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Porous medium hydraulic resistance

Storyboard

Darcy's Law considers hydraulic resistance, which in its basic form corresponds to that of a tube with a given length and radius. However, in many situations, the fluid flows through a medium containing pores rather than a single cavity. These pores act as capillaries, and their hydraulic resistance can be modeled as that of small tubes. The sum of these multiple hydraulic resistances in parallel forms the total hydraulic resistance of a porous material.

>Model

ID:(2071, 0)



Mechanisms

Iframe

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Code
Concept

Mechanisms

ID:(15730, 0)



Hydrodynamic networks in porous media

Concept

>Top


If the porous medium is modeled as a network of the hydraulic resistance ($R_h$) identical elements connected in parallel in groups of the number of equal hydraulic resistances in parallel ($N_p$), which are then summed in series as the number of identical hydraulic resistors in series ($N_s$):

In this way, the general parallel sum, which results in the total hydraulic resistance in parallel ($R_{pt}$) according to

$\displaystyle\frac{1}{ R_{pt} }=\sum_k\displaystyle\frac{1}{ R_{hk} }$

,

is transformed into

$ R_{pt} = \displaystyle\frac{ R_h }{ N_p }$

.

Similarly, the general series sum, which yields the total hydraulic resistance in series ($R_{st}$) according to

$ R_{st} =\displaystyle\sum_k R_{hk} $

,

is transformed into

$ R_t = N_s R_{pt} $

.

ID:(15908, 0)



Hydrodynamic resistance of a porous medium

Concept

>Top


By using the definition of the hydraulic resistance ($R_h$) with the values the viscosity ($\eta$), the tube length ($\Delta L$), and the tube radius ($R$) according to

$ R_h =\displaystyle\frac{8 \eta | l | }{ \pi r ^4}$

,

and calculating the number of equal hydraulic resistances in parallel ($N_p$) from the porosity ($f$) and the section Tube ($S$) using

$ N_p = \displaystyle\frac{ f S }{ \pi r ^2}$

,

as well as the number of identical hydraulic resistors in series ($N_s$) with the tube length ($\Delta L$) and the capillary length ($l$) through

$ N_s = \displaystyle\frac{ \Delta L }{ l }$

,

the final result is obtained as

$ R_t = \displaystyle\frac{ 8 \eta }{ f r ^2 }\displaystyle\frac{ \Delta L }{ S }$

.

ID:(15909, 0)



Model

Top

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Parameters

Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
$g$
g
Gravitational Acceleration
m/s^2
$R_h$
R_h
Hydraulic resistance
kg/m^4s
$\rho_w$
rho_w
Liquid density
kg/m^3
$\pi$
pi
Pi
rad
$\rho_s$
rho_s
Solid Density
kg/m^3
$R_t$
R_t
Total hydraulic resistance
kg/m^4s
$R_{pt}$
R_pt
Total hydraulic resistance in parallel
kg/m^4s
$R$
R
Tube radius
m
$\Delta p$
Dp
Variación de la Presión
Pa
$\eta$
eta
Viscosity
Pa s

Variables

Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
$l$
l
Capillary length
m
$r$
r
Capillary radius
m
$\Delta s$
Ds
Distance traveled in a time
m
$j_s$
j_s
Flux density
m/s
$\Delta h$
Dh
Height of liquid column
m
$N_p$
N_p
Number of equal hydraulic resistances in parallel
-
$N_s$
N_s
Number of identical hydraulic resistors in series
-
$V_p$
V_p
Pore volume
m^3
$f$
f
Porosity
-
$S$
S
Section Tube
m^2
$V_s$
V_s
Solid volume of a component
m^3
$\Delta t$
Dt
Time elapsed
s
$M_s$
M_s
Total Dry Mass of Sample
kg
$V_t$
V_t
Total volume
m^3
$\Delta L$
DL
Tube length
m
$J_V$
J_V
Volume flow
m^3/s

Calculations


First, select the equation: to , then, select the variable: to
Dp = rho_w * g * Dh Dp = R_t * J_V V_t = S * DL f = N_p * pi * r ^2/ S f = V_p / V_t j_s = J_V / S N_p = f * S /( pi * r ^2) N_s = DL / l rho_s = M_s / V_s R_h =8* eta * abs( l )/( pi * r ^4) R_pt = R_h / N_p R_t = N_s * R_pt R_t = 8* eta * DL /( f * r ^2* S ) S = pi * R ^2 j_s = Ds / Dt V_t = V_s + V_p lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

