A General Theory for Coupled Chemo-Electro-Thermo-Mechanical

Research Article A general theory for coupled chemo‑electro‑thermo‑mechanical heterogeneous system

Zhen‑Bang Kuang1

Received: 7 April 2020 / Accepted: 4 November 2020 / Published online: 12 December 2020 © Springer Nature Switzerland AG 2020

Abstract Many transport and rate processes in chemical, physical, mineral, material and biological felds are controlled by the coupled chemo-electro-thermo-mechanical (CETM) process. Though many literatures discussed these coupled problems, but a unifed rigorous theory and a unifed method based on the chemical thermodynamics are lacked. In this paper on the basis of electrochemistry, the non-equilibrium thermodynamics and modern continuum mechanics we modify some previous theories and give a general theory including mass conservation equation, the electric charge conservation equation, complete energy conservation equation, entropy equation, evolution equations and the complete governing equations of these couple CETM systems. An extension of Nernst–Planck equation is derived for the CETM system. This theory gives a theoretical foundation and a universal method to improve and develop engineering theories, especially for the gradual failure components and cells. In appendix we also discuss the interdifusion problems in solids with vacancies shortly as a complement of the continuum difusion.

Keywords Energy equation · Chemo-electro-thermo-mechanical systems · Entropy equation and entropy production rate · Gibbs equation · Evolution equation · Governing equation

1 Introduction are for electrically neutral system. However, in engineering a system may be worked under electromagnetic feld, In chemical, physical, material and biological systems, which may be externally imposed or internally created, many transport and rate processes are controlled by the or both. In a thermoelectric material there exist Seebeck, coupled chemo-electro-thermo-mechanical interaction. Peltier and Thomson phenomena [3, 9]. Ionic and mixed We shall abbreviate a system with coupled Chemo-Elec- ionic–electronic devices, such as solid oxide fuel cells, oxy- tro-Thermo-Mechanical interaction as CETM system. Simi- gen pumps and hydrogen production, have gained many larly, a system with coupled Chemo-Thermo-Mechanical applications [10–13]. For expansive media including clays, interaction is a CTM system and a system with coupled shales, polymers gels, corneal endothelium, immature Chemo-Electro-Mechanical interaction is a CEM system. articular cartilage and connective biological tissues, elec- All these systems are complex thermodynamic systems. trochemical interaction are also typical [14–16]. In theories A general theory for CTM system has been discussed in of these chemo-electro-mechanical (CEM) system the tem- papers [1, 2], where a complete energy conservation equa- perature efects are neglected [9–16]. Especially when one tion, an appropriate entropy equation and the govern- discusses the gradual failure of components and cells the ing equation system have been given and modifed the temperature efects may be important. So integrating the previous theories [3–8]. Most of these previous theories efects of the electromagnetic felds into the CTM system

* Zhen‑Bang Kuang, [email protected] | 1School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China.

