3. Basic Water Resources Engineering (ACiE03)

3. Basic Water Resources Engineering (ACiE03)

3.1 Fluids and Their Properties

This section delves into the fundamental characteristics and classifications of fluids, essential for understanding their behavior in engineering applications.

Types of Fluids

• Newtonian Fluids: Fluids where the shear stress is directly proportional to the rate of shear strain. The viscosity is constant and independent of the shear rate. Examples include water, air, and most common oils.
• Non-Newtonian Fluids: Fluids where the shear stress is not linearly dependent on the rate of shear strain. Their viscosity changes with the applied shear rate. Examples include blood, ketchup, and paint.
• Compressible Fluids: Fluids whose density changes significantly with pressure and temperature. Gases are typically compressible.
• Incompressible Fluids: Fluids whose density remains constant regardless of pressure and temperature changes. Liquids are generally considered incompressible for most engineering purposes.
• Ideal Fluids: Theoretical fluids that are incompressible, non-viscous, and have no surface tension. They are used for simplifying theoretical analysis.
• Real Fluids: Fluids that possess viscosity, compressibility, and surface tension, and exhibit resistance to flow.

Fluid Properties

• Mass Density (ρ): The mass of a fluid per unit volume.
  ρ = m/V
  Where:
  • ρ is mass density (kg/m³)
  • m is mass (kg)
  • V is volume (m³)
• Specific Weight (γ): The weight of a fluid per unit volume.
  γ = ρg
  Where:
  • γ is specific weight (N/m³)
  • ρ is mass density (kg/m³)
  • g is acceleration due to gravity (m/s²)
• Specific Gravity (S): The ratio of the density of a fluid to the density of a reference substance (usually water at 4°C). It is a dimensionless quantity.
  S = ρ / ρ_w
  Where:
  • S is specific gravity
  • ρ is density of the fluid (kg/m³)
  • ρ_w is density of water (approximately 1000 kg/m³)
• Specific Volume (v): The volume occupied by a unit mass of a fluid. It is the reciprocal of mass density.
  v = 1/ρ
  Where:
  • v is specific volume (m³/kg)
  • ρ is mass density (kg/m³)

Viscosity

Viscosity is a measure of a fluid's resistance to shear or tensile stress. It is often described as the "thickness" of a fluid.

• Dynamic Viscosity (μ): Also known as absolute viscosity, it is the resistance to shear.
  τ = μ * (du/dy)
  Where:
  • τ is shear stress (Pa or N/m²)
  • μ is dynamic viscosity (Pa·s or kg/(m·s))
  • du/dy is the velocity gradient (rate of shear strain) (s⁻¹)
  Example: For a Newtonian fluid, if the shear stress is 10 Pa and the velocity gradient is 5 s⁻¹, the dynamic viscosity is 2 Pa·s.
• Kinematic Viscosity (ν): The ratio of dynamic viscosity to density. It represents the fluid's resistance to flow under gravity.
  ν = μ / ρ
  Where:
  • ν is kinematic viscosity (m²/s)
  • μ is dynamic viscosity (Pa·s)
  • ρ is mass density (kg/m³)
  Example: If water has a dynamic viscosity of 1.0 x 10⁻³ Pa·s and a density of 1000 kg/m³, its kinematic viscosity is 1.0 x 10⁻⁶ m²/s.

Bulk Modulus of Elasticity (K)

A measure of a fluid's resistance to compression.

K = - dP / (dρ/ρ)

Where:

• K is bulk modulus of elasticity (Pa)
• dP is the change in pressure (Pa)
• dρ is the change in density (kg/m³)
• ρ is the original density (kg/m³)

A higher bulk modulus indicates a less compressible fluid.

Capillarity

The phenomenon by which a liquid spontaneously rises or falls in a narrow tube due to the combined effects of surface tension, adhesion, and cohesion.

