An interior heat transfer coefficient h can be defined for...

An interior heat transfer coefficient $h$ can be defined for heat conduction out of a convectively cooled slab of thickness $2 L$ as $$ h=\frac{-\left.k(\partial T / \partial x)\right|_{x=L}}{\bar{T}-T_{s}} $$ with a corresponding dimensionless Nusselt number $\mathrm{Nu}=h(2 L) / k$. Show that for $\mathrm{Fo}>0.2, \mathrm{Nu}$ has a constant value $$ \mathrm{Nu}=\frac{2 \lambda_{1}^{2} \sin \lambda_{1}}{\sin \lambda_{1}-\lambda_{1} \cos \lambda_{1}} $$ which for $\mathrm{Bi} \rightarrow \infty$ is $\pi^{2} / 2=4.934$.

An interior heat transfer coefficient $h$ can be defined for heat conduction out of a convectively cooled slab of thickness $2 L$ as
$$
h=\frac{-\left.k(\partial T / \partial x)\right|_{x=L}}{\bar{T}-T_{s}}
$$
with a corresponding dimensionless Nusselt number $\mathrm{Nu}=h(2 L) / k$. Show that for $\mathrm{Fo}>0.2, \mathrm{Nu}$ has a constant value
$$
\mathrm{Nu}=\frac{2 \lambda_{1}^{2} \sin \lambda_{1}}{\sin \lambda_{1}-\lambda_{1} \cos \lambda_{1}}
$$
which for $\mathrm{Bi} \rightarrow \infty$ is $\pi^{2} / 2=4.934$.

Key Concepts

Fourier Number

The Fourier number is a dimensionless time parameter that characterizes heat conduction in transient problems. It is defined by the ratio of diffusive transport rate to the rate of thermal energy storage and indicates how far transient effects have penetrated into a material. When the Fourier number exceeds a certain threshold (in this problem Fo > 0.2), the system’s behavior reaches a quasi-steady state which simplifies the determination of other dimensionless groups.

Eigenvalue Analysis

Eigenvalue analysis in heat conduction problems involves solving the associated eigenvalue problem that arises from the separation of variables in the heat equation, especially when boundary conditions are non-uniform (as with convective surfaces). The eigenvalues (here represented by ?1) encapsulate the characteristic modes of transient heat diffusion. The expression for Nu in terms of ?1 reflects the inherent properties of these modes and is crucial for understanding the long-time behavior of the system.

Biot Number

The Biot number is a dimensionless number that compares the internal thermal resistance within a body to the convective thermal resistance across the body's boundary. In problems involving both conduction and convection, the Biot number indicates whether the entire body responds uniformly to boundary conditions (small Biot number) or exhibits significant internal temperature gradients (large Biot number). In this case, the limit of Bi tending to infinity simplifies the analysis and leads to a particular constant value for the Nusselt number.

Heat Transfer Coefficient

The heat transfer coefficient quantifies the convective heat transfer between a solid surface and a fluid. It appears in the formulation of Newton’s law of cooling and is related to the temperature gradient at the surface. This coefficient is essential in linking the internal conduction processes within the slab to the external convective mechanisms, and its proper definition allows the conduction problem to be recast in a dimensionless form.

Nusselt Number

The Nusselt number is a dimensionless parameter representing the ratio of convective to conductive heat transfer across a boundary. It expresses how effective convection is compared to pure conduction and is critical in evaluating the performance of heat transfer systems. In this context, Nu is defined in such a way that it combines the convective heat transfer coefficient, the characteristic length (here the slab thickness), and the thermal conductivity.

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