In this paper, we studied theoretically and numerically heat losses of a at solar collector for the subsequent modelling of the solar water heating system for the Kazakhstan climate condition. For di erent climatic zones with a growing cost for energy or lack of central heating systems, promising is to nd ways to improve the energy eciency of the solar system. The solar system is simulated by the mathematical model (ODE model) of energy. To bridge the results of modelling and real values, our research investigated the important physical parameters such as loss coecient, Nu, Ra, Pr values, which are impacting on the eciency of solar at collector and for heat losses of system. The developed mathematical models, the design and composition of the software and hardware complex, automated control and monitoring systems allow solar hot water heating system to increase the energy eciency of a life support systems and heat supply of buildings, by reducing energy consumption for heat supply. KEYWORDS Solar heating system; Heat loss coecient; Dynamic simulation; Solar at collector; ODE 1. Introduction Each year renewable energy become an important role in our daily lives. Using renew- able energy can reduce the consumption of energy and decrease our dependence on fos- sil energy. The most explicit way to using renewable energy is solar energy that allows getting clean energy without pollution. In the world, there are researches concerning models, designs, and comparison of solar heating systems for domestic buildings. One of the researches is devoted to estimating solar hot water heating and analysis data in South Korea for 3 years (Yoo, 2015). There was a work for installing a solar water heating system for multi-family housing with 1179 families in 14 units. Results of the research showed that comparing with conventional boiler for heating water (eciency=85%) solar how water heating system showed a positive environmental e ect and estimated like reducing 71.9 L/year of oil and reduction of 186.3 ton CO2. CONTACT T. Merembayev Email: timur.merembayev@gmail.com Identical research was done in the article (Chow et al., 2006), The numerical research of a centralized solar hot water heating system was developed for a high-rise residential building in Hong Kong. Another type of research concerning a solar hot water heating system was described in the article (Badescu, 2005). There was a simulation of the solar system with the at collector, storage tank and circulation. It was analyzed two design of hot circulation: water { water and water { air. The result of research showed that the design water { air does not e ective. In research (Siddiqui & Said, 2015) there is a review of articles in the area of solar energy absorption by a solar system with di erent types of uid. The main focus of the research is a solar-based absorption cooling systems, di usion ab- sorption systems, ejector { based absorption systems, compression absorption systems and cogeneration / trigeneration absorption systems. The thermodynamic properties of most common working uids had been reviewed and using a ternary mixture in solar absorption systems. Author Soteris A. Kalogirou (Kalogirou, 2004) made a calculation of optical, thermal and thermodynamic analysis of solar collector, describe methods that were used for estimation of eciency. In the research (Maldonadoa et al., 2014) there was a research of construction and eciency of solar hot water heating system. Geometry and dimension of a solar system were de ned on the based of the result of the materials thermal properties. Authors (Nogueira et al., 1991) showed a software developed in Matlab and own algorithms for calibration of small a solar hot water heating system. Thermal performance can be calculated based on the rst law of thermodynamics (energy) but that does not allow estimating losses of a solar at collector (Farahat et al., 2004). The second law of thermodynamics (exergy) is used to estimation of the various losses in the solar at collector and allow to estimate the eciency of a solar hot water heating system (Luminosu & Fara, 2005; Park et al., 2014). Now in Kazakhstan, it is big attention for developing a solar hot water heating system, the price of system is expensive therefore it is not available for all costumers. To solve the issue it is to develop a new hybrid energy system with mathematical methods and computer simulation, software's and hardware's. One of the ways to the eciency of using energy is to use a new source of energy (renewable and environmentally friendly) in the fuel and energy system of Kazakhstan. So development of an energy system based on the double-circuit solar system with a heat pump is an actual and urgent problem for autonomous power supply. Kazakhstan has a solar energy potential, in the article (Amirgaliyev et al., 2018; Amirgaliyev, Kunelbayev, Merembayev, Daulbayev Irzhanova, 2019) based on actual observations and theoretical calculations generalizing there are the most favourable period and region to use the solar energy in Almaty region. In this article solar col- lector with storage tank, heat exchanger and pump stations for hot water have been considered. Simulation has been performed for di erent temperature conditions and solar irradiance for Almaty region. Simulation is done in Matlab and Simulink, the tool is often used for simulation di erent processes and in particular for simulation solar hot water heating system (Morini & Piva, 2007; Morini & Piva, 2008; Amir- galiyev, Merembayev & Kunelbayev, 2018). The simulation can allow to evaluate a technical possibility of solar water heating system and indicate system nodes which to be modi ed and improved for Kazakhstan weather condition. 2 Figure 1. Schema of solar hot water heating system. Constructed solar hot water heating system. The solar hot water heating system. 1 { solar at collector; 2 { storage tank with a heat exchanger; 3 { pump. Figure 2. Schematic diagram of solar at collector. Constructed solar at collector. The solar hot water heating system. 2. Theoretical model The solar system consists of several parts, to simplify the process of description and simulation we describe each part of the system separately. Figure 1 showed below a solar hot water heating system with a heat exchanger coil in a storage tank. The heat exchanger coil is installed inside the storage tank and it allows to heat water for consumption: heating and domestic hot water. In the central object of research, it is solar at collector, we described a mathe- matical model of solar at collector separately Figure 2 and made energy and ex- ergy analysis. Mathematical model of the collector is de ned as a function: Tcol= f(Tinc;vp;I;Tout;U), whereTcol{ temperature of outlet uid from collector, Tinc{ temperature of inlet uid into collector, vcol{ volumetric ow rate in collector cycle, I{ solar irradiance, Tout{ temperature of collector ambient, U{ heat loss coecient of collector. 3. Energy analysis Based on solar energy absorbed by the plate of collector, heat loss coecient of a collector and heat absorbed by the uid the energy balance equation can be given by (Wengang et al., 2018): 3 Table 1. Input values of solar at collector. Parameter Value Unit Plate-to-cover spacing 30 mm Plate emittance 0.95 Wind heat transfer coecient 10 W=m2C Collector tilt 45 degree Glass emittance 0.885 Area 2 m2 Tcol(t) dt=A CI(t)UA C(Tavg(t)Tout(t)) +vcol Vcol(T1(t)Tcol(t)) (1) WhereC=colccolVcol- equation for calculation overal heat capacity of the uid, Tavg(t) =T1(t)+Tcol(t) 2- average temperature of uid in a collector. Input values of solar at collector are described in Table 1. We used these parameters of the collector to make analysis of heat loss coecient. To simplify the calculation of di erential equations for calculating the heat loss, it was proposed to use common loss factors for the solar collector, similar calculations were performed in (Jafarkazemi & Ahmadifard, 2013). We consider the solar at col- lector for a system with a single glass coating. The calculation of the loss coecient for the upper surface is carried out using the formula: Ut=1 hc;pc+hr;pc+1 hw+hr;ca1 (2) The convection coecient between absorbed and glass coating hc;pcis calculated using the parameters Nusselt, Rayleigh and Prandtl. The heat transfer rate between two plates inclined at a certain angle to the horizon has an obvious signi cance in the operation of at collectors. Convective heat transfer data are usually correlated in terms of two or three dimensionless parameters: the Nusselt number Nu, the Rayleigh number Ra, and the Prandtl number Pr. Nu=hL k(3) The dependence of the parameters Nu and Ra was calculated in (Hollands et al., 1993). Nu= 1 + 1:44 11708(sin1:8 )1:6 Racos  11708 Racos + +"Racos 58301=3 1#+ (4) Ra=g 0TL3 v (5) To calculate the thermal conductivity coecient hc;pc, we use the formula (3): 4 Figure 3. The dependence of the temperature di erence and Ra. Figure 4. Dependence between Ra and Nu. h=Nuk L(6) The dependence of the temperature di erence between the receiving surface and Ra, calculated by the formula (5). From this formula, we can build a relationship between the number Ra and the temperature di erence between the receiving surface and absorbed Figure 3. From this graph, it can be noted that with a large di erence between the surface, the Ra number is minimal, which a ects the calculation of the Nu number. The correlation between Ra and Nu are calculated through the formula (4). The Figure 4 is displayed in a logarithmic format. The values of Ra were generated in the range of 5:96e+03 { 2:44e+05 and for 5 di erent angles of inclination of the collector. 5 Figure 5. Dependence between Ra and Nu in small scale In Figure 6 showed the thermal conductivity, Nu and heat transfer coecient be- tween absorbed and glass coating. The maximum values of Nu and thermal conduc- tivity gave maximum value of a heat transfer coecient; it needs to keep in mind during the design of solar at collector. Our main target is to gather maximum solar irradiance and heat water as quickly as possible. Figure 6. Dependence between the thermal conductivity, Nu and heat transfer coecient. After the thermal conductivity coecient has been calculated, it is necessary to calculate the radiation coecient from the glass coating to the absorber. hr;pc=(T2 p+T2 c)(TpTc) 1 p+1 c1(7) The calculation of the radiation coecient between the glass surface and the air is calculated by the formula: hr;ca=c(T2 c+T2 a)(Tc+Ta) (8) The calculation of the temperature of the glass surface is calculated through the formula: 6 Start Initial values Ta;T0 c;Tp;LTi c=T0 c Calculate: Radiation coecient from plate and cover. Dariation coee cient from cover to air.Calculate: Ra,Nu Calculate: Convection coecient between plate and cover Calculate: Cover glass temperature Ti c jTi cT0 cj< Calculate: Top loss coecient Finishno yes Figure 7. The ow-chart of calculation a top loss coecient. Tc=TpUt(TpTa) hc;pc+hr;pc(9) The process of calculating the loss coecient for the top surface is iterative and depends on the temperature of the glass surface, in Figure 7 showed the ow-chart of this calculation. During calculation, we get the following result for the top loss coecient in Table 2. Result of calculation is the top loss coecient from the collector plate to the ambient, W=m2Cand equal to 7.528. On the basis of formula (2) we have got the range of loss coecient for the upper surface (one surface of glass), it was calculated for di erent ambient temperature indicators [40; 20; 10], plate temperature has the range from 0 to 200C withhwwind heat transfer coecient of 10 W=m2C. In Figure 8 we got a dependence between the average temperature of plate and top loss coecient. We can notice that a temperature 7 Table 2. Result of calculation the top loss coecient. Initial Value Calculated Value Ambient temperature,C 40 Ra 5 :28e+ 05 Cover temperature,C 50 Nu 5.4279 Plate temperature,C 100Convection coecient between the plate and the cover, W=m2C4.8829 Space between cover and plate, m 0.03Radiation coecient from the plate to the cover, W=m2C8.6584 Radiation coecient for the cover to the air, W=m2C6.9555 Cover glass temperature,C 66.6 Figure 8. Dependence between the average temperature of plate and top loss coecient. and loss coecient increases almost linearly except from 0 to 60C. Our calculated coecient is 7.528, on the gure this value is equivalent to the average temperature of plate about 120C. This fact can be possible in the summertime. The calculation of the loss coecient for the lower surface of the collector is calcu- lated using the formula: Ub=k L(10) Where k is the heat conductivity of the insulating material (material - 0.04 W=(mK). L is the thickness of the insulating material. The heat loss coecient for the lateral boundaries generally has small values and often are not calculated for the total heat loss coecient. If it is necessary to calculate the exact values of heat loss, the calculation will be performed using the formula (Tabor, 1958). Ue=(UA)edge A(11) Where U is calculated using formula (11), but for lateral isolation. Collector thick- ness: 0.1 m; insulation thickness: 0.01 m. The total loss coecient will be calculated by summing all the coecients. U=Ut+Ub+Ue (12) 8 Figure 9. General block schema of the solar hot water heating system. 4. Result of simulation and discussion MatLab/Simulink has done a simulation of the model and used model was taken from research (Amirgaliyev et al., 2018). The created model has been simulated full solar hot heating water system: at collector, storage tank and pumps. Based on the mathematical model of the solar at collector formula (1) a block schema has been developed by Simulink, block schema of the collector model is showed in Figure 9. The result of the simulation is a temperature of outlet uid from the collector. We selected data for one year and for Almaty region. Input data: solar irradiance and ambient temperature. The result of the simulation is displayed in Figure 10. Based on the gure we can note that for the region the solar hot water heating system will not provide acceptable temperature in at collector and storage tank, the period from December to March. The best result (high temperature) of the system will be provided from June to the middle of October. Figure 10 shows a high correlation between solar irradiance and temperatures in the collector and storage tank. If the density of solar radiation is higher, then the temperature is more monotonous, and the standard