Single liquid flow

Now, as in the first example of the previous section, we assume a onedimensional, constant density, single liquid flow. For such flows, the constant Darcy velocity is (k/p)(Ap/L), where Ap > 0 is the usual delta p or pressure drop through the core of length L. The corresponding velocity for the present problem is v = (k/p)(Ap/> ) where k is the cake permeability, and pis the filtrate viscosity. Substitution in Equation 17-8a leads to... 

So far, we have encountered two types of Vt behavior, first for constant density, radial, single-liquid flows without mudeake, and then for lineal mudeake buildup and filtrate production without introducing any underlying... 

This equation is the basis for viscosity determination by measuring flow times through a capillary. It can also be used to describe a single liquid at two different temperatures, as required for Eq. (4.63). Combining Eqs. (4.63) and (4.64) yields... 

The two models commonly used for the analysis of processes in which axial mixing is of importance are (1) the series of perfectly mixed stages and (2) the axial-dispersion model. The latter, which will be used in the following, is based on the assumption that a diffusion process in the flow direction is superimposed upon the net flow. This model has been widely used for the analysis of single-phase flow systems, and its use for a continuous phase in a two-phase system appears justified. For a dispersed phase (for example, a bubble phase) in a two-phase system, as discussed by Miyauchi and Vermeulen, the model is applicable if all of the dispersed phase at a given level in a column is at the same concentration. Such will be the case if the bubbles coalesce and break up rapidly. However, the model is probably a useful approximation even if this condition is not fulfilled. It is assumed in the following that the model is applicable for a continuous as well as for a dispersed phase in gas-liquid-particle operations. 

A considerable amount of information has been reported regarding mass transfer between a single fluid phase and solid particles (such as those of spherical and cylindrical shape) forming a fixed bed. A recent review has been presented by Norman (N2). The applicability of such data to calculations regarding trickle-flow processes is, however, questionable, due to the fundamental difference between the liquid flow pattern of a fixed bed with trickle flow and that of a fixed bed in which the entire void volume is occupied by one fluid. 

Weekman and Myers (W3) measured wall-to-bed heat-transfer coefficients for downward cocurrent flow of air and water in the column used in the experiments referred to in Section V,A,4. The transition from homogeneous to pulsing flow corresponds to an increase of several hundred percent of the radial heat-transfer rate. The heat-transfer coefficients are much higher than those observed for single-phase liquid flow. Correlations were developed on the basis of a radial-transport model, and the penetration theory could be applied for the pulsing-flow pattern. 

These expressions have been found valid for the range 1 < [(u,lug)NRtlNs< i] < 1000. For higher values, the axial mixing is identical to that observed for the single-phase liquid flow for which the following correlation was found for the experimental data ... 

In order to solve the complete problem of liquid flow around a single spherical drop or bubble in the presence of surface-active agents it is necessary to solve the equations for viscous liquid motion simultaneously with a conservation equation for dissolved surface-active agents ... 

It contains six chapters related to the overall characteristics of the cooling systems single-phase and gas-liquid flow, heat transfer and boiling in channels of different geometries. 

The problems of micro-hydrodynamics were considered in different contexts (1) drag in micro-channels with a hydraulic diameter from 10 m to 10 m at laminar, transient and turbulent single-phase flows, (2) heat transfer in liquid and gas flows in small channels, and (3) two-phase flow in adiabatic and heated microchannels. The smdies performed in these directions encompass a vast class of problems related to flow of incompressible and compressible fluids in regular and irregular micro-channels under adiabatic conditions, heat transfer, as well as phase change. 

Qu W, Mudawar I (2002) Experimental and numerical study of pressure drop and heat transfer in a single-phase micro-channel heat sink. Int J Heat Mass Transfer 45 2549-2565 Qu W, Mudawar I (2004) Measurement and correlation of critical heat flux in two-phase micro-channel heat sinks. Int J Heat Mass Transfer 47 2045-2059 Ren L, Qu W, Li D (2001) Interfacial electro kinetic effects on liquid flow in micro-channels. Int J Heat Mass Transfer 44 3125-3134... 

In Chap. 5 the available data related to flow and heat transfer of a gas-liquid mixture in single and parallel channels of different size and shape are presented. These data concern flow regimes, void fraction, pressure drop and heat transfer. The effects of different parameters on flow patterns and hydrodynamic and thermal characteristics of gas-liquid flow are discussed. 

On the other hand Bao et al. (2000) reported that the measured heat transfer coefficients for the air-water system are always higher than would be expected for the corresponding single-phase liquid flow, so that the addition of air can be considered to have an enhancing effect. This paper reports an experimental study of non-boiling air-water flows in a narrow horizontal tube (diameter 1.95 mm). Results are presented for pressure drop characteristics and for local heat transfer coefficients over a wide range of gas superficial velocity (0.1-50m/s), liquid superficial velocity (0.08-0.5 m/s) and wall heat flux (3-58 kW/m ). 

Figure 6.35 shows dependence of the dimensionless initial liquid thickness of water and ethanol 5, on the boiling number Bo, where 5 = 5U/v, f/ is the mean velocity of single-phase flow in the micro-channel, and v is the kinematic viscosity of the... 

The details of the specific features of the heat transfer coefficient, and pressure drop estimation have been covered throughout the previous chapters. The objective of this chapter is to summarize important theoretical solutions, results of numerical calculations and experimental correlations that are common in micro-channel devices. These results are assessed from the practical point of view so that they provide a sound basis and guidelines for the evaluation of heat transfer and pressure drop characteristics of single-phase gas-liquid and steam-liquid flows. 

OS 68] [R 19] [P 50] A numbering-up of five mini mixers, tested at the pilot stage, was used [134]. Automation of the entire process was required liquid-flow splitting to the single reactors was ... by no means trivial... [134]. The capacity of one mini reactor was 30 ml s , i.e. 1081 h . The complete setup hence should be operated close to 5001 h . The micro-reactor plant was operated at intervals as the preceding step was carried out batchwise. The operation of the micro-reactor plant started in August 1998 after a period of only about 1.5 years for development. 

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