Development of an empirical model for the prediction of the sound absorption coefficient for thin and low-density fibrous materials

2.1. Thickness and bulk density experimental measurements and results

The mass of each sample was determined using a calibrated KERN PFB 2000-2 scale with an accuracy of 0.01 g and precision of ±0.03 g. Thereafter, the volume of each sample was calculated using the dimensions of the sample which were measured using a vernier calliper. From the mass and the volume, the bulk density of each sample was then determined and presented in Tables 1-4.

2.2. Airflow resistivity experimental measurements

The airflow resistivity tube presented in Fig. 2 was designed, developed and calibrated according to the ISO 9053-1 first edition 2018-10. The tube was manufactured from poly (methyl methacrylate) also known as plexiglass or acrylic glass. The air supply to the device was obtained from a central air compressing unit and was filtered before use. A pressure regulator to regulate the pressure coming into the flow meter was attached to the inlet line. A testo 512 pressure meter (0-200 Pa), with a resolution of 0.1 Pa, was used to measure the differential pressure as required by the ISO standard. A KOFLOC flow meter model RK120X series was used to regulate the inlet flow. This flow meter can measure flows as low as 5 ml/min, which is far lower than what is required by the ISO standard. A MaxiMet (GMX501) Compact Weather Station was utilised to monitor the ambient temperature, pressure and humidity in the laboratory where testing was conducted. The MaxiMet Compact Weather Station provides pressure (accuracy of 0.1 hPa), temperature (accuracy of 0.1 °C) and accurate humidity data.

The experimental testing began by preparing the local environment (the laboratory) for testing. This was achieved by ensuring that the lab temperature was constant. Hence, the air conditioning units in the laboratory were turned on at a temperature of 22 °C several hours before testing began. This allowed for the temperature of the room to stabilise. The order of testing for the specimens was random. This is important in order to eliminate the effect of any nuisance variable that may influence the observed airflow resistivity [9]. Therefore, since the testing was randomized and the environment, in which tests were performed, was as uniform as possible, this experimental design is a completely randomized design. The airflow resistivity of each sample is presented in Tables 1-4.

Fig. 2Airflow resistivity experimental setup

2.3. Sound absorption coefficient experimental measurements and results

Before the testing commenced the laboratory environment was given enough time for the ambient temperature to stabilise to approximately 22 °C. This temperature was maintained by using two air conditioning units. All experimental testing was carried out over a three-day period. The laboratory temperature, pressure, and humidity were monitored and captured for each test using the GMX501 Compact weather station. The temperature in the laboratory fluctuated no more than 2 °C during the three-day testing period. Microphone amplitude and phase calibration tests were performed according to the ISO 10534-2 standard. Furthermore, a reference test was performed using a manufacturer's sample. This was done to validate that the impedance tube using the two-microphone transfer function method was indeed capturing the data accurately. The setup is illustrated in Fig. 3.

Fig. 3Assembled impedance tube

Once the experimental setup was ready testing proceeded. The data was captured utilising a Coco-80 dynamic signal analyser and saved to an SD card. The noise source used was a 2426H JBL professional series compression driver speaker. The microphones used were ICP with model number 130E20. The captured data was then downloaded from the SD card into the Engineering Data management (EDM) software, where it was converted into excel format in order to be post-processed in MatLab using code that was developed for this research. The data for the sound absorption coefficient of the tested materials are presented in Table 1-4, in terms of the Noise Reduction Coefficient (NRC), for brevity. Also, it should be noted that all measurements below 100 Hz, were removed from the data set due to inaccuracies observed in the data below this frequency limit. The reason for these inaccuracies occurring in the data below 100 Hz are most likely due to the spacing between the microphones on the tube. This is because the measurement frequency range is dependent on the microphone spacing. As the spacing between the microphones reduces the accuracy of low-frequency measurements is limited but the accuracy of high-frequency measurements is improved.

3.1. Sound absorption coefficient collinearity check

A collinearity analysis was performed on the sound absorption coefficient data using Statistical Analysis System (SAS) software [12]. As can be seen from Table 5, all the predictor variables had a Variance Inflation Factor (VIF) of less than 10, hence no collinearity was detected. This is important since collinearity occurs when there is a correlation present among predictor variables in the model. If the predictor variables are not correlated, then there's no collinearity present in the model. Collinear predictors provide redundant information and therefore cause instability in the model by inflating the variance of the parameter estimates, which raises the p-values. Hence, it is necessary to check for collinearity between predictors and remove redundancies.

3.2. Selection criteria

There are several selection criteria that can be used for model evaluation and selection; 1. significance levels ($p$-value), 2. Information criteria (AIC – Akaike’s information criterion, AICC – corrected Akaike’s information criterion, BIC – Sawa Bayesian information criterion, SBC – Schwarz Bayesian information criterion), 3. Adjusted R-squared values. These criteria will now be applied to the various datasets to estimate which predictor variables are strongly correlated to the dependent variable. Predictor variables that are weakly correlated will be eliminated.

