﻿ 一种质子交换膜燃料电池冷启动分层数值模型
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 同济大学学报(自然科学版)  2019, Vol. 47 Issue (1): 104-112.  DOI: 10.11908/j.issn.0253-374x.2019.01.014 0

### 引用本文

XU Peng, XU Sichuan, GUO Xin, LIU Pengcheng, LIU Xingyu. A Layered Numerical Model for Cold Start in Proton Exchange Membrane Fuel Cell[J]. Journal of Tongji University (Natural Science), 2019, 47(1): 104-112. DOI: 10.11908/j.issn.0253-374x.2019.01.014

### 文章历史

1. 同济大学 汽车学院, 上海 201804;
2. 同济大学 新能源汽车工程中心，上海 201804

A Layered Numerical Model for Cold Start in Proton Exchange Membrane Fuel Cell
XU Peng 1,2, XU Sichuan 1,2, GUO Xin 1,2, LIU Pengcheng 1,2, LIU Xingyu 1,2
1. College of Automotive Studies, Tongji University, Shanghai 201804, China;
2. Clean Energy Automotive Engineering Center, Tongji University, Shanghai 201804, China
Abstract: In terms of complex physical and chemical phenomenon of proton exchange membrane fuel cell (PEMFC) in cold start, a novel numerical model of PEMFC in cold start was developed. The model comprehensively considered electrochemical reaction, thermal and water and phase change mechanism in cold start. Three factors, ionomer volume fraction in catalyst layer, porosity and membrane thickness, were investigated. The indicator of water storage capacity(WSC) was finally introduced to evaluate the cold start capability of PEMFC.
Key words: proton exchange membrane fuel cell (PEMFC)    cold start    phase change

1 PEMFC冷启动分层数值模型

 图 1 燃料电池冷启动过程水热传递 Fig.1 Thermal and water transport mechanism inside PEMFC during cold start

1.1 质子交换膜

 ${\lambda _n} = \frac{E}{{{\rho _{\rm{m}}}}}{c_{{\rm{h2o}}}}$ (1)

 ${\rho _{\rm{m}}}{C_{\rm{m}}}\frac{{{\rm{d}}{T_{\rm{m}}}}}{{{\rm{d}}t}} = {Q_{{\rm{o}} - {\rm{m}}}} + {Q_{{\rm{ac}} - {\rm{m}}}} + {Q_{{\rm{cc}} - {\rm{m}}}} + {S_{{\rm{t}} - {\rm{m}}}}$ (2)

 ${Q_{{\rm{o}} - {\rm{m}}}} = \frac{{{I^2}{A_{\rm{m}}}}}{{{\delta _{\rm{m}}}}}$ (3)

 ${A_{\rm{m}}} = \frac{{{\delta _{\rm{m}}}}}{{{\kappa _{\rm{m}}}}}$ (4)
 ${\kappa _{\rm{m}}} = \left( {0.513\;9\lambda - 0.326} \right)\exp \left[ {1\;268\left( {\frac{1}{{303.15}} - \frac{1}{{{T_{\rm{m}}}}}} \right)} \right]$ (5)

 ${Q_{{\rm{cc}} - {\rm{m}}}} = \frac{{{T_{{\rm{cc}}}} - {T_{\rm{m}}}}}{{{\delta _{{\rm{cc}}}}/2 + {\delta _{\rm{m}}}/2}}\frac{{{k_{{\rm{cc}} - {\rm{m}}}}}}{{{\delta _{\rm{m}}}}}$ (6)
 ${Q_{{\rm{ac}} - {\rm{m}}}} = \frac{{{T_{{\rm{ac}}}} - {T_{\rm{m}}}}}{{{\delta _{{\rm{ac}}}}/2 + {\delta _{\rm{m}}}/2}}\frac{{{k_{{\rm{ac}} - {\rm{m}}}}}}{{{\delta _{\rm{m}}}}}$ (7)

 ${k_{{\rm{cc}} - {\rm{m}}}} = \frac{{{\delta _{\rm{m}}}/2 + {\delta _{{\rm{cc}}}}/2}}{{{\delta _{\rm{m}}}/2/{k_{\rm{m}}} + {\delta _{{\rm{cc}}}}/2/{k_{{\rm{cc}}}}}}$ (8)
 ${k_{{\rm{ac}} - {\rm{m}}}} = \frac{{{\delta _{\rm{m}}}/2 + {\delta _{{\rm{ac}}}}/2}}{{{\delta _{\rm{m}}}/2/{k_{\rm{m}}} + {\delta _{{\rm{ac}}}}/2/{k_{{\rm{ac}}}}0}}$ (9)

