Unit - 3

Flow through Pipes

Q1) What is TEL?

A1)

Total energy Line (T.E.L. Or E.G.L)

It is known that the total head with respect to any arbitrary datums, is the sum of the elevation (potential) head, pressure head and velocity head.

Total head =

When the fluid flows along the pipe, there is loss of head and the total energy decreases in the direction of flow. If the total energy at various point along the axis of the pipe is plotted and joined by a line, the line so obtained is called the Energy gradient line (E.G.L)

Energy gradient line (E.G.L) is also known as Total energy line (T.E.L)

Q2) Define HGL.

A2)

Hydraulic gradient Line (H.G.L)

The sum of potential head and the pressure head

=

At any point is called the piezometric head.

If a line is drawn joining the piezometric levels at various points, the line so obtained is called the ‘Hydraulic gradient line.

Hydraulic gradient line (H.G.L) is always below the energy gradient line (E.G.L) and the vertical intercept between the two is equal to the velocity head

Q3) What is energy losses through pipe?

A3)

When water flows in a pipe, it experiences some resistance to its motion, due to which its velocity and ultimately the head of water available is reduced.

Major energy losses

This loss is due to friction.

Darcy – Weisbach formula

The loss of head in pipes due to friction is calculated from Darcy-Weisbach formula which is given by

hf =

Where, hf = loss of head due to friction,

f = co-efficient of friction,(a function of Reynolds number, Re)

f =   for Re varying from 4000 to 106

=  for Re < 2000 (laminar/ viscous flow)

L = Lengthof the pipe

V = Mean velocity of flow, and

D = Diameter of the pipe.

Q4) Explain minor energy loss in detail.

A4)

Minor Energy Loses

• The loss of energy due to change of velocity of the flowing fluid in magnitude or direction is called minor loss of energy.

The minor loss of energy (or head) includes the following cases:

• Loss of head due to Sudden Enlargement.

Consider a liquid flowing through a pipe that has sudden enlargement as shown in fig.

he =  

• Loss of head due to Sudden Contraction.

Consider a liquid flowing in a pipe that has a Sudden Contraction in the area as shown in fig.

hc = V22/ 2g [ 1/c -1]2

• Loss of head at the entrance of a pipe:

This is the loss of energy which occurs when a liquid enters a pipe that is connected to a large tank or reservoir.

hi = 0.5 V2/2g

• Loss of head at the exit of pipe:

This is the loss of head due to the velocity of the liquid at the outlet of the pipe which is dissipated either in the form of a free jet or it is lost in the tank or reservoir.

h0 = V2/2g

• Loss of head due to an obstruction in a pipe:

Whenever there is an obstruction in a pipe, the loss of energy takes place due to the reduction of the area of the cross-section of the pipe at the place where the obstruction is present.

Let τ0 = Shearing stress at the pipe wall

τ0 = μ× velocity gradient at the pipe wall

= μ

At y = δ let u = uδ

∴

∴τ0 =μ

∴τ0/ρ =

∴v*2 =

a = Maximum area of obstruction

A = Area of pipe

V = Velocity of liquid in the pipe

habs = V2/2g [ A/Cc(A-a) – 1]2

• Loss of head due to bend in pipe:

When there is any bend in a pipe, the velocity of the flow changes, due to which the separation of the flow from the boundary and also the formation of eddies takes place.

hb = kV2/2g

Where k is coefficient of bend

• Loss of head in various pipe fittings:

• The loss of head in various pipe fittings such as valves, couplings, etc. is expressed as

hpf = kV2/2g

Where k is coefficient of pipe fitting

Q5) Explain Darcy-Weisbach equation

A5)

Fig shows a horizontal pipe having steady flow.

Fig: Darcy wiesbach equation

Consider control volume enclosed between section 1 and 2 of the pipe.

p1 = Intensity of pressure at section 1,

p2 = Intensity of pressure at section 2,

L = Length of the pipe, between section 1 and 2,

D = Diameter of the pipe,

f = Non-dimensional factor (Whose value depends on the material and nature of      the pipe surface), and

hf= Loss of head due to friction.

Propelling force on the following fluid between the two section is = (p1 – p2) A

(Where, A = area of cross section of the pipe

Frictional resistance force = f ’ PLV

Where   P = wetted perimeter, and

V = Average flow velocity

Under Equilibrium condition,

Propelling force = Frictional resistance force

i.e. (p1 – p2) A = f PLV2.

