Comparison of 1D and 3D Heat and Mass Transfer Models

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All the above mentioned model have their own advantages and disadvantages which are vital in determining the model that is to be chosen for theoretical performance estimation and sizing of the solar collector system. Both 1D and 3D heat transfer model considers all three modes of heat transfer in all possible interaction within the solar collector components and between the solar collector components and the atmosphere. However, these models do not consider the optical imperfections and tracking errors. Due to the simplicity of this models they can be easily integrated in software tool and are the most widely used methods for estimating the thermal efficiency of the system.

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Non uniform thermal model introduces the non-uniformity of temperature created due to the uneven nature of solar flux along the angular direction of the receiver. Though this model improves the accuracy of estimation the procedure becomes very complex because of multiple equations and variables. Statistical model is useful in establishing the relationship between various components of the solar collector and can also predict the thermal performance of the receiver with consummate ease when compared to other models.

However, the main drawback of this model is that it does not consider the heat losses that occur in the system. Including the heat loss equations makes the model much more complicated that the rest. Energy and Exergy model can be used to determine optimal operating conditions and is mostly used in parametric studies. This model may be simple or complex depending on the analysis and the assumption.

Dynamic model is more accurate and can be compared with experimental results even for transient conditions. The main drawback of this model are that i) the model assumes that the volume flow rate and temperature variation is less than 2% ii) the temperature of the receiver is uniform [27]. This limits the application of this model to slight variation in insolation and to conditions when the temperature of the receiver is uniform. These conditions make the system given by [26] quasi dynamic in nature. Therefore, using energy balance conditions and open literature heat transfer correlations a 1D heat transfer model which incorporates the dynamic variation in receiver temperature as well as solar flux is studied in this work.

Heat transfer is driven by temperature gradient between two different mediums, various heat transfer mechanism taking place (fig. 3) in solar receiver are i) heat lost from absorber tube to glass cover by convection and radiation ii) heat lost from glass cover to ambient by conduction, convection and radiation iii) heat gained by HTF from absorber tube by convection. Jose I. Zapata et al [28] carried out transient modelling in once through solar receiver and concluded that proper selection of boundary conditions determines the effectiveness of the developed model. Odeh et al [29-30] proposed a heat loss model in terms of absorber wall temperature and concluded that the thermal stresses induced in the pipe was less when the exit condition is close to saturated steam condition.

Glass cover gains temperature due to absorption of solar radiation and emission of thermal radiation from the receiver tube. Since glass cover is at a lower temperature than receiver tube it gains heat from the receiver tube and losses it to ambient as ambient temperature is lower than glass cover temperature. Under ideal conditions these two heat losses will be equal. The above condition is useful in determining the cover outer temperature and HTF fluid temperature. The heat interaction model of the absorber proposed by Odeh et al is follows,There are two major heat transfer mechanisms by which the absorber tube losses its heat to the glass envelope, they are convection(q.a-g,conv) and radiation (q.a-g,t.rad). A minor portion of heat transfer occurs due conduction of heat (q.a-g,cond) through metallic bellows and supports. Convection heat transfer depends on annular pressure between absorber and glass envelope. If perfect vacuum is maintained in the annulus then heat transfer due to convection can be neglected [31], however in actual case the vacuum condition of the annulus may get altered due to broken seals and hydrogen penetration. In such case convection heat transfer becomes a predominant factor and has to be considered in overall heat loss from the absorber tube. Xiong YaXuan et al [32] carried out numerical simulation for PTC and studied the impact of selective coatings, annular pressure and wind velocity on thermal loss of parabolic trough receivers. The simulation results show that selective coatings and annular pressure influences heat loss from the receiver tube predominantly when compared to wind speed. At pressure below 0.0013 bar i.e. vacuum in annulus, the heat transfer mechanism is dominated by molecular conduction and the heat transfer coefficient is given by Hachicha et al [33] and Ratzel A [34]. The molecular diameter of air is ?=3.66*10-8 cm, thermal accommodation coefficient ?=1 could be assumed in absence of well documented data's [35].

When vacuum is lost, heat transfer in the tube annulus occurs by natural convection and the heat transfer coefficient given by Raithby and Holland [36] is considered in this analysis and the same is given below. This correlation can be used in the range of 102 ? Rac ? 107

The surface of absorber and glass envelope are considered to be as grey body, diffuse emitters and reflectors [37]. The glass envelope is assumed to be opaque to thermal radiation. The Hottel's crossed string method [38] can be used for calculating view factors with good accuracy. Since the absorber tube is not discretized in azimuthal direction we neglect these view factors and take the combined emissivity of the entire absorber tube and glass cover. The heat transfer correlation provided by Holman [39] gives a better accordance with the actual radiative heat transfer coefficient, and it can be used to determine the radiative heat loss satisfactorily.

