Supplementary Material Thermoacoustics of Solids: a Pathway to Solid State Engines and Refrigerators

Supplementary material Thermoacoustics of solids: a pathway to solid state engines and refrigerators

Derivation of the fully coupled, nonlinear, three-dimensional thermoacoustic theory. In the following derivation, we adopt an Eulerian view to write the coupled, nonlinear thermoacoustic governing equations of a solid. Using a classical Newtonian approach we enforce the equilibrium of stresses acting on the inﬁnitesimal Eulerian element. Classical index notation for stress components and sign convention are adopted. In Fig. S1, only stresses in the x direction are shown, for clarity. Enforcing the equilibrium in the x-direction gives:

∂σ ∂τ ∂τ Dv x dx(dydz) + xy dy(dxdz) + xz dz(dxdy) + X(dxdydz) = ρ(dxdydz) ∂x ∂y ∂z Dt which simpliﬁed is:

∂σ ∂τ ∂τ Dv x + xy + xz + X = ρ x (1) ∂x ∂y ∂z Dt where σ and τ represent normal and shear stress, v the particle velocity, and X are the body

D() forces in the x-direction, Dt is the material derivative, ρ is the material density. The equilibrium in the other two directions follow analogously. Using the index notation, the governing equations for a three-dimensional solid are written in compact form as:

3 Dvi X ∂σji ρ = + F (2) Dt ∂x b,i j=1 j (3)

1 From the constitutive relation for linear isotropic materials, we write the x components of the strain as:

εx = [σx − ν(σy + σz)] + α∆T (4) τ γ = xy (5) xy G τ γ = xz (6) xz G where ε and γ are normal and shear strains, ∆T is a temperature deviation from the zero- stress reference temperature Tref , ν is the Poisson’s ratio, and G is the shear modulus. Strain components in the other two directions can be found analogously. The governing equation for the temperature ﬁeld is obtained by imposing the conservation of energy (Fig. S1): DU Q˙ − W˙ = (7) net net Dt ˙ ˙ where Qnet, Wnet, and U are the net heat rate due to conduction, the rate of work, and the internal energy stored in the element, respectively. The heat transfer rate to the element is, based on the Fourier law of conduction, given ˙ ∂T by Qx,in = −κ(dydz) ∂x , where κ is the thermal conductivity of the isotropic medium, and 0 ˙ T = T0 + T is the total temperature. The heat transfer rate exiting the element is Qx,out = ˙ ˙ ∂Qx Qx,in + ∂x dx. So the net heat conduction rate in the x-direction is: ∂ ∂T Q˙ = Q˙ − Q˙ = (κ )dxdydz (8) x,net x,in x,out ∂x ∂x The total net heat conduction rate in index form is: 3 X ∂ ∂T Q˙ = κ dxdydz (9) net ∂x ∂x j=1 j j The mechanical work transfer rate is given by: 3 X Dεxj w˙ = − σ dxdydz (10) xj Dt j=1

2 The negative sign is due to the sign convention (work is negative when done on the element). ˙ All the other work done by the element is written as Wother = −q˙gdxdydz, where q˙g can be regarded as a heat source inside the element. So the net work rate is given by: 3 X Dεxj W˙ = − σ dxdydz − q˙ dxdydz (11) net xj Dt g j=1

The rate of change of energy is given by: 3 DU DT X Dεxj αET Dev = ρc dxdydz + σ dxdydz + dxdydz (12) Dt Dt xj Dt 1 − 2ν Dt j=1 where E is the Young’s modulus, α is the thermoelastic expansion coefficient, c is the specific P3 heat at constant strain, ev is the volumetric dilatation which is defined as ev = j=1 εxj . Substituting Eqns. 9, 11, and 12 into Eqn. 7, we obtain the final form of the energy equation: 3 DT αET Dev X ∂ ∂T ρc + = κ +q ˙ (13) Dt 1 − 2ν Dt ∂x ∂x g j=1 j j

Equations 2 and 13 compose the set of fully coupled, nonlinear, three-dimensional thermoacoustic equations for an isotropic solid.

