Derivation Mathematical Formulas For Tilt Angles of a Flat Plate Sun Trucking Collector with a Simulation In C++ Language

In this paper a mathematical f ormula was drive n to estimate the tilt angle of surface from horizontal Σ and the surface azimuth angle Ψ for a flat plate solar collector which gives maximum total i ncident solar radiati on on a flat plate solar collector designed for heating water. According to the formulas of estimating Σ and Ψ which driven in this study a C++ program had been built and used to simulate the performance of a sun trucking s olar collector, so the collector orientation varies with day time to give maximum incident solar radiation at each hour (in this case one hour time interval u sed) by estimating the surface azimuth angle and th e tilt angle of surface from horizontal at the desired hour (time)


Introduction
The use of renewable energy sources has a positive influence on the environmental pollution and global warming processes.The dependence on expensive traditional energy sources and the necessity to economize them promoted the creation and development of new technologies of ecological energy systems.One of them is solar energy.Orientation of solar collector in space is the main factor influencing the quantity of absorbed solar radiation energy .One of the important parameters that affects the performance of a solar collector is its tilt angle with the horizontal.This is due to the fact that the variation of tilt angle changes the amount of solar radiation reaching the collector surface [1].A review of literature sources were listed here, I.Luminosu [2] found that the thermo leading fluid temperature variation depends on the magnitudes α and β for a given value of the radiant power density in the collecting plane of the solar energy.The quantities α and β depend on: the collecting area A c , the efficiency η ((α= A c η)), the flow and the specific heat ( β = mc p ).The coordinates ΔT, α and β and the magnitudes A c , η, m, C p can be established in the parametric graph of ΔT = f (α, β).Omar Aliman and Ismail Muzamir [3] develop a new technique, by introducing a new rotational axis to the sun tracking frame, the slave mirrors of the same column or the same row can be arranged to share the same driving device.With this design, the focusing area can be as small as that of an element mirror or even smaller if the element mirrors are concave.in this technique, the alignment procedure can be simplified, thus, the engineering time can be greatly shortened.Fabio Struckmann [4] suggested a way to describe the thermal performance of a Flat Plate Solar collector.He found that the most important measure is the collector efficiency and that the overall heat loss coefficient (U) and other factors as the heat removal factor (F R ) are not constant.Can Ertkin at all [5] revealed that the optimum tilt angles exhibit a strong seasonal trend with respect to the amount of maximum daily insolation incident on the collector surface.Monthly average optimum tilt angles were reasonably well estimated as a sinusoidal function of latitude and the day of the year over Turkey.The optimum tilt angle was low in the summer and high in the autumn and winter.Jurgita Grigonienė, and Mindaugas Karnauskas [1] stated the mathematical modeling of the optimal angles of a solar collector and a solar collector with a sunray concentrator.The quantity of solar energy depends on geographical position, the trajectory of the sun, on the intensity of solar radiation energy, of sunlight duration per day and per year, the reflection coefficient of SRR, etc.Therefore, in different seasons, we will have different optimal angles of a solar collector.In Lithuania, the optimal angle of tilt of a solar collector is determined experimentally is 15 o to 60 o for the whole year: 1ｰ

Solar collector tilt angles formulas derivation
The irradiance on the surface aperture of the direct beam component I D is the product of the direct normal irradiation I DN and the cosine of the angle of incidence θ between the incoming solar rays and a line normal (perpendicular) to the surface The angle of incidence θ for any surface is defined as the angle between the incoming solar rays and a line normal to that surface.For the horizontal surface shown in Figure 1, the incident angle θ H is QOV; for the vertical surface, the incident angle θ V is QOP.For any surface, the incident angle θ is related to β, γ, and the tilt angle of the surface Σ as in equation ( 4) [6].

Mathematical model of solar collector
The net enthalpy gain The net enthalpy gain to the insolation I t (t) through the definition of collector efficiency by equation ( 9).
Where A c is the aperture area of the collector.
Where For well-designed flat plate collector,(τα) eff will have the values in range of 0.75 to0.95 .thevalue of ,(τα) eff depends on the optical design of collector .The value above 0.9 exist for simpler unglazed collectors such as those designed for swimming pool heating ,while values of 0.75 to 0.85 apply to glazed collectors used in space heating and domestic hot water service.The value of bF R depends on the thermal design of the collector ,and it determines the change of the collector efficiency with temperature .A swimming pool collector ,designed for operation near ambient temperature ,could have a value of bF R as large as 15 A measure of a flat plate collector performance is the collector efficiency (η) defined as the ratio of the useful energy gain (Q u ) to the incident solar energy over a particular time period.
The instantaneous thermal efficiency of the collector is [4].

