
1
Energy Consideration aspect in parabolic collectors:
The useful thermal energy rate delivered by the collector is [3]:
Qu  Ac[I a  U L (Tc  Ta )] ,or Qu  FR A C I a  U L Tfi  Ta 
(18)
The optical efficiency op is defined as the fraction of the solar radiation incident on
the aperture of the collector, which is absorbed at the surface of the absorber tube.
op 
Ia AC
I T A ap
(19)
Then,
Q u  op I T A ap  AcU L (Tc  Ta ) , or Q u  FR op I T A ap  A C U L Tfi  Ta 


(20)
As the instantaneous collector’s thermal efficiency p is defined as the ratio of the
useful energy delivered to the energy incident on the aperture.
Qu
I T A ap
Then, the instant thermal efficiency of the collector:
p 
p  op 

A U T  Ta 
U L TC  Ta 
, or p  FR op  C L fi

IT
A ap
IT


(21)
(22)
From equation (20) it is clear that the losses to the surroundings from the absorber
increase as the fluid inlet temperature increases and so thermal efficiency decreases.
The efficiency p decreases significantly with the inlet fluid temperature, but this
decrease is slightly non-linear. The non-linearity is due to the fact that the value of the
overall loss coefficient increases slightly as Tfi increases.
8
An increase in the mass flow rate of the thermic fluid, has as a result an increase to
the heat transfer coefficient h cf and due to this the collector efficiency factor F  , and
the collector heat removal factor FR and the efficiency increases.
An increase in the concentration ratio C, by decreasing the size of the absorber tube,
results an increase of the thermal efficiency, as the value of the optical efficiency
changes very slightly. The loss from the absorber tube which are inversely
proportional to C decreases and, hence, the thermal efficiency increases.
According to equation (22) the difference between the values of the optical efficiency
op and the thermal efficiency p is a measure of the losses due to reradiation and
convection from the absorber to the surroundings.
IT
Ia
Ac
The solar radiation intense incident on the aperture
The solar radiation intense absorbed by the absorber tube
The absorber tube receiver surface A c  Do l
Aap
The absorber surface A ap  2y s  l
Do
l
UL
Tc
Ta
Tfi
FR
The absorber tube external diameter
The length of the trough parabola
The overall heat losses coefficient
The receiver temperature
The ambient temperature
The inlet fluid temperature
The collector’s heat removal factor
Energy losses of concentrating collectors:
The losses occurring in the collector may be distinguished, in optical and thermal.
Optical losses are those, which occur in the path of the incident solar radiation before
it is absorbed at the surface of the absorber tube. The optical losses can be estimated
by simulations methods.
Thermal losses are due to convection and reradiation from the absorber tube and
conduction through the ends and supports. Thermal losses from the receiver must be
calculated, usually in terms of energy loss coefficient U L , which is based on the area
of the receiver. In principle, temperature gradients on the receiver can be accounted
for by a flow factor FR the collector heat removal factor, to allow the use of inlet
fluid temperatures in energy balance calculations.
The generalized thermal analysis of a concentrating collector is similar to that of a flat
plate collector. It is necessary to derive appropriate expressions for the collector
efficiency factor F  , the energy loss coefficient U L and the collector heat removal
factor FR . With FR and U L known, the collector useful gain can be easily calculated.
9
For an uncovered cylindrical absorbing tube that might be used as a receiver in a
linear concentrating system we calculate the thermal loss coefficient U L with the
assumption that there are no temperature gradients around to the receiver tube.
In order to have more accurate results, three models for the calculation of the heat loss
coefficient UL has been developed.
Mathematical modelling simulation:
A simulation procedure has been developed in “FORTRAN 4.0” to study theoretically
mini parabolic collectors.
The program chooses a series of equal space segment, Ns, on the parabola’s aperture
where the solar beam radiation impinges (fig. 6).
This number was set equal to 64or 128or 256or 512or 1028. The higher Ns are, the
more accurate the results are and more execution true is required.
Fig.6: Reflecting parabola and absorber on the plane ( y 1 , z 1 ).
Fig. 6 shows the reflecting parabola and the absorber section in a plane parallel to y1,
z1 at the position x1. The algebraic equation of this parabola is:
y2
Or
(23)
y 12  4f z 1  f 
z1  1  f
4f
where f is the focal distance.
The algebraic equation of the cylindrical absorber section is:
10
y 12  z 12  R 2
Where
(24)
R is the absorber radius
The program takes under consideration the direct and the diffuse component of solar
radiation.
Finally, it summarizes the effects of both components.
The scanning starts from point A(0, y s , z d ) and finishes at point A(0, y s , z d ) . AA  is
the aperture, the parallel line to the y1 axis at distance z1 = zd.
Direct component of solar radiation:
The sunrays are considered to be uniformly and parallel impinging on the parabola’s
aperture AA  2 y s (fig.20). The sun beam is considered as scanning the aperture. The
scheme is given in fig.7 from point A until point A  .
For every point of the aperture,(Ns+1) points, the program initially checks if the beam
hits the absorber directly.
Then the point and angle of incidence on the absorber is determined.
At the same time it calculates the power impinging on the relative absorber arch and
its intensity [W/m2] is calculated.
If the beam does not hit the absorber directly, the program checks if it intersects the
parabola and then checks if the reflected ray hits the absorber.
Then, it determines the point and angle of incidence on the absorber and calculates the
power hitting on the absorber relative arch and it’s intense [W/m2].
An analytic investigation about the beam directly impinging on the absorber and also,
then the beam incident on the absorber after reflection on the parabola’s surface are
followed.
Beam directly impinging the absorber:

