${{p}_{\text{c},\text{out},i}}={{\lambda }_{i}}{{p}_{\text{c},\text{in},i}}$ (1)
式中:pc,in,i和λi分别是第i级压缩机入口压力和压缩比。
${{T}_{\text{c},\text{out},i}}={{T}_{\text{c},\text{in},i}}+{{{T}_{\text{c},\text{in},i}}\left[ {{\left( {{\lambda }_{i}} \right)}^{\frac{\gamma -1}{\gamma }}}-1 \right]}/{{{\eta }_{\text{c}}}}\;$ (2)
式中:γ为绝热指数;Tc,in,i为第i级压缩机入口温度。
${{\eta }_{\text{c}}}=\eta _{\text{c}}^{0}\left[ 1-{{a}_{4}}{{\left( 1-{{{\bar{n}}}_{\text{c}}} \right)}^{2}} \right]\left( {{{{\bar{n}}}_{\text{c}}}}/{{{{\bar{m}}}_{\text{c}}}}\; \right)\left( 2-{{{{\bar{n}}}_{\text{c}}}}/{{{{\bar{m}}}_{\text{c}}}}\; \right)$ (3)
其中$\bar{m}_{c}$与$\bar{n}_{c}$的表达式分别为:
${{\bar{m}}_{\text{c}}}=\frac{{{{m}_{\text{c}}}\sqrt{T_{\text{c},\text{in},i}^{0}}}/{{{p}_{\text{c},\text{in},i}}}\;}{{m_{\text{c}}^{0}\sqrt{T_{\text{c},\text{in},i}^{0}}}/{p_{\text{c},\text{in},i}^{0}}\;}$ (4)
${{\bar{n}}_{\text{c}}}=\frac{{{{n}_{\text{c}}}}/{\sqrt{{{T}_{\text{c},\text{in},i}}}}\;}{{n_{\text{c}}^{0}}/{\sqrt{T_{\text{c},\text{in},i}^{0}}}\;}$ (5)
式中:nc和mc分别为压缩机实际转速与实际质量流量;上标0代表设计工况条件。
${{W}_{\text{c}}}=\int_{0}^{{{t}_{\text{c}}}}{{{c}_{p,\text{a}}}\sum\limits_{i=1}^{3}{{{m}_{\text{c}}}\left( {{T}_{\text{c},\text{out},i}}-{{T}_{\text{c},\text{in},i}} \right)}}\text{d}t$ (6)
$\varepsilon =\frac{1-exp\left[ \left( -{{N}_{\text{TU}}} \right)\left( 1+{{C}_{\text{r}}} \right) \right]}{1-{{C}_{\text{r}}}\exp \left[ \left( -{{N}_{\text{TU}}} \right)\left( 1+{{C}_{\text{r}}} \right) \right]}$ (7)
${{C}_{\text{r}}}=\frac{{{\left( {{c}_{p,\text{a}}}{{m}_{\text{c}}},{{c}_{p,\text{w}}}{{m}_{\text{w}}} \right)}_{\min }}}{{{\left( {{c}_{p,\text{a}}}{{m}_{\text{c}}},{{c}_{p,\text{w}}}{{m}_{\text{w}}} \right)}_{\max }}}$ (8)
${{N}_{\text{TU}}}=\frac{{{U}_{\text{A}}}}{{{\left( {{c}_{p,\text{a}}}{{m}_{\text{c}}},{{c}_{p,\text{w}}}{{m}_{\text{w}}} \right)}_{\min }}}$ (9)
式中:
cp,w和
mw分别为工作流体的比热容和质量流量。
UA为2 500 W/K
[9]。
${{T}_{\text{c},\text{in},i+1}}=\left( 1-\varepsilon \right){{T}_{\text{c},\text{out},i}}+\varepsilon {{T}_{\text{cold}}}$ (10)
${{M}_{\text{car}}}={{M}_{\text{car}}}\left( 0 \right)+\int_{0}^{t}{\left( {{m}_{\text{c}}}-{{m}_{\text{t}}} \right)}\ \text{d}t$ (11)
式中:Mcar(0)为初始状态下储存罐中的空气质量;mt为膨胀机中空气质量流量。
${{S}_{\text{ca}}}\left( t \right)=\frac{{{M}_{\text{car}}}\left( t \right)-{{M}_{\min }}}{{{M}_{\max }}-{{M}_{\min }}}$ (12)
式中:Mmax和Mmin分别为储气罐最大和最小空气质量。
${{T}_{\text{t},\text{in},j}}=\left( 1-\varepsilon \right){{T}_{\text{t},\text{out},j-1}}+\varepsilon {{T}_{\text{hot}}}$ (13)
式中:Tt,out,j-1为第j-1级膨胀机出口温度;Thot为高温储热罐的温度。
${{p}_{\text{t},\text{out},j}}=\frac{{{p}_{\text{t},\text{in},j}}}{{{\lambda }_{j}}}$ (14)
${{\eta }_{\text{t}}}=\eta _{\text{t}}^{0}\left[ 1-0.3{{\left( 1-{{{\bar{n}}}_{\text{t}}} \right)}^{2}} \right]\left( {{{{\bar{n}}}_{\text{t}}}}/{{{{\bar{m}}}_{\text{t}}}}\; \right)\left( 2-{{{{\bar{n}}}_{\text{t}}}}/{{{{\bar{m}}}_{\text{t}}}}\; \right)$ (15)
其中$\bar{m}_{t}$与$\bar{n}_{t}$的表达式分别为:
${{\bar{m}}_{\text{t}}}=\sqrt{1.4-0.4\frac{{{n}_{\text{t}}}}{n_{\text{t}}^{0}}}\sqrt{\frac{{{\left( {{{p}_{\text{t},\text{out},j}}}/{{{p}_{\text{t},\text{in},j}}}\; \right)}^{2}}-1}{{{\left( {p_{\text{t},\text{out},j}^{0}}/{p_{\text{t},\text{in},j}^{0}}\; \right)}^{2}}-1}}$ (16)
${{\bar{n}}_{\text{t}}}=\frac{{{{n}_{\text{t}}}}/{\sqrt{{{T}_{\text{t},\text{in},j}}}}\;}{{n_{\text{t}}^{0}}/{\sqrt{T_{\text{t},\text{in},j}^{0}}}\;}$ (17)
${{T}_{\text{t},\text{out},j}}={{T}_{\text{t},\text{in},j}}-{{\eta }_{\text{t}}}{{T}_{\text{t},\text{in},j}}\left[ 1-{{\left( \frac{{{p}_{\text{t},\text{out},j}}}{{{p}_{\text{t},\text{in},j}}} \right)}^{\frac{\gamma -1}{\gamma }}} \right]$ (18)
${{W}_{\text{t}}}=\int_{0}^{{{t}_{\text{dis}}}}{{{c}_{p,\text{a}}}\sum\limits_{j=1}^{3}{{{m}_{\text{t}}}\left( {{T}_{\text{t},\text{out},j}}-{{T}_{\text{t},\text{in},j}} \right)}}\text{d}t$ (19)