Johnathon Wheaton
Multi-disciplinary Senior Design I
November 9, 2014
P15484 – Solar Assisted Essential Oil Distiller
Effect of Tracking Inaccuracies on Solar Trough Performance
Summary
An inaccurate solar tracker can undermine the ability of the solar trough to produce steam. The P15484
team seeks to understand the sensitivity of the trough steam production to angular deviation from the
sun’s rays. An AMPL optimization model was used to develop optimum parabolic shapes for the trough
given changes to the tracker angular tolerance and the diameter of the receiving pipe. Using the AMPL
model a list of trough dimensions and the resulting power output was compiled allowing the team to
analyze the sensitivity of the trough’s ability to produce power and to select the dimensions most
suitable for the project. As a result of the analysis, the team has chosen to design the trough 2 meters
long with a 2 inch diameter receiver. The team will accept the resulting power output so long as some
steam is produced.
Problem
Inaccurate solar tracking leads to a significant loss in the amount of energy collected by the solar trough.
As the trough tilt deviates from normal to the sun’s rays, rays quickly being to reflect farther and farther
from the focal point of the parabola. This video illustrates the effect of deviation from normal on the
focusing of solar rays: https://www.youtube.com/watch?v=4AUhH5b2CAc.
The problem can be mitigated by specifying a solar tracking accuracy tolerance (+ or - x degrees from
normal to the sun’s rays) for the tracking mechanism. This will require the tracker to give the trough the
angle necessary to focus the required amount of energy. This accuracy, however, is a function of the
concentrator pipe diameter. The team can either specify a pipe diameter and derive the tracking
tolerance, or it can specify a tracking tolerance and determine the appropriate pipe diameter. There are
currently no known (to the team) accuracy values for tracking mechanisms.
Method
The team was interested in knowing the effects of tracking angle deviation and concentrator pipe
diameter on the power output (to water) of the trough. Having a list of options for sizing and
specification for the trough and tracker would allow the team to select the option that was most
feasible and of least risk to the project timeline and budget based off team consensus. Analyzing these
options would also allow the team to view the sensitivity of the tracker performance to various changes.
Three angle deviations were selected (1, 5, and 10 degrees) as well as five pipe diameters (0.75, 1, 1.5, 2,
2.5 in) based on benchmarked products. For each angle deviation and pipe diameter there is a maximum
distance D that the parabola can be from the focal point and still have solar ray’s hit the receiver pipe.
Goal: Maximize aperture x
given an angle deviation θ
and pipe radius r.
Figure 1: Solar Trough Cross Section
Maximizing xi with an associated a will give the maximum aperture (or collection area) for a specified
angle deviation tolerance. An AMPL model (see appendix) was developed to find this maximum xi value
with an associated a value for a given angle deviation θ and pipe radius r, which are both used to find
the maximum distance a point can be on the parabola from the focal point D.
Because the exact optical and thermal efficiency are unknown, the power output was calculated for 3
different potential efficiencies (10%, 20%, 30%). Speaking with subject matter expert Dr. Robert Stevens
and after benchmarking these efficiencies seem to be conservative values for the efficiency of a solar
trough. Two different potential trough lengths (1 and 2 meters) were also selected for analysis. Lengths
longer than 2 meters pose a higher risk to the project budget.
The power output for each combination of angle deviation, pipe diameter, length, and efficiency was
calculated from the AMPL outputs. This provided a list of design options from which the team could
select whichever they felt was most feasible and of least risk to project success.
Results
Power Output as a Result of Angle Deviation*, Thermal &
Optical Efficiency, and Pipe Diameter
1000
Angle Deviation - Efficiency
1 - 0.1
900
1 - 0.2
800
1 - 0.3
5 - 0.1
Power (W)
700
5 - 0.2
600
5 - 0.3
10 - 0.1
500
10 - 0.2
10 - 0.3
400
Target: 325W
300
200
100
0
0.75
1
1.5
2
2.5
Pipe Diameter (in)
*Angle deviation from normal to sun's rays
The data (compiled in a table in the appendix) show a dramatic reduction in power output as the angle
deviation increases. The pipe diameter also greatly influenced the power production which was
expected. The only data points that lie above the team’s original target of 325W are with 1 degree
deviation from normal to the sun’s rays.
Discussion
The team had originally planned to have 66% of the distiller’s power be provided by solar energy.
