Laminar liquid flow

Gamart G, Favre-Marinet M, Asendrych D (2005) Conduction and entrance effects on laminar liquid flow and heat transfer in rectangular micro-channels. Int J Heat Mass Transfer 48 2943-2954... 

Series Scaleup of Laminar Liquid Flows. The pressure drop is given by Equation (3.14). Taking ratios gives... 

All solutions of Eqs. (3) such as that by Yu and Sparrow (Yl) yield the velocity profiles in each phase as a function of the interfacial position h and the pressure drop. The volumetric flow rates Qt and Q are obtained by integrating each velocity profile over the respective phase cross-sectional area. The ratio of the flow rates can then be determined as a function of only the interfacial position, and since the volumetric flow rates are known, this yields an implicit fourth order equation for the interfacial position h. The holdups Rt and Rn can be calculated once the interfacial position is known. Since each equation for the volumetric flow rates is linear with respect to the pressure drop, once the interfacial position is known the pressure drop may be easily computed. An analytical procedure for determining pressure drop and holdup for turbulent gas-laminar liquid flows has been developed by Etchells (El) and verified by comparison with experimental data in horizontal systems (A7). 

In this paper we report on a detailed experimental and theoretical investigation of the influence of acoustic waves on laminar liquid flow through the Berea sandstone. We focus our investigation on Berea sandstone, which is representative for the sandstone of an oil reservoir, with permeability 100 to 300 mD and on acoustic frequencies below the critical frequency [3],... 

The following developments will be restricted to laminar liquid flow with weak gas-liquid interactions. However, this is not a limitation of the proposed methodology which could be easily applied to any other flow regime. Applications will be presented for the modelling of the irrigation rate, the dynamic liquid holdup and the apparent reaction rate in the absence of external mass transfer limitations and in the case of non volatile liquid reactants (i.e. approximatively the operating conditions of petroleum hydrotreatment). 

The local transport cell is represented by a straight pore with an inclination to the vertical characterized by angle 0 (see Figure 8). The curvatures of gas-liquid and liquid-solid interfaces are assumed to be negligible. The model also assumes that the laminar liquid flow is motivated only by gravity. 

Two forms of Eq. (6-21) have been proposed. For low values of the Reynolds number of liquid flow, liquid holdup in a packed column can be predicted by consideration of laminar liquid flow down inclined surfaces against a pressure gradient In this viscous flow regime,... 

For laminar liquid flow in circular tubes with constant wall temperamre, Sieder and Tate have recommended Equation 19.21 [51,53]... 

Nusselt provided a simple model for laminar liquid flow down an inclined plane. This assumed that the liquid had reached fully developed conditions in which drag due to viscous shear exactly balanced the weight of the film. Under these conditions, Nusselt showed that for a Newtonian fluid of kinematic viscosity, v, film thickness, /, could be written in terms of the liquid flow rate, Q, moving over a vertically inclined surface of width, w, under a gravitational acceleration, g, using the following relationship ... 

Liquid thermal conductivities, k, are required in many chemical and process engineering applications where heat transfer is prevalent. They are required to evaluate the Nusselt number hd/k, the Prandtl number cp/k, and in correlations to predict the idealized condensing film coefficient based upon laminar liquid flow over a cooled surface. The thermal conductivity of a saturated liquid is ... 

Y. S. Kadaner, Y. P. Rassadkin, and E. L. Spektor, Heat Transfer in Laminar Liquid Flow through a Pipe Cooled by Radiation, Heat Transfer-Sov. Res., (3/5) 182-188,1971. 

FIGURE 17.68 Pressure drop versus mass flow rate for a single flow passage in laminar liquid flow cooling [144]. 

The measurement and characterisation of residence time distributions for laminar liquid flow in plastic microcapillary arrays. Chem. Eng. Sci., 64 (17), 3889-3902. 

