Fluid Mechanics and Pressure Drop

An Introduction to Fluid Dynamics and Pressure Drop Calculations

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Conservation of Energy Recap: the complete conservation of energy equation is, 2 v out pout v in2 pin + + zin + Hpumps = + + z out + Hturbine + Hloss 2g ρg 2g ρg

Notes: ■ ■

This applies to a steady state scenario with one inlet and one exit. All terms have units of length and are called ‘heads’.

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Hloss is a term describing energy losses and must usually be supplied by an empirical formula.

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This is not Bernoulli’s equation – it has been derived from completely different principals – but Bernoulli can be reduced to this equation

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The power associated with an energy head is given by,

 gH = ρgQH Power = m

Energy Losses So the complete conservation of energy equation is,

H in + Hpumps = H out + Hturbine + Hloss 2 v out pout v in2 pin Hin = + + zin ; Hout = + + z out 2g ρg 2g ρg

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.

Major Loss The form of the major loss term is given by the Darcy-Weisbach equation

2

2f v L Hf = D g 2f LQ2 2 Hf = = kQ 2 D gA 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 = (π/4)D2

v in2 p in H in = + + z in 2g ρ

H out

v2 2g

p ρ

2 v out p out = + + z out 2g ρ

Hf

z Datum 2

2fL v Hf = D g

2

H f 2f v = L D g

H f H in − H out = L L

If pipe is of constant diameter PZ

H in

p in = + z in ρ

PZ

H out

p out = + z out ρ

v2 2g

p ρ

Hf

z Datum 2

2fL v Hf = D g

2

H f 2f v = L D g

Hf = L

PZ H PZ − H in out

L

Piezometric gradient

Minor Loss 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,

H min or = K Loss

v2 2g

Tables are available that show values for Kloss depending on the type of structure.

Pipe flow characteristics 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).

This he termed laminar flow.

Pipe flow characteristics As Reynolds increased the flow velocity the flow characteristics changed. At higher velocities the dye began to “wobble” and oscillate.

This was termed transition flow.

Pipe flow characteristics Finally as the pipe flow velocity was increased beyond a critical value the dye’s structure completely broke down. This is called turbulent flow.

In this state the velocity is fluctuating and randomly moving in small varied sized vortices.

Turbulent flow characteristics The structure of turbulence is extremely complex (some people have argued it is chaotic).

A

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: Velocity v

average

Time

Pipe flow characteristics 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 (ρ) 3. Pipe diameter (D) 4. Fluid viscosity (µ ) In fact he showed that the following non-dimensional number was very crucial; it was thereafter known as the Reynolds Number,

VDρ VD Re = alternatively Re = µ ν where

ν

is called the kinematic viscosity and is equal to

µ ρ

Laminar or turbulent flow

R e < 2000

Laminar flow

2000 < R e < 4000

Transitional flow

R e > 4000

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.

Friction factor for pipe flow 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,

dv τ =μ dy By employing this relationship we can show that

16 ff = Re

Friction factor for pipe flow Turbulent flow cannot be analyzed theoretically and so we must take recourse to experimentally derived correlation equations.

Blasius (~1913) was an early researcher on pipe friction. He showed that for smooth pipes (glass).

0.079 f = 0.25 Re 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.

Re f  1 = 2log  f  2.51  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.

 3.7D  1 = 2log  f  ks  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).

1.256 1 ε  = -4log +  f  Re f 3.7D  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.

1.256 1 ε  = -4log +  f  Re f 3.7D  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: mm

510 mm ID, 20 km long, roughness 0.4

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 Relative roughness

= 25188 = 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: mm

510 mm ID, 20 km long, roughness 0.4

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 Relative roughness

= 1259 = 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) 7000

Pressure Drop (kPa)

6000 5000 4000 3000 2000 1000 0 -1000

0

500

1000

1500

2000

2500

Flow Rate (m3/h) HYSYS Calc

Moody Diagram

3000

3500