For a Newtonian fluid, viscosity is

independent of shear rate, a single proportionality constant between shear stress and shear rate

a function of shear rate that must be measured at every shear rate of interest

independent of shear rate, a single proportionality constant between shear stress and shear rate

zero unless a yield stress is exceeded

only definable for gases, not liquids

Buckingham's pi theorem, applied to a problem with n physical variables spanning m independent dimensions, tells you

that the problem reduces to (n - m) independent dimensionless groups

the exact governing differential equation of the problem

that the problem reduces to (n - m) independent dimensionless groups

the numerical value of the Reynolds number for the flow

which dimensionless group is physically most important

Bernoulli's equation is obtained from the general equation of motion by

dropping viscosity entirely (Euler's equation) and integrating along a streamline for steady, incompressible flow

adding a friction-loss term to account for real viscosity

dropping viscosity entirely (Euler's equation) and integrating along a streamline for steady, incompressible flow

assuming turbulent flow throughout

including a shaft-work term for a pump

For laminar flow in a pipe, the Fanning friction factor f = 16/Re, while turbulent friction factor read off a Moody chart

depends on Reynolds number and the pipe's relative roughness

also depends only on Reynolds number

depends on Reynolds number and the pipe's relative roughness

depends only on pipe diameter, not velocity

is always exactly 16/Re regardless of regime

Which of the following are true of the minimum fluidization velocity in a packed bed?

At this velocity, the bed's pressure drop exactly equals its weight per unit cross-sectional area.

Below this velocity, the bed behaves as a fixed (packed) bed.

It marks the transition from a fixed bed to a fluidized bed.

At this velocity, the bed's pressure drop exactly equals its weight per unit cross-sectional area.

Below this velocity, the bed behaves as a fixed (packed) bed.

Above this velocity, pressure drop keeps rising sharply while bed height stays fixed.

It marks the transition from a fixed bed to a fluidized bed.

Hindered settling velocity, compared to free settling velocity of the same particle at the same fluid conditions, is

always higher, because particles push each other down

identical, since gravity acts on each particle independently

undefined at any concentration above zero

A centrifuge separates particles from a fluid more effectively than gravity settling chiefly because it

replaces gravity with a much larger, adjustable force proportional to radius and angular velocity squared

cools the fluid, increasing its density difference from the particles

replaces gravity with a much larger, adjustable force proportional to radius and angular velocity squared

converts hindered settling into free settling

eliminates the need for a density difference between particle and fluid

In constant-pressure-drop cake filtration, the filtration rate

falls with time, as the growing cake adds resistance to flow

stays constant with time, since the driving pressure does not change

rises with time, as the cake becomes more permeable

falls with time, as the growing cake adds resistance to flow

is independent of the filter medium's own resistance

Sphericity of a particle, defined as the surface area of a sphere of equal volume divided by the particle's own surface area, is

always at most 1, equal to 1 only for a perfect sphere

always exactly 1, for any particle shape

always at most 1, equal to 1 only for a perfect sphere

always greater than 1 for an irregular particle

independent of the particle's actual shape

Two geometrically similar pumps, a small test model and a large prototype, are dynamically similar when their

relevant dimensionless groups, such as the Reynolds number, are matched

Flow, Pipes and Beds, and Mechanical Operations

Section 4 of the GATE Chemical Engineering (CH) paper, and its longest — it covers two subjects the syllabus states back to back: fluid mechanics, the equations that move fluids through pipes, pumps and packed beds, and mechanical operations, the physical (not chemical) processing of solid particles. Start with fluid properties and statics, then the equations that govern flow — continuity, motion, mechanical energy — and how they play out in real pipes with friction, pumps and flow meters; then turbulence, boundary layers and flow past packed and fluidized beds; then the particulate side — how particles are sized, reduced, classified, settled and separated, and how they are mixed once suspended.

1. Fluid properties, statics and dimensional analysis

A Newtonian fluid's shear stress is directly proportional to shear rate, with viscosity as the constant of proportionality independent of the shear rate itself; a non-Newtonian fluid's apparent viscosity changes with shear rate — shear-thinning (pseudoplastic, viscosity falls as shear rate rises, most polymer solutions and slurries), shear-thickening (dilatant, the rarer opposite), and Bingham plastic (needs a yield stress before it flows at all, then behaves linearly). Surface tension, the energy per unit area of a liquid interface, is what makes a bubble spherical and drives capillary rise, and it becomes the dominant force at small length scales precisely because it scales with length while gravity scales with volume.

