Energy Recuperation in a Spiral Reactor for Lean Methane Combustion: Heat Transfer Efficiency and Design Guidelines

Abstract

Fugitive methane emissions contained in the ventilation air (VAM) from underground coal mines make a significant contribution to the global methane emissions. These methane emissions have a high global warming potential (GWP) and should be mitigated to combat climate change. This study reports on a novel integrated recuperator reactor concept designed to mitigate these low-concentration methane streams using catalytic combustion. The paper analyzes the heat recovery aspects of the novel design and illustrates a computer-aided design approach to system development. Both computational and experimental methods were used in the investigation. The double-spiral counterflow design is shown to be able to eliminate methane from the flow stream with the feed at room temperature. A methodology is illustrated that can be used to determine the operating limits of the proposed recuperative reactor system. This system is suitable for use inside a mine.

1. Introduction

The ongoing change in the world’s climate has been attributed by many to be a result of the presence in the atmosphere of the so-called greenhouse gases (GHGs), of which the two largest contributors are carbon dioxide and methane. Whilst methane is less abundant in the atmosphere than carbon dioxide, it has a much higher global warming potential (GWP). Depending on the time period, methane has a GWP of between 25 and 30 times that of carbon dioxide (100-year time scale) or about 80 times on a 20-year time scale [1]. Furthermore, because methane has a shorter life span in the atmosphere than carbon dioxide, its mitigation offers the promise of significant GWP mitigation over the shorter term. This latter observation is widely recognized, and has resulted in the Global Methane Pledge, under which many countries have agreed to a significant reduction in methane emissions [2].

The sources of methane emissions are many and varied. Whilst a significant amount of methane emissions can be defined as naturally occurring, the remainder are a result of human activity. There is a lot of extant information that details the quantities of methane that are produced from different emission sources, and recent information is given in reference [3]. Whilst these numbers should be treated with caution, because of the high degree of uncertainty in their accuracy, there is no doubt that the fossil fuel industries (coal, oil and natural gas) produce copious amounts of fugitive methane emissions, either in the production or the consumption phases, and are considered to be the second highest human-caused emission source [3,4].

Underground coal mining faces a unique challenge in managing methane emissions, which is crucial for both worker safety and environmental protection. Methane occurs naturally in coal seams. It presents a significant safety risk to miners due to its explosive properties when present in air at concentrations between 5 and 15%. During coal extraction, methane is released into the mine workings, necessitating robust ventilation systems to introduce fresh air and expel methane and other gases, thus maintaining a safe working environment. This diluted methane in the ventilation air, referred to as Ventilation Air Methane (VAM), represents the largest source of methane emissions from most underground coal mines [5,6,7]. These systems dilute the methane concentration to well below the lower explosive limit, typically maintaining levels under 1% and often below 0.5% [5]. However, due to the substantial volume of air circulated through these ventilation systems, the total quantity of methane released into the atmosphere is very large.

As noted, VAM typically contains highly dilute concentrations of methane, usually less than 1% [5]. This low concentration creates a significant challenge for utilization of mitigation technologies. Furthermore, the methane content in VAM can fluctuate substantially over time, influenced by factors such as coal production cycles and contributions from sealed areas during changes in barometric pressure [7].

The removal of methane by conversion to carbon dioxide offers the possibility of a substantial reduction in GWP. Assuming that the global warming potential of methane is about 25 times that of carbon dioxide, the combustion of 1 tonne of methane yields 2.75 tonnes of carbon dioxide, thus giving a net reduction in GWP of 87%.

Emission streams with low methane concentration and low temperature are not easily destroyed using homogeneous (thermal) combustion, and using a catalyst is advantageous. Catalytic combustion is a flameless combustion that has received considerable attention over the last few decades, although it has been reported on for over one hundred years [8,9,10]. The key advantages of catalytic combustion compared to conventional combustion are the elimination of the standard flammability limits and the ability to combust the reactants at a lower temperature.

Many catalysts have been used for hydrocarbon combustion in general, and for methane oxidation in particular [11,12]. Several metals have been reported, both non-noble (or earth common) as well as metals from the platinum group metals (PGMs), especially platinum and palladium. Earth common metals obviously lead to a lower-cost catalyst, but the reactor size tends to be larger because of their relatively low activity compared with the PGM catalysts. A good review of the use of PGM in methane combustion catalysts can be found in several references [13,14,15,16,17,18,19].

It is often desired that the catalytic reactor operates adiabatically and has autogenous operation; that is, the energy generated by the combustion reaction will maintain the reactor at a sufficiently high temperature to obtain essentially complete conversion. For the low-temperature VAM stream, autogenous operation cannot be realized without pre-heating the feed to the ignition threshold. There are many options for autogenous operation and a good review can be found in [20]. That review discusses reactor concepts such as the reverse flow reactor, as well as counterflow designs. Examples relating to methane combustion can be found in [4].

Conceptually, the thermal energy generated in the combustion reaction can be used to pre-heat the feed stream. Ideally, this would be performed in an integrated unit. Heat-integrated concepts, also referred to as thermal management systems, typically utilize a form of combined reactor heat exchanger, where thermal energy from the effluent gas is transferred to the influent. A comprehensive overview of various combined reactor heat exchanger designs is available in the literature [21]. One notable design, first proposed in the 1930s and subsequently refined by many researchers, is the catalytic flow reversal reactor (CFRR), which has been studied for the combustion of low concentration methane streams with large scale flows [22,23,24]. The CFRR uses a heat trap effect to maintain a high reactor temperature. These reactors have been employed at the large scale for the destruction of VAM, notably in China. The reactor is usually situated on the surface, not inside the mine shaft. Smaller-scale units have also been used for applications such as exhaust gas treatment from natural gas-fuelled vehicles [25,26]. A drawback of the CFRR is the number of moving parts, which might be a drawback in an in situ mine environment, especially if explosion-proof operation is required. Therefore, other concepts can be considered.

Other types of integrated reactor and heat exchanger concepts have been presented in the literature [27,28]. These designs can be broadly categorized into two groups. The first group features separate reactor and heat exchanger units, allowing for the independent control of heat transfer parameters and potentially simpler construction, conceptually illustrated in Figure 1. In the second group, reaction and heat exchange occur within the same physical unit, enabling continuous heat exchange within the reactor. One possible system is also illustrated in Figure 1. While this second approach may lead to more compact designs, the available heat transfer area becomes more dependent on the internal configuration. Additionally, material selection for reactor construction becomes a critical factor. Consequently, careful consideration of the design is essential to optimize performance and efficiency.

