Elsevier

Chemical Engineering Science

Volume 59, Issues 8–9, April–May 2004, Pages 1799-1813
Chemical Engineering Science

Design and scaling laws of ramified fluid distributors by the constructal approach

With contributions of Serge CORBEL, Hervé LE GALL and Abderrazak LATIFI
https://doi.org/10.1016/j.ces.2004.01.034Get rights and content

Abstract

The present paper contributes to theoretical advances in the conception, modelling and design of multi-scale fluidic elements, namely fluid distributors. The function of such fluid distributors is to deliver a controlled amount or rate of fluid to an array of distributing ports, in order for example to feed uniformly the channels of a multi-tubular heat exchanger, or of a catalytic monolith, or any other fluid-handling engineering apparatus. In recent years, distributors based on multi-scale channel networks, of fractal tree type, have been developed. The theoretical approach presented allows to design such distributors optimally, within certain constraints. The basis of the optimization is a compromise between “costs” related to pressure drop and viscous dissipation on one hand, and hold-up volume of the pore structure on the other hand. The calculations lead to geometric scaling laws, that is to relations between the dimensions of the channels at the different scales. Scaling relations are also established for different characteristic quantities, such as pressure drop, viscous dissipation power, volume fractions, wall surfaces, cost functions. An example of design procedure is given, and examples of distributors fabricated by stereolithography is shown.

Introduction

The controlled distribution (mostly uniform) of a fluid to the entrance surface of a process, or into the volume of an equipment, is an important issue for various aspects of process performance. The same is true at the other end of the process, for collecting the outcoming flows.

The present paper addresses this issue through three aspects: discussion of some examples, presentation of some practical implementations, and presentation of a theoretical background to handle this problem in a more or less general fashion. Earlier work which inspired the present developments and is reviewed further below, is mostly related to the modelling of the vascular or the respiratory system in living beings, and also to some recent technological developments. Let us start with some practical situations of increasing complexity where this question is posed.

As a first example, consider Fig. 1 representing a monolith of activated carbon used for example for capturing volatile organic components from an air flow. The productivity of a process based on this monolith is strongly dependent on the sharpness of the breakthrough curves for the VOC, which in turn are strongly dependent on the flow dispersion over the adsorbing bed. We want a very narrow residence-time distribution, thus a flow pattern as close as possible to plug-flow, together with fast adsorption kinetics. Deviation from plug-flow, as observed between the inlet and the outlet tubing, is due to several factors such as:

  • the existence of flow-dispersion (Taylor type) in each channel,

  • small differences in the channel geometry, entailing different average flow velocities and velocity profiles,

  • differences in the flow-rate fed to each channel, depending on the geometry of the inlet zone,

  • flow pattern and mixing in the inlet zone,

  • flow pattern and mixing in the outlet zone.

In the present approach, we consider the first two factors as fixed by the construction of the monolith itself, and plan to influence only the effects due to the inlet zone (distributor) and to the outlet zone (collector).

We are therefore concerned with the design of a distributor (and/or collector) which ensures, as much as possible, equal distribution of the flow between all the channels and minimal dispersion. These two properties, equidistribution and minimal dispersion, are clearly distinct and to some extent, independent. This can be seen by examining the classical solution of this problem, consisting in adding a large pressure drop, such as a porous plug, just upstream of the channels.

If it is well designed (i.e. homogeneous and with sufficient flow resistance), the porous plug will normally achieve satisfactorily the job of equidistribution, at the cost of a high pressure drop, and in particular it will protect the entrance of the monolith from the direct impinging of a jet emerging from the inlet tubing. However, the porous plug has to be fed through an inlet chamber, the role of which is to handle the change in diameter and in shape from the inlet tubing to the monolith. Unless carefully designed, this inlet chamber may act as a mixing chamber and may have a wide residence time distribution (RTD). One common solution is to pack the chamber with uniform inert particles, such as glass beads.

Equidistribution and minimal dispersion are not the only properties expected from a “good” distributor. Minimal void volume and minimal pressure drop are to be considered as well. The reason for minimizing void volume is best understood when one considers operations like rinsing, back-washing, displacement, elution, change of solvent or depressurization which currently occur in cyclic operations like chromatography, ion-exchange, pressure-swing adsorption, etc. Obviously, the smaller the void volume (and the lesser the dispersion) the smaller the losses if this volume is rejected, or the smaller the amount of displacing fluid, and the shorter the operation. All these issues are of primary importance in high-performance industrial chromatography for example. The minimum voidage argument holds for outlet collectors as well.

As a summary, a good distributor/collector, besides ensuring equidistribution of the flow between the streamlines of the process, should reach a compromise between void volumes, dispersion and pressure drop. Kearney (1994), and Kearney and Kochergin (1996), in a series of patents, presented an innovative solution to this problem, which is presented later, and with which the theoretical approach presented here is in strong connection.

Microporous adsorbents (activated carbons, nanotubes, zeolites, hydride forming metal powders...) are considered as potential storage media under moderate pressure for fuel gases; such as natural gas or hydrogen for use in vehicles. The optimal design of such a storage must account for the following factors:

  • High static storage capacity per unit volume, implying a large volume of nanoporous space, the nanopores being the scale at which adsorption, or condensation, or chemical reaction take place.

  • Fast transfer kinetics from the nanopores to the storage outlet (so that the gas is rapidly available upon demand from the user); reciprocally, the filling of the storage tank should be possible in a short time. Unfortunately, the mass transport at the nanoporous scale is purely diffusional, and therefore extremely slow. The properties expected therefore imply that some intermediate porous network be distributed in the storage tank, to allow rapid transport of the gas between the storage outlet and the nanopore scale. This intermediate porous network should occupy a small volume, because it contributes little to the storage capacity, and should also have low pressure drop.