SN Applied Sciences (2020) 2:2185 | https://doi.org/10.1007/s42452-020-03842-4

Vol.:(0123456789) Research Article SN Applied Sciences (2020) 2:2185 | https://doi.org/10.1007/s42452-020-03842-4 and integrating the efects of the temperature felds into to denote the serial number of the species and the serial the CEM system are necessary. Though in [3] the CETM number of the chemical reaction and the summation nota- system without chemical reaction had been discussed, tion is written in evidence, as shown in Eq. (1); but the sub- but its results are still left improvement. Therefore the scripts are used to components of a vector or tensor and CETM systems are worth to study. For a CETM system the the summation rule for the repeated indices is used. mass equation, energy equation, entropy equation and Let (k) , c(k) , v(k) , (jk) , �(jk)�̇ (j) and Θ̇ (k) be the partial den- momentum equation can be studied unitedly, but the sity or the apparent density, the mass fraction, the velocity, electric felds are produced due to various reasons and the reaction rate, the mass production rate per volume in jth every case should be researched independently. This situ- chemical reaction and in all chemical reactions of the species ation is analogous to the forces in the mechanical action. k respectively; v = u̇ and u are the barycentric velocity and The complete coupling efects of heat, difusion, chemi- the mechanical displacement vector of a representative ele- cal efect and electromagnetic feld are fully considered ment respectively, J(k) is the mass difusion fow of the species on a unifed thermodynamic foundation and a theoretical k . The mass conservation equation of the species k is [1–4]: frame of governing equations are given. An appropriate ⋅ d() �() complete governing equation system is the foundation for �ċ (k) = −∇ ⋅ J(k) + Θ̇ (k), () = = + v ⋅ ∇() solving engineering problems. The electrochemistry, the dt �t ��(k) non-equilibrium thermodynamics and modern continuum = −∇ ⋅ �(k)v(k) + Θ̇ (k); ̇� + �∇ ⋅ v = 0 mechanics allow us to construct an efcient theory for this �t N N coupling problems. �(k) c(k) = , � = �(k), c(k) = 1 We discuss an open system consisted of N (charged � k=1 k=1 (2) M or uncharged) species with total mass , total volume N N V and total density . According to the engineering cus- J(k) = �(k) v(k) − v , �v = �(k)v(k), J(k) = 0 tom we use the partial density or the mass concentration k=1 k=1 (k) (k) = M ∕V of a species k , rather than the molar con- L N (k) (k) (k) (k) (k) ̇ (k) (jk) ̇ (j) ̇ (k) centration C = ∕MM , where M and MM , are the Θ = � � , Θ = 0; k = 1, 2, ⋅⋅⋅, N total mass in V and the molecular weight of the species j=1 k=1 k respectively. where ∇ is the Eulerian gradient operator. Equation (2) shows that the mass fraction rate ċ , the mass fow J(k) and ̇ (k) 2 The mass and electric charge conservation the total mass production rate Θ of species k subject to equations the equation N N N 2.1 The mass conservation equation ċ (k) = 0, J(k) = 0, Θ̇ (k) = 0 (3) k=1 k=1 k=1 The mass conservation equation of the species k is the In the appendix, one will see that the interdifusion in same for CETM and CTM systems because the macroscopic noncontinuum solids with vacancies [21, 22] has diferent mass is independent to the electromagnetic feld. For easy mechanism with the above theory and discussed shortly to read the mass conservation equation given in literatures as the complement of the difusion phenomena in con- [1–4] is repeated here. tinuum media. Let a CETM system be consisted of N = N1 + N2 species N N M with 1 reactants and 2 products, the total mass , total 2.2 The electric charge conservation equation volume V , total density and chemical reaction number L . The jth chemical reaction equation can be written as [1, According to literature [3] one defne 2, 17–20] N N N I = (k)z(k)v(k) = I + i; I = zv, i = z(k)J(k) (jk)B(k) = 0, j = 1, 2 … , L 0 0 (1) k=1 k=1 k=1 (4) N N −1 (k) (k) (k) (k) where B(k) is the chemical formula of species k , (jk) is the z = z = c z k=1 k=1 stoichiometric constant of a species k in the chemical reaction j , which is positive if species k is a product and nega- where z(k) is the charge per mass of component k , z is the (m) th tive if k is a reactant. If B is not appear in the j chemical total charge per mass of the system, I is the total electric reaction, then (jm) = 0 . In this paper, we use superscript