However, this target poses high risk to the project timeline and budget. After a group discussion, the
team has chosen to pursue a pipe diameter of 2 inches and a trough length of 2 meters. The larger pipe
diameter will allow for greater angel deviation, which is a significant risk to the project. Larger pipes are
also substantially more expensive which is why the team has chosen a 2 in pipe over 2.5 inches. Even
with a 10 degree deviation, the trough will still (theoretically) be able to produce some steam.
The purpose of this project is proof of concept. Optimizing the power output of the trough is not the
primary goal. In the future, an accurate tracking mechanism can be designed to improve the scalability
and long-term feasibility of a solar assisted essential oil distiller.
The model does not take into consideration the effects of reflectivity of the receiving pipe. The optimal
parabolic shape found by this model may result in a high angle of incidence on the pipe causing many
rays to be reflected rather than absorbed.
The model is also simplified conservatively to make calculations easier. The receiving radius r is used in
the vertical y axis while it should, in fact, be the point on the receiver cross section perpendicular to the
incoming ray. If modeled more accurately, the y value would have been greater making the model used
more conservative than necessary.
Model used
Reality
The accuracy of the modeling used in this paper should be tested with a real trough. The team has
developed a mock-up trough to perform preliminary tests.
Appendix
AMPL Model
set DIAMETERS;
set ANGLES;
param dia;
param angle;
var
var
var
var
var
x>=0;
a;
W;
d;
l;
maximize xvalue: x;
s.t.
s.t.
s.t.
s.t.
s.t.
Const1: (dia/2)/tan((angle)*3.1416/180) = (x^2 +(a*x^2-1/(4*a))^2)^0.5;
Const2: a>=0.0001;
FocalWithin: a*x^2>=1/(4*a)+dia;
width: W = 2*x*.0254;
depth: d = a*x^2*.0254;
data;
set DIAMETERS:= .75 1 1.5 2 2.5;
set ANGLES:= 1 5 10;
AMPL Script
reset;
option solver knitro;
model msdparab.mod;
print "diameter " & "angle " & "x " & "a " & "W " & "d">output.xls;
for {diameter in DIAMETERS}
{
for{cangle in ANGLES}
{
let dia:=diameter;
let angle:= cangle;
solve;
print diameter, angle, x, a, W, d >output.xls;
}
}
Results Table
Values in red are below the 2.5 drops per second required to achieve 66% of the distillation energy
coming from solar power (the team’s original plan).
800
W/m^2
Pipe Dia
(in)
Assumed solar insolation:
Length
(m)
Deviation
(deg)
Efficiency
Width
(m)
Depth
(m)
Power
(W)
0.75
1
1
0.3
1.0907
0.2824
1
1
1
0.3
1.4543
0.3765
1.5
1
1
0.3
2.1814
2
1
1
0.3
a
Time (s)
Drops/sec
% of
target
CF
262
0.02412
0.4431
2.257
60%
57.25459
349
0.018087
0.3323
3.009
80%
57.25591
0.5647
524
0.012057
0.2216
4.513
120%
57.25459
2.9085
0.753
