An Overview on Application of Exergy and Energy for Determination of Solar Drying Efficiency

The main objective of this work is to give an overview of the different used mathematical methods for modeling and calculating the energy and exergy efficiency of a solar dry ing system. For determination of the energy efficiency of thin or thick layers, heat and mass balances are established to the different components. In the case of thick layers equations of porous media are used. These methods allow fo llowing, in particular, the variat ions of the heated air and the product temperature and humidity. As a second part, the common way for the calculus of the exergetic efficiency passes through the establishment of exergy balance and after calculating the input, output and exergy loss. The influence and illustration of several parameters are as well presented in this paper.


Introduction
Dry ing constitutes an import ant pro cess for a large variety of industries. For example, fo r foodstuffs, it is a necessary step for the preservation of the final product or to make it ready to be stored [1][2][3][4]. On the other hand and for other materials, such as woods, concretes, bricks and sludge, the passage by this process is an obligation to make these products marketable [5][6][7][8]. As it is well known, dry ing is the most energy intensive operation of the industrial processes. In most industrialized countries, between 7 and 15% of a nation's industrial energy is used for d ry ing [9]. Other estimations repo rt that nat ional energy consumption for industrial drying operations are ranging from 10 to 15% for United States, Canada, France and United Kingdom and reach 20 t o 2 5% fo r Den ma rk an d Ger man y [1 0]. Furthermo re, around 35 % o f the consumed energy in drying is used for paper and pulp industry and only 5% are used for chemicals [10]. The resulted high cost of this unit operation directed the efforts of the Scientifics to look for alternative cheap sources of energy, in particular free solar energy. Coupling the two ideas gives birth to several types o f so lar d ry ers. In general mann er, so lar d ryers are class ified in to act iv e and p ass iv e d ryers and each classificat ion is d iv ided into direct, ind irect and mixed mode typ e [11][12][13]. Ho wever and in o rder to face the intermittent character of solar energy, developed systems can be do tted with a seco nd source o f en ergy us ing electrical heater [1,14] or storage systemsfilled with packed rock-beds [15][16][17], gravels [18], desiccants [19], water [20] or phase change material [21][22][23]. We cannot have an optimu m use and design of a solar dryer without passing through simu lation and the mathematical modelling. The existing models are taking essentially two ways: the first approach studies particularly the behaviour of the product during the process represented by the drying kinetic. These developed models agree to calcu late several parameters such as coefficient of d iffusion, heat and mass coefficients and drying constants [5,[24][25][26][27][28][29][30][31][32][33][34][35]. The second approach studies the general behaviour of the solar dryer by applying heat and mass transfers. It allows fo llo wing variations of temperature and humid ity of both the dried product and the heated air. But also it permits following the different temperatures and other parameters of the solar collector and the drying chamber. By this method, efficiency of the studied system can be easily obtained. In this paper, we are focusing on this second approach.
In this last decade, exergy is introduced as an important and a powerful analysis tool for design and calculus of the performances of thermal systems. Fro m a thermodynamic point of view, exergy is defined as the maximu m amount of work which can be produced by a quantity or flow of matter, heat or wo rk as it co mes to equilibriu m with a reference environment [36]. Exergy starts finding applications in several industries and we direct our attention to the application for d rying process [37][38][39][40][41][42][43][44].
Therefore, the aim of this paper is to present a rev iew about the application of exergy and energy to solar drying with presentation of the different mathematical models that treat the general behaviour of the drying systems and calculate their energetic and exergetic efficiencies with products packed in thin or thick layers.

Solar Drying of Thin Layers
We will apply our study to a representative simple convective solar dryer, represented in Figure 1a. It is constituted of two parts; a simp le flat solar air collector (Figure 1b). It is composed of an aluminiu m plate used as an absorber, covered fro m the above by a glass plate and insulated from the exterior using polystyrene. There is a vacuum space between the absorber and the glass cover. The second part concerns the drying chamber made with bricks insulated fro m the exterior by po lystyrene and can support trays where the product is putted. Forced convection is obtained using a fan which permits to have a homogeneous distribution of the heated air inside the drying chamber; also it allows having better control of the process [1,11].
One of the frequent models used to study solar drying of thin layers (putted in perforated trays), is: step by step method [1,13,45]. It consists on taking a fictitious slice that can be noted "j", apply ing heat and mass balance to this slice, then generalizing the study to all the system by varying the step "j". This method is applied to both the solar collector ( Figure 1b) and the drying chamber ( Figure 2). Step "j" of the flat plate solar air collector [1] 2.1.1. Application to the Solar Collector A heat balance is established for the d ifferent co mponents; which are the interior and exterior side of the glass cover, the absorber, the interior and exterio r side of the insulator, resumed by Equation 1 and Table 1: Also, the air flowing between the absorber and the insulator is written in the fo llo wing form: .
The resolution of the obtained system composed of six differential equations, using any adapted method such as the iterative method of Gauss -Seidel [1], allows calcu lating the temperature of exit fro m the solar collector, necessary for the calculus of its efficiency.
The efficiency of a solar collector is generally calculated using the subsequent equation [46][47][48][49]: transformed by Hottel-Whiller-Bliss, for the steady state case and rewritten in the fo llo wing form: Where: F R is the heat removal factor of solar collector, calculated: And: U L is the global heat loss coefficient, written: This approach is widely used for experimental works as it needs to know only geometric parameters of the solar collector and just inlet and outlet temperatures wh ich can be easily obtained using thermocouples.
Studies have shown that performances of a solar collector, represented by its outlet air temperature and its efficiency, depend on several parameters like the region where the collector is installed [45], depend on their internal parameters, such as its surface ( Figure 2) and the exterior conditions like the temperature and the velocity of the ambient air [35,45]. These performances vary with time and have a general behavior as the total received radiat ions ( Figure 2).    [46] Consequently, the efficiency of a solar collector varies in time ( Figure 3) and depends strongly on the absorbed radiations and collector outlet temperature ( Figure 4). The values of the efficiency are variable and can reach 60%.

