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Model

The mass (continuity equation) balance can be written as

$\displaystyle \dot{m} = \rho A U = constant$	(9.1)
$\displaystyle \hookrightarrow \rho_1 U_1 = \rho_2 U_2$

The energy conservation (under the assumption that this model is adiabatic flow and the friction is not transformed into thermal energy) reads

$\displaystyle {T_{0}}_1 = {T_{0}}_2$

$\displaystyle \hookrightarrow T_1 + { {U_1}^2 \over 2c_p} = T_2 + { {U_2}^2 \over 2c_p}$

(9.2)

Or in a derivative form

$\displaystyle C_p dT +d \left( U^2 \over 2 \right) = 0$

(9.3)

Again for simplicity, the perfect gas model is assumed^9.3.

$\displaystyle P = \rho R T$	(9.4)
$\displaystyle \hookrightarrow {P_1 \over \rho_{1} T_{1}} = {P_2 \over \rho_{2} T_{2}}$

It is assumed that the flow can be approximated as one-dimensional. The force acting on the gas is the friction at the wall and the momentum conservation reads

$\displaystyle -AdP -\tau_{w} dA_{w} = \dot{m} dU$

(9.5)

It is convenient to define a hydraulic diameter as

$\displaystyle D_{H} = {4 \times \hbox{Cross Section Area} \over \hbox{ wetted perimeter }}$

(9.6)

Or in other words

$\displaystyle A = {\pi {D_H}^2 \over 4}$

(9.7)

It is convenient to substitute

for

and yet it still will be referred to the same name as the hydraulic diameter. The infinitesimal area that shear stress is acting on is

$\displaystyle dA_w = \pi D dx$

(9.8)

Introducing the Fanning friction factor as a dimensionless friction factor which is some times referred to as the friction coefficient and reads as the following:

$\displaystyle f = { \tau_w \over \half \rho U^{2} }$

(9.9)

By utilizing equation (9.2) and substituting equation (9.10) into momentum equation (9.6) yields

$\displaystyle - \overbrace{\pi D^2 \over 4}^{A} dP - { \pi D dx } \overbrace{ f... ...\rho U^{2} \right) }^{\tau_w} = A { \overbrace{\rho U}^{\dot {m} \over A } dU }$

(9.10)

Dividing equation (9.11) by the cross section area, $\mathbf{A}$ and rearranging yields

$\displaystyle - dP + { 4 f dx \over D} \left( \half \rho U^{2} \right) = {\rho U dU }$

(9.11)

The second law is the last equation to be utilized to determine the flow direction.

$\displaystyle s_2 \geq s_1$

(9.12)

Next: Non-Dimensionalization of the Equations Up: Fanno Flow Previous: Introduction Index

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