# A Quasi-Dimensional Model of Heat Transfer between Multi-Concentric Monolith Structures

^{*}

## Abstract

**:**

## 1. Introduction

_{h}) is much smaller than their length (D

_{h}<< L), this assumption is accurate.

^{2}K or greater in metallic monoliths, which exceeded that of cordierite by an order of magnitude. They concluded that monolith thermal conductivity caused the higher measured heat transfer rates. Similar apparent rates were measured in a reactor study by Roh et al. [18] using coated FeCrAl foil monoliths. The FeCrAl monolith demonstrated superior heat exchange when compared to a packed bed reactor operating under identical temperatures and flows.

## 2. Quasi-Dimensional Modeling Method

#### 2.1. Discrete Control-Volume Approach

_{s}is the number of constituent species in the gas mixture. Energy conservation is given with Equation (2) and accounts for the heat exchange in and out of the control volume in the radial direction. The total wall thermal resistance (R

_{th}) was formulated to include the convective conditions on both sides, the conductive resistance through the wall, and a parallel conduction term for the solid monolith foil. An increasing number of axial divisions was used to minimize the error generated using the finite difference approach, with the final division count determined with model convergence.

_{mono}). Thermal resistance due to convection (R

_{conv,i}) and conduction (R

_{cond,i}) at the walls is given with Equations (4) and (5). Total resistance between cells was modeled using Equation (3). Temperature differentials are small within a cell, making radiative effects insignificant. Because monolith cells are closed to each other, mass transfer and subsequent thermal mixing are also not present. With those considerations, individual corrugated channels were unimportant from a modeling perspective, and were ignored. To still capture radial thermal variation without modeling each cellular channel, the monolith channels were redefined as a series of annular walls and voids, which were assumed to be homogeneous in the angular direction. These voids or gas “parcels” represent all monolith channels between two concentric walls of a fixed radius. Use of this approach reduced the number of parcels needed to effectively model the monolith by several orders of magnitude. Corrugations in the monolith contributed to conductive radial heat transfer and occupied physical volume in the experimental monolith. To minimize error from this annular approximation, the radial conduction and monolith void fraction (θ) were still included in the model to decrease error. Thermal resistance in the radial direction was calculated using the series model approximation developed by Groppi and Tronconi [28] and is given with Equation (6). Each annular cell volume was multiplied by θ to achieve correct mass flow rates and thermal residence times within the model. As both gas phases and solid phases contribute to heat transfer, corresponding thermal conductivities for a gas (k

_{i}) and solid (k

_{s}) were incorporated into Equations (4)–(6).

_{h}), used in dimensionless number calculations, was determined using arbitrary duct calculations reported in Shah [16].

#### 2.2. Heat Exchange Process

^{−1}= 5 × 10

^{−1}for 50 < Re < 600. In our work, Nu was assumed to be constant. This assumption requires that most of the monolith length be under fully developed conditions. Table 1 reports our experimental ranges of Re, the Prandtl number (Pr), and Gz

^{−1}as calculated at the monolith exit. Minimum reactor-exit Gz

^{−1}was found to be one order of magnitude larger than that corresponding to fully developed Nu in prior works by Cornejo et al. [12,13]. This implies that at minimum 90% of each experimental flow in our work was fully developed. Assumption of constant Nu throughout the length of the monolith will result in minimal modeling error.

_{1}and R

_{5}are convective resistances, modeled using a constant Nu and gas properties of the parcel. R

_{2}and R

_{4}are the thermal conductivities of the coating on the monolith, which was calculated to be 6 microns thick per side of the wall based on loading data provided by the manufacturer. R

_{3}is the resistance of the monolith foil itself. The monolith foil was modeled using the thermal conductivity of its constituent FeCrAl metal, with a thermal conductivity of 16.8 W/m-K.

_{int}) and heat exchanger effectiveness (${\epsilon}_{HX})$, defined in Equations (8) and (9), were calculated based on the average enthalpy difference achieved between the entrance and exit and the log mean temperature difference (LMTD) of the fluid and the monolith mantel wall.

#### 2.3. Fluid Properties

#### 2.4. Model Division-Size Convergence

## 3. Experimental Methods

#### 3.1. Monolith Module Construction

#### 3.2. Monolith Module Instrumentation

#### 3.3. Monolith Characterization Apparatus

#### 3.4. Experimental Procedure

_{2}), argon (Ar), and clean dry air (CDA), were used in the experiments due to their availability and variation in thermal properties. CO

_{2}and Ar were obtained as compressed pure bottle gases, and CDA was generated on-site using molecular sieves. Variation in the properties of these gases resulted in varied volumetric heat capacities, providing a wider range of conditions for model calibration. Standard gas properties of these species are reported in Table 2.

