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Modelling and simulation aspects of a solar hot water system
Mathematics and Computers in Simulation 48 (1998) 33±46
Modelling and simulation of a solar thermal system J. BuzaÂs*, I. Farkas, A. BiroÂ, R. NeÂmeth Department of Physics and Process Control, University of Agricultural Sciences, GoÈdoÈlloÍ, PaÂter K. u. 1. H-2103, Hungary Abstract The paper deals with the modelling and simulation aspects of the main components of a solar hot water system. Mathematical model was developed to describe the thermal behaviour and energy balance of different solar collectors and hot water storage tanks. Block-oriented simulation technique was used to obtain the solution of the dynamic model. For the identification of the model a laboratory scale measurement was carried out. The results of the simulation could support the selection of appropriate components and sizing of a solar domestic hot water system to be built. # 1998 IMACS/Elsevier Science B.V. Keywords: Solar hot water system; Collector model; Storage tank model; Block-oriented simulation
1. Introduction Owing to the price of the traditional energy resources and the serious environmental pollution problems the use of the renewables, especially the solar energy, is continuously increasing. One of the most classical way to use the solar energy is making hot water. Such a system should consist of two main elements as solar collector and a storage tank. In order to design and study such a solar thermal system an appropriate modelling technique is required to introduce for both elements. Concerning the modelling of a collector or a storage tank their structure and nature give the way of approaches. For the coupled system a block-oriented solution seems to be reliable as it provides a high flexibility in case of changing the system layout. The evaluation of the developed models at least laboratory measurements are to be carried out to check the viability of the models and also to identify the system parameters, as well. 2. Modelling of solar collectors In this section two different kinds of solar collector models are described and analysed. ÐÐÐÐ * Corresponding author. Tel.: +36-28-410894; fax: +36-28-410804; e-mail: [email protected] 0378-4754/98/$ ± see front matter # 1998 IMACS/Elsevier Science B.V. All rights reserved PII: S 0 3 7 8 - 4 7 5 4 ( 9 8 ) 0 0 1 5 3 - 0
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Fig. 1. Heat balance of a solar collector.
2.1. Ordinary flat plate collector model Let us consider a flat plate solar collector shown in Fig. 1, where the temperature of fluid entering the collector is Tin, the temperature of fluid leaving the collector Tc, the aperture surface of the collector Ac, _ and the heat loss coefficient the irradiance in the plane of the collector I, the mass flow rate of fluid m, of the collector is U. It is intended to build a mathematical model that describes Tc as a function of Tin, _ I, and U. m, The general energy balance of the collector is given by accumulation of total energy input of total energy output of total energy ÿ time time time
(1)
The performance of a solar collector can then be described by the energy balance. The total energy is decomposed as E U K P
(2)
As the collector does not move, dK/dt0 and dP/dt0, and dE/dtdU*/dt. For solid and liquid systems dU*/dtdH/dt. In the case of steady-state conditions, the rate at which heat enters the collector is equal to the rate at which heat leaves it. Thus, the heat absorbed by the heat transfer fluid as it passes through the collector is equal to the heat gains of the collector minus the heat loss from it. The solar energy Qs absorbed by the absorber plate of the collector can be calculated as Qs IAc
(3)
Both and are dimensionless and depend on the incidence angle of the collector, both decrease as incidence angle increases. The heat loss of the collector Ql is given by Ql UAc Tabs ÿ Ta
(4)
The heat exchange between the collector and its surroundings may occur from upper and lower surfaces and the sides. Based on practical measurements and experiences in practice in the case of collectors which are insulated below the absorber plate the loss from the collector through the upper surface will dominate.
