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LECTURE – 3

THE CONTENTS OF THIS LECTURE ARE AS FOLLOWS:

1.0 EXPRESSION FOR PRESSURE LOSS DUE TO TURBULENT

FLOW IN PIPES

2.0 WORK DONE AGAINST FRICTION

3.0 WHICH TYPE OF FLOW IS FAVOURABLE IN MINES?

4.0 EDDY FORMATION

REFERENCES

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1.0 EXPRESSION FOR PRESSURE LOSS DUE TO TURBULENT FLOW IN PIPES

We have derived equation for pressure drop in case of laminar flow in pipes. In a

similar way, let us derive equation for pressure drop in case of turbulent flow in

pipes. Before we begin it is important for you to know that the expression for

pressure loss which we arrived for laminar flow is purely analytical. However in case

of turbulent flow, the expression is empirical (based on observation). Local eddies

and propagating eddies that develop in case of turbulent flow worsen the situation

and because of that till now we have not arrived at any analytical approach.

Now, let

d = diameter of pipe (m)

L = length of pipe (m)

𝜏 = shear stress (N/m2)

A = cross sectional area of the pipe (m2)

Scientist Antoine de Chezy (1719 -1798), started research on many canals and

tunnels during his period of study and came out with following conclusions for open

ducts. He noticed that the mean velocity of fluid in open ducts follow the following

trend:

u ∝ √ℎ

𝑙 , where

ℎ

𝑙 = ℎ𝑦𝑑𝑟𝑎𝑢𝑙𝑖𝑐 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑜𝑟 𝑐ℎ𝑎𝑛𝑛𝑒𝑙 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡

h= vertical distance per unit of length of channel or tunnel or ducts.

u ∝√𝐴 , 𝑤ℎ𝑒𝑟𝑒 𝐴 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎

u ∝√1

𝑤𝑒𝑡𝑡𝑒𝑑 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟(𝑝𝑒𝑟)

Combining all the three we get :

u ∝√𝐴

𝑝𝑒𝑟×

ℎ

𝑙

u = c× √𝐴

𝑝𝑒𝑟×

ℎ

𝑙 …………….(1)

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where c= Chezy coefficient

Let us discuss about the units in the above equation.

Mean velocity, u in meter per sec (𝑚

𝑠)

Channel gradient is unit less

Area, A in square meter (m2)

Wetted perimeter, (per) in meter (m)

What can be said about the unit of Chezy coefficient? As per dimensional analysis

its unit must be √𝑚

𝑠2

Now, let us see, how we arrived this unit?

We know when a viscous fluid flows, it exerts shear stress on the walls of a pipe or

channel. Let that stress be 𝜏. Force acting on the fluid is

F= τ × 𝑟𝑢𝑏𝑏𝑖𝑛𝑔 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 = 𝜏 × 𝑝𝑒𝑟 × 𝑙 …………….(2)

Where ‘per’ is the perimeter and ‘l’ is the length of channel or pipe.

Now if it is viscous, it must have head or pressure loss. Let the pressure loss be Δp.

So force acting on the fluid F= Δp× 𝐴 ………………(3)

Equating equations 2 and 3, we get:

𝜏 × 𝑝𝑒𝑟 × 𝑙 = 𝛥𝑝 × 𝐴

Thus, 𝜏 =𝛥𝑝×𝐴

𝑝𝑒𝑟×𝑙

Now we know the well-known equation Δp= ρgh. Using this in the above equation,

we get

𝜏 =𝜌𝑔ℎ×𝐴

𝑝𝑒𝑟×𝑙 =

𝜌𝑔𝐴

𝑝𝑒𝑟×

ℎ

𝑙 ………………….(4)

Now we know that, shear stress is directly proportional to the inertial stress.

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Hence, 𝜏 = 𝑓𝜌𝑢2

2 …………………..(5)

Equating equations 4 & 5 we get:

𝑓 ×𝜌𝑢2

2=

𝜌𝑔𝐴

𝑝𝑒𝑟×

ℎ

𝑙

Thus , 𝑢 = √2𝑔

𝑓× √

𝐴

𝑝𝑒𝑟×

ℎ

𝑙 …………………..(6)

Comparing equations 6 & 1 we get

𝑐 = √2𝑔

𝑓

Here,

f= friction factor called fanning friction factor or skin friction factor. It is unit less.

Thus, the unit of c is √𝑚

𝑠2

Therefore, 𝑢2 = 2𝑔𝜋𝑑2ℎ

4𝑓𝜋𝑑𝑙=

2𝑔𝑑ℎ

4𝑓𝑙

This is called Chezy- Darcy equation or simply Darcy equation or Darcy Weisbach

equation.

Therefore, ℎ =4𝑓𝑙𝑢2

2𝑔𝑑 𝑚𝑒𝑡𝑒𝑟𝑠

Here h= head loss

Therefore, pressure loss , ∆𝑝 =4𝑓𝑙𝜌𝑢2

2𝑑 pascal

Can we find any relation between f and Re?

Let us find out.

Equating pressure loss in case of laminar flow to turbulent flow, we have

4𝑓𝑙𝜌𝑢2

2𝑑=

32µ 𝑢𝑙

𝑑2

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𝑓 = 16 × (𝜇

𝜌𝑢𝑑) =

16

𝑅𝑒

In terms of Darcy friction factor, the pressure difference will be in the form

𝛥𝑝 = 𝜆𝑙𝜌𝑢2

2𝑑

Thus 4 × 𝑓𝑓𝑎𝑛𝑛𝑖𝑛𝑔 = 𝜆 =64

𝑅𝑒

In some text –books, ‘λ’ in the equation is written as ‘f’, where f is Darcy friction

factor.

Sometimes we may come across a situation wherein it may be difficult to judge,

which friction factor (of the two discussed above) is given in the problem. In order

to decide, which friction factor is given in the problem, we need to take help from

the Moody diagram (Fig. 1).

Fig. 1 Moody diagram (after McPherson, 1993)

In Fig. 1, the curve for laminar flow cuts the coefficient of friction at a value of

0.016 for which the corresponding REYNOLDS’ NUMBER is 1000 and at 0.008 for

which Reynolds’ number is 2000.

Let,

𝑓 =𝑥

𝑅𝑒

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𝑥 = 𝑓 × 𝑅𝑒

Therefore, 𝑥 = 16

And we get

𝑓 =16

𝑅𝑒

Thus, coefficient of friction factor asked in the problem is fanning friction factor or

skin friction factor.

Let me tell you that, these values of friction factors are for laminar flow in circular

pipe and it changes with change in the geometry of the opening as given in Table 1.

Table 1 Friction factor for fully developed laminar flow in pipes of various cross-sections

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We can also notice the difference in the friction factor for different types of flow.

The relationship of friction factor with geometry of the opening has been ignored to

avoid complexity at this stage.

2.0 WORK DONE AGAINST FRICTION

Fig. 2

Let us consider Fig. 2. It may be considered a typical duct or airway in mines. Then

for steady flow, the energy at point 1 must be equal to that at point 2 (applying

conservation of energy).

𝑢12

2+ 𝑔ℎ1 +

𝑝1

𝜌=

𝑢22

2+ 𝑔ℎ2 +

𝑝2

𝜌+ 𝑤12

𝐽

𝑘𝑔

All the symbols in the above equation have their standard meaning. ‘𝑤12′ in the

above expression must be equal to the energy needed to overcome the pressure

drop, if we assume no change in elevation as well as velocity.

Thus we can write, 𝑤12 =𝑝1−𝑝2

𝜌= ∆𝑝/𝜌

Hence work done against friction in case of laminar flow is

32𝑢𝜇𝑙

𝑑2×𝜌 (as ∆𝑝 𝑖𝑛 𝑐𝑎𝑠𝑒 𝑜𝑓 𝑙𝑎𝑚𝑖𝑛𝑎𝑟 𝑓𝑙𝑜𝑤 =

32𝑢𝜇𝑙

𝑑2 )

And work done against friction in case of turbulent flow is

4𝑓𝑙𝑢2

2𝑑 , (as ∆𝑝 𝑖𝑛 𝑐𝑎𝑠𝑒 𝑜𝑓 𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝑓𝑙𝑜𝑤 =

4𝑓𝑙𝑢2𝜌

𝑑2 )

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f= fanning friction factor

Note that in both the cases mentioned above we have not used mass term and

therefore the unit of w12 is 𝐽

𝑘𝑔

3.0 WHICH TYPE OF FLOW IS FAVOURABLE IN MINES?

To decide about which type of flow i.e., whether laminar or turbulent is favorable in

mines, we must first try to understand their influence on the mine ventilation.

Before deciding which one is better, let us summarize the properties of the two.

Table 1 gives the different parameters and the effect on these parameters through

laminar and turbulent flow.

Table 1 Impact of laminar and turbulent flow on different parameters

Parameter Laminar Turbulent

Eddy formation No eddies are formed Eddies are formed locally as well as they propagate

along the motion of the fluid.

Pressure drop in case of circular pipe

∆𝑝 =32𝑢𝜇𝑙

𝑑2

i.e. ∆𝑝 ∝ 𝑢.

