DISCUS ON HEAT CONDUCTION ANALYSIS IN THE ABSORBER OF SOLAR FLAT PLATE COLLECTORHEAT CONDUCTION ANALYSIS IN THE ABSORBER OF SOLAR FLAT PLATE COLLECTOR INTRODUCTION In the past recent years, the use of fossil fuels has been increasing continuously for heating and various uses which may lead to global warming and pollution. The demand of fossil fuels has been increasing continuously. People of the world have to pay attention to use of renewable energy resources due to limitation and impact of non-renewable energy resources on the environment [1]. Due to global warming, greenhouse gas emission, fluctuating oil prices, rising electricity demand in developing countries have to consider new solutions [1]. “In order to develop a poverty free world, energy security for all sectors must be ensured” [2] “In 2014, the total global primary energy consumption was about 160310 million MWh, and this value had been projected to be increased to 240318 million MWh in 2040” [3].” In 2010, the renewable energy-based electricity generation system had been raised about 20% compared with total electricity generation and it will be about 31% in 2035” [4]. ”According to the sustainable future scenario of the international energy agency by renewable energy sources, 57% of world electricity supply will be provided by 2025” [5]. Therefore scientist are focusing towards the use of renewable sources of energy in order to reduce the crisis of fossil fuel, global warming and pollution.one of the prominent renewable source of energy is “solar energy”. Solar energy is a type of renewable energy resources which has to undergo extensive-scale development for full application due to energy transmission limitation [6]. Usually solar energy has many advantage than the conventional energy resources due to reduce in carbon emission , global warming and can generate energy within lifetime [7]. When considering solar energy technology its intermittent and fluctuating characteristics, utilization, and efficiency are limited [7]. For rural and hilly areas, solar power is established. Due to randomness and fluctuation of solar power we cannot generate continuous and stable power output [6]. In general, this solar power is not adaptable to every where due to distribution of natural resources but depend on the culture of individual community [8]. When considering the fossil fuel based electricity system such as thermal power plants, then they can generate with maximum power output than the power generate by the solar energy [7]. Generally setting up solar power generation system we have to pay high initial cost. Solar energy can be used for production of both electrical and thermal energy. Generally electricity production using solar energy is overcoming across the world [7].During recent years large amount of investment has been made for the advancement of technology in developing countries to produce solar energy more effectively [7]. SOLAR COLLECTOR A solar collectors is the special type of energy exchanger which converts solar irradiation energy either to the thermal energy of the working fluid in solar thermal applications or to the electrical energy directly in PV(photovoltaic) applications [9]. For solar thermal applications, solar irradiation is absorbed by a solar collector as heat which is then transferred to the working fluid(air, water or oil) [9]. The heat carried out by the working fluid can be used to either provide domestic hot water /heating or can be stored in the large tank so that it can be used at night or cloudy days [9]. Since, 1990 Solar collectors can be categorise:- 1) Flat plate solar collector. 