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The heat absorbed by the fluid Qf as it passes through the collector can be calculated as _ Qf mc T c ÿ Tin
(5)
For simplicity, it is assumed that the liquid in the solar collector is completely mixed. Then, the energy balance of the collector based on Eqs. (3)±(5) can be given by dcVTc IAc UAc Tabs ÿ Ta Fc in cin Tin ÿ Fc out cout Tc dt where m_ Fc and V is the volume of fluid passage in the collector. The following assumptions are kept:
(6)
6(T) in the range of interest, c6c(T) in the range of interest, inout, cincoutc, V is constant. Taking into consideration the assumptions Eq. (6) can be rewritten as
dTc _ IAc ÿ UAc Tabs ÿ Ta mc T (7) in ÿ Tc dt Due to the difficulty of Tabs measurement, it is usual to express heat loss in terms of average fluid temperature Tav in the collector, as follows: cV
Tin Tc (8) 2 Between the surface of absorber plate and the fluid, it has a finite thermal resistance, when the system is in operation the absorber plate surface will be hotter than the fluid in the tubes. This yields higher radiative and convective heat loss from the absorber plate than it would occur if it was at the same temperature as that of the fluid in the tubes. Take this influence into account by a correction factor F0 which is called as the heat transfer or heat removal factor. It indicates the efficiency of heat transfer between the absorber plate surface and the heat transfer fluid. Substitution of this new term into Eq. (7) yields Tav
dTc _ IAc F 0 ÿ UAc F 0 Tav ÿ Ta mc T (9) in ÿ Tc dt In the case of collector performance tests and measurements F0 and F0 U products are used [10]. Sometimes F0 is called as `optical efficiency' 0 and F0 U is called the `overall heat loss coefficient' UL. These parameters can be determined by theoretical analysis or by measuring procedure. So Eq. (9) can be rewritten as cV
dTc _ IAc 0 ÿ UL Ac Tav ÿ Ta mc T in ÿ Tc dt Table 1 shows the typical values of 0 and UL for the different types of collector. Finally, the state equation of the collector can be written as cV
(10)
dTc Ac 0 UL A c Fc Iÿ Tav ÿ Ta Tin ÿ Tc dt C C V
(11)
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Table 1 Typical values of optical efficiency 0 and overall heat loss UL for different types of collector [8] Type of collector
Temperature range (8C)
0
UL (W mÿ2Kÿ1)
Unglazed Single glazed Double glazed Single glazed, selective Evacuated tubes
10±40 10±60 10±80 10±80 10±130
0.90 0.80 0.65 0.80 0.70
15±25 7 5 5 2
where CcV is the overall heat capacity of the fluid and the residence time is *V/Fc. The block scheme of the solar collector can be seen in Fig. 2 where the variables in Eq. (11) are the following: State variable: Tc Output variable: Tc Input variables: Disturbances: I, Ta, Tin Manipulated variable: Fc Parameters: Ac, , c, V, UL, 0 2.2. Model for a rubber made collector A simple design of solar hot water system can be used for instance in swimming pools in which case the whole collector consists of parallel connected rubber made tubes without any covering and frame. The equation for such a rubber made collector can be written in the following way: Cc
dTc F 0 Ac I ÿ UL Ta ÿ Tc ÿ Qc;s ; dt
(12)
where the right side of the equation is the heat obtained from the irradiation and reduced by the heat loss to the ambient and the transferred heat into the storage tank. 3. Modelling approaches of hot water storage tanks In this section three different kinds of storage tank models are described and analysed.
Fig. 2. Block scheme of the solar collector.
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Fig. 3. Divided hot water storage tank.
3.1. Divided hot water storage tank In this case the model is built up as a divided storage tank where the heat transfer is performed through a wall surface [9] (see Fig. 3). We assume full mixing on both sides and Td is constant. Tc is the temperature of the solar liquid which comes from the collector, T1 the temperature of the solar liquid which returns to the collector, Fc the volumetric flow rate through the collector, Td the temperature of the supplied water, T2 the temperature of the extracted water and Fl is the volumetric flow rate of the load. The energy balance equations for the two parts of the storage tank are the following: d 1 cp1 V1 T1 Fc 1 cp1 Tc ÿ Fc 1 cp1 T1 ÿ Us A T1 ÿ T2 ; dt
(13a)
d 2 cp2 V2 T2 Fl 2 cp2 Td ÿ Fl 2 cp2 T2 ÿ Us A T1 ÿ T2 ; dt
(13b)
where V1 and V2 are the volume of storage tank sections and A is the surface of the heat transfer between the storage tank sections. The following assumptions were taken into account:
12 and 6(T) in the range of interest, cp1cp2cp and cp6cp(T) in the range of interest, inout, cpin cpout cp , V1V2V are constants. Then from Eqs. (13a) and (13b) it yields cp V1
and
dT1 Fc cp Tc ÿ Fc cp T1 ÿ Us A T1 ÿ T2 ; dt
(14a)
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Fig. 4. Simulink model for Eqs. (14a) and (14b).
cp V2
dT2 Fl cp Td ÿ Fl cp T2 ÿ Us A T1 ÿ T2 dt
(14b)
The state variables in this model are T1 and T2 (equal to the outputs T1 and T2). The inputs are in principle Fc, Fl, Tc, Td, and , cp, Us are the physical parameters, and V1, V2 and A are the design parameters. For the storage tank the following simulation was carried out in Simulink (Fig. 4). The values of the parameters in the model are Tc608C, Fc0.000015 m3 sÿ1, Td108C, V1V20.075 m3, Us250 W mÿ2 Kÿ1, A0.5 m2, 1000 kg mÿ3, cp4200 J kgÿ1 Kÿ1, and 0 m3 sÿ1 for t < 5 104 s Fl t 1 10ÿ5 m3 sÿ1 for t 5 104 s The initial conditions are T1(0)16.878C and T2(0)108C. The result of the simulation can be seen in Fig. 5. 3.2. Storage tank with heat exchanger In the following case the model is built up as a hot water storage tank which contains a heat exchanger coil (Fig. 6). A completely mixed storage tank is assumed and Td is constant. Tc is the temperature of the solar liquid which comes from the collector, T1 the temperature of the solar liquid which returns to the collector, Td the temperature of the supplied water, Ts the temperature of the extracted water, Fl is the volumetric flow rate of the load and Fc is the volumetric flow rate of the collector. The energy balance equations for the storage tank are the following: d 1 cp1 VTs Fl 1 cp1 Td ÿ Ts Fc 2 cp2 Tc ÿ T1 dt
(15a)
This equation assumes that there is no change in heat storage in the coil. An additional equation is needed to find T1. Because there is no change in heat storage in the coil it can be assumed that the coil is in steady state. In this case the heat transfer can be calculated as Fc 2 cp2 Tc ÿ T1 Uc A
Tc ÿ T1 ln Tc =T1
(15b)
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Fig. 5. T1 and T2 temperature response to Fl in the case of the divided hot water storage tank.
Fig. 6. Hot water tank with heat exchanger coil.
where TcTcÿTs and T1T1ÿTs, and A is the surface of the coil. The right-hand side of Eq. (15b) can be derived from a steady state partial differential equation valid for the coil. It can be noted that it is necessary to verify whether the `no heat storage' assumption is justified. Actually, the following assumptions are taken into account:
12 and 6(T) in the range of interest, cp1cp2cp and cp6cp(T) in the range of interest, inout, cpin cpout cp , V is a constant. From Eqs. (15a) and (15b) it yields cp V
dTs Fl cp Td ÿ Ts Fc cp Tc ÿ T1 ; dt
(16a)
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Fc cp Tc ÿ T1 Uc A
Tc ÿ T1 : ln Tc =T1
(16b)
From Eq. (16b) T1 can be expressed as the function of the state (Ts) and the input (Tc, Fc): T1 Tc ÿ Ts eÿ Uc A=cp Fc Ts ;
(17)
which can be substituted into Eq. (16a). The simulation was carried out with the same values which was used in the case of divided hot water storage tank. The parameters are Tc608C, Fc0.000015 m3 sÿ1, Td108C, V0.15 m3 (V is the sum of the tank sections in the case of divided hot water storage tank), Uc250 W mÿ2 Kÿ1, A0.5 m2, 1000 kg mÿ3, cp4200 J kgÿ1 Kÿ1, and for t < 5 104 s; 0 m3 sÿ1 Fl t ÿ5 3 ÿ1 for t 5 104 s 1 10 m s (see Figs. 7 and 8) The initial conditions are Ts(0)108C and T1(0) can be calculated based on Eq. (17) with the values of the parameters: T1 0 60 ÿ 10eÿ 2500:5=42000:0000151000 10 16:87 C: 3.3. Mixed storage tank model The next equation describes the storage tank unit supposed to be fully mixed thermally: Cs
dTs Qc;s ÿ ks As Ta ÿ Ts ÿ Qs;l dt
Fig. 7. T1 and Ts temperature response to Fl in the case of storage tank with heat exchanger.
(18)
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Fig. 8. Simulink model for Eqs. (15a) and (15b).
4. Simulation of a solar hot water system (HWS) Considering a complete solar hot water system based on the described collector and storage tank models a simulation can be carried out. 4.1. System description The main components of the solar hot water system are the solar collector and the storage tank unit. The flat plate solar collector produces heat from the energy of the sun absorbed by the absorber plate. The gathered thermal energy is stored in a hot water storage tank. The heat is transferred from the collector to the storage tank by the heat transfer liquid. In the closed solar loop the liquid is circulated by a pump unit. The scheme of the solar hot water system can be seen in Fig. 9. 4.2. Simulation of the system The system has three input parameters as ambient temperature, solar irradiation and the required mass flow of the loaded hot water. These parameters have been simulated based on the information about modelling the ambient temperature and the solar irradiation found in the relevant literature [1,2,3±7]. There are also national standards for hot water usage. Using the values of the standard a step function can be calculated which gives the required mass flow as a function of time. The mathematical model used to be for the simulations based on energy flow in the collector and the storage tank. Eqs. (12) and (18) are used for such purposes. The block-oriented solution of the mathematical model can be found in Fig. 10. This way of the solution is easy to survey and its modularity gives the possibility for easily changing the set-up. The first block gives the time steps for the input parameters. The next three blocks contain the input parameters as ambient temperature, irradiation and the load. Qc,s and Qs,l give the heat transfer between the collector and the storage tank and the heat taken by the load. Tc and Ts indicate the temperature of the collector and the storage tank. They contain Eqs. (12) and (18). The graph block gives the results in graphical form as can be seen in Fig. 11. 4.3. Identification of HWS Controlling the results of the model needs comparing measurements. The input parameters of a real application are stochastic ones. For the control the inputs must be deterministic and above all
42
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Fig. 9. The set-up of the simulated solar thermal system.
Fig. 10. Block-oriented solution of the model.
repeatable. These requirements can be satisfied only in laboratory circumstances. The laboratory equipment can be seen in Fig. 12. The system consists of a data acquisition PC with a software developed for this purpose, five Pt-100 thermometers for taking the temperature of the ambient air, the water at the bottom and at the top of the storage tank, the water in the collector at the incoming and at the outgoing side. The set-up also consists of a storage tank unit, a collector, a pump, a throttle valve to set the mass flow and a measuring meter. The connection between the PC and the Pt-100s is through ADAM data logging modules. For the easier check of the temperature in the storage tank the water is mixed to avoid the stratification. The laboratory equipment can be modelled by the same equations as the real applications. The only difference seems to be the constant input parameters. The measured values of the experiment carried out are shown in Fig. 13. Comparing the simulated and measured values the system parameters were identified. Both values can be seen in Fig. 14, which show a good coincidence. The irradiation was anyway 204 W mÿ2, the ambient temperature 218C, the storage volume 25 l, and the collector area was 0.5 m2.
