Laminar flows continued pressure drop

At the small capillary numbers associated by most microfluidic applications, the pressure drop over a bubble is significant with respect to the viscous losses in the continuous liquid. For single-phase laminar flow, velocity and pressure drop are linearly related and the pressure drop is proportional to the length Az of the channel. For segmented flow, the pressure drop also contains a term that depends on n, the number of segments (i.e. bubbles or droplets) in that channel. For round channels, the two terms are... 

The opportunity to measure the dilute polymer solution viscosity in GPC came with the continuous capillary-type viscometers (single capillary or differential multicapillary detectors) coupled to the traditional chromatographic system before or after a concentration detector in series (see the entry Viscometric Detection in GPC-SEC). Because liquid continuously flows through the capillary tube, the detected pressure drop across the capillary provides the measure for the fluid viscosity according to the Poiseuille s equation for laminar flow of incompressible liquids [1], Most commercial on-line viscometers provide either relative or specific viscosities measured continuously across the entire polymer peak. These measurements produce a viscometry elution profile (chromatogram). Combined with a concentration-detector chromatogram (the concentration versus retention volume elution curve), this profile allows one to calculate the instantaneous intrinsic viscosity [17] of a polymer solution at each data point i (time slice) of a polymer distribution. Thus, if the differential refractometer is used as a concentration detector, then for each sample slice i. 

The sixth reactor design criterion requires that the pressure drop at the minimum residence time be less than 100 psi. For a small diameter channel, the flow through that channel wiU be laminar for all flow rates of interest for this particular applicahon. Neglecting end effects, the solutions to the equations of continuity and of motion for steady-state laminar flow of an incompressible Newtonian fluid are well-known, yielding a parabolic velocity distribution and the Hagen-Poiseuille equahon for pressure drop, as given in Eqs. (9) and (10) ... 

For nonisothermal systems a general differential equation of conservation of energy will be considered in Chapter 5. Also in Chapter 7 a general differential equation of continuity for a binary mixture will be derived. The differential-momentum-balance equation to be derived is based on Newton s second law and allows us to determine the way velocity varies with position and time and the pressure drop in laminar flow. The equation of momentum balance can be used for turbulent flow with certain modifications. 

Figure 10.24. Plots of the pressure drop against the flow velocity in a capillary in the laminar and turbulent flow regions for water (continuous lines) and for a drag-reducing surfactant solution (750 ppm (C TABr-f NaSal) at 27.5°C) (dashed lines)...

In both laminar and turbulent flow it is assumed that the mixture is pseudoho-mogeneous with respect to density that is, a volume average density is used. The viscosity of the continuous phase is used. Note that for very small dispersed phase drops or bubbles (under 10 p,m), the viscosity may even be higher and non-Newtonian, such as in foams and emulsions. In such cases direct measurements are required of a well-dispersed sample. Avoid correlations that average viscosities. For gas-liquid systems, the method of Lockhart and Martinelli (see Govier and Aziz, 1972) for turbulent flow is very successful and more accurate then the pseudohomogeneous method. The pressure drop for each phase flowing alone is calculated. The liquid/gas pressure drop ratio is made. This is used with an empirical correlation to get an enhancement factor for the liquid-alone pressure drop,... 

Fluid flow in small devices acts differently from those in macroscopic scale. The Reynolds number (Re) is the most often mentioned dimensionless number in fluid mechanics. The Re number, defined by pUL/jj, represents the ratio of inertial forces to viscous ones. In most circumstances involved in micro- and nanofluidics, the Re number is at least one order of magnitude smaller than unity, ruling out any turbulence flows in micro/nanochannels. Inertial force plays an insignificant role in microfluidics, and as systems continue to scale down, it will become even less important. For such small Re number flows, the convective term pu Vm) of Navier-Stokes equations can be dropped. Without this nonlinear convection, simple micro/nanofluidic systems have laminar, deterministic flow patterns. They have parabolic velocity profile in pressure-driven flows, plug-like velocity profile in elec-froosmotic flows, or a superposition of both. One of the benefits from the low Re number flow is that genomic material can be transported easily without shearing in Lab-... 

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