3. Battery Model
3.2. Elliptic-Parabolic PDE System
In this section a mathematical model which describes the transport processes in a battery [PVMI10] is introduced in one dimension. The physical and chemical introduction of this model was done by Latz and Zausch, cf. [LZI10, LZ10].
0000000000000000
Figure 3.2.: Simple battery model: the electrodes are the active particles which are connected via and separated by the electrolyte.
positive electrode
electrolyte negative electrode
Ωc Ωe Ωa
Butler-Volmer-equation
Figure 3.3.: Structure of the considered battery domain
In the parameter µ∈ D⊂RNP we summarise all input values of the following battery system like geometrical, state, material, physical and chemical parameters. Later we will specify the parameter domain D.
Let Ω⊂R be an open bounded interval, which is divided in three disjoint open subin-tervals Ωc,Ωe,Ωa ⊂ R, see Figure 3.3. For tf > 0 we define the space-time cylinder Q∶=Ω×(0, tf)what we need in Assumption 3.1. Forµ∈Dassume that the concentra-tion of lithium-ions c(⋅,⋅;µ) as well as the electric potential φ(⋅,⋅;µ) is continuous in Ω, two times continuously differentiable in the subinterval Ωiand continuously differentiable in time, i.e. c(⋅,⋅;µ), φ(⋅,⋅;µ)∈C(Ω;C1(0, tf))∩C2(Ωc∪Ω˙ e∪Ω˙ a;C(0, tf)).
The transport processes in a battery for the input parameter µ∈Dare given by
∂c
∂t − ∇ ⋅(α(c, φ;µ)∇c+β(c, φ;µ)∇φ)=0, (3.1a)
−∇ ⋅(λ(c, φ;µ)∇c+κ(c, φ;µ)∇φ)=0, (3.1b) in Ωc×(0, tf), Ωe×(0, tf)and Ωa×(0, tf), where “∇” denotes the gradient and “∇⋅” the divergence. The first equation (3.1a) describes the mass transport of the lithium-ions, the second one (3.1b) the charge transport.
The positive electrode is Ωc (cathode for discharge), the electrolyte Ωe and the positive electrode Ωa (anode for discharge).
For physical and mathematical reasons we assume that
cmax≥c(x, t)≥cmin>0∀x∈Ω, t∈[0, tf). (3.2)
3.2. Elliptic-Parabolic PDE System
Boundary, Initial and Interface Conditions The boundary conditions are
∂c
∂ν =0, φ=0 on (∂Ω∩∂Ωc)×(0, tf), (3.3a)
∂c
∂ν =0, ∂φ
∂ν =−I
σa on (∂Ω∩∂Ωa)×(0, tf), (3.3b) where ν is the outer unit normal vector, I ∈ R the current and σa ∈ R
+/{0} (electric conductivity multiplied with the cross section). So in the one-dimensional case (3.3) reads
∂c
∂x =0, φ=0 on (∂Ω∩∂Ωc)×(0, tf),
∂c
∂x =0, ∂φ
∂x =− I σa
on (∂Ω∩∂Ωa)×(0, tf).
The homogeneous Neumann boundary conditions for the concentration (3.3) mean that no flux of lithium(-ions) can pass through the exterior boundary. The inhomogeneous Neumann boundary condition for the potential is Ohm’s law, the homogeneous Dirichlet boundary condition have no physical meaning. It ensures the uniqueness of the solution if one exists.
The initial condition is denoted by
c(x, t0=0;µ)=c0(x;µ), x∈Ω, (3.4) with the initial concentration c0∶Ω→R+.
The interface conditions describe the exchange of the lithium-ions at the interfaces which are modelled by the Butler-Volmer-equation [Atk13]. By νse we denote a vector which points from the solid part, i.e. the electrodes into the electrolyte. The interface condi-tions are given by
−(α(c, φ;µ)∇c+β(c, φ;µ)∇φ)⋅νse
=⎧⎪⎪⎪
⎨⎪⎪⎪⎩
I(cec, cc, φec, φc;µ) ∣(0,tf) in (∂Ωc∩∂Ωe)
−I(cea, ca, φea, φa;µ) ∣(0,tf) in (∂Ωe∩∂Ωa) , (3.5a)
−(λ(c, φ;µ)∇c+κ(c, φ;µ)∇φ)⋅νse
=⎧⎪⎪⎪
⎨⎪⎪⎪⎩
J (cec, cc, φec, φc;µ) ∣(0,tf) in (∂Ωc∩∂Ωe)
−J(cea, ca, φea, φa;µ) ∣(0,tf) in (∂Ωe∩∂Ωa) , (3.5b) whereceadenotes the concentration in the electrolyte at the negative electrode interface:
cea(t)=lim
h↘0c(x∣∂Ωe∩∂Ωa−h, t),
and wherex−h∈Ωe holds. Analogously con-centration in the solid part, i.e. in the negative and positive electrode, and ce for the concentration in the electrolyte, φs and φe are denoted analogously. The scalar-valued functions I∶R2 circuit voltage and depends on the concentration cs in the electrodes. To the author’s knowledge the open circuit voltage is a smooth function, but not of a special type, e.g.
polynomial, cf. [DN96]. In the numerical tests of the present thesis the open circuit voltage is constant. The coefficient functions are defined as
α(c, φ;µ)∶=De(c, φ)+RT Here the transference numbert+ is zero in the electrodes and greater than zero in the electrolyte and κ is the ionic/electric conductivity; κ, t+, De∶R
+×R→R. We assume that κ, t+, De∈C1(R+×R;R). Thus it directly follows that α(⋅,⋅;µ), β(⋅,⋅;µ), λ(⋅,⋅;µ) ∈ C1(R+×R,R).
