 Terminal Velocity
 Terminal velocity if flow is due to gravitational field
 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
when flow due to centrifugal motion, terminal velocity under stoke’s law is given by
Note: At low Reynolds number, i.e., creeping flow, drag coefficient is given by
For two identically sized particles, ratio of terminal velocities is given by
In case of terminal velocity in stoke’s region (according to stoke’s law), calculation to be done as follows
 Flow equation for Orifice meter
Velocity at vena contracta of 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
 Velocity equation for laminar boundary layer
Where a1 and a2 are constants and y is distance from surface
 Pitot tube
For turbulent flow in pipe, relationship between average velocity and maximum velocity is given by
 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
Equivalent diameter in porous packed bed
 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 laminar flow
 Settling velocity of suspension
Where n = Richardson – zaki index
 Kozeny – Carman equation
Kozeny – Carman equation for a pressure drop of fluid flowing across a packed bed of solids. Conditions for KozenyCarman equation is ԑ < 0.5, Re < 10.
Pressure drop across fluidized bed for creeping flow(Re <1)
 BurkePlumer equation
For Re > 1000, BurkePlumer equation is applicable for calculating pressure drop cross fluidized/porous bed.
 Flow equation for 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. Oneseventh power law is for irrotational flow.
 Hagen – Poisuille equation
For flow of fluid in a cylindrical pipe of radius R, Hagen Poisuille equation is derived from 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., HagenPoiseuilli’s equation for laminar flows
 Bingham plastic equation
Stress equation for bingham plastic is given by
 Head loss for energy losses due to bend, sudden expansion or contraction
 Blasius equation for flat plate
 Laminar flow
 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
 Darcy friction factor(f) and fanning friction factor(f_{f})
Relationship between fanning friction factor and darcy friction factor
Fanning friction factor is onefourth of Darcy friction factor
 Laminar flow
 Turbulent flow (10^{4} < Re < 10^{7})
 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
 Relationship between pressure drop and velocity for fluid flow
 Laminar flow (Re < 2100)
 Mild turbulent flow (2100 < Re < 10^{6})
 Highly turbulent flow

 To be updated
 Reynolds number for flow of gases through packed bed
 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
 Viscosity of water
Viscosity of water is 1 cP = 0.01 Poise = 0.001 Pasec
 Incompressible fluid
Condition for fluid to be incompressible fluid
 Irrotational fluid
Condition for fluid to be 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
 Equivalent diameter and hydraulic radius
Equivalent diameter = 4 x Hydraulic radius Equivalent diameter for concentric cylinder
 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
 Centrifugal pump capacity and head dependency on impeller speed
 Three dimensional cylindrical coordinate system equation
 Equation of venturimeter
For venturimeter installed at an angle θ
For horizontal venturimeter angle θ is 0^{o} and for veritcal venturimeter angle θ is 180^{o}. 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.
 Vector form of acceleration given by velocity 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(1ϵ)
 Continuity equation, momentum balance, and energy balance
 Darcy equation
An equation that describes the flow of a fluid through a porous medium.
 Mass flow rate
 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)
 Relationship between gram per cubic meter and gram per litre
 Boundary layer
Conditions for boundary layer separation
Hydrodynamic boundary layer thickness (δ_{h})
 Volumetric change due to change in temperature
Volume change variant on either side of the central scale due to change in temperature
 Echelle grating
 Skin friction drag coefficient (C_{D,f}) and drag force due to skin friction (F_{D,f})
 Drag force
Drag force is defined as product of drag coefficient, velocity head, density and projected area. Under stoke’s law, drag coefficient C_{D} = 24/Re (when Re < 1)
 Steady state energy flow through nozzle
 Kinetic energy correction factor (∝)
 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.
Power number allows predicting drag coefficient of the agitator in fluid and hence power consumption expression.
 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
 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 = P_{o}
 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, (N_{Re} < 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 D^{3}.

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 > 10^{5}) in a pipe of diameter d for a constant pressure gradient, flow Q ∝ d^{2.5}

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 P_{g} is passed through a solvent with a saturation vapour pressure P_{s}. 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 P_{s}/P_{g}.

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 10^{21} for earth’s mantle, 100 – 40,000 for engine oil, 4 – 5 for R12 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
For filtration
where K_{p} is a constant
For 13 filter press time t for washing
 Constant pressure filtration
At constant rate filtration, linear velocity is constant. Therefore,
 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.
 Bonds law
 Equation for bonds law
 Bonds law with work index
 Rittinger’s law
Rittinger’s law: assumes energy required is directly proportional to surface area.
 Various diamters defined in solid operation and handling
 Mass mean diameter
 Volumetric mean diameter
 Arithmetic mean diameter
 SurfaceVolume mean diameter
 Thickness of pressure vessels
 Cylinder
 Sphere
 Ellipsoidal head
 Torispherical head
 Conical section enclosure
 Sphericity
 Critical rotational speed for a ball mill
 Cyclone separator
 Separation factor (s) for cyclone separator
 Collection efficiency of cyclone separator
 Angular velocity w (omega) in rad/sec (radian per second)
 Screen 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.
 Operations involving Particulate solids (Fluid Particle Mechanics)


Filter

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


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


Filter cake resistance

Specific cake resistance


For cake resistance

Tyler series

Tyler series for screens is based on 200 mesh sieve opening 0.074mm and increasing uniformly @scale ratio of 2^{1/4}. Therefore, the ratio of aperture size of screen to that smaller screen is (2^{1/4})2 = 2^{1/2}.
