8.1.1 Continuum, density, specific weight, compressibility

Fluid mechanics treats liquids and gases as continuous media, ignoring molecular discreteness. Density rho is mass per unit volume. Specific weight gamma equals rho times g. Specific gravity is density relative to water at standard state. Liquids are treated as incompressible in most refinery hydraulics because density change with pressure is small. Gases are compressible, and density change must be retained when Mach number is not small. Vapour pressure is the pressure at which liquid boils at a given temperature. When local pressure falls to vapour pressure, vapour cavities form, the root of cavitation discussed later.

Refinery framing appears at once. Crude, diesel, lube oil, water, steam, flare gas differ mainly in density, viscosity, and vapour pressure. A dense cold crude needs more pump head for the same elevation. A hot light hydrocarbon near its bubble point cavitates easily because vapour pressure is high.

8.1.2 Newton law of viscosity and meaning of viscosity

A fluid deforms continuously under any applied shear stress, unlike a solid that reaches a finite strain. Consider two parallel plates with fluid between them, the upper plate moving with velocity U. The fluid adheres to both walls. A linear velocity profile develops across gap y. Shear stress tau is proportional to the velocity gradient.

tau equals mu times du divided by dy

Here mu is dynamic viscosity. Its unit is Pascal second. It measures internal friction due to molecular momentum exchange and intermolecular attraction. Large mu means large shear stress for the same deformation rate. Kinematic viscosity nu equals mu divided by rho, with unit square metre per second. It measures momentum diffusivity, the ratio of viscous diffusion to mass density.

Newtonian fluids obey this linear relation with constant mu at fixed temperature and pressure. Water, air, light oils, petrol, kerosene, and diesel are Newtonian in normal engineering ranges. Non-Newtonian fluids do not obey a constant mu. Pseudoplastic fluids thin with shear rate. Dilatant fluids thicken with shear rate. Bingham plastics need a yield stress before flow starts. Pastes, slurries, polymer melts, heavy residues with suspended solids, and some waxy crudes below pour point show non-Newtonian behaviour.

Temperature dependence is systematic. In liquids, molecules are closely packed and attraction dominates. Higher temperature loosens attraction, so liquid viscosity falls with temperature. In gases, molecular collisions dominate. Higher temperature increases molecular activity and momentum exchange, so gas viscosity rises with temperature. This opposite trend is a repeated exam statement. In a refinery, heating a viscous fuel oil sharply lowers its viscosity and pumping power, while heating combustion air slightly raises its viscosity and pressure drop.

8.1.3 Surface tension, contact angle, capillarity rise and fall

Surface tension sigma is force per unit length along a liquid surface, arising because surface molecules lack neighbours on one side and experience net inward attraction. Its unit is Newton per metre. Contact angle theta is the angle inside the liquid where liquid meets solid and gas. It reflects adhesion between liquid and solid versus cohesion inside the liquid.

Capillarity is the rise or depression of liquid in a small bore tube due to surface tension. Force equilibrium on the liquid column gives the standard result. Upward surface tension force around the circumference balances the weight of the raised or depressed column.

h equals 4 sigma cos theta divided by rho g d

Here d is tube diameter and h is capillary rise positive upward. When liquid wets the wall, adhesion exceeds cohesion, theta is less than 90 degrees, cos theta is positive, and liquid rises. Water in clean glass is the standard case, with theta near zero, so h is positive and inversely proportional to d. When liquid does not wet the wall, cohesion exceeds adhesion, theta exceeds 90 degrees, cos theta is negative, and liquid is depressed. Mercury in glass is the standard case, so the meniscus falls below the outer level. Smaller bore gives larger magnitude in both directions because surface force scales with circumference while weight scales with cross-sectional area.

For illustration, water at 20 degrees Celsius with sigma about 0.073 Newton per metre in a 2 mm glass tube rises about 15 mm. The same formula with negative cos theta predicts mercury depression. The effect is negligible in large process piping but matters in small gauge glasses, manometer capillaries, and level bridle tubing in fuel-oil service, where meniscus correction and bore selection control reading error.

8.2.1 Hydrostatic pressure distribution and manometry

In a fluid at rest, shear stress is zero and pressure acts equally in all directions at a point. Euler equilibrium reduces to pressure gradient balancing weight. In vertical coordinate z positive upward:

dp divided by dz equals minus rho g

For constant density, integration from free surface gives p equals rho g h, where h is depth below surface. Pressure therefore grows linearly with depth. Pressure head p divided by rho g is the height of fluid column equivalent to that pressure. Absolute pressure is measured from vacuum zero. Gauge pressure is measured from local atmosphere. Vacuum is negative gauge. Atmospheric head of water is about 10.3 m, the ceiling for suction lift and siphon action discussed later.

Manometers apply the same distribution. In a U-tube, pressure difference balances the difference of fluid columns. Heavy manometer fluids such as mercury give compact readings for high pressures. Inclined manometers magnify small differences. In refinery practice, tank level by hydrostatic tapping, differential pressure across a column section, and draft pressure in heaters all rest on this linear distribution.

8.2.2 Hydrostatic force and proof that centre of pressure lies below centroid

On a plane surface of area A inclined at any angle, with centroid at vertical depth h_c and inclined distance y_c measured along the plane from the free-surface intersection, the total hydrostatic force is:

F equals rho g h_c A equals p_c A

Thus total force equals centroid pressure times area. Derivation integrates dF equals p dA equals rho g y sin phi dA over the surface, where phi is plane inclination. Centroid definition gives the stated compact result.

