# Characteristic Length

Written by Jerry Ratzlaff 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

$$\large{ l_c = \frac{ V }{ A } }$$

$$\large{ l_c = \sqrt{ \frac{ \alpha \; t_c^2 }{Fo } } }$$

$$\large{ l_c =\frac{ Nu \; k }{ h } }$$

$$\large{ l_c = \frac {Pe \; k}{ v \; \rho \; C } }$$

$$\large{ l_c = \frac{ Re \; \mu }{\rho \; v } }$$

$$\large{ l_c = \frac { Sh \; D} {K} }$$

Symbol English Metric
$$\large{ l_c }$$ = characteristic length $$\large{ ft }$$ $$\large{ m }$$
$$\large{ A }$$ = area of object surface $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ t_c }$$ = characteristic time $$\large{ sec }$$ $$\large{ sec }$$
$$\large{ \rho }$$  (Greek symbol rho) = density $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$
$$\large{ D }$$ = diffusion coefficient $$\large{\frac{ft^2}{sec}}$$  $$\large{\frac{m^2}{s}}$$
$$\large{ \mu }$$  (Greek symbol mu)  = dynamic viscosity $$\large{\frac{lbf-sec}{ft^2}}$$  $$\large{ Pa-s }$$
$$\large{ Fo }$$ = Fourier number $$\large{ dimensionless }$$
$$\large{ C }$$ = heat capacity $$\large{\frac{Btu}{F}}$$ $$\large{\frac{kJ}{K}}$$
$$\large{ h }$$ = heat transfer coefficient $$\large{\frac{Btu}{hr-ft^2-F}}$$ $$\large{\frac{W}{m^2-K}}$$
$$\large{ K }$$ = mass transfer coefficient $$\large{ dimensionless }$$
$$\large{ Nu }$$ = Nusselt number $$\large{ dimensionless }$$
$$\large{ Pe }$$ = Peclet number $$\large{ dimensionless }$$
$$\large{ Re }$$ = Reynolds number $$\large{ dimensionless }$$
$$\large{ Sh }$$ = Sherwood number $$\large{ dimensionless }$$
$$\large{ k }$$ = thermal conductivity $$\large{\frac{Btu}{hr-ft^2-F}}$$ $$\large{\frac{W}{m-K}}$$
$$\large{ \alpha }$$  (Greel symbol alpha) = thermal diffusivity  $$\large{\frac{ft^2}{sec}}$$ $$\large{\frac{m^2}{s}}$$
$$\large{ v }$$ = velocity $$\large{\frac{ft}{sec}}$$ $$\large{\frac{m}{s}}$$
$$\large{ V }$$ = $$\large{ ft^3 }$$ $$\large{ m^3 }$$ 