Derivation of the quasi-1D theory Taking advantage of the aspect ratio in a slender rod, we can derive a quasi-1D set of equations from the previous model. For a linear, isotropic solid the momentum equation becomes:

∂2u0 ∂σ0 ρ = + F (14) ∂t2 ∂x where u and σ are the displacement and stress in the x-direction, respectively. The notation ()0 denotes the ﬂuctuating component of a variable. F is a source term. Note that (subindex dropped to shorten the notation) The constitutive relation in one dimension is expressed as:

σ0 = E(ε0 − α∆T ) (15)

3 In our calculations, the reference temperature Tref is assumed coincident with the mean

0 temperature T0, so that ∆T becomes effectively the temperature ﬂuctuation T . Combining Eqns. 14 and 15 and using the kinematic relations to express all the quantities in terms of particle displacement ﬂuctuations:

∂2u0 ∂2u0 ∂T 0 ρ = E − α + F (16) ∂t2 ∂x2 ∂x

For the stability analysis, we consider the homogenous form of Eqn. 16 and we transform it to the frequency domain by assuming harmonic response in time in the form eiΛt:

iΛˆu =v ˆ (17) E d2uˆ dTˆ iΛˆv = − α (18) ρ dx2 dx

Note that the intermediate transformation in Eqn. 17 allows keeping the eigenvalue problem fully linear. The above equations are the well-known 1D form of the wave equation in a linear, isotropic, solid rod. For what concerns the energy equation, we observe that the existence of radial heat conduction in the S-segment does not allow for a pure 1D treatment of the energy equation. In order to account for this radial component while still preserving the simplicity of the one-dimensional equation, we start from the 2D axi-symmetric energy equation:

DT αET De ∂ ∂T 1 ∂ ∂T ρc + v = κ + κr +q ˙ (19) Dt 1 − 2ν Dt ∂x ∂x r ∂r ∂r g and integrate the following assumptions in order to simplify the model,

1. The problem is axi-symmetric, hence every variable is independent of the azimuthal co-

∂ ordinate θ (i.e. ∂θ = 0),

2. T0 only varies in the axial direction, i.e. T0 = T0(x),

4 3. The temperature ﬂuctuation T 0 caused by the radial deformation is neglected,

4. The axial thermal conduction is neglected.

The material derivative in Eqn. 24 can be expanded and linearized as follows: DT ∂(T + T 0) ∂u ∂(T + T 0) 1 ∂u ∂(T + T 0) ∂u ∂(T + T 0) = 0 + r 0 + θ 0 + x 0 (20) Dt ∂t ∂t ∂r r ∂t ∂θ ∂t ∂x ∂T 0 ∂u ∂T 0 ∂u ∂(T + T 0) = + r + x 0 (21) ∂t ∂t ∂r ∂t ∂x ∂T 0 ∂u ∂T = + x 0 (22) ∂t ∂t ∂x Note that the second term above results from the convective term of the material derivative and it is present only if there is a non-zero boundary gradient of the mean temperature T0. This term is the main driver for the thermoacoustic instability and one of the main difference com- pared to the classical thermoelastic formulation. The second term in Eqn. 19 can be expanded, linearized, and approximated as: αET De αET ∂e αE(T + T 0) ∂e αE(T + T 0) ∂ε αET ∂2u v ≈ v = 0 v ≈ 0 ≈ 0 (23) 1 − 2ν Dt 1 − 2ν ∂t 1 − 2ν ∂t 1 − 2ν ∂t 1 − 2ν ∂x∂t Finally, by neglecting the axial thermal conduction, Eqn. 19 becomes: ∂T 0 ∂u dT αE ∂2u 1 ∂ ∂T 0 ρc + 0 = − T + κ r (24) ∂t ∂t dx 1 − 2ν 0 ∂t∂x r ∂r ∂r Equation 24 can be transformed to the frequency domain: dT dvˆ 1 ∂ ∂Tˆ iΛTˆ = − 0 vˆ − γ + κ r (25) dx G dx r ∂r ∂r where γ = αE is the Gruneisen¨ constant. G ρc(1−2ν) The system of Eqns. 17, 18, and 25 can be solved to assess the stability of the thermoacoustic system. In order to ﬁnd a solution to the energy equation, we perform a coordinate transformation: √ r ξ = −2i (26) δk

