Week 6 (Static Pressure 2 and Conservation Laws)

Question 1

what is an equation linking mass and volume?

Answer 1

𝑚 = 𝜌V

Question 2

Flow rate into a pipe

• What do we mean by flow rate?

Can it be represented by an equation?

Answer 2

Volume of fluid entering/leaving something
per second

𝑄 = 𝑑𝑉 / 𝑑t

(Q, flow rate)

Units: m^3s^-1

Question 3

Flow rate into a pipe

Flow rate depends on: (2)

Answer 3

• How fast the water is entering the pipe
• The size of the pipe entrance

Question 4

What is an equation for flow rate involving the variables flow rate depends on:

Answer 4

𝑄 = 𝑢A

𝑢 = average velocity
Q = flow rate
A = cross sectional area, e.g of pipe
Can measure by seeing how long it takes to fill up a container that we know the volume of

Question 5

Flow of mass, 𝑚̇
Definition

Answer 5

𝑚̇ is the mass of fluid entering/leaving
something per second

Question 6

Flow of mass, 𝑚̇
Equation

Answer 6

If 𝑚𝑎𝑠𝑠 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 × 𝑣𝑜𝑙𝑢𝑚𝑒
𝑚 = 𝜌.𝑉
𝑚̇ = 𝜌.𝑄 = 𝜌.𝑢.A

Question 7

Conservation of Mass

• What do we mean by conservation of
  mass?

Answer 7

We can’t create or destroy mass, so what goes in must be balanced by stuff coming out

𝑚̇(in) = 𝑚̇ (out)

𝜌.𝑢1.𝐴1 = 𝜌.𝑢2.𝐴2

Normally density (𝜌) is constant, so we
can simplify to

𝑢1𝐴1 = 𝑢2𝐴2

Volume flow rate is constant

𝑄1 = 𝑄2

Not always the case – e.g. temperature changes – but in most cases 𝜌 is constant

Question 8

Conservation of Mass

Can have more than one entrance and exit

Equation:

Answer 8

𝑚̇1 + 𝑚̇ 2 = 𝑚̇ 3

𝑢1𝐴1 + 𝑢2𝐴2 = 𝑢3𝐴3

• In a factory or machine, we know areas (𝐴1, 𝐴2, 𝐴3). Can work out 𝑢3 if know 𝑢1and 𝑢2

Question 9

What if we don’t know if flow in a pipe is in or out?

Answer 9

• Let flow in be positive
• Let flow out be negative

Set u1A1 + u2A2 + u3A3 = 0

Isolate u3, see if +ve or -ve

Question 10

Control Volume
Definition:

Answer 10

The amount flowing in, always equal to amount flowing out

Question 11

Control Volume:
(Imagine we have a T-junction pipe, with 2 inlets and 1 outlet)

How to choose a ‘good’ control volume:

Answer 11

• Make sure the entrance to the control volume are straight and perpendicular to the fluid flow
• There is no flow through the pipe walls, so put boundary of control
  volume along the walls
  (don’t need to worry about flow in these places!)

Review: Try to pick simple control volumes that are at right angles to the fluid flow

Question 12

Flows that change with time: Up till now we’ve considered cases that don’t
change with time, but not always true

What is the difference between ‘Steady flow’ and ‘Unsteady flow’

Answer 12

‘Steady flow’ – fluid is moving, but the flow looks
the same no matter when you look

Mass flow rate in = Mass flow rate out

‘Unsteady flow’ – fluid is moving, and its velocity
changes with time

Change in mass = (mass flow rate in) - (mass flow rate out)

-Consider a tap filling a sink with a drain

Question 13

When will a container (e.g sink) fill up (to do with flow rate)?

Equation:

Answer 13

If flow rate from tap greater than flow rate down
the drain, the sink will start to fill up

(Mass coming in) - (Mass coming out) = (Change in mass of fluid in sink)

𝑚̇(in) - 𝑚̇(out) = (dm / dt)_cv

(dm / dt)_cv,
means the change in mass of fluid in the control volume

Question 14

Conservation of Energy says that energy can neither be created or destroyed

Also called the 1st Law of Thermodynamics

How does this apply to fluids?

(Deriving Bernouli’s equation)

Answer 14

• Imagine a cup of coffee,
add a small drop of milk,
As we stir the coffee the drop moves around. For simplicity, assume its shape doesn’t change.

The energy of the fluid in the droplet (the milk) should be constant 𝐸1 = 𝐸2 = 𝐸3 = 𝐸4 ..

So what is the energy of the milk droplet?• We’ll ignore temperature effects • What type of energy does it have?- Kinetic (𝐸k) - Pressure (𝐸p) - Potential (height) (𝐸z ) 𝐸total = 𝐸k + 𝐸p+ 𝐸z

Say the droplet has a mass, m

Ek = 1/2.m.V^2
(V = average velocity, square root (u^2+v^2))

• Pressure Energy (𝐸p) • Fluid is getting squashed by static pressure, 𝑝 • Pressure energy is pressure × volume
Ep = p.Vol

• Potential Energy (𝐸z ) • Droplet has potential energy. • Could be from many things, but the main one is height • Equation for potential energy is
Ez = m.g.z (z is height)

Conservation of Energy
• Putting together we get
𝐸total = 𝐸k + 𝐸p + 𝐸z
𝐸tot = 1/2 𝑚𝑉^2 + 𝑝 𝑉ol + 𝑚𝑔z

this can be simplified, say m = 𝜌.𝑉ol

𝐸tot =1/2.𝜌.𝑉ol.𝑉^2 + 𝑝.𝑉ol + 𝜌.𝑉ol.𝑔.z

Divide by volume

Etot / Vol = 1/2.𝜌.𝑉^2 + 𝑝 + 𝜌.𝑔.z

Energy and volume are constant, therefore

1/2.𝜌.𝑉^2 + 𝑝 + 𝜌.𝑔.z = constant

This is Bernoulli Equation

Question 15

The Bernoulli equation, shows what?

1/2.𝜌.𝑉^2 + 𝑝 + 𝜌.𝑔.z = constant

Answer 15

• Shows how velocity, pressure and height are related • Shows what pressure (𝑝) you would need in a pipe to get a certain velocity (𝑉)