Parametric Study of Solid-Phase Axial Heat ... - ACS Publications

9040

Ind. Eng. Chem. Res. 2008, 47, 9040–9054

Parametric Study of Solid-Phase Axial Heat Conduction in Thermally Integrated Microchannel Networks Angela Moreno,† Kevin Murphy,‡ and Benjamin A. Wilhite*,† Department of Chemical, Materials and Biomolecular Engineering and Connecticut Global Fuel Cell Center and Department of Mechanical Engineering, UniVersity of Connecticut, Storrs, Connecticut 06269-3222

A parametric study is presented to highlight design challenges of thermally integrated microchannel networks for portable chemistry and/or fuels reforming. One-dimensional modeling analysis of heat transfer in a twofluid system is presented for the case of (i) two nonreacting fluids (heat exchanger), (ii) a single exothermic reacting fluid and a second nonreacting fluid (regenerative combustor), and (iii) one exothermic reacting fluid and a second endothermic reacting fluid (heat exchanger reactor). In each case, the influence of solid-phase thermal conductivity and thermal packaging upon thermal efficiency, reaction conversion, and steady-state multiplicity is investigated. Results demonstrate the importance of both packaging and solid-phase axial thermal conduction upon system performance, with optimal performance obtained using low thermal conductivity substrates. Modeling analysis predicts steady-state multiplicity when employing low thermal conductivity materials, illustrating the need for future detailed stability analysis. Lastly, simplified mechanical analysis is presented to illustrate the value of coupled thermomechanical analysis. Introduction The use of microreactors for process intensification has been widely investigated in the past decade owing to order-ofmagnitude improvements in mass and heat transfer rates over their macroscale analogues, effectively removing fluid-solid transport limitations.1,2 Additionally, microtechnology has enabled creation of portable low weight and volume (high energy density) chemical processors for several applications.3–5 For these reasons, thermally integrated microreactors and/or microchannel networks have been widely investigated for multistage reforming of hydrocarbon fuels to hydrogen gas as part of an overall portable-power system.4,6 In these systems, two or more process fluids with opposing heat duties are allowed to exchange heat through the microreactor solid-phase (substrate). Two leading issues in the design of such microreactor systems are substrate selection and packaging methods and materials. Substrate thermal conductivity may be selected to facilitate large thermal gradients and/or localized hot-spot formation or to achieve isothermal operation by tuning the rate of axial heat conduction. Packaging materials likewise may be selected to either facilitate or hinder heat removal to further manipulate system thermal efficiency. Microreactors reported to date have been fabricated from a variety of substrates ranging from high thermal conductivity materials such as silicon7–9 and stainless steel10,11 to low thermal conductivity ceramics.12–15 Microchannel networks can be packaged with insulating materials or under vacuum conditions16–18 or interfaced with noninsulating distributors and/or housings.19,20 Several reports of microreactors employing noninsulating packaging and high thermal conductivity materials for isothermal and safe operation of highly exothermic reactions9,21,22 and isothermal kinetic studies6,20 have appeared in the literature to date. Likewise, microchannel networks employing insulating materials and/or packaging have been recently reported for efficient multistage reforming6,7,17,23,24 * To whom correspondence should be addressed. E-mail: bwilhite@ engr.uconn.edu. Tel: (860)-486-3689. Fax: (860)-486-2959. † Department of Chemical, Materials and Biomolecular Engineering and Connecticut Global Fuel Cell Center. ‡ Department of Mechanical Engineering.

of hydrocarbon fuels to hydrogen-rich mixtures for use in portable-power systems. Hydrogen gas has been identified as a promising alternative fuel in light of its relative abundance and emission-free combustion. Efficient, large-scale reactors capable of refining a variety of fuels to hydrogen are required for long-term development of a proposed hydrogen energy infrastructure,25,26 while a distributed network of smaller hydrogen reformers are required in the short term.26 Compact, multifunctional reactors have been developed for portable-power systems to produce hydrogen on site for utilization by proton-exchange membrane (PEM) or solid-oxide fuel cells (SOFC’s) in portable electronics devices7,27 and automotive applications.28–30 Portable reforming of hydrocarbon fuels is predominantly a multistage affair, generally comprised of two or more of the following processes: (i) fuel vaporization and/or preheating, (ii) endothermic and/or exothermic reforming reactions (e.g., oxidative or steam reforming), (iii) exothermic combustion to provide sufficient heat for autothermal operation, and (iv) heat exchangers for effluent or waste heat recovery. Thermally integrated reformer designs couple two or more of these processes within a single device to maximize overall thermal efficiency. The effective coupling of two processes with opposing heat duties in microsystems can be achieved through several configurations. Heat transfer from hot product gas to cold reagent gas can be accomplished by a heat-exchanger configuration.7,11,31 Heat transfer between an exothermic reacting fluid and cold reagent gas (heat removal stream) can be accomplished via a regenerative combustor configuration.10,11,32 Multiple routes exist for direct coupling of endothermic and exothermic reactions, including (i) direct coupling (directly coupled adiabatic reactor) where both reactions take place in the same bed,33–35 (ii) regenerative coupling (reverse-flow reactor) where exothermic and endothermic reactions occur periodically in the same catalyst phase,36–38 and by recuperative coupling (heat exchanger reactor configuration) where the endothermic and exothermic reactions are spatially separated.7,39–42 This latter route is the most popular in heat-integrated portable reformer designs, given the advantages of hot-spot formation and/or manipulation for maximum thermalefficiency,designflexibility,andsimplicityofoperation.43–45

10.1021/ie8001638 CCC: $40.75  2008 American Chemical Society Published on Web 07/09/2008

