Heat Transfer Fundamentals for HVAC Engineers

Heat Transfer Fundamentals for HVAC Engineers

Heat transfer governs energy exchange in all HVAC systems. Accurate prediction of heat flow through building envelopes, across heat exchanger surfaces, and within air streams determines system capacity, energy consumption, and comfort delivery. This guide provides the mathematical framework for conduction, convection, and radiation calculations essential for HVAC design.

Modes of Heat Transfer

Heat transfers through three distinct physical mechanisms, each governed by different equations and material properties.

Conduction

Heat conduction occurs through solid materials via molecular collision and electron transport. Fourier’s Law quantifies steady-state conduction:

$$q = -k A \frac{dT}{dx}$$

Where:

• $q$ = heat transfer rate (Btu/h or W)
• $k$ = thermal conductivity (Btu/(h·ft·°F) or W/(m·K))
• $A$ = cross-sectional area perpendicular to heat flow (ft² or m²)
• $dT/dx$ = temperature gradient (°F/ft or K/m)

For one-dimensional steady-state conduction through a plane wall with constant properties:

$$q = \frac{kA(T_1 - T_2)}{L}$$

Where:

• $T_1, T_2$ = surface temperatures (°F or K)
• $L$ = wall thickness (ft or m)

Thermal Resistance for Conduction:

$$R_{cond} = \frac{L}{kA}$$

Units: h·°F/Btu or K/W

graph LR
    A[Hot Surface<br/>T₁ = 75°F] -->|Heat Flow q| B[Material<br/>k, L]
    B -->|Heat Flow q| C[Cold Surface<br/>T₂ = 40°F]

    style A fill:#ffcccc
    style C fill:#ccccff
    style B fill:#f0f0f0

Convection

Convection transfers heat between a solid surface and a moving fluid (air or water). Newton’s Law of Cooling describes convective heat transfer:

$$q = hA(T_s - T_\infty)$$

Where:

• $h$ = convection heat transfer coefficient (Btu/(h·ft²·°F) or W/(m²·K))
• $T_s$ = surface temperature (°F or K)
• $T_\infty$ = bulk fluid temperature (°F or K)

Thermal Resistance for Convection:

$$R_{conv} = \frac{1}{hA}$$

The convection coefficient $h$ depends on:

• Flow regime (laminar vs. turbulent)
• Fluid properties (viscosity, thermal conductivity, density)
• Surface geometry
• Flow velocity

Typical convection coefficient ranges for HVAC applications:

Radiation

Thermal radiation transfers energy via electromagnetic waves without requiring a medium. The Stefan-Boltzmann Law governs radiation from a blackbody:

$$q = \sigma A T^4$$

Where:

• $\sigma$ = Stefan-Boltzmann constant = 0.1714×10⁻⁸ Btu/(h·ft²·R⁴) = 5.67×10⁻⁸ W/(m²·K⁴)
• $T$ = absolute temperature (R or K)

For real surfaces (gray bodies) exchanging radiation:

$$q = \epsilon \sigma A (T_1^4 - T_2^4)$$

Where:

• $\epsilon$ = emissivity (0 to 1, dimensionless)

For linearized radiation between surfaces at similar temperatures:

$$q = h_r A (T_1 - T_2)$$

Where the radiation heat transfer coefficient is:

$$h_r = \epsilon \sigma (T_1 + T_2)(T_1^2 + T_2^2)$$

Typical emissivities:

Composite Heat Transfer

HVAC systems involve multiple modes of heat transfer in series and parallel arrangements.

Series Thermal Resistances

For heat transfer through multiple layers (building envelope, heat exchanger walls):

graph LR
    A[Indoor Air<br/>T_in] -->|Convection<br/>R₁| B[Wall Layer 1<br/>R₂]
    B -->|Conduction| C[Wall Layer 2<br/>R₃]
    C -->|Conduction| D[Wall Layer 3<br/>R₄]
    D -->|Convection<br/>R₅| E[Outdoor Air<br/>T_out]

    style A fill:#ffcccc
    style E fill:#ccccff

Total thermal resistance:

$$R_{total} = R_1 + R_2 + R_3 + … + R_n = \sum_{i=1}^{n} R_i$$

Heat transfer rate:

$$q = \frac{\Delta T_{overall}}{R_{total}} = \frac{T_{in} - T_{out}}{R_{total}}$$

Overall Heat Transfer Coefficient (U-Value)

The U-value represents the overall conductance per unit area:

$$U = \frac{1}{R_{total} \cdot A} = \frac{1}{\frac{1}{h_i A} + \frac{L_1}{k_1 A} + \frac{L_2}{k_2 A} + … + \frac{1}{h_o A}}$$

Simplifying for unit area ($A = 1$):

$$U = \frac{1}{\frac{1}{h_i} + \frac{L_1}{k_1} + \frac{L_2}{k_2} + … + \frac{1}{h_o}}$$

Units: Btu/(h·ft²·°F) or W/(m²·K)

Lower U-values indicate better insulation. Building codes specify maximum U-values for envelope components.

