Fluid mechanics notes

  • Terminal Velocity
  • Terminal velocity if flow is due to gravitational field

Terminal velocity if flow is due to gravitational field

  • Terminal velocity if flow is due to centrifugal motion

Terminal velocity if flow is due to centrifugal motion

  • When stoke’s law is valid, i.e. Re <<1, for spherical particle terminal velocity due to gravitational field is given by

terminal velocity due to gravitational field for spherical particle when stoke's law is valid

when flow due to centrifugal motion, terminal velocity under stoke’s law is given by

terminal velocity due to gravitational field for spherical particle when stoke's law is valid

Note: At low Reynolds number, i.e., creeping flow, drag coefficient is given by

creeping flow drag coefficient

For two identically sized particles, ratio of terminal velocities is given by

ratio of terminal velocities for two identically sized particles

In case of terminal velocity in stoke’s region (according to stoke’s law), calculation to be done as follows

ratio of terminal velocities for two identically sized particles

  • Flow equation for Orifice meter

Velocity at vena contracta of orifice meter

Flow equation for Orifice meter

Coefficient of contraction for orifice meter

Coefficient of contraction for orifice meter

  • Weber Number

Useful in analyzing fluid flows where there is an interface between two different fluids, especially for multiphase flows with strongly curved surfaces

weber number

  • Velocity equation for laminar boundary layer

velocity equation for laminar boundary layer

Where a1 and a2 are constants and y is distance from surface

  • Pitot tube

velocity equation for pitot tube
veocity equation for pitot tube

For turbulent flow in pipe, relationship between average velocity and maximum velocity is given byrelationship between average velocity and maximum velocity for turbulent flow in pipe

  • Ergun Equation for pressure drop due to flow across a porous packed bed

It includes energy loss due to viscous force characteristic of laminar flow and energy loss at high Reynolds number due to turbulence

Ergun Equation for pressure drop due to flow across a porous packed bed

Equivalent diameter in porous packed bed

Equivalent diameter in porous packed bed

  • Pressure drop for fluidization to start or pressure drop at beginning of fluidization

Pressure drop for fluidization to start or pressure drop at beginning of fluidization

  • Power number in case of turbulent flow

Power number in case of turbulent flow

  • Power number in case of turbulent flow

Power number in case of turbulent flow

  • Power number in case of laminar flow

Power number in case of laminar flow

  • Settling velocity of suspension

Settling velocity of suspension

Where n = Richardson – zaki index

Settling velocity of suspension

  • Kozeny – Carman equation

Kozeny – Carman equation for a pressure drop of fluid flowing across a packed bed of solids. Conditions for Kozeny-Carman equation is ԑ < 0.5, Re < 10.

Kozeny – Carman equation for a pressure drop of fluid flowing across a packed bed of solids

Pressure drop across fluidized bed for creeping flow(Re <1)

Pressure drop across fluidized bed for creeping flow

  • Burke-Plumer equation

For Re > 1000, Burke-Plumer equation is applicable for calculating pressure drop cross fluidized/porous bed.

For Re > 1000, Burke-Plumer equation is applicable for calculating pressure drop cross fluidized/porous bed

  • Flow equation for rotameter

Flow equation for rotameter

Metering ratio of rotameter

Metering ratio of rotameter

For rotameter flow Q is directly proportional to x, height of the weight lifted due to flow.

  • One – seventh power law

Power law for relationships between wind speeds at one height and those at another. One-seventh power law is for irrotational flow.

One - seventh power law

  • Hagen – Poisuille equation

For flow of fluid in a cylindrical pipe of radius R, Hagen Poisuille equation is derived from equation

Hagen – Poisuille equation

Velocity of fluid element as a function of radius r is given by

Pressure drop in fluid flowing through cylindrical pipe of length l and diameter d; i.e., Hagen-Poiseuilli’s equation for laminar flows

  • Bingham plastic equation

Stress equation for bingham plastic is given by

Stress equation for bingham plastic

  • Head loss for energy losses due to bend, sudden expansion or contraction

Head loss for energy losses due to bend, sudden expansion or contraction

  • Blasius equation for flat plate
  • Laminar flow

Blasius equation for flat plate laminar flow

  • Turbulent flow

Blasius equation for flat plate turbulent flow

  • Friction factor

It is the ratio of shear force to inertial force.

