The article information
 Hongyu Shi, Jiang Zhu, Zijing Lin
 施红玉, 朱江, 林子敬
 Geometric Design of AnodeSupported MicroTubular Solid Oxide Fuel Cells by Multiphysics Simulations
 基于多物理场模拟的阳极支撑微管燃料电池的结构优化设计
 Chinese Journal of Chemical Physics, 2017, 30(4): 411417
 化学物理学报, 2017, 30(4): 411417
 http://dx.doi.org/10.1063/16740068/30/cjcp1704071

Article history
 Received on: April 14, 2017
 Accepted on: May 19, 2017
Due to their high electrical efficiency, fuel flexibility and low pollutant emission, solid oxide fuel cells (SOFCs) bear the promise of revolutionizing the fossil fuel based power generation technology [1]. For applications in the automotive field and in the auxiliary power supply sector, a new design of SOFC, i.e., microtubular SOFC (mtSOFC) with a tubular diameter typically under a few millimeters, was developed in 1990s [2]. Since its inception, mtSOFC has shown drastic improvements over the conventional SOFC designs on thermal shock resistance, fast startup and thermal cycling [3]. As a result, mtSOFC is attracting an increased attention in the research and development community [4, 5].
High volumetric power density (VPD) is a prerequisite for the commercial success of mtSOFCs in the field of vehicle applications. To meet the requirements in practice, substantial improvements in the designs of cell geometries, stack configurations, and system operations are required [6]. As mtSOFC is a relatively new design and an mtSOFC cell is the basic electricity generating unit, most research efforts are devoted to the fabrication technique for the performance improvement at the cell level. Like the cases for planar SOFCs (pSOFCs), anode supported mtSOFCs (asmtSOFCs) show higher performance than their electrolyteand cathodesupported counterparts. Logically, most researchers focused on asmtSOFCs for the performance improvement [79].
There are a large number of experimental studies on the performances of mtSOFCs with various choices of materials and geometries [4, 10, 11]. Compared with conventional SOFCs, there is hardly anything new about the material choices for mtSOFCs, e.g., YSZ or GDC for the electrolyte, NiYSZ or NiGDC for the anode, LSCF or LSM for the cathode. However, there are completely new geometric parameters for mtSOFCs, e.g., the tube diameter and length. Moreover, the anode thickness emerges as an influential design parameter as it affects the cell volume and mass that in turn affect the VPD and thermal behavior of mtSOFC. The anode thickness also affects the current collection in most mtSOFC designs that in turn affects the current output of mtSOFC [4]. Furthermore, it has been observed that the cathode location can have substantial influence on the output of mtSOFC [11]. Clearly, attention should be paid to these new geometric parameters when developing the mtSOFC technology.
There are large differences in the values of anode thickness, tube diameter and cell length of the reported asmtSOFCs. The anode thickness, though mainly in the range of 200300 μm, varies from 130 μm [12] to 2 mm [13]. The inner tube diameter is centered around 2 mm, but may vary from 0.8 mm to 22 mm [4]. The cell length varies from a few mm to 160 mm [1416]. The wide variations in the geometric parameters of the manufactured mtSOFCs may be partly attributed to the difference in the fabrication abilities of different groups. More importantly, the phenomenon reflects the fact that there is no good understanding about the correlation between the geometries and cell performance. Improved understanding is necessary for the realization of the best performing mtSOFCs to make their practical applications a reality. As experimental examination is expensive and time consuming, numerical models incorporating the physics of SOFC to predict the performance are invaluable tools for the understanding and development of mtSOFCs.
In this work, the impact of geometric parameters on VPD of asmtSOFC was examined by performing systematic multiphysics numerical simulations. The multiphysics model considers the intricate interdependency among the ionic and electronic conductions, gas transport, and electrochemical reaction. The model is validated by comparison with experimental results. Simulations with this validated numerical model provide detailed information about the dependence of VPD on geometric parameters. The optimal geometric parameters and the corresponding power output can be used to guide the design and optimization of asmtSOFCs.
Ⅱ. THEORETICAL METHODA multiphysics model was built and applied to the geometric model of mtSOFC. Simulations of the multiphysics model are carried out to examine the effect of geometric parameters on the cell output. Optimal geometric parameter sets are determined based on the optimization objective as well as practicality considerations.
A. Geometric model and optimization targetA schematic of an asmtSOFC is shown in FIG. 1(a). The mtSOFC consists of a porous anode as the inner layer, a dense electrolyte as the middle layer, and a porous cathode as the outer layer. On the cathode side