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多层平行裂隙型多孔介质通道内流体流动传热特性

李琪 王兆宇 胡鹏飞

李琪, 王兆宇, 胡鹏飞. 多层平行裂隙型多孔介质通道内流体流动传热特性. 力学学报, 2022, 54(11): 2994-3009 doi: 10.6052/0459-1879-22-285
引用本文: 李琪, 王兆宇, 胡鹏飞. 多层平行裂隙型多孔介质通道内流体流动传热特性. 力学学报, 2022, 54(11): 2994-3009 doi: 10.6052/0459-1879-22-285
Li Qi, Wang Zhaoyu, Hu Pengfei. Fluid flow and heat transfer characteristics in the multilayered-parallel fractured porous channel. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(11): 2994-3009 doi: 10.6052/0459-1879-22-285
Citation: Li Qi, Wang Zhaoyu, Hu Pengfei. Fluid flow and heat transfer characteristics in the multilayered-parallel fractured porous channel. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(11): 2994-3009 doi: 10.6052/0459-1879-22-285

多层平行裂隙型多孔介质通道内流体流动传热特性

doi: 10.6052/0459-1879-22-285
基金项目: 国家自然科学基金(51906034), 吉林省科技发展计划(20210101086JC)和吉林市科技创新发展计划(20210103087)资助项目
详细信息
    作者简介:

    胡鹏飞, 副教授, 主要研究方向: 多相流动与传热. E-mail: hupengfei@neepu.edu.cn

  • 中图分类号: O35

FLUID FLOW AND HEAT TRANSFER CHARACTERISTICS IN THE MULTILAYERED-PARALLEL FRACTURED POROUS CHANNEL

  • 摘要: 基于Brinkman-extended Darcy模型和局部热平衡模型, 对多层平行裂隙型多孔介质通道内的流动传热特性进行研究. 获得了多层平行裂隙型多孔介质通道内各区域的速度场、温度场、摩擦系数及努塞尔数解析解, 并分析了裂隙层数、达西数、空心率、有效热导率之比等对通道内流动传热特性的影响. 结果表明: 达西数较小时, 通道多孔介质层内会出现不随高度变化的达西速度, 此达西速度会随裂隙层数的增加而增大, 但却不受各裂隙层下多孔介质层位置变化的影响. 增加裂隙层数会减弱空心率对压降的影响, 会使通道内流体压降升高, 但升高程度会逐渐降低. 增大热导率之比或减小空心率会使多裂隙通道内出现阶梯式温度分布, 而在较小热导率之比或较大空心率时多裂隙情况下的温度分布曲线会趋于一致. 此外, 当热导率之比较小时, 多层裂隙通道内的传热效果在任何空心率下都要优于单裂隙情况, 当热导率之比较大时, 存在临界空心率使各裂隙层数通道内的传热效果相同, 且多裂隙通道内继续增加裂隙层数对传热强度影响不大.

     

  • 1  多层平行裂隙型多孔介质通道结构

    1.  Porous channel structure with multilayer parallel fractures

    图  2  本文计算结果与文献[14]结果比较

    Figure  2.  Comparison of the calculation results of this paper with the results of Ref. [14]

    2  本文计算结果与文献[14]结果比较(续)

    2.  Comparison of the calculation results of this paper with the results of Ref. [14] (continued)

    图  3  裂隙层数对速度场的影响

    Figure  3.  Effect of the number of fracture layers on velocity distribution

    4  孔隙率对速度分布的影响

    4.  Effect of porosity on velocity distribution

    4  孔隙率对速度分布的影响(续)

    4.  Effect of porosity on velocity distribution (continued)

    图  5  空心率对速度分布的影响

    Figure  5.  Influence of hollow ratio on velocity distribution

    图  6  不同裂隙层数和达西数下压降随空心率变化

    Figure  6.  Variation of pressure drop with hollow ratio under different fracture layer numbers and Darcy numbers

    7  Da = 1.0 × 10−3, K = 10时, 不同空心率下裂隙层数对温度分布的影响

    7.  Influence of the number of fracture layers on the temperature distribution under different hollow ratios at Da = 1.0 × 10−3, K = 10

    图  8  不同热导率之比时, 裂隙层数对温度分布的影响

    Figure  8.  Influence of the number of fracture layers on the temperature distribution under different thermal conductivity ratios

    图  9  Nusselt数与热导率之间的关系

    Figure  9.  Relationship between the Nusselt number and the thermal conductivity ratio

