Category Archives: Particle Image Velocimetry (PIV)

VERIFICATION OF THE CONFIDENCE LIMIT

Assume the case that the measured efficiency ^me is equal to the modeled efficiency ^mo during the whole test, we will get a straight line as presented in Figure

1. This line we call identity line. In the real case (Figure 2 and Figure 3) ^me and ■qmo are different. Measured and modeled efficiency can be determined with their uncertainties for a chosen confidence limit. A confidence limit of 95% is chosen here. Combining the uncertainties of ^me and ^mo we get an uncertainty ellipse for each measuring point. The uncertainty ellipse shows, with a probability of 95%, where the single measuring point can be located. If the uncertainty ellipse has an intersection with the identity line, we assume for this measuring point that the measured efficiency is equal to the modeled efficiency. To prove the value and the confidence limit of the estimated uncertainties, we tested for which percentage of the pairs (measured / modeled efficiency) the uncertainty ellipse (with 95% confidence limit) covers the identity line with:

Л measured = ‘Л modeled (see figure 1).

The intersections of the ellipse were calculated using equation (6).

SHAPE * MERGEFORMAT

SHAPE * MERGEFORMAT

i)MeSP [2] : intersection of the identity line with the elipse

VMe [2]; ЛмОо [2] : pair of the modeled and measured eficieny

UriMfe[i] ; Um [2]: measured and modeled uncertainties of the eficiencies

The modeled uncertainties for the i-th data point were calculated with:

UX1[;]….. UX6[;] : uncertainity of the variables;

UX1[;]….. UX6[;]: uncertainity of the coeficients

The uncertainty of the coefficients is taken from the vector C as discussed above for the WLS method. For the LS method we take the uncertainty of the coefficients from the result of the Excel™ spread sheet program.

We calculated the measured uncertainty for the i-th data point with equation (8).

Um [2]: Uncertainty of the collector power

Ug [2] : Uncertainty of the global radiation Ua [2] : Uncertainty of the collector area

Standard transducer uncertainties as defined by EN 12975 [ 1 ] are used for the calculation of equations (7) and (8).

For about 4,5% with the LS method and for about 7% with the WLS method the uncertainty ellipse does not cover the identity line (expected value is 5%). In the examples discussed here, we tested 134 of the efficiency pairs from one collector test. To get the 95% confidence limit result with both methods, we had to increase the uncertainty of the pyranometer from 10 W/m2 to 10 W/m2 + 1%. This indicates that the estimated uncertainty assumes the same value, with the same confidence limit (95%), with which the uncertainty was calculated using the WLS and the LS methods. The outliner points for the Ls and the WLS methods are indicated in Figure 2 and Figure 3.

THERMAL PERFORMANCE OF THE COLLECTORS

The instantaneous efficiency and the incidence angle modifier can represent the thermal performance of the collector.

4.1 INSTANTANEOUS EFFICIENCY

The instantaneous collector efficiency, q,, is a measure of collector performance that is defined as the ratio of the useful gain over some specified time period to the incident solar energy over the same time period [1]:

By introducing the Equation 3.2.5, the instantaneous efficiency becomes [1]

П; = Qu/AcGT = [Fr (та ) — FrUl (T; — T) Gt] (4.1.2)

Where the absorbed energy Sc based on the gross collector area has been replaced by

Sc = Gt (та) (4.1.3)

(та) is the effective transmittance-absorptance product based on the collector gross area. It is defined as:

(та) = SAp / GtAc = (та )avg Ap/Ac (4.1.4)

In Equation 4.1.2, two important parameters, Fr (та) and FrUl, describe how the collector works. Fr (та) indicates how the collector absorbs energy while FrUl is an indication of how energy is lost from the collector.

4.2 INCIDENCE ANGLE MODIFIER

To express the effects of the angle of incidence of the radiation on thermal performance of the flat-plate solar collector, an incidence angle modifier KTa is employed. This describes the dependence of (та ) on the angle of incidence of radiation on the collector. It is defined as [1]:

KTa = (та) /(та)п (4.2.1)

Where subscript n indicates that the transmittance-absorptance product is for the normal incidence of solar radiation.

Relevant Dynamic Processes under Real Collector Operation

During operation of real solar thermal systems, the following dynamic processes are relevant.

• Fast fluctuations of irradiance on a time scale of a few minutes,

• fast decreases of the operating temperatures caused by draw-offs (also on a time scale of minutes),

• slow increases of operating temperatures over several hours due to the warming of the contents of the storage tank,

• the switch-on process of the solar circuit pump.

Which value of an effective one-node collector capacity best describes these four processes? Is there a unique value at all for all the processes?

In order to answer these questions, the mentioned processes are (just like the procedures for determination above) analysed qualitatively, based on fundamental physical arguments. Again the dynamic behaviour of Tabs and TF is investigated, as their amplitudes correspond with the effective capacity (see also figure 2).

Fast fluctuations of irradiance are similar to the processes of the J.3-procedure. Consequently, for the amplitudes ATabs >> ATF holds. A high effective thermal capacity best describes the process.

After a draw-off during collector operation, Tabs and TF decrease. As at the same time the collector performance will increase slightly (as the point of operation on the collector
efficiency curve is shifted to the left), Tabs-TF also increases, and hence the amplitude ATabs will be smaller or almost equal to ATF; so the result is ATabs < ATF. This process is best described by a low effective capacity.

When the storage tank is charged, all the system temperatures (including Tabs and TF) will slowly rise. The relation between ATabs and ATF depends on the (slow) dynamics of the irradiance G. If G is more or less constant or decreases, the collector thermal performance will slowly decrease, and ATabs < ATF holds. For increasing irradiance, the amplitudes will be similar: ATabs « ATF. Hence the corresponding effective capacity is low.

The switch-on process shows parallels with the draw-off, since colder fluid suddenly enters the collector. Again the result is ATabs < ATF, and the best suiting effective capacity is low.