User:PMarmottant/Hydraulic resistance and capacity

We present here simple tools to compute the flow in complex network of channels, just knowing the applied pressure.

Hydrodynamic resistance

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We have seen in the previous chapter that flow rate   in a channel is proportional to the applied pressure drop  . This can be summarized in

 

with   the hydrodynamic resistance. This expression is formally the analog of the electrokinetic law between voltage difference and current,  .

The expression for the hydraulic resistance is:

  • channel of circular cross-section (total length  , radius  ):
 
  • rectangular cross-section (width   and height  )
 

In a network of channels, equivalent resistances can be computed (as in electrokinetics):

  • two channels in series have a resistance  ,
  • two channels in parallel have a resistance  

These laws provide useful tools for the design of complex networks. Actually Kirchhoff's laws for electric circuits apply, being modified in:

  • the sum of flow rates on a node of the circuit is zero
  • the sum of pressure differences on a loop is zero

Hydrodynamic capacitance

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The volume of fluid in a channel can change just because of a change in pressure: this is due either to fluid compressibility or either channel elasticity. This behavior can be summarized with

 

with   the hydrodynamic capacitance. It is the microfluidic analog of the electrokinetic law  .

Compressible fluid in a container

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A pressure increase can compress the fluid in a container. The compressibility is measured by

 

For water its value is   which is usually negligible since pressure are usually less than a bar. For air it is   which considerable if pressure attain a bar.

The flow rate entering a tube of volume  , because of fluid compression   is:

 

The hydrodynamic capacitance is therefore:

 

Elastic tubes

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We define the tube dilatability as

 

It has a positive sign, since the tube volume increases with pressure.

The tube dilatability is approximately the inverse of the Young modulus   The following table gives order of magnitude of this dilatability for different materials

 
steel  /bar
plastic  /bar
rubber  /bar

This value can be interpreted in the following way: if the pressure is increased by 1 bar the relative volume increase is  

Assuming a uniform pressure in the tube (which is not true in long tube where pressure decreases subtantially) , we find a flow rate entering the tube to inflate to be

 

The hydrodynamic capacitance is therefore

 

Modelisation of an elastic long tube with a substantial pressure drop

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We consider a tube of length   on which a pressure difference   is applied. In the tube, the pressure decreases along the tube coordinate   as  . The dilatation is therefore not homogeneous: larger near the entrance. The volume increase of the tube (compared to the rest situation at pressure  ) is

 

We have integrated the inflation of small volumes  .

 
Equivalent circuit to a long elastic tube

We obtain that the flow due to dilatation is

 

meaning that only half the pressure difference loads the volume capacitor. The capacitor is placed in the middle of the channel, where the overpressure is half, see figure.


Application: syringe injection in a microchannel

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The syringe is has a tube diameter   and a volume  , while the (cylindrical) microchannel has a diameter   and a volume  , and a length  . The resistance of the microchannel is much larger than that of the syringe  . However the capacitance of the syringe is much larger  , because of the larger volume.

The equivalent circuit is therefore

 
Equivalent circuit to a long elastic tube

The total flow is distributed in the microchannel branch and the capacitor branch:

 

If the piston is suddenly started, initially water or tube elasticity will absorb the flow, and the flow is stationary only for time larger tha a characteristic transient time

 

with   the compressibility of either water or the syringe tube.

As an example, we take a microchannel of radius 10 micrometers, length 1 cm and a syringe of volume 1cc: the characteristic time is 10 seconds, if the syringe is rigid (glass) and  , while it takes up to 1000 seconds if the syringe is in plastic  !

As a conclusion, for practical realization of microfluidic networks:

  • avoid flexible tubes and prefer metallic tubes for a faster equilibration
  • avoid flexible glues in contact with the liquid: they will compress
  • avoid bubbles in the system, their compressibility is extremely high compared to plastic!
  • impose pressure with a valve, instead of piston velocity: the pressure equilibrates at the speed of sound in the liquid and changes in pressure are very rapidly applied to the whole system.