Module 6. Flow through pipes

Lesson 16

HEAD LOSS IN FLUID FLOW:  MINOR HEAD LOSS

16.1 Introduction

As discussed in the earlier lecture, minor losses include head loss or pressure drop due to pipe fittings, entrance and exit of pipe, sudden contraction or expansion etc. These are considered as minor because pressure drop is small compared to the major or frictional head loss. Minor Head losses are:

1. Head loss at the entrance of pipe

2. Head loss at the exit of pipe

3. Head loss due to obstruction in pipe

4. Head loss due to sudden contraction in pipe

5. Head loss due to sudden expansion in pipe

6. Head loss due to pipe fittings

16.2 Head Loss at the Entrance of Pipe (h_en)

Point A shows the place where liquid is entering the piping system from the tank (Fig. 16.1).

Fig. 16.1 Head loss at the entrance of pipe

The head loss due to entrance is given as:

16.3 Head Loss at Exit from Pipe (h_ex)

Head loss at the pipe exit (point A) where liquid gets freely discharged into atmosphere is given as:

Fig. 16.2 Head loss at exit from pipe

Where,

V = velocity

16.4 Head Loss Due to Obstruction in Pipe (h_obs)

If an irregular obstruction is placed in the pipe it will cause pressure drop. Figure 16.3 shows an irregular object placed in the path of flow.

Fig. 16.3 Head loss due to obstruction in pipe

16.5 Head Loss Due to Sudden Contraction in Pipe (h_c)

After sudden contraction the streamlines contract to pass through a minimum cross sectional area (A_c) and the fluid stream widens to fill the pipe. This minimum cross section is known as vena contracta and A_c is the area of cross section at vena contracta.

	At Section 1-1´	At section 2-2´ vena contracta	At section 3-3´
Diameter of Pipe	= d1	= d2	= d2
Area of pipe	= A1	= Ac	= A2
Flow velocities	= V1	= Vc	= V2

Fig. 16.4 Sudden contraction and formation of vena contracta

Formation of eddies takes place between the vena contract and pipe wall.

Head loss due to sudden contraction=head loss upto vena contracta + head loss due to sudden enlargement.

………(i)

Velocities of flow at vena contracta=V_C

Cross sectional area of vena contracta = A_c

Velocities of flow at section 2-2´ = V₂

Cross Sectional area at section 2-2´ = A₂

………(ii)

Where C_c= Coefficient of contraction

From equation (i) and (ii)

           Where k_c = Contraction loss coefficient

In general value of K_c may be taken as 0.5

16.6 Head Loss Due to Sudden Expansion in Pipe (h_e)

For fig 16.5 we have the following assumption:

	At Section 1-1´	At section 2-2´
Diameter of Pipe	= d1	= d2
Area of pipe	= A1	= A2
Flow velocities	= V1	= V2
Pressure	= p1	= P2

Fig. 16.5 Sudden enlargement

Due to sudden enlargement turbulent eddies are formed at the corner of the enlargement of the pipe section. The formation of eddies cause loss of energy in form of heat to the surrounding. Let P_e be the pressure of the eddying fluid h_e be the head loss due to enlargement.

Applying Bernoulli’s theorem across section 1-1´ and 2-2´.

Since Z₁ = Z₂

………(i)

Net force acting between section 1-1´ and  2-2´

 Since mean pressure due to the formation of eddies (P_e) is approximately equal to the inlet pressure (P₁),

F = (P₁ –P₂)A₂ ………(ii)

Let Q be the discharge m³/sec through the pipe.

Change of momentum of liquid pressure

from continuity equation Q=A₂V₂

= ρA₂V₂ (V₂ — V₁) ………(iii)

Since net force = change of momentum of liquid per second

from equation (ii) and (iii)

(p₁ – p₂) = ρ (V₂² –V₁V₂) ………(iv)

from equation (i) and (iv)          

16.7 Head Loss Due to Pipe Fittings (h_pf)

Various pipe fittings like bend, elbow, tee etc also contribute to pressure drop. The head loss due to pipe fittings is given by the following equation:

K_L= Friction loss co-efficient/ loss co-efficient, depend on the shape, size and type of pipe fittings.

TYPE	KL
Globe Valve	10
Gate  valve(wide open)	0.2
45° Elbow	0.4
Bend	2.2

16.8 Numericals

Q 1. The velocity of water flowing through a 12 cm diameter pipe was found to be 3.5 m/s. The flow path in pipe is destructed by an iron plate of 8 cm diameter. Calculate, the loss of head due to obstruction if co-efficient of contraction C_c = 0.75.

Solution:

Q 2. At a sudden enlargement of water line from diameter of 200 mm to 350 mm, volumetric flow rate 0.5 m³ /s. Determine the head loss due to sudden enlargement.

Solution:

D₁ = 200 mm; D₂ = 350 mm; Q = 0.5 m³/s

Q 3. A horizontal water pipe has an abrupt enlargement such that the diameters of small and large cross-section are 250 mm and 450 mm respectively. The volumetric flow rate in pipe is 0.3 m³/s. Find the head loss due to abrupt enlargement.

Solution:

Q 4. At a sudden contraction of water line from diameter of 350 mm to 250 mm and volumetric flow rate 0.03 m³/s. Determine the head loss due to contraction. Consider co-efficient of contraction as 0.58.

Solution:

Q 5. The dimension of 350 mm diameter horizontal pipe is suddenly reduced to diameter of 150 mm. If water is flowing at a rate of 0.05 m³/s and co-efficient of contraction is 0.48 then determine head loss between two sections of the pipe.

Solution: