Model for Frequency Dependence of Thermal Permeability in Order to Quantify the Effects of Thermal Exchange on Wave Propagation in Multi Layered Porous Medium

The ability to quantify and predict the energy absorption/transmission characteristics of multi-layered porous medium is imperative if one is involved in the automotive, launch vehicle, commercial aircraft, architectural acoustics, petroleum exploration, or even in modeling human tissue. A case in point, the first four aforementioned fields rely on effective Noise and Vibration (NV) development for their commercial success. NV development requires the setting of NV targets at different system levels. The targets are then translated to Transmission Loss (TL), Insertion Loss (IL), and absorption (Alpha) performance for the multi-layered porous materials being utilized. Thus, it behooves to have a thorough understanding of the physics behind the energy dissipating mechanism of the material that entails the effects of the fluid meandering through the pores of the material and its interaction with the structural skeleton. In this section of the project the focus is on the thermal interchange that occurs within the porous medium. Via the acoustic modeling at the micro/macro level it is shown how this thermal exchange affects the acoustic compressibility within the porous material. In order to obtain a comprehensive approach the ensuing acoustic modeling includes the effects due to relaxation process, thus bulk viscosity and instantaneous entropy functions (effects due to vibration of diatomic molecules of air) are incorporated into the equation. The instantaneous entropy functions are explained by means of the Boltzmann’s distribution, partition function, and quantum states. The concept of thermal length and its connection to thermal permeability is clarified. Lastly, the results for TL calculations employing the aforementioned thermal exchange into the Transfer Matrix Method with finite size correction, (FTMM), pertaining to a simple multi-layered material is compared with experimentally obtained data.


Introduction
Teagle et al. [1] derived a coupled set of fluid/structure equations for a porous medium applying asymptotic and homogenization techniques. Via this modeling it is established in [1] that there are mainly 3 modes of energy transformation: 1) The first mode is through the connection between the micro and macro structural framework of the porous skeleton, 2) the second is via the viscous boundary layer, and 3) the third interaction is through thermal (entropy) boundary layer. The combination of the viscous boundary layer and how tortuous the porous material is, results in the encapsulation of the fluid medium. This, in turn, changes the apparent mass of the structural medium. Details pertaining to these encapsulating phenomena can be found in the work by Johnson, et al. [2]. In [2] the concepts of tortuosity, viscous length, and viscous permeability are explained.
This current work concentrates in the third interaction mode. There is a boundary layer where the flow experiences changes in entropy, therefore a loss/gain of energy due to thermal exchange will occur. This thermal exchange also changes the acoustic bulk modulus at certain frequencies and in turn changing the speed of sound. This is depicted through Equation (31) found in [1] * * * * * * * 1 From this equation it is understood that the compressibility factor, a β , is de- where 1 a a κ β = is the bulk modulus of the air inside the pore. k′ is the thermal permeability and is defined as , o o T P are the first terms of the asymptotic expansion (macro scale) pertaining k ω ′ . In order to attain this goal a good understanding of the physics behind the thermal exchange process has to be accomplished. This is achieved by: 1) analyzing the intensity escaping the main acoustic mode or 2) deriving a partial differential equation incorporating the porous wall impedance due to the thermal/viscous boundary layer; both are accomplished here.
This work also models an additional energy absorbing phenomena and that is the relaxation process. This is the hysteresis effect due to compression/decompresssion cycle of the molecules. The application of quantum mechanics, explained via the construct of Hilbert Spaces along with its dual space, facilitates the modeling of this hysteresis effect.
The main interest of the current authors is to calculate transmission loss (TL), Insertion loss (IL), and absorption coefficient (Alpha) for multi-layered porous medium. Therefore, calculation results for TL pertaining to a layered material, consisting of 2 different foams, are shown. This calculation is achieved by incorporating all of the aforementioned energy phenomena into the Finite-Correction Transfer Matrix Method (FTMM) [3] [4] [5].
In addition to the goals mentioned in the last paragraph a parallel effort is underway to lay down the mathematical foundation to understand how the propagating energy transfigures as it travels through the porous medium. This creates a tool that will allow design teams to make efficient absorption comparison of porous mediums consisting of different rheological substances.

Basic Statistical Mechanics: Partition Function/Boltzmann Distribution/Quantum States
One of the goals is to obtain a comprehensive set of fluid equations that incorporates any phenomena that leads to energy depletion. This means that in addition to viscous and thermal effects that transpire within their respective boundary layers, a quantification of the hysteresis effects due to the delay in thermal response should be obtained and incorporated. The most efficient tool to describe this process and for that matter any process is to apply statistical mechanic techniques. Once the Partition Function, Z, is obtained any macro level characteristic pertaining to the system can be derived. In this study Boltzmann's version of the combinatorial arrangement, is the probability of being in the i th energy state that has energy i E . The statistical mechanic definition of Entropy, s, is the measure of information plus its distribution and is defined as finding the p i s that will maximize the arrangement number, C, under the following con-A. Teagle-Hernandez et al.
J is the probability flux. For adiabatic processes the divergence of the probability flux is shown to be zero, 0 J ∇ =; therefore the probability density is stationary with respect to time. This means that the p i s are adiabatic invariant (which in turn means that entropy is constant) and thus the following expression for pressure holds The average potential between the particles is defined as 0 U and under classical mechanic conditions the partition function is is the partition function pertaining to the Ideas Gas.
In order to obtain a general expression for the average energy of the gas, Equa- A.
By applying Gaussian integrals the Partition Function for an undamped oscillator is As before, Equation (6) is used in order to obtain the average energy of the The analysis above, especially Equation (11), assumes classical mechanic conditions. Equation (13) states that the average energy of the oscillating system is T which, based on prior analysis, is connected to kinetic energy. This result is somewhat suspect since it states that regardless of how stiff the system is, the average energy is always T . In classical mechanics, the path integrals measured with respect to the Max Planck constant, , are relatively large. An observable in classical mechanics is simply a real function in Euclidean space. In quantum mechanics every possible state of a given system corresponds to a separable Hilbert Space over the complex number field. Additionally, to every physical observable there corresponds, in the Hilbert space, a linear Hermitian operator that has a complete set of orthogonal eigenvectors. The following limit applies to systems oscillating at relatively high energy levels, thus, in classical mechanics, it can be assumed that the energy forms a continuous spectrum; as compared to quantum mechanics where there is now a quantization (discretization) of the energy states [6]. In quantum mechanics a unit of energy is proportional to the natural frequency of the system by the constant, . The n th energy state is defined as Thus, the Partition Function is ( ) Therefore at high temperature levels there is enough energy such that the quantization of energy is not required. The criteria to follow are the following The cross-over point is T ω = . Lastly, it is possible to extract the Energy's standard deviation from the Partition Function. It is easy to , the specific heat constant at constant volume and The specific heat constant at constant volume is proportional to standard deviation of the energy.

Relaxation Process: Fluid Hysteresis/Bra-Ket Vectors, Schrodinger's Equation
The vibration due to the collision of molecules extract energy from the passing wave but releases the energy after some delay, thus the relaxation process. This delay, akin to mechanical damping systems leads to hysteretic energy losses. The internal energy can be partitioned into translated (defined relative to the average flow velocity), rotational kinetic energy and energy due to molecular natural frequency, , the vector space of bras is the dual space(one-to-one correspondence with the space of functionals) to the kets vector space. The Riesz representation theorem allows the following notation Ψ Ψ .
The 1-D Hamilton operator is given as where p is the Linear Momentum operator, ˆi , and x is the position operator. p and x are both observables and thus are hermitian operators on the Hilbert Space, therefore Ĥ has a purely discrete spectrum of eigenvalues, n E , that has a complete set of eigenstates, Hence, any state or ket-vector, Therefore the potential operator becomes ( ) Hamilton's operator can be pseudo separated as follows It can be shown that the commutator Based on Equation (20c) the eigenvalue equation can be written as Built on the ideas in Equation (21) the concept of annihilators, â , and creators, * a , are established; the following relationships exists Journal of Applied Mathematics and Physics Another possible solution, tied to the annihilator equation, is that The eigenstates pertaining to the higher energy levels can be generated by applying the Riesz representation notation.
As a consequence, the recurring formula for the eigenfunctions/states becomes ( ) ( ) ( ) A.
Due to the assumption of independence of the different energies the partition function pertaining to internal energies can be represented as The partition function for the vibrational portion pertaining to a particular type of molecule v (e.g. O 2 , N 2 ) is represented as  is the ground vibrational state energy (obtained by setting j = 0 in Equation (26)). Using Equation (26) in (28), the vibrational partition function can be rewritten as Applying Equation (6), an expression for the average energy is obtained Therefore the vibrational internal energy becomes Differentiating this equation with respect to temperature one obtains [7] [8] Applying the Boltzmann distribution definition for entropy Recall that , Applying the conservation of mass and the heat flux the entropy balance equ- The Navier Stokes Equation with the inclusion of the bulk viscosity becomes Applying conservation of energy results in the following expression for the vibration energy And it is this relaxation time that introduces the hysteretic energy losses [11].
The energy conservation dissipation equation is in the form of   [14]. Given these assumptions, the following asymptotic expansion is The absorption coefficients are the imaginary portion of Equation (51e) Recall that 2 c π λ ω = . At relatively low frequencies, 1 υ ω τ

