# Formulation of Mathematical Model and Neural Network Analysis of Schefler Reflector

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VSRD
-
MAPE, Vol. 1 (1
), 2011,
1
-
5

____________________________

1
Research Scholar
,
2
Principal,

12
Department

of
Mechanical Engineering
,

,
Nagpur
,
Maharashtra
, INDIA
.

3
Principal
, Department of Mechanical Engineering,
-
Patil College of
Engineering
& Technology, Nagpur
,

Maharastra
, INDIA.
*Correspondence :
rupesh.patil@gm.com

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Formulation of Mathematical Model and Neural
Network Analysis of Schefler Reflector

1
Rupesh Patil
*
,
2
Gajanan K. Awari

and
3
Mahendra P. Singh

ABSTRACT

According to a report of the United Nations, solar energy falling over an area of 800
km x 800 km harvested
with the currently available technology is enough to meet the energy needs of the whole world. The solar energy
radiation falling over India is estimated to be about 5,000 trillion kWh a year.

Scheffler reflector is one of the various

methods of harvesting the solar energy. This paper discusses about work
carried out on the Scheffler reflector. It has been focused about having scope for experimental data based
modeling to establish relationship in different variables of Scheffler refle
ctor. Scheffler reflector is studied with
a typical experimental plan of simultaneous variation of independent variables. Experimental response data is
analysed by formulating dimensional equations

and validated by using neural network analysis.

Keywords
:

Scheffler Reflector, Dimensional Equation, Generalized Model Doe, Experimental Data Based
Modeling, Neural Network Analysis.

1.

INTRODUCTION

German Scientist Wolfgang Scheffler has devised a parabolic reflector set to harness solar energy. A
concentrating
primary reflector tracks the movement of the sun, focusing sunlight on a fixed place.

The focused
light heats a very large pot, which can be used for heating, steam generation, cooking, baking breads, water
heating [1].

The Scheffler reflector can be used
for the supply of hot water for domestic purposes. These systems have one
water storage tank which performs dual function of absorbing solar radiation and preserving heat of water. The
reflector used is having an area of 8m
2
.

The sunlight that falls onto t
his reflector is reflected sideways to the
focus located at some distance of the reflector. The axis of daily rotation is located exactly in north
-
south
-
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), 2011

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direction, parallel to earth axis and runs through the centre of gravity of the reflector. That way the

reflector
always maintains its gravitational equilibrium and the mechanical tracking device (clockwork) or motor doesn't
need to be driven by much force to rotate it synchronous with the sun. The focus is located on the axis of
rotation to prevent it from

moving when the reflector rotates. During the day concentrated light rotated around
its own centre but does not move sideways in any direction and by this way the focus stays fixed. At the focus it
has a container to hold 20 liter water [2
-
8]. The reflect
or is brought to its normal position every morning by
operator. Its approximate generalized experimental data based model is focused by methodology of
experimentation proposed by H.Schank, Jr.

The scope of the research is to establish design data for Schef
fler Reflector with the help of most influencing
design parameters on functioning of Scheffler reflector. In the present work experimental study of Scheffler
Reflector water heater consisting single storage tank as an absorber mounting in side curved refle
ctor trough has
been carried out. The solar reflector of 20 liter per day capacity is designed .This will add to enhancement of
technology in the solar energy sector and Scheffler reflector can be better understood from design point of view.

2.

NEED FOR FORMU
LATING GENERALISE
D EXPERIMENTAL DATA BASED
MODEL

While working on the Scheffler Reflector it is necessary to decide the parameters that affect on the temperature
at focus point. This would be Possible when a quantitative relationship amongst various depend
ent and
independent variables of the system is established by formulating the mathematical model of Scheffler
Reflector. It is well known that such mathematical model for Scheffler Reflector cannot be formulated by
applying only logic but formulation with
an experimental data base is essential. In this experimental approach
all the independent variables are varied over a widest possible range, a response data is collected and an
analytical relationship is established. Then the technique of optimization is a
pplied to deduce the values of
independent variables at which the necessary responses can be minimized. The determination of such values of
independent variables is always the puzzle for because of complex phenomena of interaction of various
independent va
riables such as Generated Water

Temp, Incident Rays, Water Quantity, Operation time, wind
speed, ambient temperature, acceleration due to gravity whereas response variable would be generated water
temperature and heat gain. Theta (Generated Water Temperatu
re)

and heat gain

is considered while developing
model.

The methodology briefly is as follows.

Identification of Independent and Dependent variables.

Reduction of independent variable adopting dimensional analysis

Test planning comprising of determination
of test envelop, test points, test sequence and experimentation
plan.

Physical design of an experimental set up.

Execution of experimentation

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), 2011

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Purification of experimentation data

Formulation of model

Model optimization

3.

DETAILS OF FORMULATION OF APPROXIMAT
E GENERALISED EXPERIMENTAL
DATA BASED MODEL

3.1.

Design of Experimentation

Reduction of variables through dimensional analysis: The various independent and dependent variables of the
system with their symbols and dimensional formulae are given in
nomenclature.

There are several quite simple
ways in which a given test can be made compact in operating plan without loss in generality or control. The best
known and the most powerful of these is dimensional analysis. In the past dimensional analysis was primarily
us
ed as an experimental tool whereby several experimental variables could be combined to form one. The field
of fluid mechanics and heat transfer were greatly benefited from the application of this tool. Almost every major
experiment in this area was planned

with its help. Using this principle modern experiments can substantially
improve their working techniques and be made shorter requiring less time without loss of control. Deducing the
dimensional equation for a phenomenon reduces the number of independent

variables in the experiments. The
exact mathematical form of this dimensional equation is the targeted model. This is achieved by applying
Buckingham’s π theorem (Hibert, 1961).When we apply this theorem to a system involving n independent
variables, (n m
inus number of primary dimensions viz. L, M, T,

) i.e. (n
-
4) numbers of π terms are formed.
When n is large, even by applying this theorem number of π terms will not be reduced significantly than number
of all independent variables. Thus much reduction in
number of variables is not achieved. It is evident that, if we
take the product of the terms it will also be dimensionless number and hence a π term. This property is used to
achieve further reduction of the number of variables. Dimensional analysis is use
d to reduce the variables and
following Pi terms were evolved out of it.

Buckingham theorem in brief.

Generated Water Temperature [θ] ,

θ1=f ( Ib, VT, Time, D, W, θ2, g,Tr)

f (θ1, Ib, VT, Time, D,W, θ2,g ,Tr) = 0

(

1
,

2
,

3
,

4
,

5
,) = 0

N = 9 , m =4

No of

terms ( n
-

m ) = 9
-
4 =5

Repeating variables are

D [L ] ,W [ LT
-
1 ] , Ib [ MT
-
3 ] , θ2

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As

Repeating variables are

D

[L2 ] ,W [ LT
-
1 ] , Ib [ MT
-
3 ] , θ2

1 = D a1 W b1 Ib C1 * θ1 d1, θ2

[M0 L0 T0 ] =
[L2] a1[ LT
-
1 ] b1 [MT
-
3 ] C1

[θ]

Equating power of MLT on both sides we get,

Power of M,

0=C1

C1 =0

Power of L,

0=2a1+b1

2a1 =
-
b1

2a1 = 0

(as b1 = 0)

Power of T,

0=
-
b1
-
3C1

b1 = 0

(as C1 = 0)

Power of
θ
2

0=d
1
+1

d
1
=
-
1

1

=
θ
1
/

θ
2

Similarly other
Pi terms are calculated

and shown in table 2.

Table
1 :

Various Pie
Term Details

Independent Pi Terms

Sr.

Description of Pi Terms

Equation of Pi Terms

1

Pi
-
Term relating to Water quantity and Dish Diameter

2

= V
T
/ D
1.5

2

Pi
-
Term relating to wind speed

3

= W*Time / D
0.5

3

Pi
-
Term relating to g and wind speed

4

= g* D
0.5

/ W
2

4

Pi
-
Term relating to Dish Position

5

= Tr* D
0.5

/ W

Dependent Pi Terms

1

Pi
-
Term relating to Temperature

1

= Θ1/ Θ2

2

Pi
-
Term relating to Heat Gain

7

= Qh *w / D^1.5*Ib

3.2.

Test Planning

This comprises of deciding test envelope, test points, test sequence and experimentation plan for deduced set of
dimensional equations. Table 3 shows Test envelope, test points and

Table 4 shows plan of experimentation
for
all operations.

3.2.1.

Determination of Test Envelope

The test envelope comprises of complete range encompassed by the individual variables. So it is now necessary
to ascertain the complete range over which the entire experimentation is to be carried out. Th
e ranges for various
π terms are ascertained below for all processing operations.

Range of π: From the data, values of various variables can be substituted in this equation to get the values of pi
term. If these values are arranged in the descending or asc
ending order the maximum and minimum values of
this Pi term can be calculated to get range for π. These ranges for all Pi terms are given in below table.

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), 2011

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Table
2

:

Test
Envelop

Pi
Term

Equation

Test Envelope

(Summer)

Test Envelope

(Winter)

Test Envelope

(R
ainy)

Π
2

Water Quantity to Cube
of

Dish Diameter

Vt/D^1.5

(0.00088 Constant)

(0.00088 Constant)

(0.00088 Constant)

Π
3

W *Time / D^0.5

(1.76 to 537.4)

(1.76 to 143.18)

(1.76 to 470.22)

Π
4

g* D^ 0.5 / W^2 *

(0.093 to 27.746)

(0.277 to 27.46)

(0.141 to
27.746)

5

Tr* D^0.5 / W

(0.0007 to 0.024)

(0.0246 to 0.024)

(0.000868 to 0.024)

3.2.2.

Determination of Test Points

The spacing of the test points within the test envelop is selected not to get ‘symmetrical’ or ‘pleasing’ curve but
to have every part of our
experimental curve, map the same precision as every other part. Thus, the concept of
proper spacing is now replaced by permissible spacing of the test points. Similarly, for all other pi terms the test
points are decided by permissible spacing rather than
the proper spacing.

3.2.3.

Determination of Test Sequence

The choice of test sequence is decided by nature of experimentation viz reversible or irreversible. In fact all tests
basically are irreversible in the sense that no piece of apparatus returns to an ident
ical previous configuration
after same use. But if the changes brought by testing are below the level of detection such tests could be
assumed as reversible. The independent variables are varied from one extreme to another in a sequential plan or
in a perf
ectly random fashion in a random plan. The sequential plan is essential for irreversible experiments
where randomization is not practicable. In this experimentation like majority of engineering experiments partial
randomized sequence is proposed.

3.3.

Physical
Design of an Experimental Setup

It is necessary to evolve physical design of an experimental set up having provision of setting test points,
adjusting test sequence, executing proposed experimental plan, provision of necessary instrumentation for
noting do
wn the responses and independent variables. From these provisions dependent and independent Pi
Terms reduce to the dimensional equation. The experimental set up is designed considering various physical
aspects and its elements.

Procedure of design of exper
imental set up however is not totally followed in the filled experimentation because
of the use of experimentation with the available ranges of the various independent variables to assess the value
of the dependent variable.

3.4.

Selection and calibration of in
struments for measurement

The instruments are to be selected for measuring various dependent and independent variables. The
specifications of instruments include the permissible amount of error.

Instruments used for measuring various physical quantities
in experimentation are calibrated by using
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Replication techniques to have adequate predictability of measurement.

In this experimentation, the measurements done are as follows.

Temperature: For water temperature measurement thermocouple is used.

Radiations are measured by using Pyranometer.

Wind Speed, Ambient Temperature: Wind speed is measured by an Anemometer.

Dish Position is measured by using Protector.

3.5.

Experimental P
rocedure

The detail procedure for the test run is as under;

In the morning

the motor is stared which drives dish to follow the sun from morning to evening.

Water is filled in the Tank which is at focus.