One way to structure the surface is to mimic a regular porous network. Rhombic structures are used to modify and disturb the laminar liquid flow [52]. So-called streamlined fins are arranged horizontally in different rows (Figure 7.14a). The rows are shifted in such a way that one rhombus is positioned in the center of the space between two neighbored rhomb of the up- and downstream rows. This arrangement forces the redirection of the flow, thus ameliorating the mbdng of the fluid in a split and recombine manner. 

The following developments will be restricted to the laminar liquid flow with weak gas-liquid interactions. However, this is not a limitation of the proposed methodology which could be easily applied to any other flow regime. 

Kuncewics, C. (1992) Three-dimensional model of laminar liquid flow for paddle impellers and flat-blade turbines. Chem. Eng. Sci., 47, 3959-67. 

In a small-gap parallel-plate reactor with laminar liquid flow (without gas evolution), infinitely wide electrodes and a constant electrode surface concentration, the average Sherwood number over the electrode length is given by Leveque s equation [21] ... 

TAYLOR DISPERSION FOR LAMINAR LIQUID FLOW ALONG A TUBE, 57 AXIAL MIXING IN CO-CURRENT FLOW OF GAS AND LIQUID, 58 NOTATION, 64 REFERENCES, 65 INTRODUCTION... 

In this chapter, the work published by the authors is reviewed. The importance of the wakes of slugs as the primary agent in promoting mixing is considered first. Next, the problem of Taylor dispersion for laminar liquid flow along a tube is briefly outlined, and, finally, the analysis of dispersion in co-current flow of gas slugs and liquid in vertical tubes is presented. 

TAYLOR DISPERSION FOR LAMINAR LIQUID FLOW ALONG A TUBE... 

An important factor of the sample transport system is the flow rate at the location of the optical analysis. The best results can be obtained with a laminar liquid flow. At higher Reynolds numbers. 

Figure 6.2.2. Axial dispersion of (a) a solute concentration pulse and (b) a solute concentration front introduced at the inlet of a straight circular tube having a laminar liquid flow and a parabolic velocity profile.

For Reynolds numbers Rcl < 2 (Chap. 4) in laminar liquid flow, the numerical value is as follows ... 

Discussion of Experimentai Resuits - Laminar Liquid Flow... 

In the case of laminar liquid flow with Reynolds numbers of ReL < 5, the liquid holdup hL can be described reasonably well using Mersmann and Deixler s [44] method, with 85% of the experimental values verified with an accuracy of S(hL) < 20%, Fig. 4-11 a. Figure 4-11b shows deviations of 0— +50%, in which the hL,exp values are compared to the values calculated acc. to Berner and KaHs [12], see Eq. (4-20). Hence, the use of a correction factor of 3/4 in Eq. (4-20) has proved to be useful, as it leads to a considerably better conformance between the calculated and experimental values, see Table 4-3, with the exception of some experimental data taken at extremely low liquid loads, which cannot be sufficiently described by any of the methods available. 

The evaluation of the large amount of experimental hquid hold-up data leads to the conclusion that turbulent liquid flow can be expected in packed columns at Rep > 2, whilst laminar liquid flow is expected at Rcl < 2. This corresponds approximately to the model presented by Gelbe [22]. 

In the case of laminar liquid flow. Rep < 2, the model developed by Mersmann and Deixler [44], known from literature, as shown in Eq. (4-34) ... 

Analogously, it is possible to derive the correlation for determining the quotient Ap/ Apo for laminar liquid flow below the loading line, Fv/Fv,fi < 0.65. Here, it is sufficient to substitute Eq. (4-20) into Eq. (4-40). This leads to correlation (4-45) ... 

Comparing Calculated and Experimental Values for Laminar Liquid Flow, Rol <2... 

The applicabiKty of Eq. (4-50) for determining the pressure drop of irrigated random packings at laminar liquid flow for Reynolds numbers in the range 0.1 < ReL < 2, was verified based on experimental pressure drop data for 25 mm metal Pall rings, taken by... 

Figure 4-14. Comparison of pressure drop Ap/H calculated acc. to Eq. (4-50) with experimentally determined pressure drop for irrigated packed bed, valid for laminar liquid flow ReL < 2...

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