Calculations

Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

Variable Given Calculate Target : Equation To be used
Dp = rho_w * g * Dh Dp = R_t * J_V V_t = S * DL f = N_p * pi * r ^2/ S f = V_p / V_t j_s = J_V / S N_p = f * S /( pi * r ^2) N_s = DL / l rho_s = M_s / V_s R_h =8* eta * abs( l )/( pi * r ^4) R_pt = R_h / N_p R_t = N_s * R_pt R_t = 8* eta * DL /( f * r ^2* S ) S = pi * R ^2 j_s = Ds / Dt V_t = V_s + V_p lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V




Equations

#
Equation

$ \Delta p = \rho_w g \Delta h $

Dp = rho_w * g * Dh


$ \Delta p = R_t J_V $

Dp = R_h * J_V


$ V_t = S \Delta L $

DV = S * Ds


$ f = \displaystyle\frac{ N_p \pi r ^2 }{ S }$

f = N_p * pi * R ^2/ S


$ f =\displaystyle\frac{ V_p }{ V_t }$

f = V_p / V_t


$ j_s = \displaystyle\frac{ J_V }{ S }$

j_s = J_V / S


$ N_p = \displaystyle\frac{ f S }{ \pi r ^2}$

N_p = f * S /( pi * R ^2)


$ N_s = \displaystyle\frac{ \Delta L }{ l }$

N_s = DL / l


$ \rho_s = \displaystyle\frac{ M_s }{ V_s }$

rho_s = M_s / V_s


$ R_h =\displaystyle\frac{8 \eta | l | }{ \pi r ^4}$

R_h =8* eta * abs( DL )/( pi * R ^4)


$ R_{pt} = \displaystyle\frac{ R_h }{ N_p }$

R_pt = R_h / N_p


$ R_t = N_s R_{pt} $

R_st = N_s * R_h


$ R_t = \displaystyle\frac{ 8 \eta }{ f r ^2 }\displaystyle\frac{ \Delta L }{ S }$

R_t = 8* eta * DL /( f * R ^2* S )


$ S = \pi R ^2$

S = pi * r ^2


$ j_s \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$

v_m = Ds / Dt


$ V_t = V_s + V_p $

V_t = V_s + V_p

ID:(15735, 0)



Number of hydraulic resistors in series

Equation

>Top, >Model


The number of identical hydraulic resistors in series ($N_s$) can be obtained by dividing the tube length ($\Delta L$) by the capillary length ($l$):

$ N_s = \displaystyle\frac{ \Delta L }{ l }$

$l$
Capillary length
$m$
10128
$N_s$
Number of identical hydraulic resistors in series
$-$
10442
$\Delta L$
Tube length
$m$
5430
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

ID:(15904, 0)



Porosity via surface

Equation

>Top, >Model


The porosity ($f$) is the ratio of the empty section through which the liquid flows relative to the section Tube ($S$). The first is calculated by multiplying the section of each capillary, the tube radius ($R$), by the number of equal hydraulic resistances in parallel ($N_p$), so that:

$ f = \displaystyle\frac{ N_p \pi r ^2 }{ S }$

$ f = \displaystyle\frac{ N_p \pi R ^2 }{ S }$

$N_p$
Number of equal hydraulic resistances in parallel
$-$
10441
$\pi$
Pi
3.1415927
$rad$
5057
$f$
Porosity
$-$
5805
$S$
Section Tube
$m^2$
6267
$R$
$r$
Capillary radius
$m$
10444
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

ID:(15906, 0)



Number of hydraulic resistances in parallel

Equation

>Top, >Model


The number of equal hydraulic resistances in parallel ($N_p$) is calculated as the fraction given by the porosity ($f$) of the section Tube ($S$), divided by the section of a capillary, the tube radius ($R$):

$ N_p = \displaystyle\frac{ f S }{ \pi r ^2}$

$ N_p = \displaystyle\frac{ f S }{ \pi R ^2}$

$N_p$
Number of equal hydraulic resistances in parallel
$-$
10441
$\pi$
Pi
3.1415927
$rad$
5057
$f$
Porosity
$-$
5805
$S$
Section Tube
$m^2$
6267
$R$
$r$
Capillary radius
$m$
10444
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

ID:(15905, 0)



Sum of identical hydraulic resistances in parallel

Equation

>Top, >Model


The total hydraulic resistance in parallel ($R_{pt}$) is the result for the hydraulic resistance ($R_h$) identical values, obtained by dividing this by the number of equal hydraulic resistances in parallel ($N_p$):

$ R_{pt} = \displaystyle\frac{ R_h }{ N_p }$

$R_h$
Hydraulic resistance
$kg/m^4s$
5424
$N_p$
Number of equal hydraulic resistances in parallel
$-$
10441
$R_{pt}$
Total hydraulic resistance in parallel
$kg/m^4s$
5429
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

ID:(15902, 0)



