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Critical velocity

The critical velocity of the fluid is the speed at which the liquid flow transitions from streamlined to turbulent. When the velocity of the fluid in the pipe is low, the streamlines are straight parallel lines. The streamline remains straight and parallel to the pipe wall as the fluid's velocity steadily increases. When the velocity reaches a certain point, it starts to create patterns. The critical velocity will spread the streamlines throughout the pipe.

The sewer pipes are progressively slanted to allow gravity to act on the fluid flow, keeping the flow non-critical. Because solid particles are present in the flow, an excess velocity of flow can induce pipe erosion, resulting in pipe damage. Pipes damaged by the high-velocity fluid can be repaired utilising trenchless methods such as cured-in-place pipe, pipe bursting, and slip lining.

The Reynolds number, which defines the flow of streamlined or turbulent air, may be used to compute the fluid's Critical Velocity. It's a one-dimensional variable that can be computed with a formula.

The formula for Critical Velocity

The following is a mathematical depiction of critical velocity using the dimensional formula:

Vc=(Kη/rρ)

where,

K = Reynold’s number
η = coefficient of viscosity of a liquid
r = radius of the capillary tube
ρ = density of the liquid

Dimensional formula

  • Reynolds number (Re) = M⁰L⁰T⁰
  • Coefficient of viscosity (𝜂) = M¹L⁻¹T⁻¹
  • The density of the fluid (⍴) = M¹L⁻³T⁰
  • Therefore, critical velocity Vc= M⁰L¹T⁻¹

Reynolds' digit

The Reynolds number is defined as the ratio of inertial and viscous forces. Reynold's number is a pure number that may be used to determine the nature of a liquid's flow and critical velocity through a pipe.

The following is a mathematical representation of the number:

image

⍴: density of the fluid in kg.m^-3

𝜇: dynamic viscosity of the fluid in m^2s

u: velocity of the fluid in ms^-1

L: characteristic linear dimension in m

𝜈: kinematic viscosity of the fluid in m²s⁻¹

Flow type can make the following decisions based on the value of the Reynolds number:

  • The flow is streamlined or laminar if the value of Re is between 0 and 2000.
  • The flow is unstable or turbulent if the value of Re is between 2000 and 3000.
  • When Re exceeds 3000, the flow becomes extremely turbulent.

The following are the Reynolds numbers for laminar and turbulent flow regimes:

  • When the Reynolds number is low, viscous forces are dominating, resulting in laminar flow, which is defined as a smooth, continuous fluid motion.
  • When the Reynolds number is high, inertial forces become dominant, causing turbulent flow, which produces vortices, flow uncertainty, and disordered eddies.

The Reynolds number is calculated as follows:

image1

Where,

t= time

y = cross-sectional position

u =dx/dt : flow speed

τ = shear stress in Pa

A = cross-sectional area of the flow

V = volume of the fluid element

U₀= a maximum speed of the particle relative to the fluid in ms^-1

L = a characteristic linear dimension

𝜇 = fluid of dynamic viscosity in Pa.s

𝜈 = kinematic viscosity in m^2s

⍴ = density of the fluid in kg.m^-3

The Critical Velocity Ratio (CVR) is a measure of how fast something is

The concept of critical velocity was developed to prevent silting and scouring in a waterway. A relationship between critical velocity and full supply depth was created based on extensive observations.

C and n were determined to be 0.546 and 0.64, respectively, resulting in v0=0.546 D^0.64

The critical velocity in the aforementioned calculation, however, was influenced by the silt grade. As a result, a new factor (m) was introduced: the critical velocity ratio (C.V.R).

v0=0.546 D^0.64

The ratio of mean velocity 'V' to critical velocity 'Vo,' where Vo is the critical velocity ratio, is known as (C.V.R).

It is denoted by m i.e. CVR (m) = V/Vo

m = 1, there will be no silting or scouring

m>1, scouring will take place

m<1, silting will take place

By determining the value of m, the canal's state may be forecasted, indicating whether it would experience silting or scouring.

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