CELE Hydraulics & Fluid Mechanics — Properties of FluidsStudy Notes

Density and specific weight are fundamental properties that characterize how much mass (or weight) a fluid contains per unit volume. These properties are crucial in calculating hydrostatic pressure, force on submerged surfaces, and buoyancy effects in water resources engineering projects. **Density (ρ)** is defined as mass per unit volume: ρ = m/V (units: kg/m³) For water at standard conditions (4°C, 1 atm): ρ_water = 1000 kg/m³. This value is fundamental to all hydraulic calculations in the Philippines, where water is the primary fluid in irrigation systems, hydroelectric facilities, and municipal water supply networks. **Specific Weight (γ)** is the weight per unit volume, related to density through gravitational acceleration: γ = ρg (units: N/m³ or kN/m³) Using g = 9.81 m/s² for the Philippine latitude and elevation variations, the specific weight of water is: γ_water = 1000 × 9.81 = 9810 N/m³ = 9.81 kN/m³ This relationship is absolutely essential: the 9.81 kN/m³ figure appears constantly in pressure calculations, head calculations, and force determinations on dams and hydraulic structures designed under NSCP 2015 guidelines. **Physical Interpretation:** While density measures the 'compactness' of a substance in terms of mass, specific weight measures how much gravitational force that substance exerts per unit volume. In practical terms, a liter of water (1000 cm³ = 0.001 m³) has a mass of 1 kg and exerts a weight of 9.81 N. This distinction becomes critical when converting between mass-based (metric) and force-based (engineering) calculations. **Temperature Effects:** Both ρ and γ vary slightly with temperature. For water at 20°C (typical laboratory/design condition): ρ ≈ 998 kg/m³, γ ≈ 9.78 kN/m³. In Philippine tropical conditions, water temperature may reach 25–30°C in open channels and reservoirs, causing ρ to decrease slightly. However, for most civil engineering calculations, ρ_water = 1000 kg/m³ is the accepted standard unless specifically stated otherwise.

Specific gravity is a dimensionless ratio comparing the density (or specific weight) of a substance to that of water as a reference standard. This dimensionless property is invaluable in engineering because it allows quick qualitative assessment of fluid behavior and quick conversion between fluid properties. **Definition:** s = ρ_fluid / ρ_water = γ_fluid / γ_water (dimensionless) Since specific gravity is a ratio, it is independent of units and has the same numerical value whether you use SI, CGS, or US customary units. This universality makes it the preferred property for international engineering standards and for quick mental calculations on exam boards. **Practical Significance:** - s < 1: Fluid is lighter than water (floats) - s = 1: Fluid has same density as water - s > 1: Fluid is denser than water (sinks) **Common Fluids in Philippine Engineering Practice:** - Seawater (Manila Bay, Laguna de Bay intake): s ≈ 1.025 (salinity effect) - Municipal wastewater (treatment plants): s ≈ 1.02–1.03 - Industrial oils (hydraulic systems, lubricants): s ≈ 0.85–0.92 - Mercury (rarely used in Philippine labs but important for barometer equations): s ≈ 13.6 - Gasoline (hydrocarbon transport, emergency generator fuel): s ≈ 0.72 **Density Conversion Using Specific Gravity:** ρ_fluid = s × ρ_water = s × 1000 kg/m³ γ_fluid = s × γ_water = s × 9.81 kN/m³ This two-step conversion—first find s from reference tables or given data, then multiply by 1000 or 9.81—is one of the fastest calculation methods in hydraulic problem-solving. **Temperature Dependency:** While specific gravity is often treated as constant in undergraduate problems, it actually varies with temperature as both numerator (fluid) and denominator (water) densities change. For design problems requiring high accuracy, refer to fluid property tables at the design temperature. For the PRC examination, unless otherwise stated, assume standard conditions (20°C).

