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Solar Energy
journal homepage: www.elsevier.com/locate/solener
A generic solar-thermochemical reactor model with internal heat diffusion for counter-flow solid heat exchange
Christoph P. Falter a* Robert Pitz-Paalb
a Bauhaus Luftfahrt, Willy-Messerschmitt-Straße 1, 82024 Taufkirchen, Germany b DLR, Institute of Solar Research, Linder Höhe, 51147 Köln, Germany
ARTICLE INFO
ABSTRACT
Article history:
Received 18 October 2016
Received in revised form 25 January 2017
Accepted 28 January 2017
Available online 7 February 2017
Keywords: Solar fuel Redox cycle Heat recovery Model
For nonstoichiometric redox reactions that produce CO and H2 from CO2 and H2O, heat recuperation from the solid phase is a promising mechanism to improve the cycle efficiency. Many different approaches to heat recuperation and gas separation have been presented in solar thermochemical reactor concepts recently. To describe the many possible degrees of freedom in the reactor design, a generic reactor model is described for two-step redox reactions of solid pieces of reactant moving in counter flow between reduction and oxidation chambers. The reactive material is assumed to be porous ceria, where heat recuperation from the solid phase is achieved through radiation heat exchange between reduced and oxidized elements moving in opposite directions. A separation wall prevents gas cross-over and provides structural support. Heat transfer by radiation in the porous material is modeled with the Rosseland diffusion approximation and by conduction with the three resistor model. The model can be adapted to a wide range of reactor concepts.
A study of crucial design parameters shows that heat diffusion in the reactive material can have a significant influence on the performance of the heat exchanger. If the time required for heat diffusion is large with respect to the total residence time in the heat exchanger, the material thickness can be decreased to enhance the share of the material actively participating in the heat exchange process. Furthermore, at the relevant temperatures, radiation dominates the heat exchange within the porous structure, thus the overall heat exchange can be enhanced through an increase of porosity. Heat exchanger length and residence time are correlated, allowing different combinations of these two variables at constant heat exchanger efficiency. In general, efficiencies close to 70% are possible with an adequate parameter combination. However, the achievement of the maximum heat exchanger efficiency requires a minimum number of chambers and thus physical length, as irreversibilities are reduced for a larger number of intermediate temperature levels.
The presented generic model includes the description of heat diffusion within the reactive material, is a valuable tool for the design of heat exchangers, and can be used to identify technically interesting reactor concepts for the achievement of high energy conversion efficiencies.
© 2017 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND
license (http://creativecommons.org/licenses/by-nc-nd/4XI/).
1. Introduction
The transition of the energy basis from conventional sources such as coal, natural gas, and crude oil, to renewable sources is a major challenge that involves large investments and the development of new energy technologies. In case of the generation of electrical energy, a significant increase in renewable generation capacity has been achieved in the past years reaching a share of 22% of the total electricity generation (International Energy
* Corresponding author. E-mail addresses: christoph.falter@bauhaus-luftfahrt.net (C.P. Falter), robert.pitz-paal@dlr.de (R. Pitz-Paal).
Agency, 2016). In the transportation sector, however, the penetration with renewables has been considerably lower, owing to the fact that conventional fuels such as gasoline, diesel and jet fuel are ideally suited for the use in mobile applications. Their high energy and power density are requirements for the use in aviation, sea traffic and heavy-duty road transportation, while for light-duty road transportation, an electrification using batteries and electric motors is easier to implement and has seen a rising share in recent years. Given a rising demand at limited resources and concerns about climate change, the aim of a reliable and affordable energy source for transportation requires solutions such as the production of an energy-dense fuel based on renewable energy.
http://dx.doi.org/10.1016/j.solener.2017.01.063 0038-092X/© 2017 The Author(s). Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Nomenclature
A area [m2]
Ahe.ext external area of heat exchanger chamber facing the
environment [m2] C concentration ratio of solar radiation [-]
cp,ceo2 heat capacity of ceria at constant pressure [J mor1 K-1] cP;CO heat capacity of CO at constant pressure [J mor1 K-1] cp CO2 heat capacity of CO2 at constant pressure [J mor1 K-1] cPo2 heat capacity of O2 at constant pressure [J mor1 K-1] dmean mean diameter [m]
g0 parameter used in the three resistor model [-]
g1 parameter used in the three resistor model [-]
AHred reduction enthalpy [J mol-1] I intensity of solar radiation [Wm-2]
kf thermal conductivity of fluid phase [Wm-1 K-1]
ks thermal conductivity of solid phase [W m-1 K-1]
m number of computational layers in element of reactive
material [-] N number of calculations [-]
nCeO2 molar flow rate of ceria [mol s-1] nCO molar flow rate of CO [mol s-1] nCO2 molar flow rate of CO2 [mol s-1] hO2 molar flow rate of oxygen [mol s-1] pO2 oxygen partial pressure relative to standard state [-] Paux auxiliary electrical power needed to operate reactor [W] Ppump electrical power required for operation of vacuum pump [W]
Psep,CO/CO2 electrical power required for gas separation of
CO/CO2-mixture [W] Q.aux solar power needed for auxiliary processes [W] Qtieat,ceO2 solar power needed to heat ceria [W] Qjieat,cO2 solar power needed to heat CO2 [W] Qjieatexchange net rate of energy transferred between ceria elements in upper chamber half and lower chamber half of heat exchanger [W] Qjoss,conv thermal power lost by convection from heat exchanger
chamber half [W] Qjoss,rad thermal power lost by radiation from heat exchanger
chamber half [W] Qproducts thermal power recovered from gaseous products [W] qrad radiative heat flux [Wm-2] Qjed,ceO2 solar power needed to reduce ceria [W] Qrerad rate of radiation heat losses from the reduction chamber [W]
Qsolar solar power input to reactor [W] R ideal gas constant [J mol-1 K-1]