Sum of identical hydraulic resistances in series

Equation

>Top, >Model


The total hydraulic resistance in series ($R_{st}$) wird berechnet, indem the hydraulic resistance ($R_h$) mit the number of identical hydraulic resistors in series ($N_s$) multipliziert wird:

$ R_t = N_s R_{pt} $

$ R_{st} = N_s R_h $

$R_h$
$R_{pt}$
Total hydraulic resistance in parallel
$kg/m^4s$
5429
$N_s$
Number of identical hydraulic resistors in series
$-$
10442
$R_{st}$
$R_t$
Total hydraulic resistance
$kg/m^4s$
10443
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

ID:(15903, 0)



Hydraulic resistance of a tube

Equation

>Top, >Model


Since the hydraulic resistance ($R_h$) is equal to the inverse of the hydraulic conductance ($G_h$), it can be calculated from the expression of the latter. In this way, we can identify parameters related to geometry (the tube length ($\Delta L$) and the tube radius ($R$)) and the type of liquid (the viscosity ($\eta$)), which can be collectively referred to as a hydraulic resistance ($R_h$):

$ R_h =\displaystyle\frac{8 \eta | l | }{ \pi r ^4}$

$ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$

$R_h$
Hydraulic resistance
$kg/m^4s$
5424
$\pi$
Pi
3.1415927
$rad$
5057
$\Delta L$
$l$
Capillary length
$m$
10128
$R$
$r$
Capillary radius
$m$
10444
$\eta$
Viscosity
$Pa s$
5422
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

Since the hydraulic resistance ($R_h$) is equal to the hydraulic conductance ($G_h$) as per the following equation:

$ R_h = \displaystyle\frac{1}{ G_h }$



and since the hydraulic conductance ($G_h$) is expressed in terms of the viscosity ($\eta$), the tube radius ($R$), and the tube length ($\Delta L$) as follows:

$ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$



we can conclude that:

$ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$

ID:(3629, 0)



Hydraulic resistance of porous material

Equation

>Top, >Model


The total hydraulic resistance ($R_t$) is calculated from a type of hydraulic resistance density, which depends on the viscosity ($\eta$), the porosity ($f$), and the tube radius ($R$), as well as the geometric factors the tube length ($\Delta L$) and the section Tube ($S$):

$ R_t = \displaystyle\frac{ 8 \eta }{ f r ^2 }\displaystyle\frac{ \Delta L }{ S }$

$ R_t = \displaystyle\frac{ 8 \eta }{ f R ^2 }\displaystyle\frac{ \Delta L }{ S }$

$f$
Porosity
$-$
5805
$S$
Section Tube
$m^2$
6267
$R_t$
Total hydraulic resistance
$kg/m^4s$
10443
$\Delta L$
Tube length
$m$
5430
$R$
$r$
Capillary radius
$m$
10444
$\eta$
Viscosity
$Pa s$
5422
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

ID:(15907, 0)



Pressure difference between columns

Equation

>Top, >Model


The height difference, denoted by the height difference ($\Delta h$), implies that the pressure in both columns is distinct. In particular, the pressure difference ($\Delta p$) is a function of the liquid density ($\rho_w$), the gravitational Acceleration ($g$), and the height difference ($\Delta h$), as follows:

$ \Delta p = \rho_w g \Delta h $

$g$
Gravitational Acceleration
9.8
$m/s^2$
5310
$\Delta h$
Height of liquid column
$m$
5819
$\rho_w$
Liquid density
$kg/m^3$
5407
$\Delta p$
Variación de la Presión
$Pa$
6673
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

If there is the pressure difference ($\Delta p$) between two points, as determined by the equation:

$ \Delta p = p_2 - p_1 $



we can utilize the water column pressure ($p$), which is defined as:

$ p_t = p_0 + \rho_w g h $



This results in:

$\Delta p=p_2-p_1=p_0+\rho_wh_2g-p_0-\rho_wh_1g=\rho_w(h_2-h_1)g$



As the height difference ($\Delta h$) is:

$ \Delta h = h_2 - h_1 $



the pressure difference ($\Delta p$) can be expressed as:

$ \Delta p = \rho_w g \Delta h $

ID:(4345, 0)



Volume element

Equation

>Top, >Model


If we have a tube with a the section Tube ($S$) moving a distance the tube element ($\Delta s$) along its axis, having displaced the volume element ($\Delta V$), then it is equal to:

$ V_t = S \Delta L $

$ \Delta V = S \Delta s $

$S$
Section Tube
$m^2$
6267
$\Delta s$
$\Delta L$
Tube length
$m$
5430
$\Delta V$
$V_t$
Total volume
$m^3$
4946
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

ID:(3469, 0)



Volume Flow and its Speed

Equation

>Top, >Model


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_s = \displaystyle\frac{ J_V }{ S }$