As soon as dish starts moving the readings were noted at regular interval manually.

Observations: More than 600 observations we
re recorded and the sa
mple observation is shown below :

Fig.

1

:
Experimental
Set Up

4.

MODEL FORMULATION

The data of the independent and dependent parameters of the system has been gathered during experimentation.
In this case there are one dependent and five independent pi terms involved in the experimentation. It is
necessary to correlate various independen
t and dependent pi terms involved in this system quantitatively. This
correlation is nothing but a mathematical model as design tool for such workstation. Based on the
experimentation work performed on the experimental setup, using classical plan of experi
mentation, this chapter
discusses.

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Deduction of generalized Experimental Models for the dependent π terms.

To develop Artificial Neural Network simulation.

5.

Formulation of Experimental Data Based Model

It is necessary to correlate quantitatively various
independent and dependent terms involved in this very
complex phenomenon. This correlation is nothing but a mathematical model as a design tool for such situation.
The mathematical model for all processing operations is given below:

For Generated Water Tem
perature and Heat Gain:

Four independent pi terms (

2,

3,

4,

5,
) and two dependent pi terms (

1,

7
) have been in the design of

experimentation and are available for the model formulation.

Independent

terms = (

2,

3,

4,

5,
), Dependent

terms = (

1,

7
)

Each dependent

is assumed to be function of the available independent

terms,

θ

= f (

2,

3,

4,

5,
)

Where,

θ

=

1,
Dependent pi term

“f (

1,
)= function of (

2,

3,

4,

5,
) ”

A probable exact mathematical form for this phenomenon could be the

empirical relationships in between
dependent dimensionless ratio and independent dimensionless ratio and are assumed to be exponential.

Following four mathematical relationships are formed for three seasons in a year (winter, Rainy, summer).

1

= k
1

x (

2
)

a1
x

(

3
)

b1
x

(

4
)

c1
x

(

5
)

d1
x

(

6
)

e1

Considering Equation 1 to simplify

1

= k
1

x (

2
)

a1
x

(

3
)

b1
x

(

4
)

c1
x

(

5
)

d1
x

(

6
)

e1

Taking log of both the sides

Log [

1
] = Log [k
1

x (

2
)

a1
x

(

3
)

b1
x

(

4
)

c1
x

(

5
)

d1
x

(

6
)

e1
]

Log [

1
] = Log [k
1
] + Log [(

2
)

a1
]

+ Log [(

3
)

b1
]

+ Log [(

4
)

c1
]

+ Log [(

5
)

d1
]

+ Log

[(

6
)

e1
]

Log (

1
) = Log (k
1
) + a
1

x Log (

2
) + b
1

x Log (

3
) + c
1

x Log (

4
) + d
1

x Log (

5
)

+ e
1

x Log (

6
)

Let us define; Log

(

1
) = Z
,

Log (k
1
) = K
,

Log (

2
) = A
,

Log (

3
) = B
,

Log

(

4
) = C
,

Log (

5
) = D
,

Log (

6
) = E

Z = K + [a1 A] + [b1 B] + [c1 C] + [d1 D] + [e1 E]

In above equation, a1, b1, c1, d1, e1 are unknowns whose value are to be find out, while A,B,C,D,E are set of
values obtained during experimentation.

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To solve the
above equation, multiply coefficient of a
1
, b
1
, c
1
, d
1
, e
1

individually.

Multiplying by A,
(Similarly multiply to rest of the equations )

AZ = AK + [a
1

A
2
] + [b
1

AB] + [c
1

AC] + [d
1

1

AE]

Above Set of equations are valid for the number of reading ta
ken during experimentation, therefore taking
summation of these for n values,

The equations become,

(Z) = n*K + [a
1

(A)] + [b
1

(B)] + [c
1

(C)] + [d
1

(D)] + [e
1

(E)]

(AZ) = K*

A + [a
1

(A
2
)] + [b
1

(AB)] + [c
1

(AC)] + [d
1

1

(AE)]

