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Friday, 4 January 2019

January 04, 2019

VELOCITY OF SOUND IN ADIABATIC PROCESS

Till now we were discussing the various concepts and equations such as continuity equationEuler equationBernoulli’s equation and momentum equation for incompressible fluid flow. In same way we have also discussed above equations for compressible fluid flow. 

We have already seen the derivation of continuity equationBernoulli’s equationmomentum equation and velocity of sound in an isothermal process for compressible fluid flow in our previous posts. 

We will start here our discussion about the compressible fluid flow with the derivation of expression for velocity of sound in an adiabatic process. 

Expression for velocity of sound in adiabatic process 

Before understanding the process to derive the expression for velocity of sound in isothermal process, we must have to study our previous post which shows the derivation of velocity of sound wave in a fluid and velocity of sound in terms of bulk modulus

For an adiabatic process, heat must be constant. 
PVγ = Constant 
P/ ργ = Constant 
P/ ργ = Constant = C
P ρ-γ = C1  
Let us differentiate the above equation and we will have following equation as mentioned here.  
d (P ρ-γ) = 0 
P (-γ) ρ-γ-1d ρ + ρdP = 0  

We will now divide the above equation with ρ and we will have 
- γ P ρ-1d ρ + dP = 0 
γ P ρ-1d ρ = dP 
γ P/ρ = dP /d ρ 
dP /d ρ = γ RT 

Let us recall the expression for the velocity of sound wave in a fluid and we can write the above equation as mentioned here. 
Where, C is the velocity of sound 

Therefore we will have following equation, as mentioned here, which shows the expression for velocity of sound in adiabatic process. 

Further we will go ahead to find out the basic concept of stagnation pressure, 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, 27 December 2018

December 27, 2018

VELOCITY OF SOUND IN ISOTHERMAL PROCESS

Till now we were discussing the various concepts and equations such as continuity equationEuler equationBernoulli’s equation and momentum equation for incompressible fluid flow. In same way we have also discussed above equations for compressible fluid flow. 

We have already seen the derivation of continuity equationBernoulli’s equation and momentum equation for compressible fluid flow in our previous posts. We will start here our discussion about the compressible fluid flow with the derivation of expression for velocity of sound in an isothermal process. 

Expression for velocity of sound in isothermal process 

Before understanding the process to derive the expression for velocity of sound in isothermal process, we must have to study our previous post which shows the derivation of velocity of sound wave in a fluid and velocity of sound in terms of bulk modulus

For an isothermal process, temperature must be constant. 

As we know the following equation, as mentioned here, we will use this equation to derive the velocity of sound in an isothermal process. 

PV = mRT 
PV/m = RT 
P/ρ = RT 

As we are discussing here the case of an isothermal process and therefore the term RT will be constant and hence we can write the above equation as mentioned here. 

P/ρ = Constant = C1
P ρ-1= C

Let us differentiate the above equation and we will have following equation as mentioned here. 

d (P ρ-1) = 0
- P ρ-2d ρ + ρ-1dP = 0 

We will now divide the above equation with ρ-1 and we will have 

- P ρ-1d ρ + dP = 0
P ρ-1d ρ = dP
P/ ρ = dP /d ρ
dP /d ρ = RT 

Let us recall the expression for the velocity of sound wave in a fluid and we can write the above equation as mentioned here. 
Where, C is the velocity of sound 

Therefore we will have following equation, as mentioned here, which shows the expression for velocity of sound in isothermal process. 

Further we will go ahead to find out the velocity of sound in an adiabatic process, 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, 21 December 2018

December 21, 2018

VELOCITY OF SOUND IN TERMS OF BULK MODULUS

Till now we were discussing the various concepts and equations such as continuity equationEuler equationBernoulli’s equation and momentum equation for incompressible fluid flow. In same way we have also discussed above equations for compressible fluid flow. 

We have already seen the derivation of continuity equationBernoulli’s equation and momentum equation for compressible fluid flow in our previous posts. We will start here our discussion about the compressible fluid flow with the derivation of expression for velocity of sound in terms of bulk modulus. 