Dividing both sides by weight density W, we have

(P1 – P2/W) A =

hf = =

hf =

=

The ratio is called the hydraulic mean depth or hydraulic radius, denoted by m (or R).

The term () has dimensions of hf and thus the term is a non-dimensional quantity and let us replace it by another constant f.

hf = f      ------- 1

In case of a circular pipe,

Hydraulic mean depth,  m =                                   

Substituting this value in equation 1, we get

hf = f     ---------2

(The factor f is known as Darcy coefficient of friction.)

Equation 2 is known as Darcy-Weisbach equation.

Q6) Explain minor losses in pipes.

A6)

Major loss of energy refers to the loss of head or energy caused by friction in a pipe, whereas minor loss of energy refers to the loss of energy caused by a change in magnitude or direction in the velocity of the following fluid.

• Loss of head due to sudden enlargement
• Loss of head due to sudden contraction
• Loss of head at the entrance of a pipe
• Loss of head at the exit of a pipe
• Loss of head due to an obstruction in a pipe
• Loss of head due to bend in the pipe
• Loss of head in various pipe fittings

Because the above losses are minor in comparison to the loss of head due to friction in a lengthy pipe, they are referred to as minor losses and can even be overlooked without causing major inaccuracy. In the event of a short pipe, however, these losses are comparable to head loss due to friction.

Q7) Explain loss of Head Due to Sudden Enlargement.

A7)

Consider a liquid flowing through a conduit that suddenly expands, as depicted. Before and after the enlargement, consider two parts (1)-(1) and (2)-(2).

Applying Bernoulli’s Equation:

he =

Consider the control volume of liquid between sections 1-1 and 2-2. Then the force acting on the liquid in the control volume in the direction of flow is given by

Fx = p1A1 + P(A2 – A1) – p2A2

Experimentally, P = p1

Fx = p1A1 + p1(A2 – A1) – p2A2

Fx = p1A1 – p2A2

Fx = (p1 – p2) A2

Momentum of liquid/sec at section 1 – 1 = mass * velocity

= ρA1V12

Momentum of liquid/sec at section 2 – 2 = mass * velocity

= ρA2V22

Change of momentum

ρA2V22 - ρA1V12

Using continuity equation

A1V1 = A2 V2

A1 =

Change of momentum,

ρA2V22 - ρ

Now net force acting on the control volume in the direction of flow must be equal to the rate of change of momentum or change of momentum per second.

(p1 – p2)A2 = ρA2(

(p1 – p2)/ρ =

(p1 – p2)/ρg = /g

p1 /ρg – p2/ρg = /g

So,

he =

he =

he =

Q8) Explain loss of Head Due to Sudden Contraction.

A8)

Consider a liquid running through a pipe with an abrupt area contraction, as depicted. Before and after contraction, consider parts 1-1 and 2-2. As liquid flows from a large pipe to a smaller pipe, the flow area decreases until it reaches a minimum at section C-C, as depicted. Vena-contracta is the name given to this section C-C. A dramatic increase of the region occurs after section C-C. The sudden loss of head is actually owing to a sudden expansion of the Vena-contracta into a smaller channel.

Actual loss of head due to enlargement from section C-C to section 2-2 and is given by equation,

hc = =

Using continuity equation

Ac = A2 V2/ Vc

So,

hc =

hc =

Where,

k =

If,

Cc = 0.62 then k = 0.375

Then,

hc = 0.375 

If the value of Cc is not given, then the head loss due to contraction is taken as,

hc = 0.5 

Q9) Explain Loss of Head at the Entrance of a Pipe and Loss of Head at the Exit of Pipe

A9)

Loss of Head at the Entrance of a Pipe:

When a liquid enters a conduit that is connected to a huge tank or reservoir, there is a loss of energy. This is analogous to the loss of a head as a result of a rapid constriction. The amount of loss is determined by the type of entrance. This loss is significantly more for a sharp edge entrance than for a rounded or bell mouthed opening. The value of head loss at the entrance (or inlet) of a pipe with sharp cornered entrance is taken in practise as,

hi = 0.5 

Loss of Head at the Exit of Pipe:

This is the loss of head (or energy) caused by the velocity of liquid at the pipe's outlet, which is dissipated either as a free jet (if the pipe's outlet is free) or in the tank or reservoir (if the pipe's outlet is connected to the tank or reservoir).

ho = 0.5 

Q10) Explain loss of Head Due to an Obstruction in a Pipe.