• Surface Tension (σ): The tendency of liquid surfaces to shrink into the minimum surface area possible. It acts like a thin elastic membrane. Measured in N/m.
• Capillary Rise (h): The height to which a liquid rises in a narrow tube due to capillary action.
  h = (2σ cosθ) / (ρgr)
  Where:
  • h is capillary rise (m)
  • σ is surface tension (N/m)
  • θ is the contact angle between the liquid and the tube wall (degrees or radians)
  • ρ is the density of the liquid (kg/m³)
  • g is acceleration due to gravity (m/s²)
  • r is the radius of the tube (m)
  Example: Water in a glass tube (θ ≈ 0°) will rise, while mercury (θ > 90°) will depress.

Cavitation and Vapour Pressure (pv)

• Vapour Pressure (pv): The pressure exerted by the vapor of a liquid when it is in equilibrium with its liquid phase at a given temperature. If the pressure in a liquid drops below its vapour pressure, the liquid can vaporize, forming bubbles.
• Cavitation: The formation and subsequent collapse of vapor bubbles within a liquid when the local pressure drops below the vapour pressure. This can cause significant damage to machinery (e.g., pump impellers, turbine blades) due to the shock waves generated by collapsing bubbles.

3.2 Hydrostatics

This section deals with fluids at rest, focusing on pressure distribution and forces exerted by stationary fluids.

Pressure and Head

• Pressure (p): Force exerted per unit area. In fluids at rest, pressure is transmitted equally in all directions. Measured in Pascals (Pa) or N/m².
• Head: A measure of energy per unit weight. It is often expressed in terms of a height of a fluid column.
  • Pressure Head (p/γ): The height of a fluid column equivalent to the pressure. p = γh, so h = p/γ.
  • Velocity Head (V²/2g): Represents the kinetic energy per unit weight of a fluid.
  • Datum Head (z): The height of a point above a reference datum (e.g., sea level), representing potential energy per unit weight.

Pascal's Law

Pascal's law states that a pressure change at any point in a confined incompressible fluid is transmitted equally throughout the fluid. This principle is the basis for hydraulic systems.

Example: In a hydraulic jack, a small force applied to a small piston creates a pressure that is transmitted to a larger piston, resulting in a much larger output force.

Pressure-Depth Relationship

The pressure in a fluid increases with depth due to the weight of the fluid above.

p = p₀ + γh

Where:

• p is the absolute pressure at depth h (Pa)
• p₀ is the pressure at the surface (usually atmospheric pressure) (Pa)
• γ is the specific weight of the fluid (N/m³)
• h is the depth from the surface (m)

For gauge pressure (pressure relative to atmospheric), p_gauge = γh.

Manometers

Devices used to measure pressure in a fluid by balancing the fluid column against a column of another liquid (usually mercury) of known density.

• Simple Manometer: Measures the pressure at a single point.
• Differential Manometer: Measures the pressure difference between two points.
• Inverted Manometer: Used for measuring pressure differences in liquids lighter than the manometer fluid (e.g., air in pipes).

Example: A simple U-tube manometer connected to a pipe with water shows a mercury column difference of 0.1 m. If the fluid in the pipe is water (γ_w = 9810 N/m³) and mercury (γ_m = 133100 N/m³), the gauge pressure in the pipe is p_gauge = γ_m * h_m - γ_w * h_w. If the mercury is higher, p_gauge = γ_m * h_m, and if water is higher, p_gauge = γ_w * h_w.

Pressure Force and Centre of Pressure on Submerged Bodies

• Plane Surfaces:
  • Force (F): The total hydrostatic force on a submerged plane surface.
    F = γh̄A
    Where:
    • F is the hydrostatic force (N)
    • γ is the specific weight of the fluid (N/m³)
    • h̄ is the vertical distance from the fluid surface to the centroid of the submerged area (m)
    • A is the area of the submerged surface (m²)
  • Centre of Pressure (y_p): The point where the resultant hydrostatic force acts. It is always below the centroid for a submerged plane surface.
    y_p = I_xx / (Ah̄) + h̄
    Where:
    • y_p is the distance from the fluid surface to the centre of pressure (m)
    • I_xx is the moment of inertia of the submerged area about its centroidal axis parallel to the free surface (m⁴)
    • A is the area of the submerged surface (m²)
    • h̄ is the vertical distance from the fluid surface to the centroid (m)
  Example: Calculate the force on a rectangular gate submerged vertically in water.
• Curved Surfaces: The hydrostatic force on a curved surface is found by resolving it into horizontal and vertical components.
  • Horizontal Component (F_h): Equal to the hydrostatic force on the projection of the curved surface onto a vertical plane.
  • Vertical Component (F_v): Equal to the weight of the fluid directly above the curved surface up to the free surface.