Expression for velocity of sound in terms of bulk modulus 
Before understanding the process to derive the expression for velocity of sound in terms of bulk modulus, we must have to study our previous post which shows the derivation of velocity of sound wave in a fluid

As we are interested here to find the expression for velocity of sound in terms of bulk modulus, therefore let us brief here first the term bulk modulus. 

Bulk modulus of elasticity of a substance is basically defined as the ratio of compressive stress or hydro static stress to volumetric strain and it will be displayed by the symbol K. 

Bulk modulus of a substance provides the information about the resistance of substance to the uniform pressure. In simple, we can also say that Bulk modulus of a substance provides the information about the compressibility of that substance. 

Bulk modulus, K = - dP/ (dV/V) 

Where, 
dP = Change in pressure 
dV/V = Volumetric strain 

As we know that according to the conservation of mass, we will have following equation as mentioned here. 

Density x Volume = Constant 
ρ x V = C, where C is constant 

Let us differentiate the above equation and we will have following equation after differentiation 

d (ρ x V) = 0 
ρ d V + V dρ = 0 
ρ d V = - V dρ 
dV/V = -(dρ/ ρ) 

Now we will use the above value of volumetric strain in equation of bulk modulus and we will have 

K = ρ x dP/dρ 
dP/dρ = K/ ρ 

Let us recall the expression for the velocity of sound wave in a fluid and we can write the above equation as mentioned here. 

C2 = K/ ρ 

Because, 
dP/dρ = C2
Where, C is the velocity of sound 

Therefore we will have following equation, as mentioned here, which shows the expression for velocity of sound in terms of bulk modulus. 

Further we will go ahead to find out the expression for velocity of sound in isothermal process, 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, 19 December 2018

December 19, 2018

EXPRESSION FOR VELOCITY OF SOUND WAVE IN A FLUID

We were discussing the basics of drag force & lift force and drag and lift coefficient in the subject of fluid mechanics, in our recent posts. Before going in detail discussion about compressible flow, we must have basic knowledge about various equations associated with the compressible flow. 

Till now we were discussing the various concepts and equations such as continuity equationEuler equationBernoulli’s equation and momentum equation for incompressible fluid flow. In same way we have also discussed above equations for compressible fluid flow. 

We have already seen the derivation of continuity equationBernoulli’s equation and momentum equation for compressible fluid flow in our previous posts. We will start here our discussion about the compressible fluid flow with the derivation of expression for velocity of sound wave in a fluid.  

Expression for velocity of sound wave in a fluid 

Compressible flow is basically defined as the flow where fluid density could be changed during flow. 
The disturbance in a solid, liquid or gas will be transmitted from one point to other. Velocity with which the disturbance will be transmitted will be dependent over the distance between the molecules of the medium. 

If we will consider the case of solid, molecules will be very much closely packed and therefore the disturbance will be transmitted instantaneously. 

If we will consider the case of liquid, molecules will be relatively apart and therefore disturbance will be transmitted from one molecule to the next molecule. Velocity of disturbance will be less as compared to the velocity of disturbance in case of solid. 

If we will consider the case of gases, molecules will be relatively apart and disturbance will be transmitted from one molecule to the next molecule. There will be some distance between two adjacent molecules. Each molecule will have to travel a certain distance before it can transmit the disturbance. 

Therefore, velocity of disturbance in fluids (liquid and gas) will be less as compared to the velocity of disturbance in case of solids. 

This disturbance will develop the pressure waves in fluids. These pressure waves will travel with velocity of sound waves in all the directions. Let us consider here one dimensional case only. 

Following figure, displayed here, indicates the condition of one-dimensional propagation of the pressure waves. Let us consider a cylinder of having uniform cross-sectional area attached with a piston as displayed in following figure. 

Let us assume that cylinder is filled with a compressible fluid and compressible fluid is at rest initially. 

