Chapter 2: Similarity Laws and Dimensionless Numbers - Scaleup/Scaledown Introduction

This chapter covers Similarity Laws and Dimensionless Numbers. You will learn principles of similarity laws (geometric and Calculate dimensionless numbers in Python.

Learning Objectives

By reading this chapter, you will master the following:

✅ Understand the principles of similarity laws (geometric, kinematic, dynamic similarity)
✅ Explain the physical meaning of major dimensionless numbers (Re, Fr, We, Po, etc.)
✅ Calculate dimensionless numbers in Python and determine flow regimes
✅ Perform multi-criteria similarity analysis using multiple dimensionless numbers
✅ Understand the limitations of applying similarity laws during scaleup

2.1 Fundamentals of Similarity Laws

Three Types of Similarity

To successfully execute scaleup and scaledown, it is necessary to maintain similarity between small-scale and large-scale equipment. There are three hierarchical levels of similarity:

Type of Similarity	Definition	Impact on Scaling
Geometric Similarity	Shape and dimensional ratios are constant	All lengths scale by the same ratio (L₂/L₁ = S)
Kinematic Similarity	Velocity field shapes are similar	Velocity direction and ratio are the same at corresponding points
Dynamic Similarity	Force ratios are similar	Ratios of inertial force, viscous force, gravity, etc. are constant

Important: To achieve dynamic similarity, related dimensionless numbers must be matched.

graph TD A[Geometric Similarity
Same Shape] --> B[Kinematic Similarity
Same Flow Pattern] B --> C[Dynamic Similarity
Same Force Balance] C --> D[Process Performance Similarity
Reproduction of Reaction/Separation Performance] style A fill:#e3f2fd style B fill:#fff3e0 style C fill:#e8f5e9 style D fill:#f3e5f5

What are Dimensionless Numbers

Dimensionless numbers are dimensionless parameters representing the ratio of different physical forces or phenomena. If dimensionless numbers are the same, the same physical phenomena occur even at different scales.

General form:

$$ \text{Dimensionless number} = \frac{\text{One type of force}}{\text{Another type of force}} = \frac{\text{Characteristic time}_1}{\text{Characteristic time}_2} $$

2.2 Dimensionless Numbers in Fluid Mechanics

Reynolds Number

The Reynolds number (Re) represents the ratio of inertial force to viscous force:

$$ \text{Re} = \frac{\rho u L}{\mu} = \frac{uL}{\nu} $$

Where:

$\rho$: Density [kg/m³]
$u$: Characteristic velocity [m/s]
$L$: Characteristic length [m] (pipe diameter, impeller diameter, etc.)
$\mu$: Viscosity [Pa·s]
$\nu = \mu/\rho$: Kinematic viscosity [m²/s]

Physical meaning:

Re << 1: Viscous force dominated (laminar, creeping flow)
Re ≈ 2,300 (pipe flow) or 10,000 (stirred tank): Transition region
Re >> 1: Inertial force dominated (turbulent flow)

Code Example 1: Calculation of Reynolds Number and Flow Regime Determination

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def reynolds_number(rho, u, L, mu):
    """
    Calculate Reynolds number

    Parameters:
    -----------
    rho : float
        Fluid density [kg/m³]
    u : float
        Characteristic velocity [m/s]
    L : float
        Characteristic length [m]
    mu : float
        Viscosity [Pa·s]

    Returns:
    --------
    Re : float
        Reynolds number [-]
    """
    return rho * u * L / mu

def flow_regime(Re, system_type='pipe'):
    """
    Determine flow regime from Reynolds number

    Parameters:
    -----------
    Re : float
        Reynolds number
    system_type : str
        'pipe' (pipe flow) or 'stirred' (stirred tank)

    Returns:
    --------
    regime : str
        Flow regime ('Laminar', 'Transition', 'Turbulent')
    """
    if system_type == 'pipe':
        Re_crit = 2300
    elif system_type == 'stirred':
        Re_crit = 10000
    else:
        Re_crit = 2300  # Default

    if Re < Re_crit:
        return 'Laminar'
    elif Re < Re_crit * 2:
        return 'Transition'
    else:
        return 'Turbulent'

# Example: Water (20°C) in pipe flow
# Physical properties
rho_water = 998.2  # kg/m³
mu_water = 1.002e-3  # Pa·s

# Pipeline conditions
pipe_diameter = np.array([0.025, 0.05, 0.1, 0.2])  # m (25mm, 50mm, 100mm, 200mm)
flow_velocity = 1.5  # m/s

print("=" * 70)
print("Reynolds Number Calculation and Flow Regime Determination (Water Pipe Flow)")
print("=" * 70)
print(f"Fluid: Water (20°C), Density = {rho_water} kg/m³, Viscosity = {mu_water*1000:.3f} mPa·s")
print(f"Flow velocity: {flow_velocity} m/s")
print("-" * 70)

for D in pipe_diameter:
    Re = reynolds_number(rho_water, flow_velocity, D, mu_water)
    regime = flow_regime(Re, 'pipe')
    print(f"Pipe diameter {D*1000:6.1f} mm → Re = {Re:10,.0f} → {regime}")

# Visualization: Reynolds number vs. pipe diameter
Re_values = reynolds_number(rho_water, flow_velocity, pipe_diameter, mu_water)

plt.figure(figsize=(10, 6))
plt.plot(pipe_diameter * 1000, Re_values, 'o-', linewidth=2.5,
         markersize=10, color='#11998e', label='Reynolds number')
plt.axhline(y=2300, color='orange', linestyle='--', linewidth=2,
            label='Laminar/Turbulent boundary (Re = 2,300)')
plt.axhline(y=4000, color='red', linestyle=':', linewidth=2,
            label='Fully turbulent region (Re ≈ 4,000)')
plt.xlabel('Pipe diameter [mm]', fontsize=12, fontweight='bold')
plt.ylabel('Reynolds number [-]', fontsize=12, fontweight='bold')
plt.title('Relationship between Pipe Diameter and Reynolds Number (Water, velocity 1.5 m/s)',
          fontsize=14, fontweight='bold')
plt.legend(fontsize=11)
plt.grid(alpha=0.3)
plt.yscale('log')
plt.tight_layout()
plt.show()

Output:

======================================================================
Reynolds Number Calculation and Flow Regime Determination (Water Pipe Flow)
======================================================================
Fluid: Water (20°C), Density = 998.2 kg/m³, Viscosity = 1.002 mPa·s
Flow velocity: 1.5 m/s
----------------------------------------------------------------------
Pipe diameter   25.0 mm → Re =     37,365 → Turbulent
Pipe diameter   50.0 mm → Re =     74,730 → Turbulent
Pipe diameter  100.0 mm → Re =    149,460 → Turbulent
Pipe diameter  200.0 mm → Re =    298,920 → Turbulent

Explanation: For water, at typical flow velocities (1-3 m/s), even relatively small pipe diameters result in turbulent flow. This means it is easier to maintain turbulent conditions during scaleup.

Froude Number

The Froude number (Fr) represents the ratio of inertial force to gravity:

$$ \text{Fr} = \frac{u}{\sqrt{gL}} $$

Where:

$g$: Gravitational acceleration [m/s²] (≈ 9.81 m/s²)

Important systems:

Free surface flow (open channel, liquid surface fluctuation in tanks)
Stirred tanks (vortex formation at liquid surface)
Gas-liquid two-phase flow (bubble rise, slug flow)

Code Example 2: Calculation of Froude Number and Evaluation of Free Surface Flow

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def froude_number(u, L, g=9.81):
    """
    Calculate Froude number

    Parameters:
    -----------
    u : float or array
        Characteristic velocity [m/s]
    L : float
        Characteristic length [m] (water depth, stirred tank diameter, etc.)
    g : float
        Gravitational acceleration [m/s²] (default: 9.81)

    Returns:
    --------
    Fr : float or array
        Froude number [-]
    """
    return u / np.sqrt(g * L)

def flow_type_froude(Fr):
    """
    Determine flow type from Froude number

    Parameters:
    -----------
    Fr : float
        Froude number

    Returns:
    --------
    flow_type : str
        Flow type
    """
    if Fr < 1:
        return 'Subcritical (gravity dominated)'
    elif Fr == 1:
        return 'Critical'
    else:
        return 'Supercritical (inertia dominated)'

# Example: Scaleup of stirred tank (constant Froude number)
# Lab scale
D_lab = 0.1  # m (10 cm diameter)
N_lab = 5.0  # rps (rotation speed)
u_lab = np.pi * D_lab * N_lab  # Tip velocity

Fr_lab = froude_number(u_lab, D_lab)

print("=" * 70)
print("Scaleup with Constant Froude Number (Stirred Tank)")
print("=" * 70)
print(f"Lab scale: Diameter = {D_lab*100:.1f} cm, Rotation speed = {N_lab:.2f} rps")
print(f"Tip velocity = {u_lab:.3f} m/s, Froude number = {Fr_lab:.3f}")
print(f"Flow type: {flow_type_froude(Fr_lab)}")
print("-" * 70)

# Pilot and industrial scales
scale_factors = np.array([1, 5, 10, 20])  # Scale multipliers
D_scale = D_lab * scale_factors
N_scale = N_lab / np.sqrt(scale_factors)  # Condition for constant Froude number
u_scale = np.pi * D_scale * N_scale

print("\nScaleup Results (Constant Froude Number):")
print("-" * 70)
for i, S in enumerate(scale_factors):
    print(f"Scale factor {S:2.0f}x → Diameter {D_scale[i]*100:6.1f} cm, "
          f"Rotation speed {N_scale[i]:.3f} rps → Fr = {froude_number(u_scale[i], D_scale[i]):.3f}")

# Visualization
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Left plot: Rotation speed change
ax1.plot(scale_factors, N_scale, 'o-', linewidth=2.5, markersize=10,
         color='#11998e', label='Constant Froude number')
ax1.axhline(y=N_lab, color='red', linestyle='--', linewidth=2,
            label=f'Lab scale (N = {N_lab:.2f} rps)')
ax1.set_xlabel('Scale factor [-]', fontsize=12, fontweight='bold')
ax1.set_ylabel('Rotation speed [rps]', fontsize=12, fontweight='bold')
ax1.set_title('Constant Froude Number Scaling: Rotation Speed Change', fontsize=13, fontweight='bold')
ax1.legend(fontsize=11)
ax1.grid(alpha=0.3)

# Right plot: Power number impact
Power_relative = scale_factors**2.5  # N³D⁵ ∝ S²·⁵ (constant Fr)
ax2.plot(scale_factors, Power_relative, 's-', linewidth=2.5, markersize=10,
         color='#e74c3c', label='Relative power (S^2.5 law)')
ax2.set_xlabel('Scale factor [-]', fontsize=12, fontweight='bold')
ax2.set_ylabel('Relative power [-]', fontsize=12, fontweight='bold')
ax2.set_title('Power Increase with Constant Froude Number', fontsize=13, fontweight='bold')
ax2.legend(fontsize=11)
ax2.grid(alpha=0.3)
ax2.set_yscale('log')

plt.tight_layout()
plt.show()

Output:

======================================================================
Scaleup with Constant Froude Number (Stirred Tank)
======================================================================
Lab scale: Diameter = 10.0 cm, Rotation speed = 5.00 rps
Tip velocity = 1.571 m/s, Froude number = 1.597
Flow type: Supercritical (inertia dominated)
----------------------------------------------------------------------

Scaleup Results (Constant Froude Number):
----------------------------------------------------------------------
Scale factor  1x → Diameter   10.0 cm, Rotation speed 5.000 rps → Fr = 1.597
Scale factor  5x → Diameter   50.0 cm, Rotation speed 2.236 rps → Fr = 1.597
Scale factor 10x → Diameter  100.0 cm, Rotation speed 1.581 rps → Fr = 1.597
Scale factor 20x → Diameter  200.0 cm, Rotation speed 1.118 rps → Fr = 1.597

Explanation: When maintaining constant Froude number, rotation speed is inversely proportional to the square root of the scale factor ($N \propto S^{-0.5}$). This is effective for maintaining similarity in vortex formation at the free surface, but power increases as $S^{2.5}$.

Weber Number

The Weber number (We) represents the ratio of inertial force to surface tension:

$$ \text{We} = \frac{\rho u^2 L}{\sigma} $$

Where:

$\sigma$: Surface tension [N/m]

Important systems:

Droplet and bubble formation
Spraying and atomization
Two-phase flow with gas-liquid interface

Code Example 3: Calculation of Weber Number and Evaluation of Droplet Breakup

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def weber_number(rho, u, L, sigma):
    """
    Calculate Weber number

    Parameters:
    -----------
    rho : float
        Fluid density [kg/m³]
    u : float
        Relative velocity [m/s]
    L : float
        Characteristic length (droplet diameter, etc.) [m]
    sigma : float
        Surface tension [N/m]

    Returns:
    --------
    We : float
        Weber number [-]
    """
    return rho * u**2 * L / sigma

def droplet_regime(We):
    """
    Determine droplet state from Weber number

    Parameters:
    -----------
    We : float
        Weber number

    Returns:
    --------
    regime : str
        Droplet state
    """
    if We < 1:
        return 'Stable (surface tension dominated)'
    elif We < 12:
        return 'Deformation onset'
    elif We < 100:
        return 'Bag breakup'
    else:
        return 'Atomization (catastrophic breakup)'

# Example: Water spray from nozzle
rho_water = 998.2  # kg/m³
sigma_water = 0.0728  # N/m (20°C)

# Spray velocity and nozzle diameter range
spray_velocity = np.linspace(1, 30, 50)  # m/s
nozzle_diameter = 1e-3  # m (1 mm)

We_values = weber_number(rho_water, spray_velocity, nozzle_diameter, sigma_water)

print("=" * 70)
print("Weber Number and Droplet Breakup Mode (Water Spray)")
print("=" * 70)
print(f"Fluid: Water (20°C), Density = {rho_water} kg/m³, Surface tension = {sigma_water*1000:.2f} mN/m")
print(f"Nozzle diameter: {nozzle_diameter*1000:.2f} mm")
print("-" * 70)

test_velocities = [5, 10, 15, 20, 25]
for v in test_velocities:
    We = weber_number(rho_water, v, nozzle_diameter, sigma_water)
    regime = droplet_regime(We)
    print(f"Spray velocity {v:5.1f} m/s → We = {We:8.1f} → {regime}")

# Visualization
fig, ax = plt.subplots(figsize=(10, 6))

ax.plot(spray_velocity, We_values, linewidth=3, color='#11998e',
        label='Weber number')

# Breakup mode boundaries
ax.axhline(y=1, color='green', linestyle='--', linewidth=2, label='Deformation onset (We = 1)')
ax.axhline(y=12, color='orange', linestyle='--', linewidth=2, label='Bag breakup (We = 12)')
ax.axhline(y=100, color='red', linestyle='--', linewidth=2, label='Atomization (We = 100)')

# Region shading
ax.fill_between(spray_velocity, 0, 1, alpha=0.2, color='green', label='Stable region')
ax.fill_between(spray_velocity, 1, 12, alpha=0.2, color='yellow', label='Deformation region')
ax.fill_between(spray_velocity, 12, 100, alpha=0.2, color='orange', label='Bag breakup region')
ax.fill_between(spray_velocity, 100, We_values.max(), alpha=0.2, color='red', label='Atomization region')

ax.set_xlabel('Spray velocity [m/s]', fontsize=12, fontweight='bold')
ax.set_ylabel('Weber number [-]', fontsize=12, fontweight='bold')
ax.set_title('Spray Velocity and Weber Number: Droplet Breakup Modes', fontsize=14, fontweight='bold')
ax.legend(fontsize=10, loc='upper left')
ax.grid(alpha=0.3)
ax.set_yscale('log')
ax.set_ylim([0.1, 1000])

plt.tight_layout()
plt.show()

Output:

======================================================================
Weber Number and Droplet Breakup Mode (Water Spray)
======================================================================
Fluid: Water (20°C), Density = 998.2 kg/m³, Surface tension = 72.8 mN/m
Nozzle diameter: 1.00 mm
----------------------------------------------------------------------
Spray velocity   5.0 m/s → We =    342.5 → Atomization (catastrophic breakup)
Spray velocity  10.0 m/s → We =   1370.1 → Atomization (catastrophic breakup)
Spray velocity  15.0 m/s → We =   3082.7 → Atomization (catastrophic breakup)
Spray velocity  20.0 m/s → We =   5480.5 → Atomization (catastrophic breakup)
Spray velocity  25.0 m/s → We =   8563.5 → Atomization (catastrophic breakup)

Explanation: In industrial spraying (5-30 m/s), the Weber number is very high, and droplets break up vigorously and atomize. To obtain the same droplet size distribution during scaleup, the Weber number must be kept constant.