Vol:.(1234567890) SN Applied Sciences (2020) 2:2185 | https://doi.org/10.1007/s42452-020-03842-4 Research Article current density, I0 is the convective current density and i From Eq. (9) it is known that a part of electromagnetic is the conductive current density. Using Eq. (2) the electric work, zE ⋅ v − i ⋅ (v × B) , is changed to the kinetic energy. charge conservation equation is According to the Maxwell equation the balance law of electromagnetic energy [3, 23–26] is N N N (k) (k) ⋅ (k) ̇ (k) (k) ⋅ ̇ (k) (k) �ż = � ċ z = −∇ J + Θ z = −∇ i + Θ z Φ D B k=1 k=1 k=1 = E ⋅ + H ⋅ = −∇ ⋅ (E × H) − I ⋅ E (10) (5) t t t where Φ is the energy density stored in the electro-magnetic feld, E × H is the Poynting vector of energy fow, I ⋅ E 3 The momentum and energy conservation is the work on matter supplied by the electro-magnetic equations in a CETM open system feld. In I ⋅ E the part i ⋅ E produced by conductive current i is changed to Joule heat [26] and constitutes a part of the 3.1 The momentum conservation equation internal energy, but the part I0 ⋅ E , produced by convective current I0 which moves with the Centroid of medium, is The electromagnetic force f (em) applied on a moving not related to internal energy or Joule heat, but is changed charge is given by Lorentz force law, so the momentum to the kinetic energy of the system (see Eq. 9). equation in CETM system is [23–26] The energy equation in a CETM open system for a viscous fuid is ∇ ⋅ + � f (m) + f (em) = �v̇ = �ü , U̇ + K̇ = Ẇ + Q̇ + �̇ + �̇ N (6) (em) (k) (k) (k) 1 ⋅ �f = � z E + v × B = �zE + I × B U̇ = �udV̇ , K̇ = � (v ⋅ v) dV k=1 V V 2 where (k)z(k) E + v(k) × B is the Lorentz force per volume Ẇ = � f (m) + f (em) ⋅ vdV + p ⋅ vda acting on species k , f (m) is the mechanical body force per V a volume, is the stress tensor of a representative element, Q̇ =− q ⋅ nda + (i ⋅ E)dV + −�̇ dV (11) E is electric feld intensity, B is the magnetic induction. aq V V From the Maxwell equation the more general momen- L N �̇ = r(j)�̇ (j,), �̇ = �(k)Θ̇ (k)dV tum equation in CETM system for v ≪ c is [23–26] j=1 V k=1 M (m) ∇ ⋅ + + �f = �v̇ N (k) (k) M (7) �̇ =− ∇ ⋅ � J dV =(D ⊗ E + B ⊗ H)−(1∕2)(D ⋅ E + B ⋅ H) k=1 V where M is the Maxwell stress, c is the light velocity. D is where U̇ , K̇ , Ẇ , Q̇ , Π̇ and Λ̇ are the internal energy rate, the electric displacement and H is the magnetic strength, kinetic energy rate, work rate done by mechanical and is the unit tensor Kronecker delta, ⊗ is the tensor product electro–magnetic forces, external supplied heat rate, and the component of D ⊗ E is D E . For linear uniform iso- i j the energy production rate in chemical reaction and tropic medium one can prove ∇ ⋅ M = zE + I × B. the energy fow introduced by the mass fow; (k) is the chemical potential of species k ; a = a + a is the outer 3.2 The energy conservation equation q Φ surface of the volume V , aq is the part of a where heat fow is imputed, a is the part of a where mass fow is imputed, Multiplying Eq. (6) by v and using relations Φ n is the external normal of the surface, p = ⋅ n is the sur- ⋅ ⋅ ⋅ ⋅ ⋅ v (∇ ) =∇ ( v) − ∶ (∇ ⊗ v); ∶ (∇ ⊗ v) = �ijvi,j face traction, q is the heat fow vector, i E is the Joule r(j) v ⋅ (I × B) = v ⋅ (�zv + i) × B = v ⋅ (i × B) =−i ⋅ (v × B) heat source strength produced by electric current; is the ⋅ 0 v (�zv × B) = (8) one get the kinetic energy rate K̇ [3]:

d 1 K̇ = � v2 =∇⋅ ( ⋅ v) − ∶ (∇ ⊗ v) + �zE ⋅ v − i ⋅ (v × B) + �f (m) ⋅ v (9) dt 2

