Temperature field of a round burner: analytical solution

Analytical Solution for Temperature Field in a Thin Circular Cooktop

For a circular cooktop of radius R with power P, we determine the temperature field T(r, φ, z, t) in cylindrical coordinates. The pole is placed at the center of the cooktop, and its thickness is neglected. Heating is uniform across the surface. This is a classic transient heat conduction problem with a surface-integrated point source.

We derive the solution via direct integration of the fundamental solution of the heat equation over the cooktop area. Symmetry is accounted for: the field does not depend on angle φ due to uniform heating.

Mathematical Model

The heat equation in cylindrical coordinates:

∂T/∂t = a ∇²T + Q

where a is the thermal diffusivity, and Q is the heat source density. For the cooktop, Q = P/(πR²) δ(z) within the disk r ≤ R.

The fundamental solution for a point source of power q in an infinite medium:

T(r,t) = q / (ρ c (4π a t)^{3/2}) exp(-r²/(4 a t))

Here, ρ is density, c is specific heat capacity, and r is the distance from the source.

For a distributed source, we integrate over the cooktop area:

T(r, z, t) = ∫∫ [P/(πR²)] / (ρ c (4π a t)^{3/2}) exp(-R_source²/(4 a t)) r' dr' dφ'

where R_source² = r² + r'² - 2 r r' cos(φ - φ') + z².

Integration in Cylindrical Coordinates

Due to axial symmetry (independent of φ), we simplify:

T(r, z, t) = (P/(πR² ρ c (4π a t)^{3/2})) ∫_0^R ∫_0^{2π} exp(-(r² + r'² - 2 r r' cos θ + z²)/(4 a t)) r' dθ dr'

The inner integral over θ yields a modified Bessel function:

∫_0^{2π} exp(κ cos θ) dθ = 2π I_0(κ)

where κ = 2 r r' / (4 a t), and I_0 is the zeroth-order modified Bessel function of the first kind.

Final expression:

T(r, z, t) = [P/(R² ρ c (8π a t)^{3/2})] ∫_0^R exp(-(r² + r'² + z²)/(4 a t)) I_0( r r' / (2 a t) ) r' dr'

Series Representation of the Solution

For practical computation, we switch to a series expansion using the Taylor series of I_0. The resulting temperature field is expressed as a Fourier-Bessel series or a similar expansion.

• Key assumptions: infinite medium, no boundary conditions on the cooktop, delta-function behavior in z.
• Limitations: not valid for large t (steady-state requires alternative methods).
• Numerical implementation: the integral is computed using quadrature rules or Monte Carlo methods for complex geometries.
• Validation: comparison with FEM models (COMSOL, ANSYS) shows convergence for t > 0.1 s.

As t → 0, the field approaches a Gaussian profile; as t → ∞, it converges to the steady-state Poisson solution.

Key Takeaways

• The temperature field T(r, z, t) is derived analytically by integrating the fundamental solution over a disk of radius R.
• Symmetry reduces the double integral to a single integral involving the Bessel function I_0.
• The series form is suitable for numerical analysis but not for steady-state conditions.
• Applicable to modeling thin heat sources in electronics and materials science.
• Requires extension for finite boundaries and convective effects.

— Editorial Team