    图  10  Nusselt数与空心率的关系

    Figure  10.  Relationship between the Nusselt number and the hollow rate

    表  1  空心率SSn的关系

    Table  1.   The relationship between hollow ratio S and Sn

    n = 1n = 2n = 3n = 4n = 5
    S1S(1 − S)/2S/3(1 − S)/5S/5
    S2(1 + S)/2(3 − S)/6(2 + 3S)/10(5 − 2S)/15
    S3(1 + S)/2(6 − S)/10(5 + 4S)/15
    S4(3 + 2S)/5(10 − S)/15
    S5(2 + S)/3
    下载: 导出CSV

    1  裂隙层速度场系数Lj表达式

    1.   Coefficient expressions Lj of velocity field in fracture layers

    n$ {{L}}_{{1}} $$ {L}_{2} $$ {L}_{3} $$ {L}_{4} $$ {L}_{5} $$ {L}_{6} $
    n = 1$ 0 $$ \dfrac{{S}^{2}}{2} + {U}_{i} $
    n = 2$ \dfrac{{S}_{1}^{2}-{S}_{2}^{2} + 2({U}_{i1}-{U}_{i2})}{{S}_{1}-{S}_{2}} $$ \dfrac{{S}_{1}{S}_{2}\left({S}_{2}-{S}_{1}\right) + 2({U}_{i2}{S}_{1}-{U}_{i1}{S}_{2})}{2\left({S}_{1}-{S}_{2}\right)} $
    n = 3$ 0 $$ \dfrac{{S}_{1}^{2}}{2} + {U}_{i1} $$ \dfrac{2({U}_{i3}-{U}_{i2}) + {S}_{3}^{2}-{S}_{2}^{2}}{{S}_{3}-{S}_{2}} $$ {U}_{i3}-\dfrac{{S}_{2}{S}_{3}}{2}-\dfrac{{S}_{3}({U}_{i3}-{U}_{i2})}{{S}_{3}-{S}_{2}} $
    n = 4$ \dfrac{{S}_{1}^{2}-{S}_{2}^{2} + 2({U}_{i1}-{U}_{i2})}{{S}_{1}-{S}_{2}} $$ \dfrac{{S}_{1}{S}_{2}\left({S}_{2}-{S}_{1}\right) + 2({U}_{i2}\mathrm{*}{S}_{1}-{U}_{i1}\mathrm{*}{S}_{2})}{2\left({S}_{1}-{S}_{2}\right)} $$ \dfrac{2({U}_{i4}-{U}_{i3}) + {S}_{4}^{2}-{S}_{3}^{2}}{{S}_{4}-{S}_{3}} $$ {U}_{i3}-\dfrac{{S}_{3}{S}_{4}}{2}-\dfrac{{S}_{3}({U}_{i4}-{U}_{i3})}{{S}_{4}-{S}_{3}} $
    n = 5$ 0 $$ \dfrac{{S}_{1}^{2}}{2} + {U}_{i1} $$ \dfrac{2({U}_{i3}-{U}_{i2}) + {S}_{3}^{2}-{S}_{2}^{2}}{{S}_{3}-{S}_{2}} $$ {U}_{i3}-\dfrac{{S}_{2}{S}_{3}}{2}-\dfrac{{S}_{3}({U}_{i3}-{U}_{i2})}{{S}_{3}-{S}_{2}} $$ \dfrac{{S}_{4} + {S}_{5}}{2} + \dfrac{{U}_{i5}-{U}_{i4}}{{S}_{5}-{S}_{4}} $$ {U}_{i4}-\dfrac{{S}_{4}\mathrm{*}{S}_{5}}{2}-\dfrac{{S}_{4}({U}_{i5}-{U}_{i4})}{{S}_{5}-{S}_{4}} $
    下载: 导出CSV

