Chapter 8 Turbulent Flow in Circular Pipes

7/30/2019 Chapter 8 Turbulent Flow in Circular Pipes

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CHE 493Fluid Mechanics

Chapter 8:

Turbulent Flow in Circular

Pipes


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Learning Outcome

1. Derive the formula for friction factor.

2.Able to use friction factor chart.

3. Calculate head and energy loss due to frictionin pipes.


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Turbulent Flow

Most flow encountered in engineering practice areturbulent how turbulences affects wall shear stress

Turbulent flow characterized by random and rapidfluctuations of swirling regions of fluid (eddies)

throughout the flow

The swirling eddies transport mass, momentum andenergy to other regions of flow more rapidly frommolecular diffusion higher values of friction, heat and

mass transfer The chaotic fluctuations of fluid particles in turbulent flow

play a dominant role in pressure drop


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Fully Developed Pipe Flow

Wall-shear stress

Recall, for simple shear flows u=u(y), we had

= du/dy In fully developed pipe flow, it turns out that

= du/drLaminar Turbulent

w w

w,turb

> w,lam

w = shear stress at the wall,

acting on the fluid


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VelocityProfile

Turbulent flow along a wall consist of4 regions characterized by the

distance from the wall1. Viscous/laminar/linear/wall

sublayer very linear, flow isstreamlined

2. Buffer layer flow is stilldominated by viscous effect

3. Overlap/transition layer

turbulent effects more significantbut not dominant

4. Outer/turbulent layer turbulenteffects dominate over viscouseffect


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Laminarand Turbulent

Flows

For non-round pipes, define thehydraulic diameter, Dh

Dh = 4A/PA= cross-section area

P = wetted perimeter


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Darcy Weisbach Equation

Pressure drop

Where:f = friction factorL = pipe length, md= pipe diameter, m

u = average flow velocity, m/sg = acceleration of gravity, m/s2

2

u

d

LfP

2


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Darcy Weisbach Equation

Head lossThe loss of pressure in a flow system measured using a

length parameter (i.e., inches, meter etc.)

Pressure loss in terms of head loss, hL

2g

u

d

Lfh

2

L


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The Moody Chart

The friction factor in fully developed turbulent pipe flowdepends on the Reynolds number and the relativeroughness

Value of is determined experimentally by usingartificially roughened surfaces (by gluing sand grains in theinner of pipes) by Prandtls student.

Friction factor was calculated from the measurements offlow rate and pressure drop.

Colebrook equation:


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The Moody Chart

Observations from Moody chart:

For laminar flow, f with Re; independent of

surface roughness.

f minimum for smooth pipe & withroughness.

Transition region shaded area.

>> Re the line is nearly horizontal thusmaking it independent of Re


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Types of Fluid Flow Problems

Pressure drop (or head loss) pipe length &diameter given for specified flow rate (orvelocity)

Flow rate - pipe length & diameter given forspecified pressure drop (or head loss)

Pipe diameter - pipe length & flow rate givenfor specified pressure drop (or head loss)


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Minor Losses

Fluid in piping system passes through various

fittings, valves, bends, elbows, tees, inlets, exits,enlargements and contractions.

These components interrupt the smooth flow ofthe fluid inducing flow separation and mixingwhich cause additional losses.

In a long pipe system, these losses are minor

(minor losses) compared to the total head loss inthe pipes (major losses).


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Minor Losses

1. Loss coefficient, KLwhere

2. Equivalent length, Lequiv

where


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Minor Losses

Total head loss (general)

Total head loss (D = constant, V = constant)


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Example 1

Water at 15C ( = 999 kg/m3 and = 1.138 x 10-3

Pa.s) is flowing steadily in a 5 cm diameterhorizontal pipe made of stainless steel at a rate

of 0.34m3/min. Determine the pressure dropand the head loss for flow over a 61 m longsection of the pipe.

Ans: P = 88.9 kPa; hL = 9.07 m


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Calculate averagevelocity, Re to

determine flow

regime

Determine relativeroughness

Determine thefriction factor from

Moody chart

Calculate thepressure drop

Calculate the headloss

2

u

d

LfP

2

f= 0.01734


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Losses Due to Sudden

Enlargement

When the pipe diameter increases abruptly, the fluid

experience shock.

This causes the formation of eddies and some

energy is lost due to increased local turbulence. This head loss can be evaluated using the continuity,

momentum and energy principles.


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Applying the continuity equation between sections 1 & 2

Neglecting the shear forces on the walls by momentumequation

External force = Rate of change of momentum

(A)

2 2


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Applying Bernoullis equation between section 1 & 2 andtaking horizontal axis as the datum;

Where hL is the energy loss due to sudden expansion.Rearranging the above equation;

Substituting for from (A);

gvv

gvvvhL

2

2

2

2

121

2

2


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Using the continuity equation;

Where and v1

= velocity in the smaller pipe

g

AAvv

hL2

2

2

11

1


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Exit Loss

When a pipe discharges into a large reservoir, some energy isdissipated by mixing and turbulence

In this case A2 >> A1, therefore

Even if the discharge is free, all the kinetic energy is lost(converted to thermal energy causing a small temperaturerise in the fluid in the reservoir)


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Ax is expressed as

Using continuity equation:

It is assumed that all of the energy loss occurs when the jet

expands from Ax to A2

Coefficient of contraction

The expression is valid ifthe ratio of A1/A2 isbetween 0.1 and 1


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Entrance Loss

A poorly designed inlet to a pipe can cause an appreciablehead loss.

Various common inlet conditions:

A slight off will reduce the loss drastically

For a sharp entrance, provided the pipe does not protrudeinto the reservoir:


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Losses in Pipe Fittings

A pumping system will have connections whichchange the size and direction of the pipe

Pipe fittings such as valves and elbows

constrict/change the flow direction cause

additional losses

These losses are expressed as equivalent to the

friction loss in a specific length of straight pipe of

the same diameter The equivalent lengths expressed as a ratio to the

pipe diameter for typical fittings as shown on the

next table


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This approximation is valid for pipe diameters 10mm to 250mm


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Alternatively, the losses due to valves and fittings can be

expressed as:

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Chapter 8 Turbulent Flow in Circular Pipes