The equation of the line that determines the unit vector of the sun n s on the
coordinate system Ο(x1, y1, z1), which passes from an ordinary scanning point
  (0, y 1 , z d ) of the aperture:
x 1  0 y 1  y 1 z 1  z d


n x1
n y1
n z1
(25)
Solving the system of the cylinder equation (24) and the solar line equation (25) we
check if the solar line intersects the cylinders surface at point C( x 1C , y 1C , z 1C ) on the
cylinder. At position x 1  x 1C we have a circular section (fig.7).
11

ns
θid
z1
C( x 1C , y1C , z1C )

R
λ
y1
Fig.7: Beam directly impinging the absorber

Parameters n s and  id are not necessarily located in the plane y1, z1.
tan  
z 1C
y 1C
(26)
The program registers these angles λ.

The unit vector of the cylindrical radius R at point C is then given by:



R  cos  j1  sin k 1
if y 1C  0
or
(27)



R   cos  j1  sin k 1 if y 1C  0
The angle of incidence of the direct beam on the absorber at point C is given by:
 


cos  id  (n s  R )  (n s ) y1 cos   (n s ) z1 sin  if y 1C  0
or
(28)
 


cos  id  (n s  R )  (n s ) y1 (  cos  )  (n s ) z1 (  sin  ) if y 1C  0
Then the energy rate of the direct beam radiation incident directly and absorbed on the
relative arch at point C is calculated:
Pbd  I b
cos 
 s  l  cos  id  a
sin 
[W]
12
or
Pbd  I b 
where:
cos  2y s

 l  cos  id  a
cos  z N s
[W]
(29)
2y s
is the step on the aperture 2y s .
Ns
l = the length of the parabola trough
a = receiver’s absorption.
s
The ratio of the power Pbd to the relative surface of the cylinder R  d  1 gives
the intense [W/m2].
Direct beam incident on the absorber after reflection on the parabola’s surface:
If the representative line of the sun’s vector does not intersect the cylinder’s
surface then we solve the system of its equation (25) and the parabola’s equation
(23). If the solar line intersects the parabola at a point E(x1E , y1E , z1E ) we consider
the parabola of fig.8 in the position x1E .
The equation of the tangent tt  to the parabola at the point E( y1E , z1E ) on a plane
y1 , z1 at the abscissa x1E is:
z 1  z 1E  my 1  y 1E  where
 dz 1 
d  y 12  4f 2  y 1E

 
 
m  tan   
 dy 1  E dy 1  4f  2f
(30)
13
Fig.8: Schematic representation of direct beam incident after reflection.
Then, the equation of the tangent at point E is:
z 1  my 1  b  z 1 
y 1E
y 1  z 1E  2f
2f
(31)
The normal to the tangent at point E is:
2f
 1
z 1    y 1  b  
y 1  z 1E  2f
y 1E
 m
(32)
The unit vector of the normal to the parabola at point E is:



n p   sin  j1  cos  k 1 where
y 1E
2f
i  r .
For the reflection law:
So,
  arctan


 


cos i  cos r  ns  np  ns y1  sin   ns z1 cos 
(33)


where ns y1 and n s z1 obtained from equation (16)
14
The unit vector of the reflected ray is [21]:


  


r  n s  2n s  n p n p  n s  2 cos  r n p
(34)
The reflected unit vector components are:
r x
r y
r z
1
1
1
 ns x1
 ns y1  2 cos  r sin 
 ns z1  2 cos  r cos 
The equation of the reflected solar radiation’s unit vector line from point E is:
x 1  x 1E y 1  y 1E z 1  z 1E
 
 
r x1
r y1
r z1
(35)
To calculate the point where this line intersects the cylinder, we must solve the system
of equations (25) and (35). We take into account only one reflection, (this is true for
the trough parabolic concentrator).
If the reflected line intersects the cylinder at a point C( x 1C , y1C , z1C ) then we have,
equations (26) and (27)
tan  



R   cos  j1  sin  k 1
z 1C
,
y 1C
The angle of incidence of the reflected beam radiation on the absorber at point C is
given:




 


cos  ir  r  R  r y1 cos   r z1 sin  ,
Or
if y 1C  0
 


cos  ir  r  R  r y1  cos    r z1  sin   , if y 1C  0
Then, we calculate the energy rate of the direct beam radiation incident and absorbed
after reflection on the relative arch to the point C.
Pbr  I b
cos 
 s  l  r  cos  ir  a
sin 
Or
cos  2y s
Pbr  I b 

 l  r  cos  ir  a
cos  z N s
(36)
15
r = parabola’s reflectance
The ratio of the power Pbr to the relative correspondent surface of the cylinder
R  d  l gives the intense [w/m2].
Finally, the program stores the coordinates of the points of the absorber where the
beam radiation has been absorbed. The profile of these points is shown in fig. 9.
Fig. 9: Ray trace diagram for the direct beam insolation, at solar time11:00, grid
Ns=16
In order to predict the performance of the mini parabolic collector, the beam absorbed
has to be determined, always with the heat generated profile at the tubes surface.
Diffuse component of solar insolation:
The diffuse component of the solar radiation is assumed to be isotropic. Under this
hypothesis the fraction of isotropic diffuse radiation accepted by a solar collector of
concentration C is I d C [14]. Α focusing parabola with small C has a small
acceptance angle and the acceptance for diffuse radiation is somewhat smaller, [14].
For the case of the irradiance on a collector aperture one may accept:
16
a. If the collector has a high concentration ratio ( C  10 ) then it accepts only a
negligible amount of diffuse radiation and the effective irradiance on the
aperture in this case is therefore:
I eff  I bo
cos 
 I bo R b
cos  z
θ is the angle of incidence of the sun on the aperture.
b. If the collector has a low concentration ratio C  10 and do not receives
radiation reflected from the ground, the effective irradiance is [21]:
I eff  I bo R b 
I do
C
(37)
Because of the predominance of near-forward scattering in the atmosphere, the sky
radiation tends to be centred on the sun, and therefore the isotropic model
underestimates the actual acceptance for diffuse radiation.
Hence, for the solar radiation accepted by a collector of concentration C is:
I real  I bo R b 
I do
C
To determine the points where the diffuse radiation hit the absorber either directly or
reflectively and the optical and energy efficiency we use the Monte Carlo method.[8]
Fig.10: Isotropic diffuse radiation – Monte Carlo Techniques
17
In the isotropic model the angle  R is uniformly distributed between the values 0 and
2π. For the angle  R , in the isotropic hypothesis of the diffuse insolation we can’t say

that  R is uniformly distributed between the values 0 and
but cos R is uniformly
2
distributed between the values 0 and 1.
To choose values for  R and  R we use a random number generator, which provides
R 1 and R 2 , then R  2R1 and cos  R  1  R 2 [8].


