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Wednesday, 14 November 2018

November 14, 2018

WHAT IS LAMINAR BOUNDARY LAYER ?

We were discussing the basic concept of streamline and equipotential linedimensional homogeneityBuckingham pi theoremdifference between model and prototypebasic principle of similitude i.e. types of similarityvarious forces acting on moving fluid and Boundary layer theory in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to start a new topic i.e. Laminar boundary layer, in the subject of fluid mechanics with the help of this post. 

Laminar boundary layer 

We will try to understand here the basic principle and meaning of laminar boundary layer and turbulent boundary layer. First, we will see here laminar boundary layer. 

Let us consider the flow of a fluid over a smooth thin and flat plate. Let us assume that this thin and flat plate is located parallel to the direction of fluid flow as displayed here in following figure. 

Let us consider that fluid is flowing with free stream velocity U and with zero pressure gradient on one side of the stationary plate. 
As we have already discussed that when a real fluid will flow over a solid body or a solid wall, the particles of fluid will adhere to the boundary and there will be condition of no-slip. We can also conclude that the velocity of the fluid particles, close to the boundary, will have equal velocity as of the velocity of boundary. 

As we have assumed that plate is stationary and therefore the velocity of fluid flow over the surface of plate will be zero. 

If we move away from the plate, the velocity of fluid particles will also be increasing. Velocity of fluid particles will be changing from zero at the surface of stationary boundary to the free stream velocity (U) of the fluid in a direction normal to the plate. 

Therefore, there will be presence of velocity gradient, due to variation of velocity of fluid particles, near the surface of the fluid. 

This velocity gradient will develop shear resistance and this shear resistance will retard the fluid. Therefore, fluid with free stream velocity (U) is retarded in the surrounding area of the solid surface of the plate and boundary layer region will be started at the sharp leading edge. 

Once we will go away from the sharp leading edge, retardation of the fluid will be increased and therefore boundary layer region increases with increase in the retardation of the fluid. The increase in the region of boundary layer with increase in the retardation of the fluid will also be termed as growth of boundary layer. 

Near the leading edge of the surface of the plate, where thickness will be small, the flow in the boundary layer will be laminar and this layer of the fluid will be termed as laminar boundary layer. 

Let us see the above figure. AE indicates the laminar boundary layer. 

Laminar zone 

Length of the plate from the leading edge up to which laminar boundary layer exists will be termed as laminar zone. AB indicates the laminar zone in above figure. 

Length of the plate from the leading edge up to which laminar boundary layer exists i.e. laminar zone will be determined with the help of following formula as mentioned here. 
Where, 
x = Distance from leading edge up to which laminar boundary layer exists 
U = Free stream velocity of the fluid 
v = Kinematic viscosity of the fluid 

Further we will go ahead to start a new topic i.e. Turbulent boundary layer, 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, 28 October 2018

October 28, 2018

BOUNDARY LAYER THEORY IN FLUID MECHANICS

We were discussing the basic concept of streamline and equipotential linedimensional homogeneityBuckingham pi theoremdifference between model and prototypebasic principle of similitude i.e. types of similarity and various forces acting on moving fluid in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to start a new topic i.e. Boundary layer theory, in the subject of fluid mechanics with the help of this post. 

Boundary layer theory 

When a real fluid will flow over a solid body or a solid wall, the particles of fluid will adhere to the boundary and there will be condition of no-slip. 

We can also conclude that the velocity of the fluid particles, close to the boundary, will have equal velocity as of the velocity of boundary. 

If we assume that boundary is stationary or velocity of boundary is zero, then the velocity of fluid particles adhere or very close to the boundary will also have zero velocity. 

If we move away from the boundary, the velocity of fluid particles will also be increasing. Velocity of fluid particles will be changing from zero at the surface of stationary boundary to the free stream velocity (U) of the fluid in a direction normal to the boundary. 

Therefore, there will be presence of velocity gradient (du/dy) due to variation of velocity of fluid particles. 

The variation in the velocity of the fluid particles, from zero at the surface of stationary boundary to the free stream velocity (U) of the fluid, will take place in a narrow region in the vicinity of solid boundary and this narrow region of the fluid will be termed as boundary layer. 

Science and theory dealing with the problems of boundary layer flows will be termed as boundary layer theory. 

According to the boundary layer theory, fluid flow around the solid boundary might be divided in two regions as mentioned and displayed here in following figure. 

First region 
A very thin layer of fluid, called the boundary layer, in the immediate region of the solid boundary, where the variation in the velocity of the fluid particles, from zero at the surface of stationary boundary to the free stream velocity (U) of the fluid, will take place. 

There will be presence of velocity gradient (du/dy) due to variation of velocity of fluid particles in this region and therefore fluid will provide one shear stress over the wall in the direction of motion.
Shear stress applied by the fluid over the wall will be determined with the help of following equation.

𝜏 = µ x (du/dy) 

Second region
Second region will be the region outside of the boundary layer. Velocity of the fluid particles will be constant outside the boundary layer and will be similar with the free stream velocity of the fluid. 