The objective of this paper is to illustrate an autothermal reactor concept that is suitable for an in situ deployment in an underground coal mine for the destruction of VAM using catalytic combustion. The crucial factor in an integrated reactor concept is the magnitude and efficiency of the heat transfer, and that aspect is the focus of this work. Because the design and testing, that is, prototyping, of even a relatively small-scale unit is a complex and time-consuming process that has a significant cost, we show how a computer-aided design process (CAPD) that makes use of advance computational fluid dynamics (CFD) can be effectively used for preliminary design. Thus, we present a detailed workflow based on computer modelling and experimental prototyping to showcase this VAM reactor concept.

2. Materials and Methods

2.1. A Description of the Catalyst Used as a Basis for This Study

The goal of this work was to propose a system that could be progressed rapidly to a working prototype at full scale. With this objective in mind, a commercial methane oxidation catalyst was selected. The detailed kinetic analysis, water resistance and deactivation performance of this catalyst have been described elsewhere [18]. The properties of the catalyst, which is available in the form of a washcoated Cordierite monolith honeycomb, are provided in

Table 1. Properties of the washcoated catalytic Cordierite monolith used in this study.

Cell density	400 CPSI
Substrate density	1650 kg/m3
Substrate volume fraction	0.243
Washcoat density	1100 kg/m3
Washcoat volume fraction	0.12
Palladium loading in monolith (est)	100–150 g/ft3

.

The first step in the rational design of a potential catalytic combustion system must be an analysis of the catalyst to be employed, which can then be used to determine the operating window. It is essential that the catalyst activity is quantified through the development of an appropriate rate equation, for only thus can a computational model for the reactor be constructed. Referring to [18], the following rate expression was obtained. The catalyst activity was measured for the fresh catalyst, as well as after several cycles of hydrothermal ageing. The reader should refer to [18] for details. For dry feed (no water present) on the fresh catalyst, the rate expression based on the total monolith volume was

$$\left(-r_{{\mathrm{CH}}_4}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{k\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{C}_{{\mathrm{CH}}_4}}{\left(1\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}K\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{C}_{{\mathrm{H}}_2\mathrm{O}}\right)}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{5.151\times {10}^8\exp \left(\frac{-\mathrm{80,800}}{R_{g}T}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{C}_{{\mathrm{CH}}_4}}{\left(1\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}0.823\hspace{0.17em}\exp \left(\frac{\mathrm{21,290}}{R_{g}T}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{C}_{{\mathrm{H}}_2\mathrm{O}}\right)}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\mathrm{mol}}{{\mathrm{m}}^3\mathrm{s}}}$$

(1)

After three cycles of hydrothermal ageing [18], the activity for dry feed had decreased and the pre-exponential factor was reduced to a value of 2.012 × 10⁸ s⁻¹.

Note that, for illustration purposes, we only discuss in this paper the case of the dry methane feed for two levels of catalyst activity. For wet feed conditions, or for a different catalyst, the design methodology is the same, and the appropriate rate equation would be used.

2.2. The Basic Design Concept for the Recuperative Systems: Factors Influencing the Design

In previous papers we have published numerous works on the modelling of recuperative types of reactors for methane combustion with different configurations [22,23,24,25,26,29,30]. Based on our experience and some preliminary investigation, the design shown in Figure 2 was proposed. Figure 3 shows the details of the prototype that was constructed. The unit consists of a central block that contains up to three channels of monolith catalyst blocks, each row being designed for a catalyst length of 150 mm. The monolith bricks were supplied by the manufacturer as bricks of 50 mm length. The height of the catalyst channels was 50 mm, and the unit had a width of 150 mm. The central catalyst core is surrounded by two spiral-shaped “snails”, through which the feed and the effluent flow in counter-current directions, thus forming the recuperative heat exchanger. The snails were constructed from carbon steel. The internal wall spacing of the snails was 25 mm. The catalyst section consisted of nine catalyst bricks, arranged as shown in Figure 4. Sixteen thermocouples were placed in the catalyst section, the locations of which are also shown in Figure 4. Thermocouples 1 and 16 were placed in the gas stream approximately 1 cm before (TC 1) and 1 cm after (TC 16) the first and last catalyst bricks, respectively. The remainder of the thermocouples were inserted into the monolith channels to measure the catalyst temperature. Thermocouples 4, 7 and 8 to 15 were placed in the centre of the monolith bricks, as shown. Thermocouples 2, 3, 5 and 6 were placed in the first two bricks at various distances from the wall to measure the temperature distribution within the monolith sections. Two further thermocouples measured the temperature at the inlet to the reactor system (after the pre-heaters) and at the effluent of the reactor system.

2.3. Experimental Methodology

The reactor/recuperator unit was installed in a computer-controlled flow system. The system consisted of a blower and associated flow measuring device to control the flow rate. Two electric pre-heaters were used to initiate the reactor operation. They were used to heat the feed stream, which in turn heated the catalyst to the ignition temperature. They were also used as required to maintain operation at low methane concentrations. The heaters were switched off once autogenous operation was attained. The pre-heated feed was fed into the reactor inlet, where it passed through the recuperation section prior to entering the catalyst module. The computer-controlled system was monitored via the control panel. In case emergency shutdown was required, the system could be flooded with liquid nitrogen. The apparatus is shown schematically in Supplementary Information as Figure S1.

Experiments were performed to test various aspects of the systems, and each experiment is described in detail in either the Results section or the Supplementary Information. The general methodology was to set the air feed flow rate to the desired value without any added methane. The pre-heaters were then switched on and adjusted to give a temperature at the inlet to the catalyst section in the range of 575 to 600 K. Once a steady-state temperature profile was attained, the flow of methane was started at the desired concentration. Steady state was defined as the point where the change in any thermocouple reading was less than 5 K over one hour. Changes in the desired variable were made in a stepwise manner. After each step, the system was allowed to reach steady state. The inlet temperature to the unit was adjusted by changing the power to the pre-heaters. Other variables studied were the volumetric flow rate and the inlet methane concentration. For each experiment, a set of apparent steady-state points was selected for detailed modelling and analysis.

2.4. Computational Model

A detailed computational model was constructed for the experimental unit described in Section 2.2. The focus of the current paper is the heat transfer aspects of the device, and therefore the computational model consisted of the conservation equations for momentum and energy. For the open fluid sections, the equations are the standard ones for a fluid. The flow in these sections was modelled using a Reynolds-Averaged Navier–Stokes (RANS) turbulence model equation.