2) concentrating collector. FLAT PLATE SOLAR COLLECTOR :- In this type of solar collector the absorbing surface is made approximately large, so that overall collector area can intercepts the sun’s rays. Flat plate solar collectors are usually permanently fixed in position and therefore need to be oriented appropriately [9].A typical solar flat plate collector consist of glazing covers, absorber plates, insulation layers, recuperating tubes(filled with heat transfer fluids) and other auxiliaries [9]. CONCENTRATING COLLECTOR :- In this large areas of mirrors or lens focus the sunlight on the absorber plate. Concentrating collectors have much higher concentration ratio than the non-concentrating collectors [9]. They can achieve higher temperature of working fluid than the non- concentrating collector. Due to change in position of sunlight during the day, the mirrors need to be focused on the absorber plate in order to achieve higher temperature of working fluid [9]. Flat plate collector consist of :- 1) Dark flat plate absorber. 2) Transparent cover which reduces heat loss. 3) Heat carrying fluid (air, antifreeze or water) to remove heat. 4) An insulating backing. Fig.1. Components of solar flat plate collector The absorber plate consists of thin absorber sheet (of thermally stable polymers, Aluminium, steel or copper, to which a matte black or selective coating is applied) [10]. In water heat panels, fluid is usually circulated through tubing to transfer heat from the absorber to an insulated water tank [10] [10]. In residential and commercial use flat plate collectors can produce heat less than 100 degree Celsius. TYPES OF FLAT PLATE COLLECTOR 1) UNGLAZED SOLAR FLAT PLATE COLLECTOR:- The term ”unglazed water collector” refers to a solar water heating system which consists of metal absorber without glass or glazing top. The most common type of unglazed collector on the market is the transpired solar collector. The technology has been extensively monitored by these government agencies, and Natural Resources Canada developed the feasibility tool RET Screen to model the energy savings from transpired solar collectors [11]. 2) FLAT PLATE SOLAR THERMAL SYSTEMS:- This is a common type of solar collector which have been in use since 1950s. Main components of a flat plate are dark colored flat plate absorber with an insulated cover , a heat transferring liquid(antifreeze) to transfer heat from the absorber plate to the water tank. It has an advantage that it generally increases the surface area for incoming sun radiation. In this heat is transferred through copper and silicon tubes contained within the flat surface plate [10]. 