s sample standard deviation [-]
At residence time in heat exchanger chamber [s]
tCIv t-value of the student t function for the confidence
interval CI and v = N - 1 degrees of freedom T temperature [K]
T0 ambient temperature [K]
TH reduction temperature [K]
The,endj temperature of j-th layer of reactive material at the end
of the heat exchanger [K] Ti arithmetic mean temperature at position i
TL oxidation temperature [K]
Tn temperature in time step n [K] Tn+1 temperature in time step n +1 [K] Tpump temperature of vacuum pump [K] Trpc!sw temperature of RPC in lower chamber half facing separating wall [K]
TsW!RPC temperature of separating wall facing RPC in lower
chamber half [K] Twan temperature of heat exchanger wall facing surroundings [K]
Greek letters
a absorption coefficient in Monte Carlo analysis [-]
aconv convective heat transfer coefficient at heat exchanger
wall [Wm-1 K-1] b extinction coefficient [m-1]
bR Rosseland mean attenuation coefficient [m-1]
d oxygen nonstoichiometry of ceria [-]
dox oxygen nonstoichiometry of oxidized ceria [-] dred oxygen nonstoichiometry of reduced ceria [-] eCeo2 emissivity of ceria [-] esw emissivity of separating wall [-] ewan emissivity of heat exchanger wall facing the environment [-]
g efficiency (concentrated solar to chemical energy stored
in CO) [-]
gabs absorption efficiency of reactor [-]
ggasrec recuperation efficiency of gases leaving the reduction
and oxidation chambers [-] ghe heat exchanger efficiency [-]
gheat-to-eiectricity conversion efficiency of heat to electricity [-] gpump efficiency of vacuum pump [-] k thermal conductivity [W m-1 K-1]
krad radiative conductivity [W m-1 K-1] r Stefan-Boltzmann constant [W m-2 K-4]
U porosity [-]
W1 parameter used in the calculation of the extinction coefficient [-]
Due to its abundance and global availability, solar energy is an attractive primary energy source for the production of liquid fuels. The solar thermochemical pathway uses concentrated solar energy to thermochemically split water and carbon dioxide into syngas and the Fischer-Tropsch process to synthesize liquid hydrocarbon fuels. It offers the potential of a low climate impact at a large energy conversion efficiency (Falter et al., 2016; Steinfeld and Epstein, 2001). For its implementation, different materials and processes have been suggested, where in recent years, non-volatile redox reactions with cerium oxide (ceria) have shown promising experimental results (Chueh et al., 2010; Furler et al., 2014, 2012a; Marxeret al., 2015). In most implementations, ceria is only partially reduced to retain the material in its solid fluorite structure and to thus enable a process without phase changes. This facilitates the reactor design considerably, as high temperature gas separations are avoided. However, the downside is that only a
small fraction of the reactive material actually participates in the redox cycle, leading to a large thermal mass with respect to the produced amount of syngas. As the redox cycle is performed under a temperature swing, a relatively large amount of energy is required for the heating of the material to the reduction temperature. In theoretical analyses it was shown that for nonstoichiomet-ric ceria cycles, energy recuperation from the temperature swing is imperative to achieve high levels of energy conversion efficiency (Falter et al., 2015; Lapp et al., 2012).
Different reactor concepts have been presented including heat recuperation from the solid and the gaseous phase, e.g. the CR5-reactor (Diver et al., 2008), where counter-rotating rings of reactive material are reduced by solar radiation on one side and oxidized on the opposite side, exchanging heat in between the reaction zones. Another similar concept is shown in Lapp et al. (2013), where an inert cylinder rotates in the opposite direction of the reactive
cylinder surrounding it which is directly heated by solar radiation. In Ermanoski et al. (2013), a particle reactor concept is presented which includes heat exchange between the oxidized and reduced particles and an inherent gas separation through the packed bed of reduced particles. Direct heat exchange between reactive particles and heat exchanger particles is suggested in Felinks et al.
(2014), indicating a high recuperation potential. Very different approaches exist for the implementation of heat recuperation from the solid phase of nonstoichiometric redox reactions. Efficiency analyses of solar thermochemical syngas production have followed either a fundamental thermodynamic path or have focused on the detailed analysis of a specific reactor concept. However, for the description of a large number of technically interesting implementations, a generic reactor model is required which was presented with an upper-bound performance estimation for idealized internal heat transfer within the reactive medium in Falter et al.
(2015). It was noted that heat diffusion in the material may have a significant influence on the heat exchanger performance and should thus be analyzed in more detail. In this work, the presented model is used as a basis and further developed to include the modeling of heat transfer within the reactive medium to analyze its influence on the overall performance of heat exchanger concepts. A parameter study of fundamental variables then gives insight into the behavior of counter-flow heat exchangers operating with optically thick material to improve the energy conversion efficiency of solar thermochemical syngas production.
2. Model description and numerical solution algorithm
A generic and modular reactor concept for two-step thermo-chemical processes is described comprising a reduction chamber (i = 1), intermediate chambers for heat exchange (i = 2.. .n—1) and an oxidation chamber (i = n, Fig. 1). Elements of reactive material are heated by concentrated solar radiation in the reduction chamber and reoxidized with an oxidant in the oxidation chamber, i.e. CO2 or H2O or a mixture thereof, to produce syngas. In the following, without loss of generality, CO2 is used as the oxidant. An excess amount of oxidant is supplied to the oxidation for thermodynamic and kinetic reasons and the resulting mixture of CO and CO2 is separated to reduce the amount of CO2 contamination for the Fischer-Tropsch reaction. A vacuum pump reduces the oxygen partial pressure in the reduction chamber and the n-1th chamber to prevent gases from crossing over from the oxidation chamber. The heat exchanger is thus kept at the pressure of the reduction chamber.
In between the reaction chambers, heat is exchanged by radiation between the reduced and oxidized elements moving in counter-flow through the heat exchanger chambers. A wall ensures separation of the gas atmospheres between chamber halves to prevent gas cross-over, where the wall is made from HfC/SiC having a high emissivity and thermal conductivity to promote the heat exchange between the elements (Fig. 2). As SiC may undergo a chemical reaction with ceria at elevated temperatures, the wall is further separated from the elements by a layer of non-porous Al2O3. A thickness of 1 mm is chosen for the separation walls and the fluid gap between separation wall and element in the lower chamber half.
Equilibrium thermodynamics together with initial species concentration and constant temperatures are used to calculate the amount of nonstoichiometry and syngas production in the reaction chambers. An energy balance of the heat exchanger including radiation heat exchange between the chamber halves and heat losses to the surroundings then gives the temperature distribution of the elements.
As a reactive material, a reticulated porous ceramic (RPC) made from ceria undergoing nonstoichiometric redox reactions, is assumed, as it has been shown to be a suitable material with promising efficiencies in experiments (Furler et al., 2014, 2012a; Marxer et al., 2015). The description of other materials is easily possible if the respective material properties are known. Reduction is performed at an elevated temperature TH and reduced oxygen partial pressure pOa. The oxygen nonstoichiometry increases from dox after oxidation to dred after reduction, see Eq. (1).
dred — dox
dred — dox
dred — dox
dred - d
CeO2-dred + H2O
CeO2-dred + CO2
dred dox
dred — dox
H2O ! 2O2 + H2
CO2! 2O2 + CO
In Ermanoski et al. (2013), a function was fitted to the experimental data of oxygen nonstoichiometry in ceria by Panlener et al. (1975), which is used here as well. The calculation of oxygen nonstoichiometry is performed using mass conservation in a closed
Qsolar ^aux Qrerad
oss ,rad Vloss ,conv
CO,co„
Fig. 1. Schematic of modular generic rector model including n chambers, one for reduction (i = 1) and oxidation (i = n), and n-2 heat exchanger chambers (i = 2.. .n—1).
Inconel reactor wall (solid)
CeO -RPC (porous)
Qheat,co2 is the thermal power required to heat CO2 from ambient temperature to the oxidation temperature
Qheat,CO2 = nco2 / Cp,CO2 (T)dT.
Separating wall A^O (non-porous) Separating wall HfC/SiC (non-porous)
Gap (fluid)
CeO -RPC (porous)
Inconel reactor wall (solid)
The properties of CO2 and all other gases are calculated with tables from Engineering Toolbox (2015) and Kleiber and Joh (2013).
4products is the thermal power which is recovered from gases leaving the reduction and oxidation reactions (O2, CO, CO2) with an efficiency of ggasrec.
QProducts = ggasrec (nCo/ Cp,co (T)dT + nco2/ Cp,cO2 (T)dT + flO2i Cp,o2 (T)d^ .
\ J Tq JTQ JTQ J
Mechanical energy which is required to move ceria is neglected, as it is small compared to the heating value of the produced syngas. Liberated energy in the exothermic oxidation reaction is assumed to maintain the ceria temperature in the oxidation chamber and otherwise to be lost.
The vacuum pump power is
Fig. 2. Schematic of one representative heat exchanger chamber for the modeling of radiation heat exchange between chamber halves and internal heat transfer in the reactive medium and adjacent insulation.
system with defined initial gas concentrations and experimental data of enthalpy and entropy as a function of nonstoichiometry (Panlener et al., 1975), where the detailed derivation was shown in Falter et al. (2015).
2.1. Energy balance
The overall energy balance of the system is
Qsolar + Paux - Qheat,CeO2 - Qred,CeO2 - Qheat,CO2 + Qproducts - Qrerad - Ppump - Psep,CO° = 0
Ppump —
n o2 RTpumpln(p5l)
where the solar power input to the reactor is
Qsolar = g-(Qheat,CeO2 + Qred ,CeO2 + (¿heat, CO2 - Qproducts^
where the vacuum pump efficiency gpump is fitted to experimental data provided by a manufacturer (Felinks, 2014; Pfeiffer Vacuum, 2014) and is gpump = 0.8437pO23662, where the released oxygen in the reduction chamber and the oxidant lost through the periodic opening of the oxidation chamber is removed. The oxygen flow rate nO2 is derived from the stoichiometry of the overall reaction as half of the carbon monoxide flow rate, the pump temperature Tpump is assumed to be the ambient temperature and pO2 is the partial pressure of oxygen during reduction.
The CO/CO2 separation is assumed to be complete, i.e. pure streams of CO and CO2 are produced. Literature data for the CO2 capture from a flue gas stream of a fossil power plant are chosen as a reference (Zeman, 2007), where 132 kJ of heat and 9 kJ of electricity are required for the capture of one mol of CO2. The auxiliary electrical power input Paux = Ppump + Psep,co/co2 is then divided by the conversion efficiency of heat to electricity gheat-to-electricity to
gabs is the absorption efficiency of the solar reduction chamber and arrive at the auxiliary thermal power input Q^ =
is, assuming a well-insulated blackbody cavity, gabs = 1 - where I is the intensity of solar radiation and C the concentration efficiency which are assumed to be 1 kW m-2 and 3000 suns, respectively.
