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Hydraulic elements in series and parallel
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:(2109, 0)
Hydraulic resistance of series and parallel elements
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_1 = R_{pt} J_{Vt} $ |
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:(15957, 0)
Model
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Equations
$ \Delta p_t = R_{st} J_{Vt} $
Dp = R_h * J_V
$ \Delta p_1 = R_{pt} J_{Vt} $
Dp = R_h * J_V
$ \Delta p_3 = R_{h3} J_{Vt} $
Dp = R_h * J_V
$ \Delta p_1 = R_{h1} J_{V1} $
Dp = R_h * J_V
$ \Delta p_1 = R_{h2} J_{V2} $
Dp = R_h * J_V
$ \Delta p_t = \Delta p_1 + \Delta p_3 $
Dp_t = Dp_1 + Dp_2
$ J_{Vt} = J_{V1} + J_{V2} $
J_Vt = J_V1 + J_V2
$ R_{st} = R_{pt} + R_{h3} $
R_st = R_h1 + R_h2
$\displaystyle\frac{1}{ R_{pt} }=\displaystyle\frac{1}{ R_{h1} }+\displaystyle\frac{1}{ R_{h2} }$
1/ R_pt =1/ R_h1 +1/ R_h2
ID:(15956, 0)
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_{pt} + R_{h3} $ |
| $ R_{st} = R_{h1} + R_{h2} $ |
ID:(3854, 0)
Sum of resistors in parallel (2)
Equation
The parallel combination of the hydraulic Resistance 1 ($R_{h1}$) and the hydraulic Resistance 2 ($R_{h2}$) results in a total equivalent of the total hydraulic resistance in series ($R_{st}$):
| $\displaystyle\frac{1}{ R_{pt} }=\displaystyle\frac{1}{ R_{h1} }+\displaystyle\frac{1}{ R_{h2} }$ |
ID:(3858, 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_3 $ |
| $ \Delta p_t = \Delta p_1 + \Delta p_2 $ |
ID:(9943, 0)
Total flow (2)
Equation
The flujo de Volumen Total ($J_{Vt}$) represents the total sum of the individual contributions from the volume flow 1 ($J_{V1}$) and the volume flow 2 ($J_{V2}$), from the elements connected in parallel:
| $ J_{Vt} = J_{V1} + J_{V2} $ |
ID:(12800, 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_t = R_{st} J_{Vt} $ |
| $ \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_1 = R_{pt} J_{Vt} $ |
| $ \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_3 = R_{h3} J_{Vt} $ |
| $ \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 resistance (4)
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_{V1} $ |
| $ \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, 4)
Darcy's law and hydraulic resistance (5)
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_{h2} J_{V2} $ |
| $ \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, 5)