3) EVACUATED TUBE SOLAR THERMAL SYSTEMS:- It is one of the popular solar thermal system in operation. Evacuated solar thermal systems is the most common type of solar thermal energy generation with a rate of efficiency 70% [10].The rate of efficiency is achieved in the way in which the evacuated tube systems were constructed, it means that it have excellent insulation and are virtually unaffected by air temperature [10]. It contains rows of insulated glass tubes which contain copper pipes at their core in which water is heated and sent through the pipes to the water tank [10]. COMPONENTS USED OF FLAT PLATE SOLAR COLLECTOR :- 1) GLAZING COVER:- It is a type of transparent cover which is put on the top of the flat plate collector. The material of cover should be transparent to shorter wavelength of radiation and should be opaque to the larger wavelength of re-radiation from the heated absorber plate [12]. “ white glass with low-iron content” is suitable materials for Glazing cover. A portion of heat is lost from the heated absorber plate to the atmosphere either by reflection or convection. Generally convection losses can be decreased by using two or more glass cover. Similarily, reflection losses can also be reduced by coating with anti-reflective thin film of suitable materials [12]. 2) GLAZING FRAME:- It is generally used to hold the glazing material. 3) TUBING OR FLUID PIPE: It is used to facilitate the flow of working fluid. Generally water is used as a working fluid in which fluid is enter from one side of the tube and after heating leaves at the other side of the tube. Water is effective heat transport medium but it has certain disadvantages that it freeze during cold weather conditions generally “ethylene glycol is added to check freezing” Another problem is the corrosion of metal by the flow of water and leakage of water from the tubes. For better heat transfer the tubes are connected to both top and bottom. Water enters at the bottom of the header after heating it leaves from the top of the header. Generally some nano additives are added to the water in order to increase the capacity of water to absorb solar radiation [12]. 4) ABSORBER PLATE:- Absorber plate of solar collector is generally a metal plate of steel, Aluminium or copper. It is used to absorb solar radiation and then conducts this heat by conduction to the working fluid. The surface of absorber plate is generally coated with black paint with appropriate primer. The primer coat must be thin.” Alternatively selective coating is used to improve the effectiveness of the absorber plate”. Selective coatings have high absorptivity to the shorter wavelength radiation(less than 2 micrometer) and low emissivity to the longer wavelength radiation [12]. 5) INSULATOR:- Heat loss takes place when there is temperature difference between the absorber plate and the surrounding atmosphere. In order to minimize the heat loss from the casing we generally use insulator. The rear side of the absorber plate must be insulated in such a way that minimum heat loss takes place to the surroundings [12]. “Most commonly used materials are Mineral wool, Rockwool, Styrofoam etc”. 