    2  多孔介质层速度场系数Pk表达式

    2.   Coefficient expressions Pk of velocity field in porous layers

    n$ {P}_{1} $$ {P}_{2} $$ {P}_{3} $$ {P}_{4} $$ {P}_{5} $$ {P}_{6} $
    n = 1$ \dfrac{Da{\text{(}\text{e}}^{NS}-{\text{e}}^{N}) + {U}_{i}{\text{e}}^{N}}{{\text{e}}^{N(1-S)}-{\text{e}}^{N(S-1)}} $$ -\dfrac{Da{\text{(e}}^{-NS}-{\text{e}}^{-N}) + {U}_{i}{\text{e}}^{-N}}{{\text{e}}^{N(1-S)}-{\text{e}}^{N(S-1)}} $
    n = 2$ \dfrac{{U}_{i1}-Da}{{\text{e}}^{N{S}_{1}} + {\text{e}}^{-N{S}_{1}}} $$ \dfrac{{U}_{i1}-Da}{{\text{e}}^{N{S}_{1}} + {\text{e}}^{-N{S}_{1}}} $$\dfrac{\begin{array}{c}[Da{ {\rm{e} }^{N\left( { {S_2} + 1} \right)} }{ {\rm{e} }^{N{S_2} } } + \\{ {\rm{e} }^N}({U_{i2} } - Da)]\end{array} }{ { { { {\rm{e} }^{2N} }{ { - } }{ {\rm{e} }^{2N{S_2} } } } } }$$ -\dfrac{Da{\text{e}}^{N} + {\text{e}}^{N{S}_{2}}\mathrm{*}({U}_{i2}-Da)}{{\text{e}}^{2 N}-{\text{e}}^{2 N{S}_{2}}} $
    n = 3$\dfrac{\begin{array}{c}[{U}_{i1}{\text{e} }^{N\left({S}_{2}-{S}_{1}\right)}-{U}_{i2}-\\ Da{\text{e} }^{N\left({S}_{2}-{S}_{1}\right)-1}]\end{array} }{{\text{e} }^{N\left({S}_{2}-2{S}_{1}\right)}-{\text{e} }^{-N{S}_{2} }}$$\dfrac{\begin{array}{c}\left[\right({U}_{i1}-Da{\text{)e} }^{N\left({S}_{1}-{S}_{2}\right)}+\\ Da-{U}_{i2}]\end{array} }{ {\text{e} }^{N\left(2{S}_{1}-{S}_{2}\right)}{\rm{-e} }^{N{S}_{2} } }$$ \dfrac{Da{\text{(1-e}}^{N\left({S}_{3}-1\right)})-{U}_{i3}}{{\text{e}}^{N\left({S}_{3}-2\right)}-{\text{e}}^{-N{S}_{3}}} $$ \dfrac{{Da\rm{(1-e}}^{N\left(1-{S}_{3}\right)})-{U}_{i3}}{{\text{e}}^{N\left(2-{S}_{3}\right)}-{\text{e}}^{N{S}_{3}}} $
    n = 4$ \dfrac{{U}_{i1}-Da}{{\text{e}}^{N{S}_{1}} + {\text{e}}^{-N{S}_{1}}} $$ \dfrac{{U}_{i1}-Da}{{\text{e}}^{N{S}_{1}} + {\text{e}}^{-N{S}_{1}}} $$\dfrac{\begin{array}{c}[{U_{i3}} - {U_{i2}}{{\rm{e}}^{N\left( {{S_3} - {S_2}} \right)}} - \\Da(1 - {{\rm{e}}^{N\left( {{S_3} - {S_2}} \right)}})]\end{array}}{{{{\rm{e}}^{ - N{S_3}}} - {{\rm{e}}^{N\left( {{S_3} - 2{S_2}} \right)}}}}$$\dfrac{\begin{array}{c}[{U}_{i3}-{U}_{i2}{\rm{e}}^{N\left({S}_{2}-{S}_{3}\right)}-\\ Da(1-{\rm{e}}^{N\left({S}_{2}-{S}_{3}\right)})]\end{array} }{ {\rm{e}}^{N{S}_{3} }-{\rm{e}}^{N\left(2{S}_{2}{-}{S}_{3}\right)} }$$ \dfrac{-{U}_{i4}-Da({\rm{e}}^{N\left({S}_{4}-1\right)}-1)}{{\rm{e}}^{N({S}_{4}-2)}-{\rm{e}}^{-N{S}_{4}}} $$ \dfrac{-{U}_{i4}-Da({\rm{e}}^{N\left(1-{S}_{4}\right)}-1)}{{\rm{e}}^{N(2-{S}_{4})}-{\rm{e}}^{N{S}_{4}}} $
    n = 5$\dfrac{\begin{array}{c}\Big[{U}_{i1}{\rm{e}}^{N\left({S}_{2}-{S}_{1}\right)}-{U}_{i2}-\\ Da\left({\rm{e}}^{N\left({S}_{2}-{S}_{1}\right)}{-1}\right)\Big]\end{array} }{ {\rm{e}}^{N\left({S}_{2}-2{S}_{1}\right)}-{\rm{e}}^{-N{S}_{2} } }$$\dfrac{\begin{array}{c}\left[\right({U}_{i1}-Da{\text{)e} }^{N\left({S}_{1}-{S}_{2}\right)}+\\ Da-{U}_{i2}]\end{array} }{ {\text{e} }^{N\left(2{S}_{1}-{S}_{2}\right)}{\text{-e} }^{N{S}_{2} } }$$\dfrac{\begin{array}{c}\Big[{U_{i4}}{{\rm{e}}^{N\left( {{S_3} - {S_4}} \right)}} - {U_{i3}} + \\Da\left( {{{\rm{e}}^{N\left( {{S_3} - {S_4}} \right)}}{{ - 1}}} \right)\Big]\end{array}}{{{{\rm{e}}^{N\left( {{S_3} - 2{S_4}} \right)}} - {{\rm{e}}^{ - N{S_3}}}}}$$\dfrac{\begin{array}{c}[{U}_{i4}{\rm{e}}^{N\left({S}_{4}-{S}_{3}\right)}-{U}_{i3}-\\ Da({\rm{e}}^{N\left({S}_{4}-{S}_{3}\right)}-1)]\end{array} }{ {\rm{e}}^{N\left(2{S}_{4}-{S}_{3}\right)}-{\rm{e}}^{N{S}_{3} } }$$ \dfrac{-{U}_{i5}-Da({\rm{e}}^{N\left({S}_{5}-1\right)}-1)}{{\rm{e}}^{N({S}_{5}-2)}-{\rm{e}}^{-N{S}_{5}}} $$ \dfrac{-{U}_{i5}-Da({\rm{e}}^{N\left(1-{S}_{5}\right)}-1)}{{\rm{e}}^{N(2-{S}_{5})}-{\rm{e}}^{N{S}_{5}}} $
    下载: 导出CSV

