Validation of the mathematical model
For the validation of the developed mathematical model, the results of the DST tests of one thermosiphon system product line are used. Systems of this product line with different collector area and store volume were tested according to the DST method at the three laboratories CSTB, INETI and ITW. The results obtained from these tests were compared with the results that are obtained with the mathematical model. In Table 4, the test results based on the DST-test for the location of Athens and a hot water demand of 200 l/d are listed in the column fsol, DST.
In the same table, the results obtained with the mathematical model are depicted. The parameters of the different systems that have been entered in the program as “system tested” are defined in the line: “Input value fsol, DsT“. With the extrapolation tool DHWScale, the values fsol, calc are calculated for the remaining systems of the product line. In addition, the discrepancy between the calculated values of the solar fraction and the solar fractions obtained by the DST-method (Afsol = fSOl, DST - fsOi, caic ) and the relative error (srel) between both values is displayed. The relative error between the results obtained for fsol from the DST test and the calculation procedure is calculated by the following equation (5),
For the validation, three trials have been performed:
In the first trial (Trial 1) a thermosiphon system with Ac = 3.48 m2 and Vsto = 0.18 m3 (System no. 1) has been tested with the DST-method and a solar fraction of fsol, DST = 0.74 has been obtained. These values are now entered into the DHWScale program to extrapolate the results to systems of the same product line but different in size. For the systems 2 to 7 of the same product line the values of fsol can be found in the column “Trial 1, fsol, calc1”.
In a second trial it was assumed that a thermosiphon system with Ac = 3.96 m2 and Vst0 = 0.15 m3 (System no. 4) has been tested with the DST-method and a solar fraction of fsolDst = 0.70 has been obtained. Again, these values are entered in the DHWScale program. The results obtained for the other systems of the product line by means of extrapolation are listed in the column “Trial 2, fsol, calc1”.
For the third trial it is assumed that two thermosiphon systems (system 3 and system 7) have been tested with a DST-test. The parameters of both systems have been entered into the program as input data in order to obtain the solar fractions for the 5 remaining systems of the product line.
Table 4. Comparison of DST-results and the results from the mathematical model
As can be seen from Table 4, also for the system tested with DST, there is a discrepancy between the solar fraction obtained with the DST-method and the solar fraction obtained with the mathematical model. This results from the fact that there is no equation fsol = f (A, Vto) which exactly matches the default value from the DST-method.
The maximum relative error for Trial 1 is 5.9 % , 10.6 % for the second trial and 5.6 % for the third trial. The mean error, defined with
є1 +є2 +... + Є8
is 2.25 % (Trial 1), 5.9 % (Trial 2) and 3.0 % (Trial3). For the other hot water demands of 110 l/d and 300 l/d comparable results are obtained.
An extrapolation procedure including a mathematical model has been developed which can be used for determination of the solar fraction of systems which are part of a product line. By means of an extrapolation procedure, input values for only one or a few systems have to be determined by physical testing. The developed procedure has been implemented in an EXCEL based software tool named DHWScale.
As the model is based on second order polynomials, results can be obtained within seconds. The mathematical model has the advantage that only a few system tests are necessary to determine the solar fraction for a whole product line. This approach can offer the possibility to reduce the time and cost necessary to obtain Solar Keymark certification of factory made solar domestic hot water systems. The validation of the program with one thermosiphon product line tested by DST showed promising results. Depending on the desired accuracy already one system test may be sufficient (relative error of about 11 %). If two system tests are performed the relative error drops to 6 %. However, up to-date only DST-results for one product line are available. For a more profound assessment of the developed procedure more product lines have to be tested with the DST-method.
In principle it is possible to extend the DHWScale program towards other system designs such as forced circulated DHW systems with or without integrated auxiliary heating. First experiences have been gained with forced circulated DHW systems but a validation is necessary before any reliable statement about the accuracy of the results can be made. It is also expected that for other system types further parameters have to be taken into account such as the influence of the auxiliary heated volume. This will envoke a huge number of additional TRNSYS simulations.
The DHWscale program is a first approach for the determination of system test results by means of an extrapolation procedure. In principle the methodology of this approach can be extended to additional parameters (e. g. locations, loads) and other system concepts.
/1/ H. Druck, S. Fischer, H. Muller-Steinhagen,
Solar Keymark Testing of Solar Thermal Products; Proceedings of ISES 2007 Solar World Congress, September 18 to 21, 2007, Beijing, China, ISBN 978-7-302-16146-2, Tsinghua University Press, Beijing and Springer-Verlag GmbH Berlin Heidelberg, CD: ISBN 978-7-89486-623-3
/2/ DIN EN 12976, „Thermal solar systems and components - Factory made systems - Part 2: Test methods”
/3/ ISO 9459-5:1995, “Solar Heating - Domestic water heating systems - Part 2: Outdoor test methods for system performance characterisation and yearly performance prediction of solar system.
/4/ Research and experimental Validation on the DST Performance test Method for solar Domestic Water Heaters, Final Report, Contract No. SMT4-CT96-2067