In this region, there will be no velocity gradient as velocity of the fluid particles will be constant outside the boundary layer and therefore there will be no shear stress exerted by the fluid over the wall beyond the boundary layer. 

Further we will go ahead to find out the some basic concepts and definitions in the respect of boundary layer theory 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, 14 October 2018

October 14, 2018

WATER HAMMER IN PIPES

We were discussing the basics of shear stress in turbulent flowminor head losses in pipe flowhydraulic gradient and total energy line, basic concept and working of syphonflow through pipes in seriesflow through pipes in parallel, flow through branched pipes, power transmission through pipes and flow through nozzle, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to see the concept of water hammer in pipes, in the subject of fluid mechanics, with the help of this post. 

Water hammer in pipes 

A sudden change of fluid flow rate in a large pipeline, due to sudden closing of valve or pump, may involve a great mass of water moving inside the pipe. 

Pressure will be increased in the pipe, due to sudden change of flow rate, greater than the normal static pressure in the pipe. 

Excessive pressure may fracture the pipe walls or cause other damage to the pipe line system. This phenomenon will be termed as water hammer phenomenon. 

Let us understand the basic concept of water hammer in pipes 

Let us consider, as displayed here in following figure, one tank filled with water and a pipe AB which is connected with water tank at one end. Let us think that water is filled in tank up to a height of H from the centre of pipe AB. 

One valve is provided at the other end of pipe in order to regulate the flow of fluid. Let us assume that valve is fully open and water is flowing at a velocity of V through the pipe. 

If valve is closed suddenly, momentum of flowing water will be destroyed and consequently a wave of high pressure will be set up. This wave of high pressure will be transmitted along the pipe with a velocity equal to the velocity of the sound wave and may create noise called knocking. 

There will be also hammering action over the pipe walls due to this wave of high pressure and hence this phenomenon will be termed as water hammer. 

Pressure rise due to water hammer will be dependent over the following factors as mentioned here 

  • Velocity of water flow in pipe 
  • Pipe length 
  • Time taken for closing the valve 
  • Elastic properties of the material of the pipe 

The sudden rise of pressure due to water hammer may be viewed as the result of the force developed in the pipe required to stop the flowing water column. Let us think that water column has a total mass M and it is changing its velocity at the rate of dV/dT. 

Now according to the Newton’s law of motion, we will have following equation as mentioned here
F = M dV/dT

If the velocity of entire water column is reduced to zero instantly then we will have following case 

F = M dV/dT = M (V0 – 0)/0 = ∞ 
F = Infinite 

The resulting force and hence pressure will be infinite, but it is not possible because mechanical valve will take a certain amount of time for complete closure of valve. 

In addition, neither the pipe walls nor the water column involved are perfectly rigid under large pressure. The elasticities of the material of the pipe and water column will also play an important role in the phenomenon of water hammer.  

Further we will go ahead to find out the concept of boundary layer theoryin 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  

October 14, 2018

FLOW THROUGH NOZZLE

We were discussing the basics of shear stress in turbulent flowminor head losses in pipe flowhydraulic gradient and total energy line, basic concept and working of syphonflow through pipes in seriesflow through pipes in parallel, flow through branched pipes and power transmission through pipes, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to see the concept of flow through nozzle, power transmission through nozzle, efficiency of power transmission through nozzle and we will also find here the condition for maximum transmission of power through nozzle, in the subject of fluid mechanics, with the help of this post. 

What is a nozzle? 

Nozzle is an engineering device which will accelerate the fluid and hence fluid velocity or kinetic energy of fluid will be increased while pressure of fluid will be reduced. 

Let us see the nozzle which is fitted at the end of a long pipe as displayed in following figure. Total energy at the end of pipe will be summation of pressure energy and kinetic energy. Total energy will be converted in to kinetic energy by fixing a nozzle at the end of pipe and therefore velocity of fluid flowing through nozzle will be increased. 

Nozzles are used to increase the velocity of flowing fluid and let us find here the few examples of application of nozzle as mentioned below. 
  1. In case of Pelton turbine, the nozzle is used at the end of pipe for increasing the velocity. 
  2. In case of fire extinguisher, a nozzle is used at the end of hose pipe for increasing the velocity of flow. 

Let us consider the following data from above figure. 
D = Diameter of the pipe 
L = Length of the pipe 
A = Area of the pipe 
V = Velocity of flow in pipe 
H = total head at the inlet of the pipe 
d = Diameter of nozzle at outlet 
v = Velocity of flow at outlet of nozzle 
a = Area of the nozzle at the outlet 
f = Co-efficient of friction for pipe 

Velocity of flow at the outlet of nozzle

Velocity of flow at the outlet of nozzle will be given by following formula as mentioned here

Discharge through the nozzle

Discharge through nozzle will be given by following formula as mentioned here 

Power transmission through nozzle

Power transmitted through nozzle will be given by following formula as mentioned here 

Efficiency of power transmission through nozzle

Efficiency of power transmission through nozzle will be determined with the help of following formula as mentioned here. 

