Dimensional Analysis

Dimensional analysis provides a systematic method for reducing the number of variables in physical problems and establishing dimensionless parameters that govern fluid flow and heat transfer phenomena in HVAC systems. This technique enables scaling of experimental results, model testing, and correlation of complex physical processes.

Fundamental Principles

Dimensional Homogeneity

All valid physical equations must be dimensionally homogeneous—each term in the equation must have the same dimensional representation. This principle forms the foundation of dimensional analysis.

Primary Dimensions in HVAC:

• Mass [M]
• Length [L]
• Time [T]
• Temperature [θ]
• Electric current [I]

Derived Quantities:

Buckingham Pi Theorem

The Buckingham Pi theorem states that if a physical problem involves n variables and these variables contain m fundamental dimensions, the problem can be reduced to (n - m) independent dimensionless groups.

Mathematical Statement:

If a physical relationship can be expressed as:

f(x₁, x₂, x₃, …, xₙ) = 0

where the n variables involve m fundamental dimensions, then:

F(π₁, π₂, π₃, …, πₙ₋ₘ) = 0

where π terms are dimensionless groups formed from the original variables.

Application Procedure:

1. List all variables affecting the phenomenon
2. Identify fundamental dimensions (typically M, L, T, θ)
3. Determine number of dimensionless groups: (n - m)
4. Select m repeating variables containing all fundamental dimensions
5. Form dimensionless groups by combining repeating variables with remaining variables
6. Verify dimensional homogeneity of each π group

Example: Pressure Drop in Duct Flow

Variables affecting pressure drop:

• ΔP (pressure drop) [M/(L·T²)]
• ρ (density) [M/L³]
• V (velocity) [L/T]
• D (diameter) [L]
• L (length) [L]
• μ (viscosity) [M/(L·T)]
• ε (roughness) [L]

n = 7 variables, m = 3 dimensions (M, L, T) Number of π groups = 7 - 3 = 4

Selecting ρ, V, D as repeating variables:

π₁ = ΔP/(ρV²) (Euler number) π₂ = ρVD/μ (Reynolds number) π₃ = L/D (length-to-diameter ratio) π₄ = ε/D (relative roughness)

Result: π₁ = f(π₂, π₃, π₄)

This leads to the Darcy-Weisbach equation form: ΔP = f(Re, L/D, ε/D) × ρV²

Critical Dimensionless Numbers in HVAC

Reynolds Number (Re)

The Reynolds number represents the ratio of inertial forces to viscous forces in fluid flow.

Definition:

Re = (ρVD)/μ = VD/ν

where:

• ρ = fluid density (kg/m³)
• V = characteristic velocity (m/s)
• D = characteristic length (m)
• μ = dynamic viscosity (Pa·s)
• ν = kinematic viscosity (m²/s)

Physical Significance:

• Re < 2,300: Laminar flow (viscous forces dominant)
• 2,300 < Re < 4,000: Transition region
• Re > 4,000: Turbulent flow (inertial forces dominant)

HVAC Applications:

Design Implications:

• Friction factor correlation selection depends on Re
• Heat transfer coefficient calculations require Re determination
• Pressure drop predictions vary significantly with flow regime
• Transition region creates uncertainty in design calculations

Nusselt Number (Nu)

The Nusselt number represents the ratio of convective to conductive heat transfer at a boundary.

Definition:

Nu = hL/k

where:

• h = convection heat transfer coefficient (W/(m²·K))
• L = characteristic length (m)
• k = thermal conductivity (W/(m·K))

Physical Meaning:

Nu = 1 implies pure conduction Nu > 1 indicates convective enhancement

Correlations for HVAC Equipment:

Turbulent flow in tubes (Dittus-Boelter): Nu = 0.023 Re^0.8 Pr^n

where n = 0.4 for heating, 0.3 for cooling

Valid for:

• Re > 10,000
• 0.7 < Pr < 160
• L/D > 10

Laminar flow in tubes (constant surface temperature): Nu = 3.66 (fully developed)

External flow over tube banks: Nu = C Re^m Pr^0.36

where C and m depend on tube arrangement and Reynolds number range.

Application Example:

For air (Pr ≈ 0.7) flowing through a duct at Re = 50,000: Nu = 0.023 × (50,000)^0.8 × (0.7)^0.4 ≈ 118

If characteristic length = 0.3 m, k = 0.026 W/(m·K): h = Nu × k / L = 118 × 0.026 / 0.3 ≈ 10.2 W/(m²·K)

Prandtl Number (Pr)

The Prandtl number represents the ratio of momentum diffusivity to thermal diffusivity.

Definition:

Pr = ν/α = (μcp)/k

where:

• ν = kinematic viscosity (m²/s)
• α = thermal diffusivity (m²/s)
• cp = specific heat (J/(kg·K))

Physical Significance:

• Pr « 1: Thermal diffusivity dominates (liquid metals)
• Pr ≈ 1: Momentum and thermal diffusivities similar (gases)
• Pr » 1: Momentum diffusivity dominates (oils)

Values for HVAC Fluids:

Design Considerations:

Prandtl number affects the relationship between velocity and thermal boundary layers. For air (Pr ≈ 0.7), the boundary layers are approximately equal thickness. For water and glycol solutions (Pr > 1), the thermal boundary layer is thinner than the velocity boundary layer, affecting heat transfer coefficient predictions.