The monolith sections were modelled as a continuous porous medium. A pseudo-homogeneous model was selected, in which the fluid and solid temperatures are the same, and a Volume-Averaged Navier–Stokes (VANS) equation was used in these sections. In the VANS equation, the porous inertia was included by introducing a source term into the Navier–Stokes equations. We used a Darcy-type term to give the following equation [10]:

$$\nabla ·\left({\mathsf{\rho }}_f\hspace{0.17em}{v}_S\hspace{0.17em}{v}_S\right)=-\nabla p+\nabla ·\mathsf{\tau }-{\displaystyle \frac{\mathsf{\mu }}{K}}\hspace{0.17em}{v}_S$$

(2)

The monolith permeability, K, was estimated from the Hagen–Poiseuille equation assuming circular channels (reasonable for washcoated square channels) [31]:

$$K={\displaystyle \frac{\mathsf{\varphi }\hspace{0.17em}{D}_H^2}{32}}$$

(3)

The monolith substrate in the reacting sections was a Cordierite 400/6.5 CPSI (400 cells per square inch with 6.5 mil thick walls). The channels were covered with a catalytic washcoat, which gave the following disposition of fluid, washcoat and Cordierite:

$$\begin{array}{ll}\mathsf{\lambda }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.243\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hfill & \mathrm{volume} \mathrm{fraction} \mathrm{of} \mathrm{Cordierite}\hfill \\ \mathsf{\xi }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.12\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hfill & \mathrm{volume} \mathrm{fraction} \mathrm{of} \mathrm{washcoat}\hfill \\ \mathsf{\varphi }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1-\mathsf{\lambda }-\mathsf{\xi }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.637\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hfill & \mathrm{volume} \mathrm{fraction} \mathrm{of} \mathrm{fluid}\hfill \end{array}$$

(4)

The channel hydraulic diameter was 1.014 mm, to give the axial permeability as 2.04 × 10⁻⁸ m². The permeability in the direction transverse to the flow was set to 0.1% of the value in the flow direction, to eliminate the possibility of transverse flow. The gas density is computed using the ideal gas law and the viscosity from the kinetic theory of gases.

$$\mathsf{\mu }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2.67\times {10}^{-6}{\displaystyle \frac{\sqrt{M_{W}T}}{{\mathsf{\sigma }}^2{\Omega }_{\mathsf{\mu }}}}$$

(5)

The characteristic length, σ, is 3.711, and Ω_μ depends on the reduced temperature [32,33]. The gas composition was methane in air.

The steady-state energy balance in the presence of a chemical reaction is as follows:

$$\nabla \left(k\hspace{0.17em}\nabla T\right)\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{\mathsf{\rho }}_f\hspace{0.17em}{C}_{P,f}{v}_s\hspace{0.17em}\nabla T\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}(-\Delta H_R)\hspace{0.17em}\left(-r_{{\mathrm{CH}}_4}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0$$

(6)

The reaction source term applies in the reaction section. For the study of the heat transfer characteristics, we did not impose the rate equation, but rather specified thermal energy source terms in the catalyst bricks that generated thermal energy that corresponded to 100% conversion of the inlet methane at the designated flow rate, as shown below:

$$\nabla \left(k\hspace{0.17em}\nabla T\right)\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{\mathsf{\rho }}_f\hspace{0.17em}{C}_{P,f}{v}_s\hspace{0.17em}\nabla T\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\dot{q}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0$$

(7)

The source terms were imposed uniformly over a catalyst length of 150 mm. This volumetric generation rate was calculated based on the inlet flow rate and methane mole fraction. The total amount of thermal energy released, assuming 100% conversion, is as follows:

$$q_{\mathrm{gen}}=F_{\mathrm{CH}4}\times \left(-\Delta H_R\right)$$

(8)

The molar flow rate of methane, which depends on the inlet total volumetric flow rate, the total molar concentration and the mole fraction of methane, is calculated as follows:

$$F_{{\mathrm{CH}}_4}=Q\times C_T\times Y_{{\mathrm{CH}}_4}$$

(9)

The inlet temperature and pressure were taken as 300 K and 101.325 kPa, respectively. The total inlet concentration (molar density) of the feed is therefore

$$C_T={\displaystyle \frac{P}{R_{g}T}}={\displaystyle \frac{101325}{\left(8.314\right)\left(300\right)}}=40.62\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\mathrm{mol}}{{\mathrm{m}}^3}}$$

(10)

The volumetric rate of thermal energy generation is therefore given by

$$\dot{q}={\displaystyle \frac{q_{\mathrm{gen}}}{V}}={\displaystyle \frac{Q\times C_T\times Y_{{\mathrm{CH}}_4}\times \left(-\Delta H_R\right)}{V}}$$

(11)

where the enthalpy of reaction in J/mol is given by the following:

$$\left(-\Delta H_R\right)=-806.9+1.586\times {10}^{-2}T-8.485\times {10}^{-6}{T}^2-3.963\times {10}^{-9}{T}^3+2.16\times {10}^{-12}{T}^4$$

(12)

The thermal conductivity depends on the domain. For the open sections it is the thermal conductivity of the fluid, which was calculated using the kinetic theory of gases [32].

$$k_f\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{15}{4}}{\displaystyle \frac{R_g}{M_W}}\mathsf{\mu }\left[{\displaystyle \frac{4}{15}}{\displaystyle \frac{C_P{M}_W}{R_g}}+{\displaystyle \frac{1}{3}}\right]$$

(13)

For the monolith reaction sections, the value for the effective axial thermal conductivity is 0.546 W/(m·K) and the transverse value is 0.342 W/(m·K).

The temperature-dependent heat capacity of the gas is calculated using

$$C_P\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{a}_0\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{a}_{1}T\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{a}_2{T}^2\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{a}_3{T}^3\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{a}_4{T}^4\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{a}_5{T}^5\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{a}_6{T}^6\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{a}_7{T}^7$$

(14)

These equations and constant values are provided in the used computational software.

The fluid boundary conditions were zero slip at all solid/fluid interfaces, with a pressure outlet and a fixed uniform velocity at the inlet. The temperature conditions were adiabatic (zero flux) for the outlet, a fixed temperature for the inlet and continuity of flux across internal boundaries.

The computational model was solved using ANSYS Fluent version 2020 R2.

