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Showing posts with label basics of aerodynamics. Show all posts
Showing posts with label basics of aerodynamics. Show all posts

Monday, 17 April 2017

Derivation for continuity equation in integral form

Derivation for continuity equation in integral form

Integral form of continuity equation

In Aerodynamics there are three fundamental equations

1. Continuity equation
2. Momentum equation
3. Energy equation

We will study all the equations one by one. 

Let's talk about Continuity equation once.

1. Continuity equation 

Continuity equation is simply conservation of mass of the flowing fluid. Consider fluid flowing through the pipe. It is really not possible that fluid entering from one end of pipe vanishes while coming out of other end of the pipe(Except if its magical fluid, just kidding). This is a same thing which continuity equation tells us. That mass of flowing fluid is conserved.

Let 
'm' be the Mass of the fluid
'V' be the Volume of the fluid
'ρ' be the Density of the fluid
Click here to know more about fluid properties

As we know Density is equal to ration of mass and volume

hence 

ρ = m/V           (1)

So mass becomes,

m = ρ x V           (2)

Volume can be written as Area times thickness

i.e V = A x t         (3)

Where,

'A' is Cross section are of pipe
't' is thickness of fluid column in pipe

So Mass becomes,

m = ρ x A x t           (4)     (Replacing V by A x t)          

To find of mass flow rate, differentiation above equation with respect to time

'' be the mass flow rate

Hence

 = ρ x A x v         (5) (Differentiation of t with respect to time gives velocity of the fluid)           

Considering mass flow rate we got is for small section of fluid

So to find mass flow rate for entire fluid

We will write mass flow rate from equation (5) in integral form

   
From equation (2) we know that


m = ρ x V

 So taking Elemental volume '∀' instead of 'V'

 m = ρ x ∀             (6)

 To find mass flow rate integrating equation (6) with respect to time, we get



Equating both the mass flow rate equations we get,


using gauss divergence theorem,





Since the volume∀  does not change with time, the sequence of differentiation and integration in the first term of can be interchanged. Therefore

This is integral form of continuity equation

We can also write it as,


This is Continuity equation!


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Properties of fluids

Properties of fluids

Fluid Properties


Before going into complex aerodynamics it is mandatory to know about basic fluid property. Let's take a look at fluid property one by one.

Pressure:


Pressure is simply force applied on the body per unit area.
. 

Fluid property - Pressure
P= F/A
where A tends to zero
The unit of Pressure is N/m^2.
It is a point property and the pressure acting on a fluid varies from point to point.

Fluid property - Pressure




Density:


Fluid property- Density
In very simple terms, consider density as a property of fluid to measure how thick fluid is. Density is given by mass of an element in space to the volume of space occupied by it . So over here density is the ratio of the mass of the fluid element being  considered, ‘m’ to the volume of  fluid over which the mass is being considered, ‘v’.
Density = m/v
The unit of Density is kg/m^3. The density of air at sea level is about 1.2 kg/m3












Volume:



Fluid property - Volume

Lets talk about volume now. You have definitely come across this term many times before in your life. To define volume, let us consider a jar of water having a volume of 20 liters. This means that the water jar can maximum contain 20 liters of water. This signifies that volume is the measurement of the amount of a fluid that is being contained in a system. Volume is 3D quantity. If you multiply area of any shape by its thickness you get volume. If jar is in rectangular shape, it can be calculated as the product of length, breadth and height(thickness) of fluid. In other way volume is taken as the ratio between the mass (m) of the fluid being considered,  to the density(ρ) of the fluid being considered.

V = m/ρ

The unit for Volume is m^3


Temperature:

Fluid property - Temperature

In the field of aerodynamics, temperature is considered only for high speed flight. Because in high speed flight, production of heat due to friction is considerable. We will talk in depth about this in our forthcoming posts. Now coming back to temperature in relation to subsonic flow. Let us begin with a simple example, what happens to a young kid when you give him a little too much chocolate to have? He will get hyper and find a way to expend all his energy until he gets tired, yes? (Just wanted to tell about excitation) Similarly, when the molecules of a fluid are excited by an external stimulus, they start to move rapidly, colliding with one and other. Due to this rapid collision, heat is generated and the temperature of the flow increases. 
Fluid property - Temperature

High energy means high temperature!

Velocity:

                             
Fluid property - Velocity
Consider a fluid flowing with some velocity ‘u’. Now what do we mean when we say ‘flowing with some velocity’? It basically means that the fluid is being displaced from one point to another or the rate of change of displacement. When we talk about the velocity of a solid, say a ball, we can safely say that the entire ball is being displaced with the same velocity. But in the case of fluids, it is not the same, velocity is a point property and it varies from point to point. This is mainly due to the physical property of a fluid i.e. Viscosity. 


Viscosity is simply resisting force to the motion of the fluid. It is analogous to friction force in solid.

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