The Biot number, \(Bi\) is a dimensionless quantity. For a sphere, it is defined by this equation:

\[Bi = \frac{hR}{k_{solid}}\]

where \(h\) = heat transfer coefficient (W/m²-K) or (W/m²-°C),

\(R\) = radius of the sphere (m), and

\(k_{solid}\) = thermal conductivity of the solid (W/m-K) or (W/m-°C).

When the Biot number is larger than about 0.1 you cannot use lumped capacitance. You must use the following.

You can use the following equation to calculate the temperature throughout a sphere as a function of tie. You must add up several terms in the infinite series to arrive at a reasonable accurate solution. This is cumbersome without a computer.

\[T(r,t) = T_{\infty} + (T_i \,-\, T_{\infty}) \sum ^{\infty} _{n=1} 4\frac{sin(\zeta_n) – \zeta_n cos(\zeta_n)}{2\zeta_n – sin(2\zeta_n)} exp \left(-\zeta^2_n \frac{\alpha t}{R^2} \right) \frac{sin(\zeta_n \frac{r}{R})}{\zeta_n \frac{r}{R}}\]

where \(T(r,t)\) = temperature as a function of the coordinate distance from the center of the sphere, \(r\), (meters) and time, \(t\), (seconds)

\(R\) = sphere’s radius (m)

\(\alpha\) = the material’s thermal diffusivity (m²/s)

\(T_i\) = sphere’s initial temperature (°C)

\(T_{\infty}\) = surrounding fluid temperature (°C)

The values of \(\zeta_n\) (dimensionless) are calculated from the following function:

\[1 \,-\, \zeta_n cot(\zeta_n) = Bi\]

There is no analytical solution to this function. Because of the nature of the cotangent function, there are an infinite number solutions, \(\zeta_1, \zeta_2, \zeta_3,\) etc. Each solution must be looked up in a table or computed numerically.

\(Fo = \frac{\alpha t}{r^2}\) is the Fourier number. Think of it as a dimensionless time. The bigger the value of \(Fo\), the fewer number of terms you need to use in the infinite series above. When \(Fo\) is bigger than about 0.2, the solution is sufficiently accurate using only the first term (\(n=1\)). This is much less cumbersome if you are not using a computer.