Calculation and experimental study of water distributor of stratification heat accumulator of solar heating system

. The calculation method for a water distributor of a constant cross section uniformly perforated along the height for a multilayer stratified thermal storage tank used in solar heating and hot water supply systems was developed


Introduction
Water thermal storage tanks of solar heat supply systems can function at a significant degree of temperature stratification when the top of the storage tank is hotter than its bottom [1]. The principle of the layerafter-layer charging of a thermal storage tank by a solar collector, when water heated in a collector is supplied to the layer corresponding to its temperature and the mixing of layers is excluded, is widely used now in the design of solar hot water supply and heating systems [2][3][4]. The potential gain in the solar fraction for a solar plant with an ideally stratified tank and low specific water consumption through a solar collector in the range of 0.002-0.007 kg/(m 2 s) compared to a fully mixed tank and high specific water consumption through a solar collector of ~0.01-0.02 kg/(m2 s) can reach 1/3 [1] despite the fact that, under high specific consumption, the higher values of the collector heat removal factor FRare provided [1]. The increase in the solar fraction of such a plant, according to some experimental data [5][6][7], is possible from 0.48 to 0.66. In practice, such a significant gain has not yet been achieved because of the complexity of implementing good temperature stratification in storage tanks [1]. Thus, the improvement of the design of stratified thermal storage tanks and their calculation methods providing stable temperature stratification is topical.
problem for a water distributor of small length with smooth walls in the absence of friction losses is obtained. The general solution for a long-length water distributor with rough walls was obtained by numerically solving a boundary value problem. Experiments to verify the obtained solutions were carried out on an aerodynamic stand.

Results and discussion
Let us consider water delivery by a water distributor of constant cross section with uniform perforations over the length (Fig. 1a). If we replace the perforations by a nominal slot of constant width (Fig.1b) and set up the Bernoulli equation with respect to sections and + , we obtain the differential equation [8] W (3) The boundary condition are In Eqs. (1)-(5), W b and W x are the water velocity at the beginning of the distributor and in section x (m sec ⁄ ); l, P, d e , and δ are the length, perimeter, equivalent diameter, and width of the nominal slot of the distributor (m); F and f are the cross-sectional areas of distributor and slot (m 2 ); δ is the area of a perforation (m 2 ); m is the number of perforations per m of length (l m ⁄ ); p 0 and ρ are the density of the surrounding and distributed water (kg m 3 ⁄ ); λ and μ are the coefficients of resistance of friction and the flow; and g is the acceleration to gravity (m sec 2 ⁄ ). The subscripts 0 and b indicate the cross section at the stop end of the distributor and at the beginning of the distributor.
The μf ̅ are called the nominal slot parameter; λl ̅ are called the distributor parameter. The dimensionless complex Ri is the modified Richardson number, wich characterizes the relationship between the Archimedes forces and interial forces at the beginning of the distributor. In particular, Ri=0.5 means that these forces are equal (−glΔρ = 0.5ρW b 2 ). If the Archimedes forces are directed upward(∆ρ > 0), then Ri < 0; otherwise (∆ρ < 0) we have Ri > 0.Equation (1) and conditions (5) yield a complete mathematical formulation of the boundary value problem under consideration. Let us assume that it has been solved, i.e., we have determined the relative water velocity inside the distributor: In this case the relative average velocity of outflow from the slot Where V av is the average water velocity over the entire area of the slot. We should note that, in the case of uniform delivery, V ̅ x ̅ = 1. In a stratification accumulator, the distributor provides nonunifrom delivery, as a result of the Archimedes forges, the delivery being characterized by maximum outflow velocity in one extreme section where the difference ∆p ≈ 0, and minimum velocity in the opposite one. Allowing for the fact that to eliminate sucking of water into the distributor V ̅ x ̅ ≥ 0, the conditions for which it should be designed are as follows (Fig.1b): for for Ri > 0.5 V ̅ 0 = 0; A decisive quantity in calculations for the distributor is the nominal slot parameterμf ̅ . According to [7], condition (8) is satisfied for μf ̅ ≤ 1. The nominal slot parameter such that expressions (9) or (10) are satisfied will be called critical.
Let us establish how ̅ depends on Ri; for this we simplify the problem somewhat. Let us assume that the length of the distributor is small, and that it has smooth walls, i.e., ̅ = 0. Then Eq. (1) becomes For the case = 0 the solution of initial differential equation (11) has the following from. With allowance for (5) [9].
Allowing for the that small argument values ≈ , obtain The general solution of linear equation (19) is from wish we obtain, taking accountof (17) and (15), Substituting (9), (10), and (5) into (14) and (21), and also allowing for the fact that μf ̅ cr = 0, we find for Ri ≤ 0.5 1.5Arc sin 1 √2Ri for Ri ≥ 0.5 Formula (22) is approximate, and the error involved in using it increases with the variable y(x ̅); as follows from (15), it varies on the interval of existence of the arcsine functions, i.e., |y| ≤ 1. To estiateit's accuracy, differential equation (11) was solved numerically the colocation method.
Experiments aimed at checking the theoretical solutions were conducted on an aerodynamic test stand (Fig.2) with heated air. The length of the air distributor was 1.5 m; the diameter was 0.2 m. A slit 0.05 m wide, equipped with a baffle, was made along it's length. Experiments were conducted for fixed baffle conditions corresponding to ̅ from 0.3 to 1.5 with a 0.2 step, and were repeated for varying degress of roughness of the inner surface of the air distributor (for which the latter was covered with grinding-cloth).
The air distributor was carefully heat-insulated. The air velocity was measured at six points (Fig. 2) by an EA-2M semiconductor thermoanemometer, while the temperature of the heated air was measured by mercury thermometers and the ambient air temperature by an Asmanpsychrometric. To ensure accuracy, the air flowrate was determined twice: by a normal choke diaphragm and on the basis of the average outflow velocity, with the results being balanced. An experiment was regarded as satisfactory if the discrepancy of the balance did not exceed∓10%. The control and measurement instruments made it possible to measure the temperature to with in ∓1℃, the velocity to ∓0.1 / , and the pressure drop to ∓1 2 0. The essential feature of the experiment was determination of for vari ous ̅ . By varying the flowrate and temperature of the heated air, we established outflow conditions from the slot such that ̅ = 0 ( ̅ 0 = 0), i.e., conditions corresponding to (9) or (10). As follows from Fig.3, wish shows theoretical plotted on the basis of formula (21) and computer calculations, and also experimental date, the theoretical and experimental findings are in satisfactory agreement. Formula (21) can be used for making calculations for water distributors for stratification accumulators. The calculations method is similar to that of [10][11][12], and therefore we will confine ourselves only to making the sequence of calculations more precise: 1) we calculate , F, and In the calculations it is necessary to specify d e in such a way that |Ri| ≥ 1.5; otherwise the condition μf ̅ cr ≤ 1 (see Fig.3), which follows from (8), will not be observed.

