International Journal of Chemical Engineering

Volume 2019, Article ID 1635265, 11 pages

https://doi.org/10.1155/2019/1635265

## Modeling Pulse Properties near the Bubble-to-Pulse Transition in Randomly Packed Beds

Department of Chemical and Biomolecular Engineering, University of Houston, Houston, TX 77204, USA

Correspondence should be addressed to Paul Salgi; ten.labolgcbs@iglasp

Received 2 November 2018; Revised 30 November 2018; Accepted 11 December 2018; Published 8 January 2019

Academic Editor: Bhaskar Kulkarni

Copyright © 2019 Paul Salgi and Sanaa Krayem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Traveling wave analysis of a recently developed two-fluid model for bubbly flow in lab-size packed beds is used to propose a constitutive closure for the effective viscosity, a nonzero parameter that is needed in the liquid momentum balance to avoid the prediction of disturbances with an infinite growth rate. Near-solitary wave profiles are predicted over a range of velocity parameters consistent with linear stability analysis. Centimeter-scale periodic disturbances are predicted in the near-pulsing regime. Preliminary estimates of average pulse properties compare well with typically reported experimental values. Initial comparison with time integration subject to periodic boundary conditions shows agreement of the liquid saturation profiles but differences in the liquid velocity profiles.

#### 1. Introduction

The volume averaged two-fluid model [1, 2] has been successfully applied in the literature to predict key engineering quantities (average liquid saturation, pressure drop, and flow regime transitions) that are needed for the design of packed beds with cocurrent and downward gas-liquid flow. Grosser et al. [3], who were the first to use such an approach in packed beds, were able to predict with some success the observed transition from trickle to pulse flow. In a subsequent paper, Dankworth et al. [4] provided improved closure relations for the momentum transfer terms and included a Newtonian-like stress tensor with an effective viscosity in the liquid momentum balance. These authors confirmed that the location of the transition predicted from linear stability is insensitive to the value of the effective viscosity. However, they noted that when a zero effective viscosity is used, all the modes become unstable at once, with growth rates that increase monotonically to infinity. Therefore, a nonzero effective viscosity is required to formulate a working dynamic model in the near-pulsing regime. As pointed out in a recent paper [5], the scaling argument used by Dankworth et al. to obtain a closure for the effective viscosity is based on the assumption that typical variations in the interstitial liquid velocity, a mesoscale quantity, occur over distances that are of the order of the packing diameter. In this paper, we show that it is possible to extract a closure for the effective viscosity without making any *a priori* choices for the characteristic length. First, we use a traveling wave approximation to translate the problem into a 2D dynamical system; then, we impose a requirement on the eigenvalues of one of the fixed points along a special path on the bifurcation diagram. The resulting closure leads to reasonable predictions of average pulse properties near the bubble-to-pulse transition.

#### 2. The Model Equations

The continuity equations of the one-dimensional model may be written as follows [5]:

The momentum balances are given by the following equations:

Under typical simplifying assumptions, the momentum jump condition is given by the following equation:

In equations (3) and (4), the gas-liquid interaction term is written as the product of the relative velocity and a function of liquid saturation, denoted by . The derivative terms with *D* refer to material derivatives. In equation (5), is the mesoscale average value of the mean curvature of the interface. The quantities and in equations (4) and (5) refer to the average gas density in the column and the surface tension at the gas-liquid interface, respectively. In equation (3), the parameter of the liquid-solid interaction term is given by the Ergun equation:where is the superficial liquid velocity, and are the density and viscosity of the liquid phase, is the average bed porosity, and is the packing size. In this work, we use the values 150 and 1.75 for the Ergun constants *E*_{1} and *E*_{2}.

Lastly, it follows from equations (1) and (2) thatwhere

The solution of the above equations requires closure relations for the gas-liquid and liquid-solid force interaction terms ( and ), the momentum jump condition (), and the effective viscosity (). A detailed discussion of the first three closures may be found in Salgi and Balakotaiah [5]. In this work, we show that the remaining closure for the effective viscosity may be obtained from the bifurcation diagram of the dynamical problem formulated in the so-called traveling wave approximation.

To estimate the average pulse properties under given flow conditions (gas and liquid flow rates), we fix the liquid mass flux *L* and choose the gas mass flux *G* to be greater than the “transition” value predicted from our linear stability analysis [5]. We then consider a hypothetical bubbly flow (as described by our model) under these conditions. As long as we do not go far into the pulsing regime, the average properties (pressure gradient and liquid holdup) of this hypothetical bubbly flow will still be in good agreement with those of the experimental pulsing flow [5]. As discussed by Dankworth et al. [4], visible pulses may be described as solitary waves (i.e., waves where the width of the disturbance is much smaller than the wavelength). In the traveling wave approximation, which translates the model equations into a 2D dynamical problem, solitary waves are typically modeled as “homoclinic” or “heteroclinic” orbits, which consist of saddle points and the paths connecting these points [6]. In our case, the 2D dynamical problem has a single saddle point (along with another fixed point that is a focus in the relevant range of parameters). Pulsing solutions may be sought as the merging point between two paths: (i) the path along which the periodic solutions have an average liquid flux equal to the liquid flux entering the column (the “constrained path”) and (ii) a line of (homoclinic) saddle connections. We find that two such paths emerge from a third special path (the “ Path” defined in Section 3.3), which provides a connection with the linear stability results of the full model. As already pointed out by Dankworth et al. [4], the merging of the constrained path with the line of saddle connections is gradual and near-solitary wave profiles (“pulsing solutions”) may be generated over a range of velocity parameters. However, the average pulse properties extracted from these solutions are rather similar when the velocity parameter is restricted to values consistent with linear stability analysis. These predictions are also found to compare favorably with typically reported experimental values for average pulse velocity, amplitude, and frequency.

When generating our “pulsing solutions,” it was necessary to evaluate the functions and at liquid saturations that lie outside the range over which these closures can be obtained. Typically, we first generate a best polynomial fit over the range where the function is given by the closure and then use the polynomial fit to extrapolate (see Figures 1(a) and 1(b) for an example). In addition to its simplicity, the justification for such fitting and extrapolation procedure lies ultimately in the reasonable agreement of the resulting predictions with experimental values of average pulse properties.