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Hydraulic elements in series (2)
Storyboard 
When hydraulic elements are connected in series, the flow remains constant, but each hydraulic element experiences a pressure drop. The sum of these pressure drops equals the total drop, and therefore, the total hydraulic resistance is equal to the sum of all individual hydraulic resistances. On the other hand, the inverse of the total hydraulic conductivity is equal to the sum of the inverses of the hydraulic conductivities.
ID:(2105, 0)
Hydraulic resistance of elements in series (2)
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:
| $ \Delta p_2 = R_{h2} J_V $ |
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:(15953, 0)
Hydraulic conductance of elements in series (2)
Concept
In the case of a sum where the elements are connected in series, the total hydraulic conductance of the system is calculated by adding the individual hydraulic conductances of each element.
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
| $ R_{h2} = \displaystyle\frac{1}{ G_{h2} }$ |
leads to the total Series Hydraulic Conductance ($G_{st}$) can be calculated with:
| $\displaystyle\frac{1}{ G_{st} }=\displaystyle\sum_k\displaystyle\frac{1}{ G_{hk} }$ |
$\Delta p_k = \displaystyle\frac{J_{Vk}}{K_{hk}}$
So, the sum of the inverse of the hydraulic conductance in a network ($G_{hk}$) will be equal to the inverse of the total Series Hydraulic Conductance ($G_{st}$).
ID:(15951, 0)
Darcy's law and hydraulic resistance (1)
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_1 = R_{h1} 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, 1)
Darcy's law and hydraulic resistance (2)
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_2 = R_{h2} 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, 2)
Darcy's law and hydraulic resistance (3)
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_t = R_{st} 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, 3)
Darcy's law and hydraulic conductance (1)
Equation
With the introduction of the hydraulic conductance ($G_h$), we can rewrite the Hagen-Poiseuille equation with the pressure difference ($\Delta p$) and the volume flow ($J_V$) using the following equation:
| $ J_V = G_{h1} \Delta p_1 $ |
| $ J_V = G_h \Delta p $ |
If we examine the Hagen-Poiseuille law, which allows us to calculate the volume flow ($J_V$) from the tube radius ($R$), the viscosity ($\eta$), the tube length ($\Delta L$), and the pressure difference ($\Delta p$):
| $ J_V =-\displaystyle\frac{ \pi R ^4}{8 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
we can introduce the hydraulic conductance ($G_h$), defined in terms of the tube length ($\Delta L$), the tube radius ($R$), and the viscosity ($\eta$), as follows:
| $ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$ |
to arrive at:
| $ J_V = G_h \Delta p $ |
ID:(14471, 1)
Darcy's law and hydraulic conductance (2)
Equation
With the introduction of the hydraulic conductance ($G_h$), we can rewrite the Hagen-Poiseuille equation with the pressure difference ($\Delta p$) and the volume flow ($J_V$) using the following equation:
| $ J_V = G_{h2} \Delta p_2 $ |
| $ J_V = G_h \Delta p $ |
If we examine the Hagen-Poiseuille law, which allows us to calculate the volume flow ($J_V$) from the tube radius ($R$), the viscosity ($\eta$), the tube length ($\Delta L$), and the pressure difference ($\Delta p$):
| $ J_V =-\displaystyle\frac{ \pi R ^4}{8 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
we can introduce the hydraulic conductance ($G_h$), defined in terms of the tube length ($\Delta L$), the tube radius ($R$), and the viscosity ($\eta$), as follows:
| $ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$ |
to arrive at:
| $ J_V = G_h \Delta p $ |
ID:(14471, 2)
Darcy's law and hydraulic conductance (3)
