Understanding Temperature Distribution in Fins

The question asks about the specific temperature distribution formula \(\frac{{T - {T_\infty }}}{{{T_0} - {T_\infty }}} = {e^{ - mx}}\) and for which fin configuration it is valid. This formula describes how the temperature \(T\) at a distance \(x\) from the base of a fin changes relative to the base temperature \(T_0\) and the ambient temperature \(T_\infty\).

The general differential equation governing heat transfer in a fin with uniform cross-section, assuming one-dimensional heat flow and constant thermal conductivity, is given by:

\[\frac{{{d^2}T}}{{d{x^2}}} - \frac{{hP}}{{kA}}(T - {T_\infty }) = 0\]

Where:

• \(T\) is the temperature at distance \(x\)
• \(T_\infty\) is the ambient fluid temperature
• \(h\) is the convective heat transfer coefficient
• \(P\) is the fin perimeter
• \(k\) is the thermal conductivity of the fin material
• \(A\) is the cross-sectional area of the fin

This equation can be written as:

\[\frac{{{d^2}T}}{{d{x^2}}} - {m^2}(T - {T_\infty }) = 0\]

where \(m = \sqrt{\frac{{hP}}{{kA}}}\). The general solution to this second-order linear differential equation is:

\[T - {T_\infty } = {C_1}{e^{mx}} + {C_2}{e^{ - mx}}\]

The constants \(C_1\) and \(C_2\) are determined by applying boundary conditions at the base (\(x=0\)) and the tip of the fin (\(x=L\), where \(L\) is the fin length).

Boundary Conditions and Fin Types

Different boundary conditions at the fin tip lead to different temperature distributions:

• Fin of finite length with insulated end (\(x=L\)): The boundary conditions are \(T(0) = T_0\) and \(\frac{{dT}}{{dx}}\left| {_{x = L}} \right. = 0\). The solution involves hyperbolic cosine and sine functions.
• Fin of finite length with heat loss by convection at end (\(x=L\)): The boundary conditions are \(T(0) = T_0\) and \(-k\frac{{dT}}{{dx}}\left| {_{x = L}} \right. = h(T(L) - {T_\infty })\). The solution also involves hyperbolic functions.
• Fin of finite length with specified temperature at end (\(x=L\)): The boundary conditions are \(T(0) = T_0\) and \(T(L) = T_L\). The solution involves hyperbolic functions.
• Very long fins (or Infinite fins): For a very long fin, as \(x \to \infty\), the temperature \(T(x)\) should approach the ambient temperature \(T_\infty\). This means \(T(x) - T_\infty \to 0\) as \(x \to \infty\). Looking at the general solution \(T - {T_\infty } = {C_1}{e^{mx}} + {C_2}{e^{ - mx}}\), for the term \(C_1 e^{mx}\) to go to zero as \(x \to \infty\) (since \(m\) is positive), the constant \(C_1\) must be zero.

Temperature Distribution for Very Long Fins

If we set \(C_1 = 0\) for a very long fin, the general solution simplifies to:

\[T - {T_\infty } = {C_2}{e^{ - mx}}\]

The boundary condition at the base (\(x=0\)) is \(T(0) = T_0\). Substituting this into the simplified solution:

\[T_0 - {T_\infty } = {C_2}{e^{ - m \cdot 0}} = {C_2}{e^0} = {C_2}\]

So, \(C_2 = T_0 - T_\infty\). Substituting \(C_2\) back into the simplified solution, we get the temperature distribution for a very long fin:

\[T - {T_\infty } = ({T_0} - {T_\infty }){e^{ - mx}}\]

Rearranging this equation gives the exact form provided in the question:

\[\frac{{T - {T_\infty }}}{{{T_0} - {T_\infty }}} = {e^{ - mx}}\]

Therefore, this specific temperature distribution is valid for very long fins.

Comparing this result with the given options, it is clear that the formula \(\frac{{T - {T_\infty }}}{{{T_0} - {T_\infty }}} = {e^{ - mx}}\) is valid for very long fins.