T: Transient Regimes

We can try to understand the meaning of the Biot number through mathematical means, by seeing where it naturally occurs....

Consider the boundary between a fluid and a solid.

Recall that the boundary is infinitely thin, and thus cannot hold any thermal energy. therefore, even under unsteady conditions, qcond=qconv, or

$\displaystyle{-k\frac{dT}{dx} = h(T-T_\infty)}$

If we generalize this boundary condition, by making it dimensionless, we need to define a characteristic length and temperature.

Using L as our characteristic length and $T_\infty$ as our characteristic time, we get

$\displaystyle{-k\frac{d(\phi T_\infty)}{d(\xi L)} = hT_\infty(\phi-1)}$

where $\phi$ is our dimensionless T and $\xi$ is our dimensionless position. Taking our derivatives, we can pull out the constants to yield

$\displaystyle{-\frac{k T_\infty}{L}\frac{d\phi}{d\xi} = hT_\infty(\phi-1)}$

which can be easily simplified to

$\displaystyle{\frac{d\phi}{d\xi} = -\frac{hL}{k}(\phi-1) = -Bi(\phi-1)}$

We can then understand the "regimes" of transient response by asking ourselves what is the physical significance of small, large, and "order 1" Bi.

Clearly, small Bi means that the dimensionless temperature gradient must be small (i.e., there are no spatial variations in temperature), which is exactly what we reasoned before.

For order 1 Bi, we see that the dimensionless conduction and convection have roughly the same driving force (or, pessimistically, resistance).

NOTE:

In order to understand what we mean by things that are equal having different driving forces/resistances, consider running a three legged race with Yao. Who would be limiting your team's progress?!

What does it physically mean for Bi>>1?!

OUTCOME:

Identify "regimes" of transient response based on the value of the Biot number