3.2 Hydraulic circuit equivalents for fluid flow in microchannels [hydraulic circuits top]

The Hagen-Poiseuille relation is a good approximation of steady flow through long, narrow, rigid, circular microchannels, for which end effects can be ignored. However, microdevices rarely have circular microchannels–the fabrication techniques employed to make microchannels typically lead to rectangular, trapezoidal, or semicircular channel shapes. Thus, engineering application of hydraulic circuit analysis requires that we develop a more general model for the hydraulic resistance of a channel. Further, microchannels are often made of elastomers and are therefore not rigid. Even rigid channels often behave as if they were not rigid owing to bubbles trapped in the channel. Because of these factors, we define the hydraulic resistance and compliance of channels as a means for simplifying our analysis of these systems.

Hydraulic resistance The description above for acircular microchannel can be generalized (at least approximately) by replacing the tube radius R with the hydraulic radius r_h. This hydraulic radius is a tool for estimating the flow resistance of a channel without solving the fluid flow equations exactly. In this form, our 1D model of a microchannel has hydraulic resistance R_h approximated by

where r_h is the hydraulic radius of the channel, which is given by

where A is the cross-sectional area of the channel, andP  is the length of the perimeter of the channel. If the channel is circular, r_h is equal to the radius of the circle. Equation 3.8 is derived from a conservation of momentum analysis on a control volume (see Exercise  3.1) and, owing to the approximations used in its derivation, is exact only for infinitely long, circular channels. For finite channels or non-circular shapes, using hydraulic radius to predict hydraulic resistance is typically a good approximation (within 20%) of the correct result.

Electrokinetic coupling matrix The relations between hydraulic circuits and electrical circuits are discussed throughout this text, both (a) because the systems are analogous in isolation and (b) because fluid flow and current become coupled whenever we consider a system with electrically charged walls. We use an electrokinetic coupling equation, whichrelates fluid flow and current flow to applied pressure fields and electric fields. If we ignore the effects of charged surfaces, this electrokinetic coupling equationis given by

(3.10)

where Q and I are the volumetric flow rate [m³∕s] and current [A], A is the channel area [m²], σ is the electrical conductivity [S∕m], and E is the electric field in the x-direction [V∕m]. The first of these equations is the Hagen-Poiseuille law (Equation 3.6), while the second is Ohm’s law (Equation 5.61), to be discussed in Chapter 5.

Following Equation 3.10, theelectrokinetic coupling matrix χ is denoted as

(3.11)

This electrokinetic couplingmatrixbecomes useful when describing the coupling of charged surfaces (electroosmosis, electrokinetic pumping, streaming current, and streaming potential), as well as the way these phenomena change when the channels under discussion become small. At present, we focus on the Hagen-Poiseuille law, and use only the first equation in this coupling matrix.

Hydraulic capacitance: system compliance So far, we have assumed that all solid surfaces were infinitely rigid when we considered the fluid flow therein. If this is the case, then the control volume we use to analyze the tube is stationary, and the mass in the tube is constant with time. In this case, the Hagen-Poiseuille relation is a complete (albeit approximate) description of the flow-pressure relation in long narrow tubes. However, if the walls are not rigid, then the control volume we use to analyze the tube no longer has stationary boundaries, and pressure can cause the cross-sectional area of a microchannel to contract or expand. Thus the mass (or volume) inside the tube is no longer constant with time. Consider, for example, a small diameter tube made out of the same material used to make a balloon. If we apply a low pressure to one end, the tube will not expand, and the flow-pressure relation is given by the Hagen-Poiseuille law. If we apply a high pressure to one end, the tube expands to a larger diameter, and the governing flow-pressure relation for the tube is the Hagen-Poiseuille law, but with a larger channel radius. If we apply a temporally varying pressure, the Hagen-Poiseuille law is insufficient to describe the pressure-flow relation in the tube, because the tube volume varies with time. We consider the conservation of mass relation for a control volume, Equation ??:

(3.12)

For incompressible, unidirectional flow, we can draw a simple control volume (Figure 3.3) and simplify this relation to

(3.13)

If the volume of a tube changes, the time change in the tube volume is equal to the difference in the inlet and outlet volumetric flowrates. For a rigid tube, V = 0 and Q_inlet = Q_outlet.

