FLUID FLOW AND HEAT TRANSFER CHARACTERISTICS IN THE MULTILAYERED-PARALLEL FRACTURED POROUS CHANNEL
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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.Abstract: Based on the Brinkman-extended Darcy model and the local thermal equilibrium model, the fluid flow and heat transfer characteristics in the multilayered-parallel fractured porous channel are studied. The analytical solutions of velocity field, temperature field in each region of multilayered-parallel fractured porous channel, friction coefficient and Nusselt number are obtained. The effects of the fracture number, Darcy number, hollow ratio and the ratio of effective thermal conductivity on heat transfer characteristics are analyzed. The results show that when Darcy number is small, the Darcy velocity in the porous media which does not change with the porous height increases with the increase of the number of fracture layers, and is not affected by the porous layer position in multilayer porous channel with certain number of fractures. Increasing the number of fracture layers weakens the influence of hollow ratio on pressure drop and increases the fluid pressure drop in the channel, but the increase degree gradually decreases. The increase of the ratio of effective thermal conductivity or decrease of the hollow ratio leads to a stepwise temperature distribution in the multilayered fractured porous channel, while the temperature distribution curves in the multilayered fractured channel tend to be consistent when the thermal conductivity ratio is small or the hollow ratio is large. Furthermore, when the ratio of thermal conductivity is small, the heat transfer effect in multilayered fractured porous channel is better than that in single fractured porous channel at any hollow ratio. However, when the ratio of thermal conductivity is large, there is a critical hollow ratio, which makes the heat transfer effect in the channels with different numbers of fracture layers be the same, and increasing the number of fractured layers has little influence on the heat transfer effect in multilayered fractured porous channel.
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Key words:
- porous media /
- multilayered-parallel fractures /
- fluid flow /
- heat transfer
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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
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 在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S与Sn的关系
Table 1. The relationship between hollow ratio S and Sn
n = 1 n = 2 n = 3 n = 4 n = 5 S1 S (1 − S)/2 S/3 (1 − S)/5 S/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 
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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}} $
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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}}} $
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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)} $
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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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