Mathematical model

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The core of the design tool KOLEKTOR 2.2 is a mathematical model of solar flat-plate liquid collector solving one-dimensional heat transfer balances. Solar collector is defined by means of main levels: glazing exterior surface (p1), glazing interior surface (p2), absorber (abs), frame interior surface (z2) and frame exterior surface (z1). These levels are schematically outlined in Fig. 1. Detailed geometrical and physical properties of individual parts of solar collector, climatic and operation parameters are the input parameters of the model. Basic outputs of the model are usable heat gain Qu [W], efficiency ij with respect to reference collector area (gross area AG, aperture area Aa) and output heat transfer fluid temperature te.

The mathematical model of solar collector consists of external energy balance of absorber (heat transfer from absorber surface to ambient environment) and internal energy balance of absorber (heat transfer from absorber surface into heat transfer fluid). Model solves the energy balance of the solar collector under steady-state conditions according to principle Hottel-Whillier equation for usable heat gain

Qu = AaFR [ — U(tin — ta )]

Through the external energy balance of absorber (see Fig. 2) the heat transfer by radiation and by natural convection in the air gap between absorber surface and glazing (event. frame), heat conduction through glazing (event. frame) and heat transfer by convection and radiation from external glazing (event. frame) surface to ambient is solved. To calculate the heat transfer coefficients properly, temperatures for principal collector levels should be known, but on the other side the temperature distribution in the collector is dependent on the heat transfer coefficients values. Therefore, external energy balance of absorber is solved in an iteration loop starting from first temperature estimate for each main level based on given input temperature tin and ambient temperature ta. External balance loop yields in overall collector heat loss coefficient U [W/m2.K].

back side (Uz j

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hp, b2-abs hpbi-a

edge side (Up)

Fig. 2. Schematic layout of external energy balance of absorber.

 

Internal energy balance of absorber assess the heat transfer from the absorber surface into heat transfer fluid provided by fin heat conduction, by heat conduction through the bond between absorber and pipes and by forced convection from pipe internal surface to fluid. Internal balance results in determination of collector efficiency factor F’ [-] and collector heat removal factor FR on the basis of input parameters, operational and climatic conditions and results from external balance. Main outputs from internal balance are output fluid temperature te, mean heat transfer fluid temperature tm and particularly absorber temperature tabs, which governs the calculations in the external balance. Internal balance proceeds in its own iteration loop with respect to relative dependence between mean fluid temperature tm and forced convection heat transfer coefficients in absorber pipe register.

As both external and internal balances are interdependent, superior iteration loop has been introduced to transfer the results from external balance to internal (overall collector heat loss coefficient U) and from internal balance results to external balance (absorber temperature tabs).