Infinite Parallel-plate Channel

In microfluidics, the aspect ratio of a rectangular channel can often be so large that the channel is well approximated by an infinite parallel-plate configuration. Owing to large dimension of the channel in y-direction compared to that in z-direction, the y dependence drops out from equation (2.72) and the simplified governing equation is  

The solution of the governing equation (2.84) with the boundary conditions (equations (2.85) and (2.86)) is a simple parabola as 

The flow rate Q through a section of width w and height h can be expressed as 

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The influence of the Reynolds number on the velocity field of a backward-facing step is illustrated in Figure 2.1. The backward-facing step flow can be considered equivalent to two infinite parallel-plate channels with heights and /I2, and hydraulic resistances / j and R2 are... 

The dimensional analysis reported in the previous section does not include the influence of channel aspect ratio or size. We try to bring the effect of channel cross-section size in this section. In comparison to the example discussed in the last section, most systems are characterized by more than one length scale, which leads to a more involved Reynolds number analysis. As an example, consider a section of length L and width w of the infinite, parallel-plate channel with height h shown in Figure 2.5. The system is translation invariant... 

Figure 2.5 A sketch of an infinite, parallel-plate channel of height h in the x-z plane. The fluid is flowing in the j -direction because of a pressure drop AF over the section of length L...

In FFF, separation is determined by the combined action of the nonuniform flow profile and transverse field effects. The classical configuration assumes the FFF channel as two infinite parallel plates (see Figure 12.4), of which the accumulation wall lies at x=0, where x is the cross-channel axis (directed upward from the accumulation wall). Inside the channel, the carrier fluid, assumed to have a constant viscosity, has a velocity profile u(x) that takes the form... 

The infinite parallel plates construct may sound theoretical and impractical, but it is not. The flow in screw extruder channels, between the rotor and the wall of an internal mixer or between the rolls of calenders and roll-mills, to mention a few, can be considered to first approximation as taking place locally between parallel plates in relative motion. 

Comparing the present flow configuration to that in Example 2.5 of flow between two infinite parallel plates in relative motion, we note two important differences. First, the flow in the down-channel z direction is two-dimensional due to the stationary side walls created by the flight [i.e., vz (x, y), and the barrel surface has a velocity component in the x direction that results in a circulatory flow in the cross-channel direction. 

Flow in a thin rectangular channel (Figure 4.2), such as that used in field-flow fractionation, can be treated in a manner similar to that used for cylindrical capillary tubes. If the drag at the edges of the channel is neglected (infinite parallel plate model), then the force balance expression (corresponding to Eq. 4.5 for capillary tubes) becomes... 

The theoretical treatment above was based on the following assumptions (a) The channel is placed between infinite parallel plates, (b) the flow profile is parabolic, (c) a steady state concentration profile of the sample is established after action of the physical field, (d) uniform force of the physical field in the channel, and (e) absence of extraneous non-uniform forces. These assumptions are usu-... 

Field-flow fractionation experiments are mainly performed in a thin ribbonlike channel with tapered inlet and outlet ends (see Fig. 1). This simple geometry is advantageous for the exact and simple calculation of separation characteristics in FFF Theories of infinite parallel plates are often used to describe the behavior of analytes because the cross-sectional aspect ratio of the channel is usually large and, thus, the end effects can be neglected. This means that the flow velocity and concentration profiles are not dependent on the coordinate y. It has been shown that, under suitable conditions, the analytes move along the channel as steady-state zones. Then, equilibrium concentration profiles of analytes can be easily calculated. 

A field-flow fractionation (FFF) channel is normally ribbonlike. The ratio of its breadth b to width w is usually larger than 40. This was the reason to consider the 2D models adequate for the description of hydrodynamic and mass-transfer processes in FFF channels. The longitudinal flow was approximated by the equation for the flow between infinite parallel plates, and the influence of the side walls on mass-transfer of solute was neglected in the most of FFF models, starting with standard theory of Giddings and more complicated models based on the generalized dispersion theory [1]. The authors of Ref. 1 were probably the first to assume that the difference in the experimental peak widths and predictions of the theory may be due to the influence of the side walls. 

Quantitative analysis of the retention time, of a macrosolute species in the channel after sample injection may he carried out as follows (Giddings, 1991). The axial velocity profile, Vz(y), of a liquid of viscosity /r in a thin rectangular channel formed between two infinite parallel plates (Figure 7.3.21(a)) spaced a distance b apart is... 

Heat transfer in microsystems under pressure-driven flow conditions is considered here. Figure 9.8 shows two large infinite parallel plates, separated by distance, H, with uniform surface heat flux q. This configuration can be used to model flow and heat transfer in rectangular channels with large aspect ratio. Velocity and temperature are assumed to be fully developed. Inlet and outlet pressures are P,- and P , respectively. 

Calculation of the flow of a Newtonian liquid in a confining geometry may be mathematically carried out quite precisely for simplicity assuming flow between parallel infinite plates with a small gap, disregarding the thickness of the walls of the chaimel. Such an assumption is reasonable for shallow channels, when (h/b) < 10 which is generally met in practice. 

Very concise expressions can he developed for FFF if the following assumptions are made (1) the FFF channel can be modeled as the space between two parallel infinite plates, (2) the flow profile is parabolic, (3) a steady state concentration profile is established in the channel, and (4) the external field is imiform. With these assumptions, a simple mass balance across the cross section of the channel leads to the expression for the transversal flux Jx,... 

The assumption about the upstream conditions is crucial. If we assume instead that the fluid enters with parallel streamlines and that the axial temperature derivative at X = 0 is zero, we obtain the temperature map shown in Figure 8.14, which is very different qualitatively and quantitatively from the one in Figure 8.11. These boundary conditions are equivalent to assuming that the temperature profile is fully developed at the entrance (i.e., that the upstream channel is infinitely long), with the subsequent distortion in the final section caused by the recirculating flow at the exit. The computed temperature at the bottom plate at x = 0 is 196 °C, which is essentially the value obtained from Equation 3.34a with the parameters used here and in Chapter 3 if the heat transfer coefficient U is set to zero. (The limit corresponds to the case Bi oo and 11 0, with the product TlBi remaining finite.) This computation emphasizes the importance of the proper problem formulation the output from a computer simulation is only as meaningful as the input. 

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