Keywords

1 Introduction

The design, optimization and fabrication of solar desalination system have been presented. Mathematical model is developed using two main tools, dimensional analysis and regression analysis. Dimensional analysis provides a mathematical tool for analysing and making model laws [1]. The process of modeling carried out in this work is divided into two main stages – dimensional analysis and regression analysis [2].

2 Selection of Variables

Rate of evaporation, (m) of the given quantity of water in specified time (t) is the primary unknown quantity. So it is chosen as the first variable [3]. Amount of solar radiation (I) and the quantity of water (Q) are the important physical quantities which are crucial in deciding the evaporation rate [4]. They are the variables chosen next. Other factors which affect the process are Emissivity (ε) of the absorber material, Reflectivity (ρ) of the concentrator, Specific heat capacity (CP), Density (ρ1), Absorber area (Aabs) [5].

2.1 Representation in Terms of Fundamental Physical Quantities with the Units

It is necessary in dimensionless analysis to represent dimensions of all the factors in terms of fundamental quantities, M, L, T and θ1 in a table known as dimensional matrix (Table 1).

Table 1. Physical quantities in terms of fundamental dimensions
Full size table

2.2 Solution of Equations in Terms of Independent Variables and Final Development of Dimensionless Products

Now, homogeneous linear algebraic Eqs. 1 to 4 are written in terms of variables a to g whose coefficients are the numbers in the cells of the dimensional matrix corresponding to those variables.

$$ {\text{a }} + {\text{ b }} + {\text{ d }} + {\text{ f }} = \, 0 $$
(1)
$$ 2{\text{c }} + {\text{ e }} - {\text{ f }} = \, 0 $$
(2)
$$ - {\text{a }} - 3{\text{b }} - \, 2{\text{f }} = \, 0 $$
(3)
$$ - {\text{f }} + {\text{ g }} = \, 0 $$
(4)

Equations are solved in terms of independent variables and the solution is written in the form of solution matrix. Values of the dependent variables d, e, f and g are expressed in terms of 3 independent variables a, b and c using Eqs. 1 to 4 as shown below.

$$ {\text{d }} = \, - {\text{a}}/2 \, + {\text{ b}}/2 $$
(5)
$$ {\text{e }} = \, - 2{\text{c }} + {\text{ a}}/2 \, + \, 3{\text{b}}/2 $$
(6)
$$ {\text{f }} = \, - {\text{a}}/2 \, - 3{\text{b}}/2 $$
(7)
$$ {\text{g}} = {\text{f}} $$
(8)

By considering the determinant at the left hand side of the matrix of solutions, it is perceived that the rank of the matrix is invariably equal to the number of rows. Matrix of solution as given has a rank 3. So the rows are linearly independent

$$ \Pi 1 \, = {\text{ k }}\left( {\Pi 2 \, } \right){\text{b}}1 \, \times \, \left( { \,\Pi 3 \, } \right){\text{c}}1 $$
(9)

3 Forming and Solving Regression Equations

The dimensionless variables constructed by a dimensional analysis are now used as the variables of the regression to form a correct dimensional relationship, one that is homogeneous. Any coefficient in a fitted model is also dimensionless and will not change if the units of measurement are changed.

$$ { \log }\, \,\Pi _{1} = { \log }\,{\text{ k}}_{2} + {\text{ a}}_{1} { \log }\left( {\Pi _{2} } \right) \, + {\text{ b}}_{1} { \log }\left( {\Pi _{3} } \right) $$
(10)
$$ {\text{Let }}\;{\text{Y }} = {\text{ a }} + {\text{ bX}}_{1} + {\text{ cX}}_{2} $$
(11)

Three equations for three can be generated as under,

$$ {\text{Y }} = {\text{ N}}\left( {\text{a}} \right) \, + {\text{ b }}\sum {{\text{X}}_{1} } + {\text{ c }}\sum {{\text{X}}_{2} } $$
(12)
$$ \sum {{\text{X}}_{1} {\text{Y}}} \, = \, \sum {{\text{X}}_{1} \left( {\text{a}} \right)} \, + {\text{ b }}\sum {{\text{ X}}_{1}^{2} } + {\text{ c }}\sum {{\text{X}}_{1} {\text{X}}_{2} } $$
(13)
$$ \sum {{\text{X}}_{2} {\text{Y}}} \, = \, \sum {} {\text{X}}_{2} \left( {\text{a}} \right) \, + {\text{ b }}\sum {{\text{X}}_{1} {\text{X}}_{2} } + {\text{ c}}\sum {{\text{X}}_{2}^{2} } $$
(14)

3.1 Experimental Data

Experiments were conducted in different months and with different quantities of water in absorber. The first test was conducted with 2 kg mass of water using the concentrator and cookers of diameters 0.3 m, 0.25 m and 0.27 m made of stainless steel and aluminium respectively. The observations are given in Table 2.

Table 2. Readings for 2 kg
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Similar tests were conducted with 5 kg mass of water using the concentrator and cookers of diameters 0.3 m, 0.25 m and 0.27 m made of stainless steel and aluminium respectively. The constants c and k are determined using the set of observations such that overall error in t is minimized and −1 and 1/1236.3 are found to be suitable.

$$ t = \frac{{123.6X \in A_{abs} (Q^{0.395} - 1)}}{{\rho \left( {I^{0.21} d^{0.22} } \right)\left( {y\theta } \right)^{0.185} }} hours $$
(15)

All the required parameters can be calculated from the above equation

4 Validation

Time of evaporation was estimated for input data such as product of ρ1and cp equal to 4000000 J/m3K, dish diameter 2.3 m and absorber area 0.07 m2. Solution obtained is written below. The observed time for the above analysis was 6061 s. The accuracy of the generated equation is found to be 93%. Correlation coefficient was determined for experimental and calculated values using Karl Pearson’s formula (Mujumdar, 2006) (Fig. 1).

Fig. 1.
figure 1

Experimental and calculated time of evaporation

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The developed model shows that time of evaporation decreases with increase in the size of concentrator, intensity of solar radiations, surrounding temperature, emissivity of the absorber and reflectivity of the material. The time of evaporation decreases with decrease in the size of the absorber and the quantity of water to be evaporated.

5 Conclusions

In this study a mathematical model was developed to find quantity of water that was to be evaporated and time required for evaporation. The model was developed on the basis of almost 50 experiments conducted by varying different parameters. Equation involves parameters such as, size and material of the absorber, size of the concentrator, reflectivity of the concentrator and atmospheric conditions. The model was validated with experimental results value with 93% accuracy. The correlation coefficient was calculated and found to be 0.93.