  • Pressure drop across a cylindrical pipe (using fanning friction factor)

For laminar flow

Friction factor fanning friction factor

  • Darcy friction factor(f) and fanning friction factor(ff)

Relationship between fanning friction factor and darcy friction factor

Fanning friction factor is one-fourth of Darcy friction factor

darcy friction factor and fanning friction factor

  • Laminar flow

friction factor for laminar flow

  • Turbulent flow (104 < Re < 107)

friction factor for turbulent flow

  • Friction factor and Reynolds number

For flow in laminar region friction factor f should be decreasing along the length of the pipe given by equation

Reynolds number and friction factor

  • Relationship between pressure drop and velocity for fluid flow
  • Laminar flow (Re < 2100)

pressure drop verses velocity for laminar flow

  • Mild turbulent flow (2100 < Re < 106)

pressure drop verses velocity for mild turbulent flow

  • Highly turbulent flow

pressure drop verses velocity for turbulent flow

    • To be updated 

navier stoke equation

  • Reynolds number for flow of gases through packed bed

Reynolds number for flow of gases through packed bed

  • Flow of liquid sliding over a vertical wall

equation for flow of liquid sliding over a vertical wall

  • Energy balance equation

Energy balance equation involving internal energy, enthalpy, heat, work, boundary work, kinetic energy and potential energy

complete energy balance for flow of fluid

  • Viscosity of water

Viscosity of water is 1 cP = 0.01 Poise = 0.001 Pa-sec

  • Incompressible fluid

Condition for fluid to be incompressible fluid

condition for incompressible fluid

  • Irrotational fluid

Condition for fluid to be Irrotational fluid

condition for irrotational fluid

  • Energy equation for flow of fluid across pump

Energy equation for flow across pump involving velocity head, elevation head, pressure head, pump head and friction head

Energy equation for fluid flow across pump

  • Equivalent diameter and hydraulic radius

Equivalent diameter = 4 x Hydraulic radius Equivalent diameter for concentric cylinder

Equivalent diameter and hydraulic radius

  • Archimedes principle

Buoyancy Force = Weight of water displaced.

Consider a container filled with water and oil with oil on top of water. A wooden piece is immersed in it, partially in both oil and water. In this case, according to Archimedes principle, weight of wood = weight of water + weight of oil.

  • Volumetric flow rate through cylindrical pipe in laminar flow

Through cylindrical pipe with laminar flow, volumetric flow rate of fluid through the differential section dA is

Volumetric flow rate through cylindrical pipe in laminar flow

  • Centrifugal pump capacity and head dependency on impeller speed

Centrifugal pump capacity and head dependency on impeller speed

  • Three dimensional cylindrical coordinate system equation

Three dimensional cylindrical coordinate system equation

  • Equation of venturimeter

For venturimeter installed at an angle θ

equation of venturimeter installed at an angle θ

For horizontal venturimeter angle θ is 0o and for veritcal venturimeter angle θ is 180o. In both the cases the component Hsinθ would disappear.

  • Euler number

  • Froude number

Power number allows predicting drag coefficient of the agitator in fluid and hence power consumption expression. In power correlation for agitated vessels the effect of Froude number appears for unbaffled vessels when there is vortex formation and Reynolds number greater than 300. For baffled tanks, side entering propellers or for Re < 300, vortex formation does not occur and Froude number is not important.

froude number
froude number

  • Vector form of acceleration given by velocity V

vector form of acceleration

  • Relationship between velocity potential (Ψ) and velocity vector V

Relationship between velocity potential (Ψ) and velocity vector V

  • Heights of different fluidized beds

Two different fluidized bed of same volume but different height (h1, h2) and different porosity (ϵ1, ϵ2). Since total volume of solid is constant = hA(1-ϵ)

Heights of different fluidized beds

  • Continuity equation, momentum balance, and energy balance

Continuity equation, momentum balance, and energy balance

  • Darcy equation

An equation that describes the flow of a fluid through a porous medium.

darcy equation

  • Mass flow rate

mass flow rate

  • Power developed due to flow of fluid

Power developed due to flow of fluid

  • Equal division of flow rate

Flow rate in laminar flow can be divided into equal halves at radial position 0.54R from the axis.