If we apply a force through the piston in right direction, force will develop a pressure as force will be applied uniformly. Due to the application of force, piston will move by a certain distance let us say x towards right direction as displayed here in following figure. 
Due to the application of this small amount of force, there will be generation of pressure waves inside the cylinder and pressure will be applied over the fluid contained inside the cylinder. 

In simple, we can say that disturbance will be created inside the fluid due to the movement of piston by a distance x and this disturbance will move in the form of pressure wave inside the cylinder with a velocity of sound wave as discussed above. 

Let us consider following terms from above figure as mentioned here. 

x = Distance of piston from initial position 
L = Distance of sound wave from initial position 
P = Pressure applied over the piston at initial position 
P + dP = Pressure inside the cylinder at final position 
ρ = Density of the fluid at initial position 
ρ + d ρ = Density of the fluid at final position 
dt = Small amount of time taken by piston to travel distance x 
V = Velocity of piston 
C = Velocity of pressure wave or sound wave travelling in the fluid 
Distance travelled by the piston in time dt from initial position, x = v.dt 
Distance travelled by pressure wave or sound wave in time dt from initial position, L = C. dt 

Let us recall the law of conservation of mass 

Initial mass = final mass 

As we know that mass will be equal to the product of density and volume. We can write here the equation as mentioned here. 

Mass = Density x Volume 
Mass = Density x Area x Length 
Mass at initial position, M1 = ρ A L = ρ A C. dt 
Mass at final position, M2 = (ρ + dρ) A (L-x) = (ρ + dρ) A (C. dt- V. dt) 
Mass at final position, M2 = (ρ + dρ) A. dt (C - V) 

Now, considering the conservation of mass, we will have following equation as mentioned here.
Mass at initial position, M1 = Mass at final position, M
ρ A C. dt = (ρ + dρ) A. dt (C - V) 
ρ C = (ρ + dρ) (C - V) 
ρ C = ρ C - ρ V + C. dρ - V dρ 

Here, the term dρ will be very small and velocity of piston will also be very small and therefore product V dρ could be neglected. 

ρ V = C. dρ
C = ρ V/ dρ -------------------------------- Eq 1 

Let us determine the force at initial position of piston and final position of piston as mentioned here 

F1 = P.A 
F2 = (P + dP). A 
Change in force, ΔF = (P + dP). A - P.A = dP. A 

As we know that force could be written by recalling the Newton’s second law of motion and we will have the following equation 

Force = Mass x (Rate of change of velocity) 
dP. A = (ρ A C. dt) [(V-u)/dt] 

As we know that, V is the final velocity of the piston and u is the initial velocity of the piston and initial velocity of piston will be zero. 

dP. A = (ρ A C. dt) V/dt
 dP = ρ C V
C = dP / (ρ V) -------------------------------- Eq 2

We will have following equation by multiplying the equation 1 and equation 2
C2 = (ρ V/ dρ) x [dP / (ρ V)] = dρ/ dP
C2 = dρ/ dP 
Above equation represent the velocity of sound wave in fluid. 

Further, we will go ahead to find out the expression for velocity of sound in terms of bulk modulus, 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, 12 December 2018

December 12, 2018

MOMENTUM EQUATION FOR COMPRESSIBLE FLOW

We were discussing the basics of drag force &lift force and drag and lift coefficient in the subject of fluid mechanics, in our recent posts. 

We will discuss now a new topic i.e. compressible fluid flow, in the subject of fluid mechanics, with the help of this post. Before going in detail discussion about compressible flow, we must have basic knowledge about various equations associated with the compressible flow. 

Till now we were discussing the various concepts and equations such as continuity equation, Euler equation, Bernoulli’s equation and momentum equation for incompressible fluid flow. In same way we will have to discuss above equations for compressible fluid flow too. 

We have already seen the derivation of continuity equation and Bernoulli’s equation for compressible fluid flow in our previous post. We will start here our discussion about the compressible fluid flow with the basics of momentum equation for compressible fluid flow. 

Compressible flow is basically defined as the flow where fluid density could be changed during flow. 

Momentum equation for compressible fluid flow

The momentum per second of a flowing fluid will be equal to the product of mass per second and the velocity of flow. 

The momentum per second of a flowing fluid = Product of mass per second x Velocity of flow 

The momentum per second of a flowing fluid = ρ A V x V
ρ A V = Mass per second 

As we have already seen during discussion of continuity equation, term ρ A V will be constant at each section of flow. Therefore, the momentum per second of a flowing fluid will be equal to the product of mass per second which is a constant quantity and the velocity of flow. 

Therefore, we can say that momentum per second will not be affected due to compressible effect as term ρ A V is constant. In simple, we can say that momentum equation for incompressible and compressible fluid will be same. 

Momentum equation for compressible fluid for any direction will be given as mentioned here 

Momentum equation is based on the law of conservation of momentum or on the momentum principle. 

According to the law of conservation of momentum, net force acting on a fluid mass will be equivalent to the change in momentum of flow per unit time in that direction. 

Net force in a direction = Rate of change of momentum in same direction 
Net force in a direction = Mass per second x change of velocity 
Net force in a direction = ρ AV x [V2-V1

Further we will go ahead to find out the pressure wave and sound wave, 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, 10 December 2018

December 10, 2018

BERNOULLI’S EQUATION FOR COMPRESSIBLE FLUID FLOW

We were discussing the basics of drag force &lift force and drag and lift coefficient in the subject of fluid mechanics, in our recent posts. 

We will discuss now a new topic i.e. compressible fluid flow, in the subject of fluid mechanics, with the help of this post. Before going in detail discussion about compressible flow, we must have basic knowledge about various equations associated with the compressible flow. 

Till now we were discussing the various concepts and equations such as continuity equation Euler equation, Bernoulli’s equation and momentum equation for in-compressible fluid flow. In same way we will have to discuss above equations for compressible fluid flow too. 

We have already seen the derivation of continuity equation for compressible fluid flow in our previous post. We will start here our discussion about the compressible fluid flow with the derivation of Bernoulli’s equation for compressible fluid flow. 

Compressible flow is basically defined as the flow where fluid density could be changed during flow. 

Bernoulli’s equation for compressible fluid flow 

We will derive the Bernoulli’s equation for compressible fluid flow with the help of Euler’s equation. 

So, let us recall the Euler’s equation as mentioned here. 

In case of in-compressible fluid flow, the density of fluid will be constant and therefore the integral of dp/ρ will be equivalent to the P/ρ. 

We are interested here for compressible fluid flow and therefore the density of fluid will not be constant and therefore the integral of dp/ρ will not be equivalent to the P/ρ. 

In case of compressible fluid flow, the value of ρ will be changing and hence value of p will also be changing. Change in ρ and p will be dependent over the types of process during compressible fluid flow. 

We will now consider the various types of processes where pressure and temperature will be related with each other. We will secure the value of ρ in terms of p with the help of equations of these processes and we will use the value of ρ in above equation to secure the result of integral of dp/ρ. 

Bernoulli’s equation for isothermal process and for adiabatic process will be different. Let us first consider a basic process i.e. isothermal process. 

Bernoulli’s equation for compressible fluid for an isothermal process

We will secure here the value of ρ in terms of p with the help of following equation of isothermal process. 

PV = mRT, where temperature T will be constant
PV/m = RT = Constant
P/ ρ = Constant = C1
P/ ρ = C1
P / C1 = ρ

Above equation will be the Bernoulli’s equation for compressible fluid for an isothermal process. We can also write the Bernoulli’s equation for compressible fluid for an isothermal process for two points 1 and 2 as mentioned here. 

Bernoulli’s equation for compressible fluid for an adiabatic process 

We will secure here the value of ρ in terms of p with the help of following equation of adiabatic process. 

Above equation will be the Bernoulli’s equation for compressible fluid for an adiabatic process. We can also write the Bernoulli’s equation for compressible fluid for an adiabatic process for two points 1 and 2 as mentioned here. 

Further we will go ahead to find out the momentum equation for compressible fluid 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