The centre of pressure is the point where the single resultant force must act to produce the same moment as the distributed pressure. Taking moments about the free-surface axis along the plane:

y_p equals I_xx_o divided by y_c A

Here I_xx_o is second moment of area about the surface axis. By parallel axis theorem, I_xx_o equals I_xx_c plus A y_c squared, where I_xx_c is centroidal second moment. Substitution gives:

y_p minus y_c equals I_xx_c divided by y_c A

Since I_xx_c, y_c, and A are all positive for a submerged surface, the difference is positive. The centre of pressure therefore always lies below the centroid along the plane, deeper into higher pressure. Physically, pressure increases with depth, so the lower part carries more load and pulls the resultant downward. The offset shrinks as submergence increases because pressure becomes more uniform relative to its mean. Inclined tank walls, lock gates, and gate valves are standard applications. Hinge moment equals F times distance from hinge to centre of pressure, not to centroid. Using centroid by mistake underestimates the moment.

For curved surfaces, the standard method resolves force into horizontal and vertical components. Horizontal component equals force on the projected vertical area. Vertical component equals weight of fluid above the surface plus or minus displaced volume, acting through its centre of gravity.

8.2.3 Buoyancy and floatation

Archimedes principle states that the buoyant force on a submerged or floating body equals the weight of fluid displaced by the body:

F_b equals rho g V_displaced

Derivation integrates vertical pressure over the body surface. Horizontal components cancel. Net vertical force equals weight of the missing fluid column. The line of action passes through the centre of buoyancy B, the centroid of displaced volume.

A fully submerged body rises, sinks, or stays according as weight is less than, greater than, or equal to buoyant force. A floating body satisfies equilibrium weight equals buoyant force, with only part submerged. Flare knockout drums in refineries use this separation. Liquid droplets settle while vapour leaves overhead because weight and buoyancy plus drag sort phases by density.

8.2.4 Stability of floating bodies and metacentric height theory

Stability concerns response to a small angular tilt. Let G be centre of gravity of the body and B be centre of buoyancy in the upright position. After a small heel, the submerged shape changes, B shifts to B1, and the vertical through B1 intersects the original centreline at metacentre M. Metacentric height GM controls the righting moment.

GM equals I_waterplane divided by V_displaced minus BG

Here I_waterplane is second moment of the waterplane area about the tilt axis, V_displaced is submerged volume, and BG is distance from B to G positive when G is above B. Derivation equates the overturning or righting couple to the shift of buoyancy moment. The term I_waterplane divided by V_displaced equals BM, the distance from B to M. Subtracting BG gives GM.

If M lies above G, GM is positive, the couple rights the body, and equilibrium is stable. If M lies below G, GM is negative, the couple overturns further, and equilibrium is unstable. If M coincides with G, equilibrium is neutral. Large waterplane inertia improves stability, while high G degrades it. Free liquid surface inside a tank reduces effective GM because sloshing shifts mass to the low side. Subdivision, ballast low, and filled compartments restore margin.

Refinery framing is direct. Floating-roof storage tanks, barges for product movement, and floating roof drains obey this theory. Rim seals, roof drains, and level control keep G low and the roof level so that GM stays positive and the roof does not tilt and bind.

8.3.1 Eulerian and Lagrangian views, steady and uniform flow

Kinematics describes motion without forces. Lagrangian tracking follows a material particle. Eulerian observation records velocity at fixed spatial points as a field V of x, y, z, t. Refinery analysis is almost always Eulerian, using control volumes around pumps, heaters, columns, and pipes.

Steady flow means no change with time at a fixed point. Unsteady flow includes draining tanks, startup transients, and surge. Uniform flow means no change along the flow direction at an instant. Nonuniform flow includes converging nozzles, diffusers, bends, and trays. Material acceleration combines local and convective parts, which explains why velocity can change even in steady flow when area changes.

8.3.2 Continuity derived from mass conservation

Consider steady flow through a streamtube with inlet section 1 and outlet section 2, no storage inside. Mass inflow equals mass outflow:

rho_1 A_1 V_1 equals rho_2 A_2 V_2

This is the general continuity statement. For incompressible flow with constant rho, density cancels:

A_1 V_1 equals A_2 V_2 equals Q

Discharge Q is constant along the tube. Narrow section means higher velocity in exact inverse proportion to area. In differential form for incompressible flow, divergence of velocity is zero:

div V equals 0

Physically, net volume outflow from an infinitesimal element is zero because density cannot store volume. In a refinery crude heater, this relation links header velocity to branch velocities. Halving diameter at fixed Q quadruples velocity, which later quadruples velocity head and multiplies friction loss by a large factor.

8.3.3 Streamline, pathline, streakline

A streamline is a curve everywhere tangent to the velocity vector at one instant. Its differential equation is dx divided by u equals dy divided by v equals dz divided by w. No fluid crosses a streamline by definition. A streamtube is a bundle of neighbouring streamlines.

A pathline is the trajectory of one identified particle over time. A streakline is the locus at one instant of all particles that earlier passed through one fixed point, such as dye released continuously from a nozzle. In steady flow, velocity at each point does not change with time, so a particle follows the fixed tangent field, and dye accumulates along the same track. All three lines therefore coincide in steady flow. In unsteady flow they differ. Smoke in a gusty flare plume shows streaklines that are not streamlines. This coincidence condition is a repeated exam distinction.

8.3.4 Rotation, vorticity, irrotational flow and potentials

Fluid elements can translate, deform linearly, deform angularly, and rotate. Vorticity is twice the local angular velocity and equals curl of V. Flow with zero vorticity everywhere is irrotational. Flow with nonzero vorticity is rotational.

Irrotational flow admits a velocity potential phi such that V equals grad phi. For incompressible flow, continuity then gives Laplace equation grad squared phi equals 0. In two dimensions, a stream function psi also exists with u equals partial psi over partial y and v equals minus partial psi over partial x, and lines of constant psi are streamlines. Potential flow gives elegant solutions for flow around bodies, Venturi throats away from walls, and far fields. Real walls impose no slip, generate vorticity in boundary layers, and break potential behaviour near surfaces. Refinery tray hydraulics and flow around tube bundles therefore need rotational viscous treatment near walls even when the core is nearly irrotational.

