Temperature Inside Hollow Sphere At Given Radius Between Inner And Outer Radius Calculator

Formula Used:

\[ T = T_w + \frac{q_G}{6k}(r_2^2 - r^2) + \frac{q_G r_1^3}{3k}\left(\frac{1}{r_2} - \frac{1}{r}\right) \]

Surface Temperature of Wall (T_w):

Internal Heat Generation (q_G):

W/m³

Thermal Conductivity (k):

W/m·K

Outer Radius of Sphere (r₂):

Radius (r):

Inner Radius of Sphere (r₁):

Temperature (T):

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1. What is the Temperature Inside Hollow Sphere Formula?

The temperature inside hollow sphere formula calculates the temperature distribution within a hollow spherical object with internal heat generation. It accounts for the surface temperature, thermal conductivity, internal heat generation rate, and geometric parameters of the sphere.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ T = T_w + \frac{q_G}{6k}(r_2^2 - r^2) + \frac{q_G r_1^3}{3k}\left(\frac{1}{r_2} - \frac{1}{r}\right) \]

Where:

\( T \) — Temperature at given radius (K)
\( T_w \) — Surface temperature of wall (K)
\( q_G \) — Internal heat generation (W/m³)
\( k \) — Thermal conductivity (W/m·K)
\( r_2 \) — Outer radius of sphere (m)
\( r \) — Radius where temperature is calculated (m)
\( r_1 \) — Inner radius of sphere (m)

Explanation: The formula calculates temperature distribution in a hollow sphere with uniform internal heat generation, considering both conductive heat transfer and internal energy generation effects.

3. Importance of Temperature Calculation

Details: Accurate temperature calculation is crucial for thermal analysis of spherical containers, pressure vessels, insulation systems, and any spherical structure with internal heat generation to ensure structural integrity and proper thermal management.

4. Using the Calculator

Tips: Enter all values in appropriate units. Surface temperature in Kelvin, internal heat generation in W/m³, thermal conductivity in W/m·K, and all radii in meters. Ensure all values are positive and thermal conductivity is greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What types of applications use this formula?
A: This formula is used in thermal analysis of spherical pressure vessels, nuclear reactor components, spherical storage tanks, and any hollow spherical structure with internal heat generation.

Q2: What are the assumptions behind this formula?
A: The formula assumes steady-state conditions, uniform internal heat generation, constant thermal properties, and spherical symmetry.

Q3: How does internal heat generation affect temperature distribution?
A: Internal heat generation creates a temperature gradient within the sphere, with higher temperatures typically occurring toward the center where heat accumulates.

Q4: What happens if the inner radius is zero?
A: If inner radius is zero, the sphere becomes solid and the formula simplifies to the solid sphere temperature distribution equation.

Q5: Are there limitations to this formula?
A: The formula assumes constant thermal conductivity and uniform heat generation. It may not be accurate for materials with temperature-dependent properties or non-uniform heat generation patterns.