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9041

analysis of the ceramic minichannel heat-exchanger reactor did not explicitly describe the solid-phase, preventing analysis of solid-phase axial conduction or resulting solid-phase thermal profiles,36 in turn preventing mechanical analysis to predict failure. These previous studies illustrate the value of performing a parametric study focused upon solid-phase axial heat conduction to identify the influence of packaging and solid-phase properties upon the performance of heat-integrated mini- and microchannel networks. The present study aims to investigate the effects of key kinetics and design parameters, primarily the effects of solidphase heat conduction and thermal packaging, upon the performance and behavior of thermally integrated microsystems. This is accomplished through analysis of a simplified model of three heat integration schemes, specifically (i) heat transfer between two nonreacting fluids (microscale heat exchanger), (ii) heat transfer between one exothermically reacting fluid and one nonreacting fluid (microscale regenerative combustion), and (iii) heat transfer between one exothermic reacting fluid and one endothermic reacting fluid (microscale heat exchanger reactor). A simplified one-dimensional model is used to describe a system composed of two fluids in thermal contact via an intermediate solid-wall phase capable of axial heat conduction. By neglecting fluid-solid mass transfer and dispersion effects and assuming simple kinetics, the present analysis is able to focus solely upon the influence of packaging and solid-phase conduction, in turn allowing a closer examination of these key design parameters. Analysis of heat transfer in the absence of chemical reaction is reviewed to demonstrate the importance of axial conduction and packaging heat losses upon thermal efficiency in heat transfer microsystems. Subsequent analysis of reacting fluids identifies design concerns for maintaining high thermal efficiencies and identifies potential conditions for steadystate multiplicity. Lastly, mechanical analysis of thermal profiles obtained for the present study allows discussion of practical operating limits on thermally integrated microsystems.

Figure 1. Schematic of simplified one-dimensional model describing heat transfer between two parallel microchannels with solid-phase heat conduction in axial direction.

In each of these configurations, materials and packaging selection are critical to ensuring high thermal efficiencies. The importance of solid-phase axial conduction in counterflow heat exchangers has already been established in the literature. Peterson integrated thermal energy transport in a microscale counterflow heat exchanger via a one-dimensional numerical model accounting for solid-phase axial conduction46 and found that low thermal conductivity materials are necessary to achieve reasonable thermal efficiencies by reducing axial conduction losses. Similar analysis by Steif and co-workers31 suggested an optimum solid-phase thermal conductivity corresponding to glass and ceramic materials. A number of past studies have been reported in the analysis of regenerative combustors47–49 employing overall fluid heat transfer coefficients in models which neglect the influence of solid-phase axial conduction. Likewise, modeling analysis of the effective coupling of exothermic and endothermic reactions for co- and countercurrent configuration has been reported in the literature, demonstrating the importance of overall heat transfer coefficients and residence time upon reactor performance and hot-spot formation while neglecting solid-phase axial heat conduction.43 Several recent modeling studies of micro- and minichannel networks have since argued the importance of solid-phase axial heat conduction. Deshmuck and Vlachos41,42 emphasized the significance of material construction in thermally integrated devices, specifically in heat recirculation, hot-spot magnitude, and thermomechanical stability for coupled combustor-reformer devices. Frauhammer and co-workers demonstrated efficient thermal coupling between exothermic and endothermic reforming of methane40 using an alumina minichannel network (monolith) packaged such that it operated as a heat-exchanger reactor. While the low thermal conductivity ceramic substrate enabled hot-spot formation, mechanical failure was observed as a result of thermal stresses when operating at axial gradients in excess of 600 K.40 Modeling

Model Development The presented analysis is focused upon two parallel microchannels separated by an impermeable wall, shown schematically in Figure 1. A unique fluid is supplied to each channel at fixed velocity (ui)1,2) and inlet temperature (Ti)1,2) in either coor countercurrent configuration. Each fluid experiences chemical reaction and/or heat transfer with the dividing wall. The dividing

Table 1. Determination of Parameter Values for Microchannel Heat Transfer Model

fluid fluid fluid fluid

fluid properties

value

-3

1.2 1.8 × 10-5 1000 0.025

density (kg m ) viscosity (Pa · s) heat capacity (J kg-1 K-1) thermal conductivity (W m-1 K-1) microchannel properties

extruded monoliths

hydraulic diameter (m) flow length (m) surface area (m2) volume (m3) a (m-1) residence time (s) heat transfer coefficient (W m-2 K-1) NTU substrate properties thermal conductivity (W m-1 K-1)

2

CP

0.007

1.5 × 10-3 5 × 10-2 3 × 10-4 1.1 × 10-7 2700 0.1 32.5 7.2

0.01 70 1.5 200 0.7

2 0.07

200 6.61

microfabricated devices

1 15 33 2 0.7

200 66

0.5 × 10-3 1 × 10-2 2 × 10-5 2.5 × 10-9 8000 0.1 46.9 31

0.01 101 6.7 2 0.165

200 16.5

1 21.8 144

2

200

2

200

16.5

165

16.5

1651

9042 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008

Figure 2. Results for isothermal microscale heat exchanger model: (a) effectiveness and (b) conduction losses as a function of NTU and CP; corresponding thermal profiles for NTU ) 5 at (c) CP ) 0.1, (d) CP ) 1, and (e) CP ) 10.

wall in turn exchanges heat with both fluids and also with its surroundings via packaging conditions at either end of the device. A general pseudo-homogeneous one-dimensional steady-state model is developed to describe plug-flow in a uniform microchannel, assuming incompressible flow and neglecting mass transfer limitations between catalyst and fluid and internal diffusion limitations within the catalyst phase. In order to focus the present analysis solely upon the effect of solid-phase heat dispersion on heat transfer efficiency, the further assumption of negligible axial dispersion is also employed. The steadystate energy balance governing each fluid thus accounts for heat generated via chemical reaction, heat transfer between fluid and the solid-wall (eq 1). (FiuiCpi

[ ]

dTi -Eai ) hiai(Tw - Ti) + k0,iCni [-∆Hrxn,i(Ti)] exp dz RTi (1)

For the present analysis, which focuses upon the effect of solidphase heat dispersion on heat transfer efficiency, a simplified irreversible, power law kinetic model is employed. The mass balance for fluid i, accounting for reaction in the absence of mass transfer limitations and with axial convection in either co- or countercurrent operation, is given by eq 2: (

[ ]

-Eai dCi ) -k0,iCni exp dz RTi

2

dz

N

)

hidi

∑k A i)1

(Tw - Ti)

Tw ≡ fixed at z ) 0, L

(3)

w w

The resulting simplified system of ordinary differential equations allows a direct investigation of the effects of heat dissipation via solid-phase conduction upon heat transfer between two

(3a)

dTw ) 0 at z ) 0, L (3b) dz Temperature, concentration, and position within the microchannel network are rendered dimensionless as follows:

