05 Control Volume
Flow Rates¶
\[ \begin{aligned} \dot V &= vA \\ \dot m &= \rho \dot V = \rho vA \\ &= \frac{\dot V}{\nu} = \frac{vA}{\nu} \\ \Big( PV &= mRT, m = PV, \rho = \frac{P}{RT} \Big) \end{aligned} \]
Flow Work¶
\(W_\text{f} = PV\)
For non-flowing fluid (fluid that remains inside tank), Flow work = 0
Conservation of Mass¶
\[ \begin{aligned} \sum m_\text{in} - \sum m_\text{out} &= \Delta m \\ \sum \dot m_\text{in} - \sum \dot m_\text{out} &= \frac{\mathrm{d} m}{\mathrm{d} t} \\ \end{aligned} \]
Conservation of Energy¶
\[ \begin{aligned} E_\text{in} - E_\text{out} &= \Delta E_\text{cv} \\ \dot E_\text{in} - \dot E_\text{out} &= \frac{\mathrm{d} E_\text{cv}}{\mathrm{d} t} \\ \dot Q_\text{net} - \dot W_\text{net} + \dot E_\text{m, in} - \dot E_\text{m, out} &= \frac{\mathrm{d} E_\text{cv}}{\mathrm{d} t} \\ \dot E_\text{in} &= \dot m \left[ h + \frac{v^2}{2000} + gz \right] & (h = u + P\nu) \\ &= \dot m \left[ u + \frac{v^2}{2000} + gz \right] & \text{(non-flowing)} \end{aligned} \]
Steady Flow¶
Properties within the control volume remain constant with time
Mass¶
\[ \begin{aligned} \frac{\mathrm{d} m_\text{cv}}{\mathrm{d} t} &= 0 \\ \sum \dot m_\text{in} &= \sum \dot m_\text{out} \\ \dot m_1 &= \dot m_2 \\ \rho_1 v_1 A_1 &= \rho_2 v_2 A_2 \\ \frac{v_1 A_1}{\nu_1} &= \frac{v_2 A_2}{\nu_2} \end{aligned} \]
Energy¶
\[ \begin{aligned} \frac{\mathrm{d} E_\text{cv}}{\mathrm{d} t} &= 0 \\ \dot E_\text{m, in} &= \dot E_\text{m, out} \\ \dot Q_\text{net} - \dot W_\text{net} + \dot E_\text{m ,in} &= \dot E_\text{m, out} \end{aligned} \]
Steady Flow Devices¶
| Device | \(v\) | \(P\) | \(T\) | Work | |
|---|---|---|---|---|---|
| Nozzle | inc | dec | |||
| Diffuser | dec | inc | |||
| Turbine thermal \(\to\) mechanical | \(\dot W_\text{in} = 0\) | ||||
| Compressor | inc | inc | \(\dot W_\text{out} = 0\) | ||
| Throttling valve (isenthalpic) | dec | dec | \(\dot W_\text{in} = \dot W_\text{out} = 0\) | \(\begin{aligned} h_1 &= h_2 \\ u_1 + P_1 \nu_1 &= u_2 + P_2 \nu_2 \end{aligned}\) |
Unsteady/Transient Flow¶
Mass¶
\[ \begin{aligned} m_\text{in} - m_\text{out} &= \Delta m_\text{cv} \\ &= m_2 - m_1 \\ \dot m_\text{in} - \dot m_\text{out} &= \frac{\mathrm{d} m_\text{cv}}{\mathrm{d} t} \end{aligned} \]
Energy¶
\[ \begin{aligned} E_\text{in} - E_\text{out} &= \Delta E_\text{cv} \\ Q_\text{net} - W_\text{net} + E_\text{m, in} - E_\text{m, out} &= \Delta E_\text{cv} \\ \Delta E_\text{cv} &= m_2 e_2 - m_1 e_1 \\ e &= h + \frac{v^2}{2000} + gz \end{aligned} \]