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Fig. 11. Simulation results yield by the model.
Fig. 12. Set-up for the laboratory scale equipment.
5. Conclusions In this paper a mathematical model was developed and used for simulation of a solar hot water system (HWS). As a matter of models an ordinary flat plate solar collector and a simplified rubber made collector were considered and developed. Concerning to the hot water storage tanks three different cases were studied as a divided hot water tank, a storage tank with heat exchanger and a fully mixed one. For the simulation of the hot water making system a block-oriented approach was used and realised in MatlabSimulink software. To identify the system parameters a measurement was carried out. The modelling results achieved showed a fairly good coincidence with the measurements.
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Fig. 13. Results of the measuring experiment.
Fig. 14. Comparison of the measured and simulated water temperatures in the storage.
6. Nomenclature Ac As c cin cout C
aperture surface of collector (m2) outside surface of the storage tank (m2) specific heat capacity of fluid (J kgÿ1 Kÿ1) inlet fluid specific heat capacity (J kgÿ1 Kÿ1) outlet fluid specific heat capacity (J kgÿ1 Kÿ1) overall heat capacity of fluid (J Kÿ1)
J. BuzaÂs et al. / Mathematics and Computers in Simulation 48 (1998) 33±46
Cc Cs E Fc Fl F0 H I ks K m_ P Qc,s Qf Ql Qs Qs,l Ta Tabs Tav Tc Td Tin Ts T1 T2 U U* Uc UL Us V
overall heat capacity of collector (J Kÿ1) overall heat capacity of storage tank (J Kÿ1) total energy (J) volumetric flow rate of collector (m3 sÿ1) volumetric flow rate of load (m3 sÿ1) heat transfer factor between the absorber plate surface and the heat transfer fluid total enthalpy of the system (J) irradiance in plate of collector (W mÿ2) heat transfer coefficient between the storage tank and the surrounding air (W mÿ2 Kÿ1) kinetic energy (J) mass flow rate of fluid (kg sÿ1) potential energy (J) heat transferred from the collector to the storage tank (W) heat absorbed by the fluid (W) heat loss of the collector (W) solar energy absorbed by the absorber plate (W) extracted heat from the storage tank by load (W) ambient temperature (8C) temperature of the surface of the absorber plate (8C) average fluid temperature in the collector (8C) outlet temperature of the collector (8C) temperature of the supplied water (8C) inlet temperature of the collector (8C) temperature of the extracted water from the storage tank (8C) temperature of the fluid which returns from the hot water storage tank to the collector (8C) temperature of the extracted water from the divided hot water storage tank (8C) heat loss coefficient of collector (W mÿ2 Kÿ1) internal energy (J) overall heat transfer coefficient of the coil in the storage tank (W mÿ2 Kÿ1) overall heat loss coefficient of the collector (W mÿ2 Kÿ1) overall heat transfer coefficient between the two sections in the divided storage tank (W mÿ2 Kÿ1) volume (m3)
Greek symbols 0 in out *
45
absorbance of the plate of the collector (dimensionless) optical efficiency of the collector (dimensionless) density of the fluid (kg mÿ3) inlet fluid density (kg mÿ3) outlet fluid density (kg mÿ3) transmittance of the cover of the collector (dimensionless) residence time of the collector (s)
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Acknowledgements This work has been carried out within the framework of the project ``Investigation of the technical parameters and environmental influences of photovoltaic and thermal solar systems'' (Ministry of Education, MKM FKFP 1173/1997, 1997±1999). References [1] I. Farkas, Modelling and identification for control of solar and connected technological system, D.Sc. Thesis, Hungarian Academic of Science, Budapest, Hungary, 1993. [2] D.G. Erbs, S.A. Klein, W.A. Beckman, Estimation of degree dazes and ambient temperature bin data from monthly average temperatures, ASHRAE J. (1983) 60±65. [3] D.A. Kouremenos, K.A. Antonopoulos, E. Rogdakis, Performance of solar NH3/H2O absorption cycles in the Athens area, Solar Energy 39 (1987) 187±195. [4] H.P. Garg, B. Bandyopadhyay, V.K. Sharma, Investigation of rock bed solar collector cum storage system, Energy Conversion Management 21 (1981) 275±282. [5] I. Farkas, Z. Rendik, Handling of solar climatic data, Ambient Energy 14 (1993) 59±68. [6] ASHRAE Handbook, System and Application, Chapter 47, 1987. [7] J.A. Duffie, W.A. Beckman, Solar Engineering of Thermal Processes, 2nd ed., Wiley, New York, 1991. [8] European Simplified Methods for Active Solar System Designe, Bernard Bourges, Kluwer Academic Publishers for CEC, 1991. [9] J. BuzaÂs, G. van Straten, Solar hot water system study examples with MATLAB/SIMULINK, Tempus 9709-95 Report, 1996. [10] C. King, Solar Water Heating, European Commission Directorate-General XII for Science, Research and Development, 1995.
