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Isentropic Nozzle (Q=0)

The energy equation for isentropic nozzle provides

$\displaystyle dh + UdU = \overbrace{f(x)dx}^{external work}$

(7.1)

Utilizing equation (4.27) when leads to

$\displaystyle {dP \over \rho} + UdU = f(x')dx'$

(7.2)

For the isentropic process $dP = const \times k \rho^{k-1} d\rho$ when the $const = P / \rho^{k}$ at any point of the flow. The equation (7.2) becomes

$\displaystyle \overbrace{\overbrace{P \over \rho^k}^{any\;point} k {\rho^k \ove... ...\rho} + UdU= k \overbrace{P \over \rho}^{RT} { d\rho \over \rho} UdU = f(x')dx'$	(7.3)
$\displaystyle { kRT d\rho \over \rho} + UdU = {c^{2} \over \rho} d\rho + UdU = f(x')dx'$

The continuity equation as developed earlier (mass conservation equation isn't effected by the gravity)

$\displaystyle -{d\rho \over \rho} = {dA \over A} + {dU \over U} = 0$

(7.4)

Substituting $d\rho / \rho$ from equation 7.3, into equation (7.2) moving $d\rho$ to the right hand side, and diving by

yields

$\displaystyle U {dU \over dx'} = c^2 \left[ {1 \over U} {dU \over dx'} + {1 \over A } {dA \over dx'} \right] + f(x')$

(7.5)

Rearranging equation (7.5) yields

$\displaystyle {dU \over dx'} = \left[ {M^2} {dU \over dx'} + {c^2 \over AU } {dA \over dx'} \right] + {f(x') \over U}$

(7.6)

And further rearranging yields

$\displaystyle \left( 1 -M^2 \right){dU \over dx'} = {c^2 \over AU } {dA \over dx'} + {f(x') \over U}$

(7.7)

Equation (7.7) can be rearranged as

$\displaystyle {dU \over dx'} = {1 \over \left( 1 -M^2 \right) } \left[ {c^2 \over AU } {dA \over dx'} + {f(x') \over U} \right]$

(7.8)

Equation (7.8) dimensionless form by utilizing $x = x' / \ell$ and $\ell$ is the nozzle length

$\displaystyle {dM \over dx} = {1 \over \left( 1 -M^2 \right) } \left[ {1 \over AM } {dA \over dx} + {\ell f(x) \over c \underbrace{c M}_U} \right]$

(7.9)

And the final form of equation (7.9) is

$\displaystyle {d\left(M^2 \right) \over dx} = {2 \over \left( 1 -M^2 \right) } \left[ {1 \over A } {dA \over dx} + {\ell f(x) \over c^2} \right]$

(7.10)

The term ${\ell f(x) \over c^2 }$ is considered to be very small ( $0.1\times 10/100000 < 0.1$ %) for ``standard'' situations. The dimensionless number, $\ell f(x) \over c^2$ sometimes referred as Ozer number determines whether gravity should be considered in the calculations. Nevertheless, one should be aware of value of Ozer number for large magnetic fields (astronomy) and low temperature, In such cases, the gravity effect can be considerable.

As it was shown before the transition must occur when . Consequently, two zones must be treated separately. First, here the Mach number is discussed and not the pressure as in the previous chapter. For (the subsonic branch) the term ${2 \over \left( 1 -M^2 \right)}$ is positive and the treads determined by gravity and the area function.

$\displaystyle \left[ {1 \over A } {dA \over dx} + {\ell f(x) \over c^2 } \right] > 0 \Longrightarrow d(M^2) > 0$

or conversely,

$\displaystyle \left[ {1 \over A } {dA \over dx} + {\ell f(x) \over c^2 } \right] < 0 \Longrightarrow d(M^2) < 0$

For the case of

(the supersonic branch) the term ${2 \over \left( 1 -M^2 \right)}$ is negative and therefore

$\displaystyle \left[ {1 \over A } {dA \over dx} + {\ell f(x) \over c^2 } \right] > 0 \Longrightarrow d(M^2) < 0$

For the border case

, the denominator

, is zero either $d(M^2) = \infty$ or

$\displaystyle \left[ {1 \over A } {dA \over dx} + {\ell f(x) \over c^2 } \right] = 0.$

And the

is indeterminate. As it was shown in chapter (4) the flow is chocked (

) only when

$\displaystyle \left[ {dA \over dx} + {\ell f(x) \over c^2 } \right] = 0.$

(7.11)

It should be noticed that when is zero, e.g. horizontal flow, the equation (7.11) reduced into ${dA \over dx} = 0$ that was developed previously.

The ability to manipulate the location provides a mean to increase/decrease the flow rate. Yet this ability since Ozer number is relatively very small.

This condition means that the critical point can occurs in several locations that satisfies equation (7.11). Further, the critical point, sonic point is ${dA \over Ax} \neq 0$ If is a positive function, the critical point happen at converging part of the nozzle (before the throat) and if is a negative function the critical point is diverging part of the throat. For example consider the gravity, a flow in a nozzle vertically the critical point will be above the throat.

Next: Isothermal Nozzle (T=constant) Up: Nozzle Flow With External Previous: Nozzle Flow With External Index

Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21