(2)

The dividing wall separates the two reacting fluids while providing a pathway for heat transfer. The cross section of the solid-phase at any given axial position is assumed to be thermally uniform for the present analysis, owing to submillimeter wall thicknesses employed in microreactors. The resulting one-dimensional model describing the dividing wall accounts for axial thermal conduction and heat addition/removal by the two fluid-phases (eq 3): d2Tw

unique fluids, in the absence of more complex and/or secondary effects (fluid-phase dispersion, channel geometry, and wall geometry). The influence of packaging upon heat dissipation in this simplified microchannel network is modeled by appropriate selection of boundary conditions for the solid-phase. The presented analysis considers the two ideal cases of (i) isothermal and (ii) adiabatic packaging of the microchannel network. Isothermal boundary conditions assume that the wall temperature is fixed at either extrema and is defined in terms of the packaging. For countercurrent systems, wall temperatures are assumed equal to the corresponding inlet fluid temperature at each extrema. For cocurrent systems, wall temperatures are assumed equal to the weighted-average temperature of the two fluids at each extrema. Adiabatic boundary conditions assume that the first derivative of the wall temperature at each extrema is fixed and equal to zero; i.e., heat conduction out of the system is not allowed. In general, the isothermal and adiabatic boundary conditions may be written as

θi )

Ti(z) - Tmin , Tmax - Tmin

Yi )

Ci(z) , CiO

z˜ )

z L

(4)

The dimensionless temperature θi is defined in terms of the maximum and minimum achievable temperatures within the heat-exchanger system. The resulting general dimensionless expressions for M fluids exchanging heat through a single, radially uniform wall are ( (

[ ()

dYi -Ri ) -DaiYi exp dz˜ 1 + γ1θi

] [

dθi γi -Ri ) NTUi(θw - θi) + DaiYi exp dz˜ γ1 1 + γ1θi d2θw dz˜

2

M

)

∑ i)1

NTUi (θ - θi) CPi w

(1′)

]

(2′)

(3′)


Figure 3. Results for adiabatic microscale heat exchanger: (a) effectiveness as a function of NTU and CP; corresponding thermal profiles for NTU ) 5 at (b) CP ) 0.1, (c) CP ) 1, and (d) CP ) 10.

Figure 4. Results for microscale regenerative combustor model. Thermal efficiency as a function of NTU and CP for (a) isothermal packaging conditions and (b) adiabatic packaging conditions. Da ) 50, γ ) 2, R ) 10.

From eq 2′, the kinetics of the ith reacting fluid is described by the Damkohler number (Dai), dimensionless activation coefficient (Ri), and dimensionless heat of reaction (γi). Fluid-solid heat transfer is described by the number of transfer units (NTUi), while the solid-phase axial conduction is described by the conduction parameter (CPi), following previous analysis of nonreacting fluids by Peterson46 and Stief and co-workers.31 The resulting dimensionless model is used to investigate the impact of packaging and solid-phase axial heat conduction upon heat transfer applications in microchannels networks for both reacting and nonreacting fluids. Value ranges for each parameter governing this system were estimated on the basis of typical microchannel dimensions and different solid-phase material properties. Fluid-solid heat transfer coefficients are estimated assuming both gaseous and liquid phases, while fluid-solid heat transfer coefficients are estimated from the correlations of Sieder and Tate.50 Details of parameter space estimation for representation of physical and kinetic parameters are presented in Table 1.

Numerical Methods The above system of equations is solved numerically over a range of kinetic (Da, γ,R) and design parameters (NTU, CP) via a marching technique combined with a shooting method to satisfy solid-phase boundary conditions and inlet fluid conditions of the counterflow fluids (when applicable). Marching technique solver is based on a modified second-order Rosenbrock formula prepackaged in Matlab (ode23s). The shooting method employs a Nelder-Mead simplex algorithm to simultaneously search all unknown initial conditions to satisfy all upper-boundary conditions (fminsearch). All methods were carried out using Matlab (The Mathworks, Inc.) software version 7.0.1. Fluid-phase equations are solved from initial conditions defined at z ) 0, corresponding to inlet conditions (Yi,o, θi,o) for cocurrent fluids and outlet conditions (Yi,e, θi,e) for counter fluids. The secondorder equation governing heat transfer in the solid-phase is solved to satisfy either isothermal or adiabatic conditions, as detailed below. Isothermal boundary conditions allow pinning the wall temperature at either boundary, i.e., the packaging-microchannel


Figure 5. Results for microscale regenerative combustor model. Thermal profiles and efficiencies obtained for NTU ) 5, R ) 10, γ ) 2 at (a) CP ) 0.1, Da ) 960, ε ) 0.22; (b) CP ) 1, Da ) 3500, ε ) 0.056; (c) CP ) 10, Da ) 5300, ε ) 0.0067 for isothermal packaging and (d) CP ) 0.1, Da ) 575, ε ) 0.49; (e) CP ) 1, Da ) 800, ε ) 0.509; (f) CP ) 10, Da ) 800, ε ) 0.49 for adiabatic packaging.

Figure 6. Results for microscale regenerative combustor model. Effect of kinetic parameters upon thermal efficiency and conversion for isothermal packaging at NTU ) 5, CP ) 0.1.

interface, in turn allowing free heat exchange between the microchannel solid-phase and the packaging layers. These conditions correspond to the limiting case of packaging with high thermal conductivity materials (e.g., stainless steel, aluminum) which allow thermal equilibration between the packaging layer and the inlet fluid. These boundary conditions have been used by Peterson46 and Stief and co-workers31 to study heat transfer between nonreacting fluids in microchannel systems. Solution requires determination of heat flux between packaging layer and solid-phase at the lower boundary or initial condition (z ) 0) that satisfies target isothermal temperature at

the upper boundary (z ) L). For the case of countercurrent operation, this upper-bound wall temperature is equal to the inlet temperature of countercurrent flow. For the case of cocurrent operation, the upper boundary wall temperature is equal to the mass averaged temperature of the two exiting fluids. Adiabatic boundary conditions assume zero heat flux between the packaging layers and the microchannel network, in turn allowing solidphase wall temperatures to be driven purely by heat exchange between the two fluids. These conditions correspond to the limiting case of packaging with thermally insulating materials. Solution requires determination of lower boundary wall tem-


Figure 7. Results for microscale regenerative combustor model. Effect of kinetic parameters upon thermal efficiency and conversion for adiabatic packaging at NTU ) 5, CP ) 0.1.