 ${S_{{\rm{t}} - {\rm{m}}}} = {h_{{\rm{fusn}}}}\left( {{M_{{\rm{h2o}}}}{S_{{\rm{nf}} - {\rm{m}}}}} \right)$ (10)

 $\frac{{{\rho _{\rm{m}}}}}{{EW}}w\frac{{{\rm{d}}{\lambda _{\rm{n}}}}}{{{\rm{d}}t}} = {\lambda _{{\rm{m}} - {\rm{ac}}}} + {\lambda _{{\rm{m}} - {\rm{cc}}}} - {S_{{\rm{nf}} - {\rm{m}}}}$ (11)

 ${\lambda _{{\rm{m}} - {\rm{ac}}}} = \frac{{{\rho _{\rm{m}}}}}{{{\delta _{\rm{m}}}EW}}{D_{{\rm{n}} - {\rm{m}} - {\rm{ac}}}}\frac{{{\lambda _{{\rm{n}} - {\rm{ac}}}} - {\lambda _{\rm{n}}}}}{{{\delta _{\rm{m}}}/2 + {\delta _{{\rm{ac}}}}/2}}$ (12)
 ${\lambda _{{\rm{m}} - {\rm{cc}}}} = \frac{{{\rho _{\rm{m}}}}}{{{\delta _{\rm{m}}}EW}}{D_{{\rm{n}} - {\rm{m}} - {\rm{cc}}}}\frac{{{\lambda _{{\rm{n}} - {\rm{cc}}}} - {\lambda _{\rm{n}}}}}{{{\delta _{\rm{m}}}/2 + {\delta _{{\rm{cc}}}}/2}}$ (13)

 ${D_{n - {\rm{m}} - {\rm{cc}}}} = \frac{{{\delta _{\rm{m}}}/2 + {\delta _{{\rm{cc}}}}/2}}{{{\delta _{\rm{m}}}/2/{D_{{\rm{n}} - {\rm{m}}}} + {\delta _{{\rm{cc}}}}/2/{D_{n - {\rm{cc}}}}}}$ (14)
 ${D_{n - {\rm{m}} - {\rm{ac}}}} = \frac{{{\delta _{\rm{m}}}/2 + {\delta _{{\rm{ac}}}}/2}}{{{\delta _{\rm{m}}}/2/{D_{{\rm{n}} - {\rm{m}}}} + {\delta _{{\rm{ac}}}}/2/{D_{n - {\rm{ac}}}}}}$ (15)

 ${D_n} = \left\{ \begin{array}{l} 2.692\;66 \times {10^{ - 10}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\lambda _n} \le 2\\ {10^{ - 10}}\exp \left[ {2\;416\left( {\frac{1}{{303}} - \frac{1}{T}} \right)} \right]\left[ {0.87\left( {3 - {\lambda _n}} \right) + 2.95\left( {{\lambda _n} - 2} \right)} \right]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2 < {\lambda _n} \le 3\\ {10^{ - 10}}\exp \left[ {2\;416\left( {\frac{1}{{303}} - \frac{1}{T}} \right)} \right]\left[ {2.95\left( {4 - {\lambda _n}} \right) + 1.642\;454\left( {{\lambda _n} - 3} \right)} \right]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;3 < {\lambda _n} \le 4\\ {10^{ - 10}}\exp \left[ {2\;416\left( {\frac{1}{{303}} - \frac{1}{T}} \right)} \right]\left[ {2.563 - 0.33{\lambda _n} + 0.026\;4{\lambda _n} - 0.000\;671\lambda _n^3} \right.\;\;\;\;{\lambda _n} > 4 \end{array} \right.$ (16)

 ${\rho _{\rm{m}}}w\frac{{{\rm{d}}{\lambda _f}}}{{{\rm{d}}t}} = {S_{nf - {\rm{m}}}}$ (17)

1.2 催化层 1.2.1 阴极催化层

 图 2 催化层结构 Fig.2 Representation of structure in catalyst layer

 ${\overline {\rho C} _{{\rm{cc}}}}\frac{{{\rm{d}}{T_{{\rm{cc}}}}}}{{{\rm{d}}t}} = {Q_{{\rm{cg}} - {\rm{cc}}}} + {Q_{{\rm{m}} - {\rm{cc}}}} - {Q_{{\rm{ent}}}} + {Q_{{\rm{o}} - {\rm{c}}}} + {Q_{{\rm{ac}} - {\rm{c}}}} + {S_{{\rm{pc}} - {\rm{c}}}}$ (18)