Application to the Dry ing Chamber
A step ``j`` of the drying chamber is shown in Figure 5. Step "j" of the drying chamber [50] Heat balance is established between the heated air, the product and the interior face o f the brick wall leading to: . , S f represents the exchange surface between the total surface of the product and the air. It is given as a function of the dimensions of the product and its number [1,13].
The other exchanges can be represented by the next equation and table 2.
, , However, the exchanges between the brick and the polystyrene wall are g iven by: Pev is the evaporative power given as a function of the mass and the drying kinetic of the studied product.
The resolution of this system of d ifferential equatio ns allo ws calculat ing the different changing parameters inside the drying chamber especially the variations of the temperatures of the product and the air for the several horizontal trays. By this method, drying chamber efficiency can be calcu lated, well known as pick up efficiency, written [51][52][53]: This parameter determines the efficiency of moisture removal by the drying air fro m the product.
The obtained results, dealing with solar drying of mult iple trays, show that the process is done with a non-homogeneous manner, as shown in Figure 6 and 7. The product putted in tray near the heating source dries faster than products putted in the other trays.   [50] Also some studies show that drying kinetics of the dried product and its temperature is affected [1]: by the surface of the solar collector, by the characteristic dimensions of the product, by the mass of the product putted on each tray and by the air flow rate [50].

Solar Drying of Thick Layers
The unique difference between solar drying of thin and thick layers is that, for the second case, the product is filled in mu ltip le condensed layers. Ho wever, it does not affect the precedent study relative with the solar collector and the same mathematical procedure is used.
As the product is filled in thick layers, the dry ing chamber can be supposed as a porous media in the macroscopic scale. So, mathematical model describing transfer in porous media can be applied for the product, represented by the solid phase, and the heated air, presented as the gas phase. We obtain four differential equations [4,13,51].
Application of mass balance in gas phase: 2 Mass balance in solid phase: Heat balance in gas phase: Heat balance in solid phase: With: (15) Co mbinations of equations 11 with 12 and 13 with 14 give: With: These equations are obtained after considering the following simplifications [51]: the walls of the drying chamber are considered as adiabatic, the product is immob ile and the isotropic. Also, we consider that the air is flo wing in one direction. The resolution of the differential equation system permits following the variations of the temperature and humidity for both the product and the heated air with time and the deepness of the layers. In consequence, efficiency of the drying chamber can be easily calculated using equation 10 [51].  Figure 9. Influence of flow rate and initial humidity on humidity of the heated air [54] Some studies have shown that thick layer dry ing is also done in a non-homogeneous manner. The temperature and drying kinetic was affected by many parameters, such as the initial humidity of the air, the flo w rate and the height of the bed (Figure 8 and 9) [51,54]. A representation of the efficiency is shown in Figure 10, with variation in time and height of the bed.

System Efficiency
The system efficiency indicates the overall thermal performance of a dry ing system including the efficiency of a solar collector, the drying chamber and any other supplement add to the system. The global efficiency is written [52]: For natural convection solar dryers: For forced convection solar dryers that use a fan or a bowler: For hybrid solar dryers that use a second source of energy, the efficiency is calculated using the next equation: It clear (m b * LCV) represents the additional source of energy.