    3  裂隙层温度场系数Cj表达式

    3.   Coefficient expressions Cj of temperature field in fracture layers

    n$ {C}_{1} $$ {C}_{2} $$ {C}_{3} $$ {C}_{4} $$ {C}_{5} $$ {C}_{6} $
    n = 1$ 0 $$ \dfrac{{F}{S}_{1}^{4}}{12}-\dfrac{{F}{L}_{1}{S}_{1}^{3}}{6}-{F}{L}_{2}{S}_{1}^{2} + {T}_{i1} $
    n = 2$ \dfrac{\begin{array}{c}[F\left({S}_{1}^{4}-{S}_{2}^{4}\right) + 2 F{L}_{1}\left({S}_{2}^{3}-{S}_{1}^{3}\right) + \\ 12 F{L}_{2}({S}_{2}^{2}-{S}_{1}^{2}) + 12({T}_{i1}-{T}_{i2})]\end{array}}{12\left({S}_{1}-{S}_{2}\right)} $$ \dfrac{\begin{array}{c}[F{S}_{1}{S}_{2}\left({S}_{2}^{3}-{S}_{1}^{3}\right) + 2 F{L}_{1}{S}_{1}{S}_{2}\left({S}_{1}^{2}-{S}_{2}^{2}\right) + \\ 12 F{L}_{2}{S}_{1}{S}_{2}({S}_{1}-{S}_{2}) + 12({T}_{i2}{S}_{1}-{T}_{i1}{S}_{2})]\end{array}}{12\left({S}_{1}-{S}_{2}\right)} $
    n = 3$ 0 $$ \dfrac{{F}{S}_{1}^{4}}{12}-\dfrac{{F}{L}_{1}{S}_{1}^{3}}{6}-{F}{L}_{2}{S}_{1}^{2} + {T}_{i1} $$ \dfrac{\begin{array}{c}[F\left({S}_{2}^{4}-{S}_{3}^{4}\right) + 2 F{L}_{3}\left({S}_{3}^{3}-{S}_{2}^{3}\right) + \\ 12 F{L}_{4}({S}_{3}^{2}-{S}_{2}^{2}) + 12({T}_{i2}-{T}_{i3})]\end{array}}{12\left({S}_{2}-{S}_{3}\right)} $$ \dfrac{\begin{array}{c}[F{S}_{2}{S}_{3}\left({S}_{3}^{3}-{S}_{2}^{3}\right) + 2 F{L}_{3}{S}_{2}{S}_{3}\left({S}_{2}^{2}-{S}_{3}^{2}\right) + \\ 12 F{L}_{4}{S}_{2}{S}_{3}({S}_{2}-{S}_{3}) + 12({T}_{i3}{S}_{2}-{T}_{i2}{S}_{3})]\end{array}}{12\left({S}_{2}-{S}_{3}\right)} $
    n = 4$ \dfrac{\begin{array}{c}[F\left({S}_{1}^{4}-{S}_{2}^{4}\right) + 2 F{L}_{1}\left({S}_{2}^{3}-{S}_{1}^{3}\right) + \\ 12 F{L}_{2}({S}_{2}^{2}-{S}_{1}^{2}) + 12({T}_{i1}-{T}_{i2})]\end{array}}{12\left({S}_{1}-{S}_{2}\right)} $$ \dfrac{\begin{array}{c}[F{S}_{1}{S}_{2}\left({S}_{2}^{3}-{S}_{1}^{3}\right) + 2 F{L}_{1}{S}_{1}{S}_{2}\left({S}_{1}^{2}-{S}_{2}^{2}\right) + \\ 