Qheat,Ceo2 is the solar power required to raise the temperature of the oxidized ceria element to the reduction temperature TH,
rTH rTH
Qheat,CeO2 =(1 - ghe)"CeO2 / Cp,ceO2 (T)dT = nceO2 / Cp,ceO2 (T)dT,
^TL ^ The ,end j
where the heat exchanger efficiency ghe is defined as
Sm/The ,endjCp,ceO2 (T)dT
gheat-to-electricity
mf" Cp,ceO2 (T)dT
and The>end j is the temperature of the j-th layer of the ceria element at the end of the heat exchanger before entering the reduction chamber (Fig. 2).
Q_ red CeO2 is the rate of energy required to reduce the material from dox to dred
Q red , CeO2 = n CeO2 AHred = n CeO2
DHceO2 (d)dd.
The efficiency of the reactor is defined as
chemical energy stored in product ncoHHVCo
g solar power in to reactor + auxiliary power Qsolar + Qaux'
This definition accounts the energy input at the system boundary of the reactor, i.e. the concentration efficiency and the primary energy conversion efficiency for the heat used for the auxiliary power can be included depending on the chosen technologies.
The temperature of the elements in the heat exchanger is calculated with an energy balance of the single chambers (exemplary chamber shown in Fig. 2), where each chamber is subdivided into upper and lower chamber half. The domains of insulation and reactive materials are discretized with the finite volume method. Heat is transferred by radiation between the separating wall and the element in the lower chamber half, by conduction in the separating wall and the reactor wall, by conduction and radiation in the elements and in the porous insulation, and heat is lost to the environment through convection and radiation.
The radiation heat transfer between the separating wall and the lower chamber half is modeled with the expression for infinite parallel flat plates (Howell et al., 2011)
Al2O3-SiO2-msulation (porous)
Al O-SiO-insulation (porous)
eat exchange :
,Ar(T4
sw^RPC TRPC!sw)
_L + _L_1
esw eCeO?
where A is the active area of heat exchange, r is the Stephan-Boltzmann constant, Tsw!RPC and TRPC!sw are the temperatures of the separating wall facing the lower chamber half and of the RPC facing the separating wall, respectively. The emissivity of the wall is 0.85 (Ultramet, 2015) and that of ceria sCeo2 is a function of temperature, which is taken from Touloukian et al. (1971). The high temperatures in the heat exchanger cause a temperature increase of the outside wall above the temperature of the surroundings and heat is lost by radiation and convection.
Radiation heat transfer from the reactor wall to the environment is
Qloss.rad — ewallAhe,extr ^Twall — T4) >
and by convection
— aconvAhe,extDT — aAhe,ext(Twall — T0);
where the convective heat transfer coefficient aconv is conservatively assumed to be 15 W m—2 K—1 (Hischier et al., 2009) and the area facing the environment is defined to be six times the active area of heat exchange, as is the case for example for cubic elements
Ahe,ext — 6 ' A.
The temperature distribution in the heat exchanger is then derived by pseudo-transient continuation starting from a first guess (a predefined temperature distribution which depends on heat exchanger length and temperature levels, which in general increases in temperature from the heat exchanger outside to its inside and from the oxidation to the reduction chamber) until a steady state is reached, i.e. until the maximum temperature change of the elements of two consecutive time steps is smaller than a
defined value |Tn+1T1Tn| 6 10—4.
2.2. Calculation of radiation and conduction in porous domains
The radiation source term is modeled with the Rosseland diffusion approximation assuming an absorbing, emitting, and isotrop-ically scattering optically thick medium (Howell et al., 2011).
Vqrad = —V(kradVT),
where the thermal radiative conductivity krad for a refractive index of one is given by
krad — -
and where bR is the Rosseland mean attenuation coefficient and r is the Stefan-Boltzmann constant.
RPC: For the description of radiative heat transfer in reticulated porous ceramics, numerical solutions were obtained for tomo-graphic scans of material and compared with models from the literature in Suter et al. (2014). The model using multi-faced particles after Loretz et al. (2008) was found to reproduce the numerical solution best and is used here in the following. The medium is further assumed to be gray, making bR equal to the extinction coefficient b which is defined by Loretz et al. as a function of porosity
b — W
V1 — u
dmean = 2.20 X 10
3 • u + 4.59 X 10—4m
with the fitting parameter W1 — 1 :765 and the mean pore diameter dmean. The pore diameter is a function of porosity u, i.e. an increase in porosity is directly linked to an increase in pore diameter and both variables cannot be chosen independently.
Conduction through the porous medium is described with the three-resistor model which uses a linear combination of the minimum possible conductivity in the serial arrangement of phases and of the maximum possible conductivity in the parallel arrangement of the phases to derive the overall thermal conductivity of the porous medium (Suter et al., 2014).
k — (1 — Vg<) — g1U)
u + ( — u)
if + Vg0 — g1u (ukf + (1 — u)ks)
The values of the parameters g0 and g1 are 0.754 and 0.829 (Suter et al., 2014), kf and ks are the thermal conductivities of the fluid and solid material, and u is the porosity of the porous domain. The effective thermal conductivity of the porous medium k is then used in the energy conservation equation.
Insulation: The modeled porous insulation is comprised of fibers of Al2O3 and SiO2. As the pore mean diameter is not known, experimental values for the Rosseland mean attenuation coefficient and for the thermal conductivity are used to describe the thermal energy transfer across the insulation. The former are taken from Zhang et al. (2007) and the latter from the manufacturer of the type M-35 buster insulation (Zircar Zirconia, 2015) which has also been used in solar thermochemical experiments (Chueh and Haile, 2010; Furler et al., 2014, 2012b; Marxer et al., 2015).