Pressure Diagrams

Graphical representation of pressure distribution on submerged surfaces. For a vertical wall, the pressure varies linearly with depth, forming a triangular distribution. For a sloped wall or a wall with varying depth, trapezoidal or other distributions may occur.

Example: A vertical rectangular dam face of height H. The pressure at the bottom is γH. The total force is the area of the triangle (1/2 * base * height), where base is γH and height is H, giving F = 1/2 * γH * H = 1/2 γH². This force acts at H/3 from the bottom.

Buoyancy

• Archimedes' Principle: A body wholly or partially submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body.
• Buoyant Force (Fb):
  F_b = γ * V_displaced
  Where:
  • F_b is the buoyant force (N)
  • γ is the specific weight of the fluid (N/m³)
  • V_displaced is the volume of fluid displaced by the body (m³)
  If the weight of the body (W) is greater than Fb, it sinks. If W < Fb, it floats. If W = Fb, it is neutrally buoyant.

Stability of Floating/Submerged Bodies

• Metacentric Height (GM): A measure of the stability of a floating body. It is the distance between the center of gravity (G) and the metacenter (M).
  GM = I / V - BG (for a floating body)
  Where:
  • GM is the metacentric height (m)
  • I is the moment of inertia of the waterplane area about the longitudinal axis (m⁴)
  • V is the volume of displaced fluid (m³)
  • BG is the distance between the center of gravity (G) and the center of buoyancy (B) (m)
• Stability Conditions:
  • Stable: GM > 0 (Metacenter is above the center of gravity). The body will return to its original position after a small tilt.
  • Neutral: GM = 0 (Metacenter coincides with the center of gravity). The body remains in its tilted position.
  • Unstable: GM < 0 (Metacenter is below the center of gravity). The body will tilt further and capsize.

3.3 Hydro-Kinematics and Hydro-Dynamics

This section covers the study of fluid motion, considering both the geometry of flow (kinematics) and the forces causing the motion (dynamics).

Classification of Fluid Flow

• Steady/Unsteady: Steady flow has flow properties (velocity, pressure, etc.) that do not change with time at any point. Unsteady flow has properties that vary with time.
• Uniform/Non-uniform: Uniform flow has velocity that is constant in magnitude and direction at every point in the flow field. Non-uniform flow has velocity that varies in space.
• Laminar/Turbulent: Laminar flow is characterized by smooth, orderly motion in layers (viscous forces dominate). Turbulent flow is characterized by chaotic, irregular motion with eddies (inertial forces dominate).
• Compressible/Incompressible: Based on whether density changes significantly.
• Rotational/Irrotational: Rotational flow involves fluid elements rotating about their centers. Irrotational flow involves fluid elements that do not rotate.

Conservation of Mass: Continuity Equation

The continuity equation expresses the conservation of mass for a fluid flow.

General form (for 3D, unsteady, compressible flow):

∂(ρu)/∂x + ∂(ρv)/∂y + ∂(ρw)/∂z + ∂ρ/∂t = 0

Where:

• ρ is density
• u, v, w are velocity components in x, y, z directions
• t is time

For steady, incompressible flow in a pipe or conduit (1D):

A₁V₁ = A₂V₂

Where:

• A₁ and A₂ are cross-sectional areas at two points
• V₁ and V₂ are average velocities at those points

Example: Water flows from a pipe of 0.1 m² area and 2 m/s velocity into a pipe of 0.05 m² area. The velocity in the second pipe is V₂ = (A₁V₁)/A₂ = (0.1 * 2) / 0.05 = 4 m/s.

Momentum Equations and Applications

The momentum equation is based on Newton's second law (F = ma) applied to a fluid control volume. It relates the net force acting on a fluid to the change in its momentum.

Applications include calculating forces on pipe bends, jet impact on surfaces, and forces on vanes.