Vol.:(0123456789) Research Article SN Applied Sciences (2020) 2:2185 | https://doi.org/10.1007/s42452-020-03842-4 chemical reaction heat per mass in chemical reaction j , Γ N N (k) (k) (k) (k) is the total chemical reaction heat per volume in all chemi- G = U − TS + pV = � M , Ġ =−SṪ + Vṗ + � Ṁ cal reaction, ṙ > 0 for endothermal reaction and ṙ < 0 for k=1 k=1 (14) exothermal reaction. By using Eq. (6) one has where U, S, V and G are the total energy, total entropy, total volume and total Gibbs function of the system and M(k) is (m) (em) k � −v̇ + f + f ⋅ v dV + p ⋅ vda the mass of a species respectively. From the above equa- V a tion De Groot and Mazur [3] gave a specifc Gibbs equation (k) m em in c : = � −v̇ + f ( ) + f ( ) ⋅ v +∇⋅ ( ⋅ v) dV V N m em ̇ (k) (k) = ∇ ⋅ + � −v̇ + f ( ) + f ( ) ⋅ vdV �u̇ = �Tṡ − pv + � � ċ (15) V k=1 + ⋅ (∇ ⊗ v)dV = ⋅ (∇ ⊗ v)dV where s is the entropy density per mass, v = 1∕ is the spe- (12) V V cifc volume. The above equation can be directly extended From Eqs. (11 and 12) we get the local energy conserva- to CETM system by using the theory of thermodynamics of tion equation: irreversible process (TIP) [3, 23]. Using the reversible variable (r) Eq. (15) can be extended to N N �u̇ = ∶ ̇ −∇⋅ q − �̇ + i ⋅ E −∇⋅ �(k)J(k) + �(k)Θ̇ (k) N (r) (k) (k) k=1 k=1 �u̇ = �Tṡ + � ̇� ij + � � ċ ij N N k=1 (16) (k) (k) (k) (k) = � ̇� − q − �̇ + i E − � J + � Θ̇ ��u ( ) ��u ��u mn m,m m m m ,m T = , � r = , �(k) = k=1 k=1 �s ij �� c(k) (13) ij For a general medium one form of is = (r) + (i) , The formulas in second line of Eq. (16) are N + 7 constitu- where (r) is the reversible part of and (i) is the irre- tive or state equations. For nonlinear medium the state versible part of . For an ideal fuid (r) =−p, (i) = 0 or equations should be obtained by experiments and mate-

kl =−pkl , where p is the fuid pressure. For an isotropic rial microscopic theories. viscous fuid =−p + �∇ ⊗ v or ij =−pij + vi,j , where Using the frst in Eq. (2) from Eqs. (13 and 16) we get the is the viscous coefcient [3, 23]. entropy equation and entropy production rate equation:

N �Tṡ = �T ṡ (r) + ṡ (i) = −∇ ⋅ q + i ⋅ E + (i) ∶ ̇ − �̇ − J(k) ⋅ ∇�(k) k=1 (17) N q ∇T �Tṡ (r) =−T∇ ⋅ − �̇ , �Tṡ (i) =−q ⋅ + (i) ∶ ̇ + i ⋅ E − J(k) ⋅ ∇�(k) T T k=1 4 The entropy and evolution equations Usually in electromagnetic dynamics one introduces an in a CETM open system electric scalar potential and a magnetic vector potential A such that 4.1 The entropy equation E = −∇ − A∕t, B =∇×A (18) The Gibbs function and the Gibbs equation of the entire Substituting Eqs. (18 and 4) into �Tṡ (i) in Eq. (17), one system for gas are respectively [1–12] gets

Vol:.(1234567890) SN Applied Sciences (2020) 2:2185 | https://doi.org/10.1007/s42452-020-03842-4 Research Article