2.3. Material properties
In the following, the data sources for the properties of materials used in the model are shown and assumptions are described. The convective heat transfer coefficient from the reactor wall to the surroundings at 300 K is assumed to be 15 W m—1 K—1 (Hischier et al., 2009). Table 3 in the annex gives an overview of the material properties.
3. Model validation
The overall heat exchange of reactive material in counter-flow is comprised of a succession of identical heat exchange processes of RPCs at different temperatures in the single chambers of the heat exchanger. In order to validate the model, the following test case representing only one chamber is used as an assessment for the overall process. The single chamber heat exchange between two porous ceramics separated by a thin wall is solved with the proposed reactor model and with a Monte Carlo simulation, where the result of the latter is taken as a reference.
For the test case, defined initial temperatures are chosen for the RPCs and the adjacent separating wall which is made of two 1 mm-layers of Al2O3 and SiC/HfC, respectively. The former is used for the physical separation of SiC and CeO2 which may undergo carbother-mal reduction when brought into contact at high temperatures. The latter is a material used for protective coatings with a high emissivity, thermal conductivity and maximum operating temperature of over 2000 K (Ultramet, 2015). Material properties and boundary conditions of the calculation are shown in Fig. 3a and Table 1. In the Monte Carlo analysis, the resulting radiation heat term is inserted into the energy conservation equation which is then solved to obtain the temperature profile in the computational domain. Scattering is assumed to be isotropic and is described by the scattering coefficient r — qs • b, where qs is the surface reflectance of partially reduced ceria, weighted by the black-body thermal emission, and b is the extinction coefficient, both functions of temperature. The absorption coefficient a is then a — (1 — qs) • b (Furler, 2014). Heat conduction is modeled in both models with the three resistor model, so that the difference between the models is derived from the description of the radiative source term.
7Î=° = 1600 К
Separating walls
System boundary with adiabatic boundary condition
Table 2
Parameter values for example system.
Fig. 3. (a) Schematic of test case used for validation of the Rosseland diffusion model with the Monte Carlo model. At the beginning of the simulation, two porous ceramics, separated by a layer of SiC/HfC and Al2O3 (1 mm thickness each) are at temperatures of 1600 K and 1200 K, respectively, with adiabatic boundary conditions towards the environment. In the transient simulation, the temperature profile of the RPCs and the separating walls is calculated after 40 s with both models and compared. (b) Temperature profiles obtained with the Monte Carlo model (MC) and the Rosseland diffusion approximation (RDA) for two RPCs (reticulated porous ceramic) separated by a layer of SiC/HfC and Al2O3 (1 mm thickness each), after 40 s of heat exchange. The starting temperatures are 1600 K for RPC 1 (0 6 x 6 50 mm) and the walls (50 <x 6 52 mm), and 1200 K for RPC 2 (53 6 x 6 103 mm). The agreement between the models is very good except at the solid-fluid boundary, where the temperature is overpredicted by 1.6% by the RDA because of the anisotropy of radiation.
The equation solved in the first layer of the lower RPC is the following.
r(T4w -T4), kf(If -Tk) kRPC(Tk - Tk+1 ), a _ qAxcP,RPC (Tm+1 -
-L + -1--1
esw -RPC
+ Q rad =-
Parameter Label Value Unit
Concentration ratio C 3000 _
Oxidation temperature Tl 1000 K
Reduction temperature Th 1800 K
Temperature of surroundings To 300 K
Reduction pressure (relative to Pred 10-3 -
standard state of 1 atm)
Oxidation pressure (relative to Pox 1.0 atm
standard state of 1 atm)
CO2-flow (times min = dred) in f CO2 2.0 -
oxidation chamber
Number of chambers (including n 10 -
reaction chambers)
Residence time in heat exchanger At 40 s
Mass of ceria piece m 0.77 kg
Porosity of reactive material u 0.8 -
Efficiency of gas heat recovery ggasrec 0.5 -
Conversion efficiency of heat to gheat-to-electricity 0.4 -
electricity
Convective heat transfer coefficient aconv 15 Wm-2 K-1
k is thermal conductivity, Dx is the thickness of the first layer of the RPC, Dxgap is the thickness of the gap, Dt is the time step, and the subscripts "sw", "f" and "k" denote the separation wall, fluid in the gap (assumed to be oxygen) and the first layer of the RPC. The superscript ''m" refers to the time step. The radiation source term Qrad appears only in the MC model and kRPC is only the thermal conductivity in case of MC but the combined thermal and radiative conductivity in case of RDA.
The MC model is a statistical tool and uses random numbers to describe the radiation emitted in the computational domains. In order to reduce the variability of the results, the MC model is run ten times and the results are averaged and compared with the RDA solution. 95%-confidence intervals are calculated from the MC data using the student t function. All confidence intervals for the temperatures are below 1 K for a resolution of 100 layers per RPC.
In Fig. 3b), the temperature profiles for the test case are shown after 40 s of heat exchange, calculated with the Monte Carlo method for the radiation source term using 100 layers per RPC and the Rosseland diffusion approximation using 10 layers per RPC. The results of both models agree very well with an average deviation of 0.4%. The largest difference of 1.6% between the models is seen at the separating wall because the RDA can give erroneous results at boundaries where the radiation is anisotropic.
4. Parameter study
In the following, a parameter study of the thickness of the porous ceramic elements, their porosity and of the residence time and length of the heat exchanger is shown.
Table 1
Material properties and boundary conditions of Monte Carlo analysis for model validation.
