Table 4 Non-dimensional analysis of the heat and mass transfer processes in SOC

From: Discovering two general characteristic times of transient responses in solid oxide cells

Simplified governing equations for non-dimensional analysis15,40

Domains

Conservation of species

\(\frac{\partial {Y}_{i}}{\partial t}+\overrightarrow{V}\cdot \nabla {Y}_{i}-D{\nabla }^{2}{Y}_{i}=0\)

Fluid channel

\(\frac{\partial \left(\varepsilon {Y}_{i}\right)}{\partial t}+\overrightarrow{V}\cdot \nabla {Y}_{i}-{D}_{{{{{{{{\rm{eff}}}}}}}}}{\nabla }^{2}{Y}_{i}=0\)

ADL,CDL

\(\frac{\partial \left(\varepsilon {Y}_{i}\right)}{\partial t}+\overrightarrow{V}\cdot \nabla {Y}_{i}-{D}_{{{{{{{{\rm{eff}}}}}}}}}{\nabla }^{2}{Y}_{i}=\frac{{S}_{{{{{{{{\rm{m}}}}}}}},i}}{\rho }\)

AFL,CFL

Conservation of energy

\(\frac{\partial T}{\partial t}+\overrightarrow{V}\cdot \nabla T-\alpha {\nabla }^{2}T=0\)

Fluid channel

\(\frac{\partial T}{\partial t}+\frac{\rho {c}_{p}}{{\rho }_{{{{{{{{\rm{eff}}}}}}}}}{c}_{p,{{{{{{{\rm{eff}}}}}}}}}}\overrightarrow{V}\cdot \nabla T-{\alpha }_{{{{{{{{\rm{eff}}}}}}}}}{\nabla }^{2}T=\frac{{S}_{{{{{{{{\rm{h}}}}}}}}}}{{\rho }_{{{{{{{{\rm{eff}}}}}}}}}{c}_{p,{{{{{{{\rm{eff}}}}}}}}}}\)

Porousa

\(\frac{\partial T}{\partial t}-{\alpha }_{{{{{{{{\rm{s}}}}}}}}}{\nabla }^{2}T=\frac{{S}_{{{{{{{{\rm{h}}}}}}}}}}{{\rho }_{{{{{{{{\rm{s}}}}}}}}}{c}_{p,{{{{{{{\rm{s}}}}}}}}}}\)

Solidb

Scaling & definitionsc

\({t}^{*}=\frac{t}{{\tau }_{{{{{{{{\rm{m}}}}}}}}}}\) or \(\frac{t}{{\tau }_{{{{{{{{\rm{h}}}}}}}}}}\), \({Y}_{i}^{*}=\frac{{Y}_{i}-{Y}_{i,t=0}}{\Delta {Y}_{i}}=\frac{{Y}_{i}-{Y}_{i,t=0}}{{Y}_{i,t\to \infty }-{Y}_{i,t=0}}\), \({X}_{i}^{*}=\frac{{X}_{i}-{X}_{i,t=0}}{{X}_{i,t\to \infty }-{X}_{i,t=0}}\), \({V}^{*}=\frac{V}{{V}_{{{{{{{{\rm{in}}}}}}}}}}\), \({T}^{*}=\frac{T-{T}_{t=0}}{\Delta T}=\frac{T-{T}_{t=0}}{{T}_{t\to \infty }-{T}_{t=0}}\), ∇* = Lcell ∇ , \({x}^{*}=\frac{x}{{W}_{{{{{{{{\rm{cell}}}}}}}}}}\),

\({y}^{*}=\frac{y}{{L}_{{{{{{{{\rm{cell}}}}}}}}}}\), \({z}^{*}=\frac{z}{{H}_{{{{{{{{\rm{ch}}}}}}}}}}\) or \(\frac{z}{{\delta }_{{{{{{{{\rm{DL}}}}}}}}}}\) or \(\frac{z}{{\delta }_{{{{{{{{\rm{FL}}}}}}}}}}\), \({S}_{{{{{{{{\rm{m}}}}}}}},i}^{*}=\frac{{S}_{{{{{{{{\rm{m}}}}}}}},i}{\tau }_{{{{{{{{\rm{m}}}}}}}}}}{{\rho }_{0}\Delta {Y}_{i}}\), \({S}_{{{{{{{{\rm{h}}}}}}}}}^{*}=\frac{{S}_{{{{{{{{\rm{h}}}}}}}}}{\tau }_{{{{{{{{\rm{h}}}}}}}}}}{{\rho }_{{{{{{{{\rm{eff}}}}}}}}}{c}_{p,{{{{{{{\rm{eff}}}}}}}}}\Delta T}\) or \(\frac{{S}_{{{{{{{{\rm{h}}}}}}}}}{\tau }_{{{{{{{{\rm{h}}}}}}}}}}{{\rho }_{{{{{{{{\rm{s}}}}}}}}}{c}_{p,{{{{{{{\rm{s}}}}}}}}}\Delta T}\), \(\alpha=\frac{k}{\rho {c}_{p}}\), \({\alpha }_{{{{{{{{\rm{eff}}}}}}}}}=\frac{{k}_{{{{{{{{\rm{eff}}}}}}}}}}{{\rho }_{{{{{{{{\rm{eff}}}}}}}}}{c}_{p,{{{{{{{\rm{eff}}}}}}}}}}\), \({\alpha }_{{{{{{{{\rm{s}}}}}}}}}=\frac{{k}_{{{{{{{{\rm{s}}}}}}}}}}{{\rho }_{{{{{{{{\rm{s}}}}}}}}}{c}_{p,{{{{{{{\rm{s}}}}}}}}}}\)