8.4.1 Bernoulli derived from energy along a streamline

Start from Euler equation for inviscid motion along a streamline s. Pressure force plus weight accelerates the particle. For steady flow, integration with respect to s gives mechanical energy per unit weight constant along that streamline when density is constant and no shaft work or heat transfer occurs:

p divided by rho g plus V squared divided by 2 g plus z equals H constant

The three terms are pressure head, velocity head, and elevation head. Their sum H is total head. Multiplying by rho g gives pressure plus dynamic pressure plus hydrostatic term constant. Multiplying by mass flow gives power. Bernoulli is therefore mechanical energy conservation for an ideal particle. Torricelli velocity V equals sqrt of 2 g H follows as a limiting case from a large reservoir surface with near-zero velocity and atmospheric pressure to a free jet at the same atmosphere.

In refinery terms, this ideal balance explains why a constriction raises velocity and lowers pressure, why elevation rise demands pressure or pump head, and why a free jet from tank head converts elevation to velocity.

Quick illustration — Torricelli jet velocity

Given: reservoir head $H = 4.0$ m, $g = 9.81$ m/s2, large surface, free jet.

Step 1: energy balance $V = \sqrt{2gH}$.

Step 2: substitution $V = \sqrt{2 \times 9.81 \times 4.0} = \sqrt{78.48}$.

Result: $V = 8.86$ m/s.

Trap: using the same balance with loss or shaft work present inflates the jet, so reserve this form for the ideal reservoir-to-jet limiting case.

8.4.2 Four conditions, each justified

Bernoulli holds only when all of the following are satisfied.

• Steady flow. Time derivatives must vanish. In unsteady draining, startup, or water hammer, an additional temporal acceleration term appears and total head is not constant along the path.

• Incompressible flow with constant density. Density must come outside the pressure integral. When Mach number exceeds about 0.3 in gases, density change is significant, compressible relations replace Bernoulli, and flare-gas flow at high velocity needs compressible treatment.

• Frictionless flow or losses treated separately. Viscous shear dissipates mechanical energy into heat. In long pipes, valves, rough passages, and separated zones, head loss is not negligible. Plain Bernoulli overpredicts velocity and underpredicts required head unless h_L is added.

• No shaft work and no heat transfer between the two points, and both points on the same streamline in rotational flow. A pump adds energy, a turbine extracts energy, heat addition changes internal energy. Energy balance must then include those terms. In irrotational potential flow the constant is global, but in general rotational flow each streamline has its own constant.

Checking these four before calculation prevents the most common exam error.

8.4.3 Engineering energy equation with pump, turbine, and loss

Real systems include machines and friction. Between section 1 and section 2 in the flow direction:

p_1 divided by rho g plus V_1 squared divided by 2 g plus z_1 plus h_pump equals p_2 divided by rho g plus V_2 squared divided by 2 g plus z_2 plus h_turbine plus h_L

Here h_pump is head added by the pump, h_turbine is head extracted by the turbine, and h_L is irreversible head loss from friction and separation. Hydraulic grade line joins pressure plus elevation heads. Energy grade line adds velocity head and therefore lies above by V squared over 2 g. Pumps cause a sharp rise in energy grade line. Turbines cause a sharp drop. Friction causes a continuous downward slope.

A distillation reboiler thermosiphon loop is solved with this equation. Circulation rate follows from balance between thermal driving head and total h_L through exchanger, piping, and fittings. Cooling-water networks, furnace feed lines, and product transfer lines follow the same form.

8.4.4 Reason that application across a pump fails with plain Bernoulli

Applying plain Bernoulli from pump suction flange to delivery flange omits h_pump and usually h_L. Plain Bernoulli predicts nearly equal total head at both flanges, apart from small velocity and elevation changes. The real delivery head is higher by approximately the pump head minus internal loss. The across-pump application therefore violates energy conservation because shaft work crosses the control surface between the points. Correct practice places the control sections either both upstream of the pump or both downstream when using plain Bernoulli, or retains h_pump when the control spans the machine. The same logic forbids plain Bernoulli across a turbine, a fan, a compressor stage, or any valve-dominated reach unless loss is included. In exam language, the missing term is shaft head, and the symptom is apparent creation of head from nothing.

8.5.1 Common principle and discharge coefficient meaning

All constriction meters combine continuity with Bernoulli. Area reduction raises velocity and lowers pressure. Measured pressure drop therefore indicates discharge. Theoretical discharge assumes ideal one-dimensional frictionless flow. Actual discharge is smaller because of friction, nonuniform profiles, and separation. Discharge coefficient C_d is defined as:

C_d equals actual discharge divided by theoretical discharge

It is always less than unity for these meters and must be determined by calibration or standard correlations. Pitot coefficient is defined analogously for velocity. Weir coefficients likewise correct ideal nappe theory. Larger dissipation means smaller C_d and larger permanent pressure loss.

8.5.2 Venturi meter construction and working

A Venturi has a smooth converging cone, a short cylindrical throat, and a long gentle diverging diffuser, typically with total diffuser angle small enough to avoid separation. Pressure taps at inlet and throat feed a manometer. Let A_1 and V_1 be inlet, A_2 and V_2 throat, h head difference in flowing fluid column. Continuity gives V_1 A_1 equals V_2 A_2. Bernoulli between inlet and throat gives pressure drop corresponding to velocity rise. Solving yields theoretical discharge, then multiplying by C_d:

Q equals C_d A_1 A_2 sqrt of 2 g h divided by sqrt of A_1 squared minus A_2 squared

For a manometer with heavy fluid, h is converted to flowing-fluid head using density ratio. C_d for a well-made Venturi is about 0.97 to 0.99 because the streamlined shape keeps the boundary layer attached and h_L small. Permanent loss is only a small fraction of the measured drop, mainly in the diffuser. Venturis are therefore preferred in large cooling-water headers and crude charge lines where pumping cost of permanent loss matters. The diagram below shows the profile and tap positions.