Grashof Number (Gr)

The Grashof number represents the ratio of buoyancy forces to viscous forces in natural convection.

Definition:

Gr = (gβΔTL³)/ν²

where:

• g = gravitational acceleration (9.81 m/s²)
• β = thermal expansion coefficient (1/K)
• ΔT = temperature difference (K)
• L = characteristic length (m)
• ν = kinematic viscosity (m²/s)

Natural Convection Regimes:

• Gr < 10⁴: Conduction dominant
• 10⁴ < Gr < 10⁹: Laminar natural convection
• Gr > 10⁹: Turbulent natural convection

HVAC Applications:

• Heat loss from uninsulated ducts in unconditioned spaces
• Natural convection from building surfaces
• Stratification in large spaces
• Free cooling applications
• Stack effect in tall buildings

Rayleigh Number (Ra)

For natural convection problems, the product of Grashof and Prandtl numbers:

Ra = Gr × Pr = (gβΔTL³)/(να)

Critical Rayleigh number (Ra_c) determines onset of convective instability in various geometries.

Stanton Number (St)

The Stanton number relates convective heat transfer to thermal capacity of flowing fluid.

Definition:

St = h/(ρVcp) = Nu/(Re × Pr)

Applications:

• Heat exchanger effectiveness calculations
• Cooling tower performance
• Air-side heat transfer in coils

Friction Factor and Dimensionless Groups

Darcy friction factor (f):

For laminar flow (Re < 2,300): f = 64/Re

For turbulent flow, Colebrook equation: 1/√f = -2.0 log₁₀(ε/(3.7D) + 2.51/(Re√f))

Moody Chart Representation:

The Moody chart presents f as a function of Re and relative roughness (ε/D), providing a graphical solution to the Colebrook equation widely used in duct and pipe design.

Equivalent Roughness Values:

Other Relevant Dimensionless Groups

Froude Number (Fr): Fr = V/√(gL)

Ratio of inertial to gravitational forces. Relevant in:

• Open channel flow
• Hydraulic jumps
• Water features in buildings

Weber Number (We): We = (ρV²L)/σ

Ratio of inertial to surface tension forces. Applications:

• Spray nozzles
• Humidification systems
• Droplet formation

Mach Number (Ma): Ma = V/c

Ratio of flow velocity to speed of sound. Considerations:

• High-velocity duct design (Ma > 0.3)
• Pressure measurement corrections
• Compressibility effects in air distribution

Euler Number (Eu): Eu = ΔP/(ρV²)

Ratio of pressure forces to inertial forces. Direct relationship to pressure drop calculations.

Cavitation Number (Ca): Ca = (P - P_v)/(0.5ρV²)

Used to predict cavitation onset in pumps and control valves.

Similarity Theory and Model Testing

Types of Similarity

Geometric Similarity: All linear dimensions of model and prototype are related by a constant scale factor.

L_m/L_p = λ_L (length scale) A_m/A_p = λ_L² V_m/V_p = λ_L³

Kinematic Similarity: Velocity patterns are geometrically similar.

V_m/V_p = λ_V (velocity scale) t_m/t_p = (λ_L)/(λ_V)

Dynamic Similarity: Force ratios are identical between model and prototype.

Requires equality of all relevant dimensionless numbers.

Scale Model Testing in HVAC

Applications:

• Airflow patterns in large spaces
• Smoke control system validation
• Natural ventilation studies
• Wind effects on building ventilation
• Cleanroom contamination control

Scaling Considerations:

For complete dynamic similarity in isothermal flow, Reynolds number must be matched:

Re_m = Re_p (ρVD/μ)_m = (ρVD/μ)_p

If geometric scale λ_L = D_p/D_m = 10 and using same fluid:

V_m = V_p × (D_p/D_m) = 10V_p

Velocity must increase by scale factor, which may create compressibility issues or require different fluids.

Practical Compromises:

Full dynamic similarity often impossible. Engineers prioritize:

• Reynolds number similarity (flow patterns)
• Froude number similarity (buoyancy effects)
• Richardson number similarity (density-driven flows)

depending on dominant physical phenomena.

Richardson Number

For flows with both forced and natural convection:

Ri = Gr/Re² = (gβΔTL)/(V²)

• Ri « 1: Forced convection dominant
• Ri ≈ 1: Mixed convection
• Ri » 1: Natural convection dominant

ASHRAE and Code References

ASHRAE Handbook - Fundamentals:

• Chapter 3: Fluid Flow
• Chapter 4: Heat Transfer
• Dimensionless correlations for heat transfer and pressure drop

ASHRAE Research:

• RP-1515: Heat Transfer and Pressure Drop in Small Diameter Tubes
• RP-1583: Correlations for Heat Transfer and Pressure Drop in Coils

Standards:

• ASHRAE Standard 111: Practices for Measurement, Testing, Adjusting, and Balancing of Building HVAC Systems
• ASHRAE Standard 41.2: Standard Methods for Laboratory Air Flow Measurement

Engineering Applications

Duct Design

Dimensional analysis reduces duct design to relationships between:

• Friction factor (f) as function of Re and ε/D
• Pressure drop coefficient calculations
• Fitting loss coefficients scaled by velocity pressure

Equal Friction Method: Based on maintaining constant ΔP/L, inherently uses dimensionless pressure drop per length.

Heat Exchanger Design

Heat exchanger performance correlations use dimensionless groups:

Effectiveness-NTU Method:

• NTU (Number of Transfer Units) = UA/(ṁcp)_min
• Heat capacity rate ratio C = C_min/C_max
• ε = f(NTU, C, flow arrangement)

j-factor and f-factor:

Colburn j-factor: j = St × Pr^(2/3) = (h/(ρVcp)) × Pr^(2/3)

Allows comparison of heat transfer performance across different geometries and fluids.

Pump and Fan Selection

Affinity Laws derive from dimensional analysis:

Flow: Q₂/Q₁ = (N₂/N₁)(D₂/D₁)³ Head: H₂/H₁ = (N₂/N₁)²(D₂/D₁)² Power: W₂/W₁ = (N₂/N₁)³(D₂/D₁)⁵

Specific Speed:

N_s = (N√Q)/(H^(3/4))

Dimensionless parameter (when properly non-dimensionalized) characterizing pump/fan geometry and operating regime.

Cooling Tower Performance

Merkel equation uses dimensionless groups:

KaV/L = ∫(dh/(h_s - h))

where KaV/L is dimensionless tower characteristic.

Design Considerations and Best Practices

1. Correlation Validity: Always verify Reynolds number, Prandtl number, and geometric ranges for which empirical correlations were developed. Extrapolation beyond validated ranges introduces significant uncertainty.

2. Transition Region: Exercise caution in transition region (2,300 < Re < 4,000). Predictions become unreliable. Conservative design practice uses turbulent correlations for Re > 2,300 in critical applications.

3. Property Evaluation: Fluid properties in dimensionless numbers should be evaluated at appropriate reference temperature:

• Film temperature for external flow: T_f = (T_s + T_∞)/2
• Bulk temperature for internal flow with property corrections

4. Non-Circular Ducts: Use hydraulic diameter D_h = 4A/P for Reynolds number calculation. Note that friction factor and heat transfer correlations may require adjustment factors for aspect ratios significantly different from 1.0.

5. Entrance Effects: Heat transfer and friction correlations assume fully developed flow. Entrance length requirements:

Hydrodynamic: L_e/D ≈ 4.4 Re^(1/6) (turbulent) Thermal: L_t/D ≈ 10 (turbulent, Pr ≈ 0.7)

6. Surface Roughness: Duct surface condition significantly affects friction factor in turbulent flow. Specify and maintain surface finish appropriate for design calculations.

7. Computational Validation: When using CFD for complex geometries, validate turbulence models against dimensionless correlations for benchmark cases before applying to design problems.

8. Measurement and Testing: TAB procedures should verify that installed systems operate within Reynolds number ranges assumed in design. Significant deviations indicate potential performance issues.

Uncertainty Analysis

Dimensional analysis aids uncertainty quantification by reducing number of variables and identifying dominant parameters.

For pressure drop: ΔP ∝ ρV²

Error Propagation:

If velocity measurement has ±5% uncertainty, pressure drop uncertainty is approximately ±10% (squared relationship).

Understanding dimensionless scaling helps prioritize measurement accuracy for most influential parameters.

Advanced Topics

Dimensionless Governing Equations:

Navier-Stokes equations can be non-dimensionalized, revealing that solutions depend only on dimensionless parameters (Re, Fr, etc.), not absolute values of ρ, μ, V independently.

Similitude in Unsteady Flows:

Strouhal number: Sr = fL/V

Characterizes oscillating flows, vortex shedding, and pulsating phenomena relevant to:

• Variable speed drive harmonics
• Flow-induced vibration
• Acoustic resonance

Multi-Phase Flows:

Additional dimensionless groups required:

• Void fraction
• Froude number (stratification)
• Weber number (droplet formation)

Critical for:

• Refrigerant distribution
• Steam systems
• Condensate drainage

Summary

Dimensional analysis provides the theoretical framework for:

• Systematic reduction of complex problems
• Scaling experimental results to full-size systems
• Developing generalized correlations applicable across fluid types and geometries
• Understanding physical phenomena through ratios of competing forces

Proficiency in dimensional analysis enables HVAC engineers to:

• Critically evaluate published correlations
• Extend laboratory data to field conditions
• Optimize designs using validated similarity principles
• Identify dominant physical mechanisms in complex systems

The dimensionless numbers presented here form the foundation of modern HVAC system analysis and remain essential tools for equipment selection, system design, and performance prediction.