  • Adsorption (or hydride formation, or capillary condensation) being strongly exothermal, the heat evolved during the filling of the storage should be at least partially removed, to avoid both temperature and pressure increase, which affect negatively the effective storage capacity. Reciprocally, during gas withdrawal, cooling occurs and favours retention, thus diminishing the availability of the gas. Thus heat should be brought from the outside, for example through the walls of the storage and carried by conduction inside the packing. Thus, the packing should constitute a continuous, heat conducting material, and/or incorporate some “conduction promoters”.

From this description, it should be clear that a pore network with different scales should be designed to connect “equitably” the single outlet tube to the complete nanoporous space, distributed in the full volume of the storage, and this pore network should have minimal volume and pressure drop. The dispersion aspect is not a criterion in this example. In addition, the solid space, geometric dual of the void space, should be well connected and heat conductive.

We shall later introduce several theoretical aspects of network topology, but let us presently emphasize the differences between the two above examples.

A first difference lies in the flow pattern. We shall call the case of the distributor a “percolating system” or a “one-way system”. We thereby mean that the gas flows without change of direction through the pore network from inlet to outlet. By contrast, in the gas storage, the gas reverses flow and comes out the same way as it came in. We shall call this a “respiratory system”, analogous to the lungs.

A second important difference is what we shall call the “dimensionality vector” of the problem. In the first example, the distributor connects the inlet area of the monolith (a 2-D space) to the inlet port of the system, which we assimilate to a 0-D space. The dimensionality vector thus has components (0,2). Conversely, the outlet manifold has dimension (2,0). In the gas storage, on the other hand, the inlet port is connected to the full 3-D space of the storage, and the dimensionality vector is (0,3). These notions are further illustrated by the heat transfer problem, in which the 2-D surface of the storage vessel is connected to the 3-D volume, and the dimensionality vector is (2,3) or (3,2).

The three examples above share two other topological properties: full connectivity, meaning that there is a continuous path between any two points of the space or network considered (pore space, or solid space); and concavity, which is a property of the boundary, or of the “envelope” of the space considered.

In the following, we concentrate on a special family of network which satisfies these two properties: the so-called tree networks.

Section snippets

The bifurcation tree or fractal distributor

The problem of tree-like pore networks, or more generally, of multi-scale transport network geometry, has been addressed mainly by biologists or physicists modelling biological systems. Let us first mention the pioneering work of Murray (1926) which we shall recall in more detail later. Murray developed scaling relations for the dimensions of the blood vessels by assuming a tree network and minimizing a weighted sum of pressure drop and blood volume. Numerous subsequent researches were

Optimization of the channel size distribution

Let us now consider the distribution of channel radii, determined by an optimization that specifies the total flow-rate and accounts for both viscous dissipation and total pore volume. One of these constraints alone will not yield an optimal size distribution: minimizing the volume would lead to infinitely thin channels, and minimizing pressure drop would result in as large channels as possible.

Before presenting a more general approach, let us first recall Murray's demonstration, based on the

Distribution of properties between scales

In the previous section, we have established scaling relations between channels of successive generations. What is proposed now is to establish global relations between the ensembles constituted by all the channels of given scales. As a first example of this approach, let us consider the distribution of dissipation, i.e., of the first term of the cost function φ, when the “optimal” scale ratio or radii 21/3 is satisfied.

We shall first express the ratio of dissipation in a channel of radius rk

Handling the design problem

With all these relationships in hand, how does one solve a practical problem of designing a ramified distributor of this type, starting with given objectives? Let us illustrate this with an example, which will also help summarizing the essential results. Suppose we want to feed uniformly a square surface of L2=100cm2 with a flow of water f0=360l/h(100cm3s−1) under a maximal pressure drop of about 100Pa(1mbar). These specifications are all we need to design an “optimal” distributor with still

Manufacturing constructal distributors

The distributor shown on Fig. 2 was manufactured in the “Département de Chimie-Physique des Réactions”, a laboratory of the ENSIC-group in Nancy, using laser polymerization (stereolithography) (André et al., 1993; André and Corbel, 1994; Corbel et al., 2001; Dufaud and Corbel, 2004) with a photosensitive epoxy resin (RP Cure 400 AR) manufactured by RPC S.A.(Marly, Switzerland) now 3-D Systems. The design is defined using a CAD tool and the program is fed to the computer controlling the laser

Conclusions

As a conclusion, let us first cast a critical look at the approach presented and at its obvious limits, and discuss experimental aspects and future prospects.

The construction presented is obviously adapted to square outlet surfaces, and yields a number of distribution points that is a power of 2. A straightforward extension to rectangular outlet surfaces is possible, when these are decomposable into squares (length is an integer multiple of width). For surfaces of different shapes, especially

Notation

Akwall surface area of all channels of scale k, m2
Dkviscous dissipation power in all channels of scale k, W
fkflow-rate in a channel of scale k, m3s−1
f0total flow-rate into and out of the construct, m3s−1
kindex of generation or scale
lklength of channels in scale k,m
ltottotal “horizontal” path length from inlet to outlet, with m even, m
Llength of the side of the square construct, m
mtotal number of generations or scales, dimensionless
nknumber of channels in scale k, dimensionless
nCtotal number of

Acknowledgements

This research is supported by a grant of the French ADEME (Agence De l’ Environnement et de la Maı̂trise de l’ Energie).

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