🧠 Dimensional analysis says how many groups, not what the physics is

Buckingham's pi theorem says a physical problem with n variables and m independent dimensions reduces to (n - m) independent dimensionless groups — it fixes the COUNT and lets you construct the groups, but never tells you which group matters most or what the underlying equation is. Similitude then uses those same groups (Reynolds number, Froude number, Euler number) to scale a small model's test results up to a full-size prototype, provided the relevant groups are matched between the two.

2. Continuity, motion, mechanical energy, and flow through pipes

The continuity equation is mass conservation applied to a flowing fluid: for incompressible flow in a pipe, A1v1 = A2v2, so velocity rises exactly where cross-section narrows. The equation of motion (Navier-Stokes for a real fluid) balances inertia against pressure, gravity and viscous forces; dropping viscosity entirely gives Euler's equation for inviscid flow, and integrating that along a streamline for steady, incompressible, frictionless flow gives Bernoulli's equation — P/rho + v^2/2 + gz = constant. The mechanical energy balance is Bernoulli's equation generalised to real flow: it adds a friction-loss term (always positive, always a loss, never a gain) and a shaft-work term for any pump or turbine in the line.

Reading friction factor and velocity profile by flow regime
Regime (pipe flow)	Velocity profile	Friction factor f
Laminar (Re < 2100)	Parabolic	f = 16/Re (Fanning), independent of pipe roughness
Turbulent (Re > 4000)	Flatter, near-uniform core with a thin wall layer	From Moody chart / Colebrook equation, depends on Re AND relative roughness

A pump adds mechanical energy to move fluid against friction and elevation; its performance is read off a head-versus-flow-rate curve, and matching it to a system's own head-versus-flow-rate curve (static head plus friction losses, which rise roughly with flow squared) fixes the actual operating point. Flow meters — orifice, venturi, rotameter, pitot tube — all work by relating a measurable pressure drop or float position back to velocity through Bernoulli's equation, which is why every one of them needs a discharge coefficient to correct for the frictional losses Bernoulli's ideal form ignores.

3. Turbulent flow, boundary layers, and packed and fluidized beds

In turbulent flow, the instantaneous velocity is decomposed into a time-averaged mean plus a fluctuating component; it is the fluctuating component that transports momentum across the flow far more effectively than molecular viscosity alone, which is why turbulent flow mixes so much faster than laminar flow. Close to a solid wall, the velocity still has to fall to zero (no-slip), which is described by the universal velocity profile in wall units — a viscous sublayer nearest the wall, a buffer zone, and a logarithmic outer region — and turbulent-flow pressure drop, unlike laminar, depends on wall roughness because the roughness protrudes into that near-wall structure.

The boundary layer is the thin region near a submerged surface where velocity gradients — and hence viscous effects — actually matter; outside it, the flow behaves as effectively inviscid. It starts laminar near the leading edge and transitions to turbulent further along, at a Reynolds number that depends on surface roughness and free-stream turbulence.
Flow past an immersed body (a sphere settling, say) separates behind the body at high Reynolds number, creating a wake and a form-drag contribution on top of the skin friction; total drag is the sum of the two, and their relative share changes with Reynolds number.
A packed bed's pressure drop (laminar-dominated, Kozeny-Carman-type or Ergun equation) rises with flow rate until, at the minimum fluidization velocity, the bed's weight is exactly balanced by the pressure drop and the particles lift off the distributor — this is the packed-to-fluidized transition, and beyond it pressure drop stays roughly constant while bed height keeps rising with flow rate.

4. Particle characterisation, size reduction, settling and mechanical operations

A real particle's size is reported through an equivalent diameter — Sauter mean, sphericity-corrected, or a screen-sieve nominal size — because only a sphere has one unambiguous diameter; sphericity (surface area of an equal-volume sphere divided by the particle's actual surface area) is always at most 1 and captures how far a shape departs from that ideal. A particle size DISTRIBUTION, read off a sieve analysis or laser diffraction, matters more than a single average size for equipment sizing, because two powders with the same mean size but different spreads behave very differently in a hopper, a cyclone or a filter.

🎯 Free settling versus hindered settling versus centrifugal separation

A single particle settling alone under gravity reaches a terminal velocity where drag exactly balances net gravity (weight minus buoyancy) — free settling. At high particle concentration, hindered settling sets in: particles interfere with each other's flow field and displace fluid upward as they fall, so the observed settling velocity is LOWER than the free-settling value for the same particle. A centrifuge or cyclone replaces the weak, fixed force of gravity with a much larger, adjustable centrifugal force (proportional to radius and the square of angular velocity), which is exactly why it can separate particles gravity alone would take impractically long to settle, or cannot separate at all.