 $\begin{array}{l} {\overline {\rho C} _{{\rm{cc}}}} = \left( {\left( {{s_{{\rm{ice}} - {\rm{cc}}}}{\rho _{{\rm{ice}}}}{C_{{\rm{ice}}}} + {s_{{\rm{lq}} - {\rm{cc}}}}{\rho _{{\rm{lq}}}}{C_{{\rm{lq}}}}} \right){\varepsilon _{{\rm{cc}}}} + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left. {{\rho _{{\rm{cc}}}}{C_{{\rm{cc}}}}\left( {1 - {\varepsilon _{{\rm{cc}}}} - {w_{\rm{c}}}} \right) + {w_{\rm{c}}}{\rho _{\rm{m}}}{C_{\rm{m}}}} \right) \end{array}$ (19)

 ${Q_{{\rm{m}} - {\rm{cc}}}} = \frac{{{T_{\rm{m}}} - {T_{{\rm{cc}}}}}}{{{\delta _{\rm{m}}}/2 + {\delta _{{\rm{cc}}}}/2}}\frac{{{k_{{\rm{m}} - {\rm{cc}}}}}}{{{\delta _{{\rm{cc}}}}}}$ (20)
 ${Q_{{\rm{cg}} - {\rm{cc}}}} = \frac{{{T_{{\rm{cg}}}} - {T_{{\rm{cc}}}}}}{{{\delta _{{\rm{cg}}}}/2 + {\delta _{{\rm{cc}}}}/2}}\frac{{{k_{{\rm{cg}} - {\rm{cc}}}}}}{{{\delta _{{\rm{cc}}}}}}$ (21)
 ${Q_{{\rm{ent}}}} = \frac{{{T_{{\rm{cc}}}}I\Delta S}}{{2F{\delta _{{\rm{cc}}}}}}$ (22)

 ${Q_{{\rm{o}} - {\rm{cc}}}} = \frac{{{I^2}{A_{{\rm{cc}}}}}}{{3{\delta _{{\rm{cc}}}}}}$ (23)
 ${Q_{{\rm{ac}} - {\rm{c}}}} = \frac{{I\eta }}{{{\delta _{{\rm{cc}}}}}}$ (24)

 $\begin{array}{l} {S_{{\rm{pc}} - {\rm{cc}}}} = \left( {{h_{{\rm{fusn}}}}\left( {{M_{{\rm{h2o}}}}{S_{{\rm{ni}} - {\rm{cc}}}} + {S_{{\rm{li}} - {\rm{cc}}}}} \right) + \left( {{h_{{\rm{fusn}}}} + {h_{{\rm{cond}}}}} \right){S_{{\rm{vi}} - {\rm{cc}}}} + } \right.\\ \;\;\;\;\;\;\;\;\;\left. {{h_{{\rm{cond}}}}\left( {{S_{{\rm{vi}} - {\rm{cc}}}} - {S_{{\rm{nv}} - {\rm{cc}}}}{M_{{\rm{h2o}}}}} \right)} \right) \end{array}$ (25)

 $\frac{{{\rho _{\rm{m}}}}}{{EW}}w\frac{{{\rm{d}}{\lambda _{{\rm{n}} - {\rm{cc}}}}}}{{{\rm{d}}t}} = \frac{i}{{2F{\delta _{{\rm{cc}}}}}} + {\lambda _{{\rm{n}} - {\rm{m}} - {\rm{cc}}}} - {S_{{\rm{eod}}}} - {S_{{\rm{nv}} - {\rm{cc}}}} - {S_{{\rm{ni}} - {\rm{cc}}}}$ (26)

 ${\lambda _{{\rm{n}} - {\rm{m}} - {\rm{cc}}}} = \frac{{{D_{{\rm{n}} - {\rm{m}} - {\rm{cc}}}}}}{{{\delta _{{\rm{cc}}}}}}\frac{{{\lambda _{\rm{n}}} - {\lambda _{{\rm{n}} - {\rm{cc}}}}}}{{{\delta _{\rm{m}}}/2 + {\delta _{{\rm{cc}}}}/2}}\frac{{{\rho _{\rm{m}}}}}{{EW}}$ (27)
 ${S_{{\rm{eod}}}} = {n_{\rm{d}}}\frac{I}{{{\delta _{{\rm{cc}}}}F}}{\lambda _{{\rm{n}} - {\rm{ac}}}}$ (28)

 ${\rho _{{\rm{ice}}}}{\varepsilon _{{\rm{cc}}}}\frac{{{\rm{d}}{s_{{\rm{ice}} - {\rm{cc}}}}}}{{{\rm{d}}t}} = {S_{{\rm{ice}} - {\rm{cc}}}}$ (29)
 ${S_{{\rm{ice}} - {\rm{cc}}}} = {S_{{\rm{vi}} - {\rm{cc}}}} + {S_{{\rm{li}} - {\rm{cc}}}} + {S_{{\rm{ni}} - {\rm{cc}}}}{M_{{\rm{h2o}}}}$ (30)

 ${E_{{\rm{cc}}}}\frac{{{\rm{d}}{v_{{\rm{cc}}}}}}{{{\rm{d}}t}} = {S_{{\rm{nv}} - {\rm{cc}}}} - {S_{{\rm{vl}} - {\rm{cc}}}} - {S_{{\rm{vi}} - {\rm{cc}}}} - {v_{{\rm{d}} - {\rm{cc}}}}$ (31)

 ${E_{{\rm{cc}}}} = \left( {1 - {s_{{\rm{lq}} - {\rm{cc}}}} - {s_{{\rm{ice}} - {\rm{cc}}}}} \right){\varepsilon _{{\rm{cc}}}}$ (32)
 ${v_{{\rm{d}} - {\rm{cc}} - {\rm{cg}}}} = \frac{{{D_{{\rm{v}} - {\rm{cc}} - {\rm{cg}}}}}}{{{\delta _{{\rm{cc}}}}}}\frac{{{v_{{\rm{cc}}}} - {v_{{\rm{cg}}}}}}{{{\delta _{{\rm{cc}}}}/2 + {\delta _{{\rm{cg}}}}/2}}$ (33)
1.2.2 阳极催化层

 ${\overline {\rho C} _{{\rm{ac}}}}\frac{{{\rm{d}}{T_{{\rm{ac}}}}}}{{{\rm{d}}t}} = {Q_{{\rm{ag}} - {\rm{ac}}}} + {Q_{{\rm{m}} - {\rm{ac}}}} + {Q_{{\rm{o}} - {\rm{ac}}}} + {S_{{\rm{pc}} - {\rm{ac}}}}$ (34)

 $\frac{{{\rho _{\rm{m}}}}}{{EW}}w\frac{{{\rm{d}}{\lambda _{{\rm{n}} - {\rm{ac}}}}}}{{{\rm{d}}t}} = {\lambda _{{\rm{n}} - {\rm{m}} - {\rm{ac}}}} + {S_{{\rm{eod}}}} - {S_{{\rm{nv}} - {\rm{ac}}}} - {S_{{\rm{ni}} - {\rm{ac}}}}$ (35)

 ${\rho _{{\rm{ice}}}}{\delta _{{\rm{ac}}}}\frac{{{\rm{d}}{s_{{\rm{ice}} - {\rm{ac}}}}}}{{{\rm{d}}t}} = {S_{{\rm{ice}} - {\rm{ac}}}}$ (36)

 ${S_{{\rm{ice}} - {\rm{cc}}}} = {S_{{\rm{vi}} - {\rm{ac}}}} + {S_{{\rm{li}} - {\rm{ac}}}} + {S_{{\rm{ni}} - {\rm{ac}}}}{M_{{\rm{h2o}}}}$ (37)

 ${E_{{\rm{ac}}}}\frac{{{\rm{d}}{v_{{\rm{ac}}}}}}{{{\rm{d}}t}} = {S_{{\rm{nv}} - {\rm{ac}}}} - {S_{{\rm{vl}} - {\rm{ac}}}} - {S_{{\rm{vi}} - {\rm{ac}}}} - {v_{{\rm{d}} - {\rm{ac}} - {\rm{ag}}}}$ (38)

1.3 气体扩散层模型

 ${\overline {\rho C} _{\rm{g}}}\frac{{{\rm{d}}{T_{\rm{g}}}}}{{{\rm{d}}t}} = {Q_{{\rm{c}} - {\rm{g}}}} + {Q_{{\rm{ch}} - {\rm{g}}}} + {Q_{{\rm{o}} - {\rm{g}}}} + {S_{{\rm{pc}} - {\rm{g}}}}$ (39)

 ${\overline {\rho C} _{\rm{g}}} = \left( {\left( {{s_{{\rm{ice}} - {\rm{g}}}}{\rho _{{\rm{ice}}}}{C_{{\rm{ice}}}} + {s_{{\rm{lq}} - {\rm{g}}}}{\rho _{{\rm{lq}}}}{C_{{\rm{lq}}}}} \right){\varepsilon _{\rm{c}}} + {\rho _{\rm{g}}}{C_{\rm{g}}}\left( {1 - {\varepsilon _{\rm{g}}}} \right)} \right)$ (40)
 ${S_{{\rm{pc}} - {\rm{g}}}} = \left( {{h_{{\rm{fusn}}}}{S_{{\rm{li}} - {\rm{g}}}} + \left( {{h_{{\rm{fusn}}}} + {h_{{\rm{cond}}}}} \right){S_{{\rm{vi}} - {\rm{g}}}} + {h_{{\rm{cond}}}}{S_{{\rm{vl}} - {\rm{g}}}}} \right)$ (41)
 ${Q_{{\rm{ch}} - {\rm{g}}}} = \frac{{{T_{{\rm{ch}}}} - {T_{\rm{g}}}}}{{{\delta _{{\rm{ch}}}}/2 + {\delta _{\rm{g}}}/2}}\frac{{{k_{{\rm{ch}} - {\rm{g}}}}}}{{{\delta _{\rm{g}}}}}$ (42)

 ${\rho _{{\rm{ice}}}}{\varepsilon _{\rm{g}}}\frac{{{\rm{d}}{s_{{\rm{ice}} - {\rm{g}}}}}}{{{\rm{d}}t}} = {S_{{\rm{vi}} - {\rm{g}}}} + {S_{{\rm{li}} - {\rm{g}}}}$ (43)

 ${E_{\rm{g}}}\frac{{{\rm{d}}{v_{\rm{g}}}}}{{{\rm{d}}t}} = {v_{{\rm{c}} - {\rm{g}}}} - {v_{{\rm{g}} - {\rm{ch}}}} - {S_{{\rm{vi}} - {\rm{g}}}} - {S_{{\rm{vl}} - {\rm{g}}}}$ (44)

 ${v_{{\rm{c}} - {\rm{g}}}} = \frac{{{D_{{\rm{v}} - {\rm{c}} - {\rm{g}}}}}}{{{\delta _{\rm{g}}}}}\frac{{{v_{\rm{c}}} - {v_{\rm{g}}}}}{{{\delta _{\rm{g}}}/2 + {\delta _{\rm{c}}}/2}}$ (45)
 ${v_{{\rm{g}} - {\rm{ch}}}} = \frac{{{D_{{\rm{v}} - {\rm{g}} - {\rm{ch}}}}}}{{{\delta _{\rm{g}}}}}\frac{{{v_{\rm{g}}} - {v_{{\rm{ch}}}}}}{{{\delta _{\rm{g}}}/2 + {\delta _{{\rm{ch}}}}/2}}$ (46)

1.4 流场模型

 ${\rho _{{\rm{ch}}}}{C_{{\rm{ch}}}}\frac{{{\rm{d}}{T_{{\rm{ch}}}}}}{{{\rm{d}}t}} = {Q_{{\rm{g}} - {\rm{ch}}}} + {Q_{{\rm{cp}} - {\rm{ch}}}} + {Q_{{\rm{o}} - {\rm{ch}}}}$ (47)

 $\frac{{{\rm{d}}{v_{{\rm{ch}}}}}}{{{\rm{d}}t}} = {v_{{\rm{g}} - {\rm{ch}}}} - {v_{{\rm{ch}} - {\rm{out}}}}$ (48)

 ${v_{{\rm{ch}} - {\rm{out}}}} = {v_{{\rm{ch}}}}{q_{{\rm{ch}}}}$ (49)

1.5 冷却流道模型

 ${\rho _{{\rm{cp}}}}{C_{{\rm{cp}}}}\frac{{{\rm{d}}{T_{{\rm{cp}}}}}}{{{\rm{d}}t}} = {Q_{{\rm{ep}} - {\rm{cp}}}} - {Q_{{\rm{cp}} - {\rm{cc}}}}$ (50)