Viscosity is a fluid's internal resistance to shear flow, reflecting the molecular friction within the fluid. Understanding viscosity is essential for calculating head losses in pipes, selecting appropriate pump types, designing lubrication systems, and predicting flow behavior in channels and conduits throughout the Philippines' extensive irrigation and water supply networks. **Newton's Law of Viscosity:** τ = μ (dv/dy) where: - τ = shear stress (Pa = N/m²) - μ = dynamic (absolute) viscosity (Pa·s) - dv/dy = velocity gradient (s⁻¹) This equation states that shear stress is directly proportional to the rate of velocity change perpendicular to the flow direction. In a flow between two parallel plates, if the plate separation is small and velocity increases linearly from zero at the bottom plate to maximum at the top plate, then dv/dy is constant, and τ is constant. **Newtonian Fluids (Non-Newtonian Fluids):** A **Newtonian fluid** has a constant dynamic viscosity μ that is **independent of the shear rate** (dv/dy). This constancy defines a Newtonian fluid. Common Newtonian fluids include: - Water (all civil engineering applications) - Air and other gases - Light mineral oils and hydraulic oils - Glycerin **Non-Newtonian fluids** (beyond this chapter's scope but worth noting) include: - Blood plasma (used in biomedical engineering, not typical in civil works) - Sewage sludge and wastewater solids (μ increases with shear rate: pseudoplastic) - Some polymer solutions (μ decreases with shear rate: dilatant) For the PRC Civil Engineer Licensure Examination, **assume all fluids are Newtonian** unless explicitly stated otherwise. **Dynamic Viscosity (μ)** – Absolute Viscosity: μ is the proportionality constant relating shear stress to velocity gradient. Units: - SI: Pascal·second (Pa·s) = N·s/m² = kg/(m·s) - CGS: Poise (P) = 0.1 Pa·s (rarely used in modern engineering) - Centipoise (cP) = 0.001 Pa·s (still common in petroleum and food industries) **At 20°C (standard reference temperature):** - Water: μ_water ≈ 0.001 Pa·s = 1.0 cP - Air: μ_air ≈ 1.81 × 10⁻⁵ Pa·s - Glycerin: μ ≈ 1.5 Pa·s (very viscous) - Light motor oil (SAE 10W): μ ≈ 0.065 Pa·s **Temperature Dependence:** Dynamic viscosity decreases dramatically as temperature increases (liquids) or increases slightly as temperature increases (gases). For water: - At 0°C: μ ≈ 1.787 × 10⁻³ Pa·s - At 20°C: μ ≈ 1.002 × 10⁻³ Pa·s - At 50°C: μ ≈ 0.547 × 10⁻³ Pa·s In the Philippines' tropical climate, design temperatures for water systems often approach 30°C, where μ_water ≈ 0.80 × 10⁻³ Pa·s. Always verify the reference temperature when applying μ values from tables. **Kinematic Viscosity (ν) – Momentum Diffusivity:** Kinematic viscosity is the ratio of dynamic viscosity to density: ν = μ / ρ (units: m²/s) Kinematic viscosity represents the fluid's capacity to diffuse momentum (analogous to thermal diffusivity in heat transfer). It is particularly useful in fluid mechanics because it appears naturally in dimensionless numbers like the **Reynolds number**: Re = ρVD/μ = VD/ν where V is velocity and D is characteristic length (pipe diameter). The Reynolds number determines whether flow is laminar (Re < 2300 in pipes) or turbulent (Re > 4000 in pipes). **At 20°C:** - Water: ν_water ≈ 1.0 × 10⁻⁶ m²/s = 1.0 cSt (centistoke) - Air: ν_air ≈ 1.5 × 10⁻⁵ m²/s - Light oil (SAE 10W): ν ≈ 6.5 × 10⁻⁵ m²/s **Critical Distinction for Exam Success:** Examiners frequently test whether candidates understand the difference between μ and ν: - μ is a **fluid property only** (depends only on fluid and temperature) - ν is a **transport property** that combines fluid property (μ) with density (ρ) - If density changes (e.g., compressed gas), ν changes even though μ might not change significantly **Practical Application in Philippine Water Systems:** In the design of the Magat Dam or Pantabangan Dam irrigation systems, pipe friction factor depends on the Reynolds number, which depends on ν. A summer operating condition (higher T → lower μ → lower ν → higher Re for same velocity) requires different friction loss calculations than winter conditions, affecting required pump head and energy consumption. **Viscosity Index (VI):** For oils used in hydraulic systems, engineers specify both kinematic viscosity at 40°C (ISO VG grade) and the viscosity index, which measures how viscosity changes with temperature. Higher VI oils maintain more consistent viscosity over a temperature range, important for equipment operating in the Philippine tropics where ambient temperature varies 15–25°C daily.

Surface tension arises from unbalanced intermolecular forces at the interface between a liquid and a gas (or between two liquids). Molecules within a liquid are surrounded by other molecules in all directions, resulting in balanced forces. At the surface, molecules have neighbors only below and to the sides, creating a net inward force that makes the liquid surface behave like a stretched elastic membrane. This phenomenon, while often overlooked in large-scale hydraulic engineering, becomes critical in capillary tubes, soil-water interactions, and certain specialized applications. **Surface Tension (σ):** Surface tension is defined as the force per unit length acting on the perimeter of the liquid surface, or equivalently, the surface energy per unit area: σ = F/ℓ (units: N/m or mN/m) = energy/area (J/m² = N/m) At 20°C: - Water–air interface: σ ≈ 0.0728 N/m (decreases with temperature and impurities) - Mercury–air interface: σ ≈ 0.486 N/m (highly curved surfaces in capillaries) - Seawater (salinity ~35 ppt): σ ≈ 0.072 N/m (slightly lower than pure water) Unlike density or viscosity, surface tension is relatively insensitive to pressure but quite sensitive to temperature (decreases with increasing temperature) and to the presence of surfactants (soaps, detergents reduce σ significantly). In the Philippines' hot tropical climate, surface tension of water may be 2–3% lower than at 20°C, though this is usually negligible in civil engineering calculations. **Contact Angle (θ):** When a liquid meets a solid surface, the angle between the liquid–gas interface and the solid surface (measured through the liquid) is called the **contact angle**: - θ ≈ 0° (complete wetting): The liquid spreads on the surface; water on clean glass, water on concrete - θ = 90°: Neutral interaction; some oils on glass - θ > 90° (poor wetting): Liquid beads up; mercury on glass, water on paraffin-coated surfaces - θ = 180°: Complete non-wetting; mercury on glass approaches this For capillary rise calculations in soil, the contact angle of water on soil minerals is typically 0° (water completely wets soil), so cos θ = 1. **Capillary Rise (or Depression):** When a small-diameter tube (capillary tube) is inserted vertically into a liquid, the liquid either rises or falls depending on whether the liquid wets the tube walls. The height of rise (or depth of depression) is: h = (4σ cos θ) / (γd) where: - h = capillary rise (positive) or depression (negative) (m) - σ = surface tension (N/m) - θ = contact angle (°) - γ = specific weight of liquid (N/m³) - d = tube diameter (m) **Alternative form using radius r = d/2:** h = (2σ cos θ) / (γr) **Physical Interpretation:** - The numerator (2σ cos θ or 4σ cos θ) represents the vertical component of surface tension force acting along the tube's perimeter - The denominator represents the weight of the liquid column - Smaller tubes → larger ratio → greater rise/depression - Higher surface tension → greater rise - Greater specific weight (denser liquid) → less rise **Practical Significance in Civil Engineering:** 1. **Soil–Water Interaction (Geotechnical Engineering):** Water rises above the water table in soil due to capillary action. The capillary rise height depends on soil pore size (smaller pores = taller rise). In fine silts and clays, capillary rise can exceed 1–2 meters, causing soil saturation above the water table and affecting: - Bearing capacity of foundations (increased pore pressure reduces effective stress) - Slope stability (upward seepage forces reduce normal stress) - Frost heave in cold climates (ice lens formation) - Salt crystallization in arid regions (capillary rise concentrates salts) For this reason, NSCP 2015 Chapter 2 (Geotechnical Investigation) requires assessment of capillary rise in site characterization. 2. **Pipe Network Design:** In water distribution systems, capillary effects are usually negligible compared to pressure-driven flow, so they are typically ignored. However, in small-diameter measurement instruments or in very low-pressure systems, capillary rise can cause measurement errors. 3. **Hydraulic Structures:** In seepage analysis beneath dams, capillary rise in the exit region can create "piping" zones where soil is partially saturated and erodible. Modern dam design accounts for this in the drainage blanket specifications. **Advanced Note on Capillary Length:** The **capillary length** (or capillary constant) is a characteristic length scale: ℓ_c = √(σ / (ρg)) = √(σ / γ) For water at 20°C: ℓ_c = √(0.0728 N/m / 9810 N/m³) = √(7.42 × 10⁻⁶ m²) ≈ 2.7 mm This means surface tension effects dominate when geometric dimensions are comparable to or smaller than 2.7 mm. For larger dimensions, surface tension becomes negligible. In a 50 mm pipe, capillary effects are immeasurable; in a 1 mm capillary, they are dominant. **Interfacial Tension (not to be confused with surface tension):** When two immiscible liquids (e.g., oil and water) meet, the **interfacial tension** σ_oil-water describes the energy at their interface. This is generally different from surface tension and is important in: - Emulsions (oil in water, water in oil) - Separation equipment design - Petroleum pipeline transport mixed with water For the PRC examination, unless explicitly mentioned, assume liquid–gas (liquid–air) interfaces and single-liquid systems.