3. Results and Discussion

3.1. Determining the Catalyst Operating Window

An essential step in the rational design, once the catalytic rate equation is determined, is to determine an operating window for the catalyst. This operating window is typically presented as a function of catalyst volume (or mass), flow rate of feed, inlet methane concentration and the subsequent temperature required at the inlet of the reaction section to achieve complete methane conversion. The flow rate and corresponding catalyst volume can be lumped into a single parameter, called the gas hourly space velocity (GHSV), which is the volumetric flow rate of the gaseous feed, measured at a reference temperature and pressure, divided by the catalyst volume.

The operating characteristics of the catalyst can be explored using a simple one-dimensional plug flow reactor model that was solved using POLYMATH [18]. The objective is to determine a value for the temperature at the entrance of the catalytic section that will result in an essentially 100% conversion for a given GHSV and methane concentration.

Table 2. Catalyst inlet temperature required to achieve 99.9% conversion of methane at different GHSV for three methane inlet concentrations. Values with dry feed for fresh and aged catalyst.

GHSV h−1	Catalyst Entrance Temperature Required, Kelvin
	Fresh Catalyst			Aged Catalyst
	1%	0.7% t	0.4%	1%	0.7%	0.4%
180,000	646	678	715	704	739	775
120,000	625	658	692	677	712	748
90,000	610	643	677	660	694	730
72,000	600	630	667	647	680	717
60,000	592	620	658	638	670	707
48,000	582	610	648	626	657	695
36,000	570	597	633	612	642	678
24,000	555	580	615	593	622	659
18,000	544	568	603	580	608	645
12,000	531	553	587	564	590	627
9000	521	542	577	553	578	613
6000	509	528	561	539	562	597
4500	501	519	551	529	551	585
3000	490	507	538	517	537	571

summarizes the catalyst inlet temperature required to give 99.9% methane conversion for both fresh and aged catalysts over a range of GHSV. Results are given for methane concentrations of 0.4, 0.7 and 1.0 mole percent. The GHSV was changed by fixing the catalyst length and changing the inlet velocity. All inlet velocities and hence GHSV are based on a reference temperature of 300 K and a pressure of 1 atm. The data are also presented graphically in Figure 5.

These temperatures should only be used as a guide to system design. Catalyst activity changes with time, and the flow patterns in a real system make the accurate determination of an actual inlet reactor temperature subject to an additional error.

3.2. Determining the Recuperation Ability

Computational modelling was used to evaluate the recuperative ability of the twin snail recuperator. Simulations were performed over a range of inlet volumetric flow rates at effective methane concentrations of 0.4, 0.7 and 1.0 mole %. The source term in the energy balance equation for the catalytic sections was computed as explained in Section 2. A typical colour contour plot of the temperature distribution is given in Figure 6. As can be seen from this figure, the temperature in the central catalyst section is clearly the maximum, and the recuperation effect is evident.

Table 3. The simulation results from a perfectly insulated system with imposed thermal energy sources for different inlet conditions.

Exp. #	Inlet Flow Rate, m3/s	Mole Fraction Methane	Thermal Energy Generation Rate W/m3	Monolith Inlet Temperature, K
1	1.48 × 10−2	0.01	4.312 × 106	693
2	2.23 × 10−2	0.01	6.528 × 106	646
3	2.97 × 10−2	0.01	8.704 × 106	613
4	3.72 × 10−2	0.01	1.088 × 107	592
5	4.47 × 10−2	0.01	1.306 × 107	576
6	5.20 × 10−2	0.01	1.523 × 107	564
7	5.95 × 10−2	0.01	1.741 × 107	557
8	1.48 × 10−2	0.007	3.046 × 106	575
9	2.23 × 10−2	0.007	4.569 × 106	542
10	2.97 × 10−2	0.007	6.092 × 106	519
11	3.72 × 10−2	0.007	7.515 × 106	504
12	7.50 × 10−3	0.004	8.75 × 105	499
13	1.48 × 10−2	0.004	1.74 × 106	457
14	2.23 × 10−2	0.004	2.61 × 106	438
15	2.97 × 10−2	0.004	3.48 × 106	425
16	3.72 × 10−2	0.004	4.35 × 106	417

summarizes the results from this set of simulations.

We now analyze the data to develop a generalized result. We do this by the application of classical heat transfer analysis. For illustration purposes, even though the proposed system is an integrated design, we can represent it schematically according to Figure 7, as a separate reactor and heat exchanger unit.

Classical heat exchanger analysis includes the log mean temperature difference (LMTD) method, which can be applied to any heat exchanger regardless of flow pattern. Consider the generic heat transfer system illustrated in Figure 7. The box on the left represents the spiral recuperation section. Temperature T_in is the inlet temperature to the system, whilst temperature T_out is the outlet temperature. For a perfectly insulated system, it follows that the difference between these two temperatures is equal to the adiabatic temperature rise. Temperature T₁ is the temperature at the outlet of the inlet recuperator channel, or, in other words, the temperature at the inlet to the catalyst system represented by thermocouple 1 in the experimental system (see Figure 4). Temperature T₁₆ represents the temperature at the outlet of the catalyst section, which is the value recorded by thermocouple 16 (see Figure 4) of the experimental system. It also follows, although perhaps less obviously, that the difference between these two points is also approximately equal to the adiabatic temperature rise. The LMTD method of heat exchanger analysis is valid for any heat exchanger regardless of flow pattern. The transfer of heat in a heat exchanger is governed by the following LMTD design equation:

$$q=F\hspace{0.17em}\hspace{0.17em}U\hspace{0.17em}\hspace{0.17em}A\hspace{0.17em}\hspace{0.17em}\Delta {\overline{T}}_L$$

(15)

where U is the overall heat transfer coefficient, A is the area for heat transfer and F is a correction factor that depends on the flow pattern. Note that F can also be a function of temperature, but for this investigation, it was assumed to be constant. $\Delta {\overline{T}}_L$ is the LMTD:

$$\Delta {\overline{T}}_L={\displaystyle \frac{\left(T_{16}-T_1\right)-\left(T_{\mathrm{out}}-T_{\mathrm{in}}\right)}{\ln \left[\frac{\left(T_{16}-T_1\right)}{\left(T_{\mathrm{out}}-T_{\mathrm{in}}\right)}\right]}}$$

(16)

As noted in the foregoing, the two temperature differences in the numerator will be equal, both having the value of the adiabatic temperature rise, thus leading to an indeterminate solution. In such a case, it is noted that the temperature driving force is thus the same everywhere in the recuperator section, and equals the adiabatic temperature rise. If we assume constant values for the enthalpy of reaction, and the average molar heat capacity, then the relationship between T_in and T_out is

$$T_{\mathrm{out}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{T}_{\mathrm{in}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(-\Delta H_R\right){\displaystyle \frac{Y_{{\mathrm{CH}}_4}}{{\overline{C}}_P}}{X}_{{\mathrm{CH}}_4}$$

(17)

$Y_{{\mathrm{CH}}_4}$ is the mole fraction of methane in the feed and $X_{{\mathrm{CH}}_4}$ is the fractional conversion. Assuming a conversion of 100%, a constant heat capacity of 30 J/(mol K) and a constant enthalpy of reaction of (−802) kJ/mol, the adiabatic temperature rise that corresponds to complete conversion is

$$\Delta T_{\mathrm{ad}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(-\Delta H_R\right){\displaystyle \frac{Y_{{\mathrm{CH}}_4}}{{\overline{C}}_P}}{X}_{{\mathrm{CH}}_4}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(\mathrm{802,000}\right){\displaystyle \frac{Y_{{\mathrm{CH}}_4}}{30}}1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{26,733}\hspace{0.17em}\hspace{0.17em}{Y}_{{\mathrm{CH}}_4}$$

(18)

The adiabatic temperature rise for 1 mole % methane is 267 K, for 0.7% methane it is 187 K, and for 0.4% methane it is 107 K.

Note that it is not necessary to determine each of U, A and F separately, but rather, for a given geometry, we use the product as the defining feature of the system. The heat transferred to the inlet stream (which is equal to the transfer from the exit stream) can be calculated from

$$q=Q\times C_T\times C_P\left(T_1-T_{\mathrm{in}}\right)$$

(19)

We combined the two equations and substituted the values from the CFD simulations. This allowed the calculation of the product (FUA). The results are shown in

Table 4. Apparent heat transfer coefficients for the spiral recuperator design.

Exp. #	Inlet Flow Rate, m3/s	Mole Fraction CH4	Average Temperature at Monolith Inlet, K (T1)	$\mathbf{Temp} \mathbf{Increase}\left({\mathit{T}}_1-{\mathit{T}}_{\mathit{i}\mathit{n}}\right)$	Energy Transfer kW	FUA W/K
1	1.48 × 10−2	0.01	693	393	7.1	26.6
2	2.23 × 10−2	0.01	646	346	9.4	35.3
3	2.97 × 10−2	0.01	613	313	11.3	42.4
4	3.72 × 10−2	0.01	592	292	13.2	49.5
5	4.47 × 10−2	0.01	576	276	15.0	56.3
6	5.20 × 10−2	0.01	564	264	16.7	62.7
7	5.95 × 10−2	0.01	557	257	18.6	69.8
8	1.48 × 10−2	0.007	575	275	4.97	26.6
9	2.23 × 10−2	0.007	542	242	6.59	35.3
10	2.97 × 10−2	0.007	519	219	7.92	42.4
11	3.72 × 10−2	0.007	504	204	9.24	49.5
12	7.50 × 10−3	0.004	499	199	1.82	17.0
13	1.48 × 10−2	0.004	457	157	2.84	26.6
14	2.23 × 10−2	0.004	438	138	3.76	35.3
15	2.97 × 10−2	0.004	425	125	4.23	42.4
16	3.72 × 10−2	0.004	417	117	5.30	49.5
17	3.52 × 10−3	0.0096	837	537	2.30	8.95
18	5.16 × 10−3	0.0096	774	474	2.98	11.59
19	7.03 × 10−3	0.0096	737	437	3.74	14.57
20	8.67 × 10−3	0.0096	718	418	4.42	17.19
21	1.17 × 10−2	0.0096	687	387	5.53	21.51
22	1.41 × 10−2	0.0096	680	380	6.51	25.34
23	1.73 × 10−2	0.0096	662	362	7.65	29.77
24	2.18 × 10−2	0.0096	628	328	8.71	33.90
25	2.34 × 10−2	0.0096	620	320	9.14	35.57
26	3.28 × 10−2	0.0096	585	285	11.40	44.35

.

From Table 4 it can be seen that the product (FUA) depends only on the flow rate, and not on the temperature or the methane concentration. This result is quite significant, because it allows for a simple method of calculating the recuperation, once the value of (FUA) is determined for the specified configuration.

Based on the preceding observations, further simulations were performed to allow for the development of a good correlation to predict the inlet catalyst temperature for a given flow rate and methane feed concentration. These simulations, which are also given in Table 4 as Experiments 17 to 26, were performed using 0.96% methane, a concentration for which the adiabatic temperature rise was 257 K. These additional simulations added more accuracy to the correlation at lower flow rates. The results are presented graphically in Figure 8 for the volumetric flow rate referenced at 300 K and 101.325 kPa.

The data presented in Table 4 and Figure 8 can be correlated using a quartic equation, to yield

$$\left(F\hspace{0.17em}U\hspace{0.17em}A\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2.8348\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}1892.82\hspace{0.17em}Q\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\mathrm{24,902}\hspace{0.17em}{Q}^2+\mathrm{202,965}\hspace{0.17em}{Q}^3$$

(20)

where the flow rate, Q, has units of m³/s. Equation (20) can be combined with Equations (15), (18) and (19) to provide for a prediction of the inlet temperature at the catalyst:

$$T_1=T_{\mathrm{in}}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\left({\displaystyle \frac{62.181}{Q}}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\mathrm{41,519}\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\mathrm{546,225}\hspace{0.17em}Q+\mathrm{4,452,037}\hspace{0.17em}{Q}^2\right)\hspace{0.17em}\hspace{0.17em}{Y}_{{\mathrm{CH}}_4}$$

(21)

Table 5. A comparison of the inlet temperature at the catalytic monolith inlet predicted by the CFD simulation with the one predicted by the correlation, Equation (23).

Inlet Flow Rate, m3/s	Mole Fraction CH4	Temperature at Catalyst Inlet, K (T1), CFD	Temperature at Catalyst Inlet, K (T1), Correlation
1.48 × 10−2	0.01	693	686
2.23 × 10−2	0.01	646	643
2.97 × 10−2	0.01	613	613
3.72 × 10−2	0.01	592	590
4.47 × 10−2	0.01	576	574
5.20 × 10−2	0.01	564	563
5.95 × 10−2	0.01	557	558
1.48 × 10−2	0.007	575	570
2.23 × 10−2	0.007	542	540
2.97 × 10−2	0.007	519	519
3.72 × 10−2	0.007	504	503
7.50 × 10−3	0.004	499	484
1.48 × 10−2	0.004	457	454
2.23 × 10−2	0.004	438	437
2.97 × 10−2	0.004	425	425
3.72 × 10−2	0.004	417	416
3.52 × 10−3	0.0096	837	850
5.16 × 10−3	0.0096	774	788
7.03 × 10−3	0.0096	737	749
8.67 × 10−3	0.0096	718	725
1.17 × 10−2	0.0096	687	694
1.41 × 10−2	0.0096	680	675
1.73 × 10−2	0.0096	662	655
2.18 × 10−2	0.0096	628	632
2.34 × 10−2	0.0096	620	625
3.28 × 10−2	0.0096	585	591

shows the comparison of the inlet catalyst temperature obtained from the CFD simulations with those from the simple correlation. Note that this relationship does contain some simplifying assumptions, so will not provide an exact answer. Equation (21) gives an approximate value of the inlet temperature, with a maximum error of about six degrees for inlet flow rates greater than 1.0 × 10⁻² m³/s.

To facilitate analysis, an equation can be curve-fit to the data shown in Table 2 and Figure 5. The ignition temperature required can be correlated to the GHSV using an equation of the following form:

$${\left(T_1\right)}_{\mathrm{ig}}=a\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}b\hspace{0.17em}\hspace{0.17em}\ln \left({\displaystyle \frac{\mathrm{GHSV}}{1000}}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}c\right)$$

(22)

where the GHSV has units of h⁻¹. The values of the constants in Equation (22) for the six cases represented in Figure 5 are given in

Table 6. Constants used in Equation (23) for the prediction of ignition temperature.

Methane Fraction	a	b	c
Fresh
0.01	394.4	47.67	4.807
0.007	407.8	51.58	4.104
0.004	449.2	50.54	2.977
Aged
0.01	394.9	58.45	5.447
0.007	417.8	60.93	4.392
0.004	461.6	59.44	3.510

. Equation (22) can be written explicitly in terms of volumetric flow rate and catalyst length. We recall that the catalyst monolith measures 50 mm by 150 mm, to give a cross-sectional area of 7.5 × 10⁻³ m². Thus,

$${\left(T_1\right)}_{\mathrm{ig}}=a\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}b\hspace{0.17em}\hspace{0.17em}\ln \left(480{\displaystyle \frac{Q}{L}}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}c\right)$$

(23)

The catalyst length, L, has units of m and the flow rate has units of m³/s.

Equations (21) and (23) provide a useful set of design equations that can be used to evaluate the operating limits of the unit for a given system inlet temperature, catalyst length, feed flow rate and methane concentration. Figure 9 shows a plot of the inlet catalyst temperature required to achieve 99.9% methane conversion for an inlet methane mole fraction of 0.01 for the fresh catalyst at three catalyst lengths. The monolith lengths of 150, 300 and 450 mm correspond to the amount of catalyst that can be inserted into one, two and three channels, respectively. The temperatures required are shown as a function of the inlet flow rate referenced at 300 K and 101.325 kPa, with the system inlet temperature of 300 K. Also shown in Figure 9 is the temperature achieved at the catalyst inlet following the recuperation (the recuperation temperature), at a methane mole fraction of 0.01. Provided that the recuperation temperature is greater than the ignition temperature, the unit will function and the methane will be converted. The point of intersection of the line representing the necessary ignition temperature and recuperation temperature is the maximum flow rate possible to achieve methane conversion at the corresponding flow rate. The maximum flow rate for the simulated conditions are given in

Table 7. Maximum flow rate that will give 99.9% conversion of methane for various methane concentrations and catalyst activity levels.

Catalyst State	Methane Mole Fraction	Catalyst Length, mm	Maximum Flow Rate, m3/s
Fresh	0.01	150	0.0296
		300	0.0365
		450	0.0416
Aged	0.01	150	0.0216
		300	0.0272
		450	0.0311
Fresh	0.007	150	0.0105
		300	0.0137
		450	0.0159
Aged	0.007	150	0.0075
		300	0.0098
		450	0.0114

. Note that for a methane mole fraction of 0.004, the unit cannot operate at any flow rate above 1 m³/s and the results are not shown.

The trends observed are consistent with the expected behaviour. For the recuperation section, the heat transfer coefficient increases with increasing flow rate, which provides a better driving force for the heat transfer, which is beneficial. At the same time, an increasing flow rate requires more energy to be transferred to the incoming stream, and also increases the catalyst ignition temperature. The resulting trade-off determines the operating limits on the performance of the unit.

A final comment on these results is that they are valid for an inlet temperature of 300 K. Increasing this temperature through an additional heat transfer device, for example, with an external heat exchanger, will extend the operating range.

3.3. Experimnetal Validation

Seven experiments were performed using the methodology described in Section 2. Complete details of all of the experiments can be found in the Supplementary Information. The objective of the experiments was to determine and evaluate the heat transfer characteristics of the unit, and to compare the actual performance to that predicted by the CFD model.

Table 8. The temperatures at the inlet and outlet of the system and the catalyst section. The flow rates are corrected to 300 K and 101,325 Pa.

Data Point	Time, h	Mole % Methane	Flow Rate m3/s	T in K	T out K	T1 K	T16 K
Experiment 1
1	10	0.40	0.0082	576	648	728	827
2	13	0.40	0.0096	475	563	633	739
3	21	0.60	0.0072	447	580	691	833
4	24	0.60	0.0070	395	536	644	796
5	26	0.62	0.0076	368	515	620	778
6	31	0.70	0.0070	368	539	672	844
7	36	0.70	0.0036	326	505	631	810
Experiment 2
1	5.5	0.40	0.0031	579	653	743	843
2	8.8	0.40	0.0045	440	539	614	728
3	11.6	0.60	0.0076	449	588	713	860
4	16	0.60	0.0083	366	519	633	793
5	18.5	0.70	0.0064	370	543	684	856
6	23	0.70	0.0044	327	512	647	829
Experiment 3
1	14	0.41	0.0126	567	637	721	810
2	16	0.40	0.0130	526	601	679	772
3	18	0.60	0.0102	518	626	748	868
4	20	0.60	0.0118	446	568	679	809
5	23	0.70	0.0113	438	586	724	868
6	27	0.70	0.0160	314	479	588	767
Experiment 4
1	6	0.43	0.0087	602	664	764	839
2	9.5	0.70	0.0087	474	586	746	869
3	13.5	0.70	0.0097	374	517	666	807
4	16	0.90	0.0077	383	547	756	905
5	18.5	0.90	0.0083	315	508	710	873
Experiment 5
1	8	0.70	0.0116	381	551	693	841
2	11.9	0.70	0.0124	311	473	608	772
3	12.4	0.70	0.0142	311	477	604	776
4	13	0.70	0.0169	312	480	596	776
5	13.4	0.70	0.0209	313	485	589	787
Experiment 6
1	23	0.40	0.0132	477	560	636	722
2	28	0.60	0.0097	437	591	725	832
3	34	0.80	0.0091	438	579	757	885
4	43	0.80	0.0125	309	478	626	790
Experiment 7
1	13	0.40	0.0071	525	557	647	719
2	22	0.60	0.0073	392	489	625	731
3	29	0.70	0.0067	399	498	660	769
4	35	0.70	0.0084	307	442	586	736
5	39	0.90	0.0058	315	471	687	813

