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Pressure Drop


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1. What is difference between Darcy and Fanning equations for the pressure drop in a fluid flow?

Darcy friction factor is four times the Fanning friction factor. Darcy friction factor is also called as Moody friction factor or Blasius friction factor.

  •  laminar flow, Darcy friction factor = fD= 64/Re

Where Re is Reynolds number

  •  turbulent flow,

Darcy friction factor = fD= 0.06 to 0.006

Fanning friction factor (ff) is one fourth of Darcy friction factor.

For a laminar flow

ff = 16/Re = ζ /ρu2/2


ff is the local Fanning friction factor

ζ is the local shear stress

u is the local flow velocity

ρ is the density of the fluid

Δhff = 2ff u2L/gD

For turbulent flow, Colebrook equation is used to find Fanning friction factor

1/ ( ff)0.5 = -4.0 log10 ((ε/d) / 3.7 + 1.256/ ff)

Where ε, roughness of the inner surface of the pipe (dimension of length);

     d, inner pipe diameter;

ff appears on both sides of the equation and its solution can be found only by hit and trial.

Darcy–Weisbach factor, fD is more commonly used by civil and mechanical engineers, and the Fanning factor, f, by chemical engineers, but one must be careful to determine correctly the friction factor for the equation used. Fanning equation is used by chemical engineers.

  1. State the assumptions used in the Bernoulli’s theorem.

Assumptions used in the derivation of Bernoulli’s theorem are

  • Fluid is ideal i.e. there are no losses of any kind in the flow of an ideal fluid.
  • Flow is steady i.e. No changes in the flow velocity with respect to time.
  • Fluid is incompressible i.e. It is only for liquids i.e. although pressure changes are there but volume and hence density remains constant during the flow.
  • Flow is one dimensional.
  • Fluid is continuous i.e. there are no vapors in it or there are no impurities in it.
  • Only gravitational and pressure forces are acting on the fluid i.e. fluid is non viscous.

2. State Pascal’s Law.

Pressure at any point in a static fluid is same in all the directions. If there is an increase/decrease in pressure at any point in a confined fluid, there is an equal increase/decrease of pressure at every other point in the container.


3. What is a Froude’s number? What is its significance?

Froude’s number (We) is a square root of the ratio of inertia force to the gravitational force. It is a dimensionless number. Mathematically

Fr = V/ (Dg)0.5

V is the velocity in m/s, assuming full pipe flow
D is the pipe inner diameter in m
g is the gravity constant in m/s²

Experimentally it has been found that Froude number should be less than 0.3 to avoid air entrainment and ensure undisturbed flow without pulsations.

The Froude number compares the resistance to wave making between bodies of various sizes and shapes. In free surface flow, it has been found that

  • When Froude Number = 1, flow is critical.
  • If Froude Number > 1, flow is super-critical.
  • When Froude Number < 1, flow is sub-critical.

In appearance, it has similarity with Mach number. Froude’s number is used in a fluid flow around marines, over spillways or flow over the weirs in open channels.



Minor Pressure losses
Pressure drop takes place with fluids moving in pipes/channels. There are three methods for finding minor pressure loss in pipe fittings.
1. The equivalent length method——Equivalent length for each type of pipe fitting has been prescribed so that frictional pressure drop in that length can be calculated by the usual equation. This calculated pressure drop will be the pressure drop in the pipe fitting.
hf = f (leq /Dh) (v2/2g)

2. The valve flow coefficient (Cv) method
3. The resistance coefficient (K) method————Most commonly used method.

K Method—Under this method, mathematically, minor head loss for every situation is given as
hL = K (v12/2g)
where v1 is the velocity in the SMALLER pipe irrespective of enlargement or contraction of the pipe
K is called minor loss coefficient. It has no units (dimensionless). The value of K for various pipe fittings is found from a standard graph or a standard table based on experimental data.


Sr. No.                    Type of Pipe fitting                              Value of ‘K’
1.              Tee, Flanged, Dividing Line Flow                            0.2
2.             Tee, Threaded, Dividing Line Flow                          0.9
3.              Union, Threaded                                                         0.08
4.              Elbow, Flanged Regular 90o                                     0.3
5.              Elbow, Threaded Regular 90o                                             1.5
6.              Elbow, Threaded Regular 45o                                   0.4
7.              Elbow, Flanged Long Radius 90o                             0.2
8.              Elbow, Threaded Long Radius 90o                          0.7
9.              Elbow, Flanged Long Radius 45o                              0.2
10.            Globe Valve, Fully Open                                              10
11.             Angle Valve, Fully Open                                              2
12.             Gate Valve, Fully Open                                               0.15
13.             Gate Valve, 1/4 Closed                                                0.26
14.             Gate Valve, 1/2 Closed                                                2.1
15.             Gate Valve, 3/4 Closed                                                17
16.             Ball Valve, Fully Open                                                 0.05
17.             Ball Valve, 1/3 Closed                                                   5.5
18.             Ball Valve, 2/3 Closed                                                  200
19.             Diaphragm Valve, Open                                              2.3
20.            Water meter                                                                   7.0
NOTE: Values of factor K for many other fittings not covered here but can be found in literature.

