Battery Echem Thermal Coupled

def _butler_volmer(i0, eta, T):
    """Butler-Volmer current density with symmetric transfer coefficients."""
    f = _F / (_R_GAS * T)
    arg_a = np.clip(_ALPHA_BV * f * eta, -_EXP_CLAMP, _EXP_CLAMP)
    arg_c = np.clip(-_ALPHA_BV * f * eta, -_EXP_CLAMP, _EXP_CLAMP)
    return i0 * (np.exp(arg_a) - np.exp(arg_c))

def _arrhenius_rate(A, E, T):
    """Safe Arrhenius rate: A * exp(-E/(R*T)) with clamped exponent."""
    arg = np.clip(E / (_R_GAS * T), 0.0, _EXP_CLAMP)
    return A * np.exp(-arg)

def _four_stage_decomposition(alpha_sei, alpha_ae, alpha_ca, alpha_el, T):
    """Compute the four decomposition rates and total volumetric heat.

    Returns (d_sei, d_ae, d_ca, d_el, q_dot) where q_dot is W/kg (mass-specific).
    """
    k_sei = _arrhenius_rate(_A_SEI, _E_SEI, T)
    k_ae = _arrhenius_rate(_A_AE, _E_AE, T)
    k_ca = _arrhenius_rate(_A_CA, _E_CA, T)
    k_el = _arrhenius_rate(_A_EL, _E_EL, T)

    # SEI: consumed (α decreases)
    d_sei = -k_sei * alpha_sei

    # Anode-electrolyte: consumed fraction increases
    d_ae = k_ae * alpha_ae * (1.0 - alpha_ae)
    # At α_ae=0 this would give zero rate, but the seed is the
    # SEI decomposition products — use a small baseline nucleation
    # once SEI has started decomposing.
    if alpha_ae < 1e-12 and alpha_sei < 0.15 - 1e-6:
        d_ae = k_ae * 1e-6

    # Cathode: decomposed fraction increases
    d_ca = k_ca * (1.0 - alpha_ca)

    # Electrolyte: decomposed fraction increases
    d_el = k_el * (1.0 - alpha_el)

    q_dot = (_Q_SEI * _W_SEI * abs(d_sei)
             + _Q_AE * _W_AE * d_ae
             + _Q_CA * _W_CA * d_ca
             + _Q_EL * _W_EL * d_el)

    return d_sei, d_ae, d_ca, d_el, q_dot

def _radial_diffusion_3node(c, D_s, r_p):
    """Spherical Fickian diffusion on 3 uniform radial nodes (center, mid, surface).

    Returns dc/dt for the 3 nodes.  Boundary: dc/dr=0 at center, zero-flux at surface.
    """
    dr = r_p / 2.0
    dc = np.zeros(3)

    # center (symmetry BC): forward difference approximation
    dc[0] = 6.0 * D_s * (c[1] - c[0]) / (dr * dr)

    # mid-point
    r_m = dr
    flux_out = D_s * ((r_m + 0.5 * dr) ** 2) * (c[2] - c[1]) / dr
    flux_in  = D_s * ((r_m - 0.5 * dr) ** 2) * (c[1] - c[0]) / dr
    dc[1] = (flux_out - flux_in) / (r_m * r_m * dr)

    # surface (zero-flux BC at outer boundary for now)
    r_s = r_p
    flux_in_s = D_s * ((r_s - 0.5 * dr) ** 2) * (c[2] - c[1]) / dr
    dc[2] = -flux_in_s / (r_s * r_s * dr)

    return dc

def _echem_thermal_rhs(t, y):
    # Decomposition fractions
    alpha_sei = np.clip(y[0], 0.0, 1.0)
    alpha_ae  = np.clip(y[1], 0.0, 1.0)
    alpha_ca  = np.clip(y[2], 0.0, 1.0)
    alpha_el  = np.clip(y[3], 0.0, 1.0)
    T         = np.clip(y[4], 250.0, 2000.0)

    # Butler-Volmer overpotentials (treated as state-like for stiffness)
    eta_a = y[5]
    eta_c = y[6]

    # Li concentrations (3 radial nodes each)
    c_a = np.clip(y[7:10], 0.0, _CS_MAX_A)
    c_c = np.clip(y[10:13], 0.0, _CS_MAX_C)

    Q_total = y[13]
    P_gas   = y[14]
    R_int   = y[15]

    # --- 4-stage decomposition ---
    d_sei, d_ae, d_ca, d_el, q_dot_decomp = _four_stage_decomposition(
        alpha_sei, alpha_ae, alpha_ca, alpha_el, T,
    )

    # --- Butler-Volmer kinetics ---
    i_bv_a = _butler_volmer(_I0_ANODE, eta_a, T)
    i_bv_c = _butler_volmer(_I0_CATHODE, eta_c, T)

    # Overpotential relaxation toward equilibrium (τ ~ RC time constant)
    tau_relax = 1.0  # s
    # Surface concentration deviation from equilibrium drives η
    theta_a = c_a[2] / _CS_MAX_A
    theta_c = c_c[2] / _CS_MAX_C
    # OCV approximation (simplified lithium intercalation)
    U_a = 0.6 - 0.5 * theta_a
    U_c = 4.2 - 0.8 * theta_c
    V_cell = U_c - U_a
    d_eta_a = (-eta_a + (V_cell * 0.5 - U_a)) / tau_relax
    d_eta_c = (-eta_c + (U_c - V_cell * 0.5)) / tau_relax

    # --- Solid-phase diffusion ---
    dc_a = _radial_diffusion_3node(c_a, _DS_ANODE, _RP_ANODE)
    dc_c = _radial_diffusion_3node(c_c, _DS_CATHODE, _RP_CATHODE)

    # Electrode coupling: BV current consumes/produces Li at surface node
    # j_n = i_BV / F  (flux in mol/(m²·s))
    dc_a[2] += i_bv_a / _F
    dc_c[2] -= i_bv_c / _F

    # --- Temperature ---
    q_echem = abs(i_bv_a * eta_a) + abs(i_bv_c * eta_c)  # W/m² → scale to cell
    q_gen = q_dot_decomp * _M_CELL + q_echem * _A_SURF
    q_cool = _H_CONV * _A_SURF * (T - _T_AMB)
    dT = (q_gen - q_cool) / (_M_CELL * _CP)

    # Cumulative heat
    dQ = q_gen

    # Gas pressure
    n_gas_max = 0.01
    n_gas = alpha_el * n_gas_max
    dn_gas = n_gas_max * d_el
    dP = (dn_gas * _R_GAS * T + n_gas * _R_GAS * dT) / _V_HEAD

    # Internal resistance
    dR = _R0 * (0.5 * abs(d_sei) + 2.0 * d_ae)

    dy = np.empty(16)
    dy[0] = d_sei
    dy[1] = d_ae
    dy[2] = d_ca
    dy[3] = d_el
    dy[4] = dT
    dy[5] = d_eta_a
    dy[6] = d_eta_c
    dy[7:10] = dc_a
    dy[10:13] = dc_c
    dy[13] = dQ
    dy[14] = dP
    dy[15] = dR
    return dy