View source for The Q1D equations

[[Category:Compressible flow]]
[[Category:Compressible flow:Topic]]
[[Category:Quasi-one-dimensional flow]]
[[Category:Governing equations:Related]]
[[Category:Inviscid flow]]

__TOC__

=== Governing Equations ===

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\caption{Quasi-one-dimensional flow - control volume}
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In the following quasi-one-dimensional flow will be assumed. That means that the cross-section is allowed to vary smoothly but flow quantities varies in one direction only. The equations that are derived will thus describe one-dimensional flow in axisymmetric tubes. Let's assume flow in the <math>x</math>-direction, which means that all flow quantities and the cross-section area will vary with the axial coordinate <math>x</math>. 

<math display="block">
A=A(x),\ \rho=\rho(x),\ u=u(x),\ p=p(x),\ ...
</math>

We will further assume steady-state flow, which means that unsteady terms will be zero.

The equations are derived with the starting point in the governing flow equations on integral form

==== Continuity Equation ====

Applying the integral form of the continuity equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives

<math display="block">
\underbrace{\frac{d}{dt}\iiint_{\Omega}\rho d{V}}_{=0}+\iint_{\partial \Omega}\rho {\mathbf{v}}\cdot {\mathbf{n}}dS=0
</math>

<math display="block">
\iint_{\partial \Omega}\rho {\mathbf{v}}\cdot {\mathbf{n}}dS=-\rho_1 u_1 A_1+\rho_2 u_2 A_2
</math>

<math display="block">
\rho_1 u_1 A_1=\rho_2 u_2 A_2
</math>

==== Momentum Equation ====

Applying the integral form of the momentum equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives

<math display="block">
\underbrace{\frac{d}{dt}\iiint_{\Omega}\rho{\mathbf{v}} d{V}}_{=0}+\iint_{\partial \Omega}\left[\rho ({\mathbf{v}}\cdot {\mathbf{n}}){\mathbf{v}}+p{\mathbf{n}}\right]dS=0
</math>

<math display="block">
\iint_{\partial \Omega} \rho ({\mathbf{v}}\cdot {\mathbf{n}}){\mathbf{v}}dS=-\rho_1u_1^2A_1+\rho_2u_2^2A_2
</math>

<math display="block">
\iint_{\partial \Omega} p{\mathbf{n}}dS=-p_1A_1+p_2A_2-\int_{A_1}^{A_2}pdA
</math>

collecting terms

<math display="block">
\left(\rho_1u_1^2+p_1\right)A_1+\int_{A_1}^{A_2}pdA=\left(\rho_2u_2^2+p_2\right)A_2
</math>

==== Energy Equation ====

Applying the integral form of the energy equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives

<math display="block">
\underbrace{\frac{d}{dt}\iiint_{\Omega}\rho e_o d{V}}_{=0}+\iint_{\partial \Omega}\left[\rho h_o ({\mathbf{v}}\cdot{\mathbf{n}})\right]dS=0
</math>

<math display="block">
\iint_{\partial \Omega}\left[\rho h_o ({\mathbf{v}}\cdot{\mathbf{n}})\right]dS=-\rho_1u_1h_{o_1}A_1+\rho_2u_2h_{o_2}A_2
</math>

<math display="block">
\rho_1u_1h_{o_1}A_1=\rho_2u_2h_{o_2}A_2
</math>

Now, using the continuity equation <math>\rho_1u_1A_1=\rho_2u_2A_2</math> gives

<math display="block">
h_{o_1}=h_{o_2}
</math>

==== Differential Form ====

The integral term appearing the momentum equation is undesired and therefore the governing equations are converted to differential form.

The continuity equation (Eqn. \ref{eq:governing:cont:b}) is rewritten in differential form as

<math display="block">
\rho_1u_1A_1=\rho_2u_2A_2=const
</math>

<math display="block">
d(\rho uA)=0
</math>

The momentum equation (Eqn. \ref{eq:governing:mom:b}) is rewritten in differential form as

<math display="block">
\left(\rho_1u_1^2+p_1\right)A_1+\int_{A_1}^{A_2}pdA=\left(\rho_2u_2^2+p_2\right)A_2\Rightarrow d\left[(\rho u^2+p)A\right]=pdA
</math>

<math display="block">
d(\rho u^2A)+d(pA)=pdA
</math>

<math display="block">
ud(\rho uA)+\rho uAdu+Adp+\cancel{pdA}=\cancel{pdA}
</math>

From the continuity equation we have <math>d(\rho uA)</math> and thus

<math display="block">
\rho u\cancel{A}du+\cancel{A}dp=0\Rightarrow
</math>

<math display="block">
dp=-\rho udu
</math>

which is the momentum equation on differential form. Also referred to as Euler's equation. Finally, the energy equation (Eqn. \ref{eq:governing:cont:b}) is rewritten in differential form as

<math display="block">
h_{o_1}=h_{o_2}=const\Rightarrow dh_o=0
</math>

<math display="block">
h_o=h+\frac{1}{2}u^2\Rightarrow dh+\frac{1}{2}d(u^2)=0
</math>

<math display="block">
dh+udu=0
</math>

==== Summary ====

Continuity:

<math display="block">
d(\rho uA)=0
</math>

Momentum:

<math display="block">
dp=-\rho udu
</math>

Energy:

<math display="block">
dh+udu=0
</math>

The equations are valid for:


* quasi-one-dimensional flow
* steady state
* all gas models (no gas model assumptions made)
* inviscid flow


It should be noted that equations are exact but they are applied to a physical model that is approximate, i.e., the approximation that flow quantities varies in one dimension with a varying cross-section area. In reality, a variation of cross-section area would imply flow in three dimensions.