Module 6. Flow through pipes

Lesson 15

HEAD LOSS IN FLUID FLOW - MAJOR HEAD LOSS

15.1  Introduction

Liquids flow in a pipe due to pressure or gravity. In case of pressure flow, if a piezometer is connected to the pipe, the rise of liquid is suppose h meters. The pressure, P and head, h are related by the relation P = ρgh.  In SI system of units the pressure is usually measured in terms of Pa (N/m²) or kPa. In case of liquids it can also be expressed as head i.e. the rise of liquid in the piezometer. When a fluid flows through a pipe, it experiences some resistance due to which some of the energy of fluid is lost. The head loss in fluid flow is classified in two categories namely major head losses and minor head losses. In this module, reasons for such head losses and the method of estimating these head losses are discussed.

15.2  Flow through pipe

There are two cases of flow:

(i) Closed Conduit

(ii) Open Conduit

(i) Closed conduit: It is a pipe or duct through which the fluid flows by completely filling the cross-section. Since, the fluid has no free surface; its pressure may be above or below the pressure of the atmosphere.

(ii) Open conduit: It is a duct or open channel in which fluid flows with the free surface. If a closed pipe not running full, it may be treated as open channel.

15.3  Pressure/Head Loss

a. Major head losses in pipe flow: The major head losses in fluid flow are caused by friction of the conveying pipeline. The internal surface which comes in contact with the flowing fluid causes friction on the fluid layers. The extent of head loss depends on roughness of the pipeline as well as flow characteristics of the fluid. This head loss takes place continuously in the entire conveying pipeline/duct and it is characterized as major head loss.

b. Minor losses in pipe flow: Minor head losses include head loss or pressure drop due to pipe fittings, valves, entrance and exit of pipe, sudden contraction or expansion etc. This head loss is relatively small in case of very long pipeline but it may be high in case of small pipe network involving many fittings and valves.

15.4  Major Losses in Pipe Flow (Friction)

The major head losses can be estimated by the following methods:

(a) Darcy’s Formula (flow in pipe)

Fig. 15.1 Flow in pipe

Consider a uniform pipeline through which water is flowing at a uniform rate.

Let,

l = length of pipe

d = diameter of the pipe

v = velocity of water

f’ = frictional resistance per unit wetted surface per unit velocity

h_f  = loss of head due to friction

P₁ = pressure at 1----1

P₂ = pressure at 2----2

If there is no friction resistance, pressure P₁ and P₂ will be equal. Considering horizontal forces on the water at 1----1 and 2----2.

(Dividing by ω on both the sides)

It has been established from Froude’s experiments that

∴Frictional resistance = Frictional resistance per unit area per unit nelocity × area × v²

Let us introduce a co-efficient f such that

This co-efficient is called Darcy’s co-efficient or frictional co-efficient.

is a function of Reynold’s number (Re)

(b) Chezy’s Formula

Chezy’s formula is used to calculate head loss for open conduit like canal, drain etc.

Consider a long pipe through which water is flowing at a uniform rate as shown below:

Let,

l = length of the pipe

d = diameter of pipe

A = π/4 d²

P = Perimeter, πd

V = velocity

fʹ = frictional resistance per unit area per unit velocity

P₁ = pressure at 1 – 1

P₂ = pressure at 2 – 2

We know that P₁A = P₂A + frictional resistance

i.e.

We also know that frictional resistance = fʹ × πdl × V²

Substituting another term called hydraulic mean depth (also known as hydraulic radius) in the above equation, such that hydraulic mean depth

Now, Substituting two more terms in the above equation such that

15.5 Numericals

Q 1. (a) Using Darcy’s formula calculate head loss due to friction in a pipe of diameter 150 mm and length 50 m. Velocity of water is 2 m/s and co-efficient of friction is 0.004. Kinematic viscosity of water is 0.013 stoke.

(b) In the above numerical calculate the head loss using Chezy’s formula considering C = 45.

Solution:        

Given

L = 50 m

D = 0.150 m

V = 2.0 m/s

F = 0.004

v = 0.013 X 10^-4 m²/s

Q 2. In a piping system length & diameter of pipe is 100 m & 300 mm respectively, water is flowing at velocity of 4 m/s.

(a) Calculate head loss using Darcy’s formula.

      Given Kinematic viscosity = 0.013 stokes.

(b) Find head loss if Chezy’s constant C = 45.

Solution

Q 3. Water is pumped from station A to station B in a pipe of length 60 m and diameter 250 mm, at a velocity of 3 m/s. Assuming Kinematic viscosity of water is 0.013 * 10^-4 m²/s. Find the frictional loss of head by using.

(a) Darcy’s formula

(b) Chezy’s formula where C = 60.

Solution: