Druyd:Knowledge

Boltzmann's Transport Equation

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>Model

ID:(1134, 0)

Boltzmann's Transport Equation

Storyboard

Variables

Symbol

Text

Variable

Value

Units

Calculate

MKS Value

MKS Units

$R$

Constante de los gases

J/K mol

$\rho$

rho

Densidad en el espacio

kg/m^3

$f$

Función distribución de la teoría de transporte

$m$

Masa de la partícula

$T(\vec{x},t)$

Temperatura en el espacio

$\sigma_{ij}(\vec{x},t)$

sigma_ij

Tensión en el espacio

$\vec{v}$

Velocidad de las partículas (vector)

m/s

$\vec{u}$

Velocidad media (vector)

m/s

Calculations

Equation

Number

First, select the equation:

, then, select the variable:

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable

Given

Calculate

Target :

Equation

To be used

Equations

$\rho(\vec{x},t) = m\displaystyle\int f(\vec{x},\vec{v},t)d\vec{v}$

rho =int f dv

$\vec{u}(\vec{x},t) = \displaystyle\frac{m}{\rho}\int \vec{v}f(\vec{x},\vec{v},t)d\vec{v}$

u =(m/rho)int vf dv

$T(\vec{x},t) = \displaystyle\frac{m}{3R\rho}\displaystyle\int (\vec{v}\cdot\vec{v})f(\vec{x},\vec{v},t)d\vec{v}$

T =(m/3Rrho)int v^2f dv

$\sigma_{ij} = m\displaystyle\int (v_i-u_i)(v_j-u_j)f(\vec{x},\vec{v},t)d\vec{v}$

s_ij =m int (v_i-u_i)(v_j-u_j)f dv

$\displaystyle\frac{\partial f}{\partial t}+v_i\displaystyle\frac{\partial f}{\partial x_i}=C(f)$

df/dt+vdf/dv=C

Examples

Boltzmann equation

The Boltzmann function describes the transport of a particle system described by the velocity distribution function:

equation

Where the term C describes the interaction (collisions) between them.

Density

If the parameters are calculated by averaging over the speed using

equation=9075

the mass density estimation is obtained by:

equation

Speed of the Flow

If the parameters are calculated by averaging over the speed using

equation=9075

the velocity of the flow is calculated by integrating the velocity distribution function on all velocities by weighing the velocities:

equation

Temperature

If the parameters are calculated by averaging over the speed using

equation=9075

and the equipartition theorem is considered, the temperature can be estimated by integrating the kinetic energy weighted by the velocity distribution divided by the gas constant:

equation

Tension tensor

If the parameters are calculated by averaging over the speed using

equation=9075

the flow tensor is calculated by integrating the velocity distribution function on all velocities by weighing the velocity differences:

equation

>Model

ID:(1134, 0)