The Q1D equations: Difference between revisions
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=== Governing Equations === | === Governing Equations === | ||
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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>. | 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 | {{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. | ||
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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 | {{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 | {{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 | {{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 ==== | ||
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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 | {{NumEqn|<math> | ||
\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 | \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 | {{NumEqn|<math> | ||
\iint_{\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 | {{NumEqn|<math> | ||
\iint_{\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 | {{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 ==== | ||
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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 | {{NumEqn|<math> | ||
\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 | \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 | {{NumEqn|<math> | ||
\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 | \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 | {{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 | {{NumEqn|<math> | ||
h_{o_1}=h_{o_2} | h_{o_1}=h_{o_2} | ||
</math> | </math>}} | ||
==== Differential Form ==== | ==== Differential Form ==== | ||
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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 | {{NumEqn|<math> | ||
\rho_1u_1A_1=\rho_2u_2A_2=const | \rho_1u_1A_1=\rho_2u_2A_2=const | ||
</math> | </math>}} | ||
<math | {{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 | {{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 | {{NumEqn|<math> | ||
d(\rho u^2A)+d(pA)=pdA | d(\rho u^2A)+d(pA)=pdA | ||
</math> | </math>}} | ||
<math | {{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 | {{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 | {{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 | {{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 | {{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 | {{NumEqn|<math> | ||
dh+udu=0 | dh+udu=0 | ||
</math> | </math>}} | ||
==== Summary ==== | ==== Summary ==== | ||
<div style="border: 1px solid;"> | |||
{{NumEqn|<math> | |||
<math | |||
d(\rho uA)=0 | d(\rho uA)=0 | ||
</math> | </math>|nonumber=1|infobox=1|description=Continuity:|noborder=1}} | ||
<math | {{NumEqn|<math> | ||
dp=-\rho udu | dp=-\rho udu | ||
</math> | </math>|nonumber=1|infobox=1|description=Momentum:|noborder=1}} | ||
{{NumEqn|<math> | |||
<math | |||
dh+udu=0 | dh+udu=0 | ||
</math> | </math>|nonumber=1|infobox=1|description=Energy:|noborder=1}} | ||
</div> | |||
The equations are valid for: | The equations are valid for: | ||
* quasi-one-dimensional flow | * quasi-one-dimensional flow | ||