shows the temperatures at the inlet and the outlet of both the entire unit and the catalyst section for some steady-state operating points for all of the seven experiments.

The numerical simulations assumed that the system was adiabatic. In practice, the system loses energy to the surroundings in spite of the insulation. We estimated the amount of energy lost by the systems to the surroundings. Although this information is not directly transferable to a scaled-up system, it provides useful information. The energy balances are based on the temperature differences observed between the inlet and the outlet of the system. It should be noted that these temperatures are single-point temperatures measured at the centre of the duct, and thus may not be exactly the same as the average bulk temperature of the entire flowing gas stream.

The calculation of the energy lost is performed by computing the outlet temperature expected using the adiabatic temperature rise equations given earlier and then comparing them to the ones measured in the experiments. The energy loss is then computed from the temperature difference, the heat capacity and the flow rate. The results are summarized in

Table 9. Heat losses from the unit at various steady-state operating points.

Data Point	Flow Rate m3/s	Mole % Methane	Combustion Energy, W	ΔT Obs.	ΔT Adi.	Heat Loss, W	% Loss
Experiment 1
1	0.0082	0.40	1073	73	107	340	32
2	0.0096	0.40	1249	88	107	221	18
3	0.0072	0.60	1414	132	160	250	18
4	0.0070	0.60	1368	141	160	166	12
5	0.0076	0.62	1525	147	166	172	11
6	0.0070	0.70	1600	171	187	138	9
7	0.0036	0.70	821	179	187	36	4
Experiment 2
1	0.0031	0.40	397	74	107	122	31
2	0.0045	0.40	591	98	107	49	8
3	0.0076	0.60	1489	139	160	199	13
4	0.0083	0.60	1626	153	160	75	5
5	0.0064	0.70	1467	173	187	111	8
6	0.0044	0.70	1000	185	187	11	1
Experiment 3
1	0.0126	0.41	1685	70	110	609	36
2	0.0130	0.40	1699	75	107	507	30
3	0.0102	0.60	1991	108	160	650	33
4	0.0118	0.60	2303	122	160	551	24
5	0.0113	0.70	2570	148	187	537	21
6	0.0160	0.70	3649	165	187	432	12
Experiment 4
1	0.0087	0.43	1224	62	115	564	46
2	0.0087	0.70	1980	93	187	996	50
3	0.0097	0.70	2216	143	187	523	24
4	0.0077	0.90	2273	164	241	723	32
5	0.0083	0.90	2444	193	241	483	20
Experiment 5
1	0.0116	0.70	2642	169	187	256	10
2	0.0124	0.70	2832	162	187	380	13
3	0.0142	0.70	3227	166	187	364	11
4	0.0169	0.70	3866	168	187	395	10
5	0.0209	0.70	4763	172	187	385	8
Experiment 6
1	0.0132	0.40	1725	83	107	386	22
2	0.0097	0.60	1893	154	160	76	4
3	0.0091	0.80	2372	140	214	819	35
4	0.0125	0.80	3254	169	214	683	21
Experiment 7
1	0.0071	0.40	925	32	107	648	70
2	0.0073	0.60	1427	97	160	564	40
3	0.0067	0.70	1536	99	187	723	47
4	0.0084	0.70	1912	135	187	533	28

.

The energy losses vary; however, the trend is that more energy is lost at higher reactor temperatures. As the inlet methane concentration and the inlet flow rate increase, there is a trend to lower heat loss. This observation is consistent with classical heat transfer principles. The driving force for heat loss is the temperature difference between the system and the surroundings. As the methane concentration was increased, the inlet temperature was decreased, so in many experiments the average system temperature was constant. However, with a higher methane concentration, more thermal energy is generated in the reactor, and therefore the heat loss, which remains more or less constant, is a smaller percentage of the thermal energy generated. Overall, the heat loss at high flow rate and methane concentration is acceptable, and is expected to be lower in a full-sized unit.

Next, we calculated the experimental values of the constant FUA from the data. We used the LMTD based on the inlet and outlet temperatures from the system, and the inlet and outlet temperatures from the catalyst section. The heat transfer is then described by

$$q=F\hspace{0.17em}\hspace{0.17em}U\hspace{0.17em}\hspace{0.17em}A\hspace{0.17em}\hspace{0.17em}\Delta {\overline{T}}_L=F\hspace{0.17em}\hspace{0.17em}U\hspace{0.17em}\hspace{0.17em}A\hspace{0.17em}\hspace{0.17em}\left[{\displaystyle \frac{\left(T_{16}-T_1\right)-\left(T_{\mathrm{out}}-T_{\mathrm{in}}\right)}{\ln \left[\frac{\left(T_{16}-T_1\right)}{\left(T_{\mathrm{out}}-T_{\mathrm{in}}\right)}\right]}}\right]$$

(24)

The heat transfer for a given stream is computed as before; the heat transferred to the inlet and outlet streams are

$$q_{\mathrm{inlet}}=Q\times C_T\times C_P\left(T_1-T_{\mathrm{in}}\right)$$

(25)

$$q_{\mathrm{outlet}}=Q\times C_T\times C_P\left(T_{16}-T_{\mathrm{out}}\right)$$

(26)

For the adiabatic reactor, these two values are the same, as noted earlier. However, for the non-adiabatic system they are different; therefore, for the calculation of FUA we used the mean value. The results are shown in

Table 10. The experimentally determined values of the constant FUA for the steady-state points of the seven experiments. The values predicted by Equation (20) are also shown.