Variable Value Unit Source
Tt=0 (upper RPC, sep. wall) 1600 К -
F"0 (lower RPC) 1200 K -
£Al2O3 0.4 - Touloukian and DeWitt (1972)
kAl2O3 35 Wm-1 K-1 Accuratus (2016)
eSiC/HfC 0.85 - Ultramet (2015)
kSiC/HfC 80 Wm-1 K-1 Ultramet (2015)
Simulation time 40 s -
Number of rays 2 x 105 - -
Time step 0.01 s -
Extinction coefficient b 1.77-pnffi 2.2x10-3u+7.59x10-4 m-1
Scattering coefficient r (-6 x 10-5 + 0.411)b m-1 Furler (2014)
Absorption coefficient a (1 - (-6 x 10-5 + 0.411))b m-1 Furler (2014)
4.1. Thickness of elements
The thickness of the elements of reactive material is varied to find its influence on heat exchanger efficiency and cycle efficiency. Overall heat transfer between the reduced and oxidized elements is composed of (i) radiation heat exchange between the top layer of the material in the lower chamber half and the separating wall, and (ii) the internal heat transfer within the reactive materials, as well as heat losses to the surroundings. The former (i) is limited by the emis-sivities of the materials and their temperatures, while the latter (ii) is limited by the effective thermal conductivity of the material which is determined by the porosity of the material, the thermal conductivities of the solid and fluid, and the radiative properties of the porous material. An increase of material thickness does not change the effective thermal conductivity (according to Eqs. (18) and (20)), however, it increases the mass and the distance for thermal transport in the material and therefore leads to a larger temperature drop in the element. With increasing thickness of the elements and at otherwise constant heat exchanger length and residence time, it is therefore expected that the temperature drop in the material will increase, not allowing all of the material to participate to the same degree in the heat exchange process. In Fig. 4, the efficiency of the heat exchanger ghe as defined in Eq. (7) and the cycle efficiency g as defined by Eq. (12) is shown as a function of the material thickness which is varied between 0.02 m and 0.10 m at a porosity of 80% in a heat exchanger with 8 chambers at a residence time of 40 s. In general, the cycle efficiency g does not vary significantly due to the comparably low efficiency level at the chosen operating point which is limited especially by the reduction temperature and pressure. Heat exchanger efficiency increases strongly for a decreasing material thickness: at 0.10 m ghe = 22.1% and at 0.02 m thickness ghe = 71.0%. This significant improvement is due to the fact that at a defined heat exchanger length and specific residence time of the elements, there is a finite time for the thermal energy to be diffused in the material. Thinner elements have a lower mass and can be heated more quickly, while for larger thicknesses, parts of the material are participating poorly or not at all in the heat exchange between hot and cold elements which decreases heat exchanger efficiency.
In Fig. 5, the temperature profiles of the heat exchanger are shown for a material thickness of 0.02 m and 0.10 m. The heat exchanger has eight chambers, the temperatures of the reduction and oxidation chambers adjacent to the heat exchanger are 1800 K and 1000 K, respectively, and the residence time per chamber is 40 s. The temperature profiles thus show the temperature changes of the reactive material inside of the heat exchanger (compare also Fig. 1 for a general overview of the heat exchanger). Fol-
80% 70% 60%
30% 20% 10%
0.05 0.06 0.07 Thickness of RPC [m]
Fig. 4. Heat exchanger efficiency ghe and cycle efficiency g as a function of RPC thickness. The heat exchanger has eight chambers, the residence time is 40 s and the porosity of the elements is 80%. Parameter values are shown in Table 2.
lowing the path of the colder elements (i = 9 ? i = 2), the heating process can be seen, where the final temperature in the heat exchanger is reached at i = 2. Comparing the final temperature of the two elements in (a) and (b) shows that the thinner material gets heated to temperatures above 1500 K with a maximum temperature difference of about 90 K (maximum temperature: 1630 K, minimum temperature: 1538 K). The thicker material on the other hand shows a much larger maximum temperature difference of over 600 K (maximum temperature: 1610 K, minimum temperature: 990 K) since the heat diffusion process within the material is slower than the heat exchange process between the chambers with the assumed residence time. Thus, for the larger material thickness and a total residence time of 8 x 40 s = 320 s, the internal heat diffusion process is limiting the heat exchange between hot and cold elements. Either a longer total residence time (through an increase of the residence time per chamber or an increase of the heat exchanger length) or a lower material thickness has thus to be chosen to increase the efficiency of the heat exchanger. The heating process thus becomes more efficient with a reduction of material thickness as already seen in Fig. 4. The influence of material thickness on heat exchanger efficiency is of course dependent on the specific properties of the heat exchanger which were fixed here in the operating point of the baseline case (Table 1). For a different number of heat exchanger chambers and residence times, the efficiency curve is shifted, however, the general functional dependency between material thickness and efficiency remains the same, as it is defined by the relative magnitude of heat transfer between the chamber halves and within the materials.
The supposition of facilitated heat transfer in elements with a lower thickness is therefore confirmed which leads to a first recommendation for the design of heat exchangers, i.e. the consideration of both heat exchange processes (i) between hot and cold material and (ii) within the materials itself.
4.2. Porosity of elements
A change in porosity both influences radiation heat exchange through the extinction coefficient (Eq. (18)) and conduction heat exchange through the effective thermal conductivity (Eq. (20)). An increase in porosity therefore enhances radiation heat exchange through a reduction of the extinction coefficient and deteriorates conductive heat exchange within the material through a relative increase of the fluid volume which has a lower conductivity than the solid. The effect of porosity on overall heat transfer is thus a trade-off between these two mechanisms and the changed thermal mass of the element. At high temperatures above about 1000 K, radiation heat exchange dominates the thermal energy transfer within the material. Consequently, when the porosity is increased, heat can be diffused more easily within the RPCs and an enhancement of heat exchange between the hot and cold elements is expected. The porosity of the reactive material is varied between a value of 45% and 85% in order to analyze its influence on efficiency which is shown in Fig. 6. The heat exchanger efficiency increases monotonically with porosity from a value of 19.9% at 45% porosity to 49.7% at 85% porosity. Cycle efficiency rises from 7.6% to 9.9%. This result confirms the enhancement of heat exchanger efficiency with an increase of porosity due to the reduction of the thermal mass of the elements and better radiative penetration of the element volume through a reduced extinction coefficient.
