Cornell University - Visit www.cornell.edu Kirby Research Group at Cornell: Microfluidics and Nanofluidics : - Home College of Engineering - visit www.engr.cornell.edu Cornell University - Visit www.cornell.edu
Cornell University, College of Engineering Search Cornell
News Contact Info Login

Donations keep this resource free! Give here:

Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.

[Return to Table of Contents]


Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

3.3 Solution techniques [hydraulic circuits top]

Hydraulic circuit systems lead to algebraic systems of equations that can be solved directly using symbolic or matrix manipulation software. Given the hydraulic impedances of the channels, the unknowns are the pressures at the N nodes, and the flowrates in the M channels. The N equations from conservation of mass at each node and M equations from theHagen-Poiseuille law for each channel define the system. Thus, a N +M×N +M matrix equation can be written and inverted to obtain the solution.

VIDEO: Matrix formalism for solving the Hagen-Poiseuille law for channel networks.

If the channels are all purely resistive, then the pressure and flowrate phasors are real. If the channels include compliance, then the pressure and flowrate phasors are complex and the angle of the complex numbers describes the phase lag of the response.

__________________________________________________________________________________________________________________________________________________________
EXAMPLE PROBLEM 3.2: Calculate the hydraulic radius of

  1. a glass channel made using photoresist lithographically patterned to width w = 240 μm and etched isotropically in HF to a depth of d = 30 μm.
  2. a channel cast in polydimethylsiloxane made by patterning SU-8 molds of width w = 40 μm and depth d = 30 μm and casting the PDMS over the SU-8 mold.

Note that isotropic etches lead to microchannel cross sections consisting of circular side walls and flat top and bottom surfaces, while casting over SU-8 makes roughly rectangular channel cross-sections (see Figure 3.9).


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 3.9: Shapes of channels based on (a) wet etching of glass using HF and (b) casting over an SU-8 template.


Solution: Note that for circular channels, rh = r. For infinitely flat plates separated by d, rh = d. The rectangular section has a hydraulic radius approximately equal to half the length of the sides, while the isotropically etched channel, because it is wide, starts to approach rh = d.

HF etch:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

SU-8 mold:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

.

__________________________________________________________________________________________________________________________________________________________

__________________________________________________________________________________________________________________________________________________________
EXAMPLE PROBLEM 3.3: 

Consider the fluid circuit shown in Figure 3.10. Write the 7 equations in the unknowns p1, p2, p3, p4, Q1, Q2, Q3. Three of these equations are given by the problem statement, one is conservation of mass, and three are Hagen-Poiseuille relations. Write these equations as a matrix equation. Invert the matrix to solve the system for the 7 unknowns.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 3.10: A simple fluid circuit. All channels are circular in cross-section with radius R = 20 μm. L1 = 0.5 cm, L2 = 1 cm, and L3 = 2 cm. Pressures at the ports are p1 = 0.11 MPa, p3 = 0.1 MPa, and p4 = 0.1 MPa. The fluid is water, with viscosity 1×10-3 Pa s.


Solution: First we must solve for the hydraulic resistance of the three channels, using Rh = 8ηL∕πR4. This gives hydraulic resistances of 7.96×1013, 1.59×1014, and 3.18×1014 Ns m5 for the three channels.

The seven equations are
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(3.31)

microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(3.32)

microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(3.33)

microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(3.34)

microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(3.35)

microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(3.36)

and
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(3.37)

In matrix form, this is
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(3.38)

The solution is p2 = 0.1057 MPa, Q1 = 54×10-12 m3s, Q2 = 36×10-12 m3s, and Q3 = 18×10-12 m3s.

__________________________________________________________________________________________________________________________________________________________

[Return to Table of Contents]



Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.


Ad revenue from these pages is used to support student research. The presence of an advertisement on these pages does not constitute an endorsement by the Kirby Research Group or Cornell University.

Donations keep this resource free! Give here: