Numerical modeling of thermal storage tanks is often used technique to investigate thermal and mass processes. Although two or three-dimensional analysis are possible to describe flow and temperature distribution in storage tank, they are not applicable to simulation calculations of the long-term performance due to insuperable difficulties of calculation algorithms. Hence, one-dimensional modeling is a possible alternative, because of its simplicity and sufficient accuracy of computational procedures.

Simple one-dimensional numerical model has been developed for predicting the transient behavior of the vertical temperature distribution in the tank. The model describes temperature changing in different layers of the tank by means of momentary energy balance for defined quantity of water.

The stratified accumulator considered in this study is divided into m sectors (corresponding to the number of thermo sensors) with the equal volume, as depicted schematically in Fig. 1. The sectors are numbered from the bottom to the top of tank. Different sectors contain different parts of heater area (serpentine). This means that in sectors work different heat sources (heat exchange area).

Hot water is consumed from the upper sector of the tank. Consumed water quantity is compensated by injection cold water at the bottom sector. This water is assumed to mix

with the water in the sector. Some quantity of mm:,

water from bottom (1) sector enters the next upper sector (2) and mixes with the water in the sector. This process occurs in all next sectors of the tank. The temperature change in sectors by discharging process (water consumption) can be written by:

(1)

where i, n are sector and time step number; V is sector volume and T — temperature of water.

AV is quantity (volume) of consumed water for time step n. For the bottom sector (1) the temperature Ti. i,n. i is the net supply water temperature Tnet. Discharging process is simulated by consecutive passing across the sectors from the bottom to the top.

In the same time, another process is taking place — the thermal charging process (hot water accumulation). The heat from solar collectors is transferred to the water in the tank by serpentine elements. This causes temperature rise in tank. Temperature rise depends on outlet temperature from solar collectors and flow rate of the working fluid. The charging process can be considered as independent (superposition principle). Hence, a second passing across the sectors for the same time step is needed to determine the temperature rising. Energy balance in sectors gives the temperature change:

TOC o "1-5" h z K F.

T = t + ^ ‘jul(t — t )Vr, (2)

i, av i, ac

where T, n’ is the temperature in /-sector of the accumulator at n-time step, after the discharging process has passed; Tf — average fluid temperature in /-serpentine element; At — time step interval for charging process; K, serp and F, serp — heat transfer coefficient and
heat exchange area of serpentine element for /-sector; p and cp — density and specific heat capacitance of water in tank.

Overall heat transfer coefficient K/ser for serpentine element includes convective coefficients hf and hfree, corresponding to the forced circulation in serpentine pipe and free convection from external surface and conductive transfer parameters for serpentine wall:

free

where 5s and As are serpentine wall thickness and conductivity coefficient of the serpentine material.

Convective coefficients hf and hfree depend on fluid temperatures, which are unknown values in the beginning of the calculations. Known parameters for calculation start are the inlet fluid temperature for serpentine (outlet collector temperature) and water temperature in accumulator (temperature distribution in accumulator). Initial temperature distribution in accumulator must be adopted in the beginning of the calculations (initial conditions). This predestines the calculation consequence — from the top to the bottom of accumulator because the inlet of serpentine is in top region of accumulator. Calculation begins for the top accumulator sector with the known fluid temperature in entrance of serpentine element. Iteration procedure for transfer coefficient K/:Serp is adopted.

Inlet fluid temperature T/:/n in /-sector is known — the outlet temperature from the previous (upper) serpentine element (sector /+1). It stays constant in iteration process. For the top sector inlet fluid temperature in serpentine element is determined by solar collector’s performance. The outlet temperature T/:OUt depends on transferred heat energy in sector and is determined in last iteration step.

Mathematical model for solar collectors is well-established matter and detailed information can be fond in solar energy publications. Outlet temperature of working fluid for solar collectors is defined by next equaton [1]:

F

= — [9,(*-■«)e -UL(TsolM — Ta) (4)

m ■ cp

where Fr is heat removal factor, m — mass flow of working fluid, (ra)e — effective transmittance absorbing coefficient for optical part of solar collectors, qs — solar radiation flux for tilted surface [W/m2], UL — overall collector heat loss coefficient [W/m2 K], Ta — ambient temperature. Inlet temperature for solar collectors Tcol, in is the outlet temperature from the bottom serpentine element. Because of dependence between inlet temperature of working fluid for serpentine and inlet temperature for solar collectors (outlet temperature for serpentine), new iteration calculations are needed.

Special simulation algorithm binds the collector and accumulator models in a working unit. It takes into account the heat losses in pipes and accumulator. Computer program is created to manage the theoretical calculations.