Modelling and simulation of a solar thermal system J. BuzaÂs*, I. Farkas, A. BiroÂ, R. NeÂmeth Department of Physics and Process Control, University of Agricultural Sciences, GoÈdoÈlloÍ, PaÂter K. u. 1. H-2103, Hungary Abstract The paper deals with the modelling and simulation aspects of the main components of a solar hot water system. Mathematical model was developed to describe the thermal behaviour and energy balance of different solar collectors and hot water storage tanks. Block-oriented simulation technique was used to obtain the solution of the dynamic model. For the identification of the model a laboratory scale measurement was carried out. The results of the simulation could support the selection of appropriate components and sizing of a solar domestic hot water system to be built. # 1998 IMACS/Elsevier Science B.V. Keywords: Solar hot water system; Collector model; Storage tank model; Block-oriented simulation
1. Introduction Owing to the price of the traditional energy resources and the serious environmental pollution problems the use of the renewables, especially the solar energy, is continuously increasing. One of the most classical way to use the solar energy is making hot water. Such a system should consist of two main elements as solar collector and a storage tank. In order to design and study such a solar thermal system an appropriate modelling technique is required to introduce for both elements. Concerning the modelling of a collector or a storage tank their structure and nature give the way of approaches. For the coupled system a block-oriented solution seems to be reliable as it provides a high flexibility in case of changing the system layout. The evaluation of the developed models at least laboratory measurements are to be carried out to check the viability of the models and also to identify the system parameters, as well. 2. Modelling of solar collectors In this section two different kinds of solar collector models are described and analysed. ÐÐÐÐ * Corresponding author. Tel.: +36-28-410894; fax: +36-28-410804; e-mail: [email protected] 0378-4754/98/$ ± see front matter # 1998 IMACS/Elsevier Science B.V. All rights reserved PII: S 0 3 7 8 - 4 7 5 4 ( 9 8 ) 0 0 1 5 3 - 0
J. BuzaÂs et al. / Mathematics and Computers in Simulation 48 (1998) 33±46
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Fig. 1. Heat balance of a solar collector.
2.1. Ordinary flat plate collector model Let us consider a flat plate solar collector shown in Fig. 1, where the temperature of fluid entering the collector is Tin, the temperature of fluid leaving the collector Tc, the aperture surface of the collector Ac, _ and the heat loss coefficient the irradiance in the plane of the collector I, the mass flow rate of fluid m, of the collector is U. It is intended to build a mathematical model that describes Tc as a function of Tin, _ I, and U. m, The general energy balance of the collector is given by accumulation of total energy input of total energy output of total energy ÿ time time time
(1)
The performance of a solar collector can then be described by the energy balance. The total energy is decomposed as E U K P
(2)
As the collector does not move, dK/dt0 and dP/dt0, and dE/dtdU*/dt. For solid and liquid systems dU*/dtdH/dt. In the case of steady-state conditions, the rate at which heat enters the collector is equal to the rate at which heat leaves it. Thus, the heat absorbed by the heat transfer fluid as it passes through the collector is equal to the heat gains of the collector minus the heat loss from it. The solar energy Qs absorbed by the absorber plate of the collector can be calculated as Qs IAc
(3)
Both and are dimensionless and depend on the incidence angle of the collector, both decrease as incidence angle increases. The heat loss of the collector Ql is given by Ql UAc Tabs ÿ Ta
(4)
The heat exchange between the collector and its surroundings may occur from upper and lower surfaces and the sides. Based on practical measurements and experiences in practice in the case of collectors which are insulated below the absorber plate the loss from the collector through the upper surface will dominate.
