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Total Pressure difference of series resistors (2)
Equation 
In the case of hydraulic resistances in series, the pressure drops across each of them, and the sum of these pressure drops is equal to the total pressure difference across the entire series.
For two resistances in series, the hydraulic Resistance 1 ($R_{h1}$) and the hydraulic Resistance 2 ($R_{h2}$), with their respective pressure drops the pressure Difference 1 ($\Delta p_1$) and the pressure Difference 2 ($\Delta p_2$), the sum of these drops equals the total pressure difference the total pressure difference ($\Delta p_t$):
 $ \Delta p_t = \Delta p_1 + \Delta p_2 $ |
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Average speed in the Section
Concept
A flow through a section travels with a speed that can vary over it. However, an average speed can be defined simply by considering the total flow through the section.
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Liquid or Gas Flow
Concept
The flow of a liquid or gas corresponds to the volume of this flowing through a section in a given time.
The units in which it is measured is in unit of volume per unit of time such as in cubic meters per second or liters per minute.
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Mean volume flow
Equation
The volume flow ($J_V$) corresponds to the volume flowing ($\Delta V$) flowing through the channel at the time elapsed ($\Delta t$). Therefore, we have:
| $ J_V =\displaystyle\frac{ \Delta V }{ \Delta t }$ |
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Flow density
Equation
If you have a total flow
| $ j_V =\displaystyle\frac{ J_V }{ S }$ |
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Simulador
Php
El siguiente simulador logra modelar lo que es el flujo de sangre por el sistema circulatorio.
Las curvas finales muestran como se distribuyen los radios, largos, numero de vasos, como va cayendo la presión desde la sístole a la dístole y el flujo que se observa si se tiene una herida según el vaso.
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Darcy's law and hydraulic resistance
Equation
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_h J_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)
Hydraulic resistance of a tube
Equation
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 | \Delta L | }{ \pi R ^4}$ |
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}$ |
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Hydraulic resistance of elements in series
Equation
When there are multiple hydraulic resistances connected in series, we can calculate the total hydraulic resistance in series ($R_{st}$) by summing the hydraulic resistance in a network ($R_{hk}$), as expressed in the following formula:
| $ R_{st} =\displaystyle\sum_k R_{hk} $ |
One way to model a tube with varying cross-section is to divide it into sections with constant radius and then sum the hydraulic resistances in series. Suppose we have a series of the hydraulic resistance in a network ($R_{hk}$), which depends on the viscosity ($\eta$), the cylinder k radio ($R_k$), and the tube k length ($\Delta L_k$) via the following equation:
| $ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$ |
In each segment, there will be a pressure difference in a network ($\Delta p_k$) with the hydraulic resistance in a network ($R_{hk}$) and the volume flow ($J_V$) to which Darcy's Law is applied:
the total pressure difference ($\Delta p_t$) will be equal to the sum of the individual pressure difference in a network ($\Delta p_k$):
| $ \Delta p_t =\displaystyle\sum_k \Delta p_k $ |
therefore,
$\Delta p_t=\displaystyle\sum_k \Delta p_k=\displaystyle\sum_k (R_{hk}J_V)=\left(\displaystyle\sum_k R_{hk}\right)J_V\equiv R_{st}J_V$
Thus, the system can be modeled as a single conduit with the hydraulic resistance calculated as the sum of the individual components:
| $ R_{st} =\displaystyle\sum_k R_{hk} $ |
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Cylindrical Tube
Condition
One type of Borders is for example a cylindrical tube of a given radius. This can be constant or vary throughout this.
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Hydraulic Resistance
Concept
The Viscosity of a fluid causes it to resist flowing under a pressure difference. This occurs in particular in the presence of a Borders that leads to the fluid canceling its velocity on its surface.
Resistance means loss of energy that corresponds to the kinetic velocity that is lost when the fluid stops at the surface of the edges of the system.
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Hydraulic resistance of elements in series
Concept
In the case of a sum where the elements are connected in series, the total hydraulic resistance of the system is calculated by summing the individual resistances of each element.