A10)

When there is an impediment in a pipe, energy is lost due to a reduction in the area of the pipe's cross-section at the blockage. There is a dramatic increase of the flow region beyond the obstacle, resulting in loss of head.

a = Maximum area of obstruction

A = Area of pipe

V = velocity of liquid in pipe

(A-a) = Area of flow of liquid at section 1 – 1

A vena-contracta is produced beyond section 1-1 as the liquid flows, after which the stream of liquid widens again and the velocity of flow at section 2-2 becomes uniform and equal to the pipe's velocity, Vin. The flow of liquid through sudden expansion is analogous to this condition.

Vc = Velocity of liquid at venacontracta

Then loss of head due to obstruction = loss of head due to enlargement from vena-contracta to section 2-2,

= (Vc – V)2/ 2g

From continuity,

ac = AV/ Vc

Then,

Cc =

ac = Cc * (A – a)

So,

Cc × (A – a) × Vc  = A × V

Vc =

Hence,

(Vc – V)2/ 2g = = 

Q11) What is loss of Head due to Bend in Pipe?

A11)

When there is a bend c in a pipe, the velocity of the flow varies, causing the flow to separate from the boundary and the development of eddies. As a result, the energy is lost. The head loss in a pipe due to bending is given as,

hb =

The value of K depends on,

• Angle of bend
• Radius of curvature of bend
• Diameter of pipe

Q12) Define moody diagram.

A12)

In a fully developed turbulent pipe flow, the friction factor is determined by the Reynolds number and relative roughness , which is the ratio of the pipe's mean height of roughness to its diameter. All published results are acquired from tedious experiments employing intentionally roughened surfaces (typically by gluing sand grains of a given size on the inner surfaces of the pipes), and the functional form of this dependence cannot be obtained from a theoretical study.

The friction factor was determined using the flow rate and pressure drop readings.

Curve-fitting experimental data yielded tabular, graphical, and functional results, which are presented in tabular, graphical, and functional formats.

The Colebrook equation was created when Cyril.F. Colebrook (1910–1997) integrated the existing data for transition and turbulent flow in smooth and rough pipes into the following implicit relation.

Lewis F. Moody (1880–1953) redrew Rouse's diagram into the current format two years later. It is given the now-famous Moody chart. It plots the Darcy friction factor for pipe flow as a function of Reynolds number and e/D over a wide range of Reynolds numbers and  values. It is one of the most extensively accepted and utilised engineering diagrams. Although it was designed for circular pipes, it can be used for noncircular pipes by substituting the hydraulic diameter for the diameter.

Commercially available pipes differ from those used in the trials in that their roughness is not uniform, making it impossible to give an exact description. Table and the Moody chart both show equivalent roughness levels for several commercial pipes.

However, keep in mind that these figures are for fresh pipes, and as a result of corrosion, scale building, and precipitation, the relative roughness of pipes can rise over time. As a result, the friction factor could rise by 5 to 10 times. The design of piping systems must take into account actual operating circumstances. Furthermore, the Moody chart and its equivalent Colebrook equation are subject to a number of uncertainties (roughness size, experimental error, data curve fitting, and so on), therefore the findings produced should not be considered "precise." It's usually thought to be accurate to within percent of the figure's range.

Because the Colebrook equation is implicit in f, iteration is required to determine the friction factor unless an equation solver such as EES is utilised. S. E. Haaland published an approximate explicit relation for f in 1983.

The findings derived from this relationship are within 2% of the Colebrook equation's results. When using a programmable calculator or a spreadsheet to solve for f with the equation, can be utilised as a good first guess in a Newton iteration if more exact results are required.

We make the following observations from the Moody chart:

• The friction factor for laminar flow decreases with rising Reynolds number and is unaffected by surface roughness.

• For a smooth pipe, the friction factor is lowest (although not zero because of the no-slip condition) and increases with roughness. In this scenario, the Colebrook equation ε = 0 reduces to the Prandtl equation.