12 F{L}_{2}{S}_{1}{S}_{2}({S}_{1}-{S}_{2}) + 12({T}_{i2}{S}_{1}-{T}_{i1}{S}_{2})]\end{array}}{12\left({S}_{1}-{S}_{2}\right)} $$ \dfrac{\begin{array}{c}[F\left({S}_{3}^{4}-{S}_{4}^{4}\right) + 2 F{L}_{3}\left({S}_{4}^{3}-{S}_{3}^{3}\right) + \\ 12 F{L}_{4}({S}_{4}^{2}-{S}_{3}^{2}) + 12({T}_{i3}-{T}_{i4})]\end{array}}{12\left({S}_{3}-{S}_{4}\right)} $$ \dfrac{\begin{array}{c}[F{S}_{3}{S}_{4}\left({S}_{4}^{3}-{S}_{3}^{3}\right) + 2 F{L}_{3}{S}_{3}{S}_{4}\left({S}_{3}^{2}-{S}_{4}^{2}\right) + \\ 12 F{L}_{4}{S}_{3}{S}_{4}({S}_{3}-{S}_{4}) + 12({T}_{i4}{S}_{3}-{T}_{i3}{S}_{4})]\end{array}}{12\left({S}_{3}-{S}_{4}\right)} $
    n = 5$ 0 $$ \dfrac{{F}{S}_{1}^{4}}{12}-\dfrac{{F}{L}_{1}{S}_{1}^{3}}{6}-{F}{L}_{2}{S}_{1}^{2} + {T}_{i1} $$ \dfrac{\begin{array}{c}[F\left({S}_{2}^{4}-{S}_{3}^{4}\right) + 2 F{L}_{3}\left({S}_{3}^{3}-{S}_{2}^{3}\right) + \\ 12 F{L}_{4}({S}_{3}^{2}-{S}_{2}^{2}) + 12({T}_{i2}-{T}_{i3})]\end{array}}{12\left({S}_{2}-{S}_{3}\right)} $$ \dfrac{\begin{array}{c}[F{S}_{2}{S}_{3}\left({S}_{3}^{3}-{S}_{2}^{3}\right) + 2 F{L}_{3}{S}_{2}{S}_{3}\left({S}_{2}^{2}-{S}_{3}^{2}\right) + \\ 12 F{L}_{4}{S}_{2}{S}_{3}({S}_{2}-{S}_{3}) + 12({T}_{i3}{S}_{2}-{T}_{i2}{S}_{3})]\end{array}}{12\left({S}_{2}-{S}_{3}\right)} $$ \dfrac{\begin{array}{c}[F\left({S}_{4}^{4}-{S}_{3}^{4}\right) + 2 F{L}_{5}\left({S}_{5}^{3}-{S}_{4}^{3}\right) + \\ 12 F{L}_{6}({S}_{5}^{2}-{S}_{4}^{2}) + 12({T}_{i4}-{T}_{i5})]\end{array}}{12\left({S}_{4}-{S}_{5}\right)} $$ \dfrac{\begin{array}{c}[F{S}_{4}{S}_{5}\left({S}_{5}^{3}-{S}_{4}^{3}\right) + 2 F{L}_{5}{S}_{4}{S}_{5}\left({S}_{4}^{2}-{S}_{5}^{2}\right) + \\ 12 F{L}_{6}{S}_{4}{S}_{5}({S}_{4}-{S}_{5}) + 12({T}_{i5}{S}_{4}-{T}_{i4}{S}_{5})]\end{array}}{12\left({S}_{4}-{S}_{5}\right)} $
    下载: 导出CSV

