三元锂离子电池多目标热优化

三元锂离子电池多目标热优化

丁昌明, 文　华

Multi-objective thermal optimization of ternary lithium-ion battery

Changming DING, Hua WEN

表2 控制方程和边界条件

Table 2 Governing equations and boundary conditions

控制方程	边界条件
质量守恒
$\frac{? c_{1}}{? t} + \frac{1}{r^{2}} \frac{?}{? r} (- r^{2} D_{1} \frac{? c_{1}}{? r}) = 0$	${\frac{? c_{1}}{? t}\|}_{r = 0} = 0, {- D_{1} \frac{? c_{1}}{? r}\|}_{r ? = ? R_{p}} = \frac{j_{l o c}}{F}$
$ε_{2} \frac{? c_{2}}{? t} = \frac{?}{? x} (D_{2}^{e f f} \frac{? c_{2}}{? r}) + \frac{(1 - t_{+}) S_{a} j_{l o c}}{F}$	${\frac{? c_{2}}{? x}\|}_{x ? = ? 0} = {\frac{? c_{2}}{? x}\|}_{x ? = ? L} = 0$
电荷守恒
$\frac{?}{? x} (- σ_{1}^{e f f} \frac{? φ_{1}}{? x}) = - S_{a} j_{1 o c}$	${φ_{1}\|}_{x = 0} = 0, {- σ_{1}^{e f f} \frac{? φ_{1}}{? x}\|}_{x = L_{n} + L_{s} + L_{p}} = - I_{a p p}, {- σ_{1, n}^{e f f} \frac{? φ_{1}}{? x}\|}_{x = L_{n}} = {- σ_{1, p}^{e f f} \frac{? φ_{1}}{? x}\|}_{x = L_{n} + L_{s}} = 0$
$? [- σ_{2}^{e f f} ? φ_{2} + \frac{2 R T σ_{2}^{e f f}}{F} (1 + \frac{? l n f_{\pm}}{? l n c_{2}}) (1 - t_{+}) ? l n c_{2}] = S_{a} j_{l o c}$	${\frac{? φ_{2}}{? x}\|}_{x = 0} = {\frac{? φ_{2}}{? x}\|}_{x = L_{n} + L_{s} + L_{p}} = 0$
电化学反应速率
$j_{l o c} = i_{0} (e x p \frac{α_{a} F η}{R T} - e x p \frac{α_{c} F η}{R T})$ , $η = φ_{1} - φ_{2} - U$ , $i_{0} = F k_{0} c_{2}^{α_{a}} {(c_{1, m a x} - c_{1, s u r f})}^{α_{a}} c_{1, s u r f}^{α_{c}}$
能量守恒
$ρ c_{p} \frac{? T}{? t} + ? ? (- λ ? T) = Q_{r e a} + Q_{a c t} + Q_{o h m}$ $Q_{r e a} = S_{a} j_{l o c} T \frac{? U}{? T} = S_{a} j_{l o c} T \frac{? S}{F}$ , $Q_{a c t} = S_{a} j_{l o c} η$ $Q_{o h m} = σ_{1}^{e f f} ? φ_{1} ? ? φ_{1} + [σ_{2}^{e f f} ? φ_{2} - \frac{2 R T σ_{2}^{e f f}}{F} (1 + \frac{? l n f_{\pm}}{? l n c_{2}}) (1 - t_{+}) ? (l n ? c_{2})] ? ? φ_{2}$	${- λ \frac{? T}{? x}\|}_{x = 0} = {- λ \frac{? T}{? x}\|}_{x = L_{n} + L_{s} + L_{p}} = h (T - T_{a m b})$