diff --git a/mechanical/mmme1048_fluid_mechanics.md b/mechanical/mmme1048_fluid_mechanics.md index 3e1d671..4c2cfea 100755 --- a/mechanical/mmme1048_fluid_mechanics.md +++ b/mechanical/mmme1048_fluid_mechanics.md @@ -151,3 +151,95 @@ The -ve sign indicates that as $z$, height, increases, $p$, pressure, decreases. - It is usually better to use SI units - If in doubt, DA can be useful to check that your answer makes sense + +# Lecture 2 // Manometers (2021-10-13) + +![](./images/vimscrot-2021-10-13T09:09:32,037006075+01:00.png) + +$$p_{1,gauge} = \rho g(z_2-z_1)$$ + +- Manometers work on the principle that pressure along any horizontal plane through a continuous + fluid is constant +- Manometers can be used to measure the pressure of a gas, vapour, or liquid +- Manometers can measure higher pressures than a piezometer +- Manometer fluid and working should be immiscible (don't mix) + +![](./images/vimscrot-2021-10-13T09:14:59,628661490+01:00.png) + +\begin{align*} +p_A &= p_{A'} \\ +p_{bottom} &= p_{top} + \rho gh \\ +\rho_1 &= density\,of\,fluid\,1 \\ +\rho_2 &= density\,of\,fluid\,2 +\end{align*} + +Left hand side: + +$$p_A = p_1 + \rho_1g\Delta z_1$$ + +Right hand side: + +$$p_{A'} = p_{at} + \rho_2g\Delta z_2$$ + +Equate and rearrange: + +\begin{align*} +p_1 + \rho_1g\Delta z_1 &= p_{at} + \rho_2g\Delta z_2 \\ +p_1-p_{at} &= g(\rho_2\Delta z_2 - \rho_1\Delta z_1) \\ +p_{1,gauge} &= g(\rho_2\Delta z_2 - \rho_1\Delta z_1) +\end{align*} + +If $\rho_a << \rho_2$: + +$$\rho_{1,gauge} \approx \rho_2g\Delta z_2$$ + +## Differential U-Tube Manometer + +![](./images/vimscrot-2021-10-13T09:37:02,070474894+01:00.png) + +- Used to find the difference between two unknown pressures +- Can be used for any fluid that doesn't react with manometer fluid +- Same principle used in analysis + +\begin{align*} +p_A &= p_{A'} \\ +p_{bottom} &= p_{top} + \rho gh \\ +\rho_1 &= density\,of\,fluid\,1 \\ +\rho_2 &= density\,of\,fluid\,2 +\end{align*} + +Left hand side: + +$$p_A = p_1 + \rho_wg(z_C-z_A)$$ + +Right hand side: + +$$p_B = p_2 + \rho_wg(z_C-z_B)$$ + +Right hand manometer fluid: + +$$p_{A'} = p_B + \rho_mg(z_B - z_a)$$ + +\begin{align*} +p_{A'} &= p_2 + \rho_mg(z_C - z_B) + \rho_mg(z_B - zA)\\ + &= p_2 + \rho_mg(z_C - z_B) + \rho_mg\Delta z \\ +\\ +p_A &= p_{A'} \\ +p_1 + \rho_wg(z_C-z_A) &= p_2 + \rho_mg(z_C - z_B) + \rho_mg\Delta z \\ +p_1 - p_2 &= \rho_wg(z_C-z_B-z_C+z_A) + \rho_mg\Delta z \\ +&= \rho_wg(z_A-z_B) + \rho_mg\Delta z \\ +&= -\rho_wg\Delta z + \rho_mg\Delta z +\end{align*} + +## Angled Differential Manometer + +![](./images/vimscrot-2021-10-13T09:56:15,656796805+01:00.png) + +- If the pipe is sloped then + + $$p_1-p_2 = (\rho_m-\rho_w)g\Delta z + \rho_wg(z_{C2} - z_{C1})$$ + +- $p_1 > p_2$ as $p_1$ is lower +- If there is no flow along the tube, then + + $$p_1 = p_2 + \rho_wg(z_{C2} - z_{C1})$$