# A Minimum Entropy Production Approach to Optimization of Tubular Chemical Reactors with Nature-Inspired Design

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## Abstract

**:**

_{h}(z) = 2A(z)/P(z) and an equivalent surface area-based radius R

_{s}= P(z)/(2π) were computed from the cross-sectional area A(z) and perimeter P(z) measured along the nasal duct of Northern reindeer and used for shape optimization as nature-inspired design. The laminar flow in the cooling system was modeled using the Navier–Stokes equations for an incompressible liquid. In the central tube, a set of chemical reactions with temperature-dependent rates was considered. The temperature and flow velocity fields, pumping pressure, mass flow rate, and total heat flux J

_{th}were obtained by numerical methods. Comparative analyses of the efficiency of different geometries were conducted on Pareto frontiers for hydraulic resistivity Z

_{h}, thermal resistivity Z

_{th}, thermal inlet length L

_{th}, and entropy production S

_{irr}as a sum of contributions from chemical reactions, thermal, and viscous dissipation. It was shown that the tube with R

_{s}(z) as an interface between the reactor and cooler has the best Pareto efficiency using the (Z

_{h},Z

_{th},L

_{th}) objective functions. Surprisingly, this design also exhibits the lowest S

_{irr}and a more uniform distribution S

_{irr}(z) (i.e., equipartition) among other designs. This geometry is suggested for densely packed tubular reactors.

## 1. Introduction

_{2}and CO

_{2}rates, ratio of food to microbial volumes, etc.) in bioreactors [2]. Existing chemical reactors have volumes ranging from a few mm

^{3}(microreactors) in the laboratories to hundreds m

^{3}in industrial plants [3]. The chemical reactions inside can be carried out in homogenous or inhomogenous systems at low/moderate/high temperatures and pressures [2,4]. The time scale is from milliseconds (ammonia oxidation to nitric acid) to days (biochemical reactions). Physical processes as crystallization/melting, evaporation/condensation/drying, and homogenization, in continuous or periodic regimes, can also be carried out in the reactors, for transformation of the initial substances into a product. The design of a chemical reactor is very important for its efficiency which is normally defined as the ratio of the net product amount to the total operating expense (raw material costs, energy input, energy removal, technical service). Bioreactors are in addition used to maintain comfortable conditioning for optimal growth of cell populations in the laboratory, for transplantology, tissue engineering, and other biomedical purposes [2,5]. Optimization of the size and shape of the reactor as well as the setting of operating physical conditions [6], temperatures, and temperature gradients [7,8] is an essential problem in chemical and biochemical engineering.

^{2}[12] allowing a microfluidic approach to the flow description. The chemical reactions could be endo- or exothermic, and the corresponding heating or cooling system must be mounted at the sur face of the tube. The heat produced in exothermal chemical reactions can be used as an energy source for other purposes, making it an important issue for design optimization as well [2,4,6,13].

_{2}into CO

_{2}conversion and removal) [16,17], nasal ducts (heating and moistening the inhaled air, and cooling and drying the exhaled air) [18], plant leaves (water delivery over short [19] and long [20] distances, and photosynthesis [21]), mitochondria (energy accumulation and distribution), biology-inspired chemical engineering [22,23], catalysis [24] based on tree-shaped flow structures, [25] and many others. The term nature-inspired is wider than the term biomimicking. The latter means just to copy geometry and operating conditions from a system in nature, while the former implies a deep understanding of the structure-functional relationship of a natural process/system, followed by its implementation with possible variations in both geometry and operating conditions. Nature-inspired design is rather an application of an idea, that nature has a properly designed engineered system, using efficiently the available materials, energy sources, operating conditions, unit maintenance, etc. The concept of nature-inspired chemical engineering developed in [18,23,26] implies a systematic design methodology to solve engineering problems, based on the fundamental understanding of physical mechanisms and their applications in engineering.

## 2. Design Optimization in Chemical Engineering

#### 2.1. Mathematical Formulations

#### 2.2. Minimum Entropy Production Approach

_{2}oxidation reactor [30], small modular reactors [1,4], tubular steam reformer [35,56,57], the reverse water-gas shift reactors [58,59], dimethyl ether synthesis reactors [60], catalytic combustion of air pollutants with Pd/Al

_{2}O

_{3}catalyst [37], polymer electrolyte membrane fuel cells (FC) with Fermat spiral [61], biomimetic [62] and fractal-type [63] flow-fields, solid oxide FC [64], in ammonia-methane fueled micro-combustor for thermophotovoltaic applications [65], in hydrocarbon synthesis reactor with carbon dioxide and hydrogen [66], CO

