Contribution by A. Altunkaynak, Istanbul Technical University, Turkey
In this paper the authors developed a fuzzy logic model to predict hydraulic jump aeration efficiency but parts of the paper appear to be ambiguous, there are points that may need further clarification in both the application of methodology and the results sections.
First, the most important step in developing a fuzzy logic model, which is a kind of blackbox model, is to train and test the model properly. For these operations, the available data are normally split into two parts. While approximately two-thirds of the total data are used for training, the rest of the data are left for testing. The authors of the paper do not appear to have followed this procedure. It may have been better if they had used the training data to determine the fuzzy rule based membership functions of model inputs (Fr1 and Re) and output by considering the least prediction errors and then validated the model using an independent data set such as test data. The absence of these procedures makes the results of the study questionable because it is a well known fact that validating a model with the data that are used at the same time for establishing the model produces high correlations coefficients between observed and predicted values.
The authors say that they used a trial and error approach to determine the membership functions, but they failed to explain how this was done. The authors refer to some papers in the literature which explain this issue in detail (Altunkaynak et al., 2005a; Altunkaynak et al., 2005b; Şen and Altunkaynak, 2004).
Second, the number of rules in the fuzzy rule base theoretically should be 5 × 5 × 7 = 175 which is the multiplication of input and output membership functions numbers. In the paper, however, the authors use only 21 rules to express the system behaviour. It would be helpful if the authors explained why they reduced the number of rules in the fuzzy rule base. In addition there is no indication of how they decided the number of input and output membership functions.
Third, although the authors give the equations of the multiple regression model, they do not compare the results with the fuzzy model results. It would be useful if such a comparison could be made.
The fuzzy logic approach is a very flexible and reliable method to determine the relationship between input and output variables, and is easy to use. The misapplication of fuzzy logic methods may, however, lead to misinterpretations of the results. This means that every step in the application should be applied carefully and properly.
Authors' reply
As the contributor stated, the data were not trained in the model, this is because there were limited experimental data and these values were used for the model validation. In further studies, however, the data sets of Avery and Novak (1978) and Wilhelms et al. (1982) could be used for the model calibration. The fuzzy logic rules were written on the basis of experimental observations and system behaviour. This issue is also known as expert decision and this enables the fuzzy logic algorithm to mimic human thinking.
In the model, upstream Froude (Fr1) and Reynolds (Re) numbers were selected as input parameters, where Fr1 is the dominant parameter. The studies demonstrated that the gas transfer efficiency of the hydraulic jump is under the control of these parameters. The Froude number was divided into five fuzzy subsets as follows: 2·3–2·4 very low, 2·4–3·4 low, 3–7 medium, 4–9 high, and 6–15 very high. These fuzzy subsets and membership functions were created based on the physical behaviour of the system. For example, in the experiments, air entrainment took place at Fr1 > 2·3 and the influence of Fr1 strengthened at Fr1 > 4. Above Re > 1×105, the Reynolds number effect was not taken into account, which represents the fully turbulent conditions.
The fuzzy rules were sufficient to relate the parameters and they captured the dynamic behaviour of the system (Kucukali and Cokgor, 2007). There was no need to write 175 rules, which was proposed by the discusser, because it will make the model unpractical and complicated. The Avery and Novak formula also gave sufficient results for the experiments. What this equation suffers from is the lack of constraints of Fr1 and Re numbers as discussed above. In the proposed model, however, these physical constraints were taken into account.
