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. 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.

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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).

Figure 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, r_h = r. For infinitely flat plates separated by d, r_h = 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 r_h = d.

HF etch:

SU-8 mold:

.

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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.

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

The seven equations are

(3.31)

(3.32)

(3.33)

(3.34)

(3.35)

(3.36)

and

(3.37)

In matrix form, this is

(3.38)

The solution is p₂ = 0.1057 MPa, Q₁ = 54×10^-12 m³∕s, Q₂ = 36×10^-12 m³∕s, and Q₃ = 18×10^-12 m³∕s.

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