Thus pressure drop is less compared to turbulent flow

∆𝑝 =4𝑓𝑙𝜌𝑢2

2𝑑

i.e. ∆𝑝 ∝ 𝑢2

Heat transfer Heat transfer is comparatively much lesser to the underground mine

environment than that when flow is turbulent

Promotes heat flow and adds more heat to the

underground environment

Mine condition Very less dirty compared to turbulent, as suspended dust is

lesser in the mine environment

Full of dust

Methane

layering

Promotes methane layering Resist methane layering

Analysis of flow and other

considerations

Simple Complex

Velocity of air Low High

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Thus, on the basis of the above comparison, now you may be in a position to

decide, which type of flow is better for underground workings. The comparison of

Table 1 shows that laminar flow may be better for underground environment as it

creates healthy working environment with lower pressure drop thereby saving in

the power and the cost.

Though, it comes out that flow should be maintained laminar in mines. But, is it

possible to do so? Let us find it.

We know the value of Re for flow to be either turbulent or laminar. It is 2000 for

laminar and 4000 for turbulent. Viscosity of air µ has value of 1. 8 × 10−5 kg/ms .

The density of air is 1.2 𝑘𝑔/𝑚3. If we consider the value of diameter of duct to be 1

m, the lower critical velocity comes out to be 0.03 m/s and upper critical velocity

comes out to be 0.06 m/s. This value will further reduce, if we increase hydraulic

diameter of airway. This is a very low value of velocity compared to the velocity at

which air should flow in mines for comfortable working conditions. Thus, laminar

flow is ruled out. Further, as air has to move in underground mines of varying

cross-sectional areas, bends, obstructions etc, it is further disturbed thereby

making laminar flow in mines almost impossible.

Now, the question arises, how we control or overcome dust problem in mines due

to turbulent flow of air? Also, how to maintain required comfortable temperature in

underground mines as we know turbulent flow will add more heat to air from the

mine atmosphere?

Generally in mines, dust is a major issue near the face. To deal with dust problem

at the face, we may adopt a suitable ventilation system at the face. As for instance,

in case of blind headings, dust at the face can be reduced by adopting overlap

system of auxiliary ventilation. There are other technical ways also for controlling

dust in mines. Also in case of necessity in deep and hot mines, refrigeration and air

conditioning of air may be thought of for maintaining comfortable working

conditions besides increasing the air flow.

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Thus, based on the above discussion, it may be concluded that, the flow in mines in

general is turbulent in nature. May be in some part of the mine flow may be

laminar.

4.0 EDDY FORMATION

Let us try to understand the concept of eddy formation during the flow of gases. Let

us have a look at the flow of air in auxiliary ventilation system as shown in Fig. 3.

Fig. 3 Air flow pattern in auxiliary ventilation system (after McPherson, 1993)

Notice the difference in the flow near the face and away from the face both in case

of line brattice and duct system of auxiliary ventilation. It is very clear that near the

face the movement of air is not uniform. This is due to turbulence produced

because of change in direction. This effect is more in case forcing fan is used for

auxiliary ventilation. In case we use exhaust fan for the purpose of auxiliary

ventilation, the turbulence produced near the face is less because of lack of jet

effect in it, compared to that of forcing fan. This turbulent produced in the air near

the face is responsible for eddy formation, wherein the air molecules moves along

an imaginary path/zig-zag path.

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Let us now discuss about whether eddy formation is desirable or not? Does it have

any impact on mine environment?

Consider Fig. 4(a) and Fig. 4(b) wherein methane layering for level airway and

descentionally ventilated airway are shown.

Fig. 4 Methane layering in (a) a level airway (b) a descentionally ventilated

airway (after McPherson, 1993)

In case of Fig. 4(a) formation of methane layer has taken place in an efficient

manner as the airway is level and eddy formation is minimal. In Fig. 4(b), we find

that due to formation of eddy in the descentionally ventilated airway, formation of

methane layer is not in an efficient manner. Thus it reduces risk of explosion due to

methane.

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However, formation of eddies in the air at the same time is also dangerous as it

mixes coal dust and fumes at all levels of the airway. Further, because of eddy

formation, air molecules move in a highly disordered manner and increasing the

amount of heat addition to the underground mine atmosphere which is highly

undesirable. Thus eddy formation in mines should be avoided.

REFERENCES

Hartman, H. L., Mutmansky, J. M. & Wang, Y. J. (1982); “Mine Ventilation and Air

Conditioning”; John Wiley & Sons, New York.

McPherson, M. J. (1993); Subsurface Ventilation and Environmental Engineering”;

Chapman & Hall, London.

Vutukuri, V. S. & Lama, R. D. (1986); “Environmental Engineering in Mines”;

Cambridge University Press, Cambridge.