Tutorial - Page 26

Sub-problem 6.1: General prediction characteristics of 3 literature correlations on frictional pressure drop based on the whole integrated databank

1) Start a new query by selecting Start new query .

2) In the Hydrodynamics tab, check the box corresponding to Frictional pressure drop and Execute. A total of 7360 frictional pressure drop measurements were extracted from the integrated databank and will be used to analyze the literature correlations (Attou et al. (1999), Larachi et al. (1991), and TB_ANN09). Informations about the models/correlations available in the Simulation section could help the user to do the pertinent selection.

3) Select the Plot button to enable the plot creator.

4) In the Horizontal axis panel (Standard tab), click on the knob corresponding to Hydrodynamics, then on the knob corresponding to Frictional pressure drop. Highlight Experimental.

5) In the Vertical axis panel, click on Hydrodynamics, then on Frictional pressure drop. Highlight Attou et al. (1999), Larachi et al. (1991) and TB_ANN09 (N.B. To select several items, click on one item (i.e. Attou et al. (1999)), hold the Ctrl button on the keyboard and click on Larachi et al. (1991) and on TB_ANN09).

6) The diagram is composed of 3 panels, namely Parity plot, Error CDF and Statistics.

7) Parity plot offers the same options as the normal XY plots elaborated in previous case problems with the addition of two options & which are described below. The Parity plot panel contains the parity diagram with a legend, a diagonal line and error envelopes. The parity plot series for each studied correlation on the 7360 available experimental data are provided on the diagram. Here are some procedures for an efficient use of parity plots.

On the latter diagram, the upper envelope is said to be +1.0 1, which is equivalent to the square Root of the Mean Square Error (RMSE) for the model 1 (Attou et al. (1999)). To change the envelopes characteristics, select the Edit envelopes button

. The following dialog window will appear. The user may choose between No envelopes, RMSE envelopes and Relative envelopes. By default, RMSE envelopes based on the first correlation in the legend are drawn.

The RMSE value has been calculated for each correlation and presented in Statistics panel. In the present case, the RMSE based on 2720 values for Attou et al. (1999) model is 29030.9. Therefore, the RMSE envelopes correspond to (X=X; Y=X+29030.9) and (X=X; Y=X-29030.9).

The user can also modify the RMSE envelopes between 0.1 RMSE and 9.9 RMSE for his own purpose.

If the user changes the XY normal scaling into a logarithmic scaling, the RMSE envelopes will automatically disappear since it can not be represented in such a way. Instead, the Percentage envelopes based on Attou et al. (1999) Average Absolute Relative Error (AARE) will take over. This value corresponds to the average value (field) in the third table for Attou et al. (1999). This value corresponds to 0.9162 or 91.6%.

The positive Relative envelope can be settle between 1% and 200% while the negative Relative envelope varies between -1% to -99%.

In the combo box above the RMSE envelopes option, the user have the opportunity to choose between the 3 correlations. For example, it is expected that by choosing TB_ANN09 the envelopes will come closer to the diagonal (RMSE = 23137.28) as shown on the following diagram. Select TB_ANN09 and the RMSE envelopes option, then press OK (Note: envelope titles become +1.0 3 and -1.0 3 since TB_ANN09 is the third correlation in the legend).

In the Edit envelopes window dialog, choose the Relative envelopes option for the TB_ANN09. By default, the envelopes are settled at 28%, which is the AARE of the correlation. Press OK.

Change the normal scaling into logarithmic scaling. From Edit graphic properties , select the Axis tab and check the box corresponding to Logarithmic scale in the Horizontal tab. For parity plots, the logarithmic scaling for the vertical axis is automatically activated. Press OK.

Select the Edit envelopes button. The RMSE envelopes option have disappeared. Adjust the percentage envelopes to +30% and -50%. Press OK.

Another functionality, which is referred as Show only valid points , is included for parity plot analysis. By default, all simulated data even those invalid based on the correlation applicability standards are drawn on parity plots. For each series, the valid predictions are presented as full points while invalid predictions are marked as hollow points. If the user select the Show only valid points button, all the invalid predictions (hollow points) will be removed from the diagram. The application of this functionality will produce the following diagram.

CONCLUSION #1: From the parity plot point of view, the TB_ANN09 correlation seems much more homogeneous compared to the other two correlations. Based on the valid points, Larachi et al. (1991) under-predicts a majority of experimental points. It may point out to an inherent limitation of the correlation on particular reactor conditions even if the validity specifications are met for these cases. Generally speaking, TB_ANN09 produces more accurate predictions, but it is surely possible that for some operating conditions and bed specifications, the Larachi et al. (1991) or Attou et al. (1999) would become more efficient. Further investigation would have to be done for specific cases.

8) Error CDF (or Cumulative Distribution Function) diagram disposes the Absolute Relative Error (ARE) in an increasing progression (lowest ARE to highest ARE values for the studied database) on the X-axis as a function of the data rank percentage on the Y-axis (ARE rank divided by the total number of data).

From the latter diagram, select the Error CDF panel. The following diagram will be presented to the user.

The number of curves corresponds to the number of correlations/models drawn on the parity plot. In this case, three curves are presented to the user.
Based on the 7630 data sets, the studied correlations have created in some cases large ARE values. By default, the X-axis scale is adjusted in consequence to accommodate all the curves in their entirety. However, this diagram is not fully descriptive in this form, and it becomes necessary to change the X-axis scaling. To achieve this task, select the Edit graphic properties button, then select the Axis tab. Deselect the Scale automatically box in the Horizontal tab and enter a Minimum range value of 0 and Maximum range value of 100. Press OK. The user has now access to a more explicit CDF diagram.

The user should be aware that the present CDF diagram is based exclusively on the valid points since the Show only valid points button was activated previously in the parity plot. As a matter of fact, the Parity plot, Error CDF and Statistics panels are inter-connected on this aspect. In the Error CDF panel, the user can unselect the valid points option by pressing once more the Show only valid points button. After resetting the X-axis scale, the following diagram is provided.

CONCLUSION #2: Generally, the CDF diagram including the invalid predictions and the CDF diagram without the invalid predictions exhibit pretty much the same characteristics. Based on the last CDF diagram (with invalid predictions), 80% of the studied data presents an Absolute Relative Error (ARE) below 40% for the TB_ANN09 correlation. On the other hand, only 38% of data for Larachi et al. (1991) correlation and Attou et al. (1999) model were predicted below an ARE of 40%. It further corroborates Conclusion #1, establishing that TB_ANN09 generally produces better predictions compared to the other two correlations.

9) The Statistics panel provides an assortment of statistical information associated with the points drawn on the parity plot. The Attou et al. (1999) model is applicable on 2790 points (see info while the two other correlations are applied for the hole data (7360) extracted from the search result. May be this 2790 points correspond to low interaction flow regime for which the Attou et al. model is valid. This hypothesis could be verified by a discrimination on the flow regime as it is shown in the STEP 10.

The statistical calculations obtained for the three models are presented as below :

The first table, referred as Data point statistics, provides information on the correlation prediction general ability. It includes the number of Points accessed on the diagram, the square Root value of the Mean Square Error (RMSE), the correlation coefficient, , the determination coefficient, DC, and the Bias.
The second table, referred as Relative error (RE) statistics, provides information on the relative error between the predictions and the experimental values ((prediction value - experimental value) / experimental value). It includes the number of Points accessed on the diagram, the Minimum relative error value, the Maximum relative error value, the Average () of the Relative Errors (ARE) and the standard deviation () on the ARE. When the ARE () is negative, it means that the correlation globally under-predicts the experimental value while positive ARE () exhibits over-prediction characteristics.
The third table, referred as Absolute relative error (ARE) statistics, provides information on the absolute relative error between the predictions and the experimental values (positive value of (prediction value - experimental value) / experimental value). It includes the number of Points accessed on the diagram, the Minimum absolute relative error value, the Maximum absolute relative error value, the Average () of the Absolute Relative Errors (AARE) and the standard deviation () on the AARE.

10) 2790 data estimated by the Attou et al. (1999) model correspond to the low ineraction flow regime for which this model is applicable. To verify this, in Parity plot panel, press the Discriminate series icon. Activate the Code Discrimination option and highligth General flow regime.
In the statistics table, 0 points for the High interction flow and Transition low/high interaction flow means that no frictional pressure drop were estimated for these flow regimes. Then, 2720 = 2211+499 are the number of points estimated for the low interaction flow (N.B. Not available (flow) means that this information is not experimentally available in MRD_integrated databank but evaluated by the default model (i.e. TB_ANN10) during the automatic simulation using all models/correlations).

CONCLUSION #3: The statistical tabulation above, which by the way includes the invalid predictions, showcases without a doubt that the TB_ANN09 correlation predicts better the frictional pressure drop. It has the lowest RMSE at ca. 23000 Pa/m, the closest correlation coefficient to 1 (0.96) and the lowest AARE (in the third table) at 0.28. On the other hand, the Bias, the second table features and the AARE standard deviation (in the third table) are inconclusive compared to the other correlations. Moreover, the Attou et al. (1999) model is applicable only for the low interaction flow regime.

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