(BZ) =

K*

B + [a
1

(AB)] + [b
1

(B
2
)] + [c
1

(BC)] + [d
1

(BD)] + [e
1

(BE)]

(CZ) = K*

C + [a
1

(AC)] + [b
1

(BC)] + [c
1

(C
2
)] + [d
1

(CD)] + [e
1

(CE)]

(DZ) = K*

D + [a
1

1

(BD)] + [c
1

(CD)] + [d
1

(D
2
)] + [e
1

(DE)]

(EZ) = K*

E + [a
1

(AE)] +
[b
1

(BE)] + [c
1

(CE)] + [d
1

(DE)] + [e
1

(E
2
)]

To solve these equations, reducing it to matrix form

and
By putting th
e values for various parameters

in the
matrices shown above the following matrices are obtained.

For summer

119.37

360.00

-
1099.32

629.56

20.14

-
885.85

K

364.51

-
1099.32

3356.930

-
922.45

-
61.51

2705.09

a
1

239.25

=

629.56

-
1922.44

1254.44

-
75.28

-
687.15

x

b
1

3.48

20.14

-
61.51

-
75.28

228.96

59.69

c
1

312.63

-
885.85

2705.09

-
687.15

59.69

2350.31

d
1

[P
1
] = [W
1
] [X
1
]

Using Mat
lab, X
1
= W
1
\

P
1

, after solving X
1

matrix with K
1

and indices a
1
, b
1
, c
1

and d
1

are as follows

K
1

-
0.36

a
1

-
0.11

b
1

0.42

c
1

0.12

d
1

0.16

But K
1

is log value so convert into normal value take antilog of K
1

Antilog (
-
0.36) = 0.44

Hence the model for dependent term

1

1
= k
1

x (

2
)
a1
x

(

3
)
b1
x

(

4
)
c1
x

(

5
)
d1

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Θ1/ Θ2

= 0.44
*
(
VT/ D
1.5
)
-
0.11
(
W*Time / D
0.5
)
0.42
(
g* D
0.5

/ W
2
)
0.12
*
(
Tr* D
0.5

/ W
)

0.16

Details are given only for one season. Similar calculations are done for other

seasons also.

Using MATLAB7.0, following values are obtained

Pi terms

Generated Water Temperature

Heat Gain

Summer

Winter

Rainy

Summer

Winter

Rainy

K (Constant)

0.44

0.87

0.93

0.32

1.60

0.87

A

-
0.11

-
0.0007

-
0.01

-
0.15

-
0.02

-
0.01

B

0.42

0.45

0.33

-
0.05

-
0.04

-
0.18

C

0.12

0.16

0.14

-
0.69

-
0.63

-
0.55

D

0.16

0.12

0.07

0.34

0.36

.20

Table
3

:
Various Values of Indices Obtained for Various Mathematical Models

The various mathematical models for different seasons are stated below.

For Generated Water Temperature
(For Summer, winter and Rainy season respectively)

The various mathematical models for dif
ferent seasons are stated below :

Π
1

= 0.44
*

2
)
-
0.11

3
)
0.42

4
)
0.12

5
)

0.16

-

(1)

Π
1

= 0.87
*

2
)
-
0.0007

3
)
0.45

4
)
0.16

5
)

0.12

-

(2)

Π
1

= 0.93
*

2
)
-
0.01

3
)
0.33

4
)
0.14

5
)

0.07

-

(3)

For Heat Gain
(For Summer, winter and Rainy season respectively)

Π
7

= 0.33
*

2
)
-
0.15

3
)
-
0.05

4
)
-
0.69

5
)

0.34

-

(4)

Π
7

= 1.
60

*

2
)
-
0.02

3
)
-
0.0
4

4
)
-
0.63

5
)

0.36

-

(5)

Π
7

= 0.873
*

2
)
-
0.01

3
)
-
0.18

4
)
-
0.55

5
)

0.20

-

(6)

6.