The random diffuse isotropic radiation unit vector n d   has the components
on the coordinate system x1 , y 1 , z 1 :
n dx1  sin  R sin  R
n dy 1  sin  R cos  R
(38)
n dz 1  cos 
And the diffuse solar radiation equation line from point   (0, y 1 , z d ) of the
aperture is:
x 1  0 y 1  y 1 z 1  z d


n dx1
n dy 1
n dz 1
(39)
The program checks initially if the random diffuse ray presented by the above
equation hits the absorber directly. Then it determines the point and angle of
incidence on the absorber. If the diffuse solar equation line intersects the parabola the
program checks the reflected diffuse rays that hit the absorber.
The diffuse radiation directly impinging the absorber and the diffuse radiation
incident on the absorber after reflection on the parabolas surface is investigated in the
following section.
Diffuse radiation directly impinging the absorber:
Solving the system of the cylinder equation (23) and the diffuse irradiation equation
line (39) we check if the solar line intersects the cylinders surface at a point
C d ( x 1Cd , y 1Cd , z 1Cd ) on the cylinder. At position x 1  x 1Cd we have the circular
section of fig. 11
18

nd
z1

R
θidd
C d ( x 1Cd , y1Cd , z1Cd )
λ
y1
Fig.11: Isotropic diffuse impinging the absorber directly.

The quantities of the diffuse solar radiation unit vector (n d ) and the incident angle
 idd are not necessarily located in the plane z 1 , y 1 .
z
we have tan   1Cd
y 1Cd


if y 1Cd  0
R  cos  j1  sin k 1


R   cos  j1  sin k 1






if y 1Cd  0

cos  idd  (n d  R )  (n d ) y1 cos   (n d ) z1 sin 
if y 1Cd  0
or

cos  idd  (n d  R )  (n d ) y 1 (  cos  ) (n d ) z1 (  sin  )
if y 1Cd  0
And finally the energy rate of the diffuse radiation incident directly and absorbed
on the relative arch at point C d is calculated.
Pdd  I d R d 
2y s
 l  cos  idd  a [W]
Ns
(40)
The ratio of the power Pdd to the relative correspondent surface of the cylinder
R  d  l gives the intense [W/m2 ].
19
Diffuse radiation incident on the absorber after reflection on the parabola’s
surface:
If the solar line of the isotropic diffuse radiation does not intersect the cylinder’s
surface, then we solve the system of equation (39) and the equation of the parabola
(23). If the solar line intersects the parabola at a point E d ( x 1Ed , y 1Ed , z 1Ed ) we
consider the parabola of fig.12 in the position x 1E d .
Fig.12: Schematic representation of the diffuse insolation.
The equation of the tangent tt  to the parabola at the point E d ( x 1Ed , y E1d , z 1Ed ) on a
plane y 1 , z 1 at the abscissa x 1E d is:
z 1  z 1Eq  m(y 1  y 1Ed )
Where
2
 dz 1 
d  y 1  4f 2  y 1Ed


m  tan  



4f
2f
 dy 1  Ed dy 

Then, the equation of the tangent at point E d is:
y 1E
y 1  z 1E d  2f
2f
The normal to the tangent at point E d is:
z 1  my 1  b 
20
 1
2f
z 1    y 1  b  
y  z 1Ed  2f
 m
y 1E d 1
The unit vector of the normal to the parabola at point E d is:



n p   sin  j1  cos  k 1 Where
y 1E d
  arctan
2f
For the reflection law:  i   r .