In the present thesis we mainly consider a parameter set where all parameters are con-stant. Since we put the geometrical and state parameters into brackets it is defined as
D={µ∈R17∣µi,lb≤µi≤µi,ub, i=1, . . . ,17}, (3.8) where µi,lb, µi,ub ∈R for i∈{1, . . . ,17} and µi,lb >0 for i∈{2,3, . . . ,12,13,16,17}. We identify the parameters as it follows: the current isI≡µ1; the transport numbert+≡µ2;
3.2. Elliptic-Parabolic PDE System
the diffusion coefficients in the positive electrode, in the electrolyte and in the negative electrode are De,c ≡ µ3, De,e ≡µ4 and De,a ≡ µ5; the electric/ionic conductivity in the three battery domains are κc ≡µ6,κe ≡µ7 and κa ≡µ8; the start concentration in the positive electrode, in the electrolyte and in the negative electrode are given by c0c ≡µ9, c0e ≡ µ10 and c0a ≡ µ11, so we assume that the initial concentration in each subinterval is constant, cf. (3.4); the maximum concentration in the electrodes are cmaxc =µ12 and cmaxa =µ13; the open circuit voltages in the electrodes are U0,c≡µ14 and U0,a≡µ15; the reaction rates in the electrodes are kc ≡ µ16 and ka ≡ µ17. Hence the parameter set is 17-dimensional.
In a more general context we could assume that the parameters are time depend (current) or depend on the concentration and potential (e.g. diffusion coefficients). In this case we have to think about how to approximate those functions, e.g. if splines are sufficient enough. Then every parameter has coefficients which can variate and the dimension of the parameter set increases.
We remark that the coefficient functionsαandκare strictly positive for physical reasons:
the diffusivity De and the conductivity κare strictly positive. Because of the definition of the transference numbert+ the coefficient functionsβ and λare non negative.
To the author’s knowledge the only analytical investigation of the parameterised partial differential equation system (3.1) is done in the work by Seger [Seg13]: He considers the elliptic part of the system, sets all parameters equal to 1 and fixes the concentration in the solid part as well as in the electrolyte. With these settings he proves that if a weak solution to this problem exists it is unique, cf. [Seg13, Lemma 7.10]. Because we do not have this special setting in the present work we have to make the following
Assumption 3.1. Let µ ∈ D, α, β, γ, κ ∈C1(R+×R,R) and c0 ∈L∞(Ω). Then there exists a unique weak solution (c, φ)∈Yc×Yφ to (3.1), where
Yc∶=W(0, tf;Vc)∩L∞(Q), (3.9a) Yφ∶=L2(0, tf;Vφ)∩L∞(Q), (3.9b) with
Vc=H1(Ω),
Vφ={ψ∈H1(Ω)∣ψ=0 on ∂Ω∩∂Ωc}.
Remark 3.2. The weak formulation of (3.1) is given by: find c∈ Yc and φ∈ Yφ such that
∫0tf∫Ω(−c(x, t)ϕt(x, t)+α(c, φ;µ)cx(x, t)ϕx(x, t)+β(c, φ;µ)φx(x, t)ϕx(x, t))dx dt
= ∫Ωc0(x)ϕ(x,0)dx,
∫0tf∫Ω(λ(c, φ;µ)cx(x, t)ψx(x, t)+κ(c, φ;µ)φx(x, t)ψx(x, t))dx dt
= −∫0tf I Aa
ψ(x, t)∣x∈∂Ωa∩∂Ωdt,
for all ϕ ∈ W1,1(Ω) = H1(0, tf;L2(Ω))∩L2(0, tf;Vc) with ϕ(⋅, tf) = 0 in Ω and all ψ∈L2(0, tf;Vφ).
3.3. Discretisation
Analogously to Popov et al. [PVMI10] we discretise our one-dimensional parameterised partial differential equation system (3.1) with the cell centred FV method [KA00] and the backward Euler scheme, cf. Section 2.5. To follow their approach, we divide the domain of the positive electrode Ωc into Nc equidistant subintervals of the width ∆x>
0. Analogously the domain of the electrolyte Ωe is divided into Ne and the negative electrode Ωa into Na subintervals. So the battery domain Ω is divided into Nx =Nc+ Ne+Na subintervals Ωi with centrexi,i=1, . . . ,Nx.
Let tm+1, m ∈ {1, . . . ,Nt−1} be the current time step and µ ∈ D. Then cN⋅,m+1(µ) is the concentration at the current time step and φN
⋅,m+1(µ) the electrical potential. The discretised equation system of (3.1) is given by
0=! F1N(cN⋅,m+1(µ), φN
where A, B,Λ, K ∈RNx×Nx are tridiagonal matrices which are defined in the Appendix B. The boundary and interface conditions are summarised in Γ⋅,⋅. For their definition we refer to Appendix B, as well.
Calculation of the Start Value Φ0
As start vector for the damped Newton method (cf. Subsection 2.6.1) the solution for the previous time step (c(x(K), tm), φ(x(K), tm)), K ∈ N0, m ∈ {1, . . . ,Nt−1} is taken. In the computational examples the start vectorcNi,1(µ)=c0(xi;µ)=c(xi, t=0;µ),