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The heat absorbed by the fluid Qf as it passes through the collector can be calculated as _ Qf mc T c ÿ Tin
(5)
For simplicity, it is assumed that the liquid in the solar collector is completely mixed. Then, the energy balance of the collector based on Eqs. (3)±(5) can be given by dcVTc IAc UAc Tabs ÿ Ta Fc in cin Tin ÿ Fc out cout Tc dt where m_ Fc and V is the volume of fluid passage in the collector. The following assumptions are kept:
(6)
6(T) in the range of interest, c6c(T) in the range of interest, inout, cincoutc, V is constant. Taking into consideration the assumptions Eq. (6) can be rewritten as
dTc _ IAc ÿ UAc Tabs ÿ Ta mc T (7) in ÿ Tc dt Due to the difficulty of Tabs measurement, it is usual to express heat loss in terms of average fluid temperature Tav in the collector, as follows: cV
Tin Tc (8) 2 Between the surface of absorber plate and the fluid, it has a finite thermal resistance, when the system is in operation the absorber plate surface will be hotter than the fluid in the tubes. This yields higher radiative and convective heat loss from the absorber plate than it would occur if it was at the same temperature as that of the fluid in the tubes. Take this influence into account by a correction factor F0 which is called as the heat transfer or heat removal factor. It indicates the efficiency of heat transfer between the absorber plate surface and the heat transfer fluid. Substitution of this new term into Eq. (7) yields Tav
dTc _ IAc F 0 ÿ UAc F 0 Tav ÿ Ta mc T (9) in ÿ Tc dt In the case of collector performance tests and measurements F0 and F0 U products are used [10]. Sometimes F0 is called as `optical efficiency' 0 and F0 U is called the `overall heat loss coefficient' UL. These parameters can be determined by theoretical analysis or by measuring procedure. So Eq. (9) can be rewritten as cV
dTc _ IAc 0 ÿ UL Ac Tav ÿ Ta mc T in ÿ Tc dt Table 1 shows the typical values of 0 and UL for the different types of collector. Finally, the state equation of the collector can be written as cV
(10)
dTc Ac 0 UL A c Fc Iÿ Tav ÿ Ta Tin ÿ Tc dt C C V
(11)
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Table 1 Typical values of optical efficiency 0 and overall heat loss UL for different types of collector [8] Type of collector
Temperature range (8C)
0
UL (W mÿ2Kÿ1)
Unglazed Single glazed Double glazed Single glazed, selective Evacuated tubes
10±40 10±60 10±80 10±80 10±130
0.90 0.80 0.65 0.80 0.70
15±25 7 5 5 2
where CcV is the overall heat capacity of the fluid and the residence time is *V/Fc. The block scheme of the solar collector can be seen in Fig. 2 where the variables in Eq. (11) are the following: State variable: Tc Output variable: Tc Input variables: Disturbances: I, Ta, Tin Manipulated variable: Fc Parameters: Ac, , c, V, UL, 0 2.2. Model for a rubber made collector A simple design of solar hot water system can be used for instance in swimming pools in which case the whole collector consists of parallel connected rubber made tubes without any covering and frame. The equation for such a rubber made collector can be written in the following way: Cc
dTc F 0 Ac I ÿ UL Ta ÿ Tc ÿ Qc;s ; dt
(12)
where the right side of the equation is the heat obtained from the irradiation and reduced by the heat loss to the ambient and the transferred heat into the storage tank. 3. Modelling approaches of hot water storage tanks In this section three different kinds of storage tank models are described and analysed.
Fig. 2. Block scheme of the solar collector.
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Fig. 3. Divided hot water storage tank.
3.1. Divided hot water storage tank In this case the model is built up as a divided storage tank where the heat transfer is performed through a wall surface [9] (see Fig. 3). We assume full mixing on both sides and Td is constant. Tc is the temperature of the solar liquid which comes from the collector, T1 the temperature of the solar liquid which returns to the collector, Fc the volumetric flow rate through the collector, Td the temperature of the supplied water, T2 the temperature of the extracted water and Fl is the volumetric flow rate of the load. The energy balance equations for the two parts of the storage tank are the following: d 1 cp1 V1 T1 Fc 1 cp1 Tc ÿ Fc 1 cp1 T1 ÿ Us A T1 ÿ T2 ; dt
(13a)
d 2 cp2 V2 T2 Fl 2 cp2 Td ÿ Fl 2 cp2 T2 ÿ Us A T1 ÿ T2 ; dt
(13b)
where V1 and V2 are the volume of storage tank sections and A is the surface of the heat transfer between the storage tank sections. The following assumptions were taken into account:
12 and 6(T) in the range of interest, cp1cp2cp and cp6cp(T) in the range of interest, inout, cpin cpout cp , V1V2V are constants. Then from Eqs. (13a) and (13b) it yields cp V1
and
dT1 Fc cp Tc ÿ Fc cp T1 ÿ Us A T1 ÿ T2 ; dt
(14a)
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Fig. 4. Simulink model for Eqs. (14a) and (14b).
cp V2
dT2 Fl cp Td ÿ Fl cp T2 ÿ Us A T1 ÿ T2 dt
(14b)
The state variables in this model are T1 and T2 (equal to the outputs T1 and T2). The inputs are in principle Fc, Fl, Tc, Td, and , cp, Us are the physical parameters, and V1, V2 and A are the design parameters. For the storage tank the following simulation was carried out in Simulink (Fig. 4). The values of the parameters in the model are Tc608C, Fc0.000015 m3 sÿ1, Td108C, V1V20.075 m3, Us250 W mÿ2 Kÿ1, A0.5 m2, 1000 kg mÿ3, cp4200 J kgÿ1 Kÿ1, and 0 m3 sÿ1 for t < 5 104 s Fl t 1 10ÿ5 m3 sÿ1 for t 5 104 s The initial conditions are T1(0)16.878C and T2(0)108C. The result of the simulation can be seen in Fig. 5. 3.2. Storage tank with heat exchanger In the following case the model is built up as a hot water storage tank which contains a heat exchanger coil (Fig. 6). A completely mixed storage tank is assumed and Td is constant. Tc is the temperature of the solar liquid which comes from the collector, T1 the temperature of the solar liquid which returns to the collector, Td the temperature of the supplied water, Ts the temperature of the extracted water, Fl is the volumetric flow rate of the load and Fc is the volumetric flow rate of the collector. The energy balance equations for the storage tank are the following: d 1 cp1 VTs Fl 1 cp1 Td ÿ Ts Fc 2 cp2 Tc ÿ T1 dt
(15a)
This equation assumes that there is no change in heat storage in the coil. An additional equation is needed to find T1. Because there is no change in heat storage in the coil it can be assumed that the coil is in steady state. In this case the heat transfer can be calculated as Fc 2 cp2 Tc ÿ T1 Uc A
Tc ÿ T1 ln Tc =T1
(15b)
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Fig. 5. T1 and T2 temperature response to Fl in the case of the divided hot water storage tank.
