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Instant Volume Flow
Concept 
The definition of the volume flow ($J_V$) is the volume element ($\Delta V$) over the time elapsed ($\Delta t$):
| $ J_V =\displaystyle\frac{ \Delta V }{ \Delta t }$ |
which, in the limit of an infinitesimal time interval, corresponds to the derivative of the volume ($V$) with respect to the time ($t$):
| $ J_V =\displaystyle\frac{ dV }{ dt }$ |
ID:(15718, 0)
Volume Flow and its Speed
Concept
The volume ($V$) for a tube with constant the section Tube ($S$) and a position ($s$) is
| $ V = s S $ |
If the section Tube ($S$) is constant, the temporal derivative will be
$\displaystyle\frac{dV}{dt} = S\displaystyle\frac{ds}{dt}$
thus, with the volume flow ($J_V$) defined by
| $ J_V =\displaystyle\frac{ dV }{ dt }$ |
and with the flux density ($j_s$) associated with the position ($s$) via
| $ j_s =\displaystyle\frac{ ds }{ dt }$ |
it is concluded that
| $ J_V = S j_s $ |
ID:(15717, 0)
Flow for inhomogeneous flux density
Concept
In the case that the flux density ($j_s$) is constant, the volume flow ($J_V$) can be calculated using the section or Area ($S$) according to:
| $ J_V = S j_s $ |
If the flux density ($j_s$) varies, sufficiently small sectional elements $dS$ can be considered so that the equation remains valid in the sense that the contribution to flow is:
$dJ_V = j_s dS$
Integrating this expression over the entire section results in
| $ J_V =\displaystyle\int j_s dS $ |
ID:(15719, 0)
Model
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Equations
$ j_s =\displaystyle\frac{ ds }{ dt }$
j_s = @DIFF( s , t , 1 )
$ J_V =\displaystyle\frac{ dV }{ dt }$
J_V = @DIFF( V , t , 1 )
$ J_V =\displaystyle\int j_s dS $
J_V = @INT( j_s , S )
$ J_V = S j_s $
J_V = S * j_s
$ V = s S $
V = h * S
ID:(15714, 0)
Instant Volume Flow
Equation
The volume flow ($J_V$) corresponds to the quantity volume ($V$) that flows through the channel during a time ($t$). Therefore, we have:
| $ J_V =\displaystyle\frac{ dV }{ dt }$ |
The definition of the volume flow ($J_V$) is the volume element ($\Delta V$) over the time elapsed ($\Delta t$):
| $ J_V =\displaystyle\frac{ \Delta V }{ \Delta t }$ |
which, in the limit of an infinitesimal time interval, corresponds to the derivative of the volume ($V$) with respect to the time ($t$):
| $ J_V =\displaystyle\frac{ dV }{ dt }$ |
ID:(12713, 0)
Element volume
Equation
The volume ($V$) is calculated by multiplying the section Tube ($S$) with the position ($s$) along the tube:
| $ V = s S $ |
| $ V = h S $ |
ID:(4876, 0)
Instantaneous flux density
Equation
The flux density ($j_s$) is related to the position ($s$), which is the fluid position at the time ($t$), through the following equation:
| $ j_s =\displaystyle\frac{ ds }{ dt }$ |
ID:(12714, 0)
Volume Flow and its Speed
Equation
A flux density ($j_s$) can be expressed in terms of the volume flow ($J_V$) using the section or Area ($S$) through the following formula:
| $ J_V = S j_s $ |
The volume ($V$) for a tube with constant the section Tube ($S$) and a position ($s$) is
| $ V = s S $ |
If the section Tube ($S$) is constant, the temporal derivative will be
$\displaystyle\frac{dV}{dt} = S\displaystyle\frac{ds}{dt}$
thus, with the volume flow ($J_V$) defined by
| $ J_V =\displaystyle\frac{ dV }{ dt }$ |
and with the flux density ($j_s$) associated with the position ($s$) via
| $ j_s =\displaystyle\frac{ ds }{ dt }$ |
it is concluded that
| $ J_V = S j_s $ |
ID:(15716, 0)
Flow for inhomogeneous flux density
Equation
If the flux density ($j_s$) is not constant and varies across the flow tube section the volume flow ($J_V$), it is calculated as the integral over that section:
| $ J_V =\displaystyle\int j_s dS $ |
In the case that the flux density ($j_s$) is constant, the volume flow ($J_V$) can be calculated using the section or Area ($S$) according to:
| $ J_V = S j_s $ |
If the flux density ($j_s$) varies, sufficiently small sectional elements $dS$ can be considered so that the equation remains valid in the sense that the contribution to flow is:
$dJ_V = j_s dS$
Integrating this expression over the entire section results in
| $ J_V =\displaystyle\int j_s dS $ |
ID:(15712, 0)