Temperature Inside Solid Sphere At Given Radius Calculator

Formula Used:

\[ t2 = Tw + \frac{qG}{6 \times k} \times (Rs^2 - r^2) \]

Surface Temperature of Wall (Tw):

Internal Heat Generation (qG):

W/m³

Thermal Conductivity (k):

W/(m·K)

Radius of Sphere (Rs):

Radius (r):

Temperature 2 (t2):

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1. What is the Temperature Inside Solid Sphere Formula?

The formula calculates the temperature at a given radius inside a solid sphere with uniform internal heat generation. It's derived from the heat conduction equation for spherical coordinates and provides the temperature distribution within the sphere.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ t2 = Tw + \frac{qG}{6 \times k} \times (Rs^2 - r^2) \]

Where:

\( t2 \) — Temperature at given radius (K)
\( Tw \) — Surface temperature of wall (K)
\( qG \) — Internal heat generation (W/m³)
\( k \) — Thermal conductivity (W/(m·K))
\( Rs \) — Radius of sphere (m)
\( r \) — Radius at which temperature is calculated (m)

Explanation: The formula accounts for the parabolic temperature distribution within a solid sphere with uniform internal heat generation, where maximum temperature occurs at the center.

3. Importance of Temperature Calculation

Details: Accurate temperature calculation is crucial for thermal analysis of spherical objects, nuclear reactor design, chemical processing equipment, and various engineering applications involving spherical geometries with internal heat generation.

4. Using the Calculator

Tips: Enter all values in appropriate units. Surface temperature and radii must be positive values. Thermal conductivity must be greater than zero. Internal heat generation must be non-negative.

5. Frequently Asked Questions (FAQ)

Q1: What assumptions are made in this formula?
A: The formula assumes steady-state conditions, uniform internal heat generation, constant thermal properties, and spherical symmetry.

Q2: Where is this formula commonly applied?
A: This formula is used in nuclear reactor design, chemical processing equipment, geothermal studies, and any application involving spherical objects with internal heat generation.

Q3: What happens when r = 0 (center of sphere)?
A: At the center, the formula gives the maximum temperature: \( t_{max} = Tw + \frac{qG \times Rs^2}{6 \times k} \)

Q4: What are the limitations of this formula?
A: The formula assumes constant thermal conductivity, uniform heat generation, and doesn't account for temperature-dependent properties or transient effects.

Q5: How does temperature vary with radius?
A: Temperature follows a parabolic distribution, decreasing from maximum at the center to the surface temperature at the outer radius.