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6 Fanno Flow

$\displaystyle { 4 f dx \over D} = {{\left( 1 - M^2 \right) dM^2} \over {kM^4 ( 1 + {k-1 \over 2}M^2} ) }$ (61)

$\displaystyle {4 \over D} \int^{L_{max}}_{L} f dx = {1 \over k} {1 - M^2 \over M^2} + {k+1 \over 2k}\ln {{k+1 \over 2}M^2 \over 1+ {k-1 \over 2}M^2}$ (62)

A representative friction factor is defined as

$\displaystyle \bar{f} = { 1 \over L_{max}} \int ^{L_{max}} _{0} {f dx}$ (63)

$\displaystyle {4 \bar{f}L_{max}\over D} = {1 \over k} {1 - M^2 \over M^2} + {k+1 \over 2k}\ln {{k+1 \over 2}M^2 \over 1+ {k-1 \over 2}M^2}$ (64)

$\displaystyle {P \over P^{*}} = { 1 \over M} \sqrt{{k+1 \over 2} \over { 1 + {k - 1 \over 2} M^{2}} }$ (65)

$\displaystyle { T \over T^{*}} = {c^{2} \over {c^{*}}^{2} } = {{ k + 1 \over 2} \over { 1 + {k - 1 \over 2} M^{2}} }$ (66)

$\displaystyle {\rho \over \rho^{*}} = { 1 \over M} \sqrt{ { 1 + {k - 1 \over 2} M^{2}} \over {k+1 \over 2} }$ (67)

$\displaystyle { U \over U ^{*}} = \left( {\rho \over \rho^{*}} \right)^{-1} = M \sqrt{{k+1 \over 2} \over { 1 + {k - 1 \over 2} M^{2}} }$ (68)

$\displaystyle {P_{0} \over {P_{0}}^{*}} = { 1 \over M} \left({ { 1 + {k - 1 \over 2} M^{2}} \over {k+1 \over 2} } \right)^{k +1 \over 2(k -1)}$ (69)

$\displaystyle {s - s^{*} \over c_p} = \ln M^{2} \sqrt{\left({{k+1}\over 2 M^{2} \left( 1 + {k -1 \over 2 }M^{2} \right) }\right)^{ k +1 \over k} }$ (70)

$\displaystyle {T_2 \over T_1} ={ \left. T \over T^{*} \right\vert _{M_2} \over \left. T \over T^{*} \right\vert _{M_1} }$ (71)

$\displaystyle \left( {4f L_{max} \over D} \right)_{2} = \left( {4{f} L_{max} \over D} \right)_{1} - {4{f} L \over D}$ (72)


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Next: 7 RAYLEIGH FLOW Up: Free textbook pdf Fundamentals Previous: 5 Isothermal Flow
Genick Bar-Meir ||| www.potto.org
copyright Dec , 2006

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