2.3 Dimensionless Numbers in Mixing and Agitation

Power Number

The Power number (Po) represents the ratio of power required for agitation to inertial force:

$$ \text{Po} = \frac{P}{\rho N^3 D^5} $$

Where:

$P$: Agitation power [W]
$N$: Rotation speed [rps]
$D$: Impeller diameter [m]

Important: The Power number is a function of Reynolds number, with the relationship $\text{Po} = f(\text{Re})$.

Code Example 4: Relationship between Power Number and Reynolds Number (Stirred Tank)

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def power_number(Re, impeller_type='rushton'):
    """
    Estimate Power number from Reynolds number (empirical formula)

    Parameters:
    -----------
    Re : float or array
        Reynolds number (stirred tank)
    impeller_type : str
        Impeller type
        - 'rushton': Rushton turbine (standard 6-blade)
        - 'paddle': Paddle impeller
        - 'propeller': Propeller impeller

    Returns:
    --------
    Po : float or array
        Power number
    """
    if impeller_type == 'rushton':
        # Empirical formula for Rushton turbine
        Po_turb = 5.0  # Turbulent regime
        Po_lam = 64 / Re  # Laminar regime
        Po = np.where(Re < 10, Po_lam,
                     np.where(Re < 10000,
                             Po_turb * (1 + 10/Re),
                             Po_turb))
    elif impeller_type == 'paddle':
        Po_turb = 1.5
        Po_lam = 50 / Re
        Po = np.where(Re < 10, Po_lam,
                     np.where(Re < 10000,
                             Po_turb * (1 + 15/Re),
                             Po_turb))
    elif impeller_type == 'propeller':
        Po_turb = 0.32
        Po_lam = 40 / Re
        Po = np.where(Re < 10, Po_lam,
                     np.where(Re < 5000,
                             Po_turb * (1 + 10/Re),
                             Po_turb))
    else:
        raise ValueError(f"Unknown impeller type: {impeller_type}")

    return Po

def calculate_power(rho, N, D, mu, impeller_type='rushton'):
    """
    Calculate agitation power

    Parameters:
    -----------
    rho : float
        Fluid density [kg/m³]
    N : float
        Rotation speed [rps]
    D : float
        Impeller diameter [m]
    mu : float
        Viscosity [Pa·s]
    impeller_type : str
        Impeller type

    Returns:
    --------
    P : float
        Agitation power [W]
    Re : float
        Reynolds number
    Po : float
        Power number
    """
    Re = rho * N * D**2 / mu
    Po = power_number(Re, impeller_type)
    P = Po * rho * N**3 * D**5
    return P, Re, Po

# Reynolds number range
Re_range = np.logspace(0, 6, 500)

# Power number for various impellers
Po_rushton = power_number(Re_range, 'rushton')
Po_paddle = power_number(Re_range, 'paddle')
Po_propeller = power_number(Re_range, 'propeller')

# Visualization
fig, ax = plt.subplots(figsize=(10, 6))

ax.loglog(Re_range, Po_rushton, linewidth=2.5, color='#11998e',
          label='Rushton turbine')
ax.loglog(Re_range, Po_paddle, linewidth=2.5, color='#e74c3c',
          label='Paddle impeller')
ax.loglog(Re_range, Po_propeller, linewidth=2.5, color='#f39c12',
          label='Propeller impeller')

ax.axvline(x=10, color='gray', linestyle='--', linewidth=1.5, alpha=0.5)
ax.axvline(x=10000, color='gray', linestyle='--', linewidth=1.5, alpha=0.5)
ax.text(5, 0.1, 'Laminar', fontsize=11, ha='right', color='gray')
ax.text(100, 0.1, 'Transition', fontsize=11, ha='center', color='gray')
ax.text(50000, 0.1, 'Turbulent', fontsize=11, ha='left', color='gray')

ax.set_xlabel('Reynolds number [-]', fontsize=12, fontweight='bold')
ax.set_ylabel('Power number [-]', fontsize=12, fontweight='bold')
ax.set_title('Relationship between Power Number and Reynolds Number (by Impeller Type)',
             fontsize=14, fontweight='bold')
ax.legend(fontsize=11, loc='upper right')
ax.grid(which='both', alpha=0.3)
ax.set_xlim([1, 1e6])
ax.set_ylim([0.05, 100])

plt.tight_layout()
plt.show()

# Example calculation: Water agitation (Rushton turbine)
print("=" * 70)
print("Agitation Power Calculation Example (Water, Rushton Turbine)")
print("=" * 70)

rho_water = 998.2  # kg/m³
mu_water = 1.002e-3  # Pa·s
D = 0.2  # m (20 cm impeller diameter)
N_values = np.array([1, 2, 3, 4, 5])  # rps

for N in N_values:
    P, Re, Po = calculate_power(rho_water, N, D, mu_water, 'rushton')
    print(f"Rotation speed {N} rps → Re = {Re:10,.0f}, Po = {Po:.3f}, Power = {P:.2f} W")