8.5.3 Orifice meter and vena contracta

An orifice meter is a thin sharp-edged plate with a concentric hole clamped between flanges, with taps upstream and downstream. The jet contracts after the plate to a vena contracta smaller than the hole, then expands with eddies. The same Bernoulli plus continuity form applies with orifice area, but theoretical velocity uses the vena contracta implicitly through C_d. C_d is about 0.60 to 0.65 because contraction plus separated expansion dissipates a large fraction of the drop as h_L. The meter is cheap, compact, and easy to install, but permanent pressure loss is large and the edge must stay sharp. Erosion or deposits shift C_d. In refineries, orifices suit steam, fuel gas, and utility measurements where first cost matters more than pumping loss, while Venturis suit large continuous water and crude flows.

8.5.4 Pitot tube for point velocity

A Pitot tube has a forward hole sensing stagnation pressure p_0 where the stream is brought to rest, and side holes or a separate tapping sensing static pressure p. Along the stagnation streamline, Bernoulli gives:

V equals C_p sqrt of 2 times p_0 minus p divided by rho

Coefficient C_p is near unity for aligned tubes. The device measures local velocity, not bulk discharge. Traversing across a duct or stack and integrating the profile gives flow. Pitot checks suit flare stacks, ducts, and air-cooler air streams. Misalignment, pulsation, and blockage by deposits are the main errors. Static holes must be burr free and parallel to flow.

8.5.5 Notches and weirs for open channels

Sharp-crested notches measure open-channel flow such as effluent and API separator outflow. Theory integrates velocity sqrt of 2 g times depth over the nappe. Triangular V-notch with included angle theta and head H above vertex gives:

Q equals 8 over 15 times C_d sqrt of 2 g tan of theta over 2 times H to power 2.5

Rectangular weir of width B gives Q equals 2 over 3 times C_d B sqrt of 2 g times H to power 1.5. C_d is about 0.58 to 0.62, accounting for contraction, friction, and nappe aeration. V-notch gives high sensitivity at low flows because exponent 2.5 amplifies head change. Rectangular weir passes large flows. Approach velocity, crest sharpness, and full aeration control accuracy. Refinery effluent channels and oily-water separators use weirs both to measure and to skim uniformly.

8.6.1 Darcy-Weisbach equation and diameter scaling

Head loss due to wall friction in a pipe of length L, diameter D, mean velocity V is:

h_f equals f times L divided by D times V squared divided by 2 g

Here f is Darcy friction factor, dimensionless, dependent on Reynolds number and roughness. At fixed discharge Q, V equals 4 Q divided by pi D squared. Substitution gives:

h_f equals 8 f L Q squared divided by pi squared g D to power 5

Thus at fixed Q and f, loss scales as 1 divided by D to power 5. Halving diameter multiplies loss by 32. The physical chain is compounding. Smaller area raises velocity as 1 over D squared, velocity head rises as 1 over D to power 4, and L over D adds one more 1 over D. One undersized crude transfer line can therefore starve a furnace while the pump runs near shutoff. Temperature helps through viscosity because f depends on Reynolds number, and heating viscous oil lowers loss more than density change alone suggests.

Quick illustration — diameter power proof by numbers

Given: $Q = 0.05$ m3/s, $L = 100$ m, $f = 0.02$, $D_1 = 0.20$ m, $D_2 = 0.10$ m.

Step 1: loss law $h_f = 8fLQ^2/(\pi^2 g D^5)$.

Step 2: larger pipe $h_{f1} = 1.29$ m from $D_1^5$ in the denominator.

Step 3: halved pipe $h_{f2} = 41.31$ m, ratio $h_{f2}/h_{f1} = 32$.

Result: halving $D$ at fixed $Q$ multiplies loss by $32$.

Trap: scaling loss with $1/D^2$ or $1/D^4$ misses the extra $L/D$ factor, so retain the full fifth power.

8.6.2 Laminar friction factor idea and Blasius correlation

Reynolds number Re equals rho V D divided by mu is the ratio of inertia to viscous force. Below about 2000, pipe flow stays laminar. Force balance on a laminar cylinder with parabolic profile yields pressure drop proportional to V, which converts to f equals 64 divided by Re. This is exact for fully developed laminar pipe flow, not empirical.

In smooth turbulent flow up to Re about 1e5, Blasius correlation gives:

f equals 0.316 times Re to power minus 0.25

At Re 1e5, f is about 0.0178. The correlation reflects thin viscous sublayer control. At higher Re or with roughness, the viscous sublayer thins and roughness elements protrude, raising f above Blasius. Fully rough turbulence makes f independent of Re. Lube-oil lines often lie in laminar range, cooling water and product lines lie in turbulent range, so the same pipe equation uses very different f values.

8.6.3 Moody chart reading logic

Moody chart plots f on log vertical axis against Re on log horizontal axis, with curves of relative roughness k_s divided by D. Roughness k_s is equivalent sand height. Procedure in prose is fixed. Reynolds number follows from velocity, diameter, and kinematic viscosity. Relative roughness follows from pipe material and age. The chart entry is at Re on the horizontal axis, then vertical movement to the curve for that roughness, interpolating between printed curves, then horizontal movement to read f. Laminar region is the straight line f equals 64 over Re. Transition lies near Re 2000 to 4000. Smooth turbulent follows Blasius and Prandtl lines at low roughness. Rough curves flatten to horizontal in complete turbulence where f depends only on roughness. Old corroded crude lines have larger k_s, so operating point moves upward to larger f and head loss rises even at fixed Q.

8.6.4 Minor losses with table and reasoning

Bends, entrances, exits, valves, contractions, and expansions add loss h_m equals K times V squared over 2 g, where K is loss coefficient referred to local velocity head. Equivalent length L_e equals K D divided by f converts a fitting into straight pipe length for network sums.

Fitting	K approximate	Physical reasoning
Sharp entrance	0.5	vena contracta at inlet plus eddy reattachment dissipates about half a velocity head
Pipe exit to reservoir	1.0	whole kinetic energy is dissipated as mixing in the reservoir
90 degree flanged elbow	0.3	secondary flow plus separation on inner wall, smooth bend lower than miter
Fully open gate valve	0.15	gate retracts from passage, small obstruction and attached flow
Fully open globe valve	10	tortuous rise and fall with seats forces separation, very lossy
Sudden contraction	about 0.5 times contraction severity	contraction vena contracta then expansion eddy
Sudden expansion	1 minus A_1 over A_2 squared	Borda Carnot mixing loss grows with area ratio

Globe valves therefore belong only where throttling is intended. Gate valves belong in isolation duty. Long suction lines avoid strainers and sharp entrances where possible because every K directly reduces NPSH margin.

8.6.5 Series, parallel, and equivalent networks

Pipes in series carry the same Q, heads add. Total h equals h_1 plus h_2 plus minors. Composite resistance is sum of L over D to power 5 terms when f is similar. Pipes in parallel share the same head between junctions, discharges add. Total Q equals Q_1 plus Q_2. The smaller or smoother branch carries more flow in proportion to its conductance. Three-reservoir and looped networks solve by continuity at nodes plus head compatibility around loops, iterated because f depends on flow. Refinery cooling-water loops, fire-water rings, and crude distribution headers are parallel systems where one choked branch redistributes flow to others.

8.6.6 Water hammer and siphon

Sudden valve closure converts kinetic energy into pressure wave. Joukowsky rise is:

delta p equals rho c delta V

Here c is wave speed, set by fluid bulk modulus and pipe elasticity, of order 1000 m per s in water steel pipes. Even 1 m per s velocity change can generate about 10 bar surge. Slow closure relative to wave reflection time L over c, surge vessels, standpipes, and pump flywheel effect limit the peak. Long product pipelines and cooling-water discharge lines need this protection. Normal operation uses ramped valve strokes and check-valve slam control.

A siphon carries liquid over a summit above source level by subatmospheric pressure. Bernoulli from source surface to summit gives summit pressure below atmosphere by elevation plus velocity head plus loss. Absolute summit head cannot fall below vapour head without column separation and vapour lock. Practical water siphon lift is well below 10.3 m, allowing for loss and vapour margin. Air release at the summit must be vented during priming. Refinery tank drainage over bund walls and temporary transfers use siphons within this lift limit.

8.7.1 Hagen-Poiseuille parabolic profile and pressure drop

Fully developed steady laminar flow in a circular pipe of radius R, with no-slip at wall and axisymmetry, satisfies a force balance between pressure and viscous shear on concentric cylinders. Integration gives a paraboloid:

u of r equals 2 V_avg times 1 minus r squared over R squared

Centreline velocity at r equals 0 is twice the average. Wall velocity is zero. Profile shape is independent of flow rate in laminar regime, scaling only in magnitude. Volume integration gives Hagen-Poiseuille drop:

Delta p equals 32 mu L V divided by D squared

In head form, h_f equals 32 mu L V divided by rho g D squared. Loss is linear in V, unlike turbulent loss quadratic in V. This linearity is why viscous heating and pumping of lube oil, bitumen, and heavy fuel respond directly to velocity control and strongly to temperature through mu. Entrance length for development is about 0.05 Re D laminar, longer than turbulent, so short tubes may not reach the paraboloid.

8.7.2 Boundary-layer growth on a wall

Prandtl boundary layer is the thin region near a wall where velocity rises from zero at the wall to free-stream value outside. Outside, inviscid theory applies. Inside, viscosity matters despite high Re because transverse gradients are large. On a smooth flat plate, laminar thickness by Blasius scaling is:

delta equals 5 x divided by sqrt of Re_x

Here Re_x equals rho V x divided by mu with x distance from leading edge. At x 1 m and Re_x 5e5, delta is about 7 mm. Thickness grows as sqrt of x laminar because viscous diffusion spreads inward while convection carries fluid downstream. Turbulent boundary layers grow faster, about 0.37 x divided by Re_x to power 0.2, with fuller profiles and larger wall shear. Displacement thickness quantifies mass deficit, momentum thickness quantifies momentum deficit, and their ratio indicates separation tendency.

Tube bundles, air-cooler fins, and column internals all carry such layers. Fouling thickens the effective layer, raises resistance, and degrades heat transfer together.

8.7.3 Separation by adverse pressure gradient and streamlining

Along a surface, pressure gradient controls near-wall momentum. Favourable gradient with pressure falling downstream accelerates the layer and keeps it attached. Adverse gradient with pressure rising downstream decelerates the layer. Near-wall fluid has least momentum, so it stops and reverses first, while outer fluid continues. The layer then detaches as separation, forming a wake with eddies, large pressure drag, vibration, and dead zones.

Rear of blunt valve bodies, tube cross-flow, and diffuser with too wide angle all produce adverse gradients. Streamlining shapes the body with gentle contraction and slow pressure recovery so that adverse gradient stays mild and separation is delayed. A Venturi diffuser angle is kept small for the same reason. A sudden expansion separates at the corner by extreme adverse change. Refinery air-cooler rows, flare tips, and control-valve bodies use rounded entries, guided turns, and limited diffuser angles to suppress separation-induced vibration and loss.

8.8.1 Buckingham pi method in procedural form

When governing equations are complex, dimensional homogeneity still constrains the functional form. Buckingham pi theorem states that if n dimensional variables involve r fundamental dimensions, usually mass length time, the relation reduces to n minus r independent dimensionless groups.

Method in prose is fixed. List all relevant variables including geometry, fluid properties, velocity, pressure drop, and gravity or surface tension where relevant. Count fundamental dimensions r. Select r repeating variables that together contain all dimensions and do not themselves form a dimensionless group. Form each pi by combining one remaining variable with powers of repeaters to cancel dimensions. The original dimensional relation becomes a relation among pis. Experiments then vary pis, not raw variables, greatly reducing test matrix. Pipe friction analysis reduces variables rho, V, D, mu, k_s, delta p to Re, relative roughness, and pressure coefficient, which is the basis of Moody chart.

8.8.2 Meaning of Reynolds, Froude, Mach, Euler, Weber numbers

Reynolds number Re equals rho V L divided by mu, inertia over viscous force. It governs transition, friction factor, and viscous similarity. High Re means thin viscous influence.

Froude number Fr equals V divided by sqrt of g L, inertia over gravity force. It governs free-surface waves, hydraulic jumps, spillways, ship wakes, and open channels.

Mach number Ma equals V divided by c, inertia over compressibility effect, where c is sound speed. It governs density change, choking, and shocks. Below about 0.3, compressibility is small enough for incompressible treatment.

Euler number Eu equals p divided by rho V squared, pressure over inertia. It governs pressure coefficients and cavitation index forms.

Weber number We equals rho V squared L divided by sigma, inertia over surface tension. It governs droplets, atomization, capillary waves, and weir nappe aeration at small scales.

Refinery mapping is immediate. Pipes and exchangers scale by Re. Flare plumes with buoyancy and open effluent channels scale by Fr. Fuel-gas and relief headers at high velocity check Ma. Burner atomization and desalter droplets check We.

8.8.3 Model laws and ship Froude illustration

Complete similarity needs geometric, kinematic, and dynamic similarity. In practice the dominant force is matched while others are kept in noncritical range. Free-surface flows with waves match Froude, accepting Re mismatch if turbulence is fully rough. Enclosed viscous flows match Reynolds, accepting gravity mismatch where no free surface exists. High-speed gas flows match Mach. Droplet and capillary studies match Weber.

For ships, spillways, and open channels, Froude equality gives velocity scale as sqrt of length scale. A 1 to 36 ship model therefore runs at full-scale velocity divided by 6 because sqrt of 36 is 6. Resistance is then scaled by density and length ratios with frictional correction for Re mismatch. The same Froude scaling sizes weir models and cooling-water outfall models where gravity waves dominate.

8.9.1 Pelton, Francis, Kaplan construction

Pelton is an impulse turbine for very high head. A needle nozzle or multiple jets discharge at atmospheric pressure onto double-cup buckets mounted on a wheel. A spear valve regulates jet diameter. Casing only prevents splashing and does not sustain pressure. Energy conversion is pressure to jet kinetic in the nozzle, then jet impulse to runner work at nearly constant bucket pressure. No draft tube is used because exhaust is already atmospheric.

Francis is a mixed-flow reaction turbine for medium head. Water enters a spiral volute, passes stay vanes and adjustable guide vanes, flows radially inward through the runner, and exits axially through a draft tube. Pressure falls through guide vanes and runner together. Runner passages are full of pressurized water. Guide-vane angle controls discharge and power.

Kaplan is an axial-flow reaction turbine for low head and large discharge. Flow enters a spiral and guide ring, then passes an axial runner with adjustable blades like a ship propeller inside a cylindrical passage, followed by a draft tube. Both guide vanes and runner blades adjust together for high efficiency over wide load, the so-called double regulation. Fixed-blade version is called propeller turbine. Large cooling-water or hydropower recovery settings favour Kaplan geometry where head is small but volume is huge.

Refinery relevance is energy recovery. High-pressure condensate letdown, hydraulic power recovery turbines on fluid catalytic cracking units, and pressure-exchange stages follow the same impulse versus reaction choice by head and discharge.

8.9.2 Head discharge ladder

Head H and discharge Q together select the machine. High head above about 250 m with relatively small Q calls Pelton, because a free jet handles extreme pressure without pressurized runner passages and multi-jet arrangements share flow. Medium head about 40 to 600 m with medium Q calls Francis, because reaction passages digest moderate pressure efficiently. Low head below about 60 m with large Q calls Kaplan, because axial large-area passages swallow volume at small pressure. Overlap bands exist by specific speed and maker practice, but the ordering by head from high to low is Pelton then Francis then Kaplan, while ordering by discharge at fixed power is reversed. The diagram below encodes this ladder.

8.9.3 Specific speeds and unit quantities

Specific speed compresses shape into one number. Turbine specific speed is:

N_s equals N sqrt of P divided by H to power 1.25

With N in rpm, P in kW, H in m. Single-jet Pelton lies roughly 10 to 60, Francis roughly 60 to 300, Kaplan above 300. Large N_s means large discharge passage relative to head, hence axial geometry. Pump specific speed uses a different definition, N sqrt of Q divided by H to power 0.75, and the two must never be interchanged.

Unit quantities compare the same runner under different heads at fixed gate. Unit speed N_u equals N divided by sqrt of H. Unit discharge Q_u equals Q divided by sqrt of H. Unit power P_u equals P divided by H to power 1.5. They follow from similarity with velocity proportional to sqrt of H. Test data collapse onto unit curves, then scale to site head.

8.9.4 Draft tube and Thoma cavitation criterion

A draft tube is the diverging exit passage from a reaction runner to tailrace. It serves two functions. It recovers exit kinetic head by diffusion, raising effective head across the runner. It permits the runner to sit above tailrace with a suction head because the tube sustains subatmospheric pressure at runner exit without air ingress. Pelton needs no draft tube because its jet and buckets already exhaust to atmosphere.

