The Q1D equations: Difference between revisions

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[[Category:Compressible flow]]
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[[Category:Quasi-one-dimensional flow]]
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[[Category:Governing equations]]
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=== Governing Equations ===
=== Governing Equations ===


Line 20: Line 30:
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>.  
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">
{{NumEqn|<math>
A=A(x),\ \rho=\rho(x),\ u=u(x),\ p=p(x),\ ...
A=A(x),\ \rho=\rho(x),\ u=u(x),\ p=p(x),\ ...
</math>
</math>}}


We will further assume steady-state flow, which means that unsteady terms will be zero.
We will further assume steady-state flow, which means that unsteady terms will be zero.
Line 32: Line 42:
Applying the integral form of the continuity equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives
Applying the integral form of the continuity equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives


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


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


<math display="block">
{{NumEqn|<math>
\rho_1 u_1 A_1=\rho_2 u_2 A_2
\rho_1 u_1 A_1=\rho_2 u_2 A_2
</math>
</math>}}


==== Momentum Equation ====
==== Momentum Equation ====
Line 48: Line 58:
Applying the integral form of the momentum equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives
Applying the integral form of the momentum equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives


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


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


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


collecting terms
collecting terms


<math display="block">
{{NumEqn|<math>
\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
\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>
</math>}}


==== Energy Equation ====
==== Energy Equation ====
Line 70: Line 80:
Applying the integral form of the energy equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives
Applying the integral form of the energy equation on the quasi-one-dimensional flow control volume (Fig. \ref{fig:cv}) gives


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


<math display="block">
{{NumEqn|<math>
\varoiint_{\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
\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>}}


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


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


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


==== Differential Form ====
==== Differential Form ====
Line 94: Line 104:
The continuity equation (Eqn. \ref{eq:governing:cont:b}) is rewritten in differential form as
The continuity equation (Eqn. \ref{eq:governing:cont:b}) is rewritten in differential form as


<math display="block">
{{NumEqn|<math>
\rho_1u_1A_1=\rho_2u_2A_2=const
\rho_1u_1A_1=\rho_2u_2A_2=const
</math>
</math>}}


<math display="block">
{{NumEqn|<math>
d(\rho uA)=0
d(\rho uA)=0
</math>
</math>}}


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


<math display="block">
{{NumEqn|<math>
\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
\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>}}


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


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


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


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


<math display="block">
{{NumEqn|<math>
dp=-\rho udu
dp=-\rho udu
</math>
</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
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">
{{NumEqn|<math>
h_{o_1}=h_{o_2}=const\Rightarrow dh_o=0
h_{o_1}=h_{o_2}=const\Rightarrow dh_o=0
</math>
</math>}}


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


<math display="block">
{{NumEqn|<math>
dh+udu=0
dh+udu=0
</math>
</math>}}


==== Summary ====
==== Summary ====


Continuity:
<div style="border: 1px solid;">
 
{{OpenInfoBox|<math>
<math display="block">
d(\rho uA)=0
d(\rho uA)=0
</math>
</math>|description=Continuity:}}
 
Momentum:


<math display="block">
{{OpenInfoBox|<math>
dp=-\rho udu
dp=-\rho udu
</math>
</math>|description=Momentum:}}


Energy:
{{OpenInfoBox|<math>
 
<math display="block">
dh+udu=0
dh+udu=0
</math>
</math>|description=Energy:}}
</div>


The equations are valid for:
The equations are valid for:


* quasi-one-dimensional flow
* quasi-one-dimensional flow
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