Retainers are connecting elements that joins the absorber tube and glass cover. Support brackets are gripping elements linked to retainers to keep the receiver tube in position of the focal line of the reflector. Ricardo et al [16] provided an extensive analysis where heat lost was determined for both natural and forced convection. However, a much simpler approach by [39] has been adopted in this study

For long cylinders resting in still air, heat losses by natural convection are greater in horizontal position or close to horizontal, than in inclined position, because when the tubes are inclined the abnormal flow patter decreases the heat transfer, although the difference is negligible for a cylinder in vertical position.

For natural convection (no wind condition) the correlation developed by Churchill et al [40] it can be used to calculate the Nusselt number.

Churchill and Bernstein [41] correlation is used if the PTC is designed to work in exposed wind conditions i.e. under forced convection, and it is valid when RePr > 0.2. When the above condition is not satisfied, a general correlation by Kakac et al [42] can be used for all wind velocities above Vwind>0.1 m/s.Radiation heat transfer between the receiver tube and the glass cover considers diffuse reflection and radiation between the two surfaces. Furthermore, the glass envelope is considered opaque to infrared radiation [43]. The sky temperature is expressed with a simple relationship using local air temperature [44].

The heat loss that takes place in the receiver tube is a function of absorber temperature, as the absorber temperature varies with solar flux the heat loss also changes accordingly. Therefore, for every time step and control volume separate heat loss calculation has to be made once the heat loss is found out then the useful heat gained by the absorber tube can be calculated.

In direct steam generating (DSG) collectors the absorber tube will be almost parallel to the reflecting surface, however as the heat loss for horizontal cylinders are high a shallow slope is provided in the absorber tube. In a DSG system the flow domain in the absorber tube can be classified in to three major zones based on local steam quality as subcooled, two-phase and superheated flow. The real flow pattern distribution in the absorber tube is more complicated than shown in Fig 4, due to the existence of transition zones, nucleate boiling at the end of sub cooled zone, mist flow at the beginning of steam zone and possible flow dry out on the upper surface before the lower surface. However, the boiling zone occupies the major part of the absorber tube and as the lengths of these transition zones are small compared with the length of the absorber tube, excluding these zones will not have a significant effect on flow pattern evaluation [45]. In our study we will consider the entire two phase flow region as a combination of nucleate and forced convection boiling and the effect of each as the flow progresses is studied. Also here the dry steam region is not explored because of high possibility of dry out in that regime, the quality of steam is fixed to a maximum of 0.8.

If the liquid sub cooling is too high to permit bubble nucleation, the flow regime is single phase liquid and the heat transfer is by forced convection. The tube is subjected to uniform heat flux on the outer surface and varying heat loss along the length of the tube. Hence the net heat flux under which sensible heating occurs is under variable in nature. The subcooled fluid gains sensible heat as it moves along the absorber tube, this heat gain is indicated by raise in fluid temperature. This process happens until the bulk fluid temperature reaches saturation temperature of the flowing fluid at that pressure. General energy balance equation (equation: 15) can be used to determine the fluid temperature at exit, while Newton's law of cooling (equation: 16) is employed to get absorber tube wall temperature. If the flow is laminar then Nusselt number is given by [46] equation (17), if the flow is turbulent then Glinskei's correlation equation (18) can be used [39].

The pumping power required to circulate the working fluid through the absorber is a significant factor in the overall performance of a DSG system. For a DSG system mass flux of working fluid is less compared to that of oil based solar electric power generation plant, hence the pumping power required will be less for a DSG system. The pressure gradient in a DSG collector consists of three components, acceleration pressure gradient, gravitational pressure gradient, and frictional pressure gradient. The total pressure gradient is then given by

Frictional pressure drop varies significantly with mass flux, but is independent of temperature and hence independent of DNI [47]. Of all three components of total pressure drop major contribution is from frictional gradient hence other components can be ignored, the pressure drop in single phase region is generally linear and is determined by using Darcy's friction law as follows:

Heat transfer to a liquid flowing through a tube is by single phase convection as long as both the wall and liquid temperature are below the saturation temperature of the liquid at the local pressure. The rate of heat transfer in two-phase regime is higher than that of Single-phase flows under similar thermodynamic conditions of pressure and temperature. Physically heat transfer due to flow boiling is more complicated than single phase and pool boiling heat transfer phenomena, as there is a continuous change in the flow deciding parameters due to continuous vapour generation in flow direction. Boiling starts at the Onset of Nucleate Boiling (ONB) point, following ONB point the sequence of flow regimes are Bubbly-Slug-Annular-Dispersed droplet and finally single phase vapour flow [48]. Unlike Single phase flows, the heat transfer dependence on wall super heat is high for two phase flows. Similarly, the heat loss from the absorber tube varies along the length of tube in single phase regime while it remains constant in two phase regime due to constant wall temperature. Depending on the bulk fluid temperature boiling may be of two types Subcooled boiling and Saturated boiling.