Flow in Capillaries

It is instructive to consider steady fluid flow (sometimes called Poiseuille flow) in a thin capillary tube. This example has many purposes it provides (1) a model flow calculation, (2) an illustration of how velocity profiles arise, (3) an explanation of the nature of flow in capillary chromatography, and (4) a foundation for capillary flow models of packed beds. 

Flow in a capillary can be maintained by a steady pressure difference Ap applied between inlet and outlet ends. We assume gravitational (and other external) forces to be negligible (true for a horizontal tube or for any tube with a large Ap). With the application of Ap, the fluid in the tube accelerates to a flowrate at which the viscous drag forces balance the applied pressure forces. For thin tubes the Newtonian acceleration forces are significant for only a brief moment before steady flow is achieved. 

By ruling out gravitational and acceleration forces, we are left with a simple balance pressure acting against viscous forces. By considering a symmetric cylindrical tube, we have a geometry so simple that this balance can be easily formulated and the flow equations readily solved. 

This equation yields a shear rate at the wall of magnitude 

This is a differential equation whose variables v and r can be separated to give 

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Most processing methods involve flow in capillary or rectangular sections, which may be uniform or tapered. Therefore the approach taken here will be to develop first the theory for Newtonian flow in these channels and then when the Non-Newtonian case is considered it may be seen that the steps in the analysis are identical although the mathematics is a little more complex. At the end of the chapter a selection of processing situations are analysed quantitatively to illustrate the use of the theory. It must be stressed however, that even the more complex analysis introduced in this chapter will not give precisely accurate... 

Calame JP, Myers RE, Binari SC, Wood FN, Garven M (2007) Experimental investigation of micro-channel coolers for the high heat flux thermal management of GaN-on-SiC semiconductor devices. Int J Heat Mass Transfer 50 4767-4779 Celata GP, Cumo M, Zummo G (2004) Thermal-hydraulic characteristics of single- phase flow in capillary pipes. Exp Thermal Fluid Sci 28 87-95 Celata GP (2004). Heat transfer and fluid flow in micro-channels. Begell House, N.Y. 

This chapter has the following structure in Sect. 3.2 the common characteristics of experiments are discussed. Conditions that are needed for proper comparison of experimental and theoretical results are formulated in Sect. 3.3. In Sect. 3.4 the data of flow of incompressible fluids in smooth and rough micro-channels are discussed. Section 3.5 deals with gas flows. The data on transition from laminar to turbulent flow are presented in Sect. 3.6. Effect of measurement accuracy is estimated in Sect. 3.7. A discussion on the flow in capillary tubes is given in Sect. 3.8. 

Celata GP, Gumo M, Zummo G (2004) Thermal-hydraulic characteristics of single-phase flow in capillary pipes. Exp Thermal Fluid Sci 28 87-95... 

Barajas AM, Panton RL (1993) The effect of contact angle on two-phase flow in capillary tubes. Int J Multiphase Flow 19 337-346... 

Burns, J. R., Ramshaw, C., The intensification of rapid reactions in multiphase systems using slug flow in capillaries. Lab. Chip 1 (2001) 10-15. 

Takayama S, McDonald JC, Ostuni E, Liang MN, Kenis PJA, Ismagilov RF, Whitesides GM (1999) Patterning cells and their environments using multiple laminar fluid flows in capillary networks. Proc Natl Acad Sci USA 96 5545-5548... 

FIGU RE 7.12 Representation of the diffuse double layer responsible for electroosmotic flow in capillary electrophoresis. 

Effects of buffer composition on electroosmotic flow in capillary electrophoresis. /. Microcol. Sep. 2, 176-180. 

Rapp, E., and Tallarek, U. (2003). Liquid flow in capillary electrochromatography generation and control of micro- and nanoliter volumes.. Sep. Set. 26, 453-470. 

The lack of experimental data impose difficulties for modelling the processes of low-pressure moulding of thermoplastics. From this point of view, it is of interest to refer to 85> containing a wide scope of experimental material. The role played by energy dissipation as applied to flow in capillaries of viscosimeters was studied in 86>. To check the predictions of theory and to elucidate the applicability of one or another plastication unit, we have measured the pressure dynamics in the course of mould filling. Theory gives the following expression for pressure as a function of time at the head of an extrusion plasticator ... 

Thulasidas TC, Abraham MA, Cerro RL. Axial dispersion of bubble-train flow in capillaries. Chem Eng Sci 1996. 

Inputting solid particles at fixed positions, of different sizes simulates a solid phase in the fluid lattice (Fig. 4). The number of fluid particles per node and their interaction law (collisions) affect the physical properties of real fluid such as viscosity. Particle movements are divided into the so called propagation step (spatial shift) and collisions. Not all particles take part in the collisions. It strongly depends on their current positions on the lattice in a certain LGA time step. In order to avoid an additional spurious conservation law [13], a minimum of two- and three-body collisions (FHP1 rule) is necessary to conserve mass and momentum along each lattice line. Collision rules FHP2 (22 collisions) and FHP5 (12 collisions) have been used for most of the previous analyses [1],[2],[14], since the reproduction of moisture flow in capillaries, in comparison to the results from NMR tests [3], is then the most realistic. 

TWO PHASE FLOW IN CAPILLARY POROUS THERMO-ELASTIC MATERIALS... 

Two Phase Flow in Capillary Porous Thermo-Elastic Materials... 

Liu, Y., Fanguy, J.C., Bledsoe, J.M., Henry, C.S., Dynamic coating using polyelectrolyte multilayers for chemical control of electroosmotic flow in capillary electrophoresis microchips. Anal. Chem. 2000, 72(24), 5939-5944. 

There are several ways to reduce or suppress the electroosmotic flow in capillaries. These methods involve either eliminating the zeta potential across the solution-solid interface or increasing the viscosity at this interface. One approach is to coat the capillary wall, physically, with a polymer such as methylcellulose or linear polyacrylamide. Because of the difficulty in deactivating the capillary surface reproducibly, however, alternative methods employing dynamic reduction of solute-capillary interactions have been developed. Dynamic reduction of these interactions include the addition of chemical reagents such as methylhydroxyethylcellulose, S-benzylthiouro-nium chloride, and Triton X-100. 

M. Kreutzer, Hydrodynamics ofTaylor flow in Capillaries and Monolith Reactors, PhD Thesis, TU Delft, 2003. 

Monophasic fluid flow in capillary-scale ducts is characterized by a low Reynolds number, the flow in capillary-scale microreactors is generally laminar and transport... 

A.E.F. Nassar, S.V. Lucas, W.R. Jones and L.D. Hoffland, Separation of chemical warfare agent degradation products by the reversal of elec-troosmotic flow in capillary electrophoresis, Anal. Chem., 70, 1085-1091 (1998). 

Figure 4.6. Fluid flow in capillary tube driven by capillarity. This is a simple model of flow in a paper or thin-layer chromatography bed.

The velocity profile in thermal diffusion cells (see Figure 4.9) is calculated from the same balance-of-force mechanics used earlier in this chapter to analyze flow in capillaries. However, gravitational forces replace pressure forces as the driving influence balanced against viscous forces. The resulting velocity profile is found to be [15-17]... 

We note that our previous descriptions of flow processes have tacitly assumed laminar flow. For example, flow in capillaries was described by balancing pressure-derived forces against viscous forces, ignoring acceleration (inertial) effects. Darcy s law, Eq. 4.18, is also based on laminar flow. With turbulence, flow resistance increases the pressure gradient is no longer linearly related to flow (see Eqs. 4.18 through 4.20) but increases more rapidly as expressed by... 

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