N T �Aj A = A , A = + + , A = 0 �Tṡ (i) =−T ̇� i + �(i)v − J(k) �(k) + z(k) ij ij ijkl 1 ij kl ik jl il jk ijk i T ij i,j j ,j �t (21) k=1 N−1 T �Aj where Aij, Aijkl and Aijk are second order, fourth order and =−T ̇� i + �(i)v − J(k) ��(k) + z�(k) ≥ 0 i T ij i,j j ,j �t third order tensors respectively; A, and are all con- k=1 1 (k) �( k) (k) (N) stants. So for an isotropic material and under quasi-sta- � = �(k) + z(k)�, � = � − � , z�(k) = z(k) − z(N), ̇� = q ∕T j i i tionary electro-magnetic feld Eq. (20) is reduced to (19) (k) T N where is called the electrochemical potential of spe- ,i (k) (k) (k) (k) T ̇� i = qi =−L − M �, cies k ; qi, J = ̇� and vi,j are the irreversible flows, T i i i k=1 , (k) (k) (k) (i) T,j T ,j + z Aj t and J ij are the irreversible forces N T,i (22) or vice versa. In this paper we use the notation that a ̇� (k) = J(k) =−M(k) − N(km)�(m) i i T , i comma followed by index i in subscript indicates partial m=1 (i) differentiation with respect to xi, such as vj,i = vj∕ xi . �ij = �1vk,k�ij + �vi,j According to the irreversible thermodynamics the irreversible fows are functions of the irreversible forces. It is seen where L∕T = is the heat conduction coefficient. (k) (k) (k) (km) that we use J and z as the independent irreversible L, M , N , 1, are material constants. For Stokes viscous (k) ∕3 (i) 0 variables rather than i and themselves. At thermody- fuid 1 =− and kk = , i.e. only shear strain rate pro- (i) 0 0 0 �(k) 0 = 0 (i) = v namic equilibrium ṡ = , so T,j = , Aj t = , ,j = , duces irreversible viscous stress; so 1 and ij i,j v = 0 and it is usual thermodynamic equilibrium condi- for i ≠ j , (i) = 0 for i = j . The second in Eq. (22) is the i,j ij tions for electrochemical system. For quasi-stationary case extended Nernst–Planck equation for isotropic media in (k) (k) A∕ t = B∕ t = 0, one can fnd that the role in elec- CETM system. Without the term −M T,i T the second in trochemistry is equivalent to (k) in the usual chemistry. In Eq. (20) is just the usual Nernst–Planck equation [15, 27, Eq. (19) if one takes (k), z(k) and k = 1 − N , the coefcients (k) 28]. In Nernst–Planck equation some bodies prefer cj M(k) and N(km) only N − 1 are independent due to Eq. (3); if (k) ij ij (here c is the concentration) to ,j , others are quite the (k), (k) 1 1 one takes �(j) z� and k = − (N − ) the coefcients reverse. (k) (km) Mij and Nij are all independent. In the following sections From above discussions it is seen that the combination (k) this explanation will be continuously used. of chemical potential (k) (or � ) is the fundamental variable and the (N) is not the independent variable, but it can 4.2 The evolution equation be obtained from the Gibbs–Duhem relation [1–3, 17–20]:

In usual cases the linear irreversible thermodynamics is N (k) (k) , well [1–4, 23], so from Eq. (19) one has d =− sdT + dp for ideal gas k=1 N N T,j T ̇� = q =−L − M(k) �(k) + z(k)�A �t − � v (k) (k) i i ij ij , j j ijl j,l d =− sdT − ijdij, for general material T 1 k= k=1 N T,j (23) ̇� (k) (k) (k) (km) �(m) (m)� � �(k) i = Ji =−Mij − Nij , j + z Aj t − ijl vj,l T m=1 N 4.3 An example �(i) � �(k) �(m) (m)� ∕� � ij =− ijkT,k − ijk , k + z Ak t + ijklvk,l m=1 The thermoelectricity phenomena in thermoelectric- (20) ity solid are the special cases of Eqs. (20 and 22). In this , (k) (k), (kl) (kl) (lk) 2 where Lij = Lji Mij = Mji Nij = Nji = Nij , cases we have a binary system (N = ) , one is the posi- ijkl = klij = jikl are constants or the functions of T . These tive ion lattice, as species 2, another is the electron, as coefcients are determined by experiments and the micro- species 1. The fux is measured with respect to lattice, so 2 1 scopic theories. For the quasi-stationary electromagnetic J( ) = 0, J( ) = J(e) . According to Eq. (4) i = z(e)J(e) . For a feld, A∕ t = 0 , Eq. (20) is remarkable simplifed. The sec- quasi-statically electromagnetic feld without mechanical ond in Eq. (20) is the extended general Nernst–Planck efect Eqs. (19 and 20 or 22) are reduced to: equation in CETM system. For the isotropic material [23] we have.

Vol.:(0123456789) Research Article SN Applied Sciences (2020) 2:2185 | https://doi.org/10.1007/s42452-020-03842-4

(e) T,i 1 1 T,j � For the isotropic material and under quasi-stationary �Tṡ (i) =−T ̇� − J( )�( ) =−q − i �(e) ≥ 0, �(e) = � + i T j , j ,j T ,j ,j z(e) electro-magnetic feld Eq. (26) is reduced to 1 (e) 1 � (e) � 1 i =−L11 ∇� − L12∇ =−L11∇� − L12T∇ T N N T N T T T ,j (k) (k) (k) ,i (km) (m) (k) �Tṡ = L + M �, j + M + N �, i �, i 1 (e) 1 � (e) � 1 T T q =−L ∇� − L ∇ =−L ∇� − L T∇ k=1 , k=1 m=1 12 T 22 T 12 22 T j ̇ (24) + �vi,jvi,j − Γ � � � (27) where L11, L12, L22 = L11, L12, L22 T are material con- (m) stants and L12 = L21 due to the Onsager reciprocal relations Usually T,j and , j are small, so the term N N [3, 23]. Equation (24) is fully consistent with Seebeck, Pel- T M(k) ,i + N(km) (m) (k) including T (k) and tier and Thomson’s theories [3, 9]. It can be seen that the T , i , i ,j , i k=1 � m=1 � theory described in this paper is very simple and general. ∑(m) (k) ∑ In Thomson’s theory the internal points of the thermoelec- , j , i in equation �Tṡ is small terms in higher order and tric material are regular points and have continuous vary- in many practical cases can be neglected. In this case ing temperature or current. The junctions of diferent types Eq. (27) is reduced to of wire for Seebecek and Peltier efects can be considered N T,j as singular points with fnite discontinuous temperature �Tṡ = L + M(k)�(k) + �v v − Γ̇ T , j i,j i,j (28) or current. k=1 , j Therefore for the isotropic material under the quasi- 5 The governing equations for CETM System stationary electro-magnetic feld, the general governing equations, Eq. (25) is simplifed to At present the general governing equations for the CETM N system are lack. Here we attempt to give an appropriate (k) T,i (m) �ċ (k) =−J + Θ̇ (k) = M(k) + N(km)� + Θ̇ (k) discussion. The frst in Eq. (2) (the mass conservation equa- i,i T , i m=1 ,i tion), Eq. (6) (the momentum conservation equation) and T N the frst in Eq. (17) (the entropy equation) can be selected ,j (k) (k) �Tṡ = L + M � + �v , v , − Γ̇ as the governing equations, one has. T , j i j i j k=1 , j ��(k) � + �zE + e I B + �f (m) = �ü , ̇� + �v = 0 �ċ (k) =−J(k) + �̇ (k), or = −∇ ⋅ �(k)v −∇⋅ J(k) + �̇ (k) ji,j i ijk j k i i i,i j,j �t (29) (m) , M (m) �ji,j + �zEi + eijkIjBk + �f = �ü i or �ji + �ji + �f = �ü i i ,j i In Eq. (29), also in Eq. (25), the unknowns (k), , , (k), , 2 11 ̇� + �∇ ⋅ v = 0 c s T ij vi are N + , which is larger than the 4 N independent equations N + , so except Eq. (29) we should �Aj �Tṡ =− T ̇� − J(k) �(k) + z(k) + �(i)v − �̇ add N + 7 appropriate complement equations or constitu- j ,j j , j �t ij i,j k=1 tive (state) equations from Eq. (16). (25) As an example, for an ideal gas one can selects the refer-

where eijk is the permutation tensor, ence state as T0, p = p0 and the complement equations e = e = e = 1, e = e = e =−1, others = 0 . The [17–19] as: 123 231 312 321 213 132 governing Eq. (25) is general. Usually the linear evolution Eq. (20) is often appropriate and also general. Substituting Eq. (20) into the entropy equation in Eq. (25) we get

N T,j �Tṡ = L + M(k) �(k) + z(k)�A �t + � v ij T ij , j j ijl j,l k=1 , i N N T,j �A + M(k) + N(km) �(m) + z(m)�A �t + �(k)v �(k) + z(k) i (26) ij ij , j j ijl j,l , i � 1 T m=1 t k= N � �(k) �(m) (m)� ∕� � ̇ + − ijkT,k − ijk , k + z Ak t + ijklvk,l vi,j − Γ m=1

Vol:.(1234567890) SN Applied Sciences (2020) 2:2185 | https://doi.org/10.1007/s42452-020-03842-4 Research Article

