Theoretical investigation of steady state multiplicities in solid oxide fuel cells^*

Abstract

The nonlinear steady state behaviour of solid oxide fuel cells (SOFCs) is investigated. It is found that the temperature dependence of the electrolyte conductivity has a very strong influence on the occurrence of multiple steady states, instabilities and the formation of hot spots. Two correlations from the literature for the electrolyte conductivity are studied in a lumped model and in a 1D spatially distributed model of a SOFC. The cases of galvanostatic operation, potentiostatic operation, and operation under a constant ohmic load are considered. The lumped model possesses a unique steady state under galvanostatic operation and up to three steady states under potentiostatic operation or under constant load. In the distributed model, three steady states may coexist under galvanostatic operation and up to five under potentiostatic operation.

Appendices

Appendix A. Model equations of the spatially distributed model

• Anode and cathode gas channels:

  • Component material balances:

    $$ {\partial\over \partial t}\left(y_j^{A/C} c_t^{A/C}\right)= \mp{1\over H^{A/C} B}{\partial\dot{n}_j^{A/C}\over \partial z}+{\nu_j^{A/C}\over H^{A/C}}{i\over 2F} $$

    (6)
    $$y_j^A(0,t)=y_{j,in}^A(t),\quad y_j^C(L,t) = y_{j,in}^C(t) $$

    (7)

    (j=H₂, H₂O on anode side, and j=N₂, O₂ on cathode side)

  • Total material balance:

    $$ {\partial c_t^{A/C}\over \partial t}=\mp{1\over H^{A/C}B}{ \partial\dot{n}^{A/C}\over {\partial z}}+{1\over H^{A/C}} \sum_{j}\nu_j^{A/C}{i\over 2F} $$

    (8)
    $$ \dot{n}^A(0,t)=\dot{n}_{in}^A(t),\quad\dot{n}^C(L,t)=\dot{n}_{in}^C(t) $$

    (9)

  • Temperature equation on anode side

    $$ \eqalign{c_t^A c_P^A {\partial T^A\over \partial t}=&-{\dot{n}^A\over H^A B}c_P^A {\partial T^A\over \partial z}+{i\over 2F} {c_{P,{\rm H_2O}}\over H^A}\left( T^S - T^A \right)\cr &+{\alpha\over H^A}\left(T^S-T^A\right)} $$

    (10)
    $$ T^A(0,t)=T_{in}^A(t) $$

    (11)

  • Temperature equation on cathode side

    $$ c_t^C c_P^C {\partial T^C\over \partial t}={\dot{n}^C\over H^C B} c_P^C {\partial T^C\over \partial z}+{\alpha\over H^C} \left(T^S-T^C\right) $$

    (12)
    $$ T^C(L,t)=T_{in}^C(t) $$

    (13)

  • Specific molar heat capacity:

    $$ c_P^{A/C}=\sum_{j}y_j^{A/C} c_{P,j} $$

    (14)

  • Thermal equation of state:

    $$ c_t^{A/C}={p\over \mathbb{R} T^{A/C}} $$

    (15)

• Solid phase:

  • - Component material balances:

    $$ 0=D_{eff}^{A/C} c_t {y_j^{A/C}-y_j^S\over d^{A/C}}+\nu_j^{A/C}{i\over 2F} $$

    (16)

  • Anodic reaction kinetics:

    $$ i=\gamma^A y_{{\rm H}_2}^S y_{{\rm H_2O}}^S \hbox{exp}\left(- {E^A\over \mathbb{R} T^S}\right) \left\{\hbox{exp}\left(\theta_a^A{F\over \mathbb{R} T^S}\eta^A \right)-\hbox{exp}\left( -\theta_c^A{F\over \mathbb{R} T^S}\eta^A \right) \right\} $$

    (17)

  • Cathodic reaction kinetics:

    $$ i=\gamma^C {y_{{\rm O}_2}^S}^{0.25}\hbox{ exp}\left(- {E^C\over \mathbb{R}T^S}\right) \left\{\hbox{exp}\left( \theta_a^C{F\over \mathbb{R} T^S}\eta^C \right)-\hbox{exp}\left( -\theta_c^C{F\over \mathbb{R}T^S}\eta^C \right)\right\} $$

    (18)

  • Charge balances in the electrodes:

    $$ {\partial\over \partial z}\left({d^{A/C}\over \rho^{A/C}}{\partial \Phi^{A/C} \over \partial z}\right)=\pm i $$

    (19)
    $$ \Phi^A(0,t)=0, \quad {Bd^C\over \rho^C}\left.{\partial\Phi^C\over \partial z}\right|_{0,t}=I, \quad \left.{\partial\Phi^A\over \partial z}\right|_{L,t}= \left.{\partial\Phi^C\over \partial z}\right|_{L,t}=0 $$