#### 3.5. Surface Response Methodology

#### 3.6. Experimental Uncertainty Analysis

## 4. Results and Discussion

#### 4.1. Determination of Optimal Nusselt Number

#### 4.2. Parametric Study

_{wall}), and the aspect ratio of the monolith (α). The monolith aspect ratio is defined with Equation (10). The flow rate was reported as a gas-hourly space velocity (GHSV), which is a typical metric for catalyst systems. Response data from the full factorial model were processed in JMP 15 ™ to determine the relationships between the four independent variables on the thermal responses. The simple model generated using JMP for both dependent variables is shown in the scatter plots of Figure 13. Perfect linear agreement is plotted using a solid red line. Linear regression of a model output is an unconventional but useful approach in this analysis. By performing a regression, a simple and relatively accurate relationship between each variable to the response can be obtained. Fit Equations (11) and (12) are formulated such that each independent variable is normalized to vary from +1 to −1, making the slopes of all variables directly comparable. The relative importance of each variable can thus be discerned with the magnitude of its slope.

^{2}of 0.92. Most important to the heat exchange effectiveness is an aspect ratio, α. Monolith geometries with smaller diameters and longer channels result in higher heat exchange effectiveness. GHSV has the second-highest impact, with low velocities leading to the highest efficiencies. Physically, increasing flow to a heat exchanger will decrease efficiency under normal circumstances due to decreased residence time. This assumes that the flow conditions within the heat exchanger do not drastically change the convective heat transfer through turbulent transition or other means. Wall temperature does not strongly affect the heat transfer efficiency due to the temperature differential being captured in the heat exchange effectiveness term. Minor impact results from an increased thermal differential near the outer radii of the cylinder. Due to radial conduction, which scales with the natural log of the radial distance, the apparent thermal resistance along the radius increases as the radial position approaches zero. The higher absolute temperature differential increases the overall penetration of heat due to the more complete heating of these outer radial positions. Cell density was shown to have almost no effect on the heat exchange effectiveness. In the three examined densities, flow was wholly laminar and modeled as such with the terminal Nusselt number from experimental fitting. Without physical modeling of the wall structures, the effects of catalyst wash coating, or capturing minute variation in individual cells of the monolith, it is impossible to determine if cell density more strongly affects heat exchange effectiveness or plays no role whatsoever. From this regression, it is clear, however, that its effect is minor.

_{int}were also modeled in JMP 15 ™ using a simple linear fit to analyze the individual contributions of each independent variable to the overall system. The fit is reported in Equation (12), with similar scaling coefficients to indicate relative importance to the final value of h

_{int}. The best fit for the integral heat transfer coefficient was found as presented, with an R

^{2}of 0.98743 and an RMSE 0f 0.481 W m

^{−2}K

^{−1}. Like the heat exchange effectiveness, the cell density was relatively unimportant to the overall heat exchange rate. This makes sense, considering that some of the physical phenomena, which would change experimentally with cell density, such as the void fraction and conduction, were held constant in the model. These values were not varied as inputs to the model as they depend on the foil thickness chosen during monolith construction. As this variable could vary widely based on engineering requirements, the added complexity and expense of its inclusion would not yield comparable information useful for this study. Aspect ratio and temperature both show a similar scale of importance in affecting h

_{int}. As shown previously by Boger and Heibel [17], higher wall temperatures lead to higher log-mean temperature differential (LMTD) values in a monolith, which, in turn, create higher integral heat transfer coefficients. The same effect is shown in this modeling. Integral heat transfer rates are 1–2 orders of magnitude lower than those measured experimentally by Boger and Heibel. This is due to the larger diameter monolith used in this study, the lower-conductivity monolith material, and the convective condition at the monolith shell. This study used heated gas at the shell, whereas the previous study used circulating liquid water, which would provide at least an order of magnitude larger convective rate at the surface.