6) CASING:- “Generally it is a non- functional components which supports all other components of solar flat plate collector” [12]. It is a water proof box surrounds the components of the solar flat plate collector from dust and fumes. WORKING Most of the solar energy comes from sun in the form of light in short wave radiation [12].The sun radiates like a black body with surface temperature of about 57000℃ [12].When the solar radiation incident on the absorber plate of a solar flat plate collector then some of the radiation will be absorb, some will transmit and the rest of the radiation will reflect back. The absorb radiation will heat the absorber plate and by conduction it transfers the heat to the working fluid which flows through the tubes. Then the fluid can be used for domestic and commercial purposes. APPLICATION • It can be used for heating water for domestic and commercial purposes. • It can be used where the availability of electricity is less. It can be used where required temperature is less and amount of water required is more. LITERATURE REVIEW B. KUNDU [13],”performed the thermal analysis of variable thickness of the absorber plate of the flat plate absorber plate. Generally triangular profile has been taken. Taylor series of expansion were implemented for finding out the temperature distribution. Analytical method was established by DTM to find out the temperature and its relevant parameters for triangular profile of flat plate solar collector” [13]. S. a. A.-Q. Jebaraj [14], ”This work shows the performance analysis, transient analysis, energy and exergy analysis of solar flat plate collector” [14]. S. a. M. Dhariwal [15], “This work represents mathematical solution of one node transient equation of the flat plate solar collector. Analytical expression were obtained from the above mathematical solution. Test procedures for flat plate solar collector involve transient analysis” [15]. E. N. ,. J. a. S. G. Amer [16], “In this work testing of solar flat plate collector under unsteady weather conditions were developed. It has been derived from general energy balance and the solution gives the fluid temperature in terms of collector parameters and its function. it involves the direct comparison with the steady state tests. This method were validated in different weather conditions from steady to highly variable. The results were obtained close to steady state ASHRAE 93-86 standard(within 3%) and the scatter (within 2%).It was found that theoretical predictions were close to the transient conditions” [16]. S. Kalogirou [17], “This work involves the prediction of the performance of the solar flat plate collector using Artificial Neural Networks(ANN). generally six ANN were setup for the prediction of performance of solar collector both wind and non-wind conditions, incidence angle modifier coefficients both at transverse and longitudinal directions, collector time constant, collector stagnation temperature ,collector heat capacity. The data used for testing and validation were from LTS database. when unknown data were given to the network and its results were satisfactory and indicate that this proposed method can be successfully used for prediction of solar flat plate collector. This