5 where the thermal penetration thickness δk is deﬁned as: r 2κ δk = . (27) ωρc The energy equation in the transformed coordinate becomes: ∂2T˜ ∂T˜ ξ2 + ξ + ξ2T˜ = 0 (28) ∂ξ2 ∂ξ where Tˆ T˜ = − − 1 (29) dT0 duˆ (ˆu dx + γGT0 dx ) In the previous equation, uˆ was assumed independent of the radial position. The general solution to Eqn. 28 is:

˜ T (ξ) = AJ0(ξ) + BY0(ξ) (30) where A and B are constants of integration determined by imposing the boundary conditions,

J0(·) and Y0(·) are Bessel functions of the first and second kind, respectively. On the axis of the rod, the temperature fluctuation is bounded, and considering that Y0(0) = ∞ then it must be B = 0. In the S-segment, the isothermal boundary condition enforces Tˆ(R) = 0, √ ˜ R so T (ξtop) = −1, where ξtop = −2i . Substituting the constants in Eqn. 30, we find δk T˜ = − J0(ξ) . It follows that: J0(ξtop) ˆ dT0 duˆ J0(ξ) T = − uˆ + γGT0 1 − (31) dx dx J0(ξtop) Similarly, Tˆ out of the S-segment is found as: dT duˆ Tˆ = − uˆ 0 + γ T (32) dx G 0 dx By averaging Eqns. 31 and 32 over the cross-section and substituting into Eqn. 25, we obtain a quasi-1D energy equation: dT dvˆ iΛTˆ = − 0 vˆ − γ T − α Tˆ (33) dx G 0 dx H

6 where the thermo-conductive function αH is given by:  J (ξ )  1 top  ωξtop  J0(ξtop)  2 , xh < x < xc, α = J1(ξtop) R (34) H iξ −  top J (ξ ) δ2  0 top k 0, otherwise

Equations 17, 18, and 33 form the quasi-1D system of equations describing the thermoacoustic response of a rod. This system of equations can be solved numerically to ﬁnd the eigenvalues Λ. We used a second order central Euler scheme on a staggered grid (Fig. S2) for the discretization. The discretized system is given by:

iΛˆuj =v ˆj (35) E uˆ − 2ˆu +u ˆ Tˆ − Tˆ iΛˆv = ( j−1 j j+1 − α j j−1 ) (36) j ρ ∆x2 ∆x dT vˆ +v ˆ vˆ − vˆ iΛTˆ = 0 j j+1 − γ T j+1 j − α Tˆ (37) j dx 2 G 0 ∆x H j in compact form:

(iΛI − A)y = 0 (38) where I is the identity matrix, A is a matrix of coefﬁcients, Λ is the eigenvalue, and y =

[ˆu; ˆv; Tˆ] is the corresponding eigenvector (in terms of the particle displacement, particle velocity, and temperature ﬂuctuations). The mechanical boundary conditions are written as vˆ = 0

∂vˆ ∂uˆ (ﬁxed), ∂x = 0 (free), EA ∂x = −iΛˆvM (tip mass M).

Setup for the numerical solution of the 3D model The transient response of the 3D model was solved by ﬁnite element method using the commercial software Comsol Multiphysics. To facilitate the analysis, given the largely disparate time scales involved in the wave propagation and heat diffusion processes, we ﬁrst solved a static problem to calculate the elastic deformation

7 induced by the thermal boundary conditions and to achieve the steady state mean temperature distribution inside the rod. These elastic and thermal equilibrium states where imposed as initial conditions when solving the time-dependent response. In order to reduce the effect of the initial transient we also included in the initial conditions the particle displacement ﬁeld associated with the eigenfunction at the given excitation frequency (and obtained from the quasi-1D eigenvalue analysis). The linear temperature proﬁle T0 was imposed as an isothermal boundary condition

αR on the outer surface of the rod. In the damped model, Rayleigh damping ζ = 0.5( ω + βRω) was used, where αR and βR are Rayleign parameters, ζ and ω are the damping ratio and angular frequency respectively. The parameters were determined by matching ζ = 0.01 at the natural

−5 −5 frequency (i.e. αR = 3.3344×10 (rad/s), βR = 3.3344×10 (s/rad) for ω = 598(rad/s)).