• The shaded area in the Moody chart indicates the transition region from the laminar to turbulent regime (2300 < Re < 4000). Depending on flow disturbances, the flow in this region may be laminar or turbulent, or it may alternate between laminar and turbulent, causing the friction factor to fluctuate between values for laminar and turbulent flow. The information in this range is the least trustworthy. The friction factor increases in the transition area with modest relative roughness’s and approaches the value for smooth pipes.
• The friction factor curves corresponding to specified relative roughness curves are almost horizontal at very large Reynolds numbers (to the right of the dashed line in the chart), and hence the friction factors are independent of the Reynolds number. Because the thickness of the viscous sublayer drops with rising Reynolds number, it gets so thin that it is negligible in comparison to the surface roughness height, the flow in that region is referred to as fully rough, turbulent flow or simply fully rough flow. In this situation, the protruding roughness elements cause the viscous effects in the main flow, while the contribution of the laminar sublayer is insignificant.
• In the fully rough zone, the Colebrook equation reduces to the von Kármán equation stated as, which is clear in f. Although some publications refer to this zone as having wholly (or completely) turbulent flow, this is misleading because the flow to the left of the dashed blue line is likewise completely turbulent.

Q13) Explain pipe in series in detail.

A13)

Pipes in Series:

• When the pipes of different lengths and diameters are connected to each other to form a pipeline then such a pipeline is called as a compound pipe or pipes in series.
• As the pipes are in series, the discharge through each pipe will be same (continuous)

∴ Q = A1 V1 = A2 V2 = A3V3

• If a pipeline connecting two reservoirs by compound pipes, then the difference in liquid surface level is equal to the sum of the head losses in all the sections.

Let Q = Discharge through pipeline

H = Total loss of head

d₁ = Diameter of pipe 1

L₁ = Length of pipe 1

f₁ = Darcy's coefficient of friction

d2, L2, f2 =Corresponding value of pipe 2,

d3, L3, f3 =Corresponding value of pipe 3.

1. Considering all losses,

Total loss of head = Major losses + Minor losses

H=

∴Major losses = Head loss due to friction in each pipe

=

∴Minor losses = Entrance loss+ Contraction loss+ Expansion loss+ Exit loss

2. If minor losses are neglected, then Total loss of head = Major losses

H = 1/2g [

3. If discharge through the pipe is given, then

H=[                            

Q14) Explain pipes in parallel in detail.

A14)

Pipes in Parallel:

• When a pipeline divides into two or more parallel pipes which again join together at downstream as shown in Fig., then that pipe is said to be in parallel.
• In order to increase the discharge passing through the main pipe, the pipes are connected in parallel.

• The discharge in the main pipe is equal to sum of the discharge in each of the parallel pipes.

∴Q =Q1 + Q2

• When the pipes are arranged in parallel, the loss of head in each parallel pipe is same.

hf 1  =  hf 2

Q15) What is Siphons?

A15)

A syphon is a long-bent pipe is used for carrying water from a reservoir at a higher level to another reservoir at a lower level when the two reservoirs are separated by a hill or high-level ground.

The point C is highest point of syphon is called as summit.

The pressure at point C is less than atmospheric pressure as it lies above the free water surface in the tank A.

Fig: Siphon

The pressure at C can be reduced, theoretically, to -10.3 m of water but in actual practice is only 7.6 m of water (or 10.3-7.6=2.7 m of water absolute).

If the pressure becomes less than 2.7 m of water absolute, the dissolve air and other gases would come out from water and collected at the summit. It may be obstructed the flow of water.

Syphon is used in following cases:

(a) To take out water from one reservoir to another reservoir separated by a hill or ridge.

(b) To drain out water from a channel without any outlet.

(c) To take out the water from a tank which does not have any outlet.

Q16) Write down the working principle of siphon.

A16)

Negative or vacuum pressure is created in the syphon, so that liquid gets pushed into it.

The flow through syphon then remains continuous till pressure in syphon pipe remains negative but less than separation pressure.

At summit, we find the minimum pressure. Velocity or discharge through syphon can be obtained by applying Bernoulli's equation between A and Fig.as shown in.

Now applying Bernoulli's equation between section A and B,

+ZA=+ZB+Losses

But and VA=VB=0

∴ZA-ZB=Losses

H= Loss of head at entry+ Head loss due to friction+ Head loss due to exit

H= 0.5                                

Where f= Friction factor=4f=4 coefficient of friction

L = Length of syphon AB

d =Diameter of the syphon

V =Velocity of flow

Now, applying Bernoulli's equation between A and C.

+ZA=+ZB+Losses between AC

But =0, VA=0, Vc=V,

Head loss between AC=Entry loss +Frictional loss

=0.5

∴ZA=         

But ZA-ZC=-hc

= - [ hc + ( 1.5 +                            

This equation is used to find out,

=Pressure head at summit

Hc = Height of summit above the surface in the higher level

L1= Length of inlet leg

Q17) What is transmission of power?

A17)

Power is transmitted through pipe by allowing water to flow through a pipe.

Now consider a pipe AB connected to a tank as shown it.

Let,

H = Head of water at inlet

L = Length of the pipe

d = Diameter of the pipe

V = Velocity of flow in pipe

hf =Head loss due to friction

Head available at the outlet of the pipe,

=Head at inlet-Head loss due to friction

= H-hf =H -

Weight of water flowing through pipe/sec.

W =ρg x Volume of water/sec.

= ρg x Area× Velocity= ρg ×π/4 d2×V

Power transmitted at outlet of the pipe,

P = Weight of water per sec x Head at outlet

P= (ρg ×π/4 d2×V) [H -   Watt

Efficiency of power transmitted,

ɳ =

=

=

ɳ =

Q18) Give the Condition for Maximum Transmission of Power.

A18)

The condition for maximum transmission of power is obtained by differentiating Equation with respect to V and equating the same to zero.

∴(P)=0

(ρg ×π/4 d2×V) [H -

ρg ×π/4 d2 (H -

H =

∴hf=H/3

The Equation is the condition for maximum transmission of power.

Maximum Efficiency of Transmission of Power:

Efficiency of power transmission through pipe is given by,

ɳ=

For maximum power transmission, hf=H/3

Maximum efficiency,

ɳ=

For maximum power transmission, hf=H/3

Maximum efficiency,

ɳ==2/3

∴ɳ%=66.67%

Q19) What is dimensional analysis?

A19)

We make use of dimensional analysis for three prominent reasons:

• To check the consistency of a dimensional equation.
• To derive the relation between physical quantities in physical phenomena.
• To change units from one system to another.

Understanding dimensions is of utmost importance as it helps us in studying the nature of physical quantities mathematically. The basic concept of dimensions is that we can add or subtract only those quantities which have same dimensions. Also, two physical quantities are equal if they have same dimensions.

Dimensional analysis is a mathematical technique used to predict physical parameters that influence the flow in fluid mechanics, heat transfer in thermodynamics, and so forth. The analysis involves the fundamental units of dimensions MLT: mass, length, and time.

A typical fluid mechanics problem typical fluid mechanics problem in which experimentation is required consider the experimentation is required consider the steady flow of an steady flow of an incompressible Newtonian fluid through a long, smooth incompressible Newtonian fluid through a long, smooth walled, horizontal, circular pipe.

The first step in the planning of an experiment to study this problem would be to decide the factors, or variables, that will have an effect on the pressure drop.

Pressure drop per unit length p  f ( D ,  ,  , V )

Q20) Define Dimensional Homogeneity.

A20)

• A physical equation is the relationship between two or more physical quantities.
• Any correct equation expressing a physical relationship between quantities must be dimensionally homogeneous and numerically equivalent.
• Dimensional homogeneity states that every term in an equation when reduced to fundamental dimensions must contain identical powers of each dimension.
• A dimensionally homogeneous equation is applicable to all system of units.
• In a dimensionally homogeneous equation, only quantities having the same dimensions can be added, subtracted or equated.

P = w h

Dimensions of LHS = ML-1T-2

Dimensions of RHS = ML-2T-2×L = ML-1T-2

Dimensions of LHS = Dimensions of RHS

Hence, equation P= w h is dimensionally homogeneous; so it can be used in any system of units.

Applications of Dimensional Homogeneity:

• It facilitates to determine the dimensions of a physical quantity.
• It helps to check whether an equation of a physical phenomenon is dimensionally homogeneous or not.
• It facilitates conversion of units from one system to another.