Figure 8. Results for microscale regenerative combustor model. Effects of solid-phase axial conduction upon system stability, thermal efficiency, and conversion; (a) isothermal packaging and (b) adiabatic packaging for NTU ) 5, R ) 10, γ ) 2.

perature to satisfy the zero flux condition at the upper boundary, independent of fluid temperatures. Results and Discussion In the present study we first analyze the case of two nonreacting fluids for both adiabatic and isothermal cases in order to identify the effects of design parameters and packaging upon heat exchanger efficiency in the absence of kinetics. This analysis is then extended to the case of an exothermic reacting fluid with heat removal by a second nonreacting fluid to identify effects of both design and kinetic parameters alongside packaging upon heat exchanger efficiency and conversion. Analysis also focuses upon hot-spot formation and magnitude within the

microchannel network and steady state multiplicity. Lastly, analysis of the case of heat transfer between two reacting fluids (exothermic and endothermic) is presented, following the same approach. Case I: Microscale Heat Exchanger. The simplest case of the general model is the steady-state heat exchange between two fluids in absence of any chemical reaction (Ri, Dai ) 0). In this case, eqs 1′–3′ are rewritten as eqs 5–7. The resulting system of equations is cast solely in terms of the number of transfer units (NTU) and fluid conduction parameter (CP). dθ1 ) NTU1(θw - θ1) dz˜

(5)


dθ2 ) (NTU2(θw - θ2) dz˜ d2θw dz˜

2

)

NTU1 NTU2 (θ - θ1) + (θ - θ2) CP1 w CP2 w

(6) (7)

The above equations are identical to those employed by Stief and co-workers31 and Peterson46 to analyze heat transfer in isothermally packaged microchannel networks. In the present study, these equations are solved to satisfy both adiabatic and isothermal packaging conditions over a range of CP and NTU, using numerical methods detailed above for the relevant case of countercurrent operation. In this analysis, symmetric conditions (NTU1 ) NTU2, CP1 ) CP2) are assumed. The heat exchanger effectiveness, ε, is calculated from the resulting temperature profile as follows: ε)

T2|z)0 - T2,inlet ) θ2 T1,inlet - T2,inlet

(8)

Additionally, for the case of isothermal packaging conditions, the ratio of conductive (solid-phase) to convective (fluid phase) heat losses can be obtained from eq 9, as originally shown by Peterson:46 λ ) CP

(dθw/dz˜)|z˜)1 θ1|z˜)1

(9)

Results obtained for the case of isothermal packaging conditions are presented in Figure 2. Heat exchanger efficiencies and conduction losses over a range of NTU are presented for the cases of low, intermediate, and high solid-phase thermal conductivities in Figure 2a, with corresponding temperature profiles of the solid and both fluid phases in Figure 2c-e for NTU ) 5. As seen in Figure 2a, the effectiveness is a weak function of the conduction parameter for the case of isothermal

packaging conditions. However, as CP increases, substantial increases in conduction heat losses occur. For this isothermal case, the effectiveness is held at high values despite increasing conduction losses with increasing CP because the wall temperature is pinned by external conditions; λ is thus a measure of how much heat needs to be removed from the packaging to maintain these external conditions. For example, at NTU ) 5, it can be seen that conduction losses corresponding to CP ) 10 are an order of magnitude greater than those obtained at CP ) 0.1, indicating substantial packaging heat losses. These results clearly demonstrate the benefit of employing intermediate thermal conductivity wall materials to maintain rapid heat transfer while minimizing conductive heat losses via packaging. Following similar analysis, Stief and co-workers identified an optimal range of CP values between 0.1 and 0.5 which correspond to thermally insulating materials31 (see Table 1). Results obtained for the case of adiabatic packaging conditions are presented in Figure 3. Heat exchanger efficiencies are presented for the cases of low, intermediate, and high solidphase thermal conductivities, as before, in Figure 3a. From Figure 3, it can be seen that heat exchanger effectiveness is significantly reduced with increasing CP, despite the removal of conductive heat losses (via adiabatic packaging conditions). This can be understood by examining the corresponding temperature profiles of the solid and fluid phases presented in Figure 3b-d for NTU ) 5. As CP increases, the system approaches the limit of thermal equilibration between the two fluids and the wall, equivalent to cocurrent operation, and in turn imposing a maximum efficiency of 50%. Conversely, low thermal conductivity materials allow maintenance of substantial thermal gradients along the solid-phase axial length by preventing thermal equilibration. In both cases, the effectiveness is a weak function of CP at low NTU where the solution is dominated by the NTU, and

Figure 9. Results for microscale regenerative combustor model. Thermal profiles obtained for NTU ) 5, CP ) 0.1 and isothermal packaging. R ) 5, γ ) 1, ε ) 0.303; R ) 5, γ ) 2, ε ) 0.301; R ) 5, γ ) 3, ε ) 0.303; R ) 10, γ ) 1, ε ) 0.298; R ) 10, γ ) 2, ε ) 0.302; R ) 10, γ ) 3, ε ) 0.305; R ) 15, γ ) 1, ε ) 0.305; R ) 15, γ ) 2, ε ) 0.307; R ) 15, γ ) 3, ε ) 0.305.


under isothermal conditions the effectiveness is dominated by CP at large NTU owing to accumulated conduction losses. On the basis of these results and analysis presented below, an intermediate value of NTU ) 5 is employed for the remainder of this study to focus investigations upon the effect of the conduction parameter, kinetics, and packaging on system performance. From Table 1, this intermediate value for NTU falls well within the range of typical microsystem operating conditions. Case II: Microscale Regenerative Combustor. The above analysis is extended to the case of one exothermic reacting fluid exchanging heat with a second nonreacting fluid via the solidwall, for the case of a single irreversible first-order reaction. Both fluids are assumed to enter the system at identical temperatures in order to isolate the exothermic reaction as the sole heat source in the system. With this condition, the minimum temperature achievable in the system becomes the coolant/ reactant inlet temperature (Tmin ) T1,inlet) while the maximum theoretical fluid temperature (independent of heat accumulation in the solid-phase) is obtained solely from the adiabatic temperature rise of the exothermic reaction. Equations 1′–3′ can be rewritten as follows:

