Analytical fractal model of sound absorption for cellular foams with randomly distributed fully/semi-open pores

Cellular foams with randomly distributed open pores are increasingly exploited in sound management applications, where the sound absorption coefficient (SAC) typically serves as a crucial acoustic parameter for performance evaluation and design optimization. Dependent upon the processing method, the pores in a cellular foam can be either fully open or semi-open and often exhibit fractal distribution features. To facilitate engineering applications, it is imperative to analytically predict the SACs of these foams. However, predicting analytically the SAC for foams poses a challenge. Therefore, this study proposes a simplified representative structure (RS) with semi-open or fully open pores to analyze the flow properties within the foam microscopically, while the fractal theory is applied to portray the randomly distributed pores. With the extent to which the pores are open characterized using a purposely introduced parameter called the open-pore degree, both viscous and thermal characteristic lengths of the RS are analytically obtained. Subsequently, built upon the classical Johnson-Champoux-Allard (JCA) model for sound propagation in porous media, an analytical model is developed to unify the RS with the fractal theory so that the SAC can be predicted as a function of key morphological parameters of the foam having fully/semi-open pores. Compared with existing experimental measurements and numerical simulation results, the proposed analytical model predicts well the key flow properties as well as the SAC of foams having either semi-open or fully open pore topologies. In the frequency range of 0-4500 Hz, a semi-open foam can better attenuate the sound wave relative to its fully-open counterpart having the same porosity. With the porosity fixed at 0.95, the overall SAC of semi-open foam is improved by 21.2%, 57.7%, and 75.8%, respectively, as its open-pore degree is reduced from 0.75 via 0.50 to 0.25.