Although liquids are commonly treated as **incompressible** in civil engineering (a simplification valid for most applications), all real fluids compress under pressure. The degree of compression is measured by the **bulk modulus of elasticity**, which quantifies the relationship between pressure change and volume (or density) change. **Definition of Bulk Modulus:** The bulk modulus E_B (also denoted K or E_v in some texts) is defined as: E_B = −dp / (dV/V) = −(dp × V) / dV = dp / (−dV/V) = (ρ/dρ) × dp Alternatively, using density instead of volume: E_B = dp / (dρ/ρ) = ρ(dp/dρ) Units: Pascals (Pa), typically expressed in GPa for high values. **Physical Meaning:** - E_B is the slope of the pressure–volume curve (how much pressure change is needed to produce a given fractional volume change) - Higher E_B → stiffer fluid, more resistant to compression - For liquids, E_B is very large (2–2.3 GPa for water), so they are nearly incompressible - For gases, E_B is small and varies with pressure (compressibility is significant) **Isothermal vs Isentropic Bulk Modulus:** Two definitions exist: 1. **Isothermal** (T constant): ΔV/V results from pressure change while temperature held constant. This is the definition used here. 2. **Isentropic** (adiabatic, S constant): ΔV/V results from pressure change during adiabatic compression (no heat transfer). This is approximately γ times the isothermal value for gases (where γ = c_p/c_v). For liquids, the distinction is minor; isothermal is the standard used in civil engineering. **Approximate Bulk Moduli (at 20°C, 1 atm):** | Fluid | E_B (GPa) | Compressibility || |-------|-----------|----------| | Water | 2.2 | Nearly incompressible | | Seawater | 2.3 | Nearly incompressible | | Glycerin | 4.76 | Stiffer than water | | Mineral oil | 1.6 | Slightly more compressible than water | | Mercury | 27 | Very incompressible | | Air (1 atm) | 0.000101 | Highly compressible | **Compressibility (β):** Compressibility is the reciprocal of bulk modulus: β = 1/E_B = (−dV/V) / dp = (dρ/ρ) / dp For water: β_water ≈ 1/(2.2 × 10⁹ Pa) ≈ 4.55 × 10⁻¹⁰ Pa⁻¹ This means a pressure increase of 1 MPa (10 atm) causes a volume decrease of: (−dV/V) = β × dp = 4.55 × 10⁻¹⁰ × 10⁶ = 4.55 × 10⁻⁴ = 0.0455% ≈ 0.05% Such a small change is usually negligible, justifying the "incompressible fluid" assumption. **When is Water Compression Significant?** Compression effects become important when: 1. **Very high pressures** (deep ocean, high-pressure hydraulic systems >100 MPa): Accumulated volume changes become measurable. 2. **Acoustic phenomena** (water hammer, cavitation): Sound wave speed in a fluid depends on E_B: c = √(E_B/ρ). For water, c ≈ 1480 m/s (much faster than in air, 343 m/s). 3. **Elastic wave propagation** (earthquake analysis, blast waves): Wave speed reflects compressibility. 4. **Energy storage in hydraulic accumulators**: Compressible fluid stores energy; incompressible assumption would give infinite energy storage. **Sound Speed in Fluids:** The speed of sound (acoustic velocity) is related to bulk modulus by: c = √(E_B / ρ) For water: c = √(2.2 × 10⁹ Pa / 1000 kg/m³) = √(2.2 × 10⁶ m²/s²) ≈ 1483 m/s For air (1 atm): c = √(101,325 Pa / 1.225 kg/m³) ≈ 288 m/s at 0°C (increases with temperature) This explains why underwater explosions transmit energy very efficiently (water's high E_B → high c) while air is a poor conductor of shear waves (low E_B, low shear modulus). **Water Hammer (Surge):** When a valve closes suddenly in a pipe, flowing water decelerates, creating a pressure surge (water hammer). The magnitude of this pressure spike depends on: Δp ≈ ρ × c × ΔV = ρ × √(E_B/ρ) × ΔV = √(ρ × E_B) × ΔV where ΔV is the change in velocity. For water with c ≈ 1500 m/s, a sudden velocity change of 1 m/s produces a pressure surge of: Δp ≈ 1500 m/s × 1 m/s = 1500 Pa ≈ 0.015 bar (though the actual calculation is more complex) This is why rapid valve closure in large water pipelines can damage equipment; surge tanks and relief valves are designed to mitigate water hammer (refer to NSCP 