[

-R1 dY1 ) -Da1Y1 exp dz˜ 1 + γ1θ1

[

]

-R1 dθ1 ) NTUi(θw - θi) + Da1Y1 exp dz˜ 1 + γ1θ1 dθ2 ) ( NTU2(θw - θ2) dz˜ d2θw

)

(10)

]

(11) (12)

NTU1 NTU2 (θw - θ1) + (θ - θ2) CP1 CP2 w

(13) dz˜ Equations 10–13 allow a direct investigation of the effects of design (NTU, CP) and kinetic parameters (R, γ, Da) alongside both adiabatic and isothermal packaging conditions on system performance. This is gauged in terms of reaction conversion and system thermal efficiency; the is latter defined as 2

ε)

T2|z)0 - T2,inlet T2|z)0 - T2,inlet ) ) θ2 Tmax - Tmin ∆Tad

(14)

In order to investigate the effect of NTU and CP on system thermal efficiency and reactor conversion for both the isothermal and the adiabatic packaging, a set of “mild” kinetic parameters are initially employed (γ ) 2, R ) 10). Figure 4a represents a parametric study varying CP and NTU assuming isothermal packaging boundary conditions. CP is varied from 0.1 to 10 over a range of NTU’s to simulate a wide selection of solid-phase thermal conductivities. In each case presented, a sufficient value of the Damkohler number is employed to ensure complete conversion in the absence of steady-state multiplicity (discussed below) for all cases. Model results in Figure 4a clearly show that the effectiveness is a strong function of the conduction parameter, unlike the heat exchanger case discussed in Figure 2. This difference can be understood by examination of the wall boundary conditions. In the heat exchanger analysis, wall temperatures are fixed equal to fluid inlets; i.e., the driving force for heat transfer is defined solely by the boundary conditions, not by the system. For reacting fluid analysis, wall boundary temperatures are fixed corresponding to identical inlet temperatures, such that the exothermic reaction (occurring within the system) is the sole driving force for heat transfer. Increasing NTU over the range of ∼0-5 results

in increasing effectiveness regardless of CP, while further increases in NTU result in reduced effectiveness, as conduction losses accumulate. Over all values for NTU, effectiveness is a strong function of the conduction parameter, with maximum effectiveness of 0.24, 0.06, 0.03, 0.006, and 0.003 at CP values of 0.1, 0.5, 1, 5, and 10. Figure 4b shows the results for the effect of CP and NTU on the adiabatic heat exchanger reactor. At low NTU, the effectiveness is a weak function of the conduction parameter in the absence of conduction losses via packaging. However, at high NTU, CP again becomes an important parameter, reflecting the potential for localized heat accumulation in the solid-phase (i.e., hot-spot formation). For NTU of 10, effectiveness of 0.64, 0.58, 0.55, 0.51, and 0.5 corresponds to CP of 0.1, 0.5, 1, 5, and 10. For the remainder of the analysis presented, a value of NTU ) 5 is assumed, corresponding to a balance between providing sufficient contacting time for heat transfer between fluids and minimizing conduction losses and/or reactor size. The influence of solid-phase wall conduction upon hot-spot formation and thermal efficiency at fixed intermediate values of NTU is illustrated in Figure 5a–c and d–f for the cases of isothermal and adiabatic packaging conditions, respectively. In both cases, sufficient low solid-phase thermal conduction supports the formation of a localized hot-spot along the solid-phase axial profile, in addition to the expected fluid-phase hot-spot. For the case of CP ) 0.1, localized accumulation of heat within the solid-phase may result in reacting-fluid dimensionless temperatures exceeding unity for both isothermal and adiabatic conditions. This “excess enthalpy” phenomena has been previously reported for both heat-recirculating burners44,51 and in heat exchanger reactors.43,52 Under adiabatic packaging condition, this heat focusing phenomena results in thermal efficiencies upward of 50-70% (Figure 4). Under isothermal packaging conditions, heat exchanger efficiencies of 15-25% are predicted owing to substantial heat conduction losses via packaging, even in the presence of excess-enthalpy hot-spots within the microchannel network. At high rates of solid-phase thermal conduction (CP ) 10), both isothermal and adiabatic packaging cases approach the limit of a thermally uniform or isothermal solid-phase. For adiabatic packaging conditions, this “isothermal slab” condition corresponds to a maximum heat exchanger effectiveness of 50% (thermal equilibration) and a maximum exothermic fluid dimensionless temperature of unity. For the case of isothermal packaging, where wall temperatures are pinned at either extrema to the fluid inlet (minimum) temperatures, this “isothermal slab” limit corresponds to a fully quenched solid-phase with negligible heat transferred between fluids, while the majority of reaction heat is removed by the solid-phase directly to the packaging. Modeling analysis presented in Figures 4 and 5 demonstrates the importance of both materials and packaging selection for achieving microreactor design goals. Applications where hotspot formation is undesirable and heat utilization is not critical are best served by a combination of isothermal packaging and high thermal conductivity materials (e.g., silica, stainless steel) to facilitate heat sinking. Examples of the successful use of this strategy include intrinsic kinetic studies6,20 where isothermal reacting fluid conditions are desired and for safely effecting highly exothermic and /or unstable reactions.9,22 As shown in Figure 8c, sufficiently high rates of fluid-solid heat transfer (NTU . 5) are still required to avoid hot-spot formation in the reacting fluid phase. In this limiting case, selection to the coolant


Figure 10. Results for microscale regenerative combustor model. Thermal profiles obtained for NTU ) 5, CP ) 0.1 and adiabatic packaging. R ) 5, γ ) 1, ε ) 0.567; R ) 5, γ ) 2, ε ) 0.536; R ) 5, γ ) 3, ε ) 0.499; R ) 10, γ ) 1, ε ) 0.462; R ) 10, γ ) 2, ε ) 0.491; R ) 10, γ ) 3, ε ) 0.504; R ) 15, γ ) 1, ε ) 0.546; R ) 15, γ ) 2, ε ) 0.571; R ) 15, γ ) 3, ε ) 0.634.

and solid-phase energy balance becomes trivial (isothermal), and the problem reduces to the simple case of a single reacting fluid. Applications where hot-spot formation is desirable and heat utilization (specifically reaction heat recovery by a second fluid) is critical are best suited by adiabatic packaging and/or the use of low thermal conductivity substrates. The ideal case of adiabatic packaging assures thermal efficiencies greater than 50%, for two-process integration schemes, even when employing high thermal conductivity materials. Under more realistic conditions with substantial heat losses via packaging, the use of thermally insulating substrates is shown to be critical to maintaining thermal efficiency. For this reason, subsequent analysis focuses upon thermally integrated networks designed with correspondingly low values of conduction parameter (CP ) 0.1). Equations 10–13 are solved for the case of countercurrent heat exchanger reactor over a range of Da for R ∈ (5, 10, 15) and γ ∈ (1, 2, 3). Results for the isothermal and adiabatic packaging conditions are summarized in Figures 6 and 7, respectively. Circled points correspond to axial temperature gradients presented in Figures 9 and 10. For a single exothermic reaction, the general trends of increasing residence time (Da) necessary to achieve complete conversion (X > 0.99) with increasing R and γ are observed. The residence time (Da) necessary to achieve complete conversion assuming isothermal packaging is consistently higher than that required for the adiabatic packaging case under identical conditions, owing to reaction quenching by conductive heat losses. From Figures 6 and 7, multiple steady states are observed for the cases of R ) 10 and 15, regardless of packaging at NTU ) 5, CP ) 0.1.

Under these conditions, ignition/extinction behavior is expected, with important implications for microsystem operation. Figure 8 illustrates the effect of thermal conductivity (CP) on the system performance and the presence of steady-state multiplicity. For the case of isothermal packaging, the rate of heat removal from the system via packaging losses increases (independent of convective heat removal), while serves to stabilize the system at the expense of substantial losses. For the case of adiabatic packaging, only the unstable solutions change with conduction parameter, while the range of Da corresponding to multiple steady states is unaffected by CP. This analysis suggests a design tradeoff between reactor stability and competitive thermal efficiencies. The prediction of steady-state multiplicity is intriguing and bears implications for thermally efficient heat-integrated microsystems constructed from thermally insulating materials. Much has been written on the subject of steady-state multiplicity in stirred pots and empty tubes, which has often been attributed to nonlinear kinetics, fluid-phase dispersion, or transport limitations within the catalyst.53–57 It is important to note that for the case of isothermal packaging and sufficient CP and NTU corresponding to isothermal coolant and solid-phase conditions the system of equations reduces to a single plug-flow reacting fluid described by an initial-value ordinary differential equation which must yield a unique solution.53 The introduction of boundary-value equations describing the solid-phase temperature as well as any countercurrent flows in turn allows the possibility of multiple steady states.53 That the present model, which neglects several established sources of steady-state multiplicity,58–60 demonstrates multiplicity due solely to packaging and materials selection clearly shows the importance of these two design


parameters in the design of thermally integrated Microsystems. By describing an idealized microreactor, assuming perfectly uniform wall and catalyst properties and employing first-order kinetics equivalent to either (i) a single, elementary reaction or (ii) external mass transfer controlled conditions, the present model provides a useful initial guide for selecting appropriate flow configuration, solid-phase materials, and packaging methods before initiating more detailed, chemistry-specific design modeling. Likewise, this analysis provides the necessary motivation and background for subsequent studies investigating the effects of packaging and materials selection on thermally integrated systems in the presence of complex kinetics, fluidphase dispersion, and/or solid-phase diffusion. The prediction of multiple steady states and ignition/extinction behavior in thermally integrated microreactors is intriguing, as no experimental studies of multiplicity in thermally integrated microreactors have been reported to date. The majority of microsystems detailed in the literature is constructed from high thermal conductivity materials (e.g., stainless steel, silicon) and are often packaged so as to incur substantial heat losses via packaging; the present analysis indicates that multiplicity is not expected in such systems. Performance of thermally integrated microreactors constructed with adiabatic packaging7,17 and/or thermally insulating materials40 have been reported only for limited operating conditions with little information provided regarding startup/shutdown or transient behavior, although Arana61 has qualitatively reported ignition and thermal runaway behavior in an adiabatically packaged thermally integrated silicon microreactor. The present analysis clearly demonstrates the need for an experimental investigation of adiabatically sealed and/or low thermal conductivity (e.g., ceramic) thermally integrated microsystem operation over a broad range of operating conditions including transient response and startup/shutdown behavior. Figures 9 and 10 show the effect of kinetic parameters (R, γ) on the hot-spot magnitude for isothermal and adiabatic packaging, respectively. In order to study the kinetic behavior of the system, we calculate the temperature-concentration profiles at the minimum residence time that 99.9% conversion is achieved. For cases where there is multiplicity the residence time (Da) corresponds to the upper steady state. For isothermal boundary conditions and low activation energies (R ) 5), the heat exchanger effectiveness remains constant with respect to increasing γ, while the maximum temperature achieved increases slightly. At increasing activation energies, the influence of γ becomes more significant, as hot-spots become more pronounced with maximum dimensionless temperatures exceeding unity (corresponding to “excess enthalpy” conditions). Comparison of results in Figures 9 and 10 indicates that thermal efficiencies and hot-spot magnitude are consistently greater for adiabatic packaging. Case III: Heat Exchanger Reactor. Lastly, analysis is extended to the coupling of an exothermic reacting fluid with an endothermic reacting fluid. As before, the simplifying assumption of a single, irreversible first-order reaction is employed for each fluid. Analysis follows identical approach taken to study the effect of design and kinetic parameters and packaging upon thermal efficiency, reactor conversion, and system stability. The general model eqs 1′–3′ reduce to the following:

[

dY1 -R1 ) -Da1Y1 exp dz˜ 1 + γ1θ1

]

(15)

[

dY2 -R2 ) (Da2Y2 exp dz˜ 1 + γ1θ2

[

]

dθ1 -R1 ) NTUi(θw - θi) + Da1Y1 exp dz˜ 1 + γ1θ1

{

[

(16)

]

dθ2 -R2 ) ( NTU2(θw - θ2) + Da2Y2 exp dz˜ 1 + γ2θ2 d2θw dz˜

2

)

NTU1 NTU2 (θw - θ1) + (θ - θ2) CP1 CP2 w

(17)

]}

(18)

(19)

As before and for the sake of simplicity, symmetric parameters (both design and kinetic) are maintained, while employing an intermediate yet sufficient value of NTU ) 5. Symmetric kinetic parameters (R1 ) R2 ) 10; γ1 ) -γ2 ) 1) are selected on the basis of the above analysis. Thermal efficiencies are calculated on the basis of the heat released from the exothermic reaction that is utilized to (i) drive the endothermic reaction and (ii) as sensible heat, as follows: ε ) θ2 + X2γ2

(20)

Results from Figure 11 show that system efficiency and stability are complex functions of the effect of conduction parameter. Under isothermal conditions, increasing CP results in losses of thermal efficiency (at sufficient values of the Damkohler number corresponding to complete conversion of the exothermic reacting fluid) due to conduction losses via packaging, as observed in the microscale regenerative combustor case. Under adiabatic conditions, the solutions of very low and very high values of Damkohler number are independent of the conduction parameter, and thermal efficiencies are consistently higher owing to the absence of conductive losses via packaging. In both cases increasing CP results in multiple steady states for both reacting fluids. Similar analysis for the case of countercurrent operation is presented in Figure 12. Comparison of results in Figures 11 (for cocurrent) and 12 (for countercurrent) illustrates the additional influence of flow configuration on both thermal efficiency and multiplicity. While maximum efficiencies are similar for both co- and countercurrent flow, Damkohler numbers necessary for complete conversion of the exothermic fluid reduce mildly for the case of countercurrent flow, mostly noticeably at low values of the conduction parameter. Multiplicity behavior is similar for both adiabatic cases while countercurrent operation removes steady-state multiplicity in the isothermal case. Mechanical Analysis of Thermal Profiles. The above modeling analysis illustrates the value of low thermal conductivity substrates for realization of thermal gradients and/or hotspots which maximize thermal efficiency. However, a real limit to the operation of mini- and microchannel networks under such thermal loads is mechanical failure via buckling deformation. To illustrate the importance of coupling the reaction/heat transfer modeling with a mechanical analysis of the solid-phase, the following structural model (Euler-Bernoulli beam model) was developed (Appendix) to estimate the structural response (displacement) to expected thermal gradients. The mechanical model employs physical parameters of cordierite as representative of a low thermal conductivity ceramic substrate. The mechanical model parallels the one-dimensional thermal model geometry (Figures 1 and 13) by assuming a two-dimensional plate rigidly clamped at either end by packaging. Further, it is assumed that the system is initially packaged at ambient temperature. The resulting beam model employed two repre-


Figure 11. Results for microscale heat exchanger reactor modelscocurrent case. Effects of solid-phase axial heat conduction upon system stability, thermal efficiency, and conversion. (a) Conversion, isothermal case; (b) conversion, adiabatic case; (c) efficiency, isothermal case; (d) efficiency, adiabatic case. NTU1 d NTU2 ) 5, R1 ) R2 ) 10, γ1 ) -γ2 ) 1.

Figure 12. Results for microscale heat exchanger reactor modelscountercurrent case. Effects of solid-phase axial heat conduction upon system stability, thermal efficiency, and conversion. (a) Conversion, isothermal case; (b) conversion, adiabatic case; (c) efficiency, isothermal case; (d) efficiency, adiabatic case. NTU1 d NTU2 ) 5, R1 ) R2 ) 10, γ1 ) -γ2 ) 1.

sentative thermal gradient shapes, defined in terms of the maximum (adiabatic) temperature (Tad) obtained for the case of countercurrent regenerative combustor configuration (NTU ) 5; CP ) 0.1; γ ) 2; R ) 10) for both adiabatic and isothermal packaging conditions (Figures 9 and 10). Results obtained from mechanical analysis over a range of maximum temperature (Tad) are presented in Figure 13c,d. This simple model demonstrates that at 200 °C the deformation under adiabatic packaging conditions is roughly triple the deformations produced under isothermal packaging conditions. In both cases, the deformation is predicted to be the same order of magnitude as the hydraulic diameter of a typical micro- or minichannel, suggesting that wall deformation in the presence of substantial thermal gradients may be sufficient to influence fluid flow and fluid-solid heat transfer within individual channels. Thus, the preliminary analysis presented here illustrates the importance of coupling mechanical models with thermal models of the solid-phase (substrate) in thermally integrated micro- and/or minichannel

reformers fabricated from insulating ceramic substrates to achieve high thermal efficiencies. Summary The above analysis illustrates the important effects of solidphase axial heat conduction, packaging, and flow configuration upon system stability and thermal efficiency in the design of heat-integrated microchannel networks. The use of thermally conductive packaging materials (represented in the present analysis by isothermal packaging conditions) decreases reactor conversion and thermal efficiencies, owing to conductive heat losses via packaging, at the advantage of alleviating steadystate multiplicity. For thermally conductive packaging, conversion and thermal efficiency are strongly affected by solid-phase axial heat conduction regardless of Damkohler number, as the solid-phase presents the sole means of insulation against conduction losses. Under insulating packaging conditions,


Figure 13. Mechanical analysis of infinite clamped-clamped plate, using thermal profiles obtained from model.

conversion and thermal efficiency are only affected by solidphase axial heat conduction at intermediate values of Damkohler number, corresponding to incomplete conversion. In both cases, reducing axial conduction increases thermal efficiency while alleviating multiplicity. Flow configuration is shown to have negligible effect on maximum achievable thermal efficiencies (at sufficient values of Damkohler number), while potentially having a significant effect on steady-state multiplicity under isothermal packaging conditions. Results from this study clearly demonstrate the importance of both materials and packaging selection in the design of thermally integrated microchannel networks. The prediction of steady-state multiplicity under desirable (high thermal efficiency) operating conditions provides the basis for a future complete stability analysis of the governing system of differential equations, accounting for solid-phase axial conduction and thermal packaging conditions. Preliminary mechanical analysis of solid-phase deformation employing predicted thermal gradients illustrates the importance of developing coupled mechanical and thermal models of the solid-phase in heat-integrated reformer networks constructed from insulating materials. Future mechanical analysis addressing complex threedimensional structures employed in real thermally integrated microchannel systems is planned in light of these findings. Overall, the present study highlights the many challenges remaining in the design, modeling, and development of thermally integrated microsystems for portable chemistry and/or reforming.

