Fluid Mechanics for Chemical Engineers

Fluid mechanics is fundamental to chemical engineering as it governs how fluids (liquids and gases) move through process equipment. Understanding fluid behavior is essential for designing pumps, pipes, reactors, and separation equipment.

Fluid Properties

Viscosity

Viscosity ($\mu$) measures a fluid's resistance to flow:

• Dynamic viscosity: Resistance to shear stress
• Kinematic viscosity: $\nu = \mu/\rho$

Density and Specific Gravity

• Density ($\rho$): Mass per unit volume
• Specific gravity: Ratio of fluid density to water density at 4°C

Flow Regimes

Reynolds Number

The Reynolds number determines flow regime:

Where:

• $v$ = average velocity
• $D$ = characteristic length (pipe diameter)
• $\mu$ = dynamic viscosity
• $\nu$ = kinematic viscosity

Flow Classification

• Laminar flow: $Re < 2300$ (smooth, orderly)
• Transitional flow: $2300 < Re < 4000$ (unstable)
• Turbulent flow: $Re > 4000$ (chaotic, mixed)

Bernoulli Equation

For incompressible, frictionless flow:

Where:

• $P$ = pressure
• $v$ = velocity
• $z$ = elevation

Pressure Drop Calculations

Friction Factor

For laminar flow ($Re < 2300$):

For turbulent flow, use the Colebrook equation:

Darcy-Weisbach Equation

Pressure drop in pipes:

Where:

• $L$ = pipe length
• $D$ = pipe diameter
• $\epsilon$ = pipe roughness

Pumping Systems

Pump Types

• Centrifugal pumps: Most common in chemical plants
• Positive displacement pumps: For high-viscosity fluids
• Diaphragm pumps: For corrosive or abrasive fluids

Pump Performance

• Head: Energy per unit weight
• Capacity: Flow rate
• Efficiency: Power output / Power input

Net Positive Suction Head (NPSH)

Prevents cavitation:

Advanced Fluid Mechanics Concepts

Navier-Stokes Equations

The fundamental equations governing fluid motion are the Navier-Stokes equations:

Conservation of Mass (Continuity):

For incompressible flow, this simplifies to:

Conservation of Momentum:

Where:

• $\rho$ = fluid density
• $\mathbf{v}$ = velocity vector
• $p$ = pressure
• $\mu$ = dynamic viscosity
• $\mathbf{g}$ = gravitational acceleration

Boundary Layer Theory

The boundary layer is the thin region near a solid surface where viscous effects are significant:

Boundary Layer Thickness ($\delta$): For laminar flow over a flat plate:

Where $Re_x = \frac{\rho v_\infty x}{\mu}$ is the local Reynolds number.

Displacement Thickness ($\delta^*$): Represents the distance by which the external streamlines are displaced due to the boundary layer:

Turbulence Modeling

For turbulent flow, we use time-averaged equations (Reynolds-averaged Navier-Stokes):

Reynolds Decomposition:

Where $\bar{u}$ is the mean velocity and $u'$ is the fluctuating component.

Turbulent Kinetic Energy ($k$):

Non-Newtonian Fluids

Many chemical process fluids exhibit non-Newtonian behavior:

Power Law Model:

Where:

• $K$ = consistency index
• $n$ = flow behavior index
• $n < 1$: Pseudoplastic (shear thinning)
• $n > 1$: Dilatant (shear thickening)
• $n = 1$: Newtonian

Bingham Plastic Model:

Where $\tau_0$ is the yield stress.

Real-World Application: Pipeline Design

Designing a chemical pipeline requires careful fluid mechanics calculations:

Design Considerations

1. Flow rate requirements
2. Pressure limitations
3. Material compatibility
4. Energy efficiency
5. Safety factors

Advanced Pipeline Analysis

For complex pipeline systems, consider:

Minor Losses:

Where $K_L$ is the loss coefficient for fittings, valves, and expansions/contractions.

Network Analysis: For branched or looped systems, use the Hardy-Cross method or equivalent pipe length methods.

Example: Crude Oil Pipeline

Calculate pressure drop for crude oil flowing through a pipeline:

# Pipeline parameters
pipe_diameter = 0.3  # m
pipe_length = 5000   # m
pipe_roughness = 0.000045  # m (commercial steel)
flow_rate = 0.1      # m³/s

# Fluid properties (crude oil)
density = 850        # kg/m³
viscosity = 0.005    # Pa·s

# TODO: Calculate pressure drop
# Steps:
# 1. Calculate velocity from flow rate and area
# 2. Calculate Reynolds number
# 3. Determine friction factor
# 4. Calculate pressure drop using Darcy-Weisbach

velocity = 0
Re = 0
friction_factor = 0
pressure_drop = 0

print(f"Velocity: {velocity:.2f} m/s")
print(f"Reynolds number: {Re:.0f}")
print(f"Friction factor: {friction_factor:.4f}")
print(f"Pressure drop: {pressure_drop:.0f} Pa")

# Additional analysis: Check for turbulent flow transition
if Re < 2300:
    print("Flow is laminar")
elif Re < 4000:
    print("Flow is in transition region")
else:
    print("Flow is turbulent")
    
# Calculate hydraulic power required
power = pressure_drop * flow_rate  # W
print(f"Hydraulic power: {power/1000:.1f} kW")

Your Challenge: Pump Selection Analysis

In this exercise, you'll analyze pump performance and select an appropriate pump for a chemical process.

Goal: Calculate pump power requirements and evaluate system curve.

System Description

A centrifugal pump moves water from a storage tank to a reactor. The system has:

• Elevation difference: 15 m
• Pipe length: 200 m
• Pipe diameter: 0.1 m
• Pipe roughness: 0.000046 m
• Required flow rate: 0.02 m³/s
• Fluid: Water at 20°C ($\rho = 998$ kg/m³, $\mu = 0.001$ Pa·s)

Pump Performance Data

The pump has the following characteristics:

• Maximum head: 50 m
• Maximum flow: 0.05 m³/s
• Efficiency: 70% at design point

# System parameters
elevation_diff = 15  # m
pipe_length = 200    # m
pipe_diameter = 0.1  # m
pipe_roughness = 0.000046  # m
flow_rate = 0.02     # m³/s

# Fluid properties
density = 998        # kg/m³
viscosity = 0.001    # Pa·s
gravity = 9.81       # m/s²

# TODO: Calculate system head requirement
# System head = elevation head + friction head
velocity = 0
Re = 0
friction_factor = 0
friction_head = 0
system_head = 0

# TODO: Calculate pump power requirement
pump_power = 0

print(f"System head requirement: {system_head:.1f} m")
print(f"Pump power requirement: {pump_power:.1f} kW")

# Check if pump is suitable
if system_head <= 50:
    print("Pump is suitable for this application")
else:
    print("Pump cannot provide sufficient head")

What modifications could reduce the power requirement? How would changing the pipe diameter affect the system?