method when compared with the other testing methods are more fast and simple” [18]. K. a. C. R. Pandey [19], “This work represents the different techniques used to increase the efficiency of the solar flat plate collector such as use of nanofluids to increase the heat transfer rate, different design of absorber plate for more capture of radiation, different methods for heat loss reduction, use of polymers, mini and micro channels for heat transfer fluid, employing PCM, use of enhancement devices” [20] N. N. a. S. D. Dobriyal.R [21]. “This work shows an overview of the various techniques to allow flat plate collector to absorb as much radiation and minimize the losses to surrounding. use of nanofluids has increased the performance of the solar flat plate collector [21]” H. Kazeminejad [22], “In this temperature distribution of absorber plate was analyzed in 1-D and 2-D with heat generation. governing differential equation with boundary conditions were solved numerically using control-based finite difference [22]” R. Gorls [23], “In this analysis two dimensional finite element method was carried out for the performance of solar collectors. In this two dimensional model was established for the predictions of instantaneous collector performance without considering the average plat temperature [23]”. P. Raghuraman [24], “In this separate 1-D analysis were carried out for prediction of thermal and electrical performance both for air and liquid medium [24]”. From the above literature review we concluded that thermal analysis of solar flat plate collector in 1-D can be done with constant thickness of the absorber plate with constant thermal conductivity and constant heat loss coefficient. STATEMENT OF THE PROJECT • Thermal analysis of the absorber plate of a flat plate solar collector with constant thickness and constant thermal conductivity need to be solved by analytical method using two boundary conditions. • Separation of variable is to be used to solve the governing differential equation. • Result of the analytical equation is to be plotted in MATLAB and the temperature distribution for the two boundary condition is to be obtained. ANALYSIS OF SOLAR FLAT PLATE COLLECTOR IN 1-D Governing Equation for Absorber Plate Sun Radiation ＝ ? ?? Losses in the flat Plate ＝ Ul(? − ??) Where, Ul ＝ overall loss coefficient ? ?? ? Ul ?? ?? (? ??) + ?? − ?? (? − ??) ＝ ?? ?? Making these equations dimensionless ? ＝ ?−?? , ? ＝ ? , ?? ＝ ? , ?? ＝ 1 ??−?? ? ? ?? ? ?? 1 ?? ＝ ?? (?? − ??) ?? ?? ?? ?? ＝ (?? − ??) ?? ?? ?? ?? ?? ＝ (?? − ??) ?? ?? ?? ?? ?? ?? ＝ (?? − ??) ?? ?? ?? (?? − ??) ?? ＝ ?? ? ?? ?2? ? (?? − ??) ?? 2 ＝ ?? [ ? ??] ?? ?2? (?? − ??) ? ?? ?? 2 ＝ ? ?? [?? ??] ?? ?2? (?? − ??) ?? ?2? 2 ＝ ? [?? ??2] ?? ?2? (?? − ??) 1 ?2? 2 ＝ ? [? ??2] ?? ?2? (?? − ??) ?2? 2 ＝ ?2 ??2 ?? ??? ? ?(? − ??) ? ?? ? + ??? − ??? ＝ ? ?? ?? Putting in equation 1 (?? − ??) ?2? ? ?(? − ??) 1 ?? + − ＝ ?2 ??2 ??? ??? ? ?? ?2? ?2? ??2(? − ??) ?2 1 ?? + − ＝ ??2 (?? − ??)??? ???(?? − ??) ? (?? − ??) ?? Let ? ＝ ( (???−−????)) , ?′ ＝ ( ? −????)?? , ??? ＝ ? ????? ? ? ? + ? ?2?2 ′ − ??2? ＝ ?1 (?? ?−2??) ???? ?? ? ＝ ?? 1 ?? ＝ ?? (?? − ??) ?? ?? ?? ?? ＝ (?? − ??) ?? + ? ?2?2 ′ − ??2? ＝ ?1 (?? ?