Recent Updates

Sunday, 15 July 2018

July 15, 2018

TIME REQUIRED TO EMPTY A TANK THROUGH AN ORIFICE


Now we will go ahead to find out the time of emptying a tank through an orifice fitted at its bottom, in the subject of fluid mechanics, with the help of this post. 

Let us consider we have one tank filled with a liquid up to a height of H1. Let us consider that one orifice is fixed at the bottom of the tank. We are interested here to determine the time taken in emptying the tank. 
Let us consider the following data from above figure.
A = Area of the tank
a = Area of the orifice
H1= Initial height of the liquid in the tank
H2= Final height of the liquid in the tank
T = Time taken in emptying the tank from H1 to H2

Time taken in emptying the tank from H1 to H2 will be given by following formula as mentioned here. 

Let us determine the time emptying the tank, H2 will be zero. 

This is the expression for the time emptying the tank fitted with orifice at its bottom. 

Now we will go ahead to find out the method to determine the time of emptying a hemispherical tank through an orifice at its bottom, in the subject of fluid mechanics, in 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

July 15, 2018

DISCHARGE THROUGH PARTIALLY SUBMERGED ORIFICE

We were discussing the basic difference between orifice and mouthpiececlassification of orifices and mouthpiecesadvantages and disadvantages of orificeshydraulic coefficients and also experimental process to determine the hydraulic coefficients, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to find out the flow through partially submerged orifice, in the subject of fluid mechanics, with the help of this post. 

Partially submerged orifice is the one which has its outlet side partially submerged under liquid. Therefore, partially submerged orifice will have two portions i.e. upper portion and lower portion.
Upper portion of partially submerged orifice will behave like an orifice discharging freely and lower portion of partially submerged orifice will behave like a submerged orifice. 
We must note it here that only a large orifice will behave like a partially submerged orifice. Partially submerged orifice is also called as partially drowned orifice. 

Total discharge through partially submerged orifice will be determined by adding the discharge through upper portion and lower portion of partially submerged orifice. 

Total discharge through partially submerged orifice = Discharge through free portion + Discharge through submerged portion 

We have already discussed the post related with discharge through fully submerged orifice and we have following equation of discharge as mentioned here.

We have also discussed the post related with discharge through large orifice. Discharge through free portion of the orifice will be equivalent to the discharge through large orifice. We have following equation of discharge through free portion of the orifice as mentioned here.

Now we will determine the total discharge through the partially submerged orifice. 
Q = Q1 + Q2
Where, 
H1 = Height of water above the top of the orifice on the upstream side
H2 = Height of water above the bottom of the orifice
H = Difference in level of water
b = Width of the orifice
Cd = Coefficient of discharge 

This is the expression for the flow or discharge through the partially submerged orifice. 

Now we will go ahead to find out the method to determine the time of emptying a tank through an orifice at its bottom, in the subject of fluid mechanics, in 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

July 15, 2018

DISCHARGE THROUGH FULLY SUBMERGED ORIFICE

We were discussing the basic difference between orifice and mouthpiececlassification of orifices and mouthpiecesadvantages and disadvantages of orificeshydraulic coefficients and also experimental process to determine the hydraulic coefficients, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to find out the flow through fully submerged orifices, in the subject of fluid mechanics, with the help of this post. 

Fully submerged orifice is one which has its whole of the outlet side submerged under liquid so that it discharges a jet of liquid in to the liquid of the same kind. 

Fully submerged orifice is also called as totally drowned orifice. Coefficient of contraction for fully submerged orifice will be equivalent to 1. 

Let us consider the following figure displaying the fully submerged orifice. Let us consider two points i.e. 1 and 2. Point 1 will be in the reservoir on the upstream side of the orifice and point 2 will be at the vena-contracta as displayed here in following figure. 

H1 = Height of water above the top of the orifice on the upstream side
H2 = Height of water above the bottom of the orifice
H = Difference in level of water
b = Width of the orifice
Cd = Coefficient of discharge 

Height of water above the centre of orifice on upstream side
= H1 + (H2- H1)/2
= (H1+ H2)/2 

Height of water above the centre of orifice on downstream side
= (H1+ H2)/2 – H 

Now we will apply the Bernoulli’s equation at 1 and 2. 

Considering the term z1 = z2, we will have following equation as mentioned here 
This is the expression for the flow or discharge through the fully submerged orifice. 

Now we will go ahead to find out the method to determine the discharge through a partially submerged orifice, in the subject of fluid mechanics, in 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, 13 July 2018

July 13, 2018

DISCHARGE THROUGH A LARGE RECTANGULAR ORIFICE


Now we will go ahead to find out the flow through large orifices, in the subject of fluid mechanics, with the help of this post. 

First we need to understand the basic concept and meaning of large orifice

If the head of liquid is less than the five times the depth of orifice, orifice will be termed as large orifice. Mathematically, we can write here the equation for large orifice. 

Head of liquid < 5 x depth of the orifice 

When we talk about the small orifice, the velocity over the entire cross-section of the jet will be constant and we can determine the flow or discharge through the orifice by using the equation as mentioned here. 
But in case of large orifice, the velocity over the entire cross-section of the jet will not be constant. Hence, we will not be able to determine the flow or discharge through the orifice by using the above mentioned equation. 

So how we will find out the flow or discharge through a large orifice. Let us derive here one expression for the flow or discharge through a large rectangular orifice. 

Flow or discharge through a large rectangular orifice

Let us consider a tank filled with liquid and also fixed with a large rectangular orifice. Let us think that rectangular orifice is fixed at one side of tank as displayed here in following figure.
Let us consider the following data from above figure. 

