7.1 Fluid Properties, Units, and Pressure Measurement

7.1 Fluid Properties, Units, and Pressure Measurement

In the Philippine Civil Engineering Licensure Examination (CELE), fluid mechanics constitutes a core pillar of the Hydraulics and Principles of Geotechnical Engineering (HGE) cluster (30% total examination weight). Mastery of fluid statics begins with an exact understanding of physical fluid properties, thermodynamic compressibility, and the mechanics of static pressure distribution across multi-fluid manometric systems.

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1. Primary Physical Properties of Fluids

A fluid is defined as a substance that deforms continuously when subjected to shear (tangential) stress of any magnitude, encompassing both liquids (which possess a definite volume and form a free surface) and gases (which expand indefinitely to occupy their container). Under the continuum hypothesis, fluid properties are treated as point functions across continuous space.

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2. Viscosity and Newton's Law of Viscosity

Viscosity represents the internal molecular resistance of a fluid to shear deformation or flow. Consider two parallel plates separated by a small clearance $y$ filled with fluid, where the lower plate is stationary and the upper plate moves with constant velocity $v$.

Dynamic Viscosity ($\mu$)

Under laminar conditions, the shearing stress $\tau$ between fluid layers is directly proportional to the transverse velocity gradient $\frac{dv}{dy}$ according to Newton's Law of Viscosity: $\tau = \mu \frac{dv}{dy}$ where:

• $\tau = \frac{F}{A}$ is the shearing stress ($\text{Pa} = \text{N/m}^2$).
• $\mu$ is the dynamic (absolute) viscosity ($\text{Pa}\cdot\text{s} = \text{N}\cdot\text{s/m}^2 = \text{kg}/(\text{m}\cdot\text{s})$).
• In the CGS system, dynamic viscosity is expressed in Poise ($\text{P}$) or centipoise ($\text{cP}$), where $1\text{ Poise} = 0.1\text{ Pa}\cdot\text{s}$ and $1\text{ cP} = 10^{-3}\text{ Pa}\cdot\text{s} = 1\text{ mPa}\cdot\text{s}$. Water at $20^\circ\text{C}$ has $\mu \approx 1.002\text{ cP}$.

Kinematic Viscosity ($\nu$)

Kinematic viscosity is the ratio of dynamic viscosity to fluid mass density, representing momentum diffusivity: $\nu = \frac{\mu}{\rho}$

• SI unit: $\text{m}^2/\text{s}$.
• CGS unit: Stokes ($\text{St}$), where $1\text{ St} = 1\text{ cm}^2/\text{s} = 10^{-4}\text{ m}^2/\text{s}$, and $1\text{ cSt} = 10^{-6}\text{ m}^2/\text{s}$. Water at $20^\circ\text{C}$ has $\nu \approx 1.0 \times 10^{-6}\text{ m}^2/\text{s}$.

Temperature Dependence of Viscosity

• Liquids: Viscosity is dominated by cohesive intermolecular forces. As temperature increases, intermolecular bonds weaken, causing dynamic viscosity to decrease significantly.
• Gases: Viscosity is dominated by molecular collision and momentum exchange across streamlines. As temperature increases, molecular kinetic energy increases, causing gas viscosity to increase.

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3. Surface Tension, Capillarity, and Bulk Modulus

Surface Tension ($\sigma$)

Surface tension is the tensile force per unit length acting at the interface between two immiscible fluids or between a liquid and a gas, arising from unbalanced cohesive molecular attractions at the surface (SI unit: $\text{N/m}$). For clean water in air at $20^\circ\text{C}$, $\sigma \approx 0.0728\text{ N/m}$.

