Trouble with deriving Fourier Number in Unsteady Heat Conduction analysis

Question

Consider a metal ball being heated in a furnace. Thermal energy is convected to the ball while simultaneously getting conducted within.

The Fourier Number, in unsteady heat conduction analysis like the one here, is given as the ratio of conductive heat transfer rate in the ball to the rate of energy storage in the ball. I am having trouble with understanding how using this definition can we come up with the expression for Fourier Number which is

$$Fo= \frac{\alpha t}{(L_c)^2}$$

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An excerpt from the book I'm referring - Fundamentals of Heat and Mass Transfer by Incropera and Dewitt

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I don't understand, while determining the conductive heat transfer rate, what is $\Delta T$. Is it the difference between the temperatures at the surface and center of the ball? If it is, that temperature difference will vary with time. Is it time averaged temperature difference between the center and the surface?

Furthermore, what is the $\Delta T$ while determining the rate of energy storage? and how is it equal to the $\Delta T $ taken while determining the conductive heat transfer rate? How did they cancel out both the $\Delta T$s at the end to give $\frac{\alpha t}{(L_c)^2}$

Answer 1

These are scaling arguments, good for back-of-the-envelope calculations and for broadly classifying process regimes. The actual values may be uncertain by an order of magnitude of more. The important aspect is that $\Delta T$ and $3\Delta T$ and $\Delta T/2$ are all considered to be approximately equal and fundamentally different from $(\Delta T)^2$ or $\ln(\Delta T/T_0)$, for example.

Thus, is doesn't matter if the cross-sectional area is that of a rectangle ($L^2$) or a circle ($\pi L^2/4$), for instance; the important part is the $\sim\!\! L^2$ scaling.

Now to your question. $\Delta T$ is the temperature difference across the solid. Yes, this varies over time. Yes, an energy storage rate $\rho L^3c\Delta T/t$ is strictly meaningful only if the entire solid heats up by a temperature difference $\Delta T$, which is not the same as a temperature difference across the solid. In other words, a spatial gradient in temperature is being equated to a transient difference in temperature. For the reasons given above, this is acceptable for this level of analysis and further can provide insight not available otherwise. What we're checking is whether the Fourier number is, say, $10^{-5}$ or $1$ or $10^5$; it doesn't matter if some temperature estimate was inaccurate by a factor of two. Does this make sense?