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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


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 $r_0$ $r_0/2$ $r_0$

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.

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  • $\begingroup$ First, what is the "characteristic" dimension? $\endgroup$
    – Solar Mike
    Jan 25, 2022 at 14:04
  • $\begingroup$ It's not crystal clear in my mind what a characteristic dimension is, but here's what I understand at present, a characteristic dimension is some length associated with the system which affects the behavior / physics of the system the most. $\endgroup$ Jan 25, 2022 at 14:35
  • $\begingroup$ @HarshitRajput Are you sure that the characteristic length Fourier number for the cylinder is $r_0$. I have a suspicition that it should be L $\endgroup$
    – NMech
    Jan 25, 2022 at 17:42
  • $\begingroup$ @NMech Yes I'm pretty sure - en.wikipedia.org/wiki/… $\endgroup$ Jan 25, 2022 at 19:18
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    $\begingroup$ @HarshitRajput I meant to write infinite instead of infringe (I think it was the autocorrection on my mobile). :-) If it is infinite then the conductance does not make sense along the length of the cylinder, but radially instead. $\endgroup$
    – NMech
    Jan 26, 2022 at 6:54

1 Answer 1

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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 $r_0$ $2\pi r_0 L$ $\pi r_0^2\cdot L$ $r_0/2$

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.

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  • $\begingroup$ I observed something in my textbook, even for the same Biot number while checking the applicability of Lumped system analysis, the book used Lc as V/A and while using the heisler's charts it uses the Lc (for Biot number) as simply $r_o$ for cylinder and spheres. So the Lc = V/A for Biot Number is even not consistent $\endgroup$ Jan 26, 2022 at 12:10
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    $\begingroup$ @HarshitRajput this is in line of what I was saying right at the beginning, i.e. that there is no one characteristic length. What you noticed in the book makes sense, because when you are comparing between similar geometries then $r_0/2$ or $r_0/3$ isn't really that much different (also I am not certain how the book plots are drawn so there may be an implicit assumption there). The use of characteristic lengths is more evident though when you are trying to compare different geometries eg. flat and sphere or cylinder and a straightforward comparison is not possible.. $\endgroup$
    – NMech
    Jan 26, 2022 at 13:51

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