  • Volumetric flow rate through parallel planes

Volumetric flow rate through parallel planes separated by distance “B” due to constant pressure gradient (viscous flow)

Volumetric flow rate through parallel planes

  • Relationship between gram per cubic meter and gram per litre

Relationship between gram per cubic meter and gram per litre

  • Boundary layer

Conditions for boundary layer separation

Conditions for boundary layer separation

Hydrodynamic boundary layer thickness (δh)

Hydrodynamic boundary layer thickness

  • Volumetric change due to change in temperature

Volume change variant on either side of the central scale due to change in temperature

Volumetric change due to change in temperature

  • Echelle grating

echelle grating

  • Skin friction drag coefficient (CD,f) and drag force due to skin friction (FD,f)

Skin friction drag coefficient

  • Drag force

Drag force is defined as product of drag coefficient, velocity head, density and projected area. Under stoke’s law, drag coefficient CD = 24/Re (when Re < 1)

drag force and drag coefficient

  • Steady state energy flow through nozzle

Steady state energy flow through nozzle

  • Kinetic energy correction factor (∝)

Kinetic energy correction factor

  • Flow (Q) for falling film liquid on inclined wall at an angle ɸ

Flow (Q) for falling film liquid on inclined wall at an angle ɸ

  • Froude number and power number for agitation in turbulent flow

Froude number allows predicting vortex formation.

Froude number and power number for agitation in turbulent flow

Power number allows predicting drag coefficient of the agitator in fluid and hence power consumption expression.

Froude number and power number for agitation in turbulent flow

  • The shear stress – shear rate relationship

The shear stress – shear rate relationship for a liquid whose apparent viscosity decreases with shear rate is given by

shear stress – shear rate relationship

  • Surface Tension

For a soap bubble each inner and outer surface will have surface tension of σ along the entire length circumference = 𝜋D. The total force from surface tension along inner and outer film = 2σ𝜋D. If pressure inside film is P, external pressure = Po

surface tension

  • Fluid Mechanics
      • Pumps

        • NPSH

NPSH is defined as sum of velocity and pressure heads at suction minus the vapour pressure of the liquid at the suction temperature.

        • Diaphragm pumps are used for slurries and high viscosity fluids. Wet gas meter is used with system involving volumetric displacement.

        • Pumps used for pumping of edible oil (high viscous oils) – gear pump; crude oil or suspensions – airlift pump; 98% sulphuric acid – centrifugal pump; liquid containing suspensions of abrasive solids – diaphragm pump.

        • Hydraulic efficiency of pumps first increases and then decreases with discharge flow.

      • Agitaion and power law

        • For agitation, at very low rpm, (NRe < 5), follows affinity laws.

        • Affinity laws

a. At constant diameter and density

b. At constant rpm and density

        • A spherical particle is falling slowly in a viscous liquid such that Re < 1, then drag, gravitational and buoyancy forces are important.

        • 13. For low Reynolds number Re < 10, for low RPM, power number is inversely proportional to Reynolds. At low rpm, power required for agitation is proportional to D3.

        • For geometrically similar stirred tanks, the power number remains constant at high Reynolds number.

        • Drag force always increases with increasing terminal velocity.

        • In agitated vessel baffles are used to suppress vortex.

        • Fluidized beds are formed when gravity force is less than fluid friction.

        • For Similarity between model and industrial setup, (Reynolds number)model = (Reynolds number)industrial and (Power number)model = (Power number)industrial

        • With increase in superficial velocity above minimum fluidization velocity for a bed of particles, pressure drop, drag on particles, drag on column walls, bed voidage increases but bed height remains constant.

      • Fluid flow

        • For a Rheopectic fluid, the apparent viscosity increases with time under a constant applied shear stress.

        • Velocity profile for Bingham plastic is parabolic near wall and flat at center.