Dimensionless equations

Domains

Conservation of species

\(\frac{\partial {Y}_{i}^{*}}{\partial {t}^{*}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{m}}}}}}}}}}{{{{{{{{{\boldsymbol{L}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}/{{{{{{{{\boldsymbol{V}}}}}}}}}_{{{{{{{{\bf{in}}}}}}}}}}{\overrightarrow{V}}^{*}\cdot {\nabla }^{*}{Y}_{i}^{*}-\left(\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{m}}}}}}}}}}{{{{{{{{{\boldsymbol{W}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{\boldsymbol{D}}}}}}}}}\frac{{\partial }^{2}{Y}_{i}^{*}}{\partial {x}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{m}}}}}}}}}}{{{{{{{{{\boldsymbol{L}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{\boldsymbol{D}}}}}}}}}\frac{{\partial }^{2}{Y}_{i}^{*}}{\partial {y}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{m}}}}}}}}}}{{{{{{{{{\boldsymbol{H}}}}}}}}}_{{{{{{{{\bf{ch}}}}}}}}}^{2}/{{{{{{{\boldsymbol{D}}}}}}}}}\frac{{\partial }^{2}{Y}_{i}^{*}}{\partial {z}^{*2}}\right)=0\)

Fluid channel

\(\varepsilon \frac{\partial {Y}_{i}^{*}}{\partial {t}^{*}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{m}}}}}}}}}}{{{{{{{{{\boldsymbol{L}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}/{{{{{{{{\boldsymbol{V}}}}}}}}}_{{{{{{{{\bf{in}}}}}}}}}}{\overrightarrow{V}}^{*}\cdot {\nabla }^{*}{Y}_{i}^{*}-\left(\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{m}}}}}}}}}}{{{{{{{{{\boldsymbol{W}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{D}}}}}}}}}_{{{{{{{{\bf{eff}}}}}}}}}}\frac{{\partial }^{2}{Y}_{i}^{*}}{\partial {x}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{m}}}}}}}}}}{{{{{{{{{\boldsymbol{L}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{D}}}}}}}}}_{{{{{{{{\bf{eff}}}}}}}}}}\frac{{\partial }^{2}{Y}_{i}^{*}}{\partial {y}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{m}}}}}}}}}}{{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{{{{{{{{\bf{DL}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{D}}}}}}}}}_{{{{{{{{\bf{eff}}}}}}}}}}\frac{{\partial }^{2}{Y}_{i}^{*}}{\partial {z}^{*2}}\right)=0\)

ADL,CDL

\(\varepsilon \frac{\partial {Y}_{i}^{*}}{\partial {t}^{*}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{m}}}}}}}}}}{{{{{{{{{\boldsymbol{L}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}/{{{{{{{{\boldsymbol{V}}}}}}}}}_{{{{{{{{\bf{in}}}}}}}}}}{\overrightarrow{V}}^{*}\cdot {\nabla }^{*}{Y}_{i}^{*}-\left(\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{m}}}}}}}}}}{{{{{{{{{\boldsymbol{W}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{D}}}}}}}}}_{{{{{{{{\bf{eff}}}}}}}}}}\frac{{\partial }^{2}{Y}_{i}^{*}}{\partial {x}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{m}}}}}}}}}}{{{{{{{{{\boldsymbol{L}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{D}}}}}}}}}_{{{{{{{{\bf{eff}}}}}}}}}}\frac{{\partial }^{2}{Y}_{i}^{*}}{\partial {y}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{m}}}}}}}}}}{{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{{{{{{{{\bf{FL}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{D}}}}}}}}}_{{{{{{{{\bf{eff}}}}}}}}}}\frac{{\partial }^{2}{Y}_{i}^{*}}{\partial {z}^{*2}}\right)={S}_{{{{{{{{\rm{m}}}}}}}},i}^{*}\)

