add notes on pumps, turbines

uni/mmme/2047_thermodynamics_and_fluid_dynamics/pumps.md

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+---
+author: Akbar Rahman
+date: \today
+title: MMME2047 // Turbomachinery // Pumps
+tags: [ turbomachinery, pumps ]
+uuid: 11f0f745-2364-4594-8e47-127a4af39417
+lecture_slides: [ ./lecture_slides/T5 - Turbomachinery - with solutions.pdf ]
+lecture_notes: [ ./lecture_notes/turbomachinery lecture notes(H Power).pdf ]
+exercise_sheets: [ ./exercise_sheets/Turbomachinery-problems.pdf]
+---
+
+Turbomachinery are rotating devices that add (pump for liquids; fan, blower, or compressor for gases at <0.02, <1 bar, and > 1 bar respectively) or extract (turbine) energy from a fluid.
+
+# Positive Displacement (PD) Pumps
+
+- PD pumps force fluid along using volume changes (e.g. bike pumps, the heart)
+- All PD pumps deliver a periodic flow
+- They deliver any fluid regardless of viscosity (dynamic pumps struggle with viscous fluids)
+- They are self priming (will be filled automatically)
+- They can operate under high pressures (300 atm) but low flow rates (25 m$^3$h$^{-1}$)
+- flow rate can be only be changed by vary speed or displacement
+
+# Dynamics Pumps
+
+- add momentum to fluid by fast moving blades or vanes
+- classified based on direction of flow at exit:
+
+    - centrifugal
+    - axial
+    - mixed flow
+
+- fluid increases momentum while moving through open passages and extra velocity is converted to
+  pressure through exiting it into a diffuser section
+- provide high flow rates (up to 70000 m$^3$h$^{-1}$) but usually at moderate pressure rises (a few atm)
+- require priming
+
+# Centrifugal Pumps
+
+- fluid enters through eye of casing and gets caught in impeller blades
+- fluid is whirled outwards until it leaves via the expanding area section, known as the diffuser or volute
+
+## Blades
+
+- backward inclined blades - most common and efficient, intermediate pressure rise, less robust
+- straight blades - simplest geometry, high pressure rise, less robust
+- forward inclined blades - more blades but smaller, lowest pressure rise, lowest efficiency, more robust
+
+
+## Integral Analysis of Centrifugal Pumps
+
+Derivation in slides (p. 23-25).
+
+\begin{align}
+\frac{w_s}{g} - \left(\frac{u_2-_1-q}{g}\right) = \left(\frac{p_2}{\rho g} + z_2 + \frac{v_2^2}{2g}\right) -\left(\frac{p_1}{\rho g} + z_1 + \frac{v_1^2}{2g}\right) \\
+H_s - H_f &= H = H_{T,2} - H_{T,1} \nonumber
+\end{align}
+
+where $H_s$ is supplied head to pump, $H_f$ friction loss head, $H$ is head supplied to fluid, $H_{T,1}$ is total head at inlet, and $H_{T,2}$ is total head at outlet.
+
+Assuming that $z_1 \approx z_2$, $v_1 \approx v_2$ (from inlet and outlet diameters are equal) then:
+
+\begin{equation}
+H \approx \frac{p_2-p_1}{\rho g}
+\end{equation}
+
+and the power to the fluid (water horsepower) is:
+
+\begin{equation}
+P_w = \rho QgH
+\end{equation}
+
+where $Q$ is volumetric flow rate.
+
+Power supplied to the pump (brake horsepower), $P = \omega T$, lets us find the overall pump efficiency:
+
+\begin{equation}
+\eta = \frac{P_w}{P} = \frac{\rho QgH}{\omega T} = \eta_h \eta_m \eta_v
+\end{equation}
+
+where:
+
+- $\eta_h = 1 - \frac{H_f}{H_s}$ is hydraulic efficiency
+- $\eta_m = 1- \frac{P_f}{P}$ is mechanical efficiency
+- $\eta_v = \frac{Q}{Q+Q_L}$ (where $Q_L$ is loss due to leakage flow) is the volumetric efficiency
+
+## Performance
+
+![](./images/vimscrot-2023-03-13T10:07:58,750255090+00:00.png)
+
+# Cavitation
+
+Cavitation is when bubbles form in liquid by sudden pressure drop, followed by their implosion when
+original pressure is restored.
+The implosion generates a high pressure wave that can damage nearby solid surfaces.
+
+In a centrifugal pump, the fluid pressure drops at the impeller's eye, where it has the minimum
+value.