1.
For water distributors of small length with smooth walls, an analytical solution to the problem is obtained. For long-length water distributors with rough walls, a numerical solution was made and calculated curves were constructed for the design of self-regulating active elements with selective distribution of water along the height of the stratification accumulator, depending on its temperature.

2.
Experimental studies to verify the theoretical solutions of the developed physical and mathematical model of self-regulating active elements for selective distribution of the coolant depending on temperature, carried out on a universal aerodynamic stand, confirmed good agreement between the calculated and experimental data.
, , and 0 are the water velocities at the beginning of the distributor, in cross section , and at the end, respectively ( / ); ̅ ̅ is the relative water velocity; , , 0 , and are the outflow velocity from the slot at the beginning of the distributor, in cross section , at the end, and as averaged over the area of the slot, respectively ( / ); ̅ is the relative outflow velocity of the water; is the distance from the stop end of the distributor to the cross section under consideration; x ̅ is the relative distance; l, P, d e , and δ are the length, perimeter, equivalent diameter, and width of the nominal slot of the distributor, respectively (m); l ̅ is the relative length; F and f are the cross-sectional areas of the distributor and nominal slot (m 2 ); f ̅ is the relative area of the slot; δ is the area of a perforation (m 2 ); m is the number of perforationsparameter of length (1 m ⁄ ); ρ 0 and ρ are the entities of the surrounding and distributed water (kg ∕ m 3 ); ∆ρ is the difference in densities (kg ∕ m 3 ); λ and μ are the coefficients of resistance of friction and the flowrate; g is the acceleration due to gravity (m/sec); μf ̅ and μf ̅ cr are the parameter and critical parameter of the nominal slot; λl ̅ is the parameter of the water distributor; ψ, y(x ̅) are functions of x ̅; C 1 , C 2 are constants of integration; p and q are the constant coefficients of the differential equation; and Ri is the modified Richardson number.