Why different characteristic dimension for Biot Number and Fourier Number?

Question

In unsteady heat conduction analysis,

Biot Number and Fourier Number are given as follows:

$$Bi=\frac{h L_c}{K}$$

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

where $L_c$ is the characteristic dimension or length

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As written in the book I'm following, for Bi $$L_c= \frac{V}{A_s}$$ where $V$ is the volume and $A_s$ the surface area of heat convection

which respectively, for a plane wall of semithickness $L$, sphere and cylinder of radius $r_0$ turns out to be $L, r_0/3,r_0/2$

For Fo the characteristic dimensions are $L, r_0, r_0$ for plane wall, sphere and cylinder respectively.

i.e.

problem	Bi	Fo
a plane wall of semithickness $L$	L	L
sphere of radius $r_0$	$r_0/3$	$r_0$
cylinder of radius $r_0$	$r_0/2$	$r_0$

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Even though both numbers correspond to the same type of problem i.e. a plane wall, cylinder or sphere through which convection happens at the surface and conduction within, the book choses different characteristic dimensions. It never talks about why though.

Answer 1

There is no unique definition for characteristic length. The characteristic number is a number which is related to both the problem and the geometry. So the characteristic length can be defined as any number that returns length units e.g. the square root of the surface area, as the third root of the volume, as the volume-to-surface ratio, etc. At the end of the day, it all depends what on the problem (So I treat most characteristic lengths as a magical numbers that allow to compare different geometries on the same problem -- in other words is a convenience metric, and as such it has its pitfalls.)

The Biot number is defined as

$$Bi = \frac{\text{Conduction resistance within the body}}{\text{Convection resistance at the surface of the body})}$$

again for a plate of cross-section A and length $L_C$ (along the direction of heat convection, the number is derived as follows:

$$Bi = \frac{\frac{L_c}{k}}{\frac{1}{h}}= \frac{h}{k} L_C$$

In both cases the derivations deal with the conductive heat rate (or its resistance which is the inverse) but on the denominator have different quantities, namely:

• heat convective resistance which is related to a surface
• and heat capacity which is related to volume.

Hopefully the derivation above for the rectangular plate makes sense, also the cylinder and the sphere is quite easy to derive for the Biot number.

problem	Convective Area	Volume	$\frac{V}{A}$ (Bi)
a plane wall of semithickness $L$	A	$A\cdot L$	L
sphere of radius $r_0$	$4 \cdot \pi \cdot r_0^2$	$\frac{4}{3} \cdot \pi \cdot r_0^3$	$r_0/3$
cylinder of radius $r_0$	$2\pi r_0 L$	$\pi r_0^2\cdot L$	$r_0/2$

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So the Fourier number can be thought of as a ratio of the heat rate that is conducted across L of a Body of Volume, over the rate at which heat is stored in a body of volume.

$$F_0=\frac{\color{blue}{\text{the rate at which heat is conducted across L of a Body of Volume } }}{\color{red}{\text{The rate at which heat is stored in a body of volume }}}$$

For a plate the length of interest is along the direction heat convection, so assuming a cross-section $A$ and length $L_C$ along the direction of heat convection:

$$Fo=\frac{\color{blue}{\frac{k}{L_c} \cdot A \cdot \delta T}}{\color{red}{\frac{c_p \cdot \rho \cdot V \delta T}{t}}} $$

$$Fo=\frac{\color{blue}{\frac{k}{L_c}}}{\color{red}{c_p \cdot \rho }} \cdot \frac{\color{blue}{ A }}{\color{red}{V}} \cdot t$$

$$Fo=\frac{\color{blue}{k} }{ \color{blue}{L_c} \cdot \color{red}{c_p \cdot \rho } }\cdot \frac{1}{L_C} \cdot t$$

$$Fo=\frac{k}{c_p\cdot \rho \cdot L_C^2}\cdot t $$

$$Fo=\frac{α\cdot t}{L_C^2} $$

where:

• $a= \frac{k}{c_p\cdot \rho }$

The Fourier number in the case of the sphere and the infinite length cylinder is a bit different. It actually involves radial heat conduction, so in that case intuitively the representative length is the radius.