        • Viscosity of water at normal conditions = 8.90 x 10-4 Pa sec = 0.890 cP.

        • In a fully developed flow (Re > 105) in a pipe of diameter d for a constant pressure gradient, flow Q ∝ d2.5

        • For laminar flow of shear thinning fluid, if volumetric flow is doubled, the pressure gradient will increase by a factor of < 2.

        • A gas bubble at a pressure Pg is passed through a solvent with a saturation vapour pressure Ps. If the time of passage is long and gas is insoluble in the solvent, the mole fraction of solvent in the bubble will be equal to Ps/Pg.

        • When a vertical plate is heated in an infinite air environment under natural convection conditions, the velocity profile in air, normal to plate exhibits a maximum.

        • Velocity profile for bingham plastic under laminar flow conditions is plug flow at the central part of the tube. Velocity profile is flat near the center and parabolic near the wall.

      • Flow meter

        • The equilibrium position of the float in a rotameter is determined by the balance of three forces gravitational, drag and buoyancy force.

      • Prandtl number

Typical values for prandtl number: 0.7 for air and many gases, 7 for water, 7 x 1021 for earth’s mantle, 100 – 40,000 for engine oil, 4 – 5 for R-12 refrigerant, 0.015 for mercury. For mercury heat conduction is very effective compared to convection. Thermal conductivity is dominant. For engine oil, convection is very effective for transferring energy form an area compared to pure conduction, momentum diffusivity is dominant.

  • Filtration
  • Rate of filtration

rate of filteration

For filtration

equation for filtration

where Kp is a constant

For 1-3 filter press time t for washing

equation for filter press

  • Constant pressure filtration

Constant pressure filtration

At constant rate filtration, linear velocity  linear velocity at constant rate filtrationis constant. Therefore,

 linear velocity at constant rate filtration

  • Specific cake resistance

Linear velocity of filtrate is independent of L (distance from filter medium), since filtrate must pass through entire cake, V/A is same for all layers.

Specific cake resistance

  • Bonds law
  • Equation for bonds law

bonds law

  • Bonds law with work index

bonds law with work index

  • Rittinger’s law

Rittinger’s law: assumes energy required is directly proportional to surface area.

rittingers law

  • Various diamters defined in solid operation and handling
  • Mass mean diameter

mass mean diameter

  • Volumetric mean diameter

volumetric mean diameter

  • Arithmetic mean diameter

arithmetic mean diameter

  • Surface-Volume mean diameter

surface volume mean diameter

  • Thickness of pressure vessels

thickness of pressure vessel

  • Cylinder

thickness for cylindrical vessel

  • Sphere

thickness for sphere vessel

  • Ellipsoidal head

thickness for ellipsoidal head

  • Torispherical head

thickness for torispherical head

  • Conical section enclosure

thickness for conical section

  • Sphericity

sphericity

  • Critical rotational speed for a ball mill

critical rotational speed for a ball mill

  • Cyclone separator
  • Separation factor (s) for cyclone separator

separation factor for cyclone separator

  • Collection efficiency of cyclone separator

collection efficiency for a cyclone separator

  • Angular velocity w (omega) in rad/sec (radian per second)

angular veolocty

  • Screen effectiveness

sceen effectiveness

  • Cumulative mass fraction

Cumulative mass fraction of particles is fraction of weight of particles having a size smaller than a given diameter of the screen.

cumulative mass fraction

  • Operations involving Particulate solids (Fluid Particle Mechanics)
    1. Filter

      • Vacuum leaf filter is a constant pressure filter. Example of vacuum filter is rotary drum filter, precoat filter, horizontal belt filter

    2. Separating particles

      • For separating particles of different densities, the differential settling method uses a liquid sorting medium of density intermediate of those of the heavy and light particles

    3. Filter cake resistance

      • Specific cake resistance

For cake resistance

  1. Tyler series

    • Tyler series for screens is based on 200 mesh sieve opening 0.074mm and increasing uniformly @scale ratio of 21/4. Therefore, the ratio of aperture size of screen to that smaller screen is (21/4)2 = 21/2.