Condition for maximum transmission of power through nozzle   

Above expression is the condition of maximum transmission of power through nozzle. We can conclude from above equation that power transmitted through a nozzle will be maximum when the head loss due to friction will be one-third of the total head at inlet of the pipe. 

Diameter of nozzle for maximum transmission of power through nozzle  
Diameter of nozzle for maximum transmission of power through nozzle will be given by following formula as mentioned here. 

Further we will go ahead to find out the concept of water hammer in the pipesin 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, 8 October 2018

October 08, 2018

POWER TRANSMISSION THROUGH PIPE


Now we will go ahead to see the concept of power transmission through pipes, efficiency of power transmission and we will also find here the condition for maximum transmission of power, in the subject of fluid mechanics, with the help of this post. 

Power transmission through pipes 

Power is transmitted through pipes by flowing water or other liquids flowing through them. Power transmitted through pipes will be dependent over the following factors as mentioned here. 
  • Weight of the liquid flowing through the pipe 
  • Total head available at the end of the pipe 

Now we will consider a tank with which a pipe AB is connected. Let us consider the following terms from figure. 
L = Length of the pipe 
D = Diameter of the pipe 
H = Total head available at the inlet of the pipe 
V= Velocity of flow in pipe 
hf = Loss of head due to friction 
f = Co-efficient of friction 

Power transmitted at the outlet of the pipe will be determined with the help of following formula as mentioned here. 

Efficiency of power transmission 

Efficiency of power transmission will be determined with the help of following formula as mentioned here. 

Condition for maximum transmission of power 

Now we will find here the condition for maximum transmission of power and it could be secured by differentiating the equation of power transmitted at the outlet of the pipe.

Above expression is the condition of maximum transmission of power. We can conclude from above equation that power transmitted through a pipe will be maximum when the head loss due to friction will be one-third of the total head at inlet. 

Maximum efficiency of transmission of power 

As we have already secured the equation of efficiency of power transmission through pipe and it is as mentioned here. 

η = (H- hf)/H 

Now we will use the value of hf in above equation to secure the maximum efficiency of transmission of power and we have following result as mentioned here. 

η = (H- hf)/H
η = (H- H/3)/H

η = 66.7 %

Further we will go ahead to find out the concept of flow through nozzlesin 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  

October 08, 2018

FLOW THROUGH BRANCHED PIPES

We were discussing the basics of shear stress in turbulent flowminor head losses in pipe flowhydraulic gradient and total energy line, basic concept and working of syphon, flow through pipes in series and also the concept of flow through pipes in parallel, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to see the, flow through branched pipes, in the subject of fluid mechanics, with the help of this post. 

Flow through branched pipes 

Flow through branched pipes could be defined as the fluid flow through a piping system where three or more than three reservoirs will be connected with each other with the help of pipes and will have one or more than one junctions. 

Now we will see here one figure as displayed here. Three reservoirs are connected here with the help of piping system and this piping system and reservoirs are connected with one single junction i.e. junction D. 

In various branched pipe flow problems, we will have some specifications of branched piping system and usually we will have to determine the flow of fluid through each pipe. 

Let us assume that we have information about the data for length of pipes, diameter of pipes and coefficient if friction for each pipe and we need to find the data for flow of water through each pipe. 

Assumption 

Let us assumed that reservoirs are very large and levels of water surface in the reservoirs are constant in order to secure the steady conditions in the pipes. We have also assumed that minor losses are very small and could be neglected. 

Principle and equations used for solving such problems 

We will have to recall the following principles and we will have to use following equations for securing the desired data for such problems. 
  • Continuity equation 
  • Bernoulli’s equation 
  • Darcy-Weisbach equation


Let us consider that three reservoirs are A, B and C and piping system will have pipe 1, pipe 2 and pipe 3 and a single junction D as displayed above in figure. 

Flow of water from reservoir A will take place to junction D and flow of water from junction D will be towards reservoir C. 

Flow of water from junction D to reservoir B will only possible if piezometric head at D will be more than the piezometric head at B. 

Let us consider the following terms from above figure. 

ZA, ZB and ZC = Datum head at reservoir A, B and C respectively 
V1, V2 and V3= Velocity of flow of water through pipe 1, 2 and 3 respectively 
L1, L2 and L3 = length of pipe 1, 2 and 3 respectively 
hf1, hf2 and hf3 = loss of head for pipe 1, 2 and 3 respectively 

Let us now use the concept of Bernoulli’s equation for flow from A to D and we will have following equation as mentioned here. 
Let us now use the concept of Bernoulli’s equation for flow from D to B and we will have following equation as mentioned here.

Let us now use the concept of Bernoulli’s equation for flow from D to C and we will have following equation as mentioned here.
Now we will use the concept of continuity equation and we will have following equation as mentioned here. 

Discharge through AD = Discharge through DB + Discharge through DC 

Now as we can see that we are having with above four equations and we have to secure the value of V1, V2 and V3

We will use above four equations to secure the value of flow of water through each pipe. 

Further we will go ahead to find out the basic concept of power transmission through pipesin 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