_{int}were already shown to vary as a function of GHSV, the aspect ratio, and the initial temperature differential (ΔT

_{init}). This effect is best illustrated using continuous contour plots for various temperature differentials. Figure 14 shows a series of these contour plots generated from the model, which further illustrates this relationship. Increasing ΔT

_{init}consistently increases h

_{int}regardless of other input parameters. At aspect ratios below unity, the slope of contours shows a decreasing trend. The lower limit of the aspect ratio, which approaches zero, implies an infinitely flat disc of an infinite diameter. In this case, it would be expected that thermal resistance would go to infinity. It follows that these curves will rapidly collapse upon each other as the aspect ratio approaches zero. It is intuitive that a flat disc heated from the rim is not conducive to heat transfer. Moving in the other direction, towards high aspect ratios, it is shown that the contour slopes become flat, reflecting the linear fit shown earlier. An increased aspect ratio increases the surface area to volume ratio, which decreases the thermal path length from the heated outer surface to the internal gases, thereby increasing the overall heat transfer rate. For a cylinder of constant volume whose geometry varies with the aspect ratio, the surface area to volume ratio is given with Equation (13). Aspect ratio in the numerator confirms this observation. The physical significance of the heat exchange coefficient is an indication of overall resistance to heat flow.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A_{monolith} | Monolith Shared Interfacial Wall Area, m^{2} |

CFD | Computational Fluid Dynamics |

CPSI | Cell Density, Cells per Square Inch, #/in^{2} |

D | Monolith Diameter, m |

Dh | Hydraulic Diameter, m |

GCF | Gas Correction Factor |

GHSV | Gas Hourly Space Velocity, hr^{−1} |

Gz | Graetz Number |

LMTD | Log-Mean Temperature Differential |

MFC | Mass Flow Controller |

Nu | Nusselt Number |

Pr | Prandtl Number |

Q | Heat Transfer Rate, W |

Q_{max} | Maximum Heat Transfer Rate, W |

R_{cond} | Thermal Resistance due to Conduction between Gas Parcels |

R_{conv} | Thermal Resistance due to Convection |

R_{mono} | Independent Thermal Resistance due to Monolith Solid Conduction |

R_{th} | Thermal Resistance |

Re | Reynolds Number |

RMSE | Root Mean Square Error |

SA | Surface Area, m^{2} |

SCFM | Standard Cubic Feet per Minute |

SRM | Surface Response Methodology |

T_{init} | Initial Temperature, K or °C |

T_{wall} | Monolith Wall Temperature, K or °C |

V | Volume, m^{3} |

h_{int} | Integral Heat Transfer Coefficient, W/m^{2}-K |

k_{i} | Gas Thermal Conductivity of Parcel i, W/m-K |

k_{s} | Solid Thermal Conductivity, W/m-K |

r_{i} | Radial Position of Monolith Wall, m |

r_{foil} | r_{i} + t_{foil}, m |

t_{foil} | Foil Thickness, m |

x | Monolith Axial Position, m |

α | Monolith Aspect Ratio, L/D |

ΔT_{init} | Initial Temperature Differential, K or °C |

ε_{HX} | Heat Exchanger Effectiveness |

θ | Monolith Void Fraction |

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**Figure 6.**Full-scale discretized modeling of metallic monolith heat transfer along the reactor axis.

**Figure 8.**Thermally coupled monolith from Figure 7. The figure graphically depicts radial heat transfer within the annular monolith sections.

**Figure 13.**Simple linear fit models for heat exchange effectiveness and integral heat exchange coefficients.

**Figure 14.**Modeled integral heat transfer coefficient at the wall of the reactor for varied GHSV and aspect ratio.

**Figure 15.**Modeled heat exchange effectiveness at the wall of the reactor for varied GHSV and aspect ratio.

Minimum | Maximum | Average | |
---|---|---|---|

Re | 1.3 | 40.2 | 13.3 |

Pr | 0.664 | 0.744 | 0.703 |

Gz^{−1} | 5.4 | 153.2 | 33.9 |

Dry Air | Argon | CO_{2} | ||
---|---|---|---|---|

Density | g/L | 1.293 | 1.782 | 1.964 |

Specific Heat | J/g-k | 1.004 | 0.520 | 0.843 |

GCF | [-] | 1.00 | 1.39 | 0.70 |

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**MDPI and ACS Style**

Kane, S.P.; Northrop, W.F.
A Quasi-Dimensional Model of Heat Transfer between Multi-Concentric Monolith Structures. *Thermo* **2023**, *3*, 515-536.
https://doi.org/10.3390/thermo3040031

**AMA Style**

Kane SP, Northrop WF.
A Quasi-Dimensional Model of Heat Transfer between Multi-Concentric Monolith Structures. *Thermo*. 2023; 3(4):515-536.
https://doi.org/10.3390/thermo3040031

**Chicago/Turabian Style**

Kane, Seamus P., and William F. Northrop.
2023. "A Quasi-Dimensional Model of Heat Transfer between Multi-Concentric Monolith Structures" *Thermo* 3, no. 4: 515-536.
https://doi.org/10.3390/thermo3040031