3.3. Predictor identification and selection

For the sound absorption coefficient, six possible predictors were chosen i.e., frequency, thickness, porosity, fibre diameter, bulk density, and airflow resistivity. Therefore, there are $2^6=64$, possible models. This is a large number of possible models, and it is not practical to test the performance of each one, thus it is necessary to eliminate the least significant predictors. Therefore, STEPWISE model selection techniques were implemented using SAS software. The $p$-value, AIC, AICC, BIC, and SBC information criterion were all implemented for predictor selection. Fig. 4, lists the variables that were candidates for entry into the model at this step based on their significance level. It can be seen that frequency is the first predictor to enter the model.

Fig. 4p-value significance of variables in sound absorption coefficient model

In Step 2, airflow resistivity entered the model. At this point, the selection method checks whether the first predictor, frequency, has become non-significant and if so, removes it. Frequency remained significant as expected since the sound absorption of a material is highly dependent on the frequency. The stepwise selection summary, Table 6, contains each variable that was added at each step of the process.

As can be seen from Table 6, porosity is missing. Porosity was shown to be not statistically significant and therefore removed as a possible predictor. Next, the Coefficient Progression for the sound absorption coefficient is determined which is illustrated in Fig. 5.

Fig. 5Selection fit criteria for sound absorption coefficient model

Fig. 5 shows the effect each variable has on the model and how the model changes as new variables are entered. Furthermore, it can be seen that at the start of the analysis, the airflow resistivity (dark brown line) had a significant effect but as the analysis progressed and more variables are added the overall effect airflow resistivity had on the model decreased.

Thereafter, the set of fit criteria, AIC, SBC, AICC, and adjusted R-square values for each step are plotted and compared side-by-side as can be seen in Fig. 6. A model containing frequency, airflow resistivity, thickness, fibre diameter and bulk density, as predictors, is predicted by all model selection fit criteria, AIC, AICC, SBC, and the adjusted R-square value to be the best.

Fig. 6Selection fit criteria for sound absorption coefficient model

The Average Squared Error (ASE) is then calculated and plotted as illustrated in Fig. 7.

It can be seen from Fig. 7, that the ASE levels out after thickness is added to the model. Furthermore, the addition of the fibre diameter and bulk density decreases the ASE very little, this is most likely since airflow resistivity is highly dependent on these two variables.

Furthermore, several information criteria tests i.e., AIC, AICC, BIC, and SBC, were also conducted. All, the criteria tests uniformly agreed with the results obtained from the $p$-value criterion selection test, this is to say that AIC, AICC, BIC, and SBC ranked the model containing frequency, airflow resistivity, thickness, fibre diameter and bulk density as the best. The results for all the information criterion selection tests are not displayed for sake of brevity.

Fig. 7Progression of average squared error for sound absorption coefficient model

3.4. Sound absorption coefficient model selection

The model selection process requires applying the data to each model and analysing which model gives the highest adjusted R-squared value. During this process, every possible combination of the variables was checked to see which combination would give the highest adjusted R-squared value for the dataset, Microsoft Excel was utilized to perform this. The result of this process is expressed in Tables 7, 8.

From Table 7, it can be seen that the 3rd order polynomial with three interactions model gives the highest R-squared value. An interesting point to note is that this model is only dependent on frequency, airflow resistivity and thickness. This model has ten terms, however, despite the large number of terms, this model is surprisingly still simpler than current models. The final model will be selected later since further analysis for determining the best model is still necessary.

3.5. Sound absorption coefficient model comparison

Fig. 8 displays the percentage difference between the measured and predicted sound absorption coefficient models developed in this research. Note, two validation datasets were used in this research, an internal dataset which was a subset taken from the main dataset and an external dataset which was compiled using literature data. The data for Fig. 8, was obtained by running the validation dataset in Table 9 through each model and then calculating the percentage difference between the NRC measured value and the NRC predicted value. The NRC was used since it is an average sound absorption coefficient value thus making it possible to compare the performance of each model.

From Fig. 8, it is evident that the exponential model overall attained the lowest percentage difference between the measured and predicted values for the internal and external datasets.

Fig. 8Sound absorption coefficient model comparison using internal and external data dataset

Table 10 evaluates all six sound absorption coefficient models that were developed. A selection metric has been developed as seen in Eq. (1). The selection metric attempts to give an objective value that can help identify the so-called “best” model:

where $R^2$is the coefficient of determination, $k$ is the number of predictors in the model and $P_D$ is the percentage difference between the measured value and the predicted value. It can be seen that the average Selection Metric for the exponential model is 1.58, which is 28.5 % higher than the next closest, which is the 1st order polynomial model. Hence, the Exponential model is selected as the best model.