2015 Water Facilities Design Code for water distribution system surge protection requirements). **Practical Application in Philippine Water Systems:** The Magat Dam and other major hydraulic projects must account for water compressibility effects in: - Surge tank design (sudden opening/closing of turbine gates) - Pipeline vibration and noise (water hammer) - Pressure transducer accuracy (small compressions at high pressures affect calibration) - Long-term deformation analysis (small cumulative compression over decades) For routine calculations (pipe friction, open channel flow, pressure at depth), the incompressible assumption (E_B → ∞) is standard and sufficiently accurate.

Every liquid has a characteristic **vapor pressure**—the pressure exerted by vapor molecules in equilibrium with the liquid phase. When the local pressure in a flowing fluid drops to (or below) the vapor pressure, the liquid spontaneously vaporizes, forming vapor bubbles. This phenomenon, called **cavitation**, can damage pump impellers, turbine runners, and pipe walls through violent bubble collapse. **Vapor Pressure Definition:** Vapor pressure is the pressure at which a liquid and its vapor are in **thermodynamic equilibrium** at a given temperature. It is a property of the liquid and increases with temperature (following the Clausius–Clapeyron relationship). In SI units, vapor pressure is expressed in Pascals (Pa) or kPa. **Vapor Pressure of Water at Various Temperatures:** | Temperature (°C) | p_v (Pa) | p_v (kPa) | |---|---|---| | 0 | 611 | 0.611 | | 10 | 1,228 | 1.228 | | 20 | 2,337 | 2.337 | | 30 | 4,246 | 4.246 | | 40 | 7,381 | 7.381 | | 50 | 12,344 | 12.344 | | 60 | 19,932 | 19.932 | | 70 | 31,176 | 31.176 | | 80 | 47,373 | 47.373 | | 90 | 70,117 | 70.117 | | 100 | 101,325 | 101.325 | Note: At 100°C and atmospheric pressure (101.325 kPa), p_v = p_atm, and water boils. In the Philippines' tropical climate, where water temperatures in open channels and storage facilities can reach 30–35°C, vapor pressures of 4–6 kPa are relevant for pump suction conditions and downstream of turbines. **Thermodynamic Perspective:** At equilibrium, the rate of evaporation (liquid → vapor) equals the rate of condensation (vapor → liquid). The vapor pressure depends **only on temperature**, not on the amount of liquid or surrounding air pressure (though the presence of air affects total pressure). This is why boiling point depends on altitude: at higher elevations, atmospheric pressure is lower, so the vapor pressure required to boil is reached at a lower temperature. **Cavitation: Bubble Formation and Collapse:** Cavitation occurs in two steps: 1. **Bubble Inception:** When local pressure drops to or below vapor pressure, dissolved gases come out of solution and liquid molecules vaporize, forming bubbles. This typically occurs: - On pump suction side (pressure drops below atmospheric as fluid is drawn in) - Behind turbine blades (low-pressure zone behind the blade) - At inlet of siphons or gates with vena contracta - In narrow constrictions (Venturi tubes, orifice plates) 2. **Bubble Collapse:** As bubbles are carried downstream into regions of higher pressure, the pressure exceeds vapor pressure, and vapor condenses back into liquid. The bubble collapses violently, creating a high-velocity liquid jet that impacts surrounding surfaces, causing: - Material erosion (pitting on metal surfaces) - Noise and vibration - Loss of efficiency (bubbles reduce flow area) - Potential equipment failure **Cavitation Index (Sigma, σ_c):** The tendency for cavitation is quantified by the **cavitation index**: σ_c = (p − p_v) / (ρgH) = (p − p_v) / (γH) where: - p = absolute pressure at the location of interest - p_v = vapor pressure at operating temperature - γ = specific weight of the fluid - H = dynamic head (velocity head) or characteristic pressure scale Alternatively, in terms of velocity: σ_c = (p − p_v) / (½ρV²) where V is the local flow velocity. Physical meaning: - σ_c > 0: Local pressure exceeds vapor pressure; no cavitation - σ_c < 0: Local pressure below vapor pressure; cavitation occurs - Smaller σ_c: Higher risk of cavitation **Critical Cavitation Index (σ_crit):** Each hydraulic machine (pump, turbine, gate) has a critical cavitation index σ_crit below which cavitation damage becomes significant. This is determined experimentally and is a key specification provided by manufacturers. Typical values: - Centrifugal pumps: σ_crit ≈ 0.05–0.15 - Turbines: σ_crit ≈ 0.05–0.30 (turbines more susceptible than pumps) - Needle gates: σ_crit ≈ 0.10–0.20 **Net Positive Suction Head (NPSH):** For pumps, cavitation prevention is stated in terms of **Net Positive Suction Head (NPSH)**, which is the pressure head above vapor pressure available at the pump inlet: NPSH_available = (p_atm − p_v)/γ + z_inlet − h_f,inlet where: - p_atm = atmospheric pressure (at water surface) - p_v = vapor pressure at operating temperature - z_inlet = height of water surface above pump inlet (positive if inlet below surface, negative if above) - h_f,inlet = friction losses in suction line from surface to pump inlet For safe pump operation: NPSH_available > NPSH_required (specified by pump manufacturer) The manufacturer's NPSH_required accounts for the pressure drop through the pump inlet and first impeller stage needed to accelerate the fluid to the impeller speed. If NPSH_available < NPSH_required, cavitation occurs, reducing pump flow and head. **Practical Implication for Philippine Water Systems:** The Magat Dam and Pantabangan Dam hydroelectric facilities must account for cavitation in: 1. **Penstock intake** (suction region): Intake structure design avoids high-velocity regions where pressure might drop below vapor pressure 2. **Turbine runner** (wake region behind blades): Turbine runner material selection (erosion-resistant coatings or stainless steel) and blade design minimize cavitation inception 3. **Spillway gates and aerators** (downstream of gates): Aeration (introducing air) prevents cavitation by raising the local "pressure" through air cushion In municipal water supply systems, pump stations must be positioned with adequate submergence (depth below water surface) to ensure positive NPSH, or use vacuum priming systems to assist startup. **Cavitation Prevention Strategies:** 1. **Increase Local Pressure:** Raise the pressure at the critical location by: - Increasing inlet pressure (deeper submergence, higher reservoir level) - Reducing velocity (wider passages, lower flow rate) - Reducing elevation (pump lower, closer to water source) 2. **Decrease Vapor Pressure:** Lower operating temperature (cavitation more likely in warm water) - Not practical to control globally, but cool intake water sources preferred 3. **Use Cavitation-Resistant Materials:** - Stainless steel, rubber coatings, composite materials - NSCP 2015 hydraulic equipment specifications 4. **Design with Smooth Surfaces:** Rough surfaces, sharp edges, and cavities promote bubble inception 5. **Aeration (for spillways/gates):** Introduce air to prevent vacuum formation - Air slots in gates (refer to USBR design manuals and NSCP 2015) - Roughened spillway surfaces increase air entrainment **Erosion from Cavitation Collapse:** Erosion rate depends on: - Cavitation number (more negative → more aggressive) - Material properties (hardness, fatigue strength) - Time exposure - Fluid properties (viscosity affects bubble size and collapse intensity) Classic examples of cavitation damage: - Grand Coulee Dam spillway (1940s): Severe erosion required relief wells and concrete restoration - Mangla Dam spillway (Pakistan, similar climate to Philippines): Cavitation erosion led to major repairs - Industrial pumps in tropical regions: Higher water temperatures increase cavitation risk **Pressure Coefficient and Cavitation Number:** In fluid mechanics, the **pressure coefficient** is: C_p = (p − p_ref) / (½ρV²) and the **cavitation number** is: σ = (p − p_v) / (½ρV²) When σ = σ_crit (critical value), cavitation inception begins. If σ becomes negative, cavitation is fully developed and dangerous.