Exergy Analysis
Exergy is presented as a useful analysis tool in the design, assessment, optimizat ion and improvement of energy systems, that can be applied on both system and components levels [55][56]. The exergy method can help further the goal of more efficient energy resource use, because it enables the locations, types and true magnitudes of looses to be determined. As a result, exergy analysis can reveal where and by how much it is possible to design more efficient thermal systems by reducing the sources of existing inefficiencies [49].
The second law of thermodynamics introduces the useful concept of exergy in the analysis of thermal systems. Exergy analysis evaluates the available energy at d ifferent points in a system. Exergy is a measure of the quality of grade of energy and it can be destroyed in the thermal system. The second law states that part of the exergy entering a thermal system with fuel, electricity, flowing streams of matter, and so on is destroyed within the system due to irreversib ility. The second law of thermodynamics uses an exergy balance for the analysis and the design of thermal systems [49].
The general mathematical description of the exergy is presented under the follo wing equation [48][49][57][58][59]: Where: ∞represents the reference conditions. The right terms of the equations are respectively: internal energy, entropy, work, mo mentum, grav ity, chemical reactions and radiation emission.
One of the co mmon simplifications used for equation 22 is to substitute enthalpy for the internal energy and PV. Also, the exergy is used with neglecting gravitational and mo mentu m terms. The pressure term is neglected because the volume is almost kept constant and the operation is happening without any chemical reactions. Consequently, the equation is generally reduced to:

Applicati on of Exerg y for Sol ar Collector
The most presented papers, dealing with exergy analysis of solar dry ing, are p resenting solar dryers similar to one illustrated in Figure 1. The drying system contains a simp le solar air heater, and a dry ing chamber that supports horizontal trays. The system can work with natural convection (without fan) or using forced convection (with fan).
In the case of application of exergy to a simple solar air collector; the terms of mo mentum, grav ity and chemical reaction can be neglected. However, rad iation emission term that depends on temperature must be kept.
The exegy loss is the difference between the exergy in let and outlet that are respectively functions of the in let and outlet temperatures. It is exp ressed: Exergy loss = Exergy inflow-exergy outflow Mathematically determined by equation 24: The exergy in let for the collector is stated as equation 25.
And the exergy outlet for the collector is written: Then the exergy of solar radiat ion for the collector: The exergy efficiency of a solar co llector is: Ex(efficiency) = (Ex(inflo w)-Ex(loss))/Ex(inflow) (28)  Figure 11 shows the variations of the input, output and loss exergy for a solar collector with drying time. The general outline of the curves in influenced by the variation of the received solar rad iation as they have almost the same appearance. Of course, the efficiency of the solar collector decreases with the solar radiation decrease, as shown in Figure 12.

Applicati on of Exerg y to the Drying Chamber
Based on the same way, a co mprehensive procedure of the application of exergy fo r the d rying chamber was developed [36,[60][61]. A representative scheme is illustrated in Figure 13. With:   Also we have: On the other hand, the exergy efficiency is presented as:  ex = Exergy for evaporation of mo isture in product / Exergy of dry ing air supplied Or mathemat ically:   ( )] ln Some works have shown the variation of the exergy inside a drying chamber during drying time, as illustrated in Figure  14 and Figure 15.   [58] This mathematical approach can be used to study thin or packed bed drying with a media supposed as porous [56,62].

Calculation of the System Exergy Efficiency
It is determined by: Exergetic efficiency = 1-Exergy loss/Exergy input (39) Where exergy loss and exergy input can be easily calculated using similar equations to equation 25 and equation 26 with adaptation to the solar drying system.
It is important to mention that in some specific cases for transparent walls of a solar dryer, there are rad iations received by both the drying chamber and the solar collector then if the surface of the dryer is equal to the surface of the collector; the total exergy of rad iation received by the system is double of the quantity presented by equation 27 [49].
Total energy loss and exergetic efficiency of a solar dryer are presented in Figure 16.

Conclusions
Simu lation and mathematical modelling plays an important role for the design of a solar drying system. To have an idea about the general behaviour of a solar drying system and to be able to calculate its energetic efficiency; it is important to establish heat and mass balance. This method is then applied for the different components of the drying system, which means both the solar collector and the drying chamber. It allo ws following the variation, in time and space, of different temperatures and parameters in part icular temperature and humidity of the dried product and the heated air, necessary for the calculation of the energetic efficiency. For thick layers drying, the drying chamber is considered as a porous media and then mathematical treat ment by using transfers in porous media can be done. Also, this method gives the variations of the air and product temperatures and humid ity, permitting the calculat ion of the energetic efficiency for this case, too.
A detailed mathematical procedure to be adopted for calculation of the exergetic efficiency is proposed with application to a solar collector, to the drying chamber and the overall solar drying system. This mathemat ical procedure can be applied for both thin and thick layers. Exergetic efficiency can then be easily calculated by knowing some accessible parameters such as temperatures and humidity.
By means of the presented results, it is clear that the calculated efficiencies are deeply influenced by several parameters such as the variat ion of the received radiat ion, the external temperature and the wind velocity during drying time, by internal parameters and the design such as the surface of the solar collector, by the characteristics of the dried product. So, an optimu m exp loitation of the studied system with a min imu m of expenses passes through simu lating and modelling, in particular with calculation of energetic and exergetic efficiencies. →chemical potential (J/ kg) →hu midity rat io of air  0 →hu mid ity ratio of air at dead state