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Heat Equation Calculator

Describes distribution of heat over time.

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Overview

The heat equation is a parabolic partial differential equation that models the distribution and evolution of temperature in a given region over time. It establishes that the rate of temperature change at any point is proportional to the Laplacian of the temperature field, representing the flow of thermal energy from higher to lower concentrations.

Symbols

Variables

\text{Concept-only} = Note

Note

Apply it well

When To Use

When to use: Apply this equation when calculating thermal conduction in stationary solids or fluids where convection is absent. It assumes the medium is isotropic and homogeneous, meaning thermal diffusivity remains constant throughout the material and in all directions.

Why it matters: This equation is a cornerstone of thermodynamics and engineering, allowing for the design of efficient cooling systems in electronics and structural insulation in buildings. Beyond physics, its mathematical structure is used in financial modeling, such as the Black-Scholes equation, to predict the diffusion of market prices.

Avoid these traps

Common Mistakes

  • Confusing α with thermal conductivity k.
  • Boundary conditions.

One free problem

Practice Problem

A 10-meter long metal rod has its left end (x = 0) fixed at 0°C and its right end (x = 10) fixed at 100°C. Assuming the rod has reached a steady state where the temperature no longer changes over time, what is the temperature in degrees Celsius at the center of the rod (x = 5)?

Note50

Solve for:

Hint: In a 1D steady state, the temperature distribution is linear between the two boundary points.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Fundamentals of Heat and Mass Transfer by Incropera, DeWitt, Bergman, Lavine
  2. Transport Phenomena by Bird, Stewart, Lightfoot
  3. Partial Differential Equations: An Introduction by Walter A. Strauss
  4. Wikipedia: Heat equation
  5. Bird, Stewart, Lightfoot - Transport Phenomena
  6. Incropera, DeWitt, Bergman, Lavine - Fundamentals of Heat and Mass Transfer
  7. Incropera, F. P., DeWitt, D. P., Bergman, T. L., & Lavine, A. S. (2007). Fundamentals of Heat and Mass Transfer (6th ed.).
  8. Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2007). Transport Phenomena (2nd ed.). John Wiley & Sons.