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3.2 Isentropic Isothermal Flow Nozzle

$\displaystyle T_1 = T_2$ (26)

$\displaystyle {{T_0}_1 \over {T_0}_2} = {\left(1+{k -1\over2} {M_1}^2\right)\ov...
... {\left(1+{k -1\over2} {M_1}^2\right)\over \left(1+{k -1\over2}{M_2}^2\right) }$ (27)

$\displaystyle {P_2 \over P_1} = {\Huge e}^{k({M_1}^2 - {M_2}^2) \over 2} = \left( {\Huge e}^{{M_1}^2} \over {\Huge e}^{{M_2}^2}\right) ^{k\over 2}$ (28)

$\displaystyle {A_2 \over A_1} = { M_1 \over M_2} \left( \mbox{\large e}^{{M_2}^{2}} \over \mbox{\large e}^{{M_1}^{2}} \right)^{k \over 2}$ (29)

$\displaystyle {{P_0}_2 \over {P_0}_1} = { P_2 \over P_1} \left( 1 + {k -1 \over...
...ox{\large e}^{{M_1}^{2}} \over \mbox{\large e} ^{{M_1}^{2}} \right]^{k \over 2}$ (30)

The star values

$\displaystyle T = T^{*}$ (31)

$\displaystyle { P \over P^{*}} = { \rho \over \rho^{*}} =$   e$\displaystyle ^{(1-M^2) k \over 2}$ (32)

$\displaystyle { A \over A^{*}} = {1 \over M}$   e$\displaystyle ^{(1-M^2) k \over 2}$ (33)

$\displaystyle { T_0 \over {T_0}^{*}} = {2 \left( 1 + {k -1 \over 2}{M_1}^2 \right) \over k +1 } ^ {k \over k-1}$ (34)

$\displaystyle { P_0 \over {P_0}^{*}} = {\Huge e}^{(1-M)k \over 2} {2 \left( 1 + {k -1 \over 2}{M_1}^2 \right) \over k +1 } ^ {k \over k-1}$ (35)

The initial stagnation temperature is denoted as $ {T_{0}}_{int}$.

$\displaystyle {T \over{T_{0}}_{int}} = { 1 \over 1 + {k-1 \over 2} M^2}$ (36)

$\displaystyle {P \over{P_{0}}_{int}} = { 1 \over \left( 1 + {k-1 \over 2} M^2 \right) ^{k-1 \over k} }$ (37)

$\displaystyle {F_{net} \over P_0 A^{*}} = \overbrace{P_2A_2 \over P_0 A^{*}}^{f...
...1A_1 \over P_0 A^{*}}^{f(M_1)} \overbrace{\left( 1 + k{M_1}^2 \right)}^{f(M_1)}$ (38)

$\displaystyle {F \over F^{*}} = {P_1A_1 \over P^{*}A^{*}} {\left( 1 + k{M_1}^2 ...
...e function \eqref{variableArea:eq:beforeDefa}}} {1 \over \left( 1 + k \right) }$ (39)

$\displaystyle F_{net} = P_0 A^{*} (1+k) {\left( k+1 \over 2 \right)^{k \over k-1}} \left( {F_2 \over F^{*} } - { F_1 \over F^{*}}\right)$ (40)

for isothermal

$\displaystyle {F_2 \over F_1} = {P_2 A_2 \over P_1 A_1} { 1 + {{U_2}^2 \over RT} \over 1 + {{U_1}^2 \over RT }}$ (41)

$\displaystyle {F_2 \over F_1} = {M_1 \over M_2} { 1 + k {M_2}^2 \over 1 + k {M_1}^2}$ (42)

$\displaystyle {F_2 \over F^{*}} = {1 \over M_2} { 1 + k {M_2}^2 \over 1 + k }$ (43)


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Next: 4 Normal Shock Up: 3 Isentropic Flow Previous: 3.1 Relationships for Small
Genick Bar-Meir ||| www.potto.org
copyright Dec , 2006

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