BESSELI

Application:

Heat Conduction in a Cylindrical Rod

Scenario:

Imagine a long, cylindrical metal rod. Initially, the entire rod is at a uniform temperature, let's say T0​. At time t=0, we instantly raise the temperature of the curved surface of the rod to a new, constant temperature, Ts​. We want to model how the temperature inside the rod changes over time and as a function of the distance from the center.

Mathematical Model:

This problem is described by the heat equation in cylindrical coordinates. For a long rod (we can ignore the ends), the temperature T is a function of radial distance r and time t, T(r,t). The solution to this partial differential equation with the given boundary conditions involves a series of terms. A key part of the solution, especially for a specific type of heat input or for steady-state analysis in similar problems, involves the modified Bessel function of the first kind.

For simplicity, let's look at a steady-state problem where the temperature on the surface is a function of angle and time, and we're looking for the steady-state temperature distribution. A more direct and illustrative example is often found in the analysis of heat transfer with internal heat generation, or in the analysis of heat fins.

Let's consider a simpler, more direct application: the temperature profile in a cylindrical object subjected to a certain type of heat generation. A common form of the solution for problems with cylindrical symmetry is often a linear combination of Bessel functions. The modified Bessel function of the first kind, I0​(x), arises naturally when dealing with radial heat flow and certain types of boundary conditions.

For example, the steady-state temperature distribution T(r) in a long cylindrical rod of radius R with internal heat generation per unit volume q′′′ and a constant surface temperature Ts​ is given by:

$T(r)=T_s+\frac{q^{\prime \prime \prime }}{4k}(R^2-r^2)$

This is a simpler parabolic solution. However, if the heat transfer at the surface is governed by convection (Newton's Law of Cooling) and the internal heat generation is more complex, or if we are analyzing a cylindrical fin, the modified Bessel function of the first kind, I0​(x), becomes a fundamental part of the solution.

Let's consider a cylindrical fin of radius R and length L, attached to a base at temperature Tb​. The fin loses heat to the surrounding fluid at temperature T∞​ via convection. The temperature distribution T(x) along the fin (where x is the distance from the base) is given by a differential equation whose solution involves modified Bessel functions.

A simplified form of the temperature distribution is often given as:

$\frac{T(x)-T_{\infty }}{T_b-T_{\infty }}=\frac{I_1(m(L-x))K_0(mL)+K_1(m(L-x))I_0(mL)}{I_1(mL)K_0(mL)+K_1(mL)I_0(mL)}$

where m is a constant related to the fin's properties, I0​ and I1​ are modified Bessel functions of the first kind of order 0 and 1, and K0​ and K1​ are modified Bessel functions of the second kind.

Example Table:

Let's focus on the values of I0​(x), which represents the fundamental behavior of the modified Bessel function. This function grows exponentially, which is a key characteristic. The table below shows the value of I0​(x) for different values of x. In our heat conduction example, the argument x would be a dimensionless quantity like mr.

Explanation of the Table:

The values in the table show how the function I0​(x) behaves. As x increases, the value of the function increases rapidly. This exponential-like growth is a defining feature. In the context of our heat fin example, this would imply that the temperature profile along the fin is not a simple linear or exponential decay, but is shaped by this function.