# Characteristic Length

on . Posted in Classical Mechanics

Characteristic length, abbreviated as $$l_c$$, is a dimension used in physics that defines the scale of a physical system.  The length is used in 2D and 3D systems for Fluid Dynamics and Thermodynamics defining the parameter of the system.  It is usually requited by the construction of a dimensionless quantity, in the general framework of dimensionless analysis.

In computational mechanics, a characteristic length is defined to force localization of a stress softening constructive equation.  The length is associated with an integration point.  For 2D analysis, it is calculated by taking the square root of the area.  For 3D analysis, it is calculated by taking the cubic root of the volume associated to the integration point.

## Characteristic length formula

$$\large{ l_c = \frac{ V }{ A } }$$
Symbol English Metric
$$\large{ l_c }$$ = characteristic length $$\large{ ft }$$ $$\large{ m }$$
$$\large{ A }$$ = area of object surface $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ V }$$ = volume of the object $$\large{ ft^3 }$$ $$\large{ m^3 }$$

## Fourier number Characteristic length formula

$$\large{ l_c = \sqrt{ \frac{ \alpha \; t_c^2 }{Fo } } }$$
Symbol English Metric
$$\large{ l_c }$$ = characteristic length $$\large{ ft }$$ $$\large{ m }$$
$$\large{ t_c }$$ = characteristic time $$\large{ sec }$$ $$\large{ sec }$$
$$\large{ Fo }$$ = Fourier number $$\large{ dimensionless }$$
$$\large{ \alpha }$$  (Greel symbol alpha) = thermal diffusivity  $$\large{\frac{ft^2}{sec}}$$ $$\large{\frac{m^2}{s}}$$

## Nusselt number Characteristic length formula

$$\large{ l_c =\frac{ Nu \; k }{ h } }$$
Symbol English Metric
$$\large{ l_c }$$ = characteristic length $$\large{ ft }$$ $$\large{ m }$$
$$\large{ h }$$ = heat transfer coefficient $$\large{\frac{Btu}{hr-ft^2-F}}$$ $$\large{\frac{W}{m^2-K}}$$
$$\large{ Nu }$$ = Nusselt number $$\large{ dimensionless }$$
$$\large{ k }$$ = thermal conductivity $$\large{\frac{Btu}{hr-ft^2-F}}$$ $$\large{\frac{W}{m-K}}$$

## Peclet number Characteristic length formula

$$\large{ l_c = \frac {Pe \; k}{ v \; \rho \; C } }$$
Symbol English Metric
$$\large{ l_c }$$ = characteristic length $$\large{ ft }$$ $$\large{ m }$$
$$\large{ \rho }$$  (Greek symbol rho) = density $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$
$$\large{ C }$$ = heat capacity $$\large{\frac{Btu}{F}}$$ $$\large{\frac{kJ}{K}}$$
$$\large{ Pe }$$ = Peclet number $$\large{ dimensionless }$$
$$\large{ k }$$ = thermal conductivity $$\large{\frac{Btu}{hr-ft^2-F}}$$ $$\large{\frac{W}{m-K}}$$
$$\large{ v }$$ = velocity $$\large{\frac{ft}{sec}}$$ $$\large{\frac{m}{s}}$$

## Reynolds number Characteristic length formula

$$\large{ l_c = \frac{ Re \; \mu }{\rho \; v } }$$
Symbol English Metric
$$\large{ l_c }$$ = characteristic length $$\large{ ft }$$ $$\large{ m }$$
$$\large{ \rho }$$  (Greek symbol rho) = density $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$
$$\large{ \mu }$$  (Greek symbol mu)  = dynamic viscosity $$\large{\frac{lbf-sec}{ft^2}}$$  $$\large{ Pa-s }$$
$$\large{ Re }$$ = Reynolds number $$\large{ dimensionless }$$
$$\large{ v }$$ = velocity $$\large{\frac{ft}{sec}}$$ $$\large{\frac{m}{s}}$$

## Sherwood number Characteristic length formula

$$\large{ l_c = \frac{ Sh \; D}{K} }$$
Symbol English Metric
$$\large{ l_c }$$ = characteristic length $$\large{ ft }$$ $$\large{ m }$$
$$\large{ D }$$ = diffusion coefficient $$\large{\frac{ft^2}{sec}}$$  $$\large{\frac{m^2}{s}}$$
$$\large{ K }$$ = mass transfer coefficient $$\large{ dimensionless }$$
$$\large{ Sh }$$ = Sherwood number $$\large{ dimensionless }$$ 