Example: Force on a pipe bend. The change in momentum of the fluid as it changes direction requires a force from the pipe walls to cause this change.

Bernoulli's Equation

Bernoulli's equation is a statement of the conservation of energy for steady, incompressible, inviscid flow along a streamline. It relates pressure, velocity, and elevation.

p₁/γ + V₁²/2g + z₁ = p₂/γ + V₂²/2g + z₂ + h_L

Where:

• p is pressure
• γ is specific weight
• V is velocity
• g is acceleration due to gravity
• z is elevation above a datum
• h_L is the head loss due to friction and other irreversibilities

Assumptions:

• Steady flow
• Incompressible fluid
• Inviscid fluid (no viscosity)
• Flow along a streamline
• No energy added or removed (e.g., by pumps or turbines)

Applications:

• Venturimeter: Measures flow rate by measuring the pressure difference in a converging-diverging section.
• Orifice Meter: Similar to Venturimeter but uses a sharp-edged orifice plate.
• Pitot Tube: Measures local velocity by comparing stagnation pressure to static pressure.

Example: Water flows through a horizontal pipe with a diameter change. The pressure and velocity at section 1 are P1 and V1, and at section 2 are P2 and V2. Bernoulli's equation can be used to relate these parameters, neglecting head loss for ideal flow.

Flow Measurement

• Venturimeter: A device with a converging section, a throat, and a diverging section. Flow rate is calculated using the pressure difference between the inlet and throat.
• Orifice Meter: A plate with a hole inserted in a pipe. Flow rate is calculated from the pressure difference across the orifice. Less efficient than Venturimeter.
• Nozzle: Similar to Venturimeter but often used for specific applications.
• Pitot Tube: Measures velocity at a point.
• Rotameter: A variable area flow meter where a float rises in a tapered tube, indicating flow rate.

3.4 Pipe Flow

This section focuses on fluid flow through closed conduits (pipes).

Types of Flow

• Pressure Flow: Flow driven by a pressure difference, typically in closed pipes where the fluid completely fills the pipe.
• Gravity Flow: Flow driven by gravity, often in open channels or pipes that are not completely full.

Governing Equations

• Navier-Stokes Equations: The fundamental equations of fluid motion, describing conservation of momentum for viscous fluids. They are complex and often simplified for specific problems.
• Energy Equation: An extension of Bernoulli's equation that accounts for energy added by pumps and lost to turbines or friction.
• Momentum Equation: Used to calculate forces exerted by fluid flow on pipes and fittings.

Major Head Losses

Losses due to friction along the length of the pipe.

• Darcy-Weisbach Equation: The most common equation for calculating head loss due to friction.
  h_f = f * (L/D) * (V²/2g)
  Where:
  • h_f is head loss due to friction (m)
  • f is the Darcy friction factor (dimensionless, depends on Reynolds number and pipe roughness)
  • L is the length of the pipe (m)
  • D is the diameter of the pipe (m)
  • V is the average velocity of flow (m/s)
  • g is acceleration due to gravity (m/s²)
• Chezy Equation: V = C * sqrt(R*S), where C is Chezy's coefficient, R is hydraulic radius, S is slope of energy grade line.
• Manning Equation: Commonly used for open channels and gravity flow in pipes. V = (1/n) * R^(2/3) * S^(1/2), where n is Manning's roughness coefficient.
• Hazen-Williams Equation: Primarily used for water distribution systems.

Minor Losses

Losses due to fittings, valves, entrances, exits, expansions, and contractions in the pipe system. These are often expressed as a coefficient (K) multiplied by the velocity head.

h_m = K * (V²/2g)

Where:

• h_m is minor head loss (m)
• K is the loss coefficient (dimensionless, specific to the fitting)
• V is the average velocity in the pipe section where the loss occurs (m/s)

Examples: Entrance loss (K ≈ 0.5), exit loss (K = 1.0), bend loss, valve loss.

HGL and TEL

• Hydraulic Grade Line (HGL): Represents the sum of pressure head and datum head (p/γ + z). It indicates the level to which water would rise in a series of piezometers.
• Total Energy Line (TEL): Represents the sum of pressure head, datum head, and velocity head (p/γ + z + V²/2g). It indicates the total energy per unit weight of the fluid. The TEL is always above the HGL by the velocity head.

