Commit a844cfcf authored by Olivier's avatar Olivier

Chapter 2: basic equations. Moved some over from chapter 3.

parent 571f619d
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\renewcommand{\lastedityear}{2017}
\renewcommand{\lasteditmonth}{09}
\renewcommand{\lasteditday}{23}
\renewcommand{\lasteditday}{27}
\renewcommand{\numberofthischapter}{3}
\renewcommand{\titleofthischapter}{\namechapterthree}
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In this section, we apply Newton’s second law: we assert that the variation of system’s linear momentum is equal to the net force being applied to it (eq.~\ref{eq_secondlaw} p.\pageref{eq_secondlaw}). Our study of the fluid’s properties at the control surface is carried out by replacing variable $B$ by the quantity $m \vec V$: momentum. Thus, $\timederivative{B_\sys}$ becomes $\timederivative{(m \vec V_\sys)}$, which is equal to $\vec F_\net$, the vector sum of forces applied on the system as it transits the control volume.\\
In a similar fashion, $b \equiv B/m = \vec V$ and equation~\ref{eq_rtt}, the Reynolds transport theorem, becomes:
\begin{IEEEeqnarray}{CCCCCCC}
\timederivative{(m \vec V_{sys})} & = & \vec F_\net & = & \timederivative{} \iiint_\cv \rho \vec V \diff \vol & + & \iint_\cs \rho \vec V \ (\vec V_\rel \cdot \vec n) \diff A \nonumber\\\label{eq_rtt_linearmom}\\
\timederivative{(m \vec V_\text{sys})} & = & \vec F_\net & = & \timederivative{} \iiint_\cv \rho \vec V \diff \vol & + & \iint_\cs \rho \vec V \ (\vec V_\rel \cdot \vec n) \diff A \nonumber\\\label{eq_rtt_linearmom}\\
& & \begin{array}{c} {\scriptstyle \text{the vector sum of}} \\ {\scriptstyle \text{forces on the system}} \end{array} & = & \begin{array}{c} {\scriptstyle \text{the rate of change}} \\ {\scriptstyle \text{of linear momentum}} \\ {\scriptstyle \text{within the control volume}} \end{array} & + & \begin{array}{c} {\scriptstyle \text{the net flow of linear momen-}} \\ {\scriptstyle \text{tum through the boundaries}} \\ {\scriptstyle \text{of the control volume}} \end{array} \nonumber
\end{IEEEeqnarray}
......@@ -240,28 +240,6 @@
\end{IEEEeqnarray}
\end{adjustwidth}
When the control volume has well-defined inlets and outlets through which the term $\rho\ e \ (\vec V_\rel \cdot \vec n)$ can be considered uniform, this equation reduces to:
\begin{adjustwidth}{-3cm}{-0cm}%handmade
\begin{IEEEeqnarray}{rCl}
\dot Q_{\net\ \inn} + \dot W_\text{shaft, net in} + \dot W_\text{pressure, net in} & = & \timederivative{} \iiint_\cv \rho\ e \diff \vol + \sum_\out \left\{ \dot m \ e\right\} - \sum_\inn \left\{ \dot m \ e\right\} \\
\nonumber\\%
\dot Q_{\net \inn} + \dot W_\text{shaft, net in} & = & \timederivative{} \iiint_\cv \rho\ e \diff \vol + \sum_\out \left\{ \dot m (i + e_k + e_p)\right\} \nonumber\\
&&- \sum_\inn \left\{ \dot m (i + e_k + e_p)\right\} - \dot W_\text{pressure, net in} \nonumber \\
\nonumber\\%
& = & \timederivative{} \iiint_\cv \rho\ e \diff \vol + \sum_\out \left\{ \dot m \left(i + \frac{1}{2}V^2 + g z\right)\right\} \nonumber \\
&& - \sum_\inn \left\{ \dot m \left(i + \frac{1}{2}V^2 + g z\right)\right\} + \sum_\out \left\{ \dot m \frac{p}{\rho}\right\} - \sum_\inn \left\{ \dot m \frac{p}{\rho}\right\} \nonumber
\end{IEEEeqnarray}
\end{adjustwidth}
Making use of the concept of \vocab{enthalpy} defined as $h \equiv i + p/\rho$, we obtain:
\begin{adjustwidth}{-3cm}{0cm}%handmade
\begin{IEEEeqnarray}{cCcCcc0}
\dot Q_\text{net in} + \dot W_\text{shaft, net in} & = & \timederivative{} \iiint_\cv \rho\ e \diff \vol & + & \sum_\out \left\{ \dot m \left(h + \frac{1}{2}V^2 + g z \right)\right\} - \sum_\inn \left\{ \dot m \left(h + \frac{1}{2}V^2 + g z \right)\right\} \nonumber\\ ~\hspace{-2cm}\label{eq_rtt_energy_simple} \\
\begin{array}{c} {\scriptstyle \text{the net power}} \\ {\scriptstyle \text{received by the system}}\end{array} & = & \begin{array}{c} {\scriptstyle \text{the rate of change}} \\ {\scriptstyle \text{of energy inside}} \\ {\scriptstyle \text{the control volume}} \end{array} & + & \begin{array}{c} {\scriptstyle \text{the net energy flow rate}}\\ {\scriptstyle \text{exiting the control volume boundaries}} \end{array} &\nonumber
\end{IEEEeqnarray}
\end{adjustwidth}
This equation~\ref{eq_rtt_energy_simple} is particularly attractive, but it necessitates the input of a large amount of experimental data to provide useful results. It is indeed very difficult to predict how the terms $i$ and $p /\rho$ will change for a given flow process. For example, a pump with given power $\dot Q_{\net \ \inn}$ and $\dot W_\text{shaft, net in}$ will generate large increases of terms $p$, $V$ and $z$ if it is efficient, or a large increase of terms $i$ and $1/\rho$ if it is inefficient. This equation~\ref{eq_rtt_energy_simple}, sadly, does not allow us to quantify the net effect of shear and the extent of irreversibilities in a fluid flow.
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