add notes on extended bernoulli eqn

uni/mmme/1048_thermodynamics_and_fluid_mechanics/fluid_dynamics.md

@@ -147,7 +147,7 @@ Examples of this are line dye or a smoke stream which is produced from a continu
 
 ### Viscosity
 
-A fluid offers resisistance to motion due to its viscosity or internal friction.
+A fluid offers resistance to motion due to its viscosity or internal friction.
 The greater the resistance to flow, the greater the viscosity.
 
 Higher viscosity also reduces the rate of shear deformation between layers for a given shear stress.
@@ -184,10 +184,10 @@ to the velocity gradient when straight and parallel flow is involved:
 
 $$\tau = \mu\frac{\mathrm{d}v}{\mathrm{d}y}$$
 
-Where $\mu$ is the constant of proportinality and known as the dynamic viscosity, or simply the
+Where $\mu$ is the constant of proportionality and known as the dynamic viscosity, or simply the
 viscosity of the fluid.
 
-This is Newton's Law of Viscosity and fluids that ovey it are known as Newtonian fluids.
+This is Newton's Law of Viscosity and fluids that obey it are known as Newtonian fluids.
 
 ### Viscosity and Lubrication
 
@@ -393,7 +393,7 @@ Head form:
 \frac{p_s}{\rho g} + z &= \text{piezometric head}
 \end{align*}
 
-# Steady Flow Energy Equation (SFEE)
+# Steady Flow Energy Equation (SFEE) and the Extended Bernoulli Equation (EBE)
 
 SFEE is a more general equation that can be applied to **any fluid** and also is also takes
 **heat energy** into account.
@@ -486,7 +486,7 @@ Dividing everything by $\delta m$ and with a bit of rearranging we get:
 
 $$q + w_s = e_2-e_1 + \frac{p_2}{\rho_2} - \frac{p_1}{\rho_1}$$
 
-#### Substiute Back for $e$
+#### Substitute Back for $e$
 
 $$e = u + \frac{v^2}{2} + gz$$
 
@@ -534,7 +534,7 @@ $$\dot W = \dot m (h_2-h_1) = \dot m c_p(T_2-T_1)$$
 
 #### Mixing Devices
 
-e.g. hot and cold water in a shower
+e.g. Hot and cold water in a shower
 
 In these processes, work and heat transfers are not important and you can often
 neglect potential and kinetic energy terms, giving us the same equation as for the throttle valve
@@ -546,3 +546,109 @@ which you may want to write more usefully as:
 
 $$\sum \dot m h_{out} = \sum \dot m h_{in}$$
 
+## SFEE for Incompressible Fluids and Extended Bernoulli Equation
+
+$$\frac{w_s}{g} = H_{T2} - H_{T1} + \left[ \frac{(u_2-u_1)-1}{g}\ \right]$$
+
+or
+
+$$w_s = g(H_{T2}-H_{T1}+H_f$$
+
+If we assume shaft work, $w_s$, is 0, then we can get this equation:
+
+$$H_{T1}-H_{T2} = H_f$$
+
+This is very similar to the Bernoulli equation.
+The difference is that it considers friction so it can be applied to real fluids, not just ideal
+ones.
+It is called the *Extended Bernoulli Equation*.
+
+The assumptions remain:
+
+- Steady flow
+- No shaft work
+- Incompressible
+
+### $H_f$ for Straight Pipes
+
+$$H_f = \frac{4fL}{D} \frac{v^2}{2g}$$
+
+$$\Delta p = \rho g H_f \text{ (pressure form)}$$
+
+This equation applies to long, round and straight pipes.
+It applies to both laminar and turbulent flow.
+
+However be aware that in North America the equation is:
+
+$$H_f = f \frac{L}{D} \frac{v^2}{2g}$$
+
+Their $f$ (the Darcy Friction Factor) is four times our $f$ (Fanning Friction Factor).
+In mainland Europe, they use $\lambda = 4f_{Fanning}$, which is probably the least confusing version to use.
+
+### Finding $f$
+
+#### $f$ for Laminar Flow
+
+$$f = \frac{16}{\Rey}$$
+
+#### $f$ for Turbulent Flow
+
+For turbulent flow, the value defends on relative pipe roughness ($k' = \frac k d$) and Reynolds
+number.
+
+Note when calculating $k'$ that **both $k$ and $d$ are measured in mm** for some reason.
+
+A *Moody Chart* is used to find $f$:
+
+![A Moody Chart](./images/vimscrot-2022-03-08T09:28:38,519555620+00:00.png)
+
+### Hydraulic Diameter
+
+$$D_h = \frac{4 \times \text{duct area}}{\text{perimeter}}$$
+
+### Loss Factor $K$
+
+There are many parts of the pipe where losses can occur.
+
+It is convenient to represent these losses in terms of loss factor, $K$, times the velocity head:
+
+$$H_f = K \frac{v^2}{g}$$
+
+Most manufacturers include loss factors in their data sheets.
+
+#### Loss Factor of Entry
+
+![](./images/vimscrot-2022-03-08T10:01:31,557158164+00:00.png)
+
+#### Loss Factor of Expansion
+
+$$K = \left( \frac{A_2}{A_1} - 1\right)^2$$
+
+This also tells us the loss factor on exit is basically 1.
+
+For conical expansions, $K ~ 0.08$ (15 degrees cone angle), 
+$K ~ 0.25$ (30 degrees).
+For cones you use the inlet velocity.
+
+#### Loss Factor of Contraction
+
+$\frac{d_2}{d_1}$ | K
+----------------- | ----
+0                 | 0.5
+0.2               | 0.45
+0.4               | 0.38
+0.6               | 0.28
+0.8               | 0.14
+1.0               | 0
+
+#### Loss Factor of Pipe Bends
+
+On a sharp bend, $K ~ 0.9$.
+
+On a bend with a radius, $K ~ 0.16-0.35$.
+
+#### Loss Factor of Nozzle
+
+$$K ~ 0.05$$
+
+But you use the outlet velocity, increasing losses.