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37
uni/mmme/1048_thermodynamics_and_fluid_mechanics/fluid_dynamics.md
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@ -22,9 +22,9 @@ These are ideas you need to know about to know what's going on, I guess?
## Control Volumes ## Control Volumes
A control volume is a volume with an imaginary boundry to make it easier to analyze the flow of a A control volume is a volume with an imaginary boundary to make it easier to analyse the flow of a
fluid. fluid.
The boundry is drawn where the properties and conditions of the fluid is known, or where an The boundary is drawn where the properties and conditions of the fluid is known, or where an
approximation can be made. approximation can be made.
Properties which may be know include: Properties which may be know include:
@ -33,7 +33,7 @@ Properties which may be know include:
- Temperature - Temperature
- Viscosity - Viscosity
The region in the control volume is analyed in terms of enery and mass flows entering and leaving The region in the control volume is analysed in terms of energy and mass flows entering and leaving
the control volumes. the control volumes.
You don't have to understand what's going on inside the control volume. You don't have to understand what's going on inside the control volume.
@ -50,7 +50,7 @@ momentum or the air passing through the engine.
![](./images/vimscrot-2021-11-03T21:51:51,497459693+00:00.png) ![](./images/vimscrot-2021-11-03T21:51:51,497459693+00:00.png)
The control volume is drawn far enough in front of the engine that the air velocity entering can The control volume is drawn far enough in front of the engine that the air velocity entering can
be assumed to be at atmospheric pressurce and its velocity negligible. be assumed to be at atmospheric pressure and its velocity negligible.
At the exit of the engine the boundary is drawn close where the velocity is known and the air At the exit of the engine the boundary is drawn close where the velocity is known and the air
pressure atmospheric. pressure atmospheric.
@ -97,7 +97,7 @@ with respect to time, like the opposite of steady flow.
## One Dimensional Flow ## One Dimensional Flow
In one dimensional (1D) flow it is assumed that all properties are uniform over any plane In one dimensional (1D) flow it is assumed that all properties are uniform over any plane
perpedenciular to the direction of flow (e.g. all points along the cross section of a pipe have perpendicular to the direction of flow (e.g. all points along the cross section of a pipe have
identical properties). identical properties).
This means properties can only flow in one direction---usually the direction of flow. This means properties can only flow in one direction---usually the direction of flow.
@ -132,7 +132,7 @@ Same reasoning with if they are not parallel.
### Pathlines ### Pathlines
A pathline shows the route taken by a single particle during a given time interval. A pathline shows the route taken by a single particle during a given time interval.
It is equivalent to a high exposure photograph which traces the moevement of the particle marked. It is equivalent to a high exposure photograph which traces the movement of the particle marked.
You could track pathlines with a drop of injected dye or inserting a buoyant solid particle which You could track pathlines with a drop of injected dye or inserting a buoyant solid particle which
has the same density as the solid. has the same density as the solid.
@ -222,15 +222,15 @@ Between fully laminar and fully turbulent flows is a transition region.
### Development of the Reynolds Number ### Development of the Reynolds Number
In laminar flow the most influentialfactor is the magnitude of the viscous forces: In laminar flow the most influential factor is the magnitude of the viscous forces:
$$viscous\, forces \propto \mu\frac v l l^2 = \mu vl$$ $$viscous\, forces \propto \mu\frac v l l^2 = \mu vl$$
where $v$ is a characteristic velocit and $l$ is a characteristic length. where $v$ is a characteristic velocity and $l$ is a characteristic length.
In turbulent flow viscous effects are not significant but inertia effects (mixing, momentum In turbulent flow viscous effects are not significant but inertia effects (mixing, momentum
exchange, acceleration of fluid mass) are. exchange, acceleration of fluid mass) are.
Interial forces can be represented by $F = ma$ Inertial forces can be represented by $F = ma$
\begin{align*} \begin{align*}
m &\propto \rho l^3 \\ m &\propto \rho l^3 \\
@ -240,7 +240,7 @@ a &= \frac{dv}{dt} \\
&\therefore \text{Interial forces} \propto \rho l^2\frac{v^2} l = \rho l^2v^2 &\therefore \text{Interial forces} \propto \rho l^2\frac{v^2} l = \rho l^2v^2
\end{align*} \end{align*}
The ratio of internalforces to viscous forces is called the Reynolds number and is abbreviated to The ratio of internal forces to viscous forces is called the Reynolds number and is abbreviated to
Re: Re:
$$\Rey = \frac{\text{interial forces}}{\text{viscous forces}} = \frac {\rho l^2v^2}{\mu vl} = \frac {\rho vl} \mu$$ $$\Rey = \frac{\text{interial forces}}{\text{viscous forces}} = \frac {\rho l^2v^2}{\mu vl} = \frac {\rho vl} \mu$$
@ -248,12 +248,12 @@ $$\Rey = \frac{\text{interial forces}}{\text{viscous forces}} = \frac {\rho l^2v
where $\rho$ and $\mu$ are fluid properties and $v$ and $l$ are characteristic velocity and length. where $\rho$ and $\mu$ are fluid properties and $v$ and $l$ are characteristic velocity and length.