Size reduction equipment (crushers, mills) is selected by the feed's hardness and the target product size, and its energy requirement is estimated by empirical laws (Rittinger's for fine grinding, Kick's for coarse crushing, and Bond's work index in between) that all relate energy input to some power of the size-reduction ratio. Filtration separates suspended solids from a liquid by forcing flow through a porous medium, and the resistance builds up as a cake forms on that medium, which is why filtration rate falls with time at constant pressure drop (or pressure must rise with time to hold rate constant) — the same cake buildup that thickening and clarification manage on a larger, continuous scale using gravity settling in a tank rather than a filter medium; agitation and mixing then handle blending, suspension and heat transfer once a slurry or solution actually needs to move as a bulk fluid, most often with an impeller in a baffled tank chosen for the flow pattern (axial or radial) the duty needs.

Key takeaways

A Newtonian fluid's viscosity is independent of shear rate; shear-thinning, shear-thickening and Bingham-plastic fluids all have an apparent viscosity that changes with shear rate (or need a yield stress first).
Bernoulli's equation is Euler's equation integrated along a streamline for inviscid, incompressible, steady flow; the mechanical energy balance adds a friction-loss term and shaft work to make it usable for a real pump-and-pipe system.
Laminar friction factor (f = 16/Re) depends only on Reynolds number; turbulent friction factor also depends on relative wall roughness, because roughness protrudes into the near-wall turbulent structure.
At the minimum fluidization velocity, a packed bed's pressure drop exactly balances its weight and it becomes a fluidized bed; beyond that point pressure drop stays roughly constant while bed height rises with flow.
Hindered settling velocity is always lower than free-settling velocity at the same particle size, because particles interfere with each other's flow field at higher concentration; a centrifuge or cyclone separates by replacing gravity with a far larger, adjustable centrifugal force.