698
0.009043
0.1662
6.018
160%
57.25394
2.5
1
1
0.3
3.6357
0.9421
873
0.007235
0.1329
7.522
201%
57.25512
0.75
2
1
0.3
1.0907
0.2824
524
0.02412
0.2216
4.513
120%
57.25459
1
2
1
0.3
1.4543
0.3765
698
0.018087
0.1662
6.018
160%
57.25591
1.5
2
1
0.3
2.1814
0.5647
1047
0.012057
0.1108
9.026
241%
57.25459
2
2
1
0.3
2.9085
0.753
1396
0.009043
0.0831
12.035
321%
57.25394
2.5
2
1
0.3
3.6357
0.9421
1745
0.007235
0.0665
15.044
401%
57.25512
0.75
1
5
0.3
0.2144
0.064
51
0.141392
2.2544
0.444
12%
11.25459
1
1
5
0.3
0.2858
0.0853
69
0.106044
1.6912
0.591
16%
11.25197
1.5
1
5
0.3
0.4288
0.1279
103
0.070696
1.1272
0.887
24%
11.25459
2
1
5
0.3
0.5717
0.1706
137
0.053022
0.8454
1.183
32%
11.25394
2.5
1
5
0.3
0.7146
0.2132
172
0.042418
0.6764
1.478
39%
11.25354
0.75
2
5
0.3
0.2144
0.064
103
0.141392
1.1272
0.887
24%
11.25459
1
2
5
0.3
0.2858
0.0853
137
0.106044
0.8456
1.183
32%
11.25197
1.5
2
5
0.3
0.4288
0.1279
206
0.070696
0.5636
1.774
47%
11.25459
2
2
5
0.3
0.5717
0.1706
274
0.053022
0.4227
2.366
63%
11.25394
2.5
2
5
0.3
0.7146
0.2132
343
0.042418
0.3382
2.957
79%
11.25354
0.75
1
10
0.3
0.1011
0.0365
24
0.363181
4.7807
0.209
6%
5.307087
1
1
10
0.3
0.1348
0.0487
32
0.272385
3.5856
0.279
7%
5.307087
1.5
1
10
0.3
0.2022
0.0731
49
0.18159
2.3904
0.418
11%
5.307087
2
1
10
0.3
0.2696
0.0974
65
0.136193
1.7928
0.558
15%
5.307087
2.5
1
10
0.3
0.337
0.1218
81
0.108954
1.4342
0.697
19%
5.307087
0.75
2
10
0.3
0.1011
0.0365
49
0.363181
2.3904
0.418
11%
5.307087
1
2
10
0.3
0.1348
0.0487
65
0.272385
1.7928
0.558
15%
5.307087
1.5
2
10
0.3
0.2022
0.0731
97
0.18159
1.1952
0.837
22%
5.307087
2
2
10
0.3
0.2696
0.0974
129
0.136193
0.8964
1.116
30%
5.307087
2.5
2
10
0.3
0.337
0.1218
162
0.108954
0.7171
1.394
37%
5.307087
0.75
1
1
0.2
1.0907
0.2824
175
0.02412
0.6647
1.504
40%
57.25459
1
1
1
0.2
1.4543
0.3765
233
0.018087
0.4985
2.006
53%
57.25591
1.5
1
1
0.2
2.1814
0.5647
349
0.012057
0.3324
3.009
80%
57.25459
2
1
1
0.2
2.9085
0.753
465
0.009043
0.2493
4.012
107%
57.25394
2.5
1
1
0.2
3.6357
0.9421
582
0.007235
0.1994
5.015
134%
57.25512
0.75
2
1
0.2
1.0907
0.2824
349
0.02412
0.3324
3.009
80%
57.25459
1
2
1
0.2
1.4543
0.3765
465
0.018087
0.2493
4.012
107%
57.25591
1.5
2
1
0.2
2.1814
0.5647
698
0.012057
0.1662
6.018
160%
57.25459
2
2
1
0.2
2.9085
0.753
931
0.009043
0.1246
8.023
214%
57.25394
2.5
2
1
0.2
3.6357
0.9421
1163
0.007235
0.0997
10.030
267%
57.25512
0.75
1
5
0.2
0.2144
0.064
34
0.141392
3.3815
0.296
8%
11.25459
1
1
5
0.2
0.2858
0.0853
46
0.106044
2.5367
0.394
11%
11.25197
1.5
1
5
0.2
0.4288
0.1279
69
0.070696
1.6908
0.591
16%
11.25459
2
1
5
0.2
0.5717
0.1706
91
0.053022
1.2681
0.789
21%
11.25394
2.5
1
5
0.2
0.7146
0.2132
114
0.042418
1.0146
0.986
26%
11.25354
0.75
2
5
0.2
0.2144
0.064
69
0.141392
1.6908
0.591
16%
11.25459
1
2
5
0.2
0.2858
0.0853
91