 


So, cos i  cos r  n d  n p  n d y  sin   n d z cos 
1
1
The unit vector of the reflected ray in the case of isotropic diffuse radiation is [21]:




  


rd  n d  2 n d  n p n p  n d  2 cos r n p
In this case, the reflected unit vector components are:
r x d  ns x
r y d  ns y
r z d  ns z
1
1
1
1
 2 cos  r sin 
1
1
 2 cos  r cos 
The equation of the reflected solar radiation’s unit vector line from point E d now is:
x 1  x 1Ed y 1  y 1Ed z 1  z 1Ed
(41)
 
 
 r x1d
 r  y 1d
 r  z 1d
Now, to determine the point where this line intersects the cylinder, we must solve the
system of equations (23) and (41). We take into account only one reflection; this is
true for the trough parabolic concentrator.
If the reflected line intersects the cylinder at point C(x1C d , y 1Cd , z 1Cd ) then we have, as
in subchapter 5.3.1:
z 1C d
,
tan  
y 1C d



R   cos  i1  sin  k 1
 


cos ird  rd  R    r y1d cos    r z1d sin  ,
if y 1C d  0
or
21
 


cos ird  rd  R    r y1d   cos    r z1d   sin  ,
if y 1Cd  0
Finally, the energy rate Pdr of the diffuse radiation incident after reflection and
absorbed on the relative arch at the point C d is calculated using the equation:
Pdr  I d  R d 
2y s
 l  cos  ird  r  a
Ns
(42)
The ratio of the power Pdr to the relative correspondent surface of the cylinder
R  d  l gives the intense [W/m2].
The program stores the coordinates of the absorber’s points where the diffuse
radiation has hit and gives the profile of these points (fig. 13)
a
b
Fig.13a, b: Ray trace diagram for diffuse radiation. Impinging the absorber (a)
directly, (b) after reflection.
Estimation of the performance:
The developed numerical code gives three options to determine the concentrating
collector system performance:
a) an instantaneous efficiency from the number of rays incident in the
absorber and that incident on the aperture and
b) then the optical efficiency and the collector’s thermal instantaneous
efficiency and
c) the long – term performance of the system using a fluid tank.
Instantaneous efficiency based on number of rays:
We consider a number, Ntot , of rays impinging on the aperture of solar concentrator.
Then, we investigate, what happens in the two cases of direct and diffuse components
of the solar radiation:
22
a. For the direct component:
In this case, the program gives the number of rays incident directly, Nbd , on the
absorber and the number of rays which hit the absorber after reflection, N br , on
the reflecting concentrator.
We can define efficiency for the direct component:
N  Nbr
b  bd
Nb,tot
(43)
b. For the diffuse component:
In this case, we consider a number of positions, Nd,tot , on the aperture and the
program gives, using the Monte Carlo method, a number of rays, Nd,tot . From
these rays, a number, Ndd , impinges the absorber directly and a number, N dr ,
impinges the absorber after reflection.
We can define the efficiency for the diffuse component:
d 
Ndd  Ndr
Nd,tot
(44)
The numbers Nd,tot and Nb,tot are equal to each other.
Finally, we can consider the total efficiency