_{2}hydrogenation [67], isothermal crystallization processing [68], in the Fickett-Jacob cycle [69], diabatic distillation [70], in hydrogen iodide decomposition reactors heated by high-temperature helium [71], ideal reactors and practical industrial reactors [60,72], stirred tank and plug flow reactors [72], thermoelectric modules [73], heaters [13,74,75], and chillers [76]. Based on the minimum entropy production approach, a nanofluid-based tubular reactor was optimized to the elliptic shape with the axes ratio 5:3 that gave up to 16.82% reduction in the entropy production and rise in the thermal efficiency from 74% to 80% [77]. During the last few decades, the minimum entropy production approach has been recognized as an important design tool [78]. It was shown that the minimum entropy generation rate corresponds to the maximum output power for prescribed input heat and equivalent thermodynamic forces corresponded to current operating conditions [79]. Minimum entropy generation rate corresponds to maximum yield in sulfuric acid decomposition process [80].

_{2}separation from natural gas using optimal control theory has shown to be the optimal design, with total entropy reducing by 38% with respect to the reference case, and the methane losses reducing to zero [86]. The same approach was used in the application to the hydrogen production in a plug-flow chemical reactor with optimal design of its cooling/heating system [34]. Entropy-based optimization for the steam methane reforming reactor heated by molten salt allowed the energy loss reduction by 22.08% [57]. For the plug-flow reactors, it was shown that the profiles for the entropy production rate and its minimum at different wall temperatures follow the same trajectory, indicating that the reactor works at the minimum entropy conditions which is strongly recommended [97]. Also, the results indicated a positive correlation of the wall temperature and entropy production rate along the length of the tubular reactor.

_{2}oxidation [29] and exothermal ammonia reactor [55] were formulated. It was shown, by varying the reactor length and controlling the utility, that reductions in the entropy production rate were achievable, up to 25% [30].

#### 2.3. Shape Optimization and Nature-Inspired Solutions

_{2}[102] based on the anatomy of the nasal duct of the northern reindeer. Mature reindeer develop complex spiral structures along the maxiloturbinate area of their nasal duct. This structure provides fast heating of the inhaled cold air from ambient value up to the body temperature, +38.8 °C, and a moistening of up to 100% relative humidity [18]. The diameter profile d(x) along the reactor was found based on the measured cross-sectional area A(x) and perimeter P(x) of the nasal duct. Compared to a tubular reactor with constant cross-sectional are, a reduction of 11% in the total entropy production ${S}_{irr}$ was observed. The reactor length proposed from the optimization, resulted in an additional reduction in ${S}_{irr}$ by 16% [102]. Numerical computations were performed on a 1D model of the plug flow along the duct. The computed profiles of temperature, pressure, velocity and concentrations of the chemicals were used in the calculation of the entropy production, before and after introduction of nature-inspired design.

_{2}methanation into synthetic methane was, for instance, computed in a free convection multi-tubular reactor, in a CFD simulation of a single reactive channel (d = 1/40, H = 300 mm). A hexagonal shaped distribution of 23 reactive channels separated by a distance of 40 mm was proposed [103].

## 3. Problem Formulation

_{int}, R

_{ext}, are constant (Figure 1a) with a radius and length scale R

_{in}:L = 1:10. The outer cylinder has a radius according to R

_{ext}:R

_{int}= 1.5:1. This is compatible with the non-dimensional size of many different laboratory and industrial plug-flow reactors [1,6,11,39]. Transport of the reactant (fluid flow) takes place along the z-axis, while transport of heat takes place in the radial direction. Figure 1b shows the two radii R

_{int}= 1, R

_{ext}= 1.5 of the BCR, as well as two nature-inspired radii R

_{h}(z) and R

_{s}(z) (solid line and dotted line, accordingly).

^{®}, Canonsburg, PA, USA). With the circular geometry of both tubes, axisymmetric flows in (x,z)-coordinates of the reactant through the reactor (I in Figure 1a) and of the coolant through the cooler (II in Figure 1a) are assumed.

_{2}oxidation (SO

_{2}+ 0.5O

_{2}→ O

_{3}) was considered. The reaction is exothermic. The reactant liquid was modelled as passing through a porous catalyst following the work [102].

_{p}, ${r}_{S{O}_{2}}$ is the reaction rate per unit mass of the catalyst, ${H}_{r}$ is the reaction enthalpy, ${\mu}_{I}$ is the viscosity of the gas fuel, $\gamma =2\pi {R}_{I}{\left(1+0.25{({R}_{I}^{/})}^{2}\right)}^{0.5}$, ${F}_{i}$ is the molar flow rate of the chemical component i.

_{II}is the temperature, ${c}_{p,II}$ and ${\lambda}_{II}$ are the specific heat capacity at constant pressure and the heat conduction coefficient, $\Psi ={\tau}_{ik}\partial {v}_{i}/\partial {\mathrm{x}}_{k}$, is the dissipation to heat due to viscous stress, ${\tau}_{ik}$, is the shear stress tensor in the region II.

^{®}, Canonsburg, PA, USA) software provides an easy and efficient geometry modeling and mesh generation for the fluid flow in complex solid and elastic geometry with heat conduction. The semi-analytical solution of the combined fluid flow with heat transfer is available for the flow between two co-axial cylinders (i.e., base case reactor) (Figure 2a) [104]. An analytical solution for the axisymmetric flow through the base case reactor without heating is known as a flow through annulus. It has well known analytical solution [105] that will be used for validation of the numerical model (5)–(11) without thermal dissipation.

_{II}> will increase with axial distance from the inlet, and at some z = L

_{th}a maximum value <T

_{II}> will be achieved. Therefore, <T

_{II}> = const at z > L

_{th}; with L

_{th}as the thermal entry length (thermally fully developed region, with the coolant heated to maximal temperature at given operating conditions). It is worth to note that at a slow flow of the coolant, cases with L

_{th}> L could happen. In the optimal system, a correspondence L

_{th}~ L is desirable. When L

_{th}<< L, the coolant no longer (at z > L

_{th}) serves to cool the reactor. Likewise, when L

_{th}>> L, the coolant is not heated enough to be used as an energy source. The condition L

_{th}~ L can be achieved by varying the inlet temperature and the flow rate of the coolant (i.e., changing operating conditions).

_{h}, Z

_{th,}and L

_{th}were all used as objective functions for the same total system volume, $V=const$ as constraint. Since the optimization criteria could be contradictory (that was shown to be the case for the pair (Z

_{h}, Z

_{th}) [46]) the Pareto method was also applied, and three optimization problems were considered:

## 4. Numerical Method

_{FE}= (7–9) × 10

^{4}and n

_{node}= (14–20) × 10

^{4}, respectively. The mesh sizing of (0.2–2) × 10

^{−3}was chosen. Different types of chemical kinetics (5) with given values υ

_{A,B,C}were taken from the AnSys Fluent database of fuels. The most common chemical plug-flow reactors for oxidation of SO

_{2}and ethylene production had water cooling of T

_{II,0}= 273 °K. The inner tube was modeled as a uniform medium with porosity ${\theta}_{r}=0.3-0.5$. The parabolic inflow ${v}_{f0}(r)$ of the coolant and the reaction rate per unit mass of catalyst ${\chi}_{j}(T)$ were assigned in AnSys Fluent as user-defined functions.

^{−5}were tested. The mesh-independence test for the problem solution was carried out. The sizing (1–2) × 10

^{−3}and the number of nodes, n

_{node}~15 × 10

^{4}, was found reasonable from the point of view the computation time and accuracy. The test on scheme-independence of the solution was carried out. The first-order scheme for the momentum equations was found accurate enough with faster convergence.

_{I,0,}and steady reactor temperature max(T

_{I}) for different chemical reactions were taken from literature [1,39,102]. The resulting Reynolds number in the reactor varied in the range Re

_{I}= 2–20. The flow rate of the coolant was chosen based on numerical results of the relation between L

_{th}and L. At high flow rates of the coolant (Re

_{II}> 100), the thermal inlet length was too long (L

_{th}>> L), and the chosen length of the reactor was not enough for a proper heating of the coolant. Therefore, according to the estimated Reynolds numbers, the laminar flow model was taken for the coolant flow. In the tubes with varying diameters, the model produced physically relevant results with vortex formations in the convergent and divergent regions of the tube.

## 5. Results and Discussion

_{2}oxidation reactor, are shown in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. Figure 4 presents temperature contour plots for the Geom1 in four different flow parameters. The tube in Figure 4a shows the effect of the exothermal chemical reaction, initiated by the fuel flow at the inlet temperature T

_{I,0}. There is a fast increase in the temperature at the entrance to the reactor (Figure 4a) (see [29,104]). At low T

_{II,0}~ 273 K and T

_{I,0}~ 278 K, the temperatures in both tubes are much lower (Figure 4b). When T

_{II,0}is low and Q

_{II}is small, the inlet thermal length is short, and the coolant will not efficiently transfer heat from the reactor (Figure 4c). In the opposite case, the thermal inlet length could be too long, and the coolant will not be heated enough to serve as a heat source for further utilization (Figure 4d). In that way, numerical simulations allow for the selection of the best choice of the operating parameters T

_{II,0}and Q

_{II,}which are dependent on the length and type (chemical reaction) of the reactor.

_{I,II}>(z), <p(z)>, <S

_{irr}>(z), and the volume-averaged values of Z

_{h}, Z

_{th}and $\text{\hspace{0.33em}}{S}_{irr}$ have also been computed with AnSys Fluent (Ansys, Inc.

