Fundamental Heat Transfer Equations

Why This Matters

Heat transfer equations are the mathematical language describing how energy moves through the physical world. Every problem you'll encounter, whether it involves designing a heat exchanger, analyzing insulation, 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 strong exam performance from guesswork.

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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

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

Heat flux is proportional to the temperature gradient. The steeper the temperature change across a material, the faster heat flows through it.

• Thermal conductivity ($k$) measures how easily a material conducts heat. Copper has $k \approx 400 \, W/(m \cdot K)$, while fiberglass insulation sits around $k \approx 0.04 \, W/(m \cdot K)$. That four-order-of-magnitude difference is why material selection matters so much.
• The negative sign indicates heat flows from hot to cold, opposite to the direction of increasing temperature. This enforces the second law of thermodynamics within the equation itself.
• For steady, one-dimensional conduction through a flat slab of thickness $L$, the equation integrates to $q = \frac{kA(T_1 - T_2)}{L}$, which you'll use constantly.

Newton's Law of Cooling (Convection)

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

Convective heat transfer depends on the temperature difference between a surface at $T_s$ and the bulk fluid at $T_\infty$.

• The convective coefficient ($h$) captures fluid properties, flow conditions, and geometry in a single parameter. It is not a material property. Forced convection over a flat plate might give $h \approx 50 \, W/(m^2 \cdot K)$, while boiling water can push $h$ above $10{,}000 \, W/(m^2 \cdot K)$.
• Surface area ($A$) directly scales heat transfer, which is why heat exchangers use fins and extended surfaces to increase $A$ without increasing the overall device size.

Stefan-Boltzmann Law (Radiation)

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

Radiative power scales with the fourth power of absolute temperature. This makes radiation dominant at high temperatures.

• Emissivity ($\epsilon$) ranges from 0 to 1, with $\epsilon = 1$ representing a perfect blackbody. Real surfaces always have $\epsilon < 1$. Polished aluminum might have $\epsilon \approx 0.05$, while oxidized steel can reach $\epsilon \approx 0.8$.
• $\sigma = 5.67 \times 10^{-8} \, W/(m^2 \cdot K^4)$ is the Stefan-Boltzmann constant.
• Temperature must be in Kelvin. Using Celsius here will give wildly wrong answers because the equation depends on absolute temperature raised to the fourth power.
• For net radiation exchange between a surface and its surroundings: $q_{net} = \epsilon \sigma A (T_s^4 - T_{surr}^4)$.

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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 without solving differential equations.

Thermal Resistance Equation

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

This quantifies how much a material layer opposes heat flow. Higher resistance means less heat transfer for a given temperature difference.

• Resistances in series add directly: $R_{total} = R_1 + R_2 + R_3 + \cdots$, just like electrical resistors. A three-layer wall (say, brick + insulation + drywall) has a total resistance equal to the sum of each layer's resistance.
• Convective resistance takes the form $R_{conv} = \frac{1}{hA}$. You'll almost always have convective resistances on both sides of a wall, sandwiching the conduction resistances.
• Heat transfer through the entire assembly is then $q = \frac{\Delta T_{overall}}{R_{total}}$, where $\Delta T_{overall}$ is the temperature difference between the two fluids (not the two surfaces).
• Units are $K/W$ or $°C/W$.

Overall Heat Transfer Coefficient

$UA = \frac{1}{R_{total}}$

The U-value combines all resistances into a single coefficient, so you can write $q = UA \Delta T$.

• Heat exchanger design relies on U-values because they capture the complete thermal path from hot fluid to cold fluid in one number.
• Higher U means better heat transfer. This is the inverse of the R-value used for insulation ratings, where higher R means less heat transfer.
• To find $U$, build up $R_{total}$ by summing all conduction and convection resistances (and contact resistances if present), then take the reciprocal.

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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

$\frac{\partial T}{\partial t} = \alpha \nabla^2 T$

This 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 disturbances propagate through a material. High $\alpha$ means fast thermal response. Metals have high $\alpha$; wood and polymers have low $\alpha$.
• Boundary and initial conditions determine the specific solution. Without them, the equation has infinitely many solutions. Common boundary conditions include prescribed surface temperature, prescribed heat flux, and convection at the surface.
• In one dimension with no internal heat generation, this reduces to $\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}$.

Energy Balance Equation

$\dot{E}_{in} - \dot{E}_{out} + \dot{E}_{gen} = \dot{E}_{stored}$

Conservation of energy: Energy In minus Energy Out plus Generation equals Accumulation (storage).

• Steady-state problems have zero accumulation ($\dot{E}_{stored} = 0$), simplifying the balance to $\dot{E}_{in} + \dot{E}_{gen} = \dot{E}_{out}$.
• This equation is your starting point for any thermal system analysis. Write it first for your control volume, then substitute specific heat transfer expressions (Fourier's Law, Newton's Law, etc.) for each term.
• For a solid with internal heat generation (like an electrical wire carrying current), the generation term $\dot{E}_{gen} = \dot{q} \cdot V$ where $\dot{q}$ is the volumetric heat generation rate and $V$ is volume.

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Dimensionless Numbers

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

Nusselt Number

$Nu = \frac{hL}{k_f}$

The Nusselt number compares convective to conductive heat transfer in the fluid. A high $Nu$ means convection is much more effective than pure conduction at moving heat away from the surface.

• Correlations express $Nu$ as functions of Reynolds and Prandtl numbers (e.g., $Nu = 0.023 \, Re^{0.8} Pr^{0.4}$ for turbulent flow in a pipe, known as the Dittus-Boelter equation). These correlations are how you actually find $h$ in practice.
• The characteristic length $L$ depends on geometry: diameter for pipes, plate length for external flow over flat plates. Using the wrong characteristic length is a common error that throws off the entire calculation.
• The thermal conductivity $k_f$ here is that of the fluid, not the solid.

Biot Number

$Bi = \frac{hL_c}{k_s}$

The Biot number compares internal conduction resistance to external convection resistance for a solid body. It determines whether temperature gradients inside the body are significant.

• When $Bi < 0.1$, use lumped capacitance analysis. The body is nearly isothermal internally, so you can treat it as a single temperature that changes with time: $T(t) - T_\infty = (T_i - T_\infty) \exp\left(-\frac{hA}{\rho c_p V} t\right)$.
• When $Bi > 0.1$, spatial temperature variation matters, and you need the full heat diffusion equation (or Heisler charts / one-term approximations).
• $L_c$ is the characteristic length, typically defined as $V/A_s$ (volume divided by surface area) for lumped analysis. For a sphere of radius $r$, $L_c = r/3$.
• The thermal conductivity $k_s$ here is that of the solid.

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Mass Transfer Analogy

Heat and mass transfer share the same mathematical structure. Understanding this analogy lets you solve mass transfer problems using techniques you already know from heat transfer.

Fick's Law of Diffusion

$J = -D \frac{dC}{dx}$

Mass flux is proportional to the concentration gradient, directly analogous to Fourier's Law for heat.

• The diffusion coefficient $D$ (units: $m^2/s$) plays the same role as thermal conductivity. It's a transport property specific to the diffusing species and the medium it moves through.
• This analogy extends to convective mass transfer, where a mass transfer coefficient $h_m$ replaces $h$, and concentration difference replaces temperature difference: $J = h_m (C_s - C_\infty)$.
• The Sherwood number $Sh = \frac{h_m L}{D}$ is the mass transfer analog of the Nusselt number, and similar correlations apply.

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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 a problem involves 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?