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