    (20)

  • Voltage drop in the electrolyte:

    $$ \eqalign{ U^{\rm Cell}&=\Phi^C(0,t)-\Phi^A(0,t)\cr &=U^0(T^S)-\eta^A-\eta^C-\rho^E(T^S)d^E i} $$

    (21)

  • Open circuit voltage:

    $$ U^0(T^S)=-{1\over 2F}\Bigg(\Delta_R G-\Delta_R S (T^S - T_{ref}) +\mathbb{R}T^S \hbox{ln}{y_{{\rm H_2O}}^A\over y_{{\rm H}_2}^A {y_{{\rm O}_2}^C}^{0.5}}\Bigg) $$

    (22)

  • Electrical conductivity of the electrolyte:

    $$ \hbox{Variant I}:\quad\rho^E={1\over \beta_1}\hbox{exp}\left( {\beta_2\over T^S}\right) $$

    (23)
    $$ \hbox{Variant II}:\quad\rho^E={T^S\over C^{SE}}\hbox{exp}\left( {E^{SE}\over \mathbb{R}T^S}\right) $$

    (24)

  • Temperature equation for the solid phase:

    $$ \eqalign{\left(d^A + d^E + d^C \right) \left(\rho c_P\right)^S {\partial T^S\over \partial t}=&\left({(-\Delta_R H)\over 2F}- \left(\Phi^C - \Phi^A \right) \right) i\cr &+{d^A\over \rho^A} \left({\partial\Phi^A\over\partial z}\right)^2+ {d^C\over \rho^C}\left({\partial\Phi^C\over \partial z}\right)^2+\left(d^A + d^E + d^C \right)\lambda {\partial^2 T^S\over \partial z^2}\cr &+\left(\alpha+{i\over 2F}c_{P,{\rm H}_2}\right)\left(T^A - T^S \right)\cr &+\left(\alpha+{i\over 4F}c_{P,{\rm O}_2}\right)\left(T^C - T^S \right)} $$

    (25)
    $$ -\uplambda\left.{\partial T^S\over \partial z}\right|_{0,t}=\alpha\left(T_{\rm amb}-T^S(0,t)\right) $$

    (26)
    $$ \uplambda\left.{\partial T^S\over \partial z}\right|_{L,t}=\alpha \left(T_{\rm amb}-T^S(L,t)\right) $$

    (27)

  • Depending on the mode of operation, one of the following three equations is used for describing the external electrical circuit:

  • Case (1): galvanostatic operation

    $$ I={\rm const.} $$

    (28)

  • Case (2): potentiostatic operation

    $$ U^{\rm Cell}={\rm const.} $$

    (29)

  • Case (3): operation with an external ohmic load

    $$ U^{\rm Cell}=R I $$

    (30)

Appendix B. Model equations of the lumped model

• Anodic reaction kinetics:

  $$ {I\over LB}= \upgamma^A y_{{\rm H}_2}^S y_{{\rm H_2O}}^S\hbox{exp}\left(- {E^A\over \mathbb{R}T^S}\right) \left\{\hbox{ exp}\left( \theta_a^A{F\over \mathbb{R}T^S}\eta^A\right)-\hbox{exp}\left( -\theta_c^A{F\over \mathbb{R}T^S}\eta^A \right) \right\} $$

  (31)

• Cathodic reaction kinetics:

  $$ {I\over LB}= \gamma^C {y_{{\rm O}_2}^S}^{0.25} \hbox{exp}\left(- {E^C\over \mathbb{R}T^S}\right) \left\{\exp\left(\theta_a^C{F\over \mathbb{R}T^S}\eta^C \right)-\hbox{exp} \left(-\theta_c^C{F\over \mathbb{R}T^S}\eta^C \right)\right\} $$

  (32)

• Voltage drop in the electrolyte:

  $$ U^{\rm Cell}=\Phi^C-\Phi^A=U^0(T^S)-\eta^A-\eta^C- \rho^E(T^S)d^E {I\over LB} $$

  (33)

• Temperature equation for the solid phase:

  $$ \eqalign{(d^A + d^E + d^C)(\rho c_P)^S {d T^S\over dt}=&\left({(-\Delta_R H)\over 2F}- \left(\Phi^C-\Phi^A\right)\right){I\over LB}\cr &+\left(\upalpha+{c_{P,{\rm H}_2}\over 2F}{I\over LB}\right) \left(T^A - T^S \right)\cr &+\left(\upalpha+{c_{P,{\rm O}_2}\over 4F}{I\over LB}\right) \left(T^C - T^S \right)} $$

  (34)

• External load described by Equations (28), (29), or (30)