Total friction head loss
h total loss = Σ h major loss + Σ h minor loss



Pressure loss in fluid flow through pipes depends on
(i) Pipe material and its roughness
(ii) Shape and size of the pipe
(iii) Shape and size of the pipe fitting
(iv) Type of flow (laminar, turbulent and transition flow) which depends on Reynolds number

Thus pressure drop depends on density, velocity viscosity of fluid, diameter, length, and surface roughness of the pipe and the pipe fitting.

MAJOR PRESSURE LOSS IN PIPES: Pressure loss due to friction in a straight length pipe (vertical, horizontal and inclined pipes) is considered major pressure loss.

MINOR PRESSURE LOSS IN PIPES: Pressure loss caused by pipe fittings (Tees, elbows, unions, bends, valves etc.) are called minor pressure losses. These are called minor since these are small as compared to major pressure losses.

(a) Major Pressure Loss
For laminar flow——two equations are used.
      (i) Fanning equation— It uses hydraulic radius concept and not the pipe diameter. It is mostly used by Chemical Engineers.
∆pf =4 f (L/Rh )ρ V2/2
Head loss
hf = f (L/Rh) V2/2g
Fanning friction factor ‘ f ’ = 16/Re
Rh =Area of flow/Wetted Perimeter
= D/4 for full flow through the pipe

    (ii) Darcy-Weisbach-— It is being used by Civil and Mechanical Engineers.
∆p =f L ρ V2/2D
Head loss (hf ρ g =∆p)
hf = fLV2/2gD
Darcy friction factor ‘f’ = 64/Re

For turbulent flow—Only one equation (Darcy’s Equation) is used.

Darcy-Weisbach equation for turbulent flow
∆p =f L ρ V2/2D
Head loss (hf ρ g =∆pf)
hf = f LV2/2gD
But Darcy friction factor ‘ f ’ is found from Cole-brook-White equation
1/√f = –2 log 10 ( ε /3.7 Dh + 2.51/Re√f )
f is the Darcy friction factor,  dimensionless
ε is height of roughness, m
D is the inside diameter for circular pipes ,m (Dh = Di)
Dh is the hydraulic diameter for non-circular pipes ,m, and Dh = 4A/P
(A is the area of flow in the non circular pipe)
(P is the inside perimeter of the non circular pipe)
Re is the Reynolds number > 4000 for turbulent flow
There are number of empirical equations to find friction factor ‘f’ but most commonly used is the Cole-brook-White equation


Friction factor ‘ f ’ is present on both sides of the equation. Thus it can be solved only by an iterative procedure which can be very time consuming and cumbersome. Therefore for easiness the above Darcy equation has been plotted from the experimental data and has been called as Moody Diagram. From this diagram, friction factor for turbulent flow can be directly found after knowing the Reynolds number and pipe roughness.

Moody diagram

Applicable for turbulent flow only, Re > 4000, However laminar and transition region has also been shown on Moody Diagram. (Moody diagram is a graphical form of Cole-brook-White equation). Refer for Moody Diagram.
X co-ordinate —– Reynolds number
Y co-ordinate—–Darcy friction factor ‘ f ’
There are number of curves for various values of pipe roughness
Roughness of various pipe materials is also mentioned on this diagram.



PRESSURE OF A FLUID:  NORMAL force of fluid per unit area.
Its units are Pascal and bars.
Pascal = 1 N/m 2
bar = 740 mm of Hg = 10 m of water
1 atmospheric pressure  = 101.3 k Pa=760 mm of Hg = 10.3 m of water=14.7 psi = 1 ata
Different Forms of Pressure
Static Pressure: due to the height of the fluid column in the container
P = ρ g h
Dynamic pressure: due to the velocity of the fluid
P =ρ V 2/2
Datum pressure: It is with reference level where pressure is considered to zero, a reference pressure.  Normally it is zero at the ground (floor) level where the value of  Z is taken as 0.
Pressure at any height z is given by p = ρgZ
Therefore p = 0 at Z=0.
Atmospheric pressure: pressure exerted by the atmospheric air on all surfaces to which it is in contact. It is measured by a Barometer.


Absolute Pressure = 0 absolute at perfect vacuum
Therefore, absolute Pressure = 0 gage pressure at the atmospheric pressure level = 0 + 1 = 1
In general, absolute Pressure = gage pressure + 1 atmospheric pressure 
How much pressure has been decreased below atmospheric pressure

POSITIVE GAGE PRESSURE (above atmospheric): Positive pressure above atmospheric pressure  is called Positive Gage pressure.
Absolute pressure= Gage pressure + atmospheric pressure
Gage pressure = 3 bars
Absolute pressure = 3 + 1 =4 bars
Gage pressure will be applicable if water is converted into vapor above 1000C.

