Fundamental Heat Transfer Equations

Why This Matters

Heat transfer equations aren't just formulas to memorize—they're the mathematical language describing how energy moves through the physical world. Every exam problem you'll encounter, whether it involves designing a heat exchanger, analyzing insulation performance, or predicting how quickly a hot object cools, requires you to select the right equation and understand why it applies. You're being tested on your ability to recognize which mode of heat transfer dominates a situation and how to quantify it.

These equations connect three fundamental mechanisms: conduction, convection, and radiation. Beyond the basic laws, you'll need to understand how engineers simplify complex problems using concepts like thermal resistance and dimensionless numbers. Don't just memorize the formulas—know what physical phenomenon each equation describes, when to apply it, and how different equations relate to each other. That conceptual understanding is what separates students who ace FRQs from those who struggle.

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The Three Modes: Fundamental Laws

Each heat transfer mode has its own governing equation. These are your starting points for any analysis—master these first, and everything else builds from them.

Fourier's Law of Heat Conduction

• Heat flux is proportional to the temperature gradient—the steeper the temperature change across a material, the faster heat flows through it
• Thermal conductivity ($k$) determines how easily a material conducts heat; metals have high $k$, insulators have low $k$
• The negative sign in $q = -k \frac{dT}{dx}$ indicates heat flows from hot to cold, opposite to the direction of increasing temperature

Newton's Law of Cooling (Convection)

• Convective heat transfer depends on the temperature difference between surface and fluid, expressed as $q = hA(T_s - T_\infty)$
• The convective coefficient ($h$) captures fluid properties, flow conditions, and geometry in a single parameter—it's not a material property
• Surface area ($A$) directly scales heat transfer, which is why heat exchangers use fins and extended surfaces

Stefan-Boltzmann Law (Radiation)

• Radiative power scales with $T^4$—this fourth-power dependence makes radiation dominant at high temperatures
• Emissivity ($\epsilon$) ranges from 0 to 1, with $\epsilon = 1$ representing a perfect blackbody; real surfaces always have $\epsilon < 1$
• The equation $q = \epsilon \sigma A T^4$ uses absolute temperature (Kelvin), so unit errors here are catastrophic

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Resistance and Network Methods

Engineers simplify complex heat transfer problems by treating thermal systems like electrical circuits. These equations let you analyze multi-layer walls, composite systems, and combined heat transfer modes.

Thermal Resistance Equation

• Thermal resistance $R_{th} = \frac{L}{kA}$ quantifies how much a material opposes heat flow—higher resistance means less heat transfer
• Resistances in series add directly, just like electrical resistors; this simplifies multi-layer wall calculations
• Units are $\frac{K}{W}$ or $\frac{°C}{W}$, and you can calculate heat transfer as $q = \frac{\Delta T}{R_{th}}$

Overall Heat Transfer Coefficient

• The U-value combines all resistances into a single coefficient: $U = \frac{1}{R_{total}}$, where $R_{total}$ sums conduction, convection, and contact resistances
• Heat exchanger design relies on U-values because they capture the complete thermal path from hot fluid to cold fluid
• Higher U means better heat transfer—this is the opposite of R-value used for insulation ratings

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Transient and Distributed Analysis

Not all problems involve steady-state conditions. These equations govern how temperature evolves over time and space.

Heat Diffusion Equation

• The equation $\frac{\partial T}{\partial t} = \alpha \nabla^2 T$ describes how temperature changes in both time and space—it's the master equation for transient conduction
• Thermal diffusivity $\alpha = \frac{k}{\rho c_p}$ measures how quickly temperature changes propagate; high $\alpha$ means fast thermal response
• Boundary and initial conditions determine the specific solution; without them, you can't solve the equation

Energy Balance Equation

• Conservation of energy states: Energy In − Energy Out + Generation = Accumulation
• Steady-state problems have zero accumulation, simplifying the balance to Energy In + Generation = Energy Out
• This equation is your starting point for any thermal system analysis—write it first, then substitute specific heat transfer expressions

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Dimensionless Numbers

Dimensionless parameters let you characterize heat transfer behavior without solving detailed equations. They're essential for scaling, correlations, and quick engineering judgments.

Nusselt Number Correlation

• $Nu = \frac{hL}{k}$ compares convective to conductive heat transfer—a high Nusselt number means convection dominates over conduction in the fluid
• Correlations express $Nu$ as functions of Reynolds and Prandtl numbers, allowing you to find $h$ for different flow conditions
• The characteristic length $L$ depends on geometry: diameter for pipes, length for flat plates—using the wrong length is a common error

Biot Number

• $Bi = \frac{hL_c}{k}$ compares internal to external thermal resistance—it determines whether temperature gradients inside a body matter
• When $Bi < 0.1$, use lumped capacitance analysis—the body is essentially isothermal, and you can ignore internal conduction
• When $Bi > 0.1$, spatial temperature variation matters, and you need the full heat diffusion equation

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Mass Transfer Analogy

Heat and mass transfer share mathematical structures. Understanding this analogy doubles your problem-solving toolkit.

Fick's Law of Diffusion

• Mass flux is proportional to concentration gradient: $J = -D \frac{dC}{dx}$, directly analogous to Fourier's Law for heat
• The diffusion coefficient $D$ plays the same role as thermal conductivity—it's a transport property specific to the species and medium
• This analogy extends to convective mass transfer, where a mass transfer coefficient replaces $h$ and concentration difference replaces temperature difference

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Quick Reference Table

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Self-Check Questions

1. Both Fourier's Law and Fick's Law have negative signs in their equations. What physical principle do these negative signs represent, and why is it important?

2. You're analyzing heat transfer through a furnace wall with convection on both sides. Which equations would you combine, and in what order would you calculate the resistances?

3. Compare the Nusselt Number and Biot Number: they look similar mathematically, but what different questions do they answer about a heat transfer problem?

4. A small metal sphere is dropped into a cool fluid bath. What dimensionless number determines whether you can assume the sphere has uniform temperature, and what criterion must be satisfied?

5. If an FRQ asks you to analyze a high-temperature furnace where radiation dominates, why does a 10% error in temperature measurement cause a much larger error in calculated heat transfer compared to a convection problem?