Further comments on the effect of the axial conduction. The axial heat conduction was determined to be negligible in a single-stage conﬁguration. Fig. S3a and S3b show the time histories of the particle displacement at the mass end with and without axial conduction. The results correspond to the case where the single stage is mounted at [0.9 ∼ 0.95]L. A direct comparison of the numerical results conﬁrms that there is no appreciable differences when including the axial conduction.

Nevertheless, when considering the multi-stage case (Fig.S3c and d), the difference is no longer negligible. It was previously highlighted that this behavior was due to the occurrence of discontinuity of heat boundary conditions at the edges of the stage (i.e. sharp transition from isothermal to adiabatic and viceversa) that create appreciable axial heat transfer components (Fig. S4). Results were calculated for an aluminum rod of 1.8m in length and 1mm in radius. Thirty stages were distributed along [0.1 ∼ 0.9]L spaced by 20%` (`: length of the stage) and producing a 200K temperature difference. The mean temperature T0(x) had a sawtooth- like proﬁle with exponential decays between stages. The quasi-1D stability analysis for the

8 undamped rod without axial conduction returned an eigenvalue iΛ = 6.38 + i596.1(rad/s). Similarly, the 3D transient analysis with and without axial conduction (Fig. S3c and S3d)

κx=κ κx=0 provided growth rates of β3D = 5.73(rad/s) and β3D = 6.60(rad/s), respectively. These results clearly show that axial conduction can have a non-negligible impact when its effect is cumulated over multiple sections. The fact that the axial conduction contrasts the onset of the thermoacoustic instability is reasonable because it creates a mechanism where the heat provided by the stage is quickly conducted away from the S-segment.

Thermodynamic cycle. From a thermodynamic point of view, the work done by a solid ele-

1 R ∂ε 1 R 1 R ment during an acoustic/elastic cycle is hw˙ i = τ (−σ) ∂t dt = τ (−σ)dε = τ σdε¯ , where τ is the period of oscillation, σ and ε are the 1D stresses and strains, σ¯ = −σ. This value should be zero for an element outside the stage because T 0 is in phase with ε, thus σ and ε are also in phase. The rate of work is also equal to the area enclosed in a σ¯-ε cycle. Fig.S5 shows the actual σ¯-ε plot for two different elements, one inside and one outside the stage region.

9 Figure S1: Schematic of an inﬁnitesimal volume element indicating the stress and heat rate components in the x direction.

Figure S2: Notional schematic of the staggered grid. (|) locations where the fluctuations of the particle displacement uˆj and of the particle velocity vˆj are defined. (◦) locations where the ˆ temperature fluctuations Tj are defined.

10 (a) (b) 0.24 0.24 0.18 0.18 0.16 0.16 0.08 0.08 0 0 -0.08 -0.08 -0.16 -0.16 -0.18 -0.18 -0.24 -0.24 0 0.2 0.4 0.6 0 0.2 0.4 0.6 (c) (d) 4 8 3 6 2 4 1 2

0 0 -1 -2 -2 -4 -3 -6 -4 -8 0 0.2 0.4 0.6 0 0.2 0.4 0.6

Figure S3: Time histories of the particle displacement fluctuations at the mass end for an undamped fixed-mass rod. (a) single-stage configuration with axial conductivity. (b) single-stage configuration without axial conductivity. (c) multi-stage configuration with axial conductivity. (d) multi-stage configuration without axial conductivity.

11 (a)5 (b) 5

4 4

3 3

2 2

1 1

0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

∂2Tˆ Figure S4: Strength of the axial conduction term κ ∂x2 in single and multi-stage conﬁguration (arbitrary scales and units). This term is non-zero only at the edges of stages because of the discontinuity of the thermal boundary conditions. (a). Single-stage. (b). Multi-stage.

(a)3 (b) 3

2 2

1 1

0 0

-1 -1

-2 -2

-3 -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5

Figure S5: Actual thermodynamic cycles of a particle expressed in arbitrary scale. (a) A particle outside the S-segment does not perform net work during a cycle because its stress and strain are always in phase. (b) A particle inside the S-segment performs net work during a cycle because the stress is not in phase with the strain due to the T 0 component.