- During laminar flow, $\Rey$ is small as viscous forces dominate. - During laminar flow, $\Rey$ is small as viscous forces dominate.
- During turbulent flow, $\Rey$ is large as intertial forces dominate. - During turbulent flow, $\Rey$ is large as inertial forces dominate.
\textRey is a non dimensional group. \textRey is a non dimensional group.
It has no units because the units cancel out. It has no units because the units cancel out.
Non dimensional groups are very important in fluid mechancics and need to be considered when scaling Non dimensional groups are very important in fluid mechanics and need to be considered when scaling
experiments. experiments.
If \textRey is the same in two different pipes, the flow will be the same regardless of actual If \textRey is the same in two different pipes, the flow will be the same regardless of actual
@ -285,7 +285,7 @@ $$\frac 1 \rho \frac{\delta p}{\delta s} + g\frac{\delta z}{\delta s} + v\frac{\
## Assumptions / Conditions ## Assumptions / Conditions
The Euler euqation applies where the following can be assumed: The Euler equation applies where the following can be assumed:
- Steady flow - Steady flow
- The fluid is inviscid - The fluid is inviscid
@ -350,7 +350,7 @@ $$\frac p {\rho g} + z + \frac{v^2}{2g} = H_T$$
where $H_T$ is constant and: where $H_T$ is constant and:
- $\frac{p}{\rho g}$ --- static/pressure haed - $\frac{p}{\rho g}$ --- static/pressure head
- $z$ --- elevation head - $z$ --- elevation head
- $\frac{v_2}{2g}$ --- dynamic/velocity head - $\frac{v_2}{2g}$ --- dynamic/velocity head
- $H_T$ --- total head - $H_T$ --- total head
@ -399,9 +399,9 @@ SFEE is a more general equation that can be applied to **any fluid** and also is
**heat energy** into account. **heat energy** into account.
This is useful in applications such as a fan heater, jet engines, ICEs, and steam turbines. This is useful in applications such as a fan heater, jet engines, ICEs, and steam turbines.
The equation deals with 3 types of energy tranfer: The equation deals with 3 types of energy transfer:
1. Thermal energy transfer (e.g. heat tranfer from central heating to a room) 1. Thermal energy transfer (e.g. heat transfer from central heating to a room)
2. Work energy transfer (e.g. shaft from car engine that turns wheels) 2. Work energy transfer (e.g. shaft from car engine that turns wheels)
3. Energy transfer in fluid flows (e.g. heat energy in a flow, potential energy in a flow, kinetic 3. Energy transfer in fluid flows (e.g. heat energy in a flow, potential energy in a flow, kinetic
energy in a flow) energy in a flow)
@ -450,7 +450,7 @@ $$\delta E = \delta E_2 - \delta E_1 = \delta m(e_2 - e_1)$$
#### The Work Term #### The Work Term
The work term, $\delta W$, is mae up of shaft work **and the work necessary to deform the system** The work term, $\delta W$, is made up of shaft work **and the work necessary to deform the system**
(by adding $\delta m_1$ at the inlet and removing $\delta m_2$ at the outlet): (by adding $\delta m_1$ at the inlet and removing $\delta m_2$ at the outlet):
$$\delta W = \delta W_s + \text{net flow work}$$ $$\delta W = \delta W_s + \text{net flow work}$$
@ -583,7 +583,8 @@ However be aware that in North America the equation is:
$$H_f = f \frac{L}{D} \frac{v^2}{2g}$$ $$H_f = f \frac{L}{D} \frac{v^2}{2g}$$
Their $f$ (the Darcy Friction Factor) is four times our $f$ (Fanning Friction Factor). 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. In mainland Europe, they use $\lambda = 4f_{Fanning}$, which is probably the least confusing version
to use.
