F SERIES: BASIC FLUID MECHANICS Complete Fluid Mechanics ...
Fluid Mechanics and Pressure Drop
-
Upload
anon-56206 -
Category
Documents
-
view
3.750 -
download
2
Transcript of Fluid Mechanics and Pressure Drop
An Introduction to Fluid Dynamics and Pressure Drop Calculations
Recap: the complete conservation of energy equation is,
Conservation of Energy
Notes:
1. This applies to a steady state scenario with one inlet and one exit.
2. All terms have units of length and are called ‘heads’.
3. Hloss is a term describing energy losses and must usually be supplied by an empirical formula.
4. This is not Bernoulli’s equation – it has been derived from completely different principals – but Bernoulli can be reduced to this equation
5. The power associated with an energy head is given by,
lossturbineoutout
2out
pumpsinin
2in HHz
ρg
p
2g
vHz
ρg
p
2g
v
ρgQHgHmPower
So the complete conservation of energy equation is,
Energy Losses
The energy loss term, Hloss can basically come from two sources which we call:
(a) Major losses - - losses due to pipe friction, i.e. the ‘roughness’ of the pipes. This is
usually the largest energy loss in a pipeline system.
(b) Minor losses - - energy lost at local points on the pipe system such as pipe bends, pipe connections, valves, etc.
lossturbineoutpumpsin HHH HH
outout
2out
outinin
2in
in zρg
p
2g
vHz
ρg
p
2g
vH ;
The form of the major loss term is given by the Darcy-Weisbach equation
Major Loss
v is the pipe flow velocity (=Q/A)
f is called the Darcy-Weisbach friction factor and is usually calculated from an empirical formula.
L is the length of the pipe
D is the pipe diameter
A is the pipe area of flow = (/4)D2
g
Lv
D
f2H
2
f
22
2
f kQgA
LQ
D
f2H
g
v
D
fL2H
2
f g
v
D
f2
L
H 2f
Datum
z
p
g2
v2
fH
inin
2in
in zp
g2
vH
outout
2out
out zp
g2
vH
L
HH
L
H outinf
g
v
D
fL2H
2
f g
v
D
f2
L
H 2f
Datum
z
p
g2
v2
fH
ininPZ z
pH
in
outoutPZ z
pH
out
L
HH
L
HPZPZ
f outin
If pipe is of constant diameter
Piezometric gradient
The form of the minor loss term varies according to the type of structure causing the loss (e.g. valve, pipe bend etc.) but it usually takes a form like,
Minor Loss
Tables are available that show values for Kloss depending on the type of structure.
g2
vKH
2
Lossormin
Osborne Reynolds (1842-1912): observed that the flow characteristics of fluids in pipes varied with the flow velocity.
At low velocities a dye injected at the pipe center flowed in a thin straight line. Reynolds observed that the water flowed in thin laminae (sheets).
Pipe flow characteristics
This he termed laminar flow.
As Reynolds increased the flow velocity the flow characteristics changed.
At higher velocities the dye began to “wobble” and oscillate.
Pipe flow characteristics
This was termed transition flow.
Finally as the pipe flow velocity was increased beyond a critical value the dye’s structure completely broke down.
This is called turbulent flow.
Pipe flow characteristics
In this state the velocity is fluctuating and randomly moving in small varied sized vortices.
The structure of turbulence is extremely complex (some people have argued it is chaotic).
Turbulent flow characteristics
However this fluctuating, erratic velocity pattern may be thought of as being superimposed upon a mean velocity field. So if we plotted the instantaneous velocity at A versus time:
A
Time
Velocity
vaverage
Reynolds demonstrated that the type of flow that occurred depended on the interrelationship between four flow parameters:
1. Average flow velocity (V)
2. Fluid density ()
3. Pipe diameter (D)
4. Fluid viscosity ( )
In fact he showed that the following non-dimensional number was very crucial; it was thereafter known as the Reynolds Number,
Pipe flow characteristics
VD
RelyalternativVD
R ee
where is called the kinematic viscosity and is equal to
Laminar or turbulent flow
2000R e Laminar flow
4000R2000 e
4000R e
Transitional flow
Turbulent flow
Notes: (1) laminar flow rarely occurs in the oil industry, except by design. Examples include pipelines operating below design capacity, in small scale lab experiments and very close to solid boundaries; (2) these numbers are guidelines only; (3) usually we would choose a design to be fully turbulent or fully laminar since then we can analyze it.
Laminar flow is amenable to mathematical analysis by assuming that the instantaneous shear stress within the fluid can be related to the velocity gradient (or mean strain rate) by the Newtonian relationship,
Friction factor for pipe flow
By employing this relationship we can show that
dy
dvμ
e
fR
16f
Turbulent flow cannot be analyzed theoretically and so we must take recourse to experimentally derived correlation equations.
Friction factor for pipe flow
Blasius (~1913) was an early researcher on pipe friction. He showed that for smooth pipes (glass).
0.25eR
0.079f
We will define what we mean by smooth shortly.
Aside: For pipe calculations f is usually O(10-2) i.e. 0.01.
Friction factor for pipe flow
Nikuradse (~1930) took smooth pipes (glass) and artificially roughed them by sticking small sand grains of size (ks) onto the pipe wall. He performed a series of tests with pipes roughened by the addition of different sized particles.
He found that if ks was “very small” then the following friction factor equation worked. He called these pipes smooth pipes.
2.51
fR2log
f
1 e
This is an implicit equation and must be solved by iteration (trial an error).
Friction factor for pipe flow
Nikuradse found that if ks was “very large” then the following friction factor equation worked. He called these rough pipes.
sk
3.7D2log
f
1
Note that there is no dependence on Reynolds number and this is an explicit equation for f.
Rough or smooth pipes?
In fact when flow occurs in a pipe, even it it is turbulent flow, there is a very small region close to the pipe wall where turbulent fluctuations are damped out and laminar flow prevails.
This is called the laminar sub-layer. If the pipe roughness elements are contained within this layer then their effect is not felt by the gross flow field and hence the pipe flow “thinks” it is a smooth pipe. If the roughness elements protrude through the sub-layer into the flow field then they affect the gross flow as a roughness.
Smooth Rough
Colebrook and White (1937)
These researchers conducted experiments on commercially available steel pipes. They found that the following equation described the friction factor and covered all types of pipes (rough smooth and intermediate).
3.7DfR
1.256-4log
f
1
e
Notes: (1) This is really just a generalization of Nikuradse’s results into a equation for all pipe roughness; (2) for large Re the first term in brackets may tend to zero; (3) for small ks the second term vanishes; (4) in general this equation is implicit.
Problems and solutions for the implicit equation
The Colebrook-White relationship is used extensively in pipe friction calculations and design.
3.7DfR
1.256-4log
f
1
e
But the fact that it is an implicit equation has led researchers to suggest easier ways to solve it (at least in pre-computer times).
Moody Diagram
Moody plotted f versus Re for values of ks/D to produce the Moody diagram. We use Moody’s diagram as an alterative to solving the Colebrook and White equation.
Calculating Friction Pressure Loss
Procedure
1. Calculate Renolds Number
2. Determine Flow Regime
3. Determine friction factor
4. Calculate pressure drop
Calculating Friction Pressure Loss
Example 1
Given Data Oil properties: Density 847 kg/m3, Viscosity 34.3 cP, flow
velecity 2 m/s
Pipeline properties: 510 mm ID, 20 km long, roughness 0.4 mm
Calculating Friction Pressure Loss
Example 1
1. Calculate Reynolds Number Reynolds number = diameter * velocity * density / viscosity
= 0.51 * 2 * 847 / 0.0343
= 25188
Calculating Friction Pressure Loss
Example 1
2. Determine Flow Regime Reynolds number = 25188
Relative roughness = 0.4 / 510= 0.0008
Therefore, from Moody diagram flow regime is transitional
Calculating Friction Pressure Loss
Example 1
3. Determine Friction Factor From Moody diagram, friction factor is:
Moody Friction Factor (fm) = 0.0270
Calculating Friction Pressure Loss
Example 1
4. Calculate Pressure Drop Pressure Drop (kPa) = 0.5 * density * fm * length * velocity2 /
diamter
Pressure Drop (kPa) = 0.5 * 847 * 0.0270 * 20000 * 22 / 510
Pressure Drop (kPa) = 1794
Calculating Friction Pressure Loss
Example 2
Given Data Oil properties: Density 847 kg/m3, Viscosity 34.3 cP, flow
velecity 0.1 m/s
Pipeline properties: 510 mm ID, 20 km long, roughness 0.4 mm
Calculating Friction Pressure Loss
Example 1
1. Calculate Reynolds Number Reynolds number = diameter * velocity * density / viscosity
= 0.51 * 0.1 * 847 / 0.0343
= 1259
Calculating Friction Pressure Loss
Example 1
2. Determine Flow Regime Reynolds number = 1259
Relative roughness = 0.4 / 510
= 0.0008
Therefore, from Moody diagram flow regime is laminar
Calculating Friction Pressure Loss
Example 1
3. Determine Friction Factor From Moody diagram, friction factor is:
fm = 64 / Re
= 64 / 1259
= 0. 0508
Calculating Friction Pressure Loss
Example 1
4. Calculate Pressure Drop Pressure Drop (kPa) = 32000 * viscosity * length * velocity /
diamter2
Pressure Drop (kPa) = 32000 * 0.0343 * 20000 * 0.1 / 5102
Pressure Drop (kPa) = 8.43
Calculating Friction Pressure Loss
HYSYS Calculation
Pressure Drop (kPa) vs Flow Rate (m3/h)
-1000
0
1000
2000
3000
4000
5000
6000
7000
0 500 1000 1500 2000 2500 3000 3500
Flow Rate (m3/h)
Pre
ss
ure
Dro
p (
kP
a)
HYSYS Calc Moody Diagram