This conclusion is valid for the porosity formulation used in the calculations which relies on a mean pore diameter. For a further evolution of the porous reactive material, a gradual change of porosity with large pores for the inlet of direct irradiation and a decrease of porosity inside the volume to increase the mass loading could be beneficial. However, the description of this gradual change in porosity requires an adaptation of the current model.
a) Thickness: 0.02 m
£ 0.10
<1-1 o
700 900
Insulation
900 ZQQ
Insulation
b) Thickness: 0.10 m
0.40 0.35
Oil !<>■
11 00 900
- 00 -
15 00 Insulation
-1 17 )0 RPC -
- 13 11 )0 )0 RPC -
Insulation
- 700 500
1000 ^
23456789
Length of heat exchanger i
Fig. 5. Comparison of heat exchanger temperature profiles for an RPC thickness of (a) 0.02 m and (b) 0.10 m at a porosity of 80%. The heat exchanger has eight chambers (2 6 i 6 9). Shown is the temperature profile of the heat exchanger including insulation and walls. The thinner material is heated more evenly to a higher temperature and thus allows for a higher heat exchanger efficiency (compare temperature profile in lower chamber half at i = 2). Parameter values are shown in Table 2.
4.3. Length of heat exchanger
An increased number of heat exchanger chambers at a constant chamber length, i.e. increased physical length of the heat exchanger, introduces heat exchange on more intermediate temperature
20% --
Porosity of RPC [-]
Fig. 6. Heat exchanger efficiency ghe and cycle efficiency g as a function of porosity of the reactive elements for a heat exchanger with eight chambers, a residence time of 40 s and an element thickness of 0.05 m. Parameter values are shown in Table 2.
levels and thus changes the characteristic towards a continuous counter-flow heat exchanger. A larger residence time, on the other hand, intensifies heat exchange in the existing number of chambers. In the following, both the number of heat exchanger chambers and the residence time of the elements per chamber, are varied to analyze their influence on efficiency. In Fig. 7, heat exchanger efficiency is shown as a function of the number of heat exchanger chambers nhe (excluding the reaction chambers, i.e. nhe = n - 2 in Fig. 1) and residence time of the elements per heat exchanger chamber. The number of heat exchanger chambers is varied between zero and 20, and the residence time is varied between one second and 100 s. In general, a direct correlation between the number of chambers and the residence time is visible which means that a certain level of efficiency can be reached with a number of parameter combinations. For example, ghe = 0.4 can be reached with nhe = 20 and At = 14s, or nhe = 5 and At = 60s. With interest in a compact heat exchanger, a low number of chambers can be chosen and the residence time can be adjusted accordingly.
The largest efficiency that can be reached in the chosen parameter space is over 60% which requires a heat exchanger with at least 11 chambers at a residence time of 80 s.
In Fig. 7, both an increase in residence time and number of chambers increases efficiency, which indicates a dependency on the total residence time nhe x At of the elements in the heat exchanger. A given total residence time can be achieved with a
0 20 40 60 80 100
Residence time At [s]
Fig. 7. Cycle efficiency g and heat exchanger efficiency ghe as a function of the number of heat exchanger chambers (n-2) and residence time per chamber At. All other parameter values are shown in Table 2.
combination of nhe and At, however, it should be expected that there are limitations to this principle, specifically regarding the chosen number of chambers: a smaller number of chambers forces the heat exchange to take place on fewer levels of temperature, while a larger number of chambers introduces more temperature levels and thus should approach the ideal counter-flow heat exchanger more closely. The entropy change associated with the heat exchange process is proportional to the temperature difference of the chamber halves, and is thus higher for a smaller number of chambers.
In Fig. 8, heat exchanger efficiency is shown as a function of the total residence time of the elements in the heat exchanger for different heat exchanger lengths. As expected, for a given value of total residence time, efficiency increases with the number of heat exchanger chambers up to about nhe = 15. The highest heat exchanger efficiency is reached at nhe x At « 3000 s for nhe p 15. At small values of nhe x At, efficiency shows a strong increase for all nhe. If the total residence time exceeds its optimum value, efficiency decreases due to the assumed losses to the environment. With the information given about the dependency of efficiency on the number of chambers and residence time, it is possible to design the heat exchanger length to maximize efficiency.
80% 70% 60% 50%
40% 30% 20% 10% 0%
■ - - -
1000 1500 2000 2500 3000 Total residence time in heat exchanger nhexAt
Fig. 8. Heat exchanger efficiency as a function of the total residence time of the elements in the heat exchanger nhe x At. At a given value of total residence time, a larger number of chambers reduces the irreversibilities of the heat exchange process by introduction of more intermediate temperature levels.
5. Conclusions
A generic and modular model is presented for the description of solar thermochemical syngas production incorporating solid heat exchange in a counter-flow arrangement of the reactive material. Heat recuperation is achieved by the transfer of energy by thermal radiation between reduced and oxidized elements of reactive material through a thin separation wall having high thermal conductivity and emissivity. Heat diffusion in the porous domains of the reactive material and the insulation occurs by conduction and radiation, while convection is neglected due to its small influence. Radiation heat transfer is described with the Rosseland diffusion approximation and conduction with the three resistor model, where a validation is performed with Monte Carlo simulations.