MODEL OPTIMIZATION

The ultimate objective of this work is not merely developing the models but to find out best set of independent
variables, which will result in maximization/minim
ization of the objective functions. In this case there are three
different models corresponding to the summer, winter, Rainy for water temperature and three for heat gain
respectively. There are thus six objective functions corresponding to these models. T
he objective functions for
Water temperature and heat gain need to be maximized. The models have non linear form; hence it is to be
converted into a linear form for optimization purpose. This can be achieved by taking the log of both the sides of
the model
. The linear programming technique is applied which is detailed as below for Water temperature for

summer.

Π
1

= 1.39
*

2
)
-
0.006

3
)
0.43

4
)
0.06
*

5
)

0.23

Taking log of both the sides of the Equation, we get

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Log(Π
1
)=log(1.39)
-
0.006xLog(π
2
)+0.43xLog(π
3
)+

0.06xLog(π
4
)+0.23xLog(π
5
)

Z = K* a x X
1
+ b x X
2
+ c x X
3
+d x X
4

+ e x X
5

+f x X
6

+ f x X
7
and

Z=log (1.39)
-
0.006

*
log (π
2
) + 0.43

*
log (π
3
) + 0.06
*
log (π
4
) +0.23

*
log (π
5
)

Thus, equation (2, 3, 4, 5 and 6) will be objective for
maximized

for the p
urpose of purpose of formulation of
the linear programming problem. Putting the constraints equations and solving the problem,

Thus, Z
max
for Π
1

=
347.78 and values of independent terms are obtained by taking antilog of

X
1
, X
2
, X
3

and X
4

are
0.00088,537.4,27.746 and 0.24.

On the similar line models for all dependent terms were optimized for respective objective functions.

By solving
this LP problem using the MS Solver available in MS Excel and taking antilog we get corresponding optimum
value.

Optimal values of influencing parameters are

:

Pi terms

Generated water temperature

Heat gain

Summer

Winter

Rainy

Summer

Winter

Rainy

Z

347.78

222.66

259.0

7045.91

6710.35

5331.92

π
2

0.0008

0.0008

0.0008

0.0008

0.0008

0.0008

π
3

537.4

143.18

470.22

1.76

1.76

1.76

π
4

27.74

27.46

27.74

0.093

0.277

0.14

π
5

0.024

0.024

0.024

0.024

0.024

0.024

Table
4

:
Optimum values for various independent parameters for various three seasons

7.

SENSITIVITY ANALYSIS

The influence of the various
independent

term has been studied

by analyzing the indices of the various

term
in the models .Through the technique of sensitivity analysis the change in the value of a dependent

term
caused due to an introduced change in the value of individual

term is evaluated. In this case change of 10% is
introduced in the individual independent

term independently (one at a time).

Thus total range of the introduced change is

10 %

The effect of this introduced change on the change in the value of the depe
ndent

term is evaluated. The
average values of the change in the dependent

term due to the introduced change of

10 % in each
independent

terms. This is defined as sensitivity.

7.1.

Graphical Representation

Following Bar
-
charts shows the effect of introd
uced change according to the sensitivity of individual

term
involved in the phenomenon
.

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Fig.

2

:
Gr
aph Showing Sensitivity Analysis

7.2.

Effect of Introduced Change on the

te牭s fo爠桥⁳畭m敲 (Fi杵牥″

When a total range for the change of

10% introduced in the value of independent

term

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9.21% occurs in the value of

1 (computed from the model).

The change brought in the value of

1 because of change in the values of other independent

term

5 is only
3.18. Similarly
the change of about 1.31 takes place because of change in the values of

6.

It can be seen that highest change takes place because of

term

3 where as the least change takes place due to
the

term

2. Thus

1 is the most sensitive

term and

2 is the
least sensitive

term. The sequence of the
various

terms in the descending order of sensitivity is

5,

6,

4.