 ${Q_{{\rm{cp}} - {\rm{cc}}}} = \left( {{T_{{\rm{cp}}}} - {T_{{\rm{cc}}}}} \right)\frac{{{h_{\rm{c}}}}}{{{\delta _{{\rm{cp}}}}}}$ (51)

1.6 端板模型

 ${\rho _{{\rm{ep}}}}{C_{{\rm{ep}}}}\frac{{{\rm{d}}{T_{{\rm{ep}}}}}}{{{\rm{d}}t}} = - {Q_{{\rm{ep}} - {\rm{cp}}}} + {Q_{{\rm{bd}} - {\rm{ep}}}}$ (52)

 ${Q_{{\rm{bd}} - {\rm{ep}}}} = \left( {{T_{{\rm{bd}}}} - {T_{{\rm{ep}}}}} \right)\frac{{{h_{{\rm{surr}}}}}}{{{\delta _{{\rm{ep}}}}}}$ (53)

1.7 相变源项

 ${S_{{\rm{v}} - {\rm{l}}}} = \left\{ \begin{array}{l} \left\{ \begin{array}{l} {\gamma _{{\rm{cond}}}}\varepsilon \left( {1 - {s_{{\rm{lq}}}} - {s_{{\rm{ice}}}}} \right)\left( {{c_{{\rm{vp}}}} - {c_{{\rm{sat}}}}} \right)\;\;\;\;\;\left( {若\;{c_{{\rm{vp}}}} \ge {c_{{\rm{sat}}}}} \right)\\ {\gamma _{{\rm{evap}}}}\varepsilon {s_{{\rm{lq}}}}\left( {{c_{{\rm{vp}}}} - {c_{{\rm{sat}}}}} \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {若\;{c_{{\rm{vp}}}} < {c_{{\rm{sat}}}}} \right) \end{array} \right.\;\;\;\;若\;\;T \ge {T_{{\rm{freeze}}}}\\ 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;若\;\;T < {T_{{\rm{freeze}}}} \end{array} \right.$ (54)

 ${S_{{\rm{l}} - {\rm{i}}}} = \left\{ \begin{array}{l} {\gamma _{{\rm{fusn}}}}\varepsilon {s_{{\rm{lq}}}}{\rho _{{\rm{lq}}}}\;\;\;\;\;\;\;\;若\;T \le {T_{{\rm{freeze}}}}\\ - {\gamma _{{\rm{melt}}}}\varepsilon {s_{{\rm{ice}}}}{\rho _{{\rm{ice}}}}\;\;\;\;若\;T \ge {T_{{\rm{freeze}}}} \end{array} \right.$ (55)

 ${S_{{\rm{v}} - {\rm{i}}}} = \left\{ \begin{array}{l} \left\{ \begin{array}{l} {\gamma _{{\rm{desb}}}}\varepsilon \left( {1 - {s_{{\rm{lq}}}} - {s_{{\rm{ice}}}}} \right)\left( {{c_{{\rm{vp}}}} - {c_{{\rm{sat}}}}} \right)\;\;\;\;\;\left( {若\;{c_{{\rm{vp}}}} \ge {c_{{\rm{sat}}}}} \right)\\ 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {若\;{c_{{\rm{vp}}}} < {c_{{\rm{sat}}}}} \right) \end{array} \right.\;\;\;\;若\;\;T < {T_{{\rm{freeze}}}}\\ 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;若\;\;T \ge {T_{{\rm{freeze}}}} \end{array} \right.$ (56)

 ${S_{n - {\rm{v}}}} = {\zeta _{n - {\rm{v}}}}\frac{{{\rho _{\rm{m}}}}}{E}\left( {{\lambda _{{\rm{nf}}}} - {\lambda _{{\rm{equil}}}}} \right)$ (57)

 ${S_{{\rm{n}} - {\rm{i}}}} = \left\{ \begin{array}{l} {\zeta _{{\rm{n}} - {\rm{i}}}}\frac{{{\rho _{\rm{m}}}}}{E}\left( {{\lambda _{{\rm{nf}}}} - {\lambda _{{\rm{sat}}}}} \right)\;\;\;\;若\;{\lambda _{{\rm{nf}}}} \ge {\lambda _{{\rm{sat}}}}\\ 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;若\;{\lambda _{{\rm{nf}}}} \ge {\lambda _{{\rm{sat}}}} \end{array} \right.$ (58)

 ${S_{{\rm{n}} - {\rm{f}}}} = \left\{ \begin{array}{l} {\zeta _{{\rm{n}} - {\rm{f}}}}\frac{{{\rho _{\rm{m}}}}}{E}\left( {{\lambda _{{\rm{nf}}}} - {\lambda _{{\rm{sat}}}}} \right)\;\;\;\;若\;{\lambda _{{\rm{nf}}}} \ge {\lambda _{{\rm{sat}}}}\\ {\zeta _{{\rm{n}} - {\rm{f}}}}\frac{{{\rho _{\rm{m}}}}}{E}{\lambda _{\rm{f}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;若\;{\lambda _{{\rm{nf}}}} \ge {\lambda _{{\rm{sat}}}} \end{array} \right.$ (59)

1.8 边界和初始条件

 $\frac{{\partial {\lambda _{{\rm{nf}}}}}}{{\partial x}}\left| {_{{\rm{CL}}\left\| {{\rm{GDL}}} \right.}} \right. = 0$ (60)

 $\frac{{\partial {\lambda _{\rm{f}}}}}{{\partial x}}\left| {_{{\rm{CL}}\left\| {{\rm{mem}}} \right.}} \right. = 0$ (61)

 $\frac{{\partial {s_{{\rm{lq}},{\rm{ice}}}}}}{{\partial x}}\left| {_{{\rm{CL}}\left\| {{\rm{mem}}} \right.}} \right. = \frac{{\partial {s_{{\rm{lq}},{\rm{ice}}}}}}{{\partial x}}\left| {_{{\rm{GDL}}\left\| {{\rm{BP}}} \right.}} \right. = 0$ (62)

 $\frac{{\partial {c_{{\rm{vp}}}}}}{{\partial x}}\left| {_{{\rm{CL}}\left\| {{\rm{mem}}} \right.}} \right. = \frac{{\partial {c_{{\rm{vp}}}}}}{{\partial x}}\left| {_{{\rm{GDL}}\left\| {{\rm{BP}}} \right.}} \right. = 0$ (63)

 ${{\dot n}_{{\rm{cout}},{\rm{vp}}}} = {\xi _{\rm{c}}}\frac{{IA}}{{4F}}\frac{1}{{0.21}}\frac{{{p_{{\rm{sat}}}}}}{{{p_{\rm{c}}}}}$ (64)
 ${{\dot n}_{{\rm{aout}},{\rm{vp}}}} = \frac{{{\xi _{\rm{a}}}IA}}{{2F{c_{{{\rm{H}}_{\rm{2}}}}}}}{c_{{\rm{vp}}}}$ (65)

 $\dot Q = hA\left( {{T_{{\rm{amb}}}} - {T_{\rm{w}}}} \right)$ (66)

2 结果分析与讨论

 图 3 低温冷启动试验与数值模拟对比 Fig.3 Comparison of voltage between simulation results and experiment

 图 4 电压与催化层孔隙率的关系 Fig.4 Relationship between voltage and catalyst layer porosity
 图 5 冰体积分数和催化层孔隙率的关系 Fig.5 Relationship between ice volume fraction and catalyst layer porosity

 图 6 电压与质子交换膜厚度关系 Fig.6 Relationship between voltage and membrane thickness
 图 7 冰体积分数和质子交换膜厚度关系 Fig.7 Relationship between ice volume fraction and membrane thickness
 图 8 电压与催化层聚合物体积分数的关系 Fig.8 Relationship between voltage and ionomer volume fraction in catalyst layer

 图 9 冰体积分数和催化层聚合物体积分数的关系 Fig.9 Relationship between ice volume fraction and ionomer volume fraction in catalyst layer
3 结论

(1) 由于初始阶段电化学反应生成的水位于阴极催化层聚合物上，阴极催化层聚合物体积分数越大，吸收的水更多，低温冷启动能力提高

(2) 质子交换膜的吸水速率和吸水量对低温冷启动能力的影响不同.膜越薄，吸水快，初始启动电压高，膜越后，吸水量大，低温冷启动能力提高；

(3) 催化层多孔介质孔隙体积越大，容纳冰的能力增强，冷启动能力升高；

(4) WSC储水量这一指标可以用来判断燃料电池低温冷启动能力，WSC值越高有利于低温冷启动.

 [1] 张剑波, 黄福森, 黄俊, 等. 质子交换膜燃料电池零下冷启动研究进展[J]. 化学通报, 2017, 80(6): 507 ZHANG Jianbo, HUANG Fusen, HUANG Jun, et al. A review on subzero startup of proton exchange membrane fuel cell[J]. Chemistry, 2017, 80(6): 507 [2] DOE technical targets for fuel cell systems and stacks for transportation applications[EB/OL][2018-03-01] https://www.energy.gov/eere/fuelcells/doe-technical-targets-fuel-cell-systems-and-stacks-transportation-applications. [3] 许澎, 高源, 许思传. 质子交换膜燃料电池停机后吹扫仿真[J]. 同济大学学报(自然科学版), 2017, 45(12): 1873 XU Peng, GAO Yuan, XU Sichuan. Numerical simulation on gas purge after shutdown in proton exchange membrane fuel cell[J]. Journal of Tongji University (Natural Science), 2017, 45(12): 1831 [4] YANG X G, TABUCHI Y, KAGAMI, et al. Durability of membrane electrode assemblies under polymer electrolyte fuel cell cold-start cycling[J]. Journal of the Electrochemical Society, 2008, 155(7): B752 DOI:10.1149/1.2926505 [5] KONNO N, MIZUNO S, NAKAJI H, et al. Development of compact and high-performance fuel cell stack[J]. SAE International Journal of Alternative Powertrains, 2015, 4(1): 123 [6] MANABE K, IMANISHI H, OGAWA T, et al. Development of fuel cell hybrid vehicle rapid start-up from sub-freezing temperatures[J]. SAE World Congress & Exhibition, 2010, 41: 1379 [7] 李义, 许思传, 许澎. 质子交换膜燃料电池系统低温起动仿真[J]. 电源技术, 2016, 40(6): 1202 LI Yi, XU Sichuan, XU Peng. Simulation of cold start of PEM fuel cell system[J]. Chinese Journal of Power Sources, 2016, 40(6): 1202 DOI:10.3969/j.issn.1002-087X.2016.06.014 [8] SUNDARESAN M, MOORE R M. Polymer electrolyte fuel cell stack thermal model to evaluate sub-freezing startup[J]. Journal of Power Sources, 2005, 145(2): 534 DOI:10.1016/j.jpowsour.2004.12.070 [9] KHANDELWAL M, LEE S, MENCH M M. One-dimensional thermal model of cold-start in a polymer electrolyte fuel cell stack[J]. Journal of Power Sources, 2007, 172(2): 816 DOI:10.1016/j.jpowsour.2007.05.028 [10] ZHOU Y B, LUO Y, YU S, et al. Modeling of cold start processes and performance optimization for proton exchange membrane fuel cell stacks[J]. Journal of Power Sources, 2014, 247(2): 738 [11] LUO Y, JIAO K, JIA B. Elucidating the constant power, current and voltage cold start models of proton exchange membrane fuel cell[J]. International Journal of Heat and Mass Transfer, 2014, 77(4): 489 [12] DU Q, JIA B, LUO Y, et al. Maximum power cold start mode of proton exchange membrane fuel cell[J]. International Journal of Hydrogen Energy, 2014, 39(16): 8390 DOI:10.1016/j.ijhydene.2014.03.056 [13] LUO Y, JIA B, JIAO K, et al. Catalytic hydrogen-oxygen reaction in anode and cathode for cold start of proton exchange memgrane fuel cell[J]. International Journal of Hydrogen Energy, 2015, 40(32): 10293 DOI:10.1016/j.ijhydene.2015.06.094 [14] MAO L, WANG C Y. Analysis of cold start in polymer electrolyte fuel cells[J]. Journal of the Electrochemical Society, 2006, 154(2): B139 [15] JIAO K, LI X. Three-dimensional multiphase modeling of cold start processes in polymer electrolyte membrane fuel cells[J]. Electrochimica Acta, 2009, 54(27): 6876 DOI:10.1016/j.electacta.2009.06.072 [16] MENG H. A PEM fuel cell model for cold-start simulations[J]. Journal of Power Sources, 2008, 178(1): 141 DOI:10.1016/j.jpowsour.2007.12.035 [17] JIAO K. Experimental and modeling studies of cold start processes in proton exchange membrane fuel cells[D]. Waterloo: University of Waterloo, 2011. [18] JIAO K, LI X. Water transport in polymer electrolyte membrane fuel cells[J]. Progress in Energy & Combustion Science, 2011, 37(3): 221 [19] TAJIRI K, TABUCHI Y, KAGAMI F. Effects of operating and design parameters on PEFC cold start[J]. Journal of Power Sources, 2007, 165(1): 279 DOI:10.1016/j.jpowsour.2006.12.017