−2??) (?? − ??) ???? ?? ?2?2 ′ − ??2? ＝ ??2 ???? + ? ?? ?? ?? ＝ ?2 ??? ? ＝ 2 ?? ? ?2?2 ′ − ??2? ＝ ??2 ????? ????? + ? ?? ? ??2?2 + ?′ − ??2? ＝ ? ?2 ????? ??2 ?2?2 ′ − ??2? ＝ ????? + ? ?? (Governing Differential Equation for Flat Plate Absorber) Initial Condition ?(?, 0) ＝ 0 ?? ＝ 0 , ? ＝ 0 , ? ＝ 0 As, ? ＝ 0 ＝ 0 ? ＝ ?? Boundary Condition At ? ＝ 0 , ?? ＝ 0 ＝ ?? ?? ?? And at ? ＝ 1 , ? ＝ 1 , ?? ＝ −??(? − 1) ?? ?2? + ?′ − ??2? ＝ ?? ??2 ??? Applying Principle of Superposition ?(?, ??) ＝ ?(?) + ?(?, ??) ?2? ?2? ?2? 2 ＝ ??2 + ??2 ?? ?? ?? ＝ ??? ??? ?2? + ?′ − ??2? ＝ ?? ??2 ??? ?2? ′ − ??2(? + ?) ＝ ?(? + ?) + ? ??2 ??? ?2(? +2 ?) ′ − ??2(? + ?) ＝ ????? + ? ?? ???? + ?????? + ?′ − ???(? + ?) ＝ ? ???? ?? ?2?2 ′ − ??2? ＝ 0 + ? ?? ?2?2 − ??2 (? − ? ??′2) ＝ 0 ?? ?′ ? ?(?) ＝ ?? ????(???) + ?? ???(???) + ( ) ?? ? 2?2 − ??2? ＝ ? ???? ?? We can apply method of separation of variable ?(?, ??) ＝ ?(?) ?(??) Putting the boundary condition • At ?(?, 0) ＝ 0 ?? ＝ 0 ?(?) + ?(?, ??) ＝ 0 ?(?) ＝ −?(?, ??) • At ? ＝ 0 , ?? ＝ 0 ?? ? (? + ?) ＝ 0 ?? ?? ?? + ＝ 0 ?? ?? ?? ?? ＝ 0 & ＝ 0 ?? ?? • At ? ＝ 1 , ? ＝ 1 ?(?) + ?(?, ??) ＝ 1 ?????? ? ＝ 1, ? ＝ 0 ?′ ? ?(?) ＝ ?? ????(???) + ?? ????(???) + ( ) ?? Boundary condition are At ? ＝ 0 , ?? ＝ 0 ?? At ? ＝ 1 , ? ＝ 1 Putting first boundary condition at ? ＝ 0 , ?? ＝ 0 ?? ??(?) ?? ＝ ?1 sinh(???)?? + ?2 cosh(???)?? 0 ＝ ?1(0) + ?2? ?2 ＝ 0 ?′ 2 Then equation becomes ?(?) ＝ ?1 ???ℎ(???) + ?2 sinh(???) + (??) ?2 ＝ 0 ?′ 2 ?(?) ＝ ?1 ???ℎ(???) + ( ) ?? Again putting another boundary condition at ? ＝ 1 , ? ＝ 1 ?′ 2 1 ＝ ?1 ???ℎ(??) + ( ) ?? ?′ ? ? − (??) ＝ ?? ????(??) ? ? ???′ ? ????(???) + (?′ )? ?????? ( Now for the equation ? ??2?2 − ??2? ＝ ? ???? Putting the boundary condition At ?? ＝ 0 , ? ＝ 0 → ? ＝ −? At ? ＝ 0 , ?? ＝ 0 ?? At ? ＝ 1 , ? ＝ 0 Applying separation of variable ?(?, ??) ＝ ?(?) + ?(??) ?? ?? ?2? ?2? ＝ ? → 2 ＝ ? ??2 ?? ?? ?? ?? ?? ＝ ? ??? ??? Putting in the above equation ? ? ??2?2 − ??2(??) ＝ ? ? ???? Dividing whole euation by (??) ??2? ??2(??) ??? 2 − (??) ＝ (??)??? (??)?? 1 ?2?2 − ??2 ＝ ? 1 ????? ? ?? 1 ?2?2 ＝ ?1 ????? + ??2 ? ?? ? ??? ? ?? ? ? ? ??? ＝ ? ??? + ?? ＝ ±?? 1 ?? 2 2 + ?? ＝ −?? ? ??? 1 ?? ＝ (−?2 − ??2) ? ? ??? ??2)??? + ln ? ln ? ＝ (−?2? − ??2)?? + ln ? ? ＝ ??−(??? + ???)?? 1 ?2? 2 ? ??2 ＝ +?? ???2?2 2 ＝ ??? ? ??2?2 − ??2? ＝ 0 ?(?) ＝ ?? ???(???) + ?? ???(???) Putting the boundary condition At ? ＝ 0 , ?? ＝ 0 ?? ?? ＝ 0 ?? ?? ?? ＝ −?3 ???(???)?? + ?4 ???(???)?? 0 ＝ ?4?? ?4 ＝ 0 Then ?(?) ＝ ?3 ???(???) Again putting another boundary condition At ? ＝ 1 , ? ＝ 0 ? ＝ 0 ?(?) ＝ ?3 ???(???) 0 ＝ ?3 ???(??) As ?3 ≠ 0 ???(??) ＝ 0 ? ???(??) ＝ cos ( ) 2 ? ?? ＝ (?? − ?) ? (General form for n ＝ 1,2,3,… ) ?(?, ??) ＝ ?(?) + ?(??) ?(?, ??) ＝ (?3 ???(???)) + (?4?−(??2 + ??2)??) ?(?, ??) ＝ ?3?4 ???(???) ?−(?2? + ??2)?? Let ?3?4 ＝ ?? ?(?, ??) ＝ ?? ???(???) ?−(?2? + ??2)?? For finding the value of ?? ?(?, ??) ＝ ?? ???(???) ?−(??2 + ??2)?? at ?? ＝ 0 , ? ＝ −? ?? ???(???) ?−(?2? + ??2)?? ＝ −? ?? ???(???) ＝ −? Multiplying both the sides by ???(???) ?? ???2(???) ＝ −? ???(???) Integrating both the sides ?? ∫ ???2(???) ＝ − ∫ ? ???(???) ?? ?? ???(???) ??] (1 − ?∗??2)?? ?∗ ?? ＝ −2 sin(??) [ 2 + ??2) + ??2??] (?? ?∗ ?∗ ????(???) ?