+If pressure falls below saturation pressure, bubbles appear.
+Pressure grows as the fluid flows between the blades as the ducts are diverging.
+Pressure is maximum at the trailing edge of the blades, on their front side.
+This is where cavitation occurs and causes wear on the blade.
+
+<https://www.michael-smith-engineers.co.uk/resources/useful-info/pump-cavitation>  
+<https://www.youtube.com/watch?v=g1o5Z9o7b0>  
+<https://www.youtube.com/watch?v=eMDAw0TXvUo>  
+<https://www.youtube.com/watch?v=1Lbxtjfdat4>
+
+![](./images/cavitation.png)
+
+# Net Positive Suction Head (NPSH)
+
+The following conditions must be satisfied to prevent cavitation:
+
+\begin{equation}
+H_i - \frac{p_v}{\rho g} > \text{NSPSH}
+\end{equation}
+
+where $H_i = \frac{p_i}{\rho g} + \frac{v_i^2}{2g}$ is total head at inlet, $p_v$ is saturation
+pressure at $T_i$.
+
+It is important that the inlet pressure is as high as possible.
+To do this, one can reduce frictional losses (e.g. shorter smoother pipes) or install the pump lower down
+(even below the reservoir) (slides p. 36).
+
+# Dimensionless Pump Performance
+
+Output variables for a pump's performance are pump head $H$ and brake horsepower $P$.
+Input variables are discharge $Q$, impeller diameter $D$, and shaft speed $n$.
+
+The dimensional relationships that we need are
+
+$$gH = f_1(Q, D, n, \rho, \mu, \epsilon)$$
+
+$$P = f_2(Q, D, n, \rho, \mu, \epsilon)$$
+
+Pi-theorem allows the following coefficients to be derived:
+
+\begin{align}
+\Pi_1 &= C_H = \frac{gH}{n^2D^2} &\text{Head coefficient}
+\Pi_2 &= C_P = \frac{P}{\rho n^3D^5}&\text{Power coefficient}
+\Pi_3 &= C_Q = \frac{Q}{nD^3}&\text{Capacity coefficient}
+\Pi_4 &= \text{Re} = \frac{\rho nD^2}{\mu}&\text{Reynolds Number}
+\Pi_5 &= = \frac{\epsilon}{D}&\text{Roughness Parameter}
+\end{align}
+
+Therefore it can be expressed that:
+
+\begin{align*}
+C_H &= g_1(C_Q, \text{Re}, \frac{\epsilon}{D}
+C_P &= g_2(C_Q, \text{Re}, \frac{\epsilon}{D}
+\end{align*}
+
+However for pumps it is assumed that Reynolds number and roughness parameter are constant
+for a set of similar pumps:
+
+![Pump Performance Coefficients vs $C_Q$](./images/dimensionless_pump_performance.png)
+
+Similar pumps are those which have the same design, other than the dimensions.
+Therefore it can be written that:
+
+\begin{align*}
+C_H &\approx g_3(C_Q)
+C_P &\approx g_4(C_Q)
+\end{align*}
+
+Pump efficiency is already dimensionless but can be related to the other dimensionless
+groups:
+
+$$\eta = \frac{C_HC_Q}{C_P} \approx \eta(C_Q)$$
+
+A nondimensional NPSH can also be defined:
+
+$$C_{HS} = \frac{g\cdot\text{NPSH}}{n^2D^2} \approx C_{HS}(C_Q)$$
+
+However, larger impellers for similar pump designs can lead to higher efficiencies, higher Reynolds
+number, lower friction head, and lower leakage low.
+
+# Mixed and Axial Flow Pumps
+
+Centrifugal pumps are high-head, low discharge machines, and therefore not suitable when high flow
+rates are required.
+
+For high flow rates, mixed-flow and axial-flow pumps are preferred.
+The flow passes through the impeller with an axial flow component and less centrifugal component.
+
+![Types of Pumps, showing how axial pumps differ from centrifugal pumps. ](./images/pump_types.png)
+
+## Specific Speed
+
+Specific speed is a nondimensional shaft speed which is obtained by eliminating the diameter
+between $C_Q$ and $C_H$:
+
+$$N'_S = \frac{C_Q^{\frac12}}{C_H^{\frac34}}$$
+
+There is also 'lazy' dimensional version which may be used by manufacturers:
+
+$$N'_S = \frac{N_S}{2734}$$
+
+where $N_S$ is the 'lazy' dimensional version, $N'_S$ is the dimensionless version.
+
+![Chart of Specific Speeds and Efficiency of Pump Types](./images/specific_speed.png)