Alternative Empirical Formula for Predicting the Frictional Drag Penalty due to Fouling on the Ship Hull using the Design of Experiments (DOE) Method

Biofouling is known as one of the main problems in the maritime sector because it can increase the surface roughness of the ship’s hull, which will increase the hull’s frictional resistance  and consequently, the ship’s fuel consumption and emissions. It is thus important to reduce the impact of biofouling by predicting the value of . Such prediction using existing empirical methods is still a challenge today, however. Granville’s similarity law scaling method can predict accurately because it can be adjusted for all types of roughness using the roughness function  variable as the input, but it requires iterative calculations using a computer, which is difficult for untrained people. Other empirical methods are more practical to use but are less flexible because they use only one  input. The variance of  is very important to represent the biofouling roughness that grew randomly. This paper proposes an alternative formula for predicting the value of  that is more practical and flexible using the modern statistical method, the Design of Experiments (DOE), particularly two-level full factorial design. For each factor, the code translation method using nonlinear regression combined with optimization of constants was utilized. The alternative formula was successfully created and subjected to a validation test. Its error, calculated against the result of the Granville method, had a coefficient of determination R²= 0.9988 and an error rate of ±7%, which can even become ±5% based on 93.9% of 1,000 random calculations.

Added frictional resistance; Biofouling; Design of experiments; Empirical formula; Ship resistance

       The impact of fouling or biofouling on ship performance is important (Molland et al., 2014). Biofouling makes the hull’s surface rough, and hence, increases its frictional resistance ( ), which becomes a drag penalty that increases fuel consumption. As a result of biofouling, the fuel consumption could increase by up to 20% (Hakim et al., 2019); in fact, in one year, total losses from fuel waste due to biofouling reached up to $56 million (Schultz et al., 2011). By increasing fuel consumption, biofouling also contributes to increasing CO₂ emissions and global warming. Moreover, biofouling mediates the distribution of invasive species that can damage the water ecosystem structure (Ulman et al., 2019). To prevent these unwanted problems due to biofouling, a more efficient hull may be designed (Sulistyawati and Suranto, 2020) or a more efficient propeller   (Abar and Utama, 2019),  or a device may be installed (Suastika et al., 2017), but the easiest solution is to predict the impact of biofouling.

When the fluid passes through the rough surface, the turbulence boundary layer structure will be shifted downward. Mathematically, the value of the downward shift can be estimated using what is called a roughness function [ ], which is a function of the roughness length scale (k⁺). The form of the roughness function varies widely, depending on the type of roughness, including the pattern, density, geometry, and other aspects of the roughness (Chung et al., 2021). To find out accurately the form of the roughness function, the roughness must be tested first (Speranza et al., 2019) by conducting an experiment (Monty et al., 2016), a numerical simulation (Jelly and Busse, 2018; Suastika et al., 2021), or in-situ measurement (Utama et al., 2018). Then, the results of many tests can be synthesized into formulas or diagrams that can be used as an empirical method. As we know, the empirical method is the easiest, fastest, and cheapest method to use as an initial predictive tool.

Each of the existing empirical methods is challenging to use. While the similarity law scaling boundary layer method of Granville (1958, 1987) yields accurate results because it can accommodate all types of roughness by entering the  and  of the desired roughness, it requires iterative calculations on a computer, which makes it difficult for untrained people to use. The formula of Bowden and Davison (1974), and the formula of Townsin et al. (1982) and Townsin (2003), calculate  easily, but they are applicable to only one type of roughness function []. Besides, for the roughness height parameter, only a single parameter—the average hull roughness (AHR)—is used, whereas in biofouling, the roughness is very random, (especially biofouling), such that the density, shape, and pattern must also be considered to achieve an accurate prediction result (Chung et al., 2021). Finally, the method of reading the diagrams introduced by Demirel et al. (2019) is very easy to use, but if the value being determined is unavailable, it still needs to be interpolated or extrapolated. Moreover, the diagrams accommodate only one type of roughness function, that of Schultz and Flack (2007), when several types of roughness functions are most often used, namely, those of Colebrook (1939), Nikuradse (1933), and their derivatives (Grigson, 1992; Cebeci and Bradshaw, 1977; Schultz and Flack, 2007; Demirel et al., 2017a).

        Therefore, this paper proposes an alternative formula for predicting the value of  that is easy to use and flexible because it can accommodate several types of . This formula was established with the help of the Design of Experiments (DOE) method, which is a branch of modern statistics. The DOE is known to be useful for modeling with small amounts of data and even with many parameters (factors) (Lye, 2002). The type of DOE used in this study was the two-level factorial design with four factors, followed by factor code translations using the nonlinear regression and optimization method. To our knowledge, factor code translations are rarely used. Some statistical software that we often encounter also do not do factor code translations but stop at the result of a formula whose input factor is still a code (-1 or +1), which is not the actual value of the factor. Islam and Lye (2009) predicted the value of the hydrodynamic performance of the propeller without translating the factor code to the actual value, so their resulting formula became difficult to use. Therefore, in this study, we developed a different formula for predicting the impact of biofouling. We tested the result of the formula against the result of the similarity law scaling method of Granville (1958), which was used with iterative calculations. The error rate was calculated from all the error results of 1,000 random calculations.

This paper described the process of establishing an alternative formula for the prediction of the increased frictional resistance ( ) of a ship’s hull due to fouling. The design of experiments (DOE) method was used, followed by factor code translations via nonlinear regression and the optimization method. It was found that some factors and interactions of factors affected the response while others did not. The most influential factor was the roughness height k. Then, the formula was created while still inputting the code of the factor (Equation 9), after which the codes were translated into functions (Equations 10–13) that represented the actual value of each factor. The functions were substituted in Equation 9 to come up with the final alternative formula in Equation 14.

The alternative formula was validated by comparing its calculation result with that of the Granville method and computing the error. The results were quite good, with values of R²= 0.9988 and y = 0.9672 + 0.1115, as described in Figure 6. The error distribution is illustrated in Figure 7 and shows that 93.9% of the 1,000 data calculated had a ±5% error risk. The possible cause of this error is the less than perfect process of matching functions during the code translation (Figures 4–5). Of course, this equation can be refined further.

We should be grateful for the DOE, followed by the translation of factors, for allowing the creation of a formula that can calculate a response with good accuracy using minimal initial data. The initial data were generally obtained from measurements in the field, laboratory tests, or numerical simulations, all of which required resources. The resulting formula was also quite easy to use.

        Using this alternative formula, predicting the increased frictional resistance of ships due to fouling will be easier, faster, and cheaper. The formula’s error rate, which the author considers still quite good, makes the formula suitable as an initial tool for determining how much impact fouling has on ship performance. In addition, this formula has considerable flexibility in the type of roughness function it can be applied to because of its roughness constant variable C_s. The roughness constant is known to be needed because roughness (especially due to biofouling) is very diverse and even random, so it must be represented not only by the measuring height (k) but also by other factors (such as the density, shape, and concavity). Although the values of k and C_s are not easy to determine in the field case, Chung et al. (2021) can provide insights on how to do it. By predicting the impact of biofouling, it is hoped that all parties involved in maritime activities can anticipate and address problems that arise from it.

        This research project was supported by the Ministry of Research, Technology, and National Innovation and Research Agency (Kemenristek – BRIN) of the Republic of Indonesia under Master to Doctorate Program for Excellent Graduate (PMDSU) scholarship program batch III (Contract No. 1277/PKS/ITS/2020).

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