Study on differential equation models for solar heating

Introduction

There is no doubt on the importance to investigate and develop solar heating systems. Mathematical modeling is the most widely used and theoretically established tool for the purpose. In this study, a wide sort of solar heating systems are investigated having a solar collector, a heat exchanger and a storage as the main working components. The neighbouring components are connected with pipes, through which a pump circulates some heat transfer fluid as shown in Fig. 1. In many works, the thermal (heat capacity, heat loss) and delaying effects of the pipes are neglected. This is the case, when solar heating systems are modeled with conventionally used ordinary differential equations (ODEs). In [1,2] , collector-storage systems without a heat exchanger are modelled in such a way. In [2] , the collector and the storage are divided into several layers, which are characterized with homogeneous temperatures forming a system of ODEs. In [3,4] , collector-heat exchangerstorage systems are modelled with a linear ODE, the nonlinear version of which can be found in [5] .

Basic linear model

The basic linear model has been worked out in [1,3] and is the following system of four coupled ODEs of first order:

Basic nonlinear model

The basic nonlinear model (also a system of four coupled ODEs of first order) has been worked out in [5] , nevertheless, this is the first time that its derivation is detailed in literature. The standard nonlinear collector model of [6] is according to Eq. ( 2). (The wind speed dependence and the long-wave irradiance dependence of the heat loss coefficient can be neglected [7] .)

Where k 1: linear heat loss coefficient of the solar collector, W/(m 2 K); k 2: second order heat loss coefficient of the solar collector, W/(m 2 K 2 ).

Let us assume that the collector pump is off (v c = 0) and the solar irradiance intensity is zero (I c = 0). This can be attained e.g. by suitable shading of the collector. In this case only the effect of T ca -T c (that is the second and third members) plays a role in the r.h.s. of (2) and the derivative of the collector temperature is a function of the mentioned temperature difference according to curve (2) in Fig. 2. Based on curve (2), T c increases degressively from the temperature difference T 1 . It is evidently a physical unreality, since a temperature value of T ca higher than the collector temperature to a greater extent must warm up the collector faster than in case of a smaller temperature difference. It is even more unreal that temperature differences higher than T 2 not only fail to increase but even reduce T c . It should be mentioned here that generally, the value of T 1 is already pretty high and beyond the normal operation range of solar collectors, so Eq. ( 2) can be practically applied. Nevertheless, Eq. ( 2) is modified here so that it is more realistic for values higher than T 1 according to curve (3a) in Fig. 2. The original function is unchanged below T 1 (curves ( 2) and (3a) coincide below T 1 ), and is set constant from T 1 . Finally, the used nonlinear collector model, including also the irradiation and volumetric flow effects, is resulted in the following way:

The final basic nonlinear model of the solar heating system is established if Eq. ( 3a) is completed by the models of the solar storage and the heat exchangers, based on [3] with the modification that the flow rate dependence for the heat transfer coefficient of the heat exchanger is taken into account:

heat transfer coefficient of the heat exchanger, with two constants k 3 and k 4 , is according to the Dittus-Boelter correlation [8,9] .

Extended ODE models

In the improved, extended linear and nonlinear ODE models, the equations for the collector and for the storage are very similar as in ((1a) and (1d)) and ((3a) and (3d)), respectively. The equations for the heat exchanger ((1b), (1c)) and ((3b), (3c)) are replaced with the equations of the well-known effectiveness-NTU method [10] according to ((4a) and (4b)) below. This concept is well in accordance with compact heat exchangers, when A h is great and V h is small [11,12] .

Where : Heat exchanger effectiveness (-). The models are completed with four pipe, respectively, which describe the four pipes as separated working components with their own volumes and heat loss coefficients (similarly as in the case of the storage).

Results and Discussion

The simulation data of the extended model, with the above identified value U Le = 5.2 W/(m 2 K), is compared with the measured data of two independent days 19th August 2019 and 10th September 2019. (The same comparison is made in case of the basic model with U Lb = 7.3 W/(m 2 K).) Figs. 3 and4 and Table 1 compare the modelled and measured storage temperatures in case of the basic and extended models for the two days. The operating state of the pumps is also shown in the figures.

References

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