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
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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