    4  多孔介质层温度场系数Zk表达式

    4.   Coefficient expressions Zk of temperature field in porous layers

    n$ {Z}_{1} $$ {Z}_{2} $$ {Z}_{3} $$ {Z}_{4} $$ {Z}_{5} $$ {Z}_{6} $
    n = 1$ \dfrac{\begin{array}{c}[Da{{N}}^{2}\left(1-{{S}}_{1}^{2}\right) + 2{{P}}_{2}\left({\rm{e}}^{{N}}-{\rm{e}}^{{N}{{S}}_{1}}\right)+ \\ 2{{P}}_{1}({\rm{e}}^{-{N}}-{\rm{e}}^{-{N}{{S}}_{1}}) + 2 L{{N}}^{2}{T}_{i}]\end{array}}{2{L}{{N}}^{2}\left({S}_{1}-1\right)} $$ -\dfrac{\begin{array}{c}[Da{{N}}^{2}{{S}}_{1}\left(1-{{S}}_{1}\right) + 2{{P}}_{2}({{S}}_{1}{\rm{e}}^{{N}}-{\rm{e}}^{{N}{{S}}_{1}})+ \\ 2{{P}}_{1}({{S}}_{1}{\rm{e}}^{-{N}}-{\rm{e}}^{-{N}{{S}}_{1}}) + 2 L{{N}}^{2}{T}_{i}]\end{array}}{2{L}{{N}}^{2}\left({S}_{1}-1\right)} $
    n = 2$ \dfrac{{P}1-{P}2}{{L}{N}} $$ -\dfrac{\begin{array}{c}[2\left({P}_{1}{\rm{e}}^{-N{S}_{1}} + {P}_{2}{\rm{e}}^{N{S}_{1}}\right)-2 L{N}^{2}{T}_{i1}+ \\ 2 N{S}_{1}\left({P}_{1}-{P}_{2}\right) + Da{N}^{2}{S}_{1}^{2}]\end{array}}{2{L}{N}^{2}} $$ \dfrac{\begin{array}{c}[2{P}_{4}\left({\rm{e}}^{N}-{\rm{e}}^{N{S}_{2}}\right) + Da{N}^{2}{(1-S}_{2}^{2})+ \\ 2{P}_{3}\left({\rm{e}}^{-N}-{\rm{e}}^{-N{S}_{2}}\right) + 2 L{N}^{2}{T}_{i2}]\end{array}}{2{L}{N}^{2}\left({S}_{2}-1\right)} $$ -\dfrac{\begin{array}{c}[2{P}_{4}\left({S}_{2}{\rm{e}}^{N}-{\rm{e}}^{N{S}_{2}}\right) + Da{N}^{2}{({S}_{2}-S}_{2}^{2})+ \\ 2{P}_{3}\left({S}_{2}{\rm{e}}^{-N}-{\rm{e}}^{-N{S}_{2}}\right) + 2 L{N}^{2}{T}_{i2}]\end{array}}{2{L}{N}^{2}\left({S}_{2}-1\right)} $
    n = 3$\dfrac{{\begin{array}{*{20}{c}}{ - [Da{{{N}}^2}\left( {S_1^2 - S_2^2} \right) + 2{P_2}\left( {{{\rm{e}}^{N{{{S}}_1}}} - {{\rm{e}}^{N{{{S}}_2}}}} \right) + }\\\begin{array}{l}2{P_1}({{\rm{e}}^{ - N{{{S}}_1}}} - {{\rm{e}}^{ - N{{{S}}_2}}}) + \\2L{{{N}}^2}({T_{i2}} - {T_{i1}})]\end{array}\end{array}}}{{2{{L}}{{{N}}^2}\left( {{S_1} - {{{S}}_2}} \right)}}$$\dfrac{{\begin{array}{*{20}{c}}{ - \Big[Da{{{N}}^2}{{{S}}_1}{{{S}}_2}\left( {{{{S}}_2} - {{{S}}_1}} \right) + 2{P_2}\left( {{{{S}}_1}{{\rm{e}}^{N{{{S}}_2}}} - {{{S}}_2}{{\rm{e}}^{N{{{S}}_1}}}} \right) + }\\\begin{array}{l}2{P_1}({{{S}}_1}{{\rm{e}}^{ - N{{{S}}_2}}} - {{{S}}_2}{{\rm{e}}^{ - N{{{S}}_1}}}) + \\2L{{{N}}^2}({{{S}}_2}{T_{i1}} - {{{S}}_1}{T_{i2}})\Big]\end{array}\end{array}}}{{2{{L}}{{{N}}^2}\left( {{S_1} - {{{S}}_2}} \right)}}$$ \dfrac{\begin{array}{c}\Big[2{P}_{4}\left({\rm{e}}^{N}-{\rm{e}}^{N{S}_{3}}\right) + Da{N}^{2}{(1-S}_{3}^{2})+ \\ 2{P}_{3}\left({\rm{e}}^{-N}-{\rm{e}}^{-N{S}_{3}}\right) + 2 L{N}^{2}{T}_{i3}\Big]\end{array}}{2{L}{N}^{2}\left({S}_{3}-1\right)} $$ -\dfrac{\begin{array}{c}[2{P}_{4}\left({S}_{3}{\rm{e}}^{N}-{\rm{e}}^{N{S}_{3}}\right) + Da{N}^{2}{({S}_{3}-S}_{3}^{2})+ \\ 2{P}_{3}\left({S}_{3}{\rm{e}}^{-N}-{\rm{e}}^{-N{S}_{3}}\right) + 2 L{N}^{2}{T}_{i3}]\end{array}}{2{L}{N}^{2}\left({S}_{3}-1\right)} $
    n = 4$ \dfrac{{P}1-{P}2}{{L}{N}} $$ -\dfrac{\begin{array}{c}\Big[2\left({P}_{1}{\rm{e}}^{-N{S}_{1}} + {P}_{2}{\rm{e}}^{N{S}_{1}}\right)-2 L{N}^{2}{T}_{i1}+ \\ 2 N{S}_{1}\left({P}_{1}-{P}_{2}\right) + Da{N}^{2}{S}_{1}^{2}\Big]\end{array}}{2{L}{N}^{2}} $$ \dfrac{\begin{array}{c}-\Big[2{P}_{4}\left({\rm{e}}^{N{S}_{2}}-{\rm{e}}^{N{S}_{3}}\right) + {D}{a}{N}^{2}{(S}_{2}^{2}{-S}_{3}^{2}) + \\ 2{P}_{3}\left({\rm{e}}^{-N{S}_{2}}-{\rm{e}}^{-N{S}_{3}}\right) + 2 L{N}^{2}({T}_{i3}-{T}_{i2})\Big]\end{array}}{2{L}{N}^{2}\left({S}_{2}-{S}_{3}\right)} $$\dfrac{{\begin{array}{*{20}{c}}\begin{array}{l} - \Big[ {2{P_4}\left( {{S_2}{{\rm{e}}^{N{S_3}}} - {S_3}{{\rm{e}}^{N{S_2}}}} \right)} + \\\left. {{{Da}}{N^2}({S_2}S_3^2 - {S_3}S_2^2} \right) + \end{array}\\\begin{array}{l}2{P_3}\left( {{S_2}{{\rm{e}}^{ - N{S_3}}} - {S_3}{{\rm{e}}^{ - N{S_2}}}} \right) + \\2L{N^2}({S_3}{T_{i2}} - {S_2}{T_{i3}})\Big]\end{array}\end{array}}}{{2{{L}}{N^2}\left( {{S_2} - {S_3}} \right)}}$${\dfrac{{\begin{array}{*{20}{l}}{\Big[2{P_6}\left( {{{\rm{e}}^N} - {{\rm{e}}^{N{S_4}}}} \right) + }\\{Da{N^2}(1 - S_4^2) + }\\{2{P_3}\left( {{{\rm{e}}^{ - N}} - {{\rm{e}}^{ - N{S_4}}}} \right) + }\\{2L{N^2}{T_{i4}}\Big]}\end{array}}}{{2L{N^2}\left( {{S_4} - 1} \right)}}}$$\dfrac{ {\begin{array}{*{20}{c} }\begin{array}{l} - \Big[2{P_6}\left( { {S_4}{{\rm{e}}^N} - {{\rm{e}}^{N{S_4} } } } \right) + \\Da{N^2}({S_4} - S_4^2) + \end{array}\\\begin{array}{l}2{P_5}\left( { {S_4}{{\rm{e}}^{ - N} } - {{\rm{e}}^{ - N{S_4} } } } \right) + \\2L{N^2}{T_{i4} }\Big]\end{array}\end{array} } }{ {2{{L} }{N^2}\left( { {S_4} - 1} \right)} }$