Discussion
The most important result in this study was the derivation of the formulas ( 5) and ( 7) that give the values of the angles Σ and ψ which gives the maximum value of the incident solar radiation received by the collector (optimum case).A C++ simulation program had built to study the performance of the sun trucking flat plate solar collector in city of Baghdad in January at which the angles Σ and ψ varies hourly.Calculations focus on the comparison between two cases one stationary solar collector (normal case) in this case solar collector oriented to the south, the angle with the altitude Σ have the value of latitude angle ,in Baghdad this angle have the value of L=33.3 o (so Σ= 33.3 o ) .The other case was the sun trucking solar collector (optimum case due to optimal tilt angles which gives maximum solar radiation incident on collector), in this case the angles Ψ and Σ varies with time (with one hour time interval).Two dimensions of solar collector were used, one of 20m 2 of area with 2KW of maximum Load supplied and other of 30m 2 of area with 3KW of maximum load supplied Fig ( 2) shows total solar radiation in two cases of stationary (normal) and sun trucking (optimum) solar collector installation, the large difference between two values can note clearly, at 8 A.M the difference approximate to 300 W/m 2 , this value decrease to less than 100 W/m 2 at 12 P.M .The difference in two values of solar radiation result from the varying the values of the angles Σ and ψ.Morning the difference in values of the angle ψ in stationary and sun trucking cases was at maximum, so the difference between the incident solar radiations in two cases was at maximum also.At midday the difference in values of the angle ψ in two cases close to zero because of orientation of the stationary case which was to the south ( ψ = 0) which is the same at the sun trucking case at midnight and the difference in solar radiations in two cases come just from the difference in values of the angle Σ in two cases.Fig (3) shows the difference between storage temperature in two cases .Again sun trucking system case record the higher temperatures (from simulation program which build in this study) with collector area of 20m 2 and 2KW Load supplied and the reason was the solar radiation in sun trucking case was higher than form the other case for all simulation times.For example the difference in temperatures in two cases was 22 C o at the hour 66 of the simulation time.In fig ( 4) the stability of system with load of 2KW in sun trucking case was clear with respect to normal case that because of the high capacity of sun trucking collector with respect to normal case.Finally Fig ( 6) illustrates the value of instantaneous collector efficiency in cases of stationary and sun trucking solar collector.The efficiency in sun trucking case was higher than the efficiency in normal case for intervals (8-9) A.M and (2-4) P.M , this because of net enthalpy gain Q u (t) which greater than its in normal case because the high storage temperature .In interval 9 A.M to 2 P.M the efficiency in sun trucking case was lower than the efficiency in normal case because of the value of I t in sun trucking case would greater than its in normal case where η = Q u (t)/( I t A c ) and the relation between η and I t is inverse relation.On other hand the load was limited so the value of Q u (t) still at constant value.

Conclusions
The use of sun trucking flat plate solar collector technique for domestic water heating in the city of Baghdad By varying each angles Σ and ψ hourly according to formulas (5) and ( 7) which derived in this study enable the flat plate solar collector to receive more energy from sun and so we have higher water temperature in storage tank so its more useful than other types of flat plate collectors .This mean more depending on green energy , saving for electrical energy and reducing the environmental pollution and global warming processes.

References
1. Jurgita Grigoniene, Mindaugas Karnauskas ,"Mathematical modeling of optimal tilt angles of solar collector and sunray reflector", Energetica,T.55.Nr.1.p. 41-46,2009 use the two equations (5) and (7), (which derived in this research) to estimate the maximum total solar radiation I t with respect to PDF created with pdfFactory Pro trial version www.pdffactory.com γ 1 is a measure of the ratio of the ability of the working fluid to remove energy from collector to the reference solar energy.The value of γ 1 can range from zero to very large values depending on the relative values of the mass flow rate through the collector ,the collector area , and the reference insolation.The non flow or stagnation temperature of the plate is given divided by the convective loss per degree above ambient.

Fig ( 5 )
illustrates the same result with solar collector of 30m 2 and 3KW of PDF created with pdfFactory Pro trial version www.pdffactory.comIn this figure we can see the difference in temperatures of two cases for example the difference equal to 20 C o at hour 44 of simulation time.
Figure (1) Solar Angles For Vertical And Horizontal Surfaces

com Eng. & Tech. Journal, Vol.28, No.18, 2010 Derivation Mathematical Formulas For Tilt Angles of a Flat Plate Sun Trucking Collector with a Simulation In C++ Language 5685
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Tech. Journal, Vol.28, No.18, 2010 Derivation Mathematical Formulas For Tilt Angles of a Flat Plate Sun Trucking Collector with a Simulation In C++ Language 5688
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Tech. Journal, Vol.28, No.18, 2010 Derivation Mathematical Formulas For Tilt Angles of a Flat Plate Sun Trucking Collector with a Simulation In C++ Language 5690 Table (1) Designed values used in simulation program. Table (2) Ambient temperature for the city of Baghdad in January [8].
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c Collector heat transfer coefficient, W/m 2 C o 0.2 U st Value of storage tank 6.0 A st , Storage tank surface area , m 2 300 Initial storage temperature ,K o 300 Minimum usable temperature ,K o 380 Maximum safe temperature ,K o 0.1 Load mass flow rate kg/min
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