Review of conservation principles
1 Extensive vs. intensive properties
An extensive property is dependent of the size of the system. For example, the mass of a fluid system.
An intensive property is independent of the size of the system. For example, the density of the same system.
2 Specifications of the flow field
2.1 Lagrangian specification of the flow field
The observer follows a system of interest (the same fluid particles) in space and time.
2.2 Eulerian specification of the flow field
However, we usually make observations at a fixed region (control volume).
A control volume is a fixed volume in space of interest and its surrounding closed surface is a control surface.
3 Reynolds transport theorem
How can we relate these two specifications of the flow field?
The Reynolds transport theorem!
Change within the system = Change within the control volume + Change across the control surface
3.1 Change of what?
Let $B$ be an extensive property of a fluid.
At time $t$, \begin{equation}B_\text{sys}(t)=B_\text{cv}(t)\end{equation}
At time $t+dt$, \begin{equation}B_\text{sys}(t+dt)=B_\text{cv}(t+dt)+B_\text{out}(t+dt)-B_\text{in}(t+dt)\end{equation}
3.2 Change within the system
\begin{equation} \begin{aligned} \frac{dB_\text{sys}}{dt} &=\frac{B_\text{sys}(t+dt)-B_\text{sys}(t)}{dt}\\ &=\frac{B_\text{cv}(t+dt)+B_\text{out}(t+dt)-B_\text{in}(t+dt)-B_\text{cv}(t)}{dt}\\ &=\frac{B_\text{cv}(t+dt)-B_\text{cv}(t)}{dt}+\frac{B_\text{out}(t+dt)-B_\text{in}(t+dt)}{dt}\\ &=\frac{B_\text{cv}(t+dt)-B_\text{cv}(t)}{dt}+\frac{\left[B_\text{out}(t+dt)-B_\text{in}(t+dt)\right]-\left[B_\text{out}(t)-B_\text{in}(t)\right]}{dt}\\ &=\frac{dB_\text{cv}}{dt}+\dot{B}_\text{net out}\\ \end{aligned} \end{equation} because $B_\text{out}(t)=B_\text{in}(t)$.
By letting $B\equiv B_\text{sys}$ (system property), \begin{equation} \frac{dB}{dt}=\frac{dB_\text{cv}}{dt}+\dot{B}_\text{net out} \end{equation}
3.3 Change within the control volume
\[ \newcommand{\volsym}{\rlap{\kern.08em–}V} \newcommand{\volsubsym}{\rlap{\scriptsize\kern.08em–}V} \] \begin{equation} \frac{dB_\text{cv}}{dt} =\frac{d}{dt}\int\limits_\text{cv}dB =\frac{d}{dt}\int\limits_\text{cv}d(bm) =\frac{d}{dt}\int\limits_\text{cv}b\,dm =\frac{d}{dt}\int\limits_\text{cv}b\rho\,d\volsym \end{equation} where $m$ is the mass, $\rho$ is the density, $b=\frac{B}{m}$ is the intensive property, and $\volsym$ is the volume.
Remember, $b$ doesn’t change with mass (a fixed quantity regardless of the mass), so we can take $b$ out of $d(bm)$.
3.4 Change across the control surface
For net out, outflux positive and influx negative
\begin{equation} \dot{B}_\text{net out} =b\dot{m}_\text{net out} =b\rho\dot{\volsym}_\text{net out} =\int\limits_\text{cs}b\rho V\,dA =\int\limits_\text{cs}b\rho(\vec{V}\cdot\vec{n})\,dA \end{equation}
3.5 Reynolds transport theorem equation
\begin{equation} \frac{dB}{dt} =\frac{d}{dt}\int\limits_\text{cv}b\rho\,d\volsym +\int\limits_\text{cs}b\rho(\vec{V}\cdot\vec{n})\,dA \end{equation}
4 Intensive property
Let’s take a closer look at $b$ before we move on. By definition, $b=\frac{B}{m}$, so $B=bm$. Taking the derivative of both sides with respect to $m$, we obtain $\frac{dB}{dm}=b$. Oh! $b=\frac{B}{m}$ and $b=\frac{dB}{dm}$.