3.6. Validating model assumptions

Before a model can be used for future predictions the model assumptions must be validated. The assumptions of linearity, independence, normality, and homogeneity of variances were all validated. Furthermore, the influential observations were tested and analysed. No influential data points were found when applying the Cook’s distance, using the software NumXL [13], in the dataset and hence no model adjustment was necessary.

Model linearity was validated using scatter plots of the response versus the predictor variables. The normality assumptions were validated by making use of a histogram plot of the residuals. The independence and equal variance assumptions were validated by making use of a residual plot of the errors.

4.1. Derivation of sound absorption coefficient empirical equation

From the above analysis, it was shown that the variables frequency, airflow resistivity, thickness, bulk density, and fibre diameter are all statistically significant and therefore should be included in the sound absorption coefficient model. Therefore, the regression function in Microsoft Excel was utilized to do a regression analysis on the predictors selected. From the previous section, it was found that overall, the exponential model performed the best and hence was selected as the model of choice. Also, it must be noted that upon further analysis, it was found that including the airflow resistivity in the exponential model yielded no improvement and therefore was removed as a predictor. The derivation of the sound absorption coefficient regression model will now be carried out.

The exponential model to be derived is nonlinear and therefore needs to be transformed into a linear form. The nonlinear form of the equation is given in Eq. (2):

Simplifying the right side:

Then applying the law of logs:

So that in the linear form:

Then substituting the independent variables that were selected i.e., frequency, fibre diameter, bulk density, and thickness, into Eq. (5), the formulation appears as follows:

Applying the natural log to the independent data in order to transform the data to a linear form is now applied. Since there is a lot of data an example only using data from one sample will be given.

Applying this transformation to the data and running a regression analysis in Microsoft Excel yields the regression model data illustrated in Table 13.

It can be seen from Table 13, that all variables are significant with $p$-values much less than the significance level criteria of ${\alpha }_p=$ 0.05. Also, the adjusted R-squared value is 0.792, this is rather low, since using a 3rd order polynomial regression model to predict the sound absorption coefficient yields an R-squared value of 0.931 which is significantly higher. Nevertheless, it was shown in the previous section that the exponential model outperformed the 3rd order polynomial regression model when it came to prediction accuracy, hence the reason it was chosen. Now substituting the coefficients from Table 13 into Eq. (8), the following equation is derived:

Then transforming back to the nonlinear form, the following equation is arrived at:

where $\alpha$ is the sound absorption coefficient, $f$ is the frequency in Hz, $t$ is the material thickness in mm, $d_f$ is the fibre diameter in μm, and ${\rho }_B$ is the bulk density in kg/m³.

4.2. Model comparison using validation datasets

In this section, the model developed in this research is compared to the currently existing models using both the internal dataset and an external dataset. From this, the model performance is analysed.

Table 14 gives the model performance of the currently available sound absorption coefficient models for nonwoven fibrous materials. It can be seen that overall, the Mechel model performed best when compared to current models, with an average noise reduction coefficient percentage difference between the actual and predicted value of $30.86\%$. MatLab was used to perform all the calculations of the various models.

The average prediction error for the developed sound absorption coefficient exponential model compared with the two current best-performing models can be seen in Fig. 9. It is immediately evident that the developed sound absorption coefficient exponential model outperforms the currently available models on both datasets.

Fig. 9Viable model comparison on internal and external data

The literature data used to produce the external data graph in Fig. 9, was obtained from the following reference: Ballagh [23], Garai and Pompoli [19], Li et al. [24], Liy et al. [21], and Yang et al. [25].

Fig. 10 visually demonstrates the prediction accuracy of the developed sound absorption coefficient exponential model on four different materials from the internal validation dataset. As can be seen, the exponential model closely follows the experimental data.

Fig. 10Prediction accuracy on validation dataset (solid lines represent experimental data and dotted lines represent model prediction)

Furthermore, Figs. 11-12, demonstrates the prediction accuracy of the developed sound absorption coefficient exponential model against the currently existing best two models.

The experimental data plotted in Fig. 12, was obtained from reference [21]. The data was captured from a nonwoven specimen that was manufactured from 77 % kapok fibre and 23 % polypropylene fibre. The nonwoven had a thickness of 6 mm a bulk density of 45.07 kg/m³ and a mean fibre diameter of 17.9 μm. It must be noted that it is very difficult to find data in the literature with similar properties to the material that was developed in this research. This is simply because no regression models have been developed for this range of materials.

Thus, it can be seen from Figs. 11-12, that the sound absorption coefficient exponential model outperforms the current best models. Furthermore, the sound absorption coefficient exponential model also shows great flexibility in being able to accurately predict the sound absorption coefficient of a material that is made up of two different fibres, one being natural and the other being synthetic, as seen in Fig. 12.

Fig. 11Model accuracy comparison using internal data

Fig. 12Model accuracy comparison using external data