# FM 2

1.Laminar flow

2.Turbulent flow (10^4 < Re < 10^7)

3.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

1.Laminar flow (Re < 2100)

2.Mild turbulent flow (2100 < Re < 10^6)

3.Highly turbulent flow

• Navier Stoke Equation

• 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 Pa-sec

• Incompressible fluid

Condition for fluid to be incompressible fluid

• Irrotational fluid

Condition for fluid to be Irrotational fluid

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 the water. A wooden piece is immersed in it, partially in both oil and water. In this case, according to Archimedes principle, a weight of wood = weight of water + weight of oil.

• Volumetric flow rate through cylindrical pipe in laminar flow

Through cylindrical pipe with laminar flow, a 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 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.

• Vector form of acceleration given by velocity V

• A relationship between velocity potential (Ψ) and velocity vector V

• Heights of different fluidized beds

• 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

• Mass flow rate

• The power developed due to a 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

• The volumetric flow rate through parallel planes

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

Q α B³

• Relationship between gram per cubic meter and gram per litre
• Boundary layer

Hydrodynamic boundary layer thickness (δh)

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

• Drag force

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

• Steady-state energy flow through a nozzle

• Kinetic energy correction factor (∝)

• Flow (Q) for falling film liquid on an 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

• Surface Tension

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

2.Fluid Mechanics

• Pumps
• NPSH

NPSH is defined as a 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. The wet gas meter is used with a 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; a liquid containing suspensions of abrasive solids – diaphragm pump.
• Hydraulic efficiency of pumps first increases and then decreases with discharge flow.
• Agitation and power law
• For agitation, at very low rpm, (NRe < 5), follows affinity laws
• Affinity laws

a. At constant diameter 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, the 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 the 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 constant applied shear stress.
• Velocity profile for Bingham plastic is parabolic near a 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 a 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 vapor 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 the 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. The 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 from an area compared to pure conduction, momentum diffusivity is dominant.