    n = 5$\dfrac{{\begin{array}{*{20}{c}}{ - \Big[Da{{{N}}^2}\left( {S_1^2 - S_2^2} \right) + 2{P_2}\left( {{{\rm{e}}^{N{{{S}}_1}}} - {{\rm{e}}^{N{{{S}}_2}}}} \right) + }\\\begin{array}{l}2{P_1}({{\rm{e}}^{ - N{{{S}}_1}}} - {{\rm{e}}^{ - N{{{S}}_2}}}) + 2L{{{N}}^2}({T_{i2}} - {T_{i1}})\Big]\end{array}\end{array}}}{{2{{L}}{{{N}}^2}\left( {{S_1} - {{{S}}_2}} \right)}}$$ \dfrac{\begin{array}{c}-\Big[Da{{N}}^{2}{{S}}_{1}{{S}}_{2}\left({{S}}_{2}-{{S}}_{1}\right) + 2{P}_{2}\left({{S}}_{1}{\rm{e}}^{N{{S}}_{2}}-{{S}}_{2}{\rm{e}}^{N{{S}}_{1}}\right) + \\ 2{P}_{1}({{S}}_{1}{\rm{e}}^{-N{{S}}_{2}}-{{S}}_{2}{\rm{e}}^{-N{{S}}_{1}}) + 2 L{{N}}^{2}({{S}}_{2}{T}_{i1}-{{S}}_{1}{T}_{i2})\Big]\end{array}}{2{L}{{N}}^{2}\left({S}_{1}-{{S}}_{2}\right)} $$ \dfrac{\begin{array}{c}-\Big[Da{{N}}^{2}\left({S}_{3}^{2}-{S}_{4}^{2}\right) + 2{P}_{4}\left({\rm{e}}^{N{{S}}_{3}}-{\rm{e}}^{N{{S}}_{4}}\right) + \\ 2{P}_{3}({\rm{e}}^{-N{{S}}_{3}}-{\rm{e}}^{-N{{S}}_{4}}) + 2 L{{N}}^{2}({T}_{i4}-{T}_{i3})\Big]\end{array}}{2{L}{{N}}^{2}\left({S}_{3}-{{S}}_{4}\right)} $$\dfrac{ {\begin{array}{*{20}{c} }\begin{array}{l} - \Big[Da{ {{N} }^2}{ {{S} }_3}{ {{S} }_4}\left( { { {{S} }_4} - { {{S} }_3} } \right) + \\2{P_4}\left( { { {{S} }_3}{{\rm{e}}^{N{ {{S} }_4} } } - { {{S} }_4}{{\rm{e}}^{N{ {{S} }_3} } } } \right) + \end{array}\\\begin{array}{l}2{P_3}({ {{S} }_3}{{\rm{e}}^{ - N{ {{S} }_4} } } - { {{S} }_4}{{\rm{e}}^{ - N{ {{S} }_3} } }) + \\2L{ {{N} }^2}({ {{S} }_4}{T_{i3} } - { {{S} }_3}{T_{i4} })\Big]\end{array}\end{array} } }{ {2{{L} }{ {{N} }^2}\left( { {S_3} - { {{S} }_4} } \right)} }$$\dfrac{{\begin{array}{*{20}{c}}\begin{array}{l}\Big[2{P_6}\left( {{{\rm{e}}^N} - {{\rm{e}}^{N{S_5}}}} \right) + \\Da{N^2}(1 - S_5^2) + \end{array}\\\begin{array}{l}2{P_5}\left( {{{\rm{e}}^{ - N}} - {{\rm{e}}^{ - N{S_5}}}} \right) + \\2L{N^2}{T_{i5}}\Big]\end{array}\end{array}}}{{2{{L}}{N^2}\left( {{S_5} - 1} \right)}}$$\dfrac{{\begin{array}{*{20}{c}}\begin{array}{l} - \Big[2{P_6}\left( {{S_5}{{\rm{e}}^N} - {{\rm{e}}^{N{S_5}}}} \right) + \\Da{N^2}({S_5} - S_5^2) + \end{array}\\\begin{array}{l}2{P_5}\left( {{S_5}{{\rm{e}}^{ - N}} - {{\rm{e}}^{ - N{S_5}}}} \right) + \\2L{N^2}{T_{i5}}\Big]\end{array}\end{array}}}{{2{{L}}{N^2}\left( {{S_5} - 1} \right)}}$
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  • 收稿日期:  2022-06-04
  • 录用日期:  2022-09-16
  • 网络出版日期:  2022-09-17
  • 刊出日期:  2022-11-18

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