Heat Exchanger Effectiveness

Heat exchangers transfer thermal energy between two fluid streams without mixing. Effectiveness ($\epsilon$) quantifies actual heat transfer relative to maximum possible transfer.

Heat Exchanger Energy Balance

For a counterflow or parallel-flow heat exchanger:

$$q = \dot{m}h c{p,h} (T_{h,in} - T_{h,out}) = \dot{m}c c{p,c} (T_{c,out} - T_{c,in})$$

Where:

• $\dot{m}$ = mass flow rate (lb/h or kg/s)
• $c_p$ = specific heat (Btu/(lb·°F) or J/(kg·K))
• Subscripts $h$ and $c$ denote hot and cold streams

Effectiveness-NTU Method

Heat exchanger effectiveness:

$$\epsilon = \frac{q_{actual}}{q_{max}}$$

Where $q_{max} = C_{min}(T_{h,in} - T_{c,in})$

Heat capacity rate:

$$C = \dot{m} c_p$$

Minimum heat capacity rate:

$$C_{min} = \min(C_h, C_c)$$

Capacity ratio:

$$C_r = \frac{C_{min}}{C_{max}}$$

Number of Transfer Units (NTU):

$$NTU = \frac{UA}{C_{min}}$$

Where:

• $U$ = overall heat transfer coefficient
• $A$ = heat transfer surface area

For counterflow heat exchangers:

$$\epsilon = \frac{1 - e^{-NTU(1-C_r)}}{1 - C_r \cdot e^{-NTU(1-C_r)}} \quad \text{for } C_r < 1$$

$$\epsilon = \frac{NTU}{1 + NTU} \quad \text{for } C_r = 1$$

graph TB
    A[Hot Fluid In<br/>T_h_in] --> B[Heat Exchanger]
    C[Cold Fluid In<br/>T_c_in] --> B
    B --> D[Hot Fluid Out<br/>T_h_out]
    B --> E[Cold Fluid Out<br/>T_c_out]

    F[Effectiveness ε<br/>= q_actual / q_max] -.-> B

    style A fill:#ffcccc
    style D fill:#ffdddd
    style C fill:#ccccff
    style E fill:#ddddff

Thermal Conductivity of Common Materials

Practical Applications

Building Envelope Design

1. Wall U-Value Compliance: Calculate composite U-values for code compliance (ASHRAE 90.1, IECC)
2. Thermal Bridging: Account for studs, joists, and fasteners that bypass insulation
3. Window Performance: Evaluate NFRC-rated U-values and SHGC for fenestration selection
4. Roof Insulation: Size rigid insulation to meet climate zone requirements

HVAC Equipment Sizing

1. Heat Loss/Gain: Apply U-values to envelope areas for load calculations
2. Duct Heat Loss: Calculate heat loss/gain through uninsulated or insulated duct runs
3. Pipe Heat Loss: Size pipe insulation to limit heat loss in hot water distribution
4. Heat Exchanger Selection: Use effectiveness ratings to evaluate energy recovery equipment

System Performance Analysis

1. Furnace Heat Exchanger: Convection coefficients on combustion gas and air sides
2. Cooling Coil: Combined convection and condensation on evaporator surfaces
3. Boiler Flue Loss: Conduction through refractory and metal surfaces
4. Radiant Panel Output: Radiation and convection from floor/ceiling surfaces

Common Design Pitfalls

• Ignoring air films: Convection resistances at surfaces contribute 10-30% of total R-value
• Using center-of-glass U-values: Frame and edge effects increase whole-window U-values by 20-50%
• Neglecting thermal bridging: Steel studs can reduce nominal wall R-value by 50%
• Oversimplifying radiation: Linearized radiation equations only accurate for small temperature differences
• Confusing R-value and U-value: R-value is resistance (higher is better), U-value is conductance (lower is better)

Summary

Heat transfer analysis underpins all HVAC system design:

• Conduction through solids follows Fourier’s Law: $q = kA(T_1 - T_2)/L$
• Convection between surfaces and fluids follows Newton’s Law: $q = hA(T_s - T_\infty)$
• Radiation between surfaces follows Stefan-Boltzmann Law: $q = \epsilon \sigma A (T_1^4 - T_2^4)$
• Composite heat transfer uses series/parallel thermal resistance networks
• U-values quantify overall heat transfer through building assemblies (lower is better)
• Heat exchanger effectiveness measures recovery efficiency using the ε-NTU method

Accurate heat transfer calculations enable proper equipment sizing, energy code compliance, and energy-efficient system design.

Related Technical Guides:

• Psychrometric Fundamentals
• Heating Load Calculations
• Cooling Load Calculations
• Building Envelope Heat Transfer

References:

• ASHRAE Handbook of Fundamentals, Chapter 4: Heat Transfer
• ASHRAE Handbook of Fundamentals, Chapter 26: Heat, Air, and Moisture Control in Building Assemblies
• Incropera, F.P., DeWitt, D.P., Fundamentals of Heat and Mass Transfer, 7th Edition
• ASHRAE Standard 90.1: Energy Standard for Buildings Except Low-Rise Residential Buildings