Energy Loss: Thermal Boundary Layer-Bessel Functions
In [1], multi-scales technique is used on the coupled fluid/structure set of equations. Equation (16), in that publication, represents the partial differential equation that connects fluctuations of temperature and pressure in the fluid medium.
The dimensional form of this equation is Before deriving the thermal exchange for a general porous medium (with pores of arbitrary geometry), the thermal exchange for a typical cylindrical pore with cross-sectional radius R will be analyzed. The form this solution takes will be used as a pattern to emulate when considering solutions for the general case.

Equation (52) in cylindrical form is
The Bessel function generator is defined as The ensuing 2 equations are obtained by combining the equations in (55a) Equation (55c) is integrated from 0 to R, this generates an equation that will be utilized shortly Teagle [1] showed that 0 P is independent of the micro (fast) scale y. Therefore the particular solution for Equation (53) is The homogeneous solution is For statistical energy analysis calculations, average temperature of the pore cross-section suffices, therefore For the more general case, the focus is on analyzing the energy flux escaping the acoustic mode into the entropy boundary layer. Johnson et al. [2] accomplished similar work for the viscous boundary layer.
Trilling [13] gave a detailed explanation of the separation technique utilized to obtain 3 dispersion relations from the Navier-Stokes Equation.
Applying asymptotic expansion for each of these relationships, it is possible to derive 3 sets of simple partial differential equations along with their pertinent conditions. The process polarized the linear Navier-Stokes equation into 3 different flow regimes.
The superposition principle can be applied to obtain the total velocity. The flow regimes or modes can be characterized by the following: It is important to point out that Equation (58) A.
The dispersion for the vorticity and entropy modes are Near the solid surface it can be assumed that the solution has the form where this relation satisfies the following equation and applying the T ∇ ⋅ (horizontal divergence) along with Equation (67) Remember that Additionally, using Equation (68) Recall that the relative velocity at the wall is equal to zero. From the first relation in Equation (60) it can be deduced that The second term in the right hand side of Equation (73) will average out to zero given enough surface area. Equation (74) will be applied in the third term of Equation (73) to finally obtain the following relation ( ) ( ) For the mechanical dissipation the first term in the right hand side is used by [2]. The second term will be utilized in this study to treat the thermal dissipation and derive the acoustic bulk modulus.
The thermal counterpart of dynamic permeability and tortuosity are where o T and o P satisfy the following 2 0 in and 0 Since the membrane is adiabatic and since conditions listed in Equation (78) are being satisfied the integral in Equation (80) The first term in the right hand side of Equation (84) goes to zero because of boundary condition found in Equation (80). Inserting this result in Equation (81) and solving for ω the following is obtained In [1] it is shown that o P is independent of the micro scale variable and therefore it is considered as an external source of power for this thermal element.
Applying Equation (82), (83), and the Green's theorem to the first term in Equation (86), the same result of Equation (85) is obtained; again, the singularities of ( ) α ω ′ also lie in the negative imaginary axis.
Following the same steps performed in [2] in deriving the relation for viscous length, the last term in the right hand side of Equation (73) A.
The second term in Equation (90) is obtained by applying the definition of ent δ . This term can asymptotically (as ω → ∞ )be represented as where, 0 k is the thermal permeability as 0 ω → . A combined asymptotic re- Method with finite size correction via Green's functions techniques (FTMM) [16]. The parameters to this study are listed in Table 1 in [1]. Figure 1 shows the comparison between experimental data and the FTMM calculations for the multi-layer configuration shown in Figure 2. Further analysis, especially a parameter study on the different relaxation and thermal components is currently being done. This study mainly assumes air within the porous medium but there is current interest in applying these to pores with different rheological make up. FTMM v9