Practice questions (11)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

For a Newtonian fluid, viscosity is
1. a function of shear rate that must be measured at every shear rate of interest
2. independent of shear rate, a single proportionality constant between shear stress and shear rate
3. zero unless a yield stress is exceeded
4. only definable for gases, not liquids
Show answer
Answer: B — independent of shear rate, a single proportionality constant between shear stress and shear rate
The defining property of a Newtonian fluid is that shear stress is directly proportional to shear rate with a single constant of proportionality, viscosity, that does not itself change with the shear rate — that constancy is exactly what distinguishes it from shear-thinning, shear-thickening or Bingham-plastic behaviour.
Buckingham's pi theorem, applied to a problem with n physical variables spanning m independent dimensions, tells you
1. the exact governing differential equation of the problem
2. that the problem reduces to (n - m) independent dimensionless groups
3. the numerical value of the Reynolds number for the flow
4. which dimensionless group is physically most important
Show answer
Answer: B — that the problem reduces to (n - m) independent dimensionless groups
The theorem is purely a counting and construction result: it guarantees (n - m) independent dimensionless groups exist and shows how to build them, but says nothing about which is most physically significant or what the governing equation actually is — that has to come from the physics itself, e.g. via experiment or a derived equation.
Bernoulli's equation is obtained from the general equation of motion by
1. adding a friction-loss term to account for real viscosity
2. dropping viscosity entirely (Euler's equation) and integrating along a streamline for steady, incompressible flow
3. assuming turbulent flow throughout
4. including a shaft-work term for a pump
Show answer
Answer: B — dropping viscosity entirely (Euler's equation) and integrating along a streamline for steady, incompressible flow
Bernoulli's equation is the inviscid (Euler) equation of motion integrated along a streamline under steady, incompressible flow — it is the mechanical energy balance, not Bernoulli's equation itself, that adds the friction-loss and shaft-work terms back in for a real system.
For laminar flow in a pipe, the Fanning friction factor f = 16/Re, while turbulent friction factor read off a Moody chart
1. also depends only on Reynolds number
2. depends on Reynolds number and the pipe's relative roughness
3. depends only on pipe diameter, not velocity
4. is always exactly 16/Re regardless of regime
Show answer
Answer: B — depends on Reynolds number and the pipe's relative roughness
Laminar friction factor is a clean function of Re alone because the parabolic profile has no wall-roughness interaction to speak of; turbulent flow's near-wall structure is thin enough that surface roughness protrudes into it, so turbulent f depends on BOTH Reynolds number and relative roughness (epsilon/D), which is exactly what the Moody chart plots.
Which of the following are true of the minimum fluidization velocity in a packed bed?
1. At this velocity, the bed's pressure drop exactly equals its weight per unit cross-sectional area.
2. Below this velocity, the bed behaves as a fixed (packed) bed.
3. Above this velocity, pressure drop keeps rising sharply while bed height stays fixed.
4. It marks the transition from a fixed bed to a fluidized bed.
Show answer
Answer: A — At this velocity, the bed's pressure drop exactly equals its weight per unit cross-sectional area.; B — Below this velocity, the bed behaves as a fixed (packed) bed.; D — It marks the transition from a fixed bed to a fluidized bed.
The third statement is false: beyond minimum fluidization, pressure drop stays roughly CONSTANT (the bed's weight per area does not change) while bed height keeps RISING with further increase in flow — the opposite of what the option claims. The first, second and fourth are exactly the definition of the fixed-to-fluidized transition.
Hindered settling velocity, compared to free settling velocity of the same particle at the same fluid conditions, is
1. always higher, because particles push each other down
2. always lower, because particles interfere with each other's flow field and displace fluid upward
3. identical, since gravity acts on each particle independently
4. undefined at any concentration above zero
Show answer
Answer: B — always lower, because particles interfere with each other's flow field and displace fluid upward
At appreciable particle concentration, each settling particle must displace fluid that has to flow upward past its neighbours, and the neighbours' own settling motion works against it too — both effects slow the observed settling velocity below the free (dilute, isolated-particle) value.
A centrifuge separates particles from a fluid more effectively than gravity settling chiefly because it
1. cools the fluid, increasing its density difference from the particles
2. replaces gravity with a much larger, adjustable force proportional to radius and angular velocity squared
3. converts hindered settling into free settling
4. eliminates the need for a density difference between particle and fluid
Show answer
Answer: B — replaces gravity with a much larger, adjustable force proportional to radius and angular velocity squared
A centrifuge's separating force is proportional to r*omega^2, which for typical rotational speeds and radii vastly exceeds g and can be turned up simply by spinning faster — that adjustable, much larger driving force, not any change in the settling regime or the need for a density difference, is why it separates particles gravity alone would take impractically long to settle.
In constant-pressure-drop cake filtration, the filtration rate
1. stays constant with time, since the driving pressure does not change
2. rises with time, as the cake becomes more permeable
3. falls with time, as the growing cake adds resistance to flow
4. is independent of the filter medium's own resistance
Show answer
Answer: C — falls with time, as the growing cake adds resistance to flow
As filtration proceeds, more solid deposits on the medium and the cake thickens, adding resistance in series with the medium's own; at constant pressure drop, that rising total resistance means the flow rate must fall with time (the standard alternative is to raise pressure with time to hold rate constant instead).
Sphericity of a particle, defined as the surface area of a sphere of equal volume divided by the particle's own surface area, is
1. always exactly 1, for any particle shape
2. always at most 1, equal to 1 only for a perfect sphere
3. always greater than 1 for an irregular particle
4. independent of the particle's actual shape
Show answer
Answer: B — always at most 1, equal to 1 only for a perfect sphere
A sphere has the minimum surface area for a given volume among all shapes, so any equal-volume irregular particle has MORE surface area than the equivalent sphere, which makes the ratio (equal-volume sphere's area over the particle's actual area) at most 1, reaching exactly 1 only when the particle is itself a perfect sphere.
For laminar flow of a Newtonian fluid in a circular pipe, the Fanning friction factor is given by f = 16/Re. At Re = 800, what is the value of f?
Numerical answer — type the value.
Show answer
Answer: 0.02
f = 16/Re = 16/800 = 0.02. This laminar relation holds only below the transitional Reynolds number (roughly 2100 for a smooth circular pipe); above it, f must instead be read from the Moody chart or the Colebrook equation.
Two geometrically similar pumps, a small test model and a large prototype, are dynamically similar when their
1. absolute flow rates are equal
2. relevant dimensionless groups, such as the Reynolds number, are matched
3. absolute pressures are equal
4. physical sizes are identical
Show answer
Answer: B — relevant dimensionless groups, such as the Reynolds number, are matched
Dynamic similitude between a scaled model and a full-size prototype requires the relevant dimensionless groups the flow depends on (Reynolds number, Froude number, and for a pump also flow and head coefficients) to match between the two, not the absolute sizes, flow rates or pressures, which necessarily differ by the scale factor.