### Finding $f$ ### Finding $f$

36
uni/mmme/1048_thermodynamics_and_fluid_mechanics/fluid_mechanics.md
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@ -17,22 +17,20 @@ uuid: 43e8eefa-567f-438b-b93d-63ae08e61d8f
## What is a Fluid? ## What is a Fluid?
- A fluid may be liquid, vapor, or gas - A fluid may be liquid, vapour, or gas
- No permanent shape - No permanent shape
- Consists of atoms in random motion and continual collision - Consists of atoms in random motion and continual collision
- Easy to deform - Easy to deform
- Liquids have fixed volume, gasses fill up container - Liquids have fixed volume, gasses fill up container
- **A fluid is a substance for wich a shear stress tends to produce unlimited, continuous - **A fluid is a substance for which a shear stress tends to produce unlimited, continuous
deformation** deformation**
## Shear Forces ## Shear Forces
- For a solid, application of shear stress causes a deformation which, if not too great (elastic), - For a solid, application of shear stress causes a deformation which, if not too great (elastic),
is not permanent and solid regains original positon is not permanent and solid regains original position
- For a fluid, continuious deformation takes place as the molecules slide over each other until the - For a fluid, continuous deformation takes place as the molecules slide over each other until the
force is removed force is removed
- **A fluid is a substance for wich a shear stress tends to produce unlimited, continuous
deformation**
## Density ## Density
@ -51,7 +49,7 @@ uuid: 43e8eefa-567f-438b-b93d-63ae08e61d8f
- Matter is not continuous on molecular scale - Matter is not continuous on molecular scale
- For fluids in constant motion, we take a time average - For fluids in constant motion, we take a time average
- For most practical purposes, matter is considered to be homogenous and time averaged - For most practical purposes, matter is considered to be homogeneous and time averaged
## Pressure ## Pressure
@ -77,7 +75,7 @@ uuid: 43e8eefa-567f-438b-b93d-63ae08e61d8f
- A fluid at rest has constant pressure horizontally - A fluid at rest has constant pressure horizontally
- That's why liquid surfaces are flat - That's why liquid surfaces are flat
- But fluids at rest do have a vertical gradient, where lower parts have higher presure - But fluids at rest do have a vertical gradient, where lower parts have higher pressure
### How Does Pressure Vary with Depth? ### How Does Pressure Vary with Depth?
@ -116,11 +114,11 @@ The -ve sign indicates that as $z$, height, increases, $p$, pressure, decreases.
### Absolute and Gauge Pressure ### Absolute and Gauge Pressure
- Absolute Pressure is measured relative to zero (a vacuum) - Absolute Pressure is measured relative to zero (a vacuum)
- Guage pressure = absolute pressure - atmospheric pressure - Gauge pressure = absolute pressure - atmospheric pressure
- Often used in industry - Often used in industry
- If abs. pressure = 3 bar and atmospheric pressure is 1 bar, then gauge pressure = 2 bar - If absolute pressure = 3 bar and atmospheric pressure is 1 bar, then gauge pressure = 2 bar
- Atmospheric pressure changes with altitude - Atmospheric pressure changes with altitude
## Compressibility ## Compressibility
@ -132,7 +130,7 @@ The -ve sign indicates that as $z$, height, increases, $p$, pressure, decreases.
## Surface Tension ## Surface Tension
- In a liquid, molecules are held together by molecular attraction - In a liquid, molecules are held together by molecular attraction
- At a boundry between two fluids this creates "surface tension" - At a boundary between two fluids this creates "surface tension"
- Surface tension usually has the symbol $$\gamma$$ - Surface tension usually has the symbol $$\gamma$$
## Ideal Gas ## Ideal Gas
@ -155,7 +153,7 @@ The -ve sign indicates that as $z$, height, increases, $p$, pressure, decreases.
- Pressure always in Pa - Pressure always in Pa
- Temperature always in K - Temperature always in K
## Units and Dimentional Analysis ## Units and Dimensional Analysis
- It is usually better to use SI units - It is usually better to use SI units
- If in doubt, DA can be useful to check that your answer makes sense - If in doubt, DA can be useful to check that your answer makes sense
@ -289,7 +287,7 @@ p_1 - p_2 &= \rho_wg(z_C-z_B-z_C+z_A) + \rho_mg\Delta z \\
the upper surface (figure 1.4). The tank and riser are filled with the upper surface (figure 1.4). The tank and riser are filled with
water such that the water level in the riser pipe is 3.5 m above the water such that the water level in the riser pipe is 3.5 m above the
Calulate: Calculate:
i. The gauge pressure at the base of the tank. i. The gauge pressure at the base of the tank.
@ -299,7 +297,7 @@ p_1 - p_2 &= \rho_wg(z_C-z_B-z_C+z_A) + \rho_mg\Delta z \\
> $$\rho gh = 1000\times9.81\times3.5 = 34 \text{ kPa}$$ > $$\rho gh = 1000\times9.81\times3.5 = 34 \text{ kPa}$$
iii. The force exercted on the base of the tank due to gauge water pressure. iii. The force exerted on the base of the tank due to gauge water pressure.