Equation
With the introduction of the hydraulic conductance ($G_h$), we can rewrite the Hagen-Poiseuille equation with the pressure difference ($\Delta p$) and the volume flow ($J_V$) using the following equation:
| $ J_V = G_{st} \Delta p_t $ |
| $ J_V = G_h \Delta p $ |
If we examine the Hagen-Poiseuille law, which allows us to calculate the volume flow ($J_V$) from the tube radius ($R$), the viscosity ($\eta$), the tube length ($\Delta L$), and the pressure difference ($\Delta p$):
| $ J_V =-\displaystyle\frac{ \pi R ^4}{8 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
we can introduce the hydraulic conductance ($G_h$), defined in terms of the tube length ($\Delta L$), the tube radius ($R$), and the viscosity ($\eta$), as follows:
| $ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$ |
to arrive at:
| $ J_V = G_h \Delta p $ |
ID:(14471, 3)
Hydraulic resistance of a tube (1)
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_{h1} =\displaystyle\frac{8 \eta | \Delta L_1 | }{ \pi R_1 ^4}$ |
| $ 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}$ |
ID:(3629, 1)
Hydraulic resistance of a tube (2)
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_{h2} =\displaystyle\frac{8 \eta | \Delta L_2 | }{ \pi R_2 ^4}$ |
| $ 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}$ |
ID:(3629, 2)
Hydraulic conductance (1)
Equation
In the context of electrical resistance, there exists its inverse, known as electrical conductance. Similarly, what would be the hydraulic conductance ($G_h$) can be defined in terms of the hydraulic resistance ($R_h$) through the expression:
| $ R_{h1} = \displaystyle\frac{1}{ G_{h1} }$ |
| $ R_h = \displaystyle\frac{1}{ G_h }$ |
ID:(15092, 1)
Hydraulic conductance (2)
Equation
In the context of electrical resistance, there exists its inverse, known as electrical conductance. Similarly, what would be the hydraulic conductance ($G_h$) can be defined in terms of the hydraulic resistance ($R_h$) through the expression:
| $ R_{h2} = \displaystyle\frac{1}{ G_{h2} }$ |
| $ R_h = \displaystyle\frac{1}{ G_h }$ |
ID:(15092, 2)
Hydraulic conductance (3)
Equation
In the context of electrical resistance, there exists its inverse, known as electrical conductance. Similarly, what would be the hydraulic conductance ($G_h$) can be defined in terms of the hydraulic resistance ($R_h$) through the expression:
| $ R_{st} = \displaystyle\frac{1}{ G_{st} }$ |
| $ R_h = \displaystyle\frac{1}{ G_h }$ |
ID:(15092, 3)
Hydraulic Conductance of a Pipe (1)
Equation
With the tube radius ($R$), the viscosity ($\eta$) and the tube length ($\Delta L$) we have that a hydraulic conductance ($G_h$) is:
| $ G_{h1} =\displaystyle\frac{ \pi R_1 ^4}{8 \eta | \Delta L_1 | }$ |
| $ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$ |
ID:(15102, 1)
Hydraulic Conductance of a Pipe (2)
Equation
With the tube radius ($R$), the viscosity ($\eta$) and the tube length ($\Delta L$) we have that a hydraulic conductance ($G_h$) is:
| $ G_{h2} =\displaystyle\frac{ \pi R_2 ^4}{8 \eta | \Delta L_2 | }$ |
| $ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$ |
ID:(15102, 2)
Sum of resistors in series (2)
Equation
The series combination of the hydraulic Resistance 1 ($R_{h1}$) and the hydraulic Resistance 2 ($R_{h2}$) results in a total sum of the total hydraulic resistance in series ($R_{st}$):
| $ R_{st} = R_{h1} + R_{h2} $ |
ID:(3854, 0)
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 $ |
ID:(9943, 0)
Hydraulic conductance in series (2)
Equation
The series combination of the hydraulic conductance 1 ($G_{h1}$) and the hydraulic conductance 2 ($G_{h2}$) results in a total sum of the total Series Hydraulic Conductance ($G_{st}$):
| $\displaystyle\frac{1}{ G_{st} }=\displaystyle\frac{1}{ G_{h1} }+\displaystyle\frac{1}{ G_{h2} }$ |
ID:(3860, 0)