N 1 �v 1 �� foods through certain chemical reaction or “organic work”, �Tṡ = �C Ṫ − � Tṗ + � � ċ , � = =− ̇ (k) p i i v �T � �T rather than chemical reaction heat Γ . In this case Θ in i=1 p p Eq. (2) should be understood as the mass growth of the �(k) � �(k) ln (k) � �(k) ln ln (k) = + 0 + RT p = + 0 + RT p + RT c biological tissues, and in Eq. (11) Γ=̇ 0 in formula Q̇ , but N �(k) in formula Ẇ a term “organic work rate” produced by foods p = p(k), p(k) = RT, pv = RT, � =−p� (k) ij ij etc. should be added. However the biological processes k=1 M M and the mechanical loadings correspond to largely difer- (30) ent time scales, so in the energy rate Eq. (11) the biological In Eq. (30) is the dilatation coefcient, Cp is the specifc growth rate can often be neglected. But the initial states heat per mass. solving the diferential governing equations correspond- Substituting the first of Eq. (30) into the second of ing to diferent time are diferent due to the biological Eq. (29), we get the energy equation or the temperature growth. The electromagnetic effects may also be fully (heat) equation for an ideal gas: taken into account in the biological processes. N N T,j �C Ṫ − � Tṗ + � � ċ = L + M(k)�(k) + �v v − Γ̇ p i i T , j i,j i,j i=1 k=1 , j 6 Conclusions (31) where the chemical reaction heat is a part of tem- In this paper a complete energy equation including the perature equation and the electric potential is included in variations of the structure energy and the reaction heat the electrochemical potential. are derived for coupled heterogeneous chemo-electro- In order to solve the practical engineering problems the thermo-mechanical diffusive systems. Applying this boundary conditions are needed. The boundary condi- energy equation, the corresponding entropy and entropy tions in CETM system are complex and diferent in various production rate equations are obtained. The evolution cases and it should be discussed case by case. equations derived from the entropy production rate equa- The rest of this section some comments will be given. tions are fully consistent with current classical theories. Current theories in literatures for CETM system are all The extended Nernst–Planck equation in CETM system is approximate theories. The general governing Eq. (29) can obtained. Some interesting questions are proposed. The be used to estimate which elements are neglected in these present theory modifes and simplifes existing theories present theories and can also be regarded as reference and can be used as the foundation to improve the practical theory to improve them. When one discusses the gradual engineering theories. In appendix we also discuss interdif- failure of components and cells the temperature efects fusion problems in the solids with vacancies shortly as a may be important which is included in Eq. (29). complement of the continuum difusion. The theory and In electric battery the general governing Eq. (29) can opinions in this paper should be tested in practice. directly apply to the liquid electrolyte region. Natural and forced convection and the difusion, the electric, chemical, temperature and mechanical effects are all taken Compliance with ethical standards into account. In ion-exchange membranes, such as fuel cells, the Chlor–Alkali process, and water electrolysis Conflict of interest The authors declare that they have no confict of interest. etc., in the electrode region the ion/molecule reaction needs absorb heat from the environment or release heat to the environment. The molecule (or atom) and ion can Appendix: The interdifusion problems be considered as two components of the system. But in in solids existing theories these efects are all neglected [29–31]. In the liquid electrolyte region the temperature efect is Other than fuids, in solids difusion is the only way to also neglected. In [15] Loret and Simões, discussed the move atoms. So for a solid generally v = 0 and = const. deformation, difusion, mass transfer and growth in multi- However the Kirkendall efect shows that the interface S species multi-phase biological tissues detailely. They pro- int (the inert marker) between two metals with diferent difu- posed that the mass change due to the tissue growth is sion rates of the metal atoms can be moved. This interdif- produced by two physical phenomena, namely difusion of fusion phenomenon in a difusion couples of binary sys- the species through the solid skeleton, and mass transfer tem is related to vacancies in noncontinuum media and is or organic growth supplied by the environment, such as