> $$F = p\times A = 49\times10^3\times6\times3 = 8.8\times10^5 \text{ N}$$ > $$F = p\times A = 49\times10^3\times6\times3 = 8.8\times10^5 \text{ N}$$
@ -345,7 +343,7 @@ p_1 - p_2 &= \rho_wg(z_C-z_B-z_C+z_A) + \rho_mg\Delta z \\
## Submerged Surfaces ## Submerged Surfaces
### Prepatory Maths ### Preparatory Maths
#### Integration as Summation #### Integration as Summation
@ -370,7 +368,7 @@ Take the following lamina:
1. Split the lamina into elements parallel to the chosen axis 1. Split the lamina into elements parallel to the chosen axis
2. Each element has area $\delta A = w\delta y$ 2. Each element has area $\delta A = w\delta y$
3. The moment of area ($\delta M$) of the element is $\delta Ay$ 3. The moment of area ($\delta M$) of the element is $\delta Ay$
4. The sum of moments of all the elements is equal to the moment $M$ obtained by assuing all the 4. The sum of moments of all the elements is equal to the moment $M$ obtained by assuming all the
area is located at the centroid or: area is located at the centroid or:
$$Ay_c = \int_{area} \! y\,\mathrm{d}A$$ $$Ay_c = \int_{area} \! y\,\mathrm{d}A$$
@ -426,7 +424,7 @@ Determine the location of the centroid of a rectangular lamina.
</details> </details>
### Horizontal Submereged Surfaces ### Horizontal Submerged Surfaces
![](./images/vimscrot-2021-10-20T10:33:16,783724117+01:00.png) ![](./images/vimscrot-2021-10-20T10:33:16,783724117+01:00.png)
@ -492,12 +490,12 @@ Where $\rho$ is the density of the fluid, and $V$ is the volume of displaced flu
### Immersed Bodies ### Immersed Bodies
As pressure increases with depth, the fluid exerts a resultant upward force on a body. As pressure increases with depth, the fluid exerts a resultant upward force on a body.
There is no horizontal component of the buoyancy force because the vertiscal projection of the body There is no horizontal component of the buoyancy force because the vertical projection of the body
is the same in both directions. is the same in both directions.
### Rise, Sink, or Float? ### Rise, Sink, or Float?
- $F_B = W$ \rightarrow equilirbrium (floating) - $F_B = W$ \rightarrow equilibrium (floating)
- $F_B > W$ \rightarrow body rises - $F_B > W$ \rightarrow body rises
- $F_B < W$ \rightarrow body sinks - $F_B < W$ \rightarrow body sinks

12
uni/mmme/1048_thermodynamics_and_fluid_mechanics/thermodynamics.md
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@ -23,12 +23,12 @@ Thermodynamics deals with the transfer of heat energy and temperature.
A region of space, marked off by its boundary. A region of space, marked off by its boundary.
It contains some matter and the matter inside is what we are investigating. It contains some matter and the matter inside is what we are investigating.
There are two types of sysems: There are two types of systems:
- Closed systems - Closed systems
- Contain a fixed quantity of matter - Contain a fixed quantity of matter
- Work and heat cross bounaries - Work and heat cross boundaries
- Impermeable boundaries, some may be moved - Impermeable boundaries, some may be moved
- Non-flow processes (no transfer of mass) - Non-flow processes (no transfer of mass)
@ -125,7 +125,7 @@ c_p &= \frac{\gamma}{\gamma -1} R
</details> </details>
### The Specfic and Molar Gas Constant ### The Specific and Molar Gas Constant
The molar gas constant is represented by $\tilde R = 8.31 \text{JK}^{-1}\text{mol}^{-1}$. The molar gas constant is represented by $\tilde R = 8.31 \text{JK}^{-1}\text{mol}^{-1}$.
@ -141,10 +141,10 @@ An example of a process is expansion (volume increasing).
A *cycle* is a process or series of processes in which the end state is identical to the beginning. A *cycle* is a process or series of processes in which the end state is identical to the beginning.
And example of this could be expansion followed by a compression. And example of this could be expansion followed by a compression.
### Reversible and Irreversible Proccesses ### Reversible and Irreversible Processes
During reversible processes, the system undergoes a continuous succession of equilibrium states. During reversible processes, the system undergoes a continuous succession of equilibrium states.
Changes in the system can be defined and reversed to restore the intial conditions Changes in the system can be defined and reversed to restore the initial conditions
All real processes are irreversible but some can be assumed to be reversible, such as controlled All real processes are irreversible but some can be assumed to be reversible, such as controlled
expansion. expansion.
@ -225,7 +225,7 @@ These properties are the *properties of state* and they always have the same val
state. state.
A *property* can be defined as any quantity that depends on the *state* of the system and is A *property* can be defined as any quantity that depends on the *state* of the system and is
independant of the path by which the system arrived at the given state. independent of the path by which the system arrived at the given state.
Properties determining the state of a thermodynamic system are referred to as *thermodynamic Properties determining the state of a thermodynamic system are referred to as *thermodynamic
properties* of the *state* of the system. properties* of the *state* of the system.