I bo R b
I R
b  bo d d
IT
IT
(45)
I bo R b
is the fraction of the total radiation impinging on the aperture which
IT
I R
consider the direct component and do d is the fraction of the total radiation
IT
impinging on the aperture which considers the diffuse component.
Where
As it is well known, I T  Ibo Rb  Ido Rd
Instantaneous Optical Efficiency:
The optical efficiency is defined as the ratio between the energy rate absorbed [W]
and the energy rate incident on the aperture I T A ap [W]: [2]
23
op 
Q abs
I T A ap
(46)
The program calculates the energy rate absorbed in four steps:
1.
2.
3.
4.
The energy rate Pbd of the direct beam radiation incident directly and
absorbed on the receiver (29).
The energy rate Pbr of the direct beam radiation incident and absorbed on
the receiver after reflection (36).
The energy rate Pdd of the diffuse radiation incident directly and absorbed
on the receiver (40).
The energy rate Pdr of the diffuse radiation incident after reflection and
absorbed on the receiver (42).
The program distinguishes between optical efficiency for the direct beam radiation
and optical efficiency for the diffuse radiation.
Then,
opb 
Ns
Ns
i 1
i 1
 Pbd   Pbr
I bo R b A ap
Ns
opd 
(47)
Ns
 Pdd   Pdbr
i 1
i 1
I do R d A ap
(48)
The total optical efficiency:
op 
total energy rate absorbed (diffuse and direct )
total energy rate incident on the aperture (diffuse and direct )
That is,
k
 op 
 P
i 1
bd
 Pbr  Pda  Pdr 
I bo R b  I do R d A
(49)
Or
op  opb
I bo R b
I R
 opd do d
IT
IT
(50)
24
Notice: As far concerning the trough parabolic with a tracking system with reflectivity
r and absorptivity α of the absorber for the direct beam radiation the optical
efficiency  opb is approximately opb  r  a we can check this with the program
results only at noon
Instantaneous thermal collector efficiency:
The simulation code estimates the instantaneous thermal efficiency using two
different paths. The first one is via equation (21).
Qu
p 
A ap I T
Where, ground reflected radiation is neglected and useful heat gain rate is:
 C p Tfo  Tfi  [W]
Qu  m
 :
m
Cp :
Tfo , Tfi :
(51)
Fluid mass flow rate [kg/sec]
Specific heat capacity of the fluid [J/ kgK]
Fluid outlet and inlet temperatures respectively
The second path for the estimation of the instantaneous thermal efficiency can be
described by the equations (22)
p  op 
U L (Tc  Ta )
IT

A U T  Ta 
, p  FR op  C L fi

A ap
IT


Long – term performance:
In order to have the option to calculate the collectors’ long - term performance we
constructed a fluid tank.
 in the absorber tube is every time selected to take 54 liter
The fluid flow rate mass m
2
of fluid per hour and m of aperture surface
Fig. 14: Tank used on our experiment
 
m
54  A aptot
 3600  1000
[m3/s]
(52)
25
Where A apto t  2y s  l  n ch
And n ch is the number of parabolas channels.
The volume of the tank (fig. 14) is equal to the volume of the fluid that comes throw
the collector in an hour with the above flow rate.
  3600 [m3]
Vm
(53)
To execute the software, we supposed Tsi = Tfi = Tc = Ta then with iteration after the
calculation of FR and UL we take.
Tc  Ta 
Qu
A c I a  Qu
, Tfo 
 Tf i
 Cp
m
Ac UL
(54)
And
Tf 
Qu
 Tfi
 Cp
2m
The program calculate the tank’s useful energy rate
Qus  FR ' Ac [I a  U L (Tsi  Ta )]
We assume
FR’= 0.9 FR
For a time period of an hour and for unit absorber surface we have:
Q us  3600
 (mc p ) S (Tsf  Tfi )
Ac
(55)
And then
Tsf 
3600  Qus
 Tsi
A c (54Cp )s
(56)
26
References:
1. Duffie, J. A. and W. A Beckman, “Solar Energy Thermal Processes”, John
Wiley and sons, New York (1991).
2. Rabl, A, “Active solar collectors and their applications”.Oxford University
Press, New York, (1985)
3. Kreith, F and J. F. Kreider, “principles of solar Engineering”, Mc Graw – Hill,
New York (1978
4. Suhas P. Sukhatme, “Solar Energy” Tata Mc Graw – Hill, New Delhi (1996)
J.R.
5. Sir H. W. Massey and H. Kestelman, ‘Ancillary Mathematics’, Sir Isaac
Pitman & Son, London (1964).
6. Grant R. Fowles and G.L.Cassiday “Analytical Mechanics” Saunders Golden
Sunburst Series, Orlando (1993
7. D. E. Prapas, “Optics of
Parabolic –Trough Solar Energy Collectors
Possessing Small Concentrating Ratios”, Solar Energy, v39, No 6 (1987).
8. S. N. Kaplanis “ The Monte Carlo Method” , Thessaloniki, (1978), (in Greek)
27