Recent Updates

Sunday, 23 September 2018

September 23, 2018

FLOW THROUGH SYPHON PIPE

We were discussing the concept of laminar and turbulent flowReynolds experimentfrictional loss in pipes, derivation of expression for loss of head due to friction in pipesco-efficient of friction in terms of shear stressbasics of shear stress in turbulent flow,  minor head losses in pipe flow and also the  concept of hydraulic gradient and total energy line, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to see the basic concept and working of syphon, in the subject of fluid mechanics, with the help of this post. 

Flow through Syphon 

Syphon is basically defined as a long pipe which is basically used to transfer liquid from a reservoir at a higher elevation to another reservoir at a lower level when the two reservoirs are separated by a hill or high level ground. 

Let us consider that we have two reservoirs i.e. A and B separated by a hill as displayed here in following figure. A tube i.e. syphon is connected with reservoir A and reservoir B. Point C which will be at the highest of the syphon will be termed as summit. 

Flow through the syphon will be only possible if pressure at the point C is below than the atmospheric pressure. Pressure at point C will be less than atmospheric pressure because point C is above the free surface of the water in the tank A. This difference in pressure will cause the flow of liquid through the syphon. 

Theoretically pressure at point C may be reduced to -10.3 m of water, but in actual practice this pressure will be -7.6 m of water. 

Application of Syphon 

Syphon is basically used in following cases. 
  1. To transport water from one reservoir to another reservoir separated by a hill or ridge. 
  2. To take out the liquid from a tank, which is not having any outlet. 
  3. To empty a channel not provided with any outlet sluice. 

Further we will go ahead to find out the flow through pipes in series or flow through compound pipes, in the subject of fluid mechanics, with the help of our next post. 

Do you have any suggestions? Please write in comment box. 

Reference: 

Fluid mechanics, By R. K. Bansal 
Image courtesy: Google  

Also read  

Monday, 17 September 2018

September 17, 2018

HYDRAULIC GRADIENT LINE AND TOTAL ENERGY LINE

We were discussing the concept of laminar and turbulent flowReynolds experimentfrictional loss in pipes, derivation of expression for loss of head due to friction in pipesco-efficient of friction in terms of shear stress, basics of shear stress in turbulent flow and also the minor head losses in pipe flow, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to see the concept of hydraulic gradient and total energy line, in the subject of fluid mechanics, with the help of this post. 

Concepts of hydraulic gradient line and total energy line will be quite useful when we analyze the problems of fluid flow through pipes. Now we will understand here the concept of hydraulic gradient line and total energy line. 

Hydraulic gradient line and total energy line are the graphical representation for the longitudinal variation in piezometric head and total head. 

Hydraulic gradient line 

Hydraulic gradient line is basically defined as the line which will give the sum of pressure head and datum head or potential head of a fluid flowing through a pipe with respect to some reference line. 

Hydraulic gradient line = Pressure head + Potential head or datum head 
H.G.L = P/ρg + Z 
Where, 
H.G.L = Hydraulic gradient line 
P/ρg = Pressure head 
Z = Potential head or datum head 

Total Energy Line 

Total energy line is basically defined as the line which will give the sum of pressure head, potential head and kinetic head of a fluid flowing through a pipe with respect to some reference line. 

Total energy line = Pressure head + Potential head + Kinetic head 
H.G.L = P/ρg + Z + V2/2g 
Where, 
T.E.L = Total energy line 
P/ρg = Pressure head 
Z = Potential head or datum head 
V2/2g = Kinetic head or velocity head 

Relation between hydraulic gradient line and total energy line 

H.G.L = E.G.L - V2/2g 

Let us see the following figure, there is one reservoir filled with water and also connected with one pipe of uniform cross-sectional diameter. 

Hydraulic gradient and energy lines are displayed in figure. 

At Velocity V = 0, Kinetic head will be zero and therefore hydraulic gradient line and energy gradient line will be same. 

At Velocity V = 0, EGL = HGL 

Further we will go ahead to find out the basic concept of flow through syphon, in the subject of fluid mechanics, with the help of our next post.  

Do you have any suggestions? Please write in comment box.  

Reference: 

Fluid mechanics, By R. K. Bansal 
Image courtesy: Google  

Also read  

Sunday, 9 September 2018

September 09, 2018

MINOR HEAD LOSSES IN PIPE FLOW

We were discussing the concept of laminar and turbulent flowReynolds experimentfrictional loss in pipes, derivation of expression for loss of head due to friction in pipesco-efficient of friction in terms of shear stress and also the basics of shear stress in turbulent flow, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to see minor head losses in pipe flow, in the subject of fluid mechanics, with the help of this post. 

Even we have already seen earlier one post based on major and minor losses in pipes earlier. Today, we will be interested to secure the detailed information about the minor head losses in pipe flow only. 

Minor head losses in pipe flow 
Head loss in pipe flow system due to viscous effect i.e. due to friction will be termed as major head loss and will be indicated by h L-Major

Head loss in pipe flow system due to various piping components such as valves, fittings, elbows, contractions, enlargement, tees, bends and exits will be termed as minor head loss and will be indicated by h L-Minor

We will see here the following cases of minor head losses in pipe flow. 
  1. Loss of head due to sudden enlargement 
  2. Loss of head due to sudden contraction 
  3. Loss of head at the entrance of a pipe 
  4. Loss of head at the exit of a pipe 
  5. Loss of head due to an obstruction in a pipe 
  6. Loss of head due to bend in the pipe 
  7. Loss of head in various pipe fitting 

If we want to evaluate the loss of head in long pipe, above losses will be small as compared with major loss of head i.e. loss of head due to friction. That’s why above losses are considered as minor losses and we can also neglect such losses for long pipe. 

But if we want to evaluate the loss of head in small pipe, above losses must be considered as such losses will be comparable with the loss of head due to friction. 

Loss of head due to sudden enlargement 

Let us assume that a liquid is flowing through a pipe which has a sudden enlargement as displayed here in following figure. 
Loss of head due to sudden enlargement will be given by following equation. 
Where, 
V1 = Velocity of fluid flow at section 1-1 
V2 = Velocity of fluid flow at section 2-2 
he = Loss of head due to sudden enlargement 
g = Acceleration due to gravity 

Loss of head due to sudden contraction 

Let us assume that a liquid is flowing through a pipe which has a sudden contraction as displayed here in following figure. 
Loss of head due to sudden contraction will be given by following equation. 
Where, 
V2 = Velocity of fluid flow at section 2-2 
hC = Loss of head due to sudden contraction 
k = Minor loss co-efficient 

Value of k, for different fittings or pipe components, might be secured from the post “major and minor losses in pipes”. 

Loss of head at the entrance of a pipe 

Loss of head at the entrance of a pipe is the loss of head when a liquid enters a pipe which is connected with a large tank or reservoir. 

Loss of head at the entrance of a pipe will be given by the following equation as mentioned here. 
Where, 
V = Velocity of fluid flow in the pipe 
hi = Loss of head at the entrance of a pipe 

Loss of head at the exit of a pipe 

Loss of head at the exit of a pipe will be given by the following equation as mentioned here. 
Where, 
V = Velocity of fluid flow at the outlet of pipe 
hO = Loss of head at the exit of a pipe 

Loss of head due to an obstruction in a pipe 

Whenever there will be an obstruction in a pipe, loss of head will take place due to the reduction of area of the cross-section of the pipe at the location where obstruction is located. There will be sudden enlargement of the flow area beyond the obstruction and due to this reason there will be loss of head. 

Let us consider a pipe of cross-sectional area A with an obstruction as displayed here in following figure. 
Loss of head due to an obstruction in a pipe will be given by the following equation as mentioned here. 

Where, 
V = Velocity of fluid flow in the pipe 
A = Area of pipe 
a = maximum area of the obstruction 
CC = Co-efficient of contraction 

Loss of head due to bend in pipe 

Loss of head due to bend in pipe will be given by the following equation as mentioned here. 
Where, 
V = Velocity of fluid flow 
hb = Loss of head due to bend in pipe 
k = Co-efficient of bend 

Value of k will be dependent over angle of bend, radius of curvature of bend and diameter of the pipe. 

Loss of head in various pipe fitting 

Loss of head in various pipe fitting will be given by the following equation as mentioned here. 
Where, 
V = Velocity of fluid flow 
hb = Loss of head due to bend in pipe 
k = Co-efficient of bend 

Further we will go ahead to see the basics of hydraulic gradient and total energy line, in the subject of fluid mechanics, with the help of our next post. 

Do you have any suggestions? Please write in comment box. 

Reference: 

Fluid mechanics, By R. K. Bansal 
Image courtesy: Google  

Also read  

Friday, 7 September 2018

September 07, 2018

SHEAR STRESS IN TURBULENT FLOW

We were discussing the concept of laminar and turbulent flow, Reynolds experiment, frictional loss in pipes, derivation of expression for loss of head due to friction in pipes and  also co-efficient of friction in terms of shear stress, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to see the basics of shear stress in turbulent flow, in the subject of fluid mechanics, with the help of this post. 

As we know that shear stress in case of viscous flow is provided by Newton’s law of viscosity and it is as mentioned here. 

Similarly, J Boussinesq has explained the turbulent shear as mentioned here 
Where,
τt = Shear stress due to turbulence
η = Eddy viscosity
ữ = Average velocity at a distance y from boundary 

Ratio of eddy viscosity and mass density will be called as kinematic eddy viscosity and will be displayed by ε. We can write kinematic eddy viscosity as mentioned here. 

ε = η /ρ 

If shear stress due to viscous flow will also be considered then we will have following equation for the total shear stress and it is as mentioned here. 
For laminar flow, η = 0 

For any other case, the value of η might be several thousand times the value of μ. 

Further we will go ahead to see the minor head losses in pipe flow, in the subject of fluid mechanics, with the help of our next post. 

Do you have any suggestions? Please write in comment box. 

Reference: 

Fluid mechanics, By R. K. Bansal  
Image courtesy: Google  

Also read 

Thursday, 6 September 2018

September 06, 2018

COEFFICIENT OF FRICTION IN TERMS OF SHEAR STRESS

We were discussing the concept of laminar and turbulent flow, Reynolds experiment, frictional loss in pipes and also the derivation of expression for loss of head due to friction in pipes, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to find out the expression for the coefficient of friction in terms of shear stress, in the subject of fluid mechanics, with the help of this post. 

Expression for coefficient of friction in terms of shear stress 

We will determine here the expression for coefficient of friction in terms of shear stress.
Let us consider that fluid is flowing through a uniform horizontal pipe with steady flow as displayed here in following figure. 

Now we will assume two sections of pipe i.e. section 1-1 and section 2-2. 
Let us consider the following terms to derive the required expression of coefficient of friction in terms of shear stress. 
P1 = Pressure intensity at section 1-1 
V1 = Velocity of flow at section 1-1 
P2 = Pressure intensity at section 2-2 
V2 = Velocity of flow at section 2-2 
L = Length of pipe between section 1-1 and section 2-2 
hf = Loss of head due to friction 
d = Diameter of the pipe 
A = Area of pipe = (П /4) x d
τ0 = Shear stress 
F1= Force due to shear stress (τ0
F1 = τ0 x П d x L 
Now we will apply the Bernoulli’s equations between section 1-1 and section 2-2. 
 
Because, 
Pipe is horizontal and hence, Z1 = Z
Diameter of uniform pipe is same at both sections and hence, V1 = V2  
Where hf  is the Darcy-Weisbach equation which is commonly used to determine the loss of head due to friction in pipes. 

Now we will write here the equation of equilibrium of forces
P1A – P2A – F1 =0
(P1 – P2)A = F1
(P1 – P2)A = Force due to shear stress (τ0)
(P1 – P2) (П /4) x d2 = τ0 x П d x L
(P1 – P2) = 4 τ0 x L/d

Now we will equate the expression for (P1 – P2) and we will have following equation as mentioned here. 
Above equation is the expression for the coefficient of friction in terms of shear stress. 

Further we will go ahead to derive the expression of shear stress in turbulent flow, in the subject of fluid mechanics, with the help of our next post. 

Do you have any suggestions? Please write in comment box. 

Reference: 

Fluid mechanics, By R. K. Bansal 
Image courtesy: Google 

Also read 

Wednesday, 5 September 2018

September 05, 2018

DERIVE AN EXPRESSION FOR THE LOSS OF HEAD DUE TO FRICTION IN PIPES

We were discussing the concept of laminar and turbulent flow, Reynolds experiment and also frictional loss in pipes, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to find out the derivation of expression for loss of head due to friction in pipes, in the subject of fluid mechanics, with the help of this post. 

Expression for loss of head due to friction in pipes

As we are quite aware that when liquid flows through a pipe, velocity of the liquid layer adjacent to the pipe wall will be zero. Velocity of liquid will be increasing from the pipe wall and therefore there will be produced velocity gradient and shear stress in the liquid due to viscosity of liquid. 

This viscous action will cause the loss of energy which will be termed as frictional loss or loss of head due to friction. 

Now we will determine the expression for loss of head due to friction in pipes 

Let us consider that fluid is flowing through a uniform horizontal pipe with steady flow as displayed here in following figure. 

Now we will assume two sections of pipe i.e. section 1-1 and section 2-2. 
Let us consider the following terms to derive the required expression of loss of head due to friction in pipe. 

P1 = Pressure intensity at section 1-1 
V1 = Velocity of flow at section 1-1 
P2 = Pressure intensity at section 2-2 
V2 = Velocity of flow at section 2-2 
L = Length of pipe between section 1-1 and section 2-2
f ' = Frictional resistance per unit wetted area per unit velocity  
hf = Loss of head due to friction
A = Area of the pipe
d = Diameter of the pipe  

Now we will apply the Bernoulli’s equations between section 1-1 and section 2-2. 
Because, 

Pipe is horizontal and hence, Z1 = Z2
Diameter of uniform pipe is same at both sections and hence, V1 = V2

Above equation of loss of head due to friction i.e. hf shows that there will be loss of head due to friction or intensity of pressure will be dropped in the direction of flow. 

Frictional resistance = Frictional resistance per unit wetted area per unit velocity x wetted area x velocity 2

F1 = f ' x ПdL x V2
F1 = f ' x P x L x V2

Where, 

P = Perimeter = Пd 

Now we will consider the forces acting on the fluid between section 1-1 and section 2-2

Pressure force at section 1-1 = P1 x A
Pressure force at section 2-2 = P2 x A

Let us write here the equation of equilibrium of forces 

Where,
P/A = Пd / (Пd2/4)
P/A = 4/d
And
f '/ρg = f/2, where f will be called as co-efficient of friction 

Above equation will be called as Darcy-Weisbach equation and commonly used to determine the loss of head due to friction in pipes. 

There is one more expression of loss of head due to friction in pipes and this expression could be written as mentioned here. 

Where, f is the friction factor  

Further we will go ahead to derive the expression for coefficient of friction in terms of shear stress, in the subject of fluid mechanics, with the help of our next post. 

Do you have any suggestions? Please write in comment box. 

Reference: 

Fluid mechanics, By R. K. Bansal 
Image courtesy: Google 

Also read