Both external and internal heat exchange processes between the elements as well as within the elements have to be considered for heat exchanger design. For a large material thickness and low porosity, the element mass is high and internal heat diffusion can be slow compared to the residence time in the heat exchanger which leads to a challenge to dissipate the exchanged heat within the material. As only a part of the material is effectively participating in the heat exchange, this negatively influences the heat exchange process and therefore heat exchanger efficiency. A better performance can be achieved through a decrease of the material mass by reducing the thickness, an increase of the residence time, or an increase of material porosity.
An increase in material porosity (at a proportionally larger mean pore diameter) achieves a better penetration of the radiation into the material volume and at the same time decreases thermal conductivity and mass. The effects of higher volumetric radiation penetration and reduced thermal mass outweigh the reduced thermal conductivity and thus a higher porosity increases the heat exchanger efficiency.
A correlation exists between heat exchanger length and residence time, allowing different combinations of these two variables at constant heat exchanger efficiency. Within certain limits and with respect to a simpler concept, it is therefore advantageous to design the heat exchanger with a shorter length and a correspondingly longer residence time of the elements per chamber. In principle, ghe close to 70% is possible with an adequate combination of length and residence time. However, the achievement of the maximum heat exchanger efficiency requires a certain number of chambers and thus physical length, as irreversibilities are reduced for a larger number of intermediate temperature levels. The efficiency of the thermodynamic cycle is derived under the assumptions of a concentration ratio of 3000, vacuum pumping, separation of the resulting CO/CO2 gas mixture, radiation losses from the reduction chamber, heat recuperation from the gas phase, and an efficiency of 40% for the conversion of heat to electricity. The energy requirements for vacuum pumping, gas separation, and the energy lost by reradiation limit the cycle efficiency to values of about 12%. Higher cycle efficiencies of over 20% are possible using more favorable assumptions of decreased energy penalties and higher operating temperatures. This may require progress in the field of material design, possibly with a different redox material, to increase the reduction temperature and to reduce the reduction enthalpy or increase the nonstoichiometry.
The presented model including the description of heat diffusion within the reactive material is a valuable tool for the design of heat exchangers and helps to identify technically interesting reactor concepts for the achievement of high energy conversion efficiencies.
Acknowledgements
The authors gratefully acknowledge the contribution of Valentin Batteiger, Arne Roth and Parthasarathy Pandi. The research
Table 3
Material properties used in the simulations.
Inconel
Source
eInc = 0.69 kinc = 15.9 Cpinc = 465 Pinc = 8470 Ceria CeO2
cPjCeo2 =(67.95 - 9.9 x 105 • T-2 + 0.01 • T)/MCeo2 kceü2 = 4.61 • (i)4 - 26.64 • (i)3 + 58.30 • (i)2 -£ceO2 = 0.5 + (13T001_1101°00) • 0.4; 1100K< T <1300K
Ceria-RPC
bRPC = 1.765 • ^T-ffi/(2.2 x 10-3 • u + 7.59 x 10-4)
3RPC = kCeÜ2 • [(1 1 pg0 - g1 eCe02 )
g0 = 0.754; g1 = 0.829
PRPC = PceÜ2 • (1 - u) Separating wall made from SiCa75/HfCa25
eSiC/HfC = 0.85
59.28 • (i) + 25.52
(eCe02 +(1 1 eCeÜ2 ) 3COO2) +Pg0 - g 1 eCe02 (eCe02 3^ +(1 1 eCe02
3SiC/HfC
cP;SiC/HfC = 0.75 • 670 + 0.25 • 200 PSiC/HfC = 0.75 • 3210 + 0.25 • 12700
Separating wall made from solid Al2O3
3Al2 03 = 35 cp ,Al203 = 88 0
Pai203 = 3890 Al2O3-insulation
kins = 6 x 10-8T2 - 2 x 10-5T + 0.08 + Mg3
bIns = 1.73 x 10-8T4 - 5.00 x 10-5T3 + 6.13 x 10-2T2 - 31.53T + 1.08 x 104 Cp ins = 4 x 10-7 • T3 - 1.38 x 10-3 • T2 + 1.60 • T + 477.70 Gases
302 =-1.29 x 10-3 + 0.11 x 10-3 • T - 0.05 x 10-6 • T2 + 0.03 x 10-9 • T3 - 0.01 x 10-Cp;C0 = -30.1 • (^)5 + 250.8 • yy 4 - 77°.9 • yy3 +1008.9 • yy2 - 349.2 • yy +1072.8 Cp,C02 = 14.2 • (i)5 - 146.6 • (i)4 + 615.6 • (i)3 - 1353.5 • (i)2 + 1646.2 • + 458.5 MCe02 = 0.172
• T4
Wm-1 K-1 J kg-1 K-1 kgm-3
J kg-1 K-1 Wm-1 K-1
Wm-1 K-1
Wm-1 K-1 J kg-1 K-1 kgm-3
Wm-1 K-1 J kg-1 K-1 kgm-3
Wm-1 K-1 m-1
J kg-1 K-1
Wm-1 K-1 J kg-1 K-1 J kg-1 K-1 kg mol-1
Special Metals (2015) Special Metals (2015) Special Metals (2015) Special Metals (2015)
Riess et al. (1986) Touloukian et al. (1971) Touloukian and DeWitt (1972)
Suter et al. (2014) Suter et al. (2014)
Ultramet (2015) Desmaison-Brut et al. (2007) Ultramet (2015) Ultramet (2015)
Accuratus (2016) Accuratus (2016) Accuratus (2016)
Zircar Zirconia (2015) Zhang et al. (2007) Furler (2014)
Kleiber and Joh (2013) Engineering Toolbox (2015) Engineering Toolbox (2015) Lapp (2013)
leading to these results has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement no. 654408.
Appendix A
See Table 3.
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