Similar conclusions can be made for finding Influencing Pie terms on Dependent π1

Table
5

: Sequence of influence of independent pi terms on
dependent pi terms for three seasons

Dependent Pi terms

Sequence of independent pi terms according

to intensity of influence

(High to Low)

Summer:
Π
1

Π
3

Π
5

Π
4

Π
2

Winter:
Π
1

Π
3

Π
4

Π
5

Π
2

Rainy:
Π
1

Π
3

Π
5

Π
4

Π
2

Table
6

: Sequence of influence of
independent pi terms on de
pendent pi terms: For Heat Gain

Dependent Pi terms

Sequence of independent pi terms according to intensity of influence

Summer:
Π
7

Π
5

Π
3

Π
2

Π
4

Winter:
Π
7

Π
5

Π
2

Π
3

Π
4

Rainy:
Π
7

Π
5

Π
2

Π
3

Π
4

8.

COMPUTATION

OF

THE

PREDICTED

VALUES

BY

‘ANN

In this research the main issue is to predict the future result. In such complex phenomenon involving non
-
linear
system it is also planned to develop Artificial Neural Network (ANN). The output of this network can be
evaluated by
comparing it with observed data and the data calculated from the mathematical models. For
development of ANN the designer has to recognize the inherent patterns. Once this is accomplished training the
network is mostly a fine
-
tuning process.

An ANN consist
s of three layers (representing the synapses) and the output layer .It uses nodes to represent the
brains neurons and these layers are connected to each other in layers of processing. The specific mapping
performed by ANN depends on its architecture and va
lues of synaptic weights between the neurons. ANN as
such is highly distributed representation and transformation that operate in parallel and has distributed control
through many highly interconnected nodes. ANN were developed utilizing this black box con
cepts. Just as
human brain learns with repetition of similar stimuli, an ANN trains itself within historical pair of input and
output data usually operating without a priory theory that guides or restricts a relationship between the inputs
and outputs. the

ultimate accuracy of the predicted output , rather than the description of the specific path(s) or
relationship(s) between the input and output , is the goal of the model .The input data is passed through the
nodes of the hidden layer(s) to the output lay
er , a non linear transfer function assigns weights to the information
as it passes through the brains synapses. The role of ANN model is to develop a response by assigning the
weights in such a way that it represents the true relationship that really exis
ts between the input and output.
During training, the ANN effectively interpolates as function between the input and output neurons. ANN does
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not an explicit description of this function. The prototypical use of ANN is in structural pattern recognition. In

such a task, a collection of features is presented to the ANN; it must be able to categories the input feature
pattern as belonging to one or more classes . In such cases the network is presented with all relevant information
simultaneously. The results o
f ANN are shown and discussed below.

Water Temperature operation: (summer, winter
, Rainy) :
This training stopped when the validation error
increased, which occurred after 16 iterations. Refer figure 5 .Training produces a plot of the training errors,
vali
dation errors, and test errors, as shown in the following figure. In this example, the result is reasonable
because the final mean square error is small, the test set error and the validation set error has similar
characteristics, and no significant over f
itting has occurred.

Fig.

3

: Topography for ANN

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Fig.

4

: Graphs showing ANN Results For Summer (Generated water temperature)

9.

RESULT AND DISCUSSION

From equation 1, 2 and 3, the absolute index of π
3
is highest 0.42, 0.45, and 0.33 of all π
terms. The factor π
3
is
related to ratio of Wind speed* time to Dish area. The value of this index is positive indicating involvement of
this ratio has strong impact on Π
1
for summer, winter and Rainy season on generated water quantity.

The value of this i
ndex is positive indicate that theta increases with increase in the ratio and or otherwise.

The absolute index of π
2

is lowest index
-
0.11,
-
0.0007,
-
0.01 of all π terms for summer, winter and rainy season.
This factor is related to water quantity to Dish ar
ea and is the least influencing term in this model. This is the
least influencing term in this model for generated water quantity.

From equation 1, the relative influence of pi terms on dependent pi can be calculated by dividing power of
respective pi term
s. For summer season (theta) relative influence on dependent Pi term of π
3
and π
4
is 3.66
whereas of π
3
and π
5

is 2.67.

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Similar calculation for remaining 5 models gives relative influence as below. For winter season (theta) relative
influence on dependent
Pi term of π
3
and π
4
is 2.73 whereas of π
3
and π
5

is 3.58. For rainy season (theta) relative
influence on dependent Pi term of π
3
and π
4
is 2.41 whereas of π
3
and π
5

is 4.93.

From equation 4, 5 and 6, relative influence of π
2
and π
3

as well as π
2
and π
4
is

calculated on dependent pi term.
For summer this is 3.12 and 0.21.For winter is 0.58 and 0.035.And for rainy season relative influence of π
2
and
π
3

is 0.046 and relative influence of π
2
and π
4
is 0.014.

From equation 4, 5 and 6, the absolute index of π
5
i
s highest 0.34, 0.36, and 0.20 of all π terms. The factor π
5
is
related to ratio of Dish Position* Dish area to Wind Speed. The value of this index is positive indicating
involvement of this ratio has strong impact on Π
7
for summer, winter and Rainy season

on Heat gain.

The value of this index is positive indicate that heat gain increases with increase in the ratio and or otherwise.

The absolute index of π
4

is lowest index
-
0.69,
-
0.63,
-
0.55 of all π terms for summer, winter and rainy season.
This factor is
related to ratio of Acceleration due to gravity *Dish Area to Wind speed and is the least
influencing term in this model.

The equations show magnification factors (K) as 0.44, 0.87, 0.93 for theta and 0.33, 1.60, 0.87 for heat gain.

Optimal values of infl
uencing parameters are also determined. Optimal

parameters in Scheffler reflector

are
usually selected to achieve a high water temperature and high heat gain with minimum process time which is
directly inclined towards economy of the process.

The various
optimal value calculated are

Pi Terms

Generated Water Temperature

Heat Gain

Summer

Winter

Rainy

Summer

Winter

Rainy

Z

347.78

222.66

259.0

7045.91

6710.35

5331.92

π
2

0.0008

0.0008

0.0008

0.0008

0.0008

0.0008

π
3

537.4

143.18

470.22

1.76

1.76

1.76

π
4

27.74

27.46

27.74

0.093

0.277

0.14

π
5

0.024

0.024

0.024

0.024

0.024

0.024

Following table
s

shows the coefficient of correlation between dependent variable

and various independent
parameters for various s
easons
.

Generated Water Temperature

Parameters

Training Validation

Validation

Testing

Test

Summer

0.9944

0.9898

0.9927

Winter

0.9771

0.7572

0.7165

Rainy

0.9109

0.8046

0.8222

Heat Gain

Parameters

Training Validation

Validation

Testing

Test

Summer

0.
77

0.
76

0.
68

Winter

0.
89

0.7
4

0.
66

Rainy

0.75

0.59

0.60

10.

CONCLUSION

This study has developed dimensionless correlations for analyzing the performanance of Scheffler reflector. The
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major contributions and analytical findings of this study can be summarized as follows.

Dimensional correlations for generated water temperature, heat gain and various other independent variables
have been developed based on experimental results. The dimensional analysis shows that generated water
temperature is determined primarily by ratio
of product of wind speed and time of operation to Dish size.

The models have been formulated mathematically for the local

conditions. After training the Artificial Neural
Network it is found that every case of experimental results are in good agreement wi
th the predicted values
obtained by ANN. The coefficient of correlation (R) for every case tested is found to be in the range of 0.60 to
0.99 which suggest the validation of experimentation carried out in the present work
.

From the results it is seen that
the mathematical models can be successfully used for the computation of
dependent terms for a given set of independent terms.

The trends for the behavior of the models demonstrated by the graphical analysis and sensitivity analysis are
found complimentary

to each other.

Nomenclature

θ
1

Generated Water Temp.( Celsius)

Qh

Heat Gain (KJ)

Ib

Incident Rays (
W/m
2)

VT

Water Quantity (Litre )

Time

Operation time (Minutes)

D

Dish area (
m
2)

W

Wind Speed( Kmph)

θ
2

Ambient Temp. (Celsius)

g

Acceleration due

to
Gravity

Tr

Angular velocity of dish reached after time interval T (rad/sec)

D

Dish diameter [m2]

11.

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