$j_s$
Flux density
$m/s$
7220
$S$
$S$
Section Tube
$m^2$
6267
$J_V$
Volume flow
$m^3/s$
5448
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

Flow is defined as the volume the volume element ($\Delta V$) divided by time the time elapsed ($\Delta t$), which is expressed in the following equation:

$ J_V =\displaystyle\frac{ \Delta V }{ \Delta t }$



and the volume equals the cross-sectional area the section Tube ($S$) multiplied by the distance traveled the tube element ($\Delta s$):

$ \Delta V = S \Delta s $



Since the distance traveled the tube element ($\Delta s$) per unit time the time elapsed ($\Delta t$) corresponds to the velocity, it is represented by:

$ j_s =\displaystyle\frac{ \Delta s }{ \Delta t }$



Thus, the flow is a flux density ($j_s$), which is calculated using:

$ j_s = \displaystyle\frac{ J_V }{ S }$

ID:(4349, 0)



Darcy's law and hydraulic resistance

Equation

>Top, >Model


Darcy rewrites the Hagen Poiseuille equation so that the pressure difference ($\Delta p$) is equal to the hydraulic resistance ($R_h$) times the volume flow ($J_V$):

$ \Delta p = R_t J_V $

$ \Delta p = R_h J_V $

$R_h$
$R_t$
Total hydraulic resistance
$kg/m^4s$
10443
$\Delta p$
Variación de la Presión
$Pa$
6673
$J_V$
Volume flow
$m^3/s$
5448
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

The volume flow ($J_V$) can be calculated from the hydraulic conductance ($G_h$) and the pressure difference ($\Delta p$) using the following equation:

$ J_V = G_h \Delta p $



Furthermore, using the relationship for the hydraulic resistance ($R_h$):

$ R_h = \displaystyle\frac{1}{ G_h }$



results in:

$ \Delta p = R_h J_V $

ID:(3179, 0)



Porosity

Equation

>Top, >Model


The porosity ($f$) expresses the relationship between the pore volume ($V_p$) and the total volume ($V_t$), allowing us to define the equation as follows:

$ f =\displaystyle\frac{ V_p }{ V_t }$

$V_p$
Pore volume
$m^3$
5806
$f$
Porosity
$-$
5805
$V_t$
Total volume
$m^3$
4946
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

ID:(4245, 0)



Surface of a disk

Equation

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The surface of a disk ($S$) of ($$) is calculated as follows:

$ S = \pi R ^2$

$ S = \pi r ^2$

$r$
$R$
Tube radius
$m$
5417
$\pi$
Pi
3.1415927
$rad$
5057
$S$
$S$
Section Tube
$m^2$
6267
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

ID:(3804, 0)



Average Speed

Equation

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The mean Speed ($\bar{v}$) can be calculated from the distance traveled in a time ($\Delta s$) and the time elapsed ($\Delta t$) using:

$ j_s \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$

$ \bar{v} \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$

$\Delta s$
Distance traveled in a time
$m$
6025
$\bar{v}$
$j_s$
Flux density
$m/s$
7220
$\Delta t$
Time elapsed
$s$
5103
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

ID:(3152, 0)



Total volume with general porosity

Equation

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The total volume ($V_t$) is the sum of the pore volume ($V_p$), which includes both the micropores and the macropores in the soil, and the total Dry Mass of Sample ($M_s$), so that:

$ V_t = V_s + V_p $

$V_p$
Pore volume
$m^3$
5806
$V_s$
Solid volume of a component
$m^3$
6038
$V_t$
Total volume
$m^3$
4946
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

ID:(4726, 0)



Solid density

Equation

>Top, >Model


Since we already know the total Dry Mass of Sample ($M_s$) and the solid volume ($V_s$) from the sample, we can introduce the solid Density ($\rho_s$) and calculate it using the following equation:

$ \rho_s = \displaystyle\frac{ M_s }{ V_s }$

$\rho_s$
Solid Density
$kg/m^3$
4944
$V_s$
Solid volume of a component
$m^3$
6038
$M_s$
Total Dry Mass of Sample
$kg$
5987
j_s = Ds / Dt Dp = R_t * J_V V_t = S * DL R_h =8* eta * abs( l )/( pi * r ^4) S = pi * R ^2 f = V_p / V_t Dp = rho_w * g * Dh j_s = J_V / S V_t = V_s + V_p rho_s = M_s / V_s R_pt = R_h / N_p R_t = N_s * R_pt N_s = DL / l N_p = f * S /( pi * r ^2) f = N_p * pi * r ^2/ S R_t = 8* eta * DL /( f * r ^2* S )lrDsj_sgDhR_hrho_wN_pN_spiV_pfSrho_sV_sDtM_sR_tR_ptV_tDLRDpetaJ_V

ID:(15073, 0)