The seven fundamental fluid properties—density ρ, specific weight γ, specific gravity s, dynamic viscosity μ, kinematic viscosity ν, surface tension σ, bulk modulus E_B, and vapor pressure p_v—are interconnected and together define how a fluid responds to external forces and internal stresses. Understanding these properties is the prerequisite for every subsequent topic in hydraulics: pressure calculations, flow analysis, energy equations, and hydraulic machinery design. **Hierarchy of Fluid Properties (from most to least commonly used in civil engineering):** 1. **Density (ρ) and Specific Weight (γ):** Used in nearly every calculation (buoyancy, pressure, force). 2. **Viscosity (μ, ν):** Essential for flow regime classification (Reynolds number), friction factor, and head loss. 3. **Specific Gravity (s):** Used for quick density determination and relative comparisons. 4. **Vapor Pressure (p_v):** Critical for cavitation analysis and pump suction conditions. 5. **Surface Tension (σ):** Important for capillary action in soil (geotechnical) and small-diameter tubes; less common in large-scale open channel or pipe flow. 6. **Bulk Modulus (E_B):** Used for water hammer analysis and compressibility effects; usually negligible for incompressible flow assumption. **Dimensional Analysis (Buckingham π-Theorem Perspective):** Fluid properties form the basis for dimensionless numbers that govern flow behavior: - **Reynolds number:** Re = ρVD/μ = VD/ν (inertial vs viscous forces) - **Froude number:** Fr = V/√(gD) (inertial vs gravitational forces) - **Weber number:** We = ρV²D/σ (inertial vs surface tension forces) - **Mach number:** Ma = V/c (where c = √(E_B/ρ)) (velocity vs sound speed) Mastery of these dimensionless groups requires deep understanding of the base properties. **Temperature Dependence Summary:** Property | At 20°C | At 30°C | At 40°C | Trend | |---|---|---|---|---| ρ (kg/m³) | 998 | 996 | 992 | Decreases slightly | μ (Pa·s) | 1.002 × 10⁻³ | 0.801 × 10⁻³ | 0.656 × 10⁻³ | Decreases significantly | ν (m²/s) | 1.004 × 10⁻⁶ | 0.804 × 10⁻⁶ | 0.661 × 10⁻⁶ | Decreases significantly | p_v (kPa) | 2.34 | 4.25 | 7.38 | Increases exponentially | σ (N/m) | 0.0728 | 0.0712 | 0.0696 | Decreases slightly | E_B (GPa) | 2.20 | 2.19 | 2.18 | Decreases slightly | For **PRC exam purposes:** Warm water (25–35°C typical in Philippines) has noticeably lower viscosity and higher vapor pressure than the 20°C reference values. This affects head loss calculations (lower μ → lower head loss for same flow) and cavitation margins (higher p_v → lower NPSH available). **Common Exam Pitfalls and How to Avoid Them:** 1. **Confusing γ and ρ:** Remember γ = ρg. Pressure calculations use γ (specific weight); mass calculations use ρ (density). 2. **Unit conversion errors:** Always verify units carefully. Pa·s for μ (not cP without conversion), m²/s for ν (not cm²/s or cSt without conversion). 3. **Capillary formula confusion:** h = 4σ cos θ / (γd) uses **diameter d**, not radius. Some texts use h = 2σ cos θ / (γr) with radius r—know both forms. 4. **Specific gravity as ratio:** s = ρ_fluid / ρ_water is dimensionless and the same in all unit systems. Do not assign units to s. 5. **Viscosity temperature dependence:** Cold water is more viscous (higher μ). Remember: μ **decreases** with temperature for liquids, **increases** for gases. 6. **NPSH calculation sign conventions:** (p_atm − p_v) in NPSH formula: both must be absolute pressures. If you use gauge pressures (p_gauge = p_absolute − p_atm), the formula changes. Use absolute pressures consistently. 7. **Cavitation index σ_c vs σ_crit:** σ_c = (p − p_v)/(½ρV²) is calculated from local conditions. σ_crit is a property of the machine (given by manufacturer). **Cavitation occurs when σ_c < σ_crit**. 8. **Compressibility assumption:** In 99% of civil engineering problems, water is incompressible. Never use E_B unless the problem explicitly involves high pressure, water hammer, or surge analysis. **Exam Problem-Solving Strategy:** **Step 1: Identify the fluid.** Is it water? Oil? Seawater? What is the temperature? Look up or calculate properties from given data. **Step 2: Determine which properties are relevant.** Is the problem about: - Pressure/force? → Use γ and ρ - Flow regime (laminar/turbulent)? → Use ν (for Reynolds number) - Head loss in pipes? → Use ν (for friction factor) - Pump cavitation risk? → Use p_v and ρ (for NPSH or cavitation index) - Capillary rise in soil? → Use σ, γ, and d (or r) - Surge/water hammer? → Use c = √(E_B/ρ) **Step 3: Set up the equation.** Write the relevant formula clearly, identifying all variables. **Step 4: Verify units.** Before calculating, ensure all terms have consistent units (SI: kg, m, s, Pa, N, etc.). **Step 5: Calculate and interpret.** Perform the arithmetic, check reasonableness, and explain the physical meaning of the result. **Example Integrated Problem (Synthesis):** A small hydroelectric power station intakes water from a mountain stream (2°C, p_v ≈ 0.7 kPa) through a 0.5 m diameter penstock. The intake elevation is 1500 m above sea level, and the intake is submerged 2 m below the stream surface. The penstock is 500 m long with friction factor f ≈ 0.025. During peak output, the flow rate is 1.5 m³/s. At the turbine inlet (50 m below intake elevation), determine: (a) The Reynolds number of the flow (is it turbulent?) (b) The friction head loss in the penstock (c) The absolute pressure at the turbine inlet (d) Calculate the cavitation index; assess cavitation risk (σ_crit ≈ 0.08) **Solution Outline:** (a) V = Q/A = 1.5 m³/s / (π × 0.25² m²) ≈ 7.64 m/s ν = 1.56 × 10⁻⁶ m²/s (at 2°C, colder than standard 20°C) Re = VD/ν = 7.64 × 0.5 / 1.56 × 10⁻⁶ ≈ 2.45 × 10⁶ >> 4000 → **Highly turbulent** (b) h_f = f(L/D)(V²/2g) = 0.025 × (500/0.5) × (7.64²/(2 × 9.81)) ≈ 3.8 m (c) At intake: p_abs = p_atm + ρg(submerged depth) ≈ 101.3 + 9.81 × 2 ≈ 121.1 kPa At turbine inlet (50 m below intake): p_abs ≈ 121.1 + 9.81 × 50 − friction − velocity head = 121.1 + 490.5 − 37.3 − 2.98 ≈ 571 kPa (approximately) (d) σ_c = (p − p_v) / (½ρV²) ≈ (571 − 0.7) / (½ × 1000 × 58.4) ≈ 570.3 / 29,200 ≈ **0.0195** Since σ_c (0.0195) < σ_crit (0.08), **cavitation risk exists**. The design needs improvement: deeper intake submergence or higher intake elevation relative to turbine. This synthesis problem tests: - Property selection (viscosity for Reynolds number) - Temperature sensitivity (viscosity at 2°C differs from 20°C) - Dimensional consistency - Pressure calculations with elevation change - Head loss (friction + dynamic) - Cavitation index calculation and interpretation - Design judgment (recognizing cavitation risk) Such comprehensive problems are typical of PRC licensure exams and require integrated understanding of all fluid properties.