AFL,CFL

Conservation of energy

\(\frac{\partial {T}^{*}}{\partial {t}^{*}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{L}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}/{{{{{{{{\boldsymbol{V}}}}}}}}}_{{{{{{{{\bf{in}}}}}}}}}}{\overrightarrow{V}}^{*}\cdot {\nabla }^{*}{T}^{*}-\left(\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{W}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{\boldsymbol{\alpha }}}}}}}}}\frac{{\partial }^{2}{T}^{*}}{\partial {x}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{L}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{\boldsymbol{\alpha }}}}}}}}}\frac{{\partial }^{2}{T}^{*}}{\partial {y}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{H}}}}}}}}}_{{{{{{{{\bf{ch}}}}}}}}}^{2}/{{{{{{{\boldsymbol{\alpha }}}}}}}}}\frac{{\partial }^{2}{T}^{*}}{\partial {z}^{*2}}\right)=0\)

Fluid channel

\(\frac{\partial {T}^{*}}{\partial {t}^{*}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{L}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}/{{{{{{{{\boldsymbol{V}}}}}}}}}_{{{{{{{{\bf{in}}}}}}}}}}\frac{\rho {c}_{p}}{{\rho }_{{{{{{{{\rm{eff}}}}}}}}}{c}_{p,{{{{{{{\rm{eff}}}}}}}}}}{\overrightarrow{V}}^{*}\cdot {\nabla }^{*}{T}^{*}-\left(\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{W}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{\alpha }}}}}}}}}_{{{{{{{{\bf{eff}}}}}}}}}}\frac{{\partial }^{2}{T}^{*}}{\partial {x}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{L}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{\alpha }}}}}}}}}_{{{{{{{{\bf{eff}}}}}}}}}}\frac{{\partial }^{2}{T}^{*}}{\partial {y}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{{{{{{{{\bf{DL}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{\alpha }}}}}}}}}_{{{{{{{{\bf{eff}}}}}}}}}}\frac{{\partial }^{2}{T}^{*}}{\partial {z}^{*2}}\right)={S}_{{{{{{{{\rm{h}}}}}}}}}^{*}\)

ADL, CDL

\(\frac{\partial {T}^{*}}{\partial {t}^{*}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{L}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}/{{{{{{{{\boldsymbol{V}}}}}}}}}_{{{{{{{{\bf{in}}}}}}}}}}\frac{\rho {c}_{p}}{{\rho }_{{{{{{{{\rm{eff}}}}}}}}}{c}_{p,{{{{{{{\rm{eff}}}}}}}}}}{\overrightarrow{V}}^{*}\cdot {\nabla }^{*}{T}^{*}-\left(\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{W}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{\alpha }}}}}}}}}_{{{{{{{{\bf{eff}}}}}}}}}}\frac{{\partial }^{2}{T}^{*}}{\partial {x}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{L}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{\alpha }}}}}}}}}_{{{{{{{{\bf{eff}}}}}}}}}}\frac{{\partial }^{2}{T}^{*}}{\partial {y}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{{{{{{{{\bf{FL}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{\alpha }}}}}}}}}_{{{{{{{{\bf{eff}}}}}}}}}}\frac{{\partial }^{2}{T}^{*}}{\partial {z}^{*2}}\right)={S}_{{{{{{{{\rm{h}}}}}}}}}^{*}\)

AFL, CFL

\(\frac{\partial {T}^{*}}{\partial {t}^{*}}-\left(\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{W}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{\alpha }}}}}}}}}_{{{{{{{{\bf{s}}}}}}}}}}\frac{{\partial }^{2}{T}^{*}}{\partial {x}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{L}}}}}}}}}_{{{{{{{{\bf{cell}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{\alpha }}}}}}}}}_{{{{{{{{\bf{s}}}}}}}}}}\frac{{\partial }^{2}{T}^{*}}{\partial {y}^{*2}}+\frac{{{{{{{{{\boldsymbol{\tau }}}}}}}}}_{{{{{{{{\bf{h}}}}}}}}}}{{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{{{{{{{{\bf{s}}}}}}}}}^{2}/{{{{{{{{\boldsymbol{\alpha }}}}}}}}}_{{{{{{{{\bf{s}}}}}}}}}}\frac{{\partial }^{2}{T}^{*}}{\partial {z}^{*2}}\right)={S}_{{{{{{{{\rm{h}}}}}}}}}^{*}\)

Solidb

  1. aPorous media includes ADL, AFL, CDL, and CFL.
  2. bSolid domain includes interconnect and solid oxide electrolyte.
  3. cThe subscript ‘eff’ denotes the effective properties of porous media, which are calculated by averaging the fluid and solid properties volumetrically15. For example, the effective thermal conductivity of porous media is calculated as keff = εk + (1 − ε)ks.
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