In pipe flow, the HGL and TEL slope downwards in the direction of flow due to energy losses.

Design of Pipes

• Economic Diameter: The pipe diameter that minimizes the total cost (initial cost of pipe + annual cost of energy losses).
• Velocity Criteria: Limiting velocities to prevent excessive friction loss, erosion, or sedimentation.

Pipe Network Problems

Systems of interconnected pipes. Solving for flow rates and pressures can be complex.

• Hardy Cross Method: An iterative method for analyzing flow in pipe networks by balancing flow at junctions and head losses in loops.

Unsteady Flow in Pipes

• Water Hammer: A pressure surge that occurs in pipelines when the flow velocity is suddenly changed (e.g., by rapid valve closure). It can cause very high pressures.
• Surge Tanks: Devices used to absorb pressure surges and prevent water hammer in long pipelines, especially in hydropower systems.

Relief Devices

• Pressure Relief Valves: Automatically open to release excess pressure, protecting the pipe system.
• Vacuum Breakers: Prevent vacuum conditions in pipes, which can cause pipe collapse or siphoning of unwanted substances.

3.5 Open Channel Flow

This section deals with fluid flow in open conduits, where the fluid surface is exposed to atmospheric pressure (e.g., rivers, canals, sewers).

Geometrical Properties

• Wetted Area (A): The cross-sectional area of the flow.
• Wetted Perimeter (P): The length of the boundary of the flow in contact with the channel.
• Hydraulic Radius (R): R = A/P. A measure of the efficiency of the channel cross-section.
• Hydraulic Depth (D): D = A/T, where T is the top width of the flow.

Types of Flows

• Uniform Flow: Depth and velocity are constant along the channel.
• Non-uniform Flow: Depth and velocity vary along the channel.
• Steady/Unsteady: As defined previously.
• Subcritical Flow: Low velocity, large depth (Froude number Fr < 1). Disturbances can propagate upstream.
• Supercritical Flow: High velocity, small depth (Fr > 1). Disturbances cannot propagate upstream.
• Critical Flow: The flow condition where specific energy is minimum for a given discharge (Fr = 1).

Energy and Momentum Principles

• Specific Energy (E): The energy per unit weight of fluid relative to the channel bed.
  E = y + V²/2g
  For a given discharge per unit width (q = Q/B), E = y + q²/(2gy²).
• Specific Force (F_s): The force per unit width exerted by the fluid on the channel bed. F_s = Q²/gA + A*ȳ, where ȳ is the depth to the centroid of the area. For rectangular channels, F_s = Q²/gA + A*y/2.

Specific Energy Diagram

A plot of Specific Energy (E) versus flow depth (y) for a constant discharge. It shows that for a given discharge, there can be two depths (alternate depths) with the same specific energy. The minimum specific energy occurs at critical depth (yc).

Critical Depth (yc): The depth at which specific energy is minimum for a given discharge.

For a rectangular channel of width B, yc = (q²/g)^(1/3), where q = Q/B is discharge per unit width.

Gradually Varied Flow Profiles

Describes the variation of flow depth along a channel when the flow is not uniform but changes gradually. Profiles are classified by their shape and the relative position of the actual depth to the normal depth (yn) and critical depth (yc).

Profile Types:

• Mild (M): Normal depth (yn) > critical depth (yc)
• Mild (S): Normal depth (yn) < critical depth (yc)
• Critical (C): Normal depth (yn) = critical depth (yc)
• Sloped (H): Normal depth (yn) > channel slope

Combined with depth relative to critical and normal depths: M1, M2, M3, S1, S2, S3, C1, C3, H2, H3.

Hydraulic Jump

A phenomenon where supercritical flow abruptly transitions to subcritical flow, accompanied by a significant loss of energy and an increase in depth.

• Sequent Depth Ratio: For a rectangular channel, y₂/y₁ = 0.5 * (sqrt(1 + 8Fr₁²) - 1), where y₁ and y₂ are depths before and after the jump, and Fr₁ is the Froude number before the jump.
• Energy Loss: Significant energy is dissipated as turbulence.
• Length of Jump: Typically 4 to 6 times the larger sequent depth (y₂).