^{®}, Canonsburg, PA, USA). The detailed plots are presented in the Supplementary Materials. All computed dependences were found physically relevant, as demonstrated by Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Tubular design of a chemical reactor: (

**a**) Base case reactor design with two coaxial tubes of constant radii R

_{int}= 1, R

_{ext}= 1.5; (

**b**) nature-inspired radius distributions, ${R}_{h}(z)=2A(z)/P(z)$ and R

_{s}(z) = P(z)/(2π), along the axis of the tube together with R

_{int}= 1 and R

_{ext}= 1.5.

**Figure 2.**Chosen geometry of the reactor (internal tube) and its cooling system (external tube): (

**a**) Geom1 (base case reactor); (

**b**) Geom2; (

**c**) Geom3; (

**d**) Geom4; (

**e**) Geom5; (

**f**) Geom6; (

**g**) Geom7; (

**h**) Geom8; (

**i**) Geom9.

**Figure 3.**Refined uniform mesh for the geometry pictured in Figure 2b.

**Figure 4.**Contour plots of the temperature distributions in the basic case reactor: (

**a**) inside the reactor; (

**b**) 3D in the reactor and cooler; (

**c**) 2D in the cooler with short L

_{th}; (

**d**) 2D in the cooler with long L

_{th}.

**Figure 5.**Pareto frontiers for the optimal problem solution (1) in (14): (

**a**) best geometries; (

**b**) flow patterns in the best geometries.

**Figure 6.**Pareto frontiers for the optimal problem solution (2) in (14): (

**a**) best geometries; (

**b**) flow patterns in the best geometries (the temperature scale is the same as in Figure 5b).

**Figure 7.**Pareto frontiers for the optimal problem solution (3) in (14): (

**a**) best geometries; (

**b**) flow patterns in the best geometries (the temperature scale is the same as in Figure 5b).

**Figure 8.**Pareto frontiers for the pair (${S}_{irr}$,${Z}_{h}$) of objective functions: (

**a**) best geometries; (

**b**) flow patterns in the best geometries (the temperature scale is the same as in Figure 5b).

**Figure 9.**Pareto frontiers for the pair (${S}_{irr}$,${Z}_{th}$) of objective functions: (

**a**) best geometries; (

**b**) flow patterns in the best geometries (the temperature scale is the same as in Figure 5b).

**Figure 10.**Distributions in different geometries: (

**a**) entropy production ${S}_{irr}(z)$; (

**b**) coolant temperature T

_{c}(z).

**Table 1.**Geometric details for non-dimensional radii of the internal and external tubes in the system.

Name | R_{int} | R_{ext} | Image |
---|---|---|---|

Geom1 (base case reactor) | 1 | 1.5 | Figure 2a |

Geom2 | R_{h}(z) scaled to 1 | R_{h}(z) scaled to 1.5 | Figure 2b |

Geom3 | R_{s}(z) scaled to 1 | R_{s}(z) scaled to 1.5 | Figure 2c |

Geom4 | 1 | R_{h}(z) scaled to 1.5 | Figure 2d |

Geom5 | R_{h}(z) scaled to 1 | 1.5 | Figure 2e |

Geom6 | 1 | R_{s}(z) scaled to 1.5 | Figure 2f |

Geom7 | R_{s}(z) scaled to 1 | 1.5 | Figure 2g |

Geom8 | R_{h}(z) scaled to 1 | R_{s}(z) scaled to 1.5 | Figure 2h |

Geom9 | R_{s}(z) scaled to 1 | R_{h}(z) scaled to 1.5 | Figure 2i |

${\mathit{Z}}_{\mathit{h}}$ | ${\mathit{Z}}_{\mathit{t}\mathit{h}}$ | ${\mathit{L}}_{\mathit{t}\mathit{h}}$ | ${\mathit{S}}_{\mathit{i}\mathit{r}\mathit{r}}$ | |

${Z}_{h}$ | - | 1, 2, 7 | 3, 4, 6, 7 | 1, 3 |

${Z}_{th}$ | - | 4, 7, 9 | 1, 2, 3, 7 | |

${L}_{th}$ | - | 6 |

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**MDPI and ACS Style**

Kizilova, N.; Shankar, A.; Kjelstrup, S.
A Minimum Entropy Production Approach to Optimization of Tubular Chemical Reactors with Nature-Inspired Design. *Energies* **2024**, *17*, 432.
https://doi.org/10.3390/en17020432

**AMA Style**

Kizilova N, Shankar A, Kjelstrup S.
A Minimum Entropy Production Approach to Optimization of Tubular Chemical Reactors with Nature-Inspired Design. *Energies*. 2024; 17(2):432.
https://doi.org/10.3390/en17020432

**Chicago/Turabian Style**

Kizilova, Natalya, Akash Shankar, and Signe Kjelstrup.
2024. "A Minimum Entropy Production Approach to Optimization of Tubular Chemical Reactors with Nature-Inspired Design" *Energies* 17, no. 2: 432.
https://doi.org/10.3390/en17020432