E.2 CFD study with the four turbulence models selected
E.2.4 Solution algorithm and procedure, discretization schemes
The solution alghorithm used and the numerical schemes were the same as in Section 0.
The solution for the pressure was obtained with the PCG (Preconditioned Conjugate Gradients) method. The preconditioner was a GAMG solver. For the velocities and the turbulence quantities GAMG solvers were used.
During the calculations the under relaxation factors were changed two times. For the first 100 iterations the under relaxation factor for the pressure was 0.2, for the velocity 0.3 and for the turbulence quantities 0.4. These were changed to 0.3, 0.4 and 0.5 , respectively, for further 900 iterations. The last 19000 iterations were performed 0.3, 0.5 and 0.6.
It was ensured, that the variables monitored became stable and the residuals decreased by 5 orders of magnitude. All the variables were monitored at selected location with probes.
At the end of the calculations the global conservation of mass was checked.
E.2.5 MEASUREMENT METHODS
The geometry of the measurement stand was shown Subchapter 3.4. In overall three different measurement series were done:
- to determine the velocity profiles inside the duct, - to determine the static pressure inside the ducts, - to determine the flow rates through each nozzle.
APPENDICES
10 different measurement locations were made on the duct. These locations are shown in Fig. E.21. The measurement of the flow rates through the nozzles was detailed in Subchapter 3.4.
Fig. E.21. Measuring locations.
E.2.5.1 INLET FLOW RATE
The inlet flow rate of the rectangular duct was measured with the Venturi-tube shown in Subchapter 3.4. The uncertainty and the flow rate was calculated similarly as in the mentioned Subchapter.
E.2.5.2 VELOCITY PROFILES
The velocity profiles were measured according to the method described in the standard [94]
for flow rate measurements. The rectangular cross section was divided to equal areas, and the measurement were conducted in the centre point of these small surfaces (as shown in Fig. E.22). For most of the locations the measurements were performed in 25 points.
Fig. E.22. Measuring grid used for the measurement of the velocity profile in the 1st location.
As the width of the duct was decreasing, this number had to be decreased in some locations too. The actual number is shown in:
Table E.6. The width of the duct and the number of measuring points used at different locations.
Locations 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Width of the duct [mm] 285 259 230 201 174 141 112 81 50 19 Re Number of measurement points in different locations (verticalxhorizontal)
34000 5x5 5x5 5x5 5x5 5x5 5x5 5x5 5x5 6x4 11x1
APPENDICES
In most details the schema of the measurements was the same as shown in Fig. 3.10. In addition, the velocities were measured with a Testo hot wire anemometer (with a diameter of 7.5 mm). The accuracy of the hot wire probe is ±(0.03 m/s+5%·measured value). The systematic uncertainty of the velocity measurements with 95% confidence is therefore estimated to be -#./ = (0.03 0/1 + 5% · 02314526 73842)/ √3, according to [94]. In addition to that, the measurements have random errors, as well. The probe was connected to a Testo 435-4 multifunction indoor air quality meter and it was continuously logged with a PC, connected to the instrument with a USB cable. The sampling frequency was the highest possible, 1 Hz. The mean velocity is then calculated from the logged values, the logging was equal or longer than 120 s, which was found to be long enough. The random error of the mean velocity is calculated with:
n n σ
λ w
δ r st ⋅ w
+ −
= , 1
2 95 . 0
1 (E.8)
σw is the standard deviation, λst is the inverse value of the Student-distribution at the confidence level of 0.95 and degrees of freedom equal to n-1. n is the number of the samples.
The total uncertainty of the mean velocity is:
2
2 r
s δw
w δ w
δ = + (E.9)
The standard deviation has a physical meaning, as Tu can directly be calculated with:
w
Tu = σw (E.10)
The uncertainty of the standard deviation is calculated with:
n n σ
λ
δσw st ⋅ w
⋅
+ −
= 3
1 2 ,
95 . 0
1 (E.11)
The uncertainty of the turbulence intensity is thus:
2 2
1 2
⋅
+
⋅
= δw
w δσ σ
Tu w
δ w w (E.12)
E.2.5.3 STATIC PRESSURE PROFILES
For the static pressure a different measuring grid was used, as the Prandtl-tube used for these measurements was not long enough to reach the bottom of the duct. This grid for the first location can be seen in Fig. E.23.
Fig. E.23. Measuring grid used for the measurement of the static pressure in the 1st location.
APPENDICES
In this case only five locations (1,3,5,7,9) were investigated. The number of measuring points was 5x5 for each location. As the head part of the Prandtl-tube is long, in reality the static pressure was not measured in the locations shown in Fig. E.21 but 30 mm upstream. This was taken into account when the simulation results were exported.
A Prandtl-tube was used for the measurement of the static pressure was connected to a transducer, a Datcon DT700, which can be used for pressure differences up to 250 Pa with an accuracy of ±5%. The transducer was connected to an Alhborn ALMEMO 2890-9 data logger and the sampling frequency was 1 Hz.
The Prandtl-tube had a diameter of 7 mm. The main problem with the static pressure measurements is, that it is not known how the crossflow present in the supply duct due the continuous outflow of the air influences the results. Therefore the results have to be interpreted carefully.
As the Datcon instrument was inaccurate for these measurements, according to the factory default values, it was calibrated with a Betz micromanometer produced by ACIN instruments, capable to measure a maximum of 2500 and a minimum of -50 Pa. The calibration curves were determined in Matlab with linear-least squares and the lowest order best fitting polynomial was determined according to [147]. The simultaneous confidence bounds of the fit evaluated at the measured values were computed with the “polyconf”
function. The calibration curves were only determined for pressures below 100 Pa. The results can be seen in Fig. E.24.
Fig. E.24. Calibration results with 6th order polynomials from different dates for the Datcon DT700.
The confidence interval of the curve fit is magnified by a factor of 10!
The measurement uncertainty for the pressure measurements was calculated similarly to Eq. (E.9). The systematic error was derived from the confidence interval determined during the curve fitting.
E.2.5.4 TEMPERATURE MEASUREMENTS
The Testo hot wire can be used for air temperature measurements too, for which its accuracy
APPENDICES
temperature probe, which accuracy is ±0.1 °C. The probe was connected to an Alhborn ALMEMO 2890-9 data logger and the sampling frequency was 1 Hz. The mean temperature was calculated from the logged values and the uncertainty was estimated similarly to Eq.
(E.9).