 NEGATIVE GAGE PRESSURE  (below atmospheric): Negative pressure (how much below atmospheric pressure) is called vacuum.
How much below atmospheric pressure is vacuum.
Vacuum pressure = atmospheric pressure – Actual  gage pressure
Atmospheric pressure = 760 mm Hg
Say Gage pressure is 700 mm Hg
∴ Vacuum = 760 –700=40 mm Hg

Vacuum is applicable where water is to be converted into vapor below 100C.
Such applications are water vapor in atmospheric air.
Water vapor in air is under vacuum pressure since the pressure of water vapor in air is much below atmospheric pressure.
Total pressure of air and water vapors in atmosphere is atmospheric pressure.
Vacuum condition causes evaporation at temperature lower than normal boiling point.

Measurement of Pressure, Mechanical and Electrical Transducers

Moderate pressures are measured with sufficient accuracy by simple pressure gauges like Piezometer and U-Tube manometers etc., while very low or very high pressures are measured by advanced gauges.

Two Types Of Pressure Measuring Devices
1.Mechanical Devices ( Pressure Transducers)
(i) Manometer—a column of liquid is balanced by same or different column of liquid.
(ii) Bourdon tube—A column of liquid is balanced by a spring or dead weight
(iii) Spiral and Helix Bourdon Tubes
(iv) Spring and bellows
(v) Diaphragm
(vi) Single and double inverted bell
2. Electrical Devices (Pressure Transducers)
(i) Strain gauge
(ii) Vibrating wire
(iii) Piezoelectric
(iv) Capacitance
(v) Linear variable differential transformer
(vi) Optical



It is difference of pressure between two points in a fluid flow. These two points can be at horizontal level or at different levels.

(i)Pressure drop is the driving force for the flow of a fluid.
(ii) Likewise temperature difference is the driving force for heat transfer.
(iii) Voltage difference is the driving force for current flow.
(iv) Concentration difference is the driving force for mass transfer
(v) Momentum difference is the driving force for momentum transfer.


Three types of Pressure drop

  1. Frictional pressure drop
  2. Gravitational pressure drop (two points at different elevations)
  3. Acceleration pressure drop in an evaporator (During phase change only i.e. liquid changing into vapor)


De-acceleration pressure drop in a condenser (During phase only i.e. vapor changing into liquid)

 Frictional Pressure Drop In Liquids

It can be found in two ways.
(i)  First–Darcy – Weisbach Equation which is applicable only for liquids with fully developed and steady flow. First head loss (Δh) is calculated and then pressure drop (Δp) is found.
Frictional head loss ∆hf —–Darcy – Weisbach equation

∆hf = 4f (L/D) (V2/2g) = 4fLV2/2gD

∆hf =Frictional head loss in a pipe of length ‘L’ with average velocity of flow, V
f = Fanning friction factor = 16/Re (dimensionless)
L = Length of the pipe (m)
D = internal diameter of a circular pipe (m)
D = Hydraulic diameter (4A / P) in case of a non-circular pipe (m)
A = area of flow in a non circular pipe (m2)

P = Perimeter of the non-circular pipe (m)
V = Average velocity of flow of fluid (m/s)
g = Local acceleration due to gravity (m/s2)

(ii) Second Method of finding frictional pressure drop

∴ Frictional pressure drop    ∆pf
∆pf = f (ρL V2/ 2D)= fLρV2/ 2D
(a) For a laminar flow, friction factor is a function of only Reynolds number  and  f = 64 / Re
(b) For the turbulent flow Friction factor is a function of Reynolds number and pipe roughness and is found from the Cole Brook relation
f -1 = –2 Log((2.51/(Re f1/2) + (k/D) x0.269))
‘f’ exist on both sides of the equation and can be found only by an iterative procedure. k is the roughness of the pipe.

The value of ‘f ‘ can also be found from Moody Chart for the turbulent flow since Moody chart is graphic form of Cole Brook Equation.

∆p = ρg(Z1 –Z2)

∆p = G2 ∆v
Where ∆v is the change in specific volume between the two points under consideration
G is the mass flux = mass flow rate per unit area
In most practical problems where the change in elevation is not extremely large, atmospheric pressure can be assumed to be constant. Otherwise there will be another pressure drop due to difference in atmospheric pressure between the two points. If the two points under consideration are widely separated. For example, pipe line is from an Arab country to India.

In compressible fluids (gases and vapors), there is  significant increase in velocity due to pressure decrease (Expansion) and hence requires a different equation as compared to that of liquids.
(p12—p22) / 2p1 = f (L/D) ρ1(Vav2/ 2) (T av / T1)
p1 = pressure at entrance, is known
p2 = pressure at exit, to be found
T av = (T1 + T2) / 2
NOTE: If the change in density between points 1 and 2 is less than 2% (i.e. almost in-compressible) ,  then pressure drop in gases and vapors can be found by DARCY EQUATION used for liquids as liquids being in-compressible.