Fig. 6. Hot water tank with heat exchanger coil.
where TcTcÿTs and T1T1ÿTs, and A is the surface of the coil. The right-hand side of Eq. (15b) can be derived from a steady state partial differential equation valid for the coil. It can be noted that it is necessary to verify whether the `no heat storage' assumption is justified. Actually, the following assumptions are taken into account:
12 and 6(T) in the range of interest, cp1cp2cp and cp6cp(T) in the range of interest, inout, cpin cpout cp , V is a constant. From Eqs. (15a) and (15b) it yields cp V
dTs Fl cp Td ÿ Ts Fc cp Tc ÿ T1 ; dt
(16a)
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Fc cp Tc ÿ T1 Uc A
Tc ÿ T1 : ln Tc =T1
(16b)
From Eq. (16b) T1 can be expressed as the function of the state (Ts) and the input (Tc, Fc): T1 Tc ÿ Ts eÿ Uc A=cp Fc Ts ;
(17)
which can be substituted into Eq. (16a). The simulation was carried out with the same values which was used in the case of divided hot water storage tank. The parameters are Tc608C, Fc0.000015 m3 sÿ1, Td108C, V0.15 m3 (V is the sum of the tank sections in the case of divided hot water storage tank), Uc250 W mÿ2 Kÿ1, A0.5 m2, 1000 kg mÿ3, cp4200 J kgÿ1 Kÿ1, and for t < 5 104 s; 0 m3 sÿ1 Fl t ÿ5 3 ÿ1 for t 5 104 s 1 10 m s (see Figs. 7 and 8) The initial conditions are Ts(0)108C and T1(0) can be calculated based on Eq. (17) with the values of the parameters: T1 0 60 ÿ 10eÿ 2500:5=42000:0000151000 10 16:87 C: 3.3. Mixed storage tank model The next equation describes the storage tank unit supposed to be fully mixed thermally: Cs
dTs Qc;s ÿ ks As Ta ÿ Ts ÿ Qs;l dt
Fig. 7. T1 and Ts temperature response to Fl in the case of storage tank with heat exchanger.
(18)
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Fig. 8. Simulink model for Eqs. (15a) and (15b).
4. Simulation of a solar hot water system (HWS) Considering a complete solar hot water system based on the described collector and storage tank models a simulation can be carried out. 4.1. System description The main components of the solar hot water system are the solar collector and the storage tank unit. The flat plate solar collector produces heat from the energy of the sun absorbed by the absorber plate. The gathered thermal energy is stored in a hot water storage tank. The heat is transferred from the collector to the storage tank by the heat transfer liquid. In the closed solar loop the liquid is circulated by a pump unit. The scheme of the solar hot water system can be seen in Fig. 9. 4.2. Simulation of the system The system has three input parameters as ambient temperature, solar irradiation and the required mass flow of the loaded hot water. These parameters have been simulated based on the information about modelling the ambient temperature and the solar irradiation found in the relevant literature [1,2,3±7]. There are also national standards for hot water usage. Using the values of the standard a step function can be calculated which gives the required mass flow as a function of time. The mathematical model used to be for the simulations based on energy flow in the collector and the storage tank. Eqs. (12) and (18) are used for such purposes. The block-oriented solution of the mathematical model can be found in Fig. 10. This way of the solution is easy to survey and its modularity gives the possibility for easily changing the set-up. The first block gives the time steps for the input parameters. The next three blocks contain the input parameters as ambient temperature, irradiation and the load. Qc,s and Qs,l give the heat transfer between the collector and the storage tank and the heat taken by the load. Tc and Ts indicate the temperature of the collector and the storage tank. They contain Eqs. (12) and (18). The graph block gives the results in graphical form as can be seen in Fig. 11. 4.3. Identification of HWS Controlling the results of the model needs comparing measurements. The input parameters of a real application are stochastic ones. For the control the inputs must be deterministic and above all
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Fig. 9. The set-up of the simulated solar thermal system.
Fig. 10. Block-oriented solution of the model.
repeatable. These requirements can be satisfied only in laboratory circumstances. The laboratory equipment can be seen in Fig. 12. The system consists of a data acquisition PC with a software developed for this purpose, five Pt-100 thermometers for taking the temperature of the ambient air, the water at the bottom and at the top of the storage tank, the water in the collector at the incoming and at the outgoing side. The set-up also consists of a storage tank unit, a collector, a pump, a throttle valve to set the mass flow and a measuring meter. The connection between the PC and the Pt-100s is through ADAM data logging modules. For the easier check of the temperature in the storage tank the water is mixed to avoid the stratification. The laboratory equipment can be modelled by the same equations as the real applications. The only difference seems to be the constant input parameters. The measured values of the experiment carried out are shown in Fig. 13. Comparing the simulated and measured values the system parameters were identified. Both values can be seen in Fig. 14, which show a good coincidence. The irradiation was anyway 204 W mÿ2, the ambient temperature 218C, the storage volume 25 l, and the collector area was 0.5 m2.
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Fig. 11. Simulation results yield by the model.
Fig. 12. Set-up for the laboratory scale equipment.
5. Conclusions In this paper a mathematical model was developed and used for simulation of a solar hot water system (HWS). As a matter of models an ordinary flat plate solar collector and a simplified rubber made collector were considered and developed. Concerning to the hot water storage tanks three different cases were studied as a divided hot water tank, a storage tank with heat exchanger and a fully mixed one. For the simulation of the hot water making system a block-oriented approach was used and realised in MatlabSimulink software. To identify the system parameters a measurement was carried out. The modelling results achieved showed a fairly good coincidence with the measurements.
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Fig. 13. Results of the measuring experiment.
Fig. 14. Comparison of the measured and simulated water temperatures in the storage.
6. Nomenclature Ac As c cin cout C
aperture surface of collector (m2) outside surface of the storage tank (m2) specific heat capacity of fluid (J kgÿ1 Kÿ1) inlet fluid specific heat capacity (J kgÿ1 Kÿ1) outlet fluid specific heat capacity (J kgÿ1 Kÿ1) overall heat capacity of fluid (J Kÿ1)
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Cc Cs E Fc Fl F0 H I ks K m_ P Qc,s Qf Ql Qs Qs,l Ta Tabs Tav Tc Td Tin Ts T1 T2 U U* Uc UL Us V
overall heat capacity of collector (J Kÿ1) overall heat capacity of storage tank (J Kÿ1) total energy (J) volumetric flow rate of collector (m3 sÿ1) volumetric flow rate of load (m3 sÿ1) heat transfer factor between the absorber plate surface and the heat transfer fluid total enthalpy of the system (J) irradiance in plate of collector (W mÿ2) heat transfer coefficient between the storage tank and the surrounding air (W mÿ2 Kÿ1) kinetic energy (J) mass flow rate of fluid (kg sÿ1) potential energy (J) heat transferred from the collector to the storage tank (W) heat absorbed by the fluid (W) heat loss of the collector (W) solar energy absorbed by the absorber plate (W) extracted heat from the storage tank by load (W) ambient temperature (8C) temperature of the surface of the absorber plate (8C) average fluid temperature in the collector (8C) outlet temperature of the collector (8C) temperature of the supplied water (8C) inlet temperature of the collector (8C) temperature of the extracted water from the storage tank (8C) temperature of the fluid which returns from the hot water storage tank to the collector (8C) temperature of the extracted water from the divided hot water storage tank (8C) heat loss coefficient of collector (W mÿ2 Kÿ1) internal energy (J) overall heat transfer coefficient of the coil in the storage tank (W mÿ2 Kÿ1) overall heat loss coefficient of the collector (W mÿ2 Kÿ1) overall heat transfer coefficient between the two sections in the divided storage tank (W mÿ2 Kÿ1) volume (m3)
Greek symbols 0 in out *
45
absorbance of the plate of the collector (dimensionless) optical efficiency of the collector (dimensionless) density of the fluid (kg mÿ3) inlet fluid density (kg mÿ3) outlet fluid density (kg mÿ3) transmittance of the cover of the collector (dimensionless) residence time of the collector (s)
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Acknowledgements This work has been carried out within the framework of the project ``Investigation of the technical parameters and environmental influences of photovoltaic and thermal solar systems'' (Ministry of Education, MKM FKFP 1173/1997, 1997±1999). References [1] I. Farkas, Modelling and identification for control of solar and connected technological system, D.Sc. Thesis, Hungarian Academic of Science, Budapest, Hungary, 1993. [2] D.G. Erbs, S.A. Klein, W.A. Beckman, Estimation of degree dazes and ambient temperature bin data from monthly average temperatures, ASHRAE J. (1983) 60±65. [3] D.A. Kouremenos, K.A. Antonopoulos, E. Rogdakis, Performance of solar NH3/H2O absorption cycles in the Athens area, Solar Energy 39 (1987) 187±195. [4] H.P. Garg, B. Bandyopadhyay, V.K. Sharma, Investigation of rock bed solar collector cum storage system, Energy Conversion Management 21 (1981) 275±282. [5] I. Farkas, Z. Rendik, Handling of solar climatic data, Ambient Energy 14 (1993) 59±68. [6] ASHRAE Handbook, System and Application, Chapter 47, 1987. [7] J.A. Duffie, W.A. Beckman, Solar Engineering of Thermal Processes, 2nd ed., Wiley, New York, 1991. [8] European Simplified Methods for Active Solar System Designe, Bernard Bourges, Kluwer Academic Publishers for CEC, 1991. [9] J. BuzaÂs, G. van Straten, Solar hot water system study examples with MATLAB/SIMULINK, Tempus 9709-95 Report, 1996. [10] C. King, Solar Water Heating, European Commission Directorate-General XII for Science, Research and Development, 1995.