Internal pressure intensity $\Delta P = P_{\text{internal}} - P_{\text{external}}$ depends on geometric curvature:

1. Spherical Liquid Droplet (1 interface): $\Delta P = \frac{4\sigma}{d} = \frac{2\sigma}{R}$
2. Soap Bubble / Hollow Sphere (2 interfaces): $\Delta P = \frac{8\sigma}{d} = \frac{4\sigma}{R}$
3. Cylindrical Liquid Jet: $\Delta P = \frac{2\sigma}{d} = \frac{\sigma}{R}$

Capillary Rise and Depression

When a narrow glass tube of internal diameter $d$ is inserted into a liquid, the meniscus elevates or depresses due to the balance between adhesive forces (liquid-to-solid) and cohesive forces (liquid-to-liquid). By equating the vertical surface tension component around the tube perimeter to the weight of the liquid column: $(\sigma \cos \theta)(\pi d) = \gamma \left(\frac{\pi d^2}{4}\right) h \implies h = \frac{4\sigma \cos \theta}{\gamma d}$ where:

• $h$ is capillary height ($\text{m}$).
• $\theta$ is the contact angle (wetting angle). For clean glass and pure water, $\theta \approx 0^\circ$ ($\cos \theta = 1.0$), resulting in capillary rise ($h > 0$).
• For clean glass and mercury, $\theta \approx 130^\circ\text{–}140^\circ$ ($\cos \theta < 0$), resulting in capillary depression ($h < 0$).

Bulk Modulus of Elasticity ($E_v$) & Compressibility ($\beta$)

The bulk modulus of elasticity quantifies a fluid's resistance to volumetric compression under uniform hydrostatic pressure: $E_v = -\frac{\Delta P}{\Delta V / V} = \rho \frac{dP}{d\rho}$

• SI unit: $\text{Pa}$ or $\text{GPa}$. For water at standard atmospheric conditions, $E_v \approx 2.20\text{ GPa} = 2.20 \times 10^9\text{ Pa}$.
• Compressibility is the reciprocal of the bulk modulus: $\beta = \frac{1}{E_v}$ ($\text{Pa}^{-1}$). Because $E_v$ is extremely high, liquids are treated as practically incompressible in standard civil engineering statics.

Vapor Pressure and Cavitation Risk

The saturation vapor pressure ($P_v$) is the partial pressure exerted by vapor molecules in thermodynamic equilibrium with liquid at a given temperature. If the local absolute pressure within a hydraulic system (such as at the crest of a siphon, a pipe bend, or near pump suction impellers) drops to or below the fluid's vapor pressure ($P_v = 2.34\text{ kPa}$ absolute for water at $20^\circ\text{C}$), the liquid rapidly vaporizes at ambient temperature, producing vapor bubbles. When these bubbles travel into higher-pressure zones, they collapse violently, generating microjets and localized shock pressures exceeding hundreds of megapascals—a damaging phenomenon known as cavitation.

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4. Hydrostatic Pressure Fundamentals

Pascal's Principle and Pressure Direction

Pascal's Principle dictates that external pressure applied to an enclosed static fluid is transmitted undiminished in all directions. Fluid pressure at any point is isotropic ($P_x = P_y = P_z$) and always acts perpendicular (normal) to any contacting boundary surface.

Absolute vs. Gauge vs. Vacuum Pressure

Static pressure is referenced against two distinct baselines:

1. Absolute Pressure ($P_{\text{abs}}$): Measured relative to an absolute perfect vacuum (zero pressure datum).
2. Gauge Pressure ($P_{\text{gauge}}$): Measured relative to local atmospheric pressure ($P_{\text{atm}}$).
3. Vacuum (Negative Gauge) Pressure ($P_{\text{vac}}$): Pressure below local atmospheric pressure.

$P_{\text{abs}} = P_{\text{gauge}} + P_{\text{atm}}$ $P_{\text{vac}} = P_{\text{atm}} - P_{\text{abs}} = -P_{\text{gauge}}$

Pressure Variation with Depth

Equating vertical static forces on an elemental fluid prism of cross-sectional area $dA$ and height $dh$: $\frac{dP}{dh} = \gamma = \rho g \implies dP = \gamma dh$ Integrating for an incompressible homogeneous fluid with constant specific weight $\gamma$: $P_2 - P_1 = \gamma (h_2 - h_1) = \gamma h$ where $h$ is the vertical depth below the reference level. The quantity $\frac{P}{\gamma} = h$ is designated as the pressure head (expressed in meters of fluid).

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5. Manometry and Pressure Measurement Devices

Manometers utilize columns of static liquid to measure pressure differences.