H1 = Height of liquid surface from the top edge of the rectangular orifice
H2 = Height of liquid surface from the bottom edge of the rectangular orifice
Depth of rectangular orifice = H2- H1
b = Breadth of the rectangular orifice
d = Depth of rectangular orifice
d = H2- H1
Cd = Coefficient of discharge 

Let us consider one elementary horizontal strip of depth dh at a depth of h, below the free surface of the liquid in the tank, in rectangular orifice as displayed in above figure. 

Discharge through elementary horizontal strip,
dQ = Coefficient of discharge x Area of elementary horizontal strip x Velocity

In order to secure the expression for the flow or discharge through the entire rectangular orifice, we will integrate the above equation between the limits H1 and H2.

This is the expression for the flow or discharge through the large rectangular orifice. 

Now we will go ahead to find out the method to determine the discharge through a fully submerged orifice, in the subject of fluid mechanics, in 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, 11 July 2018

July 11, 2018

EXPERIMENTAL DETERMINATION OF HYDRAULIC COEFFICIENTS

We were discussing the basic difference between orifice and mouthpiececlassification of orifices and mouthpiecesadvantages and disadvantages of orifices and also hydraulic coefficients, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to find out the experimental process to determine the hydraulic coefficients, in the subject of fluid mechanics, with the help of this post. 

Experimental process to determine the hydraulic coefficients 

Coefficient of discharge (Cd)

Let us consider a tank filled with water and fitted with an orifice of area a as displayed here in following figure. Let us think that water is flowing through the orifice under a constant head H. 

Water flowing through the orifice will be collected in a measuring tank for a time t and we will also measure the height of water collected in the measuring tank in time t. 
Image: Tank with orifice and measuring tank

Actual discharge and theoretical discharge through the orifice will be determined by the following formulas as mentioned here. 

Coefficient of discharge (CV)

Let us think that water, flowing through the orifice, is developing a liquid jet whose cross-sectional area is smaller than the cross-sectional area of the circular orifice. Area of liquid jet is decreasing and area is minimum at section CC. 

Section CC will be approximately at a distance of half of diameter of the circular orifice. At section CC, the streamlines are straight and parallel with each other and perpendicular to the plane of the orifice. This section CC will be termed as Vena-contracta. 

Beyond the section CC, liquid jet diverges and will be attracted towards the downward direction due to gravity. 

Let us consider that a liquid particle which is at vena-contracta at any time and takes the position at P along the jet in time t. Let us assume following data as mentioned here. 

x = Horizontal distance travelled by the particle in time t
y = Vertical distance between P and section CC
V = Actual velocity of jet at vena-contracta 

Coefficient of discharge (CV)

Let us recall the relation between hydraulic coefficient and we will secure the value of coefficient of contraction by using the value of coefficient of discharge and coefficient of velocity. 

Now we will go ahead to find out the method to determine the flow through a large orifice, in the subject of fluid mechanics, in 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

Tuesday, 10 July 2018

July 10, 2018

RELATION BETWEEN HYDRAULIC COEFFICIENTS

We were discussing the basic difference between orifice and mouthpiececlassification of orifices and mouthpieces, advantages and disadvantages of orifices and also hydraulic coefficients, in the subject of fluid mechanics, in our recent posts. 

Now we will go ahead to find out the relation between hydraulic coefficients, in the subject of fluid mechanics, with the help of this post. 

We will first briefly explain the various types of hydraulic coefficients and after that we will secure here the relation between hydraulic coefficients. 

Co-efficient of velocity, CV

Co-efficient of velocity is basically defined as the ratio of actual velocity of liquid jet at vena-contracta to the theoretical velocity of the liquid jet. 

Co-efficient of velocity is denoted by CV and will be given as mentioned here. 

Co-efficient of velocity = Actual velocity of liquid jet at vena-contracta / theoretical velocity 

Co-efficient of contraction, CC

Co-efficient of contraction is basically defined as the ratio of area of liquid jet at vena-contracta to the area of the orifice. 

Co-efficient of contraction is denoted by CC and will be given as mentioned here. 

Co-efficient of contraction = Area of liquid jet at vena-contracta / Area of the orifice 

Co-efficient of discharge, Cd

Co-efficient of discharge is basically defined as the ratio of actual discharge from an orifice to the theoretical discharge from the orifice. 

Co-efficient of discharge is denoted by Cd and will be given as mentioned here 

Co-efficient of discharge = Actual discharge from an orifice / Theoretical discharge from the orifice 

Relation between hydraulic coefficients

We can secure the relation between hydraulic coefficients by elaborating the formula of coefficient of discharge. 

As we have seen above that
Co-efficient of discharge = Actual discharge from an orifice / Theoretical discharge from the orifice 

Cd = Q/Qth
Where,
Q = Actual discharge from an orifice
Qth = Theoretical discharge from an orifice
Q = Actual velocity x Actual area
Qth = Theoretical velocity x Theoretical area 

Co-efficient of discharge = (Actual velocity x Actual area) / (Theoretical velocity x Theoretical area) 
Co-efficient of discharge = (Actual velocity / Theoretical velocity) X (Actual area / Theoretical area) 
Co-efficient of discharge = Co-efficient of velocity X Co-efficient of contraction 

Cd = CV x C

Therefore, we can also define the co-efficient of discharge as the product of Co-efficient of velocity and Co-efficient of contraction. 

Now we will go ahead to find out the method to determine the various types of hydraulic co-efficients, in the subject of fluid mechanics, in 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