Differential Pressure Flow Meter Working Principle (Derivation and Equation)

differential_pressure_flow_meter_working_principle

General theory for restriction type flow meter

Fig shows schematic representation of a one-dimensional flow system with restriction, Showing upstream 1 and downstream 2.

Assume a one-dimensional flow in through a restriction (venturi) fitted in a pipe. p, a,v,z are pressure, area, velocity, and elevation of fluid at a point respectively. According to the Bernoulli’s principle of conservation of energy total energy (total head) remains constant.
p/ρg+v^2/2g+z=constant
The first term of the above equation is called potential head or potential energy. The second term is called kinetic head or KE energy. This relationship between velocity and pressure provides the basis for the operation of all head type meters.

p_1/ρg+〖v_1〗^2/2g+z_1=p_2/ρg+〖v_2〗^2/2g+z_1
As the pipe is horizontal z1 = z2 or considering, the elevation is negligible z2-z1 = 0 the Bernoulli’s equation can be written as
p_1/ρg+〖v_1〗^2/2g=p_2/ρg+〖v_2〗^2/2g  (p_1-p_2)/ρg=  〖v_2〗^2/2g-〖v_1〗^2/2g
The difference of the pressure head at section 1 and 2 is equal to “h.”
(p_1-p_2)/ρg=h
Substitute this value in the above equation we get
h =  〖v_2〗^2/2g-〖v_1〗^2/2g
Now apply continuity equation at section 1 and 2
a_1 v_1=a_2 v_2  v_1=(a_2 v_2)/a_1
Substitute value of v1 in equation of h
h =  〖v_2〗^2/2g-((a_2 v_2)/a_1 )^2/2g  h =  〖v_2〗^2/2g (1-〖a_2〗^2/〖a_1〗^2 )  〖v_2〗^2=2gh 〖a_1〗^2/〖〖〖a_1〗^2-a〗_2〗^2   v_2=√(2gh 〖a_1〗^2/〖〖〖a_1〗^2-a〗_2〗^2 )  =  a_1/√(〖〖〖a_1〗^2-a〗_2〗^2 )  √2gh
Discharge Q=a_2 v_2 Q=(a_1 a_2)/√(〖〖〖a_1〗^2-a〗_2〗^2 )  √2gh
The above equation gives theoretical discharge in ideal condition. The actual discharge will be less than theoretical discharge.
Q=C_d  (a_1 a_2)/√(〖〖〖a_1〗^2-a〗_2〗^2 )  √2gh
Cd is known as the coefficient of discharge and its value always less than 1

- Differential Pressure Flow Meter Advantages and Limitation
Difference between Differential Pressure Flow Meter and Positive Displacement Flow Meter

Conversion of manometer deflection x to Pressure head difference h

Case 1: differential manometer contains liquid heavier than the liquid flowing through the pipeline.
Let
Sm = specific gravity of the heavier fluid
Sp = specific gravity of the fluid in the pipe
X = manometer deflection
Then
Pressure head,h=x(s_m/s_p -1)

Case 2: if the differential manometer contains lighter than the fluid in the pipeline.
Pressure head,h=x(1-s_m/s_p )

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