deviation of the temperature in the storage is close to the average value, for example, from July to August. The working process of the pump in the collector cycle depends on reached temperature in the collector cycle, we used same a limit values for activation the pump like in research (Amirgaliyev et al., 2018). Figure 10 showed that pump turned to operate regularly in summertime to transfer reached temperature from the collector into storage tank. In wintertime, the pump does not evidently work. 9 Figure 10. The result of simulation for one year. To analyze simulated results in detail we scaled simulated data to 3 days (1st July { 3rd July). The result of simulation showed the maximum temperature of water in storage tank 56.9C in Almaty region and in collector is 60.5C (Figure 11 and 12). In gure temperatures in the collector, storage tank and pump operation mode are displayed. We can see a correlation between ambient temperature and heating water temperature. When temperature in collector reaches 40C then the pump switches and starts to transfer a heated uid to the exchanger coil inside the storage tank and receives the cooled uid in the collector cycle for heating. The second reason for the cooling of uid in the cycle is due to weather conditions (decrease in ambient temperature, sunset). Figure 11. Ambient temperature, temperature in collector and storage tank. 10 Figure 12. Temperature in collector and storage tank with pump operational mode. 5. Conclusion Before performing a water heating simulation, it is necessary to analyze the technical characteristics of the solar at collector, such as the heat loss of the collector. This analysis takes more time but allows simulating results that are more accurate. After the calculated values, we can proceed to energy analysis. The main purpose of the simulation model is to determine the installation limit values and identify weak instal- lation points for improvement. The developed simulation model makes it possible to simulate the operation of a solar hot water heating system in various climatic condi- tions (temperature and solar radiation) and various individual system parameters to increase the system eciency in various climatic conditions of Kazakhstan. The result of the simulation showed that during the daytime, the temperature of the water in the collector is 70C and during this period, it can be used at home. The main problem is the storage of heated water since at night the heated water is cooled. Another task we can de ne from analysis of the simulation is to nd a way to extend a work period of the system with high eciency (April { October), maybe it will be engineering, technical or programming of controller solutions. This point needs to investigate and improve. The next stage of the research will be the numerical simulation of the reser- voir design. It is necessary to determine the design with high and low eciency. We will determine the optimal system parameters for di erent climatic conditions using the machine learning algorithms to adjust the parameters. We will conduct an experiment and a comparative analysis of simulated and experimental data. 6. Acknowledgement This work is supported by the program { targeted funding from the Ministry of Edu- cation and Science of the Republic of Kazakhstan within the framework of the Project BR05236693 "Mathematical and computer models, hardware and software tools and experimental development on creation of network combined e ective dual-circuit solar collectors with thermosiphon circulation and monitoring of their functioning". 11 Nomenclature Thermal di usivity ( m2=s); 0Volume expansion coecient; T Temperature di erence between the boards; c Absorbent emission (0.88); p Glass emission (0.95);  Eciency (percentage); col Density of uid in colletor ( kg=m3);  Stefan - Boltzmann constant (5 :670367105Wm2K4) ccol Speci c heat capacity in a collector cycle ( J=kgK ); g Gravitational constant (g=9.81 m=s2); h Heat transfer coecient ( W=m2K); hc;pcThermal conductivity coecient ( W=m2C); hr;caRadiation coecient between the glass surface and the air ( W=m2C); hr;pcRadiation coecient from the glass coating to the absorber ( W=m2C); hw Wind heat transfer coecient ( W=m2C); I Solar irradiance ( W=m2); k Heat conductivity of the insulating material ( W=m2K); L Distance between the absorbent and the glass; Nu Nusselt number; Pr Prandtl number; T1 Temperature of output uid from a heat exchanger (C); Tc Glass surface temperature (C); Tp Absorbent temperature (C); Tcol Temperature of outlet uid from collector (C); Tinc Temperature of inlet uid into collector (C); Tout Temperature of collector ambient (C); U Heat loss coecient of collector ( W=m2C); v Kinematic viscosity ( m2=s); Vcol Volume of uid in a collector cycle ( m3); vcol Volumetric ow rate in collector cycle ( m3=s); A Area ( m2); 12