Nemours Corp. (through award of a DuPont Young Professor Grant). Benjamin A. Wilhite gratefully acknowledges a Junior Faculty Summer Fellowship by the University of Connecticut. The authors thank George L. Assard II and the School of Engineering-Engineering Computing Services at the University of Connecticut for providing computational software and technical support. Appendix To illustrate the importance of mechanical modeling of nonisothermal heat integrated microchannel networks, a simple model for the deformation of the wall as a result of the generated thermal gradient is developed. The present model assumes a specified temperature profile a priori resulting from solution of governing transport equations in the absence of deformation. The wall is described using a nonlinear Euler-Bernoulli beam model. This model permits bending, where the strains are linearly distributed through the thickness of the beam (i.e., linear bending theory) and nonlinear stretching of the neutral axis. This latter effect allows the model to predict large lateral deformations, including postbuckled behavior. A Ritz approach is used to obtain the governing equations.62 In short, this requires minimizing the total strain energy (to satisfy condition of structural equilibrium) and making the following assumptions regarding deformation geometry. The total strain in the beam consists of a mechanical strain and a thermal strain, i.e. ε ) εmech + εthermal

Acknowledgment The authors acknowledge Professor Arvind Varma for his substantial contributions to the field of Chemical Reaction Engineering and his valuable correspondences in the development of this work. This research was funded by the Office of Naval Research (Grant N000140710828) and the DuPont de

(A1)

The mechanical strain is obtained from the Lagrangian strain tensor, retaining first-order nonlinear effects63 as follows: 1 εmech ) ux + wx2 - ywxx 2

(A2)

where u(x) is the axial deformation of the neutral axis, w(x) is


the lateral deformation of the neutral axis, and y is the distance above the neutral axis. The thermal strain is defined as follows: εthermal ) RT

(A3)

where R is the coefficient of thermal expansion and T(x) is the temperature rise above ambient. The total strain energy is obtained from the product of the mechanical stress and total strain U)

1 2

∫ σε dy dz dx

(A4)

For a one-dimensional elastic continuum Hooke’s law may be used, reducing the strain energy to U)

E 2

∫∫∫ L

0

b

0

+h/2

(u + 21 w x

-h/2

2

x

1 - ywxx - RT ux + wx2 2

)(

)

ywxx dy dz dx(A5) Equilibrium positions can then be obtained via minimization of the structure strain energy with respect to the displacement fields, u(x) and w(x). Before proceeding with this minimization procedure, the scope of the problem can be reduced somewhat by limiting consideration to the case of the clamped-clamped (rigid) geometry at either end of the supported plate, shown schematically in Figure 13a. An assumed form of the deformation of the neutral axis (shown in Figure 13b) can then be employed. This can be expressed as

( πxL )

u(x) ) p1 sin

( πxL )]

[

and w(x) ) q1 1 - cos2

(A6)

where p1 and q1 are unknown deflection amplitudes. The strain energy, U, can then be minimized upon substitution of the above expressions into eq A5. Minimizing the energy is accomplished by the following: ∂U ) 0 and ∂p1

∂U )0 ∂q1

(A7)

The resulting system of two nonlinear algebraic equations can be solved for the unknown amplitudes p1 and q1 using a multidimensional Newton-Raphson technique. Note that the temperature profiles employed in eq A5 are obtained from the thermal analysis. These profiles were fit to fourth-order polynomials for both adiabatic and isothermal cases as follows: Tiso(x) x 4 x 3 x 2 ) 1.2341 - 2.4593 + 0.9462 + Tmax L L L x 0.2809 + 0.0015(A8) L

()

() () T (x) x x x ) 1.2286( ) - 2.0614( ) + 0.6498( ) + T L L L x 0.1434( ) + 0.4874(A9) L ad

4

()

3

2

max

This permits the integrals of eq A5 to be evaluated, prior to the minimization step in eq A7. Notation a ) surface area/volume ratio (m-1) C ) concentration of reactive fluids (mol m-3) CO ) inlet concentration of reactive fluids (mol m-3) Cp ) fluid specific heat capacity (J kg-1 K-1) d ) hydraulic diameter (m) D ) mass diffusivity (m2 s-1) Ea ) activation energy (J mol-1)

h ) heat transfer coefficient (W m-2 K-1) k ) thermal conductivity (W m-1 K-1) k0 ) pre-exponential factor of the rate constant of reaction (s-1) L ) reactor length (m) r ) reaction rate (mol s-1) R ) universal gas constant (J mol-1 K-1) T ) temperature (K) u ) fluid velocity (m s-1) z ) axial position along the reactor (m) Symbols ∆Hrxn ) heat of reaction (J mol-1) ∆Tad,i ) -∆Hrx,iC0,i/FiCpi ) adiabatic temperature rise for fluid i (K) F ) density (kg m-3) Subscripts/Superscripts i ) ith fluid M ) number of channels max ) maximum min ) minimum n ) reaction order ref ) reference w ) wall Dimensionless Groups and Variables z˜ ) z/L ) dimensionless axial position Y ) C/CO ) dimensionless concentration θ ) (Ti(z) - Tmin)/(Tmax - Tmin) ) dimensionless temperature τ ) L/u ) residence time ε ) (T2|z)0 - T2,inlet)/(Tmax - Tmin) ) heat exchanger effectiveness λ ) CP[(dθw/dz˜)|z˜)1]/θ1|z˜)1 ) conduction to convective heat loss ratio NTUi ) hiaiτi/FiCpi ) fluid-solid heat transfer parameter for fluid i CPi ) kw/LFiuiCpi ) solid-phase axial conduction parameter for fluid i Ri ) Eai/RTref ) dimensionless activation energy for fluid i Dai ) k0,iCion-1τi ) Damkohler number for fluid i γi ) ∆Tadi/Tref ) dimensionless heat of reaction for fluid i

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ReceiVed for reView January 29, 2008 ReVised manuscript receiVed May 4, 2008 Accepted May 16, 2008 IE8001638

Parametric Study of Solid-Phase Axial Heat ... - ACS Publications

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