(?, ??) ＝ [ ? + (? − ? ??) ????(??) ] ?? ∞ ∗? ? − ? ?∑＝? ???(???) ???(??) ???[−(??? + ???)??] [( ? −??+ ????))?? + ????∗??] (?? Taking the 2nd boundary condition ????? ?? + ?′ ＝ ???? − ?? ? Boundary conditions are At X＝0, ?? ＝ 0 ?? At X＝1, ?? ＝ −??̇(? − 1) ?? Converting it into homogenous By applying principle of superposition ?(?, ?) ＝ ?(?) + ?(?, ?) ?2? ?2? ?2? Putting in equation 1 ?2?2 ???2? 2(? + ?) + ?′ ＝ ?? ?? + 2 − ?0 ?? ?2?2 + ???2?2 2? − ?02? + ?′ ＝ ???? − ?0 ?? ???? − ???? + ?′ ＝ ?…..equation(2) ?? ???? ?? ＝ ??…equation(3) ?? − ?? ?? Boundary conditions are At X＝0, ?? ＝ 0 ⇒ ?(?+?) ⇒ ?? + ?? ＝ 0 ?? ?? ?? ?? At X＝1, ?? ?? ＝ −??(? − 1) ＝ ???? + ???? ＝ −??(? + ? − 1) ?? +?? ? ? ?? ＝ −??(? − 1) − ??? ? ??? ＝ −??(? − 1) & ???? ＝ −??? ?2?2 2? + ?′ ＝ 0 − ?0 ?? ???2?2 2 (? − ?2′) ＝ 0… − ?0 ?0 Solution of this equation is ?′ ?(?) ＝ ?? ????(???) + ?? ????(???) + ? ?? Applying boundary conditions at X＝0, ?? ＝ 0 ?? ?′ ?(?) ＝ ?1 ???ℎ(?0?) + ?2 ???ℎ(?0?) + ? 02 ?? ? ? ＝ (?1 ??? ℎ(?0?) + ?2 ???ℎ(?0?))?0 0 ＝ (?1 ???ℎ(?0 × 0) + ?2 ???ℎ(?0 × 0))?0 ?2 ＝ 0 Then the equation becomes ?′ ?(?) ＝ ?1 ???ℎ(?0?) + ? 02 Putting the 2nd boundary condition At X＝1, ??? ? ＝ −??(? − 1) ?? ? ? ＝ ?1 ???ℎ(?0?)?0 −??(? − 1) ＝ ?1 ???ℎ(?0) ⋅ ?0 ?′ −?? (?1 ???ℎ(?0) + ? 02 − 1) ＝ ?1 ???ℎ(?0). ?0 ?′ ?1 ???ℎ(?0)?0 + ??(?1 cosh ?0) ＝ −?? (? 02 − 1) ?′ ?1 ＝ ? 0 ???ℎ?(?? 0−)?+? ?(?? 0???2) ℎ(?0) ?′ ?1 ＝ ? ???ℎ?(??0()1+−???02???) ℎ(?0) 0 Therefore A(X)＝? ???ℎ(?0?) + ? ?02′ ?′ A(X)＝? 0?????(1ℎ−(??020))+????? ℎ???(?0ℎ?(?)0) + ??02′ Now we will solve ? 2?2 − ?02? ＝ ?? ?? ?? Boundary conditions are At x＝0,?? ＝ 0 ?? At x＝1,???? ＝ −??? And initial conditions are At F＝0,? ＝ 0, ? + ? ＝ 0, ? ＝ −? Applying separation of variable ?(?, ?) ＝ ?(?) ⋅ ?(?) ?? ?? ＝ ? ?? ?? ?2? ?2? 2 ＝ ? ??2 ?? ?? ?? ＝ ? ?? ?? Putting in the above equation ?2?2 2? ＝ ???? − ?0 ?? ?2? ?? ? ??2 − ?02(? ⋅ ?) ＝ ? ? ? Divide whole equation by (??) 1 ?2?2 − ?02 ＝ ? 1 ???? ? ?? 1 ?2?2 ＝ +?2? ??? ? 1 ???? + ?02 ＝ −?2? ? ?? Taking the value of plus eigen value ?2?2 2? ＝ ?? ?? ?2?2 2? ＝ 0 − ?? ?? Then the solution becomes ?(?) ＝ ?? ???(???) + ?? ???(???) Similarily for the equation of ? Taking the second part 1 ?? 2 − ?02 ＝ −?? ? ?? ∫ ? ? ＝ −(?2? + ?02) ?? ? ?? ? ＝ −(?2? + ?02)?0 + ln ? ?? ? − ln ? ＝ −(?2? + ?02)?0 ?? (?) ＝ −(?2? + ?02)? ? ? ＝ ?6?−(?2?+?02)? From the equation of ?(?) ?(?) ＝ ?3 ???(???) + ?4 ???(???) Putting the boundary conditions At X＝0, ?? ＝ 0, ?(?⋅?) ＝ 0, ?? ＝ 0 ?? ?? ?? ?? ?? ＝ (−?3 ???(??) + ?4 ???(??))? ＝ 0 (−?3 ???(0) + ?4 ???(0))? ＝ 0 ?4 ＝ 0 ?(?) ＝ ?3 ???(??) + 0 ?(?) ＝ ?? ???(??) 2nd boundary conditions At x＝1,???? ＝ −???, ? (????) ＝ −??(? ⋅ ?), ? ??? ＝ −???, ? ??? ＝ ? At X＝1, ? ??? ＝ −??? ?? ?? ＝ −?3 ???(??)? −??(?3 ???(?)) ＝ −?3 ???(??)? ?3(−? ??? ? + ?? ??? ?) ＝ 0 Since ?3 ≠ 0 ??????? ???????? ?????? ?? ????? ?? ???? Therefore, −? ??? ? + ?? ??? ? ＝ 0 ? ??? ? ＝ ?? ??? ? ? ??? ? ＝ ?? −1 (??) ? ＝ ??? ? Applying separation of variable ?(?, ?) ＝ ?(?)?(?) ?(?, ?) ＝ ?3?6 ???(? ⋅ ?)?−(?2?+?02)? ? ? ?＝1 ?? ??? ?? ????? ?? ??????? ????? Applying last boundary conditions at F＝0, ? ＝ 0 ?? ? ＝ −? ∞ ∑ ?? ???(??)?−(??2+?02)?＝-A ?＝1 ∞ ∑ ?? ???(??) ＝ −? ?＝1 Multiply both sides by ???(??) ?? ??? (???) ? ???