Angular Heat Flux Equation Calculator

Formula first

Overview

Derived from Fourier's Law of heat conduction, this expression relates the angular heat flux $q_{\theta }^{\prime \prime }$ to the thermal conductivity $k$ and the partial derivative of temperature $T$ with respect to the angular coordinate $\theta$. The term $1/r$ accounts for the geometric scaling required in spherical coordinate systems, where the arc length changes with the radial distance $r$. This component is essential for analyzing multidimensional heat conduction problems in spherical geometries, such as those involving non-uniform surface heating or cooling.

Apply it well

When To Use

When to use: Use this equation when analyzing heat conduction in spherical systems where temperature varies along the angular coordinate θ.

Why it matters: It allows for the accurate calculation of heat flow in non-radial directions, which is critical for modeling complex thermal distributions in spherical shells or solid spheres.

Avoid these traps

Common Mistakes

• Forgetting the 1/r geometric factor when calculating the flux.
• Confusing the angular coordinate θ with the azimuthal angle φ in spherical coordinate systems.

One free problem

Practice Problem

In a spherical coordinate system, if the temperature gradient with respect to θ is zero, what does this imply about the angular heat flux in that direction?

Solve for: $q_{\theta }^{\prime \prime }$

Hint: Look at the relationship between the flux and the temperature gradient in the formula.

The full worked solution stays in the interactive walkthrough.

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Related Formulas

References

Sources

1. Incropera, F. P., & DeWitt, D. P. (2007). Fundamentals of Heat and Mass Transfer (6th ed.). John Wiley & Sons.
2. Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2002). Transport Phenomena (2nd ed.). John Wiley & Sons.
3. NIST CODATA
4. IUPAC Gold Book
5. Heat Transfer by Yunus A. Cengel
6. Fundamentals of Heat and Mass Transfer by Incropera and DeWitt
7. Incropera, Frank P.; DeWitt, David P.; Bergman, Theodore L.; Lavine, Adrienne S. Fundamentals of Heat and Mass Transfer
8. Çengel, Yunus A.; Ghajar, Afshin J. Heat and Mass Transfer: Fundamentals and Applications