Complete lecture 2 on mmme1048 fluid mechanics

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
+
+  $$p_1 = p_2 + \rho_wg(z_{C2} - z_{C1})$$