Heat Transfer Between Concentric Spheres Calculator

Heat Transfer Formula:

\[ q = \frac{A_1 \cdot \sigma \cdot (T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \left(\frac{1}{\varepsilon_2} - 1\right) \cdot \left(\frac{r_1}{r_2}\right)^2} \]

Surface Area of Body 1 (A₁):

m²

Temperature of Surface 1 (T₁):

Temperature of Surface 2 (T₂):

Emissivity of Body 1 (ε₁):

Emissivity of Body 2 (ε₂):

Radius of Smaller Sphere (r₁):

Radius of Larger Sphere (r₂):

Heat Transfer (q):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Heat Transfer Between Concentric Spheres?

Heat transfer between concentric spheres refers to the radiative heat exchange between two spherical surfaces that share the same center. This configuration is common in various engineering applications such as thermal insulation systems, spherical containers, and astronomical bodies.

2. How Does the Calculator Work?

The calculator uses the radiative heat transfer formula for concentric spheres:

\[ q = \frac{A_1 \cdot \sigma \cdot (T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \left(\frac{1}{\varepsilon_2} - 1\right) \cdot \left(\frac{r_1}{r_2}\right)^2} \]

Where:

\( q \) — Heat transfer rate (W)
\( A_1 \) — Surface area of inner sphere (m²)
\( \sigma \) — Stefan-Boltzmann constant (5.670367×10⁻⁸ W/m²K⁴)
\( T_1 \) — Temperature of inner surface (K)
\( T_2 \) — Temperature of outer surface (K)
\( \varepsilon_1 \) — Emissivity of inner sphere
\( \varepsilon_2 \) — Emissivity of outer sphere
\( r_1 \) — Radius of inner sphere (m)
\( r_2 \) — Radius of outer sphere (m)

Explanation: The formula accounts for radiative heat exchange between two concentric spherical surfaces, considering their temperatures, emissivities, and geometric configuration.

3. Importance of Heat Transfer Calculation

Details: Accurate calculation of heat transfer between concentric spheres is crucial for thermal system design, insulation analysis, energy efficiency optimization, and temperature control in various engineering applications.

4. Using the Calculator

Tips: Enter all required parameters with appropriate units. Surface areas and temperatures must be positive values. Emissivity values range from 0 to 1. Ensure consistent units for all inputs.

5. Frequently Asked Questions (FAQ)

Q1: What is the Stefan-Boltzmann constant?
A: The Stefan-Boltzmann constant (σ = 5.670367×10⁻⁸ W/m²K⁴) is a fundamental physical constant that relates the total energy radiated per unit surface area of a black body to the fourth power of its temperature.

Q2: How does emissivity affect heat transfer?
A: Emissivity (ε) ranges from 0 to 1, where 1 represents a perfect black body. Lower emissivity values reduce the radiative heat transfer between surfaces.

Q3: What are typical emissivity values?
A: Polished metals: 0.02-0.2, oxidized metals: 0.3-0.7, non-metallic surfaces: 0.7-0.95, black body: 1.0.

Q4: When is this formula applicable?
A: This formula applies to concentric spherical surfaces where radiation is the dominant heat transfer mode and the surfaces are diffuse gray bodies.

Q5: What are the limitations of this calculation?
A: The formula assumes steady-state conditions, uniform temperatures, diffuse gray surfaces, and no participating medium between the spheres.