(???) ?? ?′ ? ?? ?? ?′ Let M＝ ? 0 ??? ℎ?(??0()1+−???20???) ℎ(?0) ??? ? ＝ ? ?02′ (? ??? ?) (???) ?? ?? ?? (? ??? ?) ?? ?? ??? ?? ( ) ? ???(???) ?? ?? ) ? ?2? sin ?? cosh(? ?? ＝ −? (?2? + ?02) 2???+? sin 20)??+ ??? cos ?? sinh(?0) −? (sin1????) 4?? Putting the Value of M ?? ??????? ?? ＝ 2 ??+sin2?? (?0)+?? cosℎ?0 ?? sin ?? cosh??2+??002+?0 cos?? sinh ?0) + ? ?02′ sin ??. ?0 sinh ?(?, ?) ＝ ? + ? ?′ ?(?, ?) ?? ?) ?′ ? ? ?? ??? ? +(???) ? ?′ ??(?− ?) ????(???) ?′ ?(?, ?) ＝ ?? ????(????)+?? ????(??) + ??? + ? ? ? ? ? { ?? ????? ?? ?? ????? ?? ????′??? ??? ?? ??? ?? ???? ???? ??? ?? ???? ??) + ???′ ??? ?? ? ???? ??? RESULT • The analytical methodology presented above is solve by separation of variable with two boundary condition are used. • The above equation is coded in MATLAB and the results are obtained with considering the solar energy incident on the plate at constant rate S＝1, and absorber thermo-geometric Parameter Z＝0.5. • The results obtained are:- ➢ Temperature distribution of absorber plate with respect to the length of the absorber plate. ➢ Graph between ? ?? ? was obtained for the two equation. . Table 1: value of Temperature distribution (?) and length of the absorber plate(X) of equation considering first boundary condition Temperature distribution(?) Length of collector(X) 0.003 0.0 0.004 0.1 0.0042 0.2 0.0046 0.3 0.0047 0.4 0.0049 0.5 0.005 0.6 0.0052 0.7 0.010 0.8 0.768 0.9 2.222 1.0 Table 2: value of Temperature distribution(?) and length of absorber plate(X) considering 2nd boundary condition with Bi＝0.5 Temperature distribution(?) Length of collector(X) 0.00121 0.0 0.0012 0.1 0.00123 0.2 0.00132 0.3 0.00133 0.4 0.00136 0.5 0.0014 0.6 0.187 0.7 0.302 0.8 0.430 0.9 0.530 1.0 GRAPHS Fig. theta (?) vs length of the absorber plate(x) for first boundary condition • From the fig it can be concluded that the value of theta(?) is zero upto the value of x＝0.8. After x＝0.8 there is gradual increase in the temperature with respect to the length of the absorber plate. Fig. (?) vs length of the absorber plate (X) for the 2nd boundary condition with Bi＝0.5 • From the fig it can be concluded that the value of theta(?) is zero upto x＝0.6 it means that the temperature of absorber plate is equal to the temperature of surrounding. After x＝0.6 ,there is gradual increase in the value of theta(?) with respect to x it means that there is increased in solar flux on the absorber plate. • The difference between these two graph is that the temperature variation due to conduction in the absorber plate is less compared to the convective heat transfer at the boundary of the tubes carrying fluid. CONCLUSION • Thermal analysis of the absorber plate in a solar flat plate collector is carried out analytically with two separate boundary conditions at the surface where the heat transfer takes place to the tubes carrying fluid. In this analytical method separation of variable was used in both the cases. Finally, it can be seen that the temperature variation due to conduction in the absorber plate is less compared to the convective heat transfer at the boundary of the tubes carrying fluid.
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