0.106044
1.2684
0.788
21%
11.25197
1.5
2
5
0.2
0.4288
0.1279
137
0.070696
0.8454
1.183
32%
11.25459
2
2
5
0.2
0.5717
0.1706
183
0.053022
0.6341
1.577
42%
11.25394
2.5
2
5
0.2
0.7146
0.2132
229
0.042418
0.5073
1.971
53%
11.25354
0.75
1
10
0.2
0.1011
0.0365
16
0.363181
7.1711
0.139
4%
5.307087
1
1
10
0.2
0.1348
0.0487
22
0.272385
5.3783
0.186
5%
5.307087
1.5
1
10
0.2
0.2022
0.0731
32
0.18159
3.5856
0.279
7%
5.307087
2
1
10
0.2
0.2696
0.0974
43
0.136193
2.6892
0.372
10%
5.307087
2.5
1
10
0.2
0.337
0.1218
54
0.108954
2.1513
0.465
12%
5.307087
0.75
2
10
0.2
0.1011
0.0365
32
0.363181
3.5856
0.279
7%
5.307087
1
2
10
0.2
0.1348
0.0487
43
0.272385
2.6892
0.372
10%
5.307087
1.5
2
10
0.2
0.2022
0.0731
65
0.18159
1.7928
0.558
15%
5.307087
2
2
10
0.2
0.2696
0.0974
86
0.136193
1.3446
0.744
20%
5.307087
2.5
2
10
0.2
0.337
0.1218
108
0.108954
1.0757
0.930
25%
5.307087
0.75
1
1
0.1
1.0907
0.2824
87
0.02412
1.3294
0.752
20%
57.25459
1
1
1
0.1
1.4543
0.3765
116
0.018087
0.9970
1.003
27%
57.25591
1.5
1
1
0.1
2.1814
0.5647
175
0.012057
0.6647
1.504
40%
57.25459
2
1
1
0.1
2.9085
0.753
233
0.009043
0.4985
2.006
53%
57.25394
2.5
1
1
0.1
3.6357
0.9421
291
0.007235
0.3988
2.507
67%
57.25512
0.75
2
1
0.1
1.0907
0.2824
175
0.02412
0.6647
1.504
40%
57.25459
1
2
1
0.1
1.4543
0.3765
233
0.018087
0.4985
2.006
53%
57.25591
1.5
2
1
0.1
2.1814
0.5647
349
0.012057
0.3324
3.009
80%
57.25459
2
2
1
0.1
2.9085
0.753
465
0.009043
0.2493
4.012
107%
57.25394
2.5
2
1
0.1
3.6357
0.9421
582
0.007235
0.1994
5.015
134%
57.25512
0.75
1
5
0.1
0.2144
0.064
17
0.141392
6.7631
0.148
4%
11.25459
1
1
5
0.1
0.2858
0.0853
23
0.106044
5.0735
0.197
5%
11.25197
1.5
1
5
0.1
0.4288
0.1279
34
0.070696
3.3815
0.296
8%
11.25459
2
1
5
0.1
0.5717
0.1706
46
0.053022
2.5363
0.394
11%
11.25394
2.5
1
5
0.1
0.7146
0.2132
57
0.042418
2.0291
0.493
13%
11.25354
0.75
2
5
0.1
0.2144
0.064
34
0.141392
3.3815
0.296
8%
11.25459
1
2
5
0.1
0.2858
0.0853
46
0.106044
2.5367
0.394
11%
11.25197
1.5
2
5
0.1
0.4288
0.1279
69
0.070696
1.6908
0.591
16%
11.25459
2
2
5
0.1
0.5717
0.1706
91
0.053022
1.2681
0.789
21%
11.25394
2.5
2
5
0.1
0.7146
0.2132
114
0.042418
1.0146
0.986
26%
11.25354
0.75
1
10
0.1
0.1011
0.0365
8
0.363181
14.3422
0.070
2%
5.307087
1
1
10
0.1
0.1348
0.0487
11
0.272385
10.7567
0.093
2%
5.307087
1.5
1
10
0.1
0.2022
0.0731
16
0.18159
7.1711
0.139
4%
5.307087
2
1
10
0.1
0.2696
0.0974
22
0.136193
5.3783
0.186
5%
5.307087
2.5
1
10
0.1
0.337
0.1218
27
0.108954
4.3027
0.232
6%
5.307087
0.75
2
10
0.1
0.1011
0.0365
16
0.363181
7.1711
0.139
4%
5.307087
1
2
10
0.1
0.1348
0.0487
22
0.272385
5.3783
0.186
5%
5.307087
1.5
2
10
0.1
0.2022
0.0731
32
0.18159
3.5856
0.279
7%
5.307087
2
2
10
0.1
0.2696
0.0974
43
0.136193
2.6892
0.372
10%
5.307087
2.5
2
10
0.1
0.337
0.1218
54
0.108954
2.1513
0.465
12%
5.307087