5 Conservation principles
- Conservation of mass
- also known as the continuity equation
- Conservation of energy
- also known as the first law of thermodynamics
- Conservation of momentum
- also known as Newton’s second law of motion
5.1 Conservation of mass
- $B=m$
- $b=\frac{B}{m}=\frac{m}{m}=1$
Mass doesn’t change over time. What is $\frac{dB}{dt}$? 0
\begin{equation} 0 =\frac{d}{dt}\int\limits_\text{cv}\rho\,d\volsym +\int\limits_\text{cs}\rho(\vec{V}\cdot\vec{n})\,dA \end{equation}
Change within the control volume is balanced by the net outflux through the control surface.
5.1.1 One-dimensional steady flow of an incompressible fluid
In a given time period, the mass of the control volume remains the same ($\frac{d}{dt}\int\limits_\text{cv}\rho\,d\volsym=0$) and what’s going into the control volume should come out of it.
\begin{equation} \int\limits_\text{cs}\rho(\vec{V}\cdot\vec{n})\,dA =Q_\text{out}-Q_\text{in} =0 \end{equation}
By the Reynolds transport theorem, \begin{equation} \begin{aligned} \frac{dB}{dt} &=\frac{d}{dt}\int\limits_\text{cv}b\rho\,d\volsym +\int\limits_\text{cs}b\rho(\vec{V}\cdot\vec{n})\,dA\\ &=0+Q_\text{out}-Q_\text{in}=0 \end{aligned} \end{equation} and we obtain Eq. (3.32) in the text: \begin{equation} Q_\text{in}=Q_\text{out}=A_1V_1=A_2V_2 \end{equation} and $Q=AV$ is constant.
5.2 Conservation of energy
- $\vec{B}=E$
- $\vec{b}=\frac{E}{m}$
\begin{equation} \frac{dE}{dt} =\frac{d}{dt}\int\limits_\text{cv}\frac{E}{m}\rho\,d\volsym +\int\limits_\text{cs}\frac{E}{m}\rho(\vec{V}\cdot\vec{n})\,dA \end{equation}
5.2.1 One-dimensional steady flow of an incompressible fluid
\begin{equation} \frac{p_1}{\gamma}+z_1+\alpha_1\frac{V_1^2}{2g} =\frac{p_2}{\gamma}+z_2+\alpha_2\frac{V_2^2}{2g}+h_f \end{equation}
5.3 Conservation of momentum
- $\vec{B}=m\vec{V}$
- $\vec{b}=\frac{\vec{B}}{m}=\frac{m\vec{V}}{m}=\vec{V}$
- $\frac{d\vec{B}}{dt}=\frac{d(m\vec{V})}{dt}=m\frac{d\vec{V}}{dt}=m\vec{a}=\vec{F}$
- $\vec{F}$ is the net force of the system acting on the control volume
- Can be written as $\sum\vec{F}$
\begin{equation} \sum\vec{F} =\frac{d}{dt}\int\limits_\text{cv}\vec{V}\rho\,d\volsym +\int\limits_\text{cs}\vec{V}\rho(\vec{V}\cdot\vec{n})\,dA \end{equation}
5.3.1 One-dimensional steady flow of an incompressible fluid
\begin{equation} \frac{d}{dt}\int\limits_\text{cv}\vec{V}\rho\,d\volsym=0 \end{equation}
\begin{equation} \sum F_s =\int\limits_\text{cs}\rho v_s(\vec{V}\cdot\vec{n})\,dA =\sum(\beta\rho QV_s)_\text{out}-\sum(\beta\rho QV_s)_\text{in} \end{equation} where the momentum flux correction coefficient $\beta=\frac{\int_Av_s^2\,dA}{V_s^2A}$.
- $\beta=1$ in a uniform flow
- $\beta$ is not significantly greater than 1 in turbulent flow in prismatic channels
- $\beta\neq 1$ in a nonuniform flow