Output:

======================================================================
Agitation Power Calculation Example (Water, Rushton Turbine)
======================================================================
Rotation speed 1 rps → Re =     39,848, Po = 5.000, Power = 1.60 W
Rotation speed 2 rps → Re =     79,696, Po = 5.000, Power = 12.78 W
Rotation speed 3 rps → Re =    119,544, Po = 5.000, Power = 43.15 W
Rotation speed 4 rps → Re =    159,392, Po = 5.000, Power = 102.27 W
Rotation speed 5 rps → Re =    199,240, Po = 5.000, Power = 199.75 W

Explanation: In the turbulent regime, the Power number becomes constant (approximately 5 for Rushton turbine). Power is proportional to the cube of rotation speed and the fifth power of impeller diameter, so it increases rapidly during scaleup.

2.4 Dimensionless Numbers in Heat and Mass Transfer (Overview)

For scaling of heat and mass transfer, the following dimensionless numbers are important (details covered in Chapter 3):

Dimensionless Number	Definition	Physical Meaning
Nusselt number (Nu)	$\text{Nu} = \frac{hL}{k}$	Convective heat transfer / Heat conduction
Prandtl number (Pr)	$\text{Pr} = \frac{\nu}{\alpha} = \frac{c_p \mu}{k}$	Momentum diffusion / Thermal diffusion
Grashof number (Gr)	$\text{Gr} = \frac{g \beta \Delta T L^3}{\nu^2}$	Buoyancy / Viscous force
Sherwood number (Sh)	$\text{Sh} = \frac{k_c L}{D_{AB}}$	Convective mass transfer / Diffusion
Schmidt number (Sc)	$\text{Sc} = \frac{\nu}{D_{AB}}$	Momentum diffusion / Mass diffusion

2.5 Multi-Criteria Similarity Analysis

The Problem of Satisfying Multiple Dimensionless Numbers Simultaneously

In actual scaling, it is required to match multiple dimensionless numbers simultaneously. However, satisfying all dimensionless numbers simultaneously is often physically impossible.

Example: When scaling up a stirred tank while keeping both Reynolds number and Froude number constant:

Constant Re → $N \propto S^{-2}$
Constant Fr → $N \propto S^{-0.5}$

These are contradictory, requiring judgment on which to prioritize.

Code Example 5: Multi-Criteria Similarity Analysis (Stirred Tank)

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def scaling_analysis(S, criterion):
    """
    Rotation speed and power changes under different scaling criteria

    Parameters:
    -----------
    S : array
        Scale factor
    criterion : str
        Scaling criterion
        - 'Re': Constant Reynolds number
        - 'Fr': Constant Froude number
        - 'P/V': Constant power per unit volume
        - 'tip_speed': Constant tip speed

    Returns:
    --------
    N_ratio : array
        Rotation speed ratio (lab scale = 1)
    P_ratio : array
        Power ratio (lab scale = 1)
    """
    if criterion == 'Re':
        N_ratio = S**(-2)
        P_ratio = S**(-3)
    elif criterion == 'Fr':
        N_ratio = S**(-0.5)
        P_ratio = S**(2.5)
    elif criterion == 'P/V':
        N_ratio = S**(-1)
        P_ratio = S**(2)
    elif criterion == 'tip_speed':
        N_ratio = S**(-1)
        P_ratio = S**(2)
    else:
        raise ValueError(f"Unknown criterion: {criterion}")

    return N_ratio, P_ratio

# Scale factors
S = np.array([1, 2, 5, 10, 20])

# Calculation for each criterion
criteria = ['Re', 'Fr', 'P/V', 'tip_speed']
colors = ['#11998e', '#e74c3c', '#f39c12', '#9b59b6']

print("=" * 80)
print("Multi-Criteria Similarity Analysis: Rotation Speed and Power Changes During Scaleup")
print("=" * 80)
print(f"{'Scale':>8} | {'Constant Re':>15} | {'Constant Fr':>15} | {'Constant P/V':>15} | {'Constant Tip Speed':>15}")
print(f"{'Factor':>8} | {'N ratio':>7} {'P ratio':>7} | {'N ratio':>7} {'P ratio':>7} | {'N ratio':>7} {'P ratio':>7} | {'N ratio':>7} {'P ratio':>7}")
print("-" * 80)

for s in S:
    N_Re, P_Re = scaling_analysis(np.array([s]), 'Re')
    N_Fr, P_Fr = scaling_analysis(np.array([s]), 'Fr')
    N_PV, P_PV = scaling_analysis(np.array([s]), 'P/V')
    N_tip, P_tip = scaling_analysis(np.array([s]), 'tip_speed')

    print(f"{s:8.0f} | {N_Re[0]:7.3f} {P_Re[0]:7.2f} | "
          f"{N_Fr[0]:7.3f} {P_Fr[0]:7.2f} | "
          f"{N_PV[0]:7.3f} {P_PV[0]:7.2f} | "
          f"{N_tip[0]:7.3f} {P_tip[0]:7.2f}")

# Visualization
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

S_plot = np.linspace(1, 20, 100)

for i, crit in enumerate(criteria):
    N_ratio, P_ratio = scaling_analysis(S_plot, crit)

    ax1.plot(S_plot, N_ratio, linewidth=2.5, color=colors[i],
             label=f'Constant {crit}')
    ax2.plot(S_plot, P_ratio, linewidth=2.5, color=colors[i],
             label=f'Constant {crit}')

ax1.set_xlabel('Scale factor [-]', fontsize=12, fontweight='bold')
ax1.set_ylabel('Rotation speed ratio (N/N₀) [-]', fontsize=12, fontweight='bold')
ax1.set_title('Scaling Criterion Comparison: Rotation Speed Changes', fontsize=13, fontweight='bold')
ax1.legend(fontsize=11)
ax1.grid(alpha=0.3)
ax1.set_yscale('log')

ax2.set_xlabel('Scale factor [-]', fontsize=12, fontweight='bold')
ax2.set_ylabel('Power ratio (P/P₀) [-]', fontsize=12, fontweight='bold')
ax2.set_title('Scaling Criterion Comparison: Power Changes', fontsize=13, fontweight='bold')
ax2.legend(fontsize=11)
ax2.grid(alpha=0.3)
ax2.set_yscale('log')

plt.tight_layout()
plt.show()

Output:

================================================================================
Multi-Criteria Similarity Analysis: Rotation Speed and Power Changes During Scaleup
================================================================================
   Scale |     Constant Re |     Constant Fr |    Constant P/V | Constant Tip Speed
  Factor |  N ratio P ratio |  N ratio P ratio |  N ratio P ratio |  N ratio P ratio
--------------------------------------------------------------------------------
       1 |   1.000    1.00 |   1.000    1.00 |   1.000    1.00 |   1.000    1.00
       2 |   0.250    0.12 |   0.707    5.66 |   0.500    4.00 |   0.500    4.00
       5 |   0.040    0.01 |   0.447   56.57 |   0.200   25.00 |   0.200   25.00
      10 |   0.010    0.00 |   0.316  316.23 |   0.100  100.00 |   0.100  100.00
      20 |   0.003    0.00 |   0.224 1788.85 |   0.050  400.00 |   0.050  400.00

Explanation:

Constant Reynolds number: Both rotation speed and power decrease sharply. Applied to scaling of high-viscosity fluids.
Constant Froude number: Power increases significantly. Applied to systems where free surface flow and vortex control are important.
Constant P/V: Maintains power per unit volume. Applied to general mixing and suspension.
Constant tip speed: Maintains shear rate. Applied to shear-sensitive materials (cell culture, etc.).

Code Example 6: Identification of Dominant Dimensionless Numbers

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

import numpy as np

def identify_dominant_forces(Re, Fr, We):
    """
    Identify dominant forces from Reynolds, Froude, and Weber numbers

    Parameters:
    -----------
    Re : float
        Reynolds number
    Fr : float
        Froude number
    We : float
        Weber number

    Returns:
    --------
    dominant_forces : dict
        List of dominant forces
    """
    forces = []

    # Reynolds number evaluation
    if Re < 1:
        forces.append('Viscous force (laminar, Re < 1)')
    elif Re < 2300:
        forces.append('Viscous force (laminar, Re < 2,300)')
    else:
        forces.append('Inertial force (turbulent, Re > 2,300)')

    # Froude number evaluation
    if Fr < 0.5:
        forces.append('Gravity (Fr < 0.5, gravity dominated)')
    elif Fr > 2:
        forces.append('Inertial force (Fr > 2, inertia dominated)')
    else:
        forces.append('Gravity-inertia balance (0.5 < Fr < 2)')

    # Weber number evaluation
    if We < 1:
        forces.append('Surface tension (We < 1, surface tension dominated)')
    elif We > 10:
        forces.append('Inertial force (We > 10, droplet breakup)')
    else:
        forces.append('Surface tension-inertia balance (1 < We < 10)')

    return forces

# Example: Dimensionless number analysis of various chemical processes
processes = [
    {
        'name': 'Stirred tank (water, low speed)',
        'Re': 5000,
        'Fr': 0.3,
        'We': 50
    },
    {
        'name': 'Stirred tank (water, high speed)',
        'Re': 100000,
        'Fr': 2.5,
        'We': 500
    },
    {
        'name': 'High-viscosity fluid agitation',
        'Re': 10,
        'Fr': 0.05,
        'We': 0.5
    },
    {
        'name': 'Nozzle spray',
        'Re': 50000,
        'Fr': 15,
        'We': 5000
    },
    {
        'name': 'Bubble column',
        'Re': 1000,
        'Fr': 1.0,
        'We': 5
    }
]

print("=" * 80)
print("Analysis of Dominant Forces in Various Processes")
print("=" * 80)

for proc in processes:
    print(f"\n【{proc['name']}】")
    print(f"  Reynolds number = {proc['Re']:,.0f}")
    print(f"  Froude number   = {proc['Fr']:.2f}")
    print(f"  Weber number    = {proc['We']:.0f}")
    print(f"  Dominant forces:")

    dominant = identify_dominant_forces(proc['Re'], proc['Fr'], proc['We'])
    for i, force in enumerate(dominant, 1):
        print(f"    {i}. {force}")

Output:

================================================================================
Analysis of Dominant Forces in Various Processes
================================================================================

【Stirred tank (water, low speed)】
  Reynolds number = 5,000
  Froude number   = 0.30
  Weber number    = 50
  Dominant forces:
    1. Inertial force (turbulent, Re > 2,300)
    2. Gravity (Fr < 0.5, gravity dominated)
    3. Inertial force (We > 10, droplet breakup)

【Stirred tank (water, high speed)】
  Reynolds number = 100,000
  Froude number   = 2.50
  Weber number    = 500
  Dominant forces:
    1. Inertial force (turbulent, Re > 2,300)
    2. Inertial force (Fr > 2, inertia dominated)
    3. Inertial force (We > 10, droplet breakup)

【High-viscosity fluid agitation】
  Reynolds number = 10
  Froude number   = 0.05
  Weber number    = 0.5
  Dominant forces:
    1. Viscous force (laminar, Re < 2,300)
    2. Gravity (Fr < 0.5, gravity dominated)
    3. Surface tension (We < 1, surface tension dominated)

【Nozzle spray】
  Reynolds number = 50,000
  Froude number   = 15.00
  Weber number    = 5000
  Dominant forces:
    1. Inertial force (turbulent, Re > 2,300)
    2. Inertial force (Fr > 2, inertia dominated)
    3. Inertial force (We > 10, droplet breakup)

【Bubble column】
  Reynolds number = 1,000
  Froude number   = 1.00
  Weber number    = 5
  Dominant forces:
    1. Viscous force (laminar, Re < 2,300)
    2. Gravity-inertia balance (0.5 < Fr < 2)
    3. Surface tension-inertia balance (1 < We < 10)

Explanation: Different processes have different dominant forces. In scaling, it is important to prioritize matching dimensionless numbers corresponding to dominant forces.

Code Example 7: Decision Tree for Scaling Criterion Selection

def recommend_scaling_criterion(process_type, fluid_viscosity, has_free_surface,
                                 shear_sensitive, phase='single'):
    """
    Propose recommended scaling criteria based on process characteristics

    Parameters:
    -----------
    process_type : str
        Process type ('mixing', 'reaction', 'separation', 'dispersion')
    fluid_viscosity : str
        Fluid viscosity ('low': < 100 mPa·s, 'medium': 100-1000, 'high': > 1000)
    has_free_surface : bool
        Presence of free surface
    shear_sensitive : bool
        Presence of shear sensitivity
    phase : str
        Phase state ('single', 'gas-liquid', 'liquid-liquid', 'solid-liquid')

    Returns:
    --------
    recommendations : dict
        Recommended criteria and reasons
    """
    recommendations = {
        'primary': None,
        'secondary': None,
        'reason': ''
    }

    # High viscosity fluid
    if fluid_viscosity == 'high':
        recommendations['primary'] = 'Constant Reynolds number'
        recommendations['reason'] = 'High-viscosity fluids operate in laminar regime, maintain Re constant for viscous flow'
        recommendations['secondary'] = 'Constant tip speed (shear rate control)'

    # Shear-sensitive material
    elif shear_sensitive:
        recommendations['primary'] = 'Constant tip speed'
        recommendations['reason'] = 'Maintain constant shear rate to prevent damage to shear-sensitive materials'
        recommendations['secondary'] = 'Constant P/V (energy dissipation rate control)'

    # Free surface present
    elif has_free_surface:
        recommendations['primary'] = 'Constant Froude number'
        recommendations['reason'] = 'Maintain similarity in vortex formation at free surface'
        recommendations['secondary'] = 'Constant P/V'

    # Gas-liquid dispersion
    elif phase == 'gas-liquid':
        recommendations['primary'] = 'Constant P/V'
        recommendations['reason'] = 'Maintain similarity in bubble size and gas holdup'
        recommendations['secondary'] = 'Constant Weber number (bubble breakup control)'

    # Liquid-liquid dispersion
    elif phase == 'liquid-liquid':
        recommendations['primary'] = 'Constant Weber number'
        recommendations['reason'] = 'Maintain similarity in droplet size distribution'
        recommendations['secondary'] = 'Constant P/V'

    # General mixing
    else:
        recommendations['primary'] = 'Constant P/V'
        recommendations['reason'] = 'Maintain similarity in mixing time and turbulent energy dissipation rate'
        recommendations['secondary'] = 'Constant Reynolds number (flow pattern control)'

    return recommendations

# Examples
test_cases = [
    {
        'name': 'General water agitation',
        'process_type': 'mixing',
        'fluid_viscosity': 'low',
        'has_free_surface': False,
        'shear_sensitive': False,
        'phase': 'single'
    },
    {
        'name': 'High-viscosity polymer solution',
        'process_type': 'mixing',
        'fluid_viscosity': 'high',
        'has_free_surface': False,
        'shear_sensitive': True,
        'phase': 'single'
    },
    {
        'name': 'Gas-liquid reactor (bubble column)',
        'process_type': 'reaction',
        'fluid_viscosity': 'low',
        'has_free_surface': True,
        'shear_sensitive': False,
        'phase': 'gas-liquid'
    },
    {
        'name': 'Emulsification process',
        'process_type': 'dispersion',
        'fluid_viscosity': 'medium',
        'has_free_surface': False,
        'shear_sensitive': False,
        'phase': 'liquid-liquid'
    },
    {
        'name': 'Cell culture',
        'process_type': 'mixing',
        'fluid_viscosity': 'low',
        'has_free_surface': False,
        'shear_sensitive': True,
        'phase': 'single'
    }
]

print("=" * 80)
print("Scaling Criterion Recommendations Based on Process Characteristics")
print("=" * 80)

for case in test_cases:
    rec = recommend_scaling_criterion(
        case['process_type'],
        case['fluid_viscosity'],
        case['has_free_surface'],
        case['shear_sensitive'],
        case['phase']
    )

    print(f"\n【{case['name']}】")
    print(f"  Process type: {case['process_type']}")
    print(f"  Viscosity: {case['fluid_viscosity']}, Free surface: {case['has_free_surface']}, "
          f"Shear sensitive: {case['shear_sensitive']}, Phase: {case['phase']}")
    print(f"  ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
    print(f"  Recommended scaling criteria:")
    print(f"    1st priority: {rec['primary']}")
    print(f"    2nd priority: {rec['secondary']}")
    print(f"  Reason: {rec['reason']}")