One way to model a tube with varying cross-section is to divide it into sections with constant radius and then sum the hydraulic resistances in series. Suppose we have a series of the hydraulic resistance in a network ($R_{hk}$), which depends on the viscosity ($\eta$), the cylinder k radio ($R_k$), and the tube k length ($\Delta L_k$) via the following equation:
| $ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$ |
In each segment, there will be a pressure difference in a network ($\Delta p_k$) with the hydraulic resistance in a network ($R_{hk}$) and the volume flow ($J_V$) to which Darcy's Law is applied:
the total pressure difference ($\Delta p_t$) will be equal to the sum of the individual pressure difference in a network ($\Delta p_k$):
| $ \Delta p_t =\displaystyle\sum_k \Delta p_k $ |
therefore,
$\Delta p_t=\displaystyle\sum_k \Delta p_k=\displaystyle\sum_k (R_{hk}J_V)=\left(\displaystyle\sum_k R_{hk}\right)J_V\equiv R_{st}J_V$
Thus, the system can be modeled as a single conduit with the hydraulic resistance calculated as the sum of the individual components:
| $ R_{st} =\displaystyle\sum_k R_{hk} $ |
ID:(3630, 0)
Viscosity
Concept
Viscosity can be understood as the tendency of the fluid to redistribute momentum and its corresponding velocity.
In a high viscosity liquid, a high speed zone is slowed down by dragging the liquid from surrounding areas with a low speed that therefore gains speed.
In a low viscosity liquid a high speed zone is not affected mostly by lower speed zones by displacing these and continuing the flow without further speed reduction.
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Pressure difference
Equation
When two liquid columns are connected with the pressure in column 1 ($p_1$) and the pressure in column 2 ($p_2$), a the pressure difference ($\Delta p$) is formed, which is calculated according to the following formula:
| $ \Delta p = p_2 - p_1 $ |
the pressure difference ($\Delta p$) represents the pressure difference that will cause the liquid to flow from the taller column to the shorter one.
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Parallel hydraulic conductivity
Concept
If we have three hydraulic resistances $R_{h1}$, $R_{h2}$, and $R_{h3}$, the series sum of the resistances will be:
| $ K_{pt} = \displaystyle\sum_k K_{hk}$ |
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Hydraulic Resistance in Series (N)
Equation
If you have
| $ R_{st} = N R_h $ |
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Hydraulic resistance of parallel elements
Equation
The total hydraulic resistance in parallel ($R_{pt}$) can be calculated as the inverse of the sum of the hydraulic resistance in a network ($R_{hk}$):
| $\displaystyle\frac{1}{ R_{pt} }=\sum_k\displaystyle\frac{1}{ R_{hk} }$ |
The parallel total hydraulic conductance ($G_{pt}$) combined with the hydraulic conductance in a network ($G_{hk}$) in
| $ G_{pt} =\displaystyle\sum_k G_{hk} $ |
and along with the hydraulic resistance in a network ($R_{hk}$) and the equation
leads to the total hydraulic resistance in parallel ($R_{pt}$) via
| $\displaystyle\frac{1}{ R_{pt} }=\sum_k\displaystyle\frac{1}{ R_{hk} }$ |
ID:(3181, 0)
Hydraulic Resistance in Parallel (N)
Equation
If you have
| $ R_{pt} =\displaystyle\frac{1}{ N } R_h $ |
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Hydraulic conductancia of elements in series
Equation
In the case of hydraulic resistances in series, the inverse of the total Series Hydraulic Conductance ($G_{st}$) is calculated by summing the inverses of each the hydraulic conductance in a network ($G_{hk}$):
| $\displaystyle\frac{1}{ G_{st} }=\displaystyle\sum_k\displaystyle\frac{1}{ G_{hk} }$ |
The total hydraulic resistance in series ($R_{st}$), along with the hydraulic resistance in a network ($R_{hk}$), in
| $ R_{st} =\displaystyle\sum_k R_{hk} $ |
and along with the hydraulic conductance in a network ($G_{hk}$) and the equation
leads to the total Series Hydraulic Conductance ($G_{st}$) can be calculated with:
| $\displaystyle\frac{1}{ G_{st} }=\displaystyle\sum_k\displaystyle\frac{1}{ G_{hk} }$ |
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Hydraulic conductance of elements in parallel
Equation
The parallel total hydraulic conductance ($G_{pt}$) is calculated with the sum of the hydraulic conductance in a network ($G_{hk}$):
| $ G_{pt} =\displaystyle\sum_k G_{hk} $ |
With the total flow ($J_{Vt}$) being equal to the volume flow in a network ($J_{Vk}$):
| $ J_{Vt} =\displaystyle\sum_k J_{Vk} $ |
and with the pressure difference ($\Delta p$) and the hydraulic conductance in a network ($G_{hk}$), along with the equation
for each element, it leads us to the conclusion that with the parallel total hydraulic conductance ($G_{pt}$),
$J_{Vt}=\displaystyle\sum_k J_{Vk} = \displaystyle\sum_k G_{hk}\Delta p = G_{pt}\Delta p$
we have
| $ G_{pt} =\displaystyle\sum_k G_{hk} $ |
.
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