Example: A hydraulic jump occurs in a rectangular channel. If the upstream Froude number is 3, the sequent depths can be calculated.

Flow in Mobile Boundary Channels (Alluvial Channels)

Channels where the bed and banks are composed of erodible material (sediment). Flow characteristics are influenced by sediment transport.

• Design Principles/Approaches: Regime theory, tractive force method.
• Inception Motion Condition: The condition at which sediment particles begin to move.
• Shields Diagram: A plot showing the critical Shields parameter (τ*c) versus the particle Reynolds number (Re*), used to determine the condition for incipient motion of sediment particles.

3.6 Hydrology

The study of water on Earth, its movement, distribution, and quality.

Hydrologic Cycle and Water Balance

• Hydrologic Cycle: The continuous movement of water on, above, and below the surface of the Earth.
• Water Balance Components:
  • Precipitation (P): Rain, snow, hail, etc.
  • Evaporation (E): Water changing from liquid to gas from surfaces.
  • Transpiration (T): Water vapor released from plants.
  • Infiltration (I): Water seeping into the ground.
  • Runoff (R): Water flowing over the land surface or in channels.
  P - E - T - I - R = ΔS (Change in storage)

Flow Measurement and Rating Curves

• Stage-Discharge Relationship: The relationship between the water level (stage) in a river and the corresponding flow rate (discharge).
• Rating Curve: A graphical representation of the stage-discharge relationship.
• Curve Shifting: Adjustments to the rating curve due to changes in channel geometry or flow conditions.

Hydrograph Analysis

• Limnograph: A device for continuously recording the water level.
• Unit Hydrograph: A hydrograph representing the direct runoff from a storm of unit duration (e.g., 1 hour) and unit depth (e.g., 1 cm or 1 inch) over a catchment area. It's a fundamental tool for predicting runoff from rainfall.
• S-curve: A cumulative hydrograph used in unit hydrograph theory.

Synthetic Unit Hydrograph

Methods to derive a unit hydrograph when historical storm data is limited.

• Snyder's Method: A method based on catchment characteristics like time to peak and basin lag.
• SCS Method (NRCS): A widely used method that uses dimensionless hydrographs and catchment parameters.

Rainfall-Runoff Analysis

• Rational Method: A simplified method for estimating peak runoff rate for small catchments.
  Q_peak = C * i * A
  Where:
  • Q_peak is the peak runoff rate (m³/s or cfs)
  • C is the runoff coefficient (dimensionless, depends on land use and soil type)
  • i is the average rainfall intensity for the time of concentration (mm/hr or in/hr)
  • A is the catchment area (hectares or acres)
• Runoff Coefficients: Values representing the fraction of rainfall that becomes runoff.

Flood Hydrology

• Flood Frequency Analysis: Statistical analysis to estimate the probability of floods of different magnitudes occurring.
• Gumbel Method: A method for fitting extreme value distributions to flood data.
• Log-Pearson Type III Method: Another common method for flood frequency analysis.
• Design Flood: The flood magnitude used for designing hydraulic structures, typically based on a chosen return period.

Groundwater Hydrology

• Darcy's Law: Describes the flow of groundwater through porous media.
  Q = -K * A * (dh/dl)
  Where:
  • Q is the volumetric flow rate (m³/s)
  • K is the hydraulic conductivity (m/s)
  • A is the cross-sectional area perpendicular to flow (m²)
  • dh/dl is the hydraulic gradient (dimensionless)
• Aquifer Types: Confined, unconfined, semi-confined.
• Well Hydraulics: Analysis of flow towards wells.
• Theis Equation: Used for analyzing transient flow to wells in unconfined or confined aquifers.
• Thiem Equation: Used for analyzing steady-state flow to wells in confined aquifers.
  Q = 2π * K * b * (h₁ - h₂) / ln(r₂/r₁)
  Where:
  • Q is discharge
  • K is hydraulic conductivity
  • b is aquifer thickness
  • h₁ and h₂ are water surface elevations at radii r₁ and r₂.