Cavitation is vapour bubble formation where local pressure falls to vapour pressure, followed by violent collapse downstream. Symptoms are pitting on blades, noise, vibration, and efficiency loss. Thoma number sigma judges margin:

sigma equals H_atm minus H_v minus H_s all divided by H

Here H_atm is atmospheric head, H_v vapour head, H_s suction head of runner above tailrace, H net head. Critical sigma_c depends on machine type and specific speed. Safe design needs sigma above sigma_c. Lowering the runner, cooling the fluid, and selecting lower specific speed all raise margin. Refinery hydraulic recovery turbines and boiler-feed settings apply the same criterion as pumps, only with turbine head in the denominator.

8.10.1 Centrifugal and reciprocating construction compared

A centrifugal pump is a rotary dynamic machine. An impeller with curved vanes spins inside a volute or diffuser casing. Liquid enters axially at the eye, gains whirl and pressure through the impeller, and diffuses to delivery pressure in the casing. Flow is continuous. Euler head for infinite vanes is H equals V_w2 u_2 minus V_w1 u_1 all divided by g, where u is blade speed and V_w tangential fluid velocity. Backward vanes give stable drooping characteristics.

A reciprocating pump is a positive displacement machine. A piston or plunger in a cylinder with suction and delivery check valves displaces a fixed volume each stroke driven by crank and connecting rod. Flow pulsates as sine-like pulses. Slip is the difference between theoretical and actual discharge due to leakage and valve lag.

Comparison in tabular form encodes exam duty choice.

Feature	Centrifugal	Reciprocating
Working action	Continuous rotary dynamic flow	Positive displacement pulsating strokes
Head and discharge	Medium head, large discharge	High head, small discharge
Starting practice	Against closed delivery valve to limit power	Against open delivery valve, never closed, relief valve essential
Priming need	Required, not self priming because air density is too low to sustain suction	Self priming by displacement
Viscosity suitability	Clean low viscosity liquids, performance falls with viscous friction	Viscous liquids, metering, high pressure dosing
Maintenance and cost	Simple, low first cost, uniform flow	More parts, valves and seals wear, pulsation control needed
Refinery post	Cooling water, crude charge, reflux circulation	High pressure chemical injection, dosing, desalter feed at high pressure

Refinery framing is direct. Cooling-water and crude-charge duties need bulk movement at moderate head and favour centrifugal machines in parallel. Chemical injection and high-pressure wash need exact small flow at high pressure and favour reciprocating or other displacement machines.

8.10.2 Static, manometric, and total heads plus efficiencies

Suction head is elevation plus pressure at pump inlet. Delivery head is elevation plus pressure at outlet. Static head is their sum against gravity. Manometric head H_m adds friction in suction and delivery lines plus velocity head at delivery. It is the useful head actually imparted to liquid. Total dynamic head adds instrument corrections where needed.

Efficiencies chain multiply. Hydraulic or manometric efficiency equals H_m divided by Euler head, accounting for impeller and casing losses. Volumetric efficiency accounts for leakage through clearances. Mechanical efficiency accounts for bearings, seals, and disc friction. Overall efficiency equals rho g Q H divided by shaft power P_shaft. Operating point is the intersection of pump H-Q curve with system curve static head plus k Q squared. Throttling delivery raises system k and moves duty to lower Q and higher head. Speed change shifts the whole pump curve by affinity laws.

8.10.3 Affinity laws derived from similarity

At fixed impeller diameter D and varying speed N, velocity triangles remain similar when flow coefficient is preserved. Blade speed u is proportional to N D. Radial flow component is proportional to Q divided by D squared. Head is proportional to u squared. Power is proportional to rho g Q H. Eliminating geometry at fixed D gives:

Q proportional to N, H proportional to N squared, P proportional to N cubed

Full diameter form is Q proportional to N D cubed, H proportional to N squared D squared, P proportional to N cubed D to power 5. Doubling speed therefore doubles flow, quadruples head, and demands eight times the power. The cubic power law is the motor-burnout mechanism. A drive and motor must be rated for the higher speed, and higher speed also lowers NPSH margin and raises noise. Variable-speed drives in cooling-water service exploit the same laws in reverse. Small speed reduction gives large power saving at slightly lower flow.

Quick illustration — affinity doubling

Given: speed ratio $N_2/N_1 = 2$, base $Q_1 = 0.04$ m3/s, $H_1 = 25$ m, $P_1 = 12$ kW.

Step 1: flow $Q_2 = Q_1 \times 2 = 0.08$ m3/s from $Q \propto N$.

Step 2: head $H_2 = H_1 \times 2^2 = 25 \times 4 = 100$ m from $H \propto N^2$.

Step 3: power $P_2 = P_1 \times 2^3 = 12 \times 8 = 96$ kW from $P \propto N^3$.

Result: $Q$ doubles, $H$ quadruples, $P$ rises eightfold.

Trap: budgeting motor power linearly with speed invites overload, so rate the drive for the cubic rise plus NPSH check.

8.10.4 NPSH theory and prevention logic with schematic

Net positive suction head available at pump inlet centreline is suction stagnation head minus vapour head:

NPSH_a equals p_s divided by rho g plus V_s squared divided by 2 g minus p_v divided by rho g

Required NPSH_r is a maker curve, the NPSH at which head drops by a defined amount due to cavitation inception, rising with Q and N. Safe operation needs NPSH_a above NPSH_r with margin, typically with allowance for transients and fouling. When margin vanishes, bubbles form at impeller eye, collapse violently downstream, and cause pitting, rattling, swinging pressure, and head loss.

Prevention logic follows directly from the definition. Raise static suction head by elevating sump level or lowering pump elevation. Reduce suction loss by shorter wider suction pipe, smooth bends, clean strainer, and fully open suction valve. Cool the liquid where possible to lower p_v. Reduce speed to lower NPSH_r and raise NPSH_a through lower suction loss. Throttle the delivery valve, not the suction valve, to move duty to lower Q where NPSH_r is smaller. Suction throttling is prohibited because it directly lowers p_s and NPSH_a and worsens cavitation. Operation far from best efficiency point is avoided because recirculation lowers local pressure. The schematic below shows the head margins.

Quick illustration — NPSH margin call

Given: $p_s/(\rho g) = 6.0$ m, $V_s^2/(2g) = 0.5$ m, $p_v/(\rho g) = 2.3$ m, $NPSH_r = 3.0$ m.

Step 1: available head $NPSH_a = 6.0 + 0.5 - 2.3 = 4.2$ m.

Step 2: margin $NPSH_a - NPSH_r = 4.2 - 3.0 = 1.2$ m.

Result: margin $1.2$ m positive, so the duty runs clear of cavitation inception.

Trap: throttling the suction valve to cut flow lowers $p_s$ and erases this margin, so throttle the delivery side only.

8.10.5 Priming need and air vessels for pulsation

Priming means filling casing and suction line fully with liquid and venting air before start. A centrifugal pump with air inside cannot develop suction because air density is about a thousandth of liquid density, so the same impeller head in metres of air gives negligible pressure rise. Foot valves, priming funnels, vacuum priming, and flooded suction layouts ensure liquid-filled start. Reciprocating pumps displace air directly and are self priming, though priming still eases valve seating.

Air vessels are closed chambers with trapped air on reciprocating pump lines. On delivery side, the vessel stores liquid during peak stroke and releases during trough, smoothing flow toward uniform and reducing acceleration head and power fluctuation. On suction side, a smaller vessel near the cylinder reduces suction acceleration and improves filling. The air cushion compresses and expands elastically. Periodic air recharge compensates dissolved air loss. Long chemical injection and boiler-feed displacement lines use vessels to protect piping from pulsation fatigue. Refinery cooling-water centrifugal systems do not need air vessels for pulsation but use surge vessels for water hammer, a distinct transient function.

• Fluid behaviour starts from Newton law tau equals mu du over dy. Dynamic viscosity mu and kinematic viscosity nu separate friction from density. Liquids thin with heating while gases thicken. Capillary rise or depression follows h equals 4 sigma cos theta over rho g d, with wetting liquids rising and nonwetting liquids depressed.
• Hydrostatics gives linear pressure p equals rho g h, resultant F equals rho g h_c A, and centre of pressure below centroid by I_xx_c over y_c A because deeper pressure carries more load. Buoyancy equals displaced weight. Floating stability needs positive GM equals I_waterplane over V minus BG.
• Kinematics gives incompressible continuity A_1 V_1 equals A_2 V_2 and div V equals 0. Streamline, pathline, and streakline coincide only in steady flow. Zero curl defines irrotational flow with velocity potential, broken near walls by boundary layers.
• Bernoulli p over rho g plus V squared over 2 g plus z constant is energy conservation for steady incompressible frictionless flow with no shaft work along a streamline. Each condition has a physical reason tied to unsteady, compressible, viscous, or machine effects. Real systems use the energy equation with h_pump added and h_turbine plus h_L subtracted. Plain Bernoulli across a pump omits shaft head and violates energy balance.
• Constriction meters convert pressure drop to discharge through Bernoulli plus continuity. Venturi C_d about 0.97 to 0.99 reflects attached streamlined flow. Orifice C_d about 0.60 to 0.65 reflects vena contracta plus eddy loss. Pitot gives point velocity from stagnation minus static. V-notch and rectangular weirs integrate nappe velocity with C_d about 0.58 to 0.62.
• Pipe loss follows Darcy-Weisbach h_f equals f L over D times V squared over 2 g, scaling as 1 over D to power 5 at fixed Q. Laminar f equals 64 over Re follows from Poiseuille. Smooth turbulence follows Blasius to Re 1e5. Moody chart reads f from Re and relative roughness. Minors use K times velocity head with globe valves very lossy and gate valves mild. Series add heads, parallel add discharges. Hammer follows rho c delta V. Siphon lift is limited by atmospheric minus vapour head.
• Laminar pipe profile is parabolic u equals 2 V_avg times 1 minus r squared over R squared, centre twice mean, drop Delta p equals 32 mu L V over D squared. Boundary layer grows as 5 x over sqrt Re_x laminar. Adverse pressure rise causes separation and wake drag. Gentle shaping and small diffuser angles keep flow attached.
• Buckingham pi reduces n variables with r dimensions to n minus r groups. Re is inertia over viscous, Fr inertia over gravity, Ma inertia over compressibility, Eu pressure over inertia, We inertia over surface tension. Free-surface models match Froude with velocity ratio sqrt of length ratio, enclosed viscous models match Reynolds.
• Turbines order by head as Pelton high, Francis medium, Kaplan low, with discharge ordering reversed. Constructions differ as buckets and nozzle, spiral with guide vanes and runner, axial adjustable blades. Turbine N_s equals N sqrt P over H to power 1.25 selects shape. Unit N_u, Q_u, P_u scale one runner across heads. Draft tube recovers kinetic head and allows suction setting. Thoma sigma must exceed critical to avoid cavitation.
• Centrifugal pumps suit bulk moderate-head flow with priming needed, reciprocating pumps suit high-head small viscous or metering flow with self priming. Heads separate static, manometric, and loss parts. Efficiencies chain to overall rho g Q H over shaft power. Affinity Q with N, H with N squared, P with N cubed follows from velocity-triangle similarity and explains cubic power penalty. NPSH_a minus vapour head must exceed NPSH_r. Margin is protected by higher sump or lower pump, clean wide short suction, cooler liquid, lower speed, and delivery throttling, never suction throttling. Priming fills air out of centrifugal casings. Air vessels smooth reciprocating pulsation.
• Refinery translation stays constant. Heater circulation is head loss, knockout separation is buoyancy, tank farms are hydrostatics and GM stability, cooling-water reliability is Venturi metering plus Moody loss plus NPSH discipline, and recovery power is turbine ladder plus Thoma margin.