Data Point	LMTD K	Flow Rate m3/s	q Inlet W	q Outlet W	q Mean W	FUA Exp	FUA Model
Experiment 1
1	85.1	0.0082	1527	1803	1665	19.6	16.8
2	96.6	0.0096	1842	2053	1947	20.2	18.9
3	137.2	0.0072	2150	2235	2193	16.0	15.3
4	146.4	0.0070	2123	2212	2168	14.8	14.9
5	152.4	0.0076	2319	2420	2370	15.5	15.8
6	171.4	0.0070	2599	2607	2603	15.2	15.0
7	179.1	0.0036	1340	1338	1339	7.5	9.3
Experiment 2
1	86.1	0.0031	610	707	658	7.6	8.4
2	105.9	0.0045	962	1049	1006	9.5	10.9
3	142.3	0.0076	2449	2521	2485	17.5	15.9
4	156.6	0.0083	2705	2773	2739	17.5	17.0
5	172.5	0.0064	2455	2455	2455	14.2	14.0
6	183.0	0.0044	1709	1696	1703	9.3	10.7
Experiment 3
1	79.0	0.0126	2375	2649	2512	31.8	23.2
2	83.4	0.0130	2430	2714	2572	30.8	23.7
3	114.2	0.0102	2853	3007	2930	25.7	19.8
4	125.8	0.0118	3346	3460	3403	27.1	22.0
5	146.2	0.0113	3917	3868	3893	26.6	21.3
6	172.0	0.0160	5353	5612	5483	31.9	27.6
Experiment 4
1	68.2	0.0087	1720	1863	1791	26.3	17.6
2	117.5	0.0087	2882	2996	2939	25.0	17.5
3	141.7	0.0097	3456	3439	3448	24.3	19.1
4	156.2	0.0077	3524	3372	3448	22.1	16.1
5	177.0	0.0083	4017	3707	3862	21.8	17.0
Experiment 5
1	158.4	0.0116	4404	4107	4255	26.9	21.7
2	163.4	0.0124	4499	4529	4514	27.6	22.9
3	168.5	0.0142	5051	5156	5103	30.3	25.2
4	174.2	0.0169	5857	6107	5982	34.3	28.7
5	184.9	0.0209	7029	7678	7353	39.8	33.4
Experiment 6
1	85.0	0.0132	2573	2615	2594	30.5	24.0
2	129.1	0.0097	3403	2845	3124	24.2	19.0
3	134.1	0.0091	3537	3394	3465	25.8	18.1
4	166.3	0.0125	4817	4738	4777	28.7	23.0
Experiment 7
1	48.9	0.0071	1057	1401	1229	25.1	15.1
2	101.9	0.0073	2072	2149	2111	20.7	15.4
3	103.8	0.0067	2136	2220	2178	21.0	14.5
4	142.0	0.0084	2854	3006	2930	20.6	17.1
5	140.7	0.0058	2647	2431	2539	18.0	13.1

and plotted in Figure 10.

At the lowest flow rates, the difference between the predicted and measured FUA values is small, but at higher flow rates the experimental values are up to 20% larger than the CFD predictions.

We now consider the CFD simulations that replicate Experiments 1 and 5. These two experiments represent between them a range of methane concentrations (Experiment 1) and a wide range of flow rates at a constant methane concentration (Experiment 5). We consider the steady-state operating points. Computational simulations were performed for the experimental conditions of flow rate and inlet temperature. We compared the results obtained from the CFD simulations (which are insulated) and then the experimental results, which had some heat loss to the surroundings.

Table 11. The steady-state points of operation used for the analysis of Experiments 1 and 5. The results shown were obtained from CFD analysis of the reactor, assuming that it is perfectly insulated.

Data Point	Flow Rate m3/s	Mole % Methane	System Inlet T, K	Catalyst Inlet T, K, Exp	Catalyst Inlet T K, CFD
Experiment 1
1	8.23 × 10−3	0.4	575	728	743
2	9.58 × 10−3	0.4	475	633	635
3	7.23 × 10−3	0.6	447	691	706
4	7.00 × 10−3	0.6	395	644	656
5	7.55 × 10−3	0.62	368	620	635
6	7.02 × 10−3	0.7	368	672	670
7	3.60 × 10−3	0.7	326	631	695
Experiment 5
1	1.16 × 10−2	0.70	381	693	655
2	1.24 × 10−2	0.70	311	608	580
3	1.42 × 10−2	0.70	311	604	575
4	1.70 × 10−2	0.70	312	595	563
5	2.09 × 10−2	0.70	313	589	545

shows the results obtained from these CFD simulations.

Comparing the results from the experiments and the CFD simulations, some observations can be made. For Experiment 1, the experimental and simulated temperatures agree well. The experimental values typically fall below the predicted ones, which is expected because there is heat loss in the experimental unit. The monolith inlet temperature predicted by the correlation is within the error of the correlation. For Experiment 5, the experimental temperatures observed at the inlet to the catalyst section is consistently higher than the value predicted by the CFD. These observations are reasonable. For Experiment 1, the flow rate was low and the percentage of heat loss was relatively high; thus, a lower than predicted inlet catalyst temperature is not surprising. For Experiment 5, there was a lower percentage of heat loss because of the higher feed flow rate, and at these flow rates the FUA value is observed to be higher than the model; thus, a higher inlet catalyst temperature is also reasonable.

4. Concluding Remarks

The detailed results presented in Section 3 clearly demonstrate the efficacy of the proposed design for the application of combustion of ventilation air methane. The agreement between the experimental operation and the computational model predictions is very good, and certainly justifies this approach. The construction of such a prototype is quite time-consuming and costly, and therefore the use of CFD to reduce the prototyping is a very useful and cost-saving measure.

The CFD simulations are not very time-consuming to perform, although setting up the problem correctly is important. We believe that using a set of complex CFD simulations to develop simplified design equations is a good approach. Clearly, being able to predict the operation of the prototype does require a detailed knowledge of the catalyst characteristics, and that is likely the most time-consuming part of the investigation. However, this step must be performed in any design workflow.

A final point to note is that we have only considered one arrangement for the recuperator. Having demonstrated that the computational model is able to give reliable results, the method can be used to perform more in-depth shape optimization, as well as provide scale-up guidelines. These topics will be addressed in future papers concerning this reactor concept.