We describe the tendency of the liquid volume contained within a channel to contract under pressure as itscompliance or hydraulic capacitance C_h [m³∕Pa]. The compliance of a system is given by the decrease in system volume V per change in pressure.

and the volumetric flow rate stored by a compliant system is given by Q = -C_h. Here, the system refers to the working fluid, typically an aqueous solution, and the compliance measures the compressibility of the system. By substituting in the definition of compliance, we can see that this relation gives Q = -C_h = = , which recapitulates the conservation of mass relation in Equation 3.12. The compliance is related to both the physical rigidity of the liquid as well as the physical rigidity of its encapsulating material. Two effects are important. First, most materials are reduced in volume when their pressure increases. This leads to a positive contribution to compliance. Second, an increase in pressure pushes out on the material that is encapsulating the fluid, and thus the volume available for that fluid to fill increases. This second factor leads to a negative contribution to compliance. In microfluidic systems with water as the working fluid, the compression of the water itself is usually safely ignored. However, the flexibility of the material encapsulating the fluid is widely variable depending on the microdevice material. Water in an infinitely rigid channel has low positive compliance, because the water is hard to compress and the channel is impossible to expand. If our working fluid were air in an infinitely rigid channel, it would have a high positive compliance, because air is rather easy to compress. Water enclosed within a flexible polymer channel has a large negative compliance, because the compliant channel inflates like a balloon when the pressure is increased, so the volume taken up by the water actually expands when the pressure is increased. Also, if a liquid system has bubbles, the bubbles play the role of a flexible encapsulating material, and from the standpoint of water flow, a microdevice with bubbles inside it has large negative compliance. Overall, then, incompressible fluid systems (like the aqueous systems we are concerned with) generally have zero or negative compliance—the fluid volume expands when the pressure increases.

As mentioned earlier, the most compliant component of a microdevice is usually any bubbles that have accidentally worked their way into the system, owing to the compliant nature of an ideal gas. The compliance of rigid microchannel substrates can often be ignored (for example, for glass, silicon, Zeonor, polycarbonate, and plexiglas). However, the compliance of PDMS (polydimethylsiloxane) can typically not be ignored. Figure 3.4 illustrates the mechanical compliance of rigid and non-rigid walls, and the effect of bubbles in these systems.

We typically represent compliance with a hydraulic capacitor, with symbol and pressure-flowrate relation shown in Figure 3.5. A compliant channel is represented as a hydraulic resistor and hydraulic capacitor in parallel.

3.2.1 Analytic representation of sinusoidal pressures and flowrates

If the time dependence of our pressures or flowrates are sinusoidal, the relations between the pressure/flowrate, their integrals, and their derivatives take on a special form. These harmonic functions (sines and cosines) are also closely related to complex exponentials. Because of this, we usethe analytic or complex representation of real quantities like pressure and flowrate, discussed in detail in Section G.3 in the Appendix.

3.2.2 Hydraulic impedance

The impedanceof a hydraulic element is a complex quantity that extends the Hagen-Poiseuille law (Δp = QR_h) to handle sinusoidally time-varying fluid flows, as long as we assume that the flow is always well approximated by the steady-state, infinite-tube Poiseuille pressure-flowrate relationship. We can write sinusoidal pressures and flowrates as follows (note Re means the real part of):

(3.18)

(3.19)

In this case, we have defined the analytic representations  and  of the pressure and flowrate, respectively, using phasors p₀ and Q₀. With these definitions, for circuit elements we can define a hydraulic impedance , which describes the voltage-current relationship through the complex equation = or phasor equation p₀ = ₀. Each hydraulic circuit element has a complex impedance corresponding to its circuit properties:

• hydraulic resistors:

• hydraulic capacitors:

3.2.3 Hydraulic circuit relations

The Hagen-Poiseuille law and the relations for hydraulic impedance describe the volumetric flowrate through an element or the pressure drop across that element. For a circuit that is composed of a network of these elements, we need a relation that links these components. That relation is conservation of mass. Conservation of mass requires that the net volumetric flow into or out of a node is zero (Figure 3.6):

where Q in this case is defined positive into the node. This, combined with the Hagen-Poiseuille law and hydraulic impedance, prescribes the approximate flow solution with a system of algebraic equations.

3.2.4 Series and parallel component rules

Conservation of mass is all that is needed to calculate the flowrate and pressures in a general network of circuit elements. In a number of special cases, though, we have developed rules of thumb by applying conservation of mass to fundamental network components. In particular, the results of conservation of mass for parallel and series circuits are shown below.

Series hydraulic circuit rules The hydraulic resistance of two hydraulic resistors in series is equal to the sum of the hydraulic resistances:

the reciprocal of the compliance of two hydraulic capacitors in series is equal to the sum of the reciprocals of the compliances:

and the hydraulic impedance of two hydraulic impedances in series is equal to the sum of the hydraulic impedances:

Series component relations for hydraulic circuits are depicted in Figure 3.7.

Parallel hydraulic circuit rules The reciprocal of the hydraulic resistance of two hydraulic resistors in parallel is equal to the sum of the reciprocals of the hydraulic resistances:

the compliance of two hydraulic capacitors in parallel is equal to the sum of the compliances:

and the reciprocal of the hydraulic impedance of two hydraulic impedances in parallel is equal to the sum of the reciprocals of the hydraulic impedances:

Parallel component relations are depicted in Figure 3.8.

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