Vol.:(0123456789) Research Article SN Applied Sciences (2020) 2:2185 | https://doi.org/10.1007/s42452-020-03842-4 diferent with above textuary theory. For the continuum References media in the mass conservation Eq. (2) v is the barycentric velocity, but in the interdifusion problems the interface 1. Kuang Z-B (2015) Energy and entropy equations in coupled nonequilibrium thermal mechanical difusive chemical heterogeneous sys- Sint movement speed v is introduced by the vacancy dif- tem. Sci Bull 60: 952–957. Corrigenda: Sci. Bull. (2018) 63: 732 fusion mechanism in the metallic materials. 2. Kuang Z-B (2019) A general theory for heterogeneous coupled (k) Let be an atomic concentration or the number of chemical reaction-thermal difusion systems. Chem Phys Lett atom particles per volume of species k . Let a metal k(1) with 715:268–272 1 1 a kind of actual atom particle difusion fow J( ) = (1)v( ) 3. De Groot S, Mazur RP (1969) Nonequilibrium Thermodynamics, R Dover Publications. INC, New York in a fxed coordinate system, be in the left of Sint and a 4. Demirel Y (2009) Thermodynamically coupled heat and mass (2) (2) (2) (2) metal k fow JR = v be in the right of Sint . Similar fows in a reaction-transport system with external resistances. to Eqs. (2) and (23) under isothermal condition the mass Int J Heat and Mass transf 52:2018–2025 (k) 5. Deen WM (1998) Analysis of transport phenomena. Oxford Uni- difusion fow J relative to interface Sint is versity Press, Oxford 2 6. Schmidt LD (2005) The engineering of chemical reactions. J(k) = (k) v(k) − v , J(k) = N(km)∇(m) Oxford University Press, New York 7. De Zarate JMO, Sengers JV, Bedeaux D, Kjelstrup S (2007) Con- m=1 (32) centration fuctuations in nonisothermal reaction-difusion sys- (k) (k) (k) (k) (k), 1, 2 JR = v = v + J k = tems. J Chem Phys 127:034501 8. Hu S, Shen S (2013) Non-equilibrium thermodynamics and varia- where N(km) is the diffusion coefficient for isotropic tional principles for fully coupled thermal-mecganical-chemical materials. The total number of atom particles in two met- processes. Acta Mech 224:2895–2910 9. Goldsmid HJ (2010) Introduction to thermoelectricity. Springer, als should be constant. So one has Heidelberg 2 10. Swaminathany N, Qu J, Sun Y (2007) An electrochemomechanical (1) (2) (1m) (2m) (m) (1) (2) theory of defects in ionic solids. I Theory. Philos Mag 87:1705–1721 JR + JR = N + N ∇ + + v = 0 11. Maier J (2004) Physical chemistry of ionic materials, ions and m=1 electrons in solids. Wiley, Hoboken (33) 12. Singhal SC, Kendall K (2003) High temperature solid oxide For the isotropic material N(im) =−D(i) if m = i; fuel cells fundamentals design and applications. Elsevier, N(im) = 0, if m ≠ i , where D(i) is diffusion coefficient of Amsterdam (i) 13. 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26. Landau LD, Lifshitz EM (1960) Electrodynamics of continuum 31. Reddy RD, Cameselle C (2009) Electrochemical remediation Media. Pergamon Press, Oxford technologies for polluted solids Sediments and Groundwater. 27. Strathmann H (2004) Ion-exchange membrane separation pro- Wiley, Hoboken cesses, 1st edn. Elsevier, Amsterdam 32. Okino T (2012) Theoretical evidence for revision of fckian frst 28. Moshtarikhah S et al (2017) Nernst-planck modeling of multi- law and new understanding of difusion problems. J Mod Phys component ion transport in a nafon membrane at high current 3:1388–1393 density. J Appl Electroche 47:51–62 33. Okino T (2013) Ending of darken equation and intrinsic difusion 29. Dao T-S et al (2012) Simplifcation and order reduction of lith- concept. J Mod Phys 4:1495–1498 ium-ion battery model based on porous-electrode theory. J Power Sour 198:329–337 Publisher’s Note Springer Nature remains neutral with regard to 30. Cha Q (2002) Introduction to kinetics of electrode process. Sci- jurisdictional claims in published maps and institutional afliations. ence press, Beijing (in Chinese)

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