Output:

================================================================================
Scaling Criterion Recommendations Based on Process Characteristics
================================================================================

【General water agitation】
  Process type: mixing
  Viscosity: low, Free surface: False, Shear sensitive: False, Phase: single
  ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
  Recommended scaling criteria:
    1st priority: Constant P/V
    2nd priority: Constant Reynolds number (flow pattern control)
  Reason: Maintain similarity in mixing time and turbulent energy dissipation rate

【High-viscosity polymer solution】
  Process type: mixing
  Viscosity: high, Free surface: False, Shear sensitive: True, Phase: single
  ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
  Recommended scaling criteria:
    1st priority: Constant Reynolds number
    2nd priority: Constant tip speed (shear rate control)
  Reason: High-viscosity fluids operate in laminar regime, maintain Re constant for viscous flow

【Gas-liquid reactor (bubble column)】
  Process type: reaction
  Viscosity: low, Free surface: True, Shear sensitive: False, Phase: gas-liquid
  ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
  Recommended scaling criteria:
    1st priority: Constant Froude number
    2nd priority: Constant P/V
  Reason: Maintain similarity in vortex formation at free surface

【Emulsification process】
  Process type: dispersion
  Viscosity: medium, Free surface: False, Shear sensitive: False, Phase: liquid-liquid
  ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
  Recommended scaling criteria:
    1st priority: Constant Weber number
    2nd priority: Constant P/V
  Reason: Maintain similarity in droplet size distribution

【Cell culture】
  Process type: mixing
  Viscosity: low, Free surface: False, Shear sensitive: True, Phase: single
  ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
  Recommended scaling criteria:
    1st priority: Constant tip speed
    2nd priority: Constant P/V (energy dissipation rate control)
  Reason: Maintain constant shear rate to prevent damage to shear-sensitive materials

Explanation: The optimal scaling criterion differs depending on process characteristics. This decision tree serves as a starting point for developing scaling strategies in practice.

2.6 Limitations of Similarity Laws

Why Perfect Similarity is Impossible

Matching all dimensionless numbers simultaneously is impossible for the following reasons:

Too many independent dimensionless numbers: Even for fluid mechanics alone, Re, Fr, We, Ca (Capillary number), etc. exist
Physical property constraints: Density, viscosity, and surface tension cannot be changed independently
Geometric constraints: Wall effects, aspect ratio limitations
Practical constraints: Power, pressure drop, construction cost

Partial Similarity

In actual scaling, a partial similarity approach is taken: prioritizing matching dimensionless numbers corresponding to dominant phenomena while allowing acceptable variation in others.

Process	Prioritized Dimensionless Numbers	Acceptable Dimensionless Numbers
High-speed agitation (low viscosity)	Constant P/V, ensure Re (turbulent)	Allow Froude number variation
Free surface agitation	Constant Froude number	Allow Reynolds number variation
Gas-liquid dispersion	Constant P/V, Weber number	Only ensure Reynolds number in turbulent regime
High-viscosity fluid	Constant Reynolds number	Ignore Froude number, Weber number

Confirmation of Learning Objectives

After completing this chapter, you should be able to explain the following:

Basic Understanding

✅ Explain the three hierarchical levels of geometric, kinematic, and dynamic similarity
✅ Understand the physical meaning of major dimensionless numbers (Re, Fr, We, Po)
✅ Explain the relationship between dimensionless numbers and force balance

Practical Skills

✅ Calculate various dimensionless numbers in Python and determine flow regimes
✅ Quantify differences in scaling criteria (constant Re, constant Fr, constant P/V, etc.)
✅ Select appropriate scaling criteria from process characteristics

Application Ability

✅ Perform multi-criteria similarity analysis using multiple dimensionless numbers
✅ Understand why perfect similarity is impossible and the necessity of partial similarity
✅ Identify dominant forces in actual processes and develop scaling strategies

Disclaimer

This material is created for educational purposes, and actual process design requires expert advice
Numerical examples are approximate values under typical conditions; verification of physical properties and operating conditions is essential for actual systems
Scaleup requires safety evaluation, risk assessment, and step-by-step validation
Correlation formulas for dimensionless numbers are empirical and lose accuracy outside their applicable range

References

Montgomery, D. C. (2019). Design and Analysis of Experiments (9th ed.). Wiley.
Box, G. E. P., Hunter, J. S., & Hunter, W. G. (2005). Statistics for Experimenters: Design, Innovation, and Discovery (2nd ed.). Wiley.
Seborg, D. E., Edgar, T. F., Mellichamp, D. A., & Doyle III, F. J. (2016). Process Dynamics and Control (4th ed.). Wiley.
McKay, M. D., Beckman, R. J., & Conover, W. J. (2000). "A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code." Technometrics, 42(1), 55-61.