# The use of mixture model theory in CFD for the chemical reaction between CO_{2} and soda lime in closed circuit rebreather scrubbers

- Shona Cunningham
^{1, 2}Email author, - Aoife Burke
^{1}and - Ger Kelly
^{1}

**2**:578

https://doi.org/10.1186/2193-1801-2-578

© Cunningham et al.; licensee Springer. 2013

**Received: **19 April 2013

**Accepted: **18 September 2013

**Published: **30 October 2013

## Abstract

A mixture model simulation is presented by modeling the axial scrubber in a Closed Circuit Rebreather (CCR). The mixture model is a good substitute for the full Eulerian multiphase model because the interphase laws are unknown in this case. Analysis of mesh size, mesh type and inflation are made to independently characterize their accuracy by means of convergence before further comparisons with experimental data. The importance of mesh refinement is demonstrated near the wall with satisfactory results seen on the near grid wall of the boundary where a finer mesh is utilized. The contribution of inflation and grid independence to the accuracy of the model is presented in the results section.

## Keywords

_{2}Mixture model theory

## Introduction

_{2}laden exhaled breath leaving the mouthpiece of the diver in which it is absorbed by a chemical compound, generally soda lime, lithium hydroxide or Baralyme (Wang 1975) in the scrubber, the latter not seen in rebreathers for over 35 years. The development of a transient model simulating the chemical reactions which occur between CO

_{2}and soda lime would greatly enhance the fundamental understanding of these systems and may lead to an optimum design in these systems.

Prior to Clarke (Clarke 2001), the kinetics of CO_{2} absorption in scrubbers was poorly understood. A stochastic method simulating a bed containing a minimum of 200,000 volume elements or cells was employed and within each cell, the temperature and quantity of CO_{2} stored for each time increment was defined. The model was constrained by physics, the chemical absorption within each cell with its resulting temperature are probabilistic as opposed to the chemical properties of the absorbent or CO_{2} which means mass and heat transfer are also determined stochastically. The outputs of the analysis were comprised of a model simulating CO_{2} absorption and thermal fluctuations however the model is only applied to axial design based scrubbers.

Dongsik, Fumin and West (Dongsik et al. 2011) analyzed imperfect CO_{2} removal mechanisms of CO_{2} scrubbers. Their work introduced a stochastic model for three CO_{2} related rebreather faults: (i) CO_{2} bypass, (ii) scrubber exhaustion and (iii) scrubber breakthrough. The authors proposed a stochastic process driven by a Poisson counter to characterize the concept of CO_{2} channeling and the three stated rebreather faults. In probability theory, a Poisson process is a stochastic process which counts the number of events and the time that these events occur in a given time interval. The aforementioned work also advances the understanding of breathing dynamics associated with CCRs and how to maximize performance in terms of breathing and peak to peak pressure. This model is constrained as it does not model the chemical reactions occurring within the scrubber.

*m*which can be expressed in terms of density of gas in the node and the node’s volume as,

where ∆V is the change in volume during a time increment ∂ and ∆*ρ* is the change in density during time increment ∂.

_{2}and soda lime are analyzed. The chemical reaction between soda lime and CO

_{2}is an exothermic reaction and generates humidity (Nuckols et al. 1985). A liquid and gas phase process occurring within the reaction is identified which must be taken into account when modeling the reaction (Olutoye & Eterigho 2005). The absorption of CO

_{2}by sodium hydroxide is accompanied by a chemical reaction to form sodium carbonate as a by-product (Olutoye & Eterigho 2005). To validate the proposed methodology of the mixture model theory in this paper, a detailed study of the dependence of the results on mesh density, convergence criteria, mesh type, inflation, aspect ratio and skewness is presented. The simulated temperature rise during the exothermic reaction is benchmarked against measurements taken from an experimental design. The transient temperature rise throughout the scrubber canister is analogous to a moving temperature front. The CO

_{2}gas is absorbed by the first layer of soda lime granules and once the soda lime reaches its limit to absorb CO

_{2}, the reaction continues to the next layer. In order to compare the temperature results directly to the CFD results a thermocouple rig illustrated in Figure 2 was placed into the axial canister.

The average of (T1, T5, T9), (T2, T6, T10), (T3, T7, T11) and (T4, T8, T12) are taken to obtain a temperature for that layer of soda lime granules for any given time.

### Proposed modeling approach

^{th}century and have no known general analytical solution but can be discretized and solved numerically. The derivation of the Navier–Stokes equations begins with an application of Newton’s second law: conservation of momentum (often alongside mass and energy conservation) being written for an arbitrary portion of the fluid. In an inertial frame of reference, the general form of the equations of fluid motion is:

*ρ*is the fluid density,

*p*is the pressure,

*T*is the (deviatoric) stress tensor, and f represents body forces (per unit volume) acting on the fluid (Manninen et al. 1996). There are a number of different solution methods that are used in commercial CFD codes. The method used in Ansys CFX 13.0 is the finite volume technique. This technique divides the area of interest into smaller sub-regions, called control volumes. The equations are discretized and solved iteratively for each control volume (ANSYS CFX 2006). The result gives an approximation of the value of each variable at specific points throughout the domain. This describes a full picture of the behaviour of the flow. Mean particle Reynolds number is used to verify the decision of the laminar model, a calculation is carried out where;

*ρ*is the gas density of CO

_{2}, $\overline{V}$ is the superficial velocity,

*e*is the particle diameter of the soda lime and

*μ*. is the dynamic viscosity of CO

_{2}in Eq. 3 (Rhodes 1989).

Taking the density of CO_{2} gas as 1.87 kg/m^{3}, the Re number can be shown to be 0.189 for a particle diameter of 0.0011 m, velocity of 0.0126 m/s and a dynamic viscosity of 0.0001372 kg/ms. The laminar conditions apply up to Re=10 (Rhodes 1989). Within the laminar model, mixture model theory and Henry’s law were applied in the simulation.

### Mixture model theory

The mixture model is a type of multiphase system fined as a mixture of the phases of solid, liquid and gas. Multi-phase flow phenomena are typically dominated by one phase and another non-dominating phase e.g. dust in air (Manninen et al. 1996. However in the case presented in this paper the secondary phase or non-dominating phase cannot be neglected due to the influence on the fluid dynamic behavior of the mixture. The model contains an air|liquid pairing where the air is the inlet gas and the liquid is a reacting component. The decision of modeling threaction as a liquid-particle mixture is based on literature which states absorption as “*the removal of one or more selected components from a mixture of gases by absorption into a suitable liquid is the second major operation of chemical engineering that is based on interface mass transfer controlled largely by rates of diffusion”*(Sinnott 1996). Gas absorption occurs when a mixture of gas comes into contact with a liquid for the purpose of dissolving one or more components of the gas mixture in the liquid. Thus the absorption of CO_{2} occurs with the NaOH component of soda lime in the liquid phase (Physical and Engineering Data 1978).

_{2}required for the chemical reaction. Strong coupling between the phases of CO

_{2}and soda lime is necessary in this model for liquid-particle mixtures (Ishii 1975). The motions of individual components are treated in terms of diffusion through the mixture. A homogeneous flow within the mixture model is applicable when the phases are strongly coupled in drag dominated flow and their velocities equalize over short spatial length scales (Bowen 1976; Joseph et al. 1990; Johnson et al. 1991). All phases are assumed to move at the same velocity. The volume fraction of the soda lime granule is assigned in the porosity media. The continuity equation for the mixture is denoted in Eq. 6,

where *α*
_{
K
} the volume fraction of the phase *K*, *ρ*
_{
K
} is the average material density, *u*
_{
K
} is the local instant velocity of phase *K* and *ρ*
_{
m
} is the local density of the mixture (Ishii 1975).

### Henry’s Law

_{2}is a simple gas to describe the equilibrium between vapour and liquid. This law has been employed previously (Farajzadeh et al. 2009; Farajzadeh et al. 2007) in the sequestration/capture of CO

_{2}. The application of Henry’s law to calculate CO

_{2}yields a value 29.41 Latm/mol from Eq. 7,

where *p* is the partial pressure of the solute in the gas above the solution, c is the concentration of the solute and k_{H} is a constant.

### The three step reaction

_{2}is a three step exothermic reaction producing water vapour. Eq. 8 describes how gaseous CO

_{2}dissolves in water which is the first of the three step reaction (Reid et al. 1987).

_{2}(W.R. Grace and Co. 1986).

Water is required to initiate the CO_{2} absorption (Eq. 8). However water is a by-product of the chemical reaction that takes place within the canister (Eq. 9). If the incoming gas stream is saturated with water vapour, an excess of water vapour will remain in the canister. This excess water coats the soda lime granules and cause blockages in the pores. The CO_{2} does not absorb as efficiently and this may also cause caustic vapour in the loop which could burn the diver’s throat. Conversely if the incoming gas stream is too dry, the commencement of the reaction may be limited or the absorbent bed may be too dried out, thereby preventing absorption. Moisture levels of the incoming gas stream should be maintained above 70% RH when using soda lime.

### Boundary conditions and assumptions

The simulation was performed using the CFD software program Ansys 13.0 CFX and the following assumptions are made for the model; (i) the CO_{2} is absorbed fully without loss until breakthrough, where breakthrough is defined as the time until the canister effluent/CO_{2} passes through the soda lime granules unscrubbed, (ii) the CO_{2} gas is uniformly distributed throughout and (iii) a constant CO_{2} injection rate is employed. The axial scrubber is analyzed using CFX and is comprised of a packed bed of soda lime granules modeled as a porous media in which exhaled breath (5% CO_{2}, 16% O_{2}, 78% N_{2}) and traces of water vapour (William et al. 2009) is passed through the scrubber at an inlet velocity of 0.0126m/s. When the composition of exhaled breath passes through this porous media, a series of chemical reactions take place absorbing the CO_{2} and producing water vapour and heat. The inlet gas has a static temperature of 25°C initially at atmospheric conditions.

The continuous governing equations are converted into algebraic equations by using the finite volume method. These subdomains or boxes allow the analysis of flow in each box individually and then these fluid portions can be collated to yield a complete picture of fluid flow in the entire domain of the scrubber.

### Modelling the geometry

### Mesh type

^{-4}is given as a prescribed tolerance (Pordal 2006a). The accuracy of the solution requires a balance between the number of elements used in the grid, the type of mesh and the time taken for the iterative computation of the model. The automatic hex mesh illustrated in Figure 4 (i) generated an unstructured mix of hex and tet elements with their characteristics given in Table 1.

**Table of initial mesh details**

Hexahedron | Tetrahedron | Sweep/Automatic | |
---|---|---|---|

| 4129 | 4815 | 2852 |

| 5697 | 13678 | 1342 |

| 10 h 52 mins | 17 h 54 mins | 4 h 18 mins |

In order to initiate this comparison the different types of mesh, the number of elements and the duration of computation are given in Table 1. The results varied in computational duration with the tet mesh the most computationally expensive.

^{-15}and is computationally less expensive in terms of the time to reach convergence. A structured grid was therefore chosen as the optimum mesh.

**Table of modified mesh details**

Hexahedron | Tetrahedron | Sweep/Automatic | |
---|---|---|---|

| 15050 | 4815 | 15686 |

| 12735 | 13678 | 13420 |

| 19 h 2 3mins | 17 h 54 mins | 16 h 48 mins |

The computational run time using this hex mesh was recorded at 19 h 23 mins. This mesh is the least accurate and the most time consuming. The tet and swept mesh both exhibit convergence and accurate similar values in comparison with the experimental data. The swept mesh is computationally less time expensive and structured so it was deemed the best system.

There is also a strong interaction between modeling errors and the time and space resolution of the grid. The quality of the mesh can be determined by many different factors (i) mesh type, (ii) convergence criteria, (iii) inflation, (iv) aspect ratio and (v) skewness of the mesh.

### Inflation test

**Details of modified inflation layers with a constant quad grid**

No Inflation | Inflation3 | Inflation5 | Inflation8 | Inflation10 | |
---|---|---|---|---|---|

| Near convergence | Near convergence | Near convergence | Near convergence | Converged |

| 11.33 | 6.16 | 5.042 | 6.9746 | 10.978 |

| 5.56e-3 | 0.6 | 0.6 | 0.6 | 0.6 |

| 4350 | 13200 | 14800 | 16000 | 16800 |

| 9312 | 26867 | 30075 | 32080 | 34085 |

Figure 14 illustrates the need for inflation in laminar flow against the walls of the axial scrubber segment as the temperature values particularly in 'No Inflation 2’ are not only too high but there is a greater time lag observed. Due to the slow nature of heat transfer, there is no difference between 3, 5, 8 or 10 inflation layers as the results collapse onto each other in both graphs. It is shown there is a need for inflation against the walls when analyzing heat transfer. Due to the difference in convergence, 10 inflation layers are used for the grid independence test even though it is 0.978 more than a good aspect ratio from Table 3.

### Grid independence test

**Details of modified edge-sizing grid with constant inflation layers**

Grid200Inflation10 | Grid400Inflation10 | Grid600Inflation10 | Grid800Inflation10 | |
---|---|---|---|---|

| Near convergence | Converged | Converged | Converged |

| 10.978 | 10.978 | 9.2437 | 12.3 |

| 0.6 | 0.6 | 0.6 | 0.6 |

| 8400 | 16800 | 31800 | 42400 |

| 17085 | 34085 | 57095 | 76095 |

### Grid convergence study using Richardson extrapolation

**Details of the grids used for GCI**

Grid 1 | Grid 2 | Grid 3 | |
---|---|---|---|

| 34085 | 57095 | 99567 |

| 16800 | 31800 | 60420 |

*Fs*=1.25 (Wilcox 1998) for comparisons over three or more grids. Due to the monotonic convergence condition of the results, Eq. 16 is used.

*R*gave a result which made the solution monotonic. This is the first indication of good convergence. The GCI solution is calculated at 5% which indicates the chosen grid is an acceptable grid (Roache 1994).

**Order of accuracy and grid convergence index**

ϵ | ϵ | R | p | GCI(%) | |
---|---|---|---|---|---|

| 3.87 | 0.385 | 0.099 | 3.59 | 5 |

## Discussion

The paper presented a CFD model of a CCR axial scrubber using the mixture model theory to analyze the chemical reaction between soda lime and CO_{2} as an alternative technique to current methods. The mesh density influences the accuracy of the results and thus a benchmark with experimental data of the final mesh was conducted. The first parameter influencing the mesh is the type of mesh chosen for the model. A structured quad mesh was identified as being the optimum as it showed satisfactory convergence and comparison with the experimental data. It was also less computationally expensive to run. The analysis of inflation was conducted on the boundary walls of the model. The level of inflation was varied for the different models and compared against experimental data where it is shown that inflation on the walls will contribute to the accuracy of the model. A grid independence test was carried out to analyze how the fineness or coarseness of the mesh grid influenced the results. The finer the mesh, the more accurate the solution becomes however at significant computational cost. The point at which similar results are seen between two meshes acts as validation in the choice of selecting a 600 grid with 10 inflation layers as the optimum grid. This mesh density is a close match in temperature and produces better results over the time on the x-axis. The aspect ratio and skewness of the cell are also used as a means of validating the mesh independent of experimental results. For the final mesh both the aspect ratio and skewness are acceptable values at 9.24 and 0.6. The relationship between the actual mesh and the experimental data show that the predicted results lag behind the actual experimental profile. This may be attributed to a lag needed in the simulation where the model needs to reach a steady state phase before the temperatures correlate to the experimental data as is seen with the second set of monitor points. The overall trend of the model prediction agrees well with the experimental data. The mesh density; including type, grid size and inflation coupled with the aspect ratio and skewness provide a method of characterizing the mesh. The GCI value of 5% is also an acceptable result. The method presented allows an independent validation of the mesh quality which is further validated with the experimental results.

## Declarations

## Authors’ Affiliations

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