Classification of Manometers

1. Piezometer Tube: A simple vertical glass tube open to the atmosphere at the top, tapped directly into a pipe or container. It measures positive liquid gauge pressure directly: $P = \gamma h$. It cannot measure gas pressures (gas escapes) or vacuum pressures (air is sucked into the vessel).
2. Open U-Tube Manometer: A U-shaped tube containing a heavy, immiscible gage fluid (typically mercury, $SG = 13.6$). It can measure high liquid or gas pressures as well as vacuum pressures.
3. Differential Manometer: Connects two points in a piping system or two different fluid tanks to measure the differential pressure $\Delta P = P_A - P_B$.
4. Inverted U-Tube Manometer: An upside-down U-tube containing a lighter fluid (such as air or light oil) at the top, used for measuring small pressure differences between two liquid lines with high precision.

The Step-by-Step Continuous Meniscus Method

To write the governing hydrostatic equation for any multi-fluid manometer:

1. Start at one terminal point (e.g., container $A$ or open atmosphere), writing down its pressure symbol ($P_A$ or $P_{\text{atm}} = 0\text{ kPa gauge}$).
2. Traverse through the tube step-by-step from interface to interface:
   • Moving downward through a vertical column of fluid of specific weight $\gamma_i$ and height $h_i$ adds pressure: $+ \gamma_i h_i$.
   • Moving upward through a vertical column subtracts pressure: $- \gamma_i h_i$.
3. Jump horizontally across identical continuous fluids: Two points at the exact same horizontal elevation in the same continuous, homogeneous, static liquid have identical pressures. You can transition directly between legs at this datum.
4. Equate the running sum to the pressure at the final terminal point (e.g., $= P_B$ or $= P_{\text{atm}}$).

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6. Worked Example: Multi-Fluid Differential Manometer

Problem Statement: A differential U-tube manometer connects Pipe A carrying freshwater ($\gamma_w = 9.81\text{ kN/m}^3$) and Pipe B carrying hydraulic oil ($SG_o = 0.82$, $\gamma_o = 0.82 \times 9.81 = 8.044\text{ kN/m}^3$). The gage liquid in the bottom loop is mercury ($SG_m = 13.6$, $\gamma_m = 13.6 \times 9.81 = 133.42\text{ kN/m}^3$).

• Centerline elevation of Pipe A: $\text{Elev } 12.00\text{ m}$
• Centerline elevation of Pipe B: $\text{Elev } 13.50\text{ m}$
• Water descends from Pipe A to the water-mercury interface in the left leg at $\text{Elev } 10.80\text{ m}$.
• The mercury-oil interface in the right leg rests at $\text{Elev } 11.45\text{ m}$.
• Oil fills the right leg from the mercury interface up to Pipe B.

Determine the pressure difference $P_A - P_B$ in kilopascals.

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Step-by-Step Solution:

1. Calculate vertical column heights:

   • Freshwater column in left leg: $h_w = 12.00\text{ m} - 10.80\text{ m} = 1.20\text{ m}$ (downward traverse from Pipe A).
   • Mercury differential column: $h_m = 11.45\text{ m} - 10.80\text{ m} = 0.65\text{ m}$ (upward traverse in right leg above the left meniscus level).
   • Hydraulic oil column in right leg: $h_o = 13.50\text{ m} - 11.45\text{ m} = 2.05\text{ m}$ (upward traverse to Pipe B).

2. Set up the continuous meniscus manometer equation: $P_A + \gamma_w h_w - \gamma_m h_m - \gamma_o h_o = P_B$

3. Isolate differential pressure $(P_A - P_B)$: $P_A - P_B = \gamma_m h_m + \gamma_o h_o - \gamma_w h_w$

4. Substitute numerical values: $P_A - P_B = (133.42\text{ kN/m}^3)(0.65\text{ m}) + (8.044\text{ kN/m}^3)(2.05\text{ m}) - (9.81\text{ kN/m}^3)(1.20\text{ m})$ $P_A - P_B = 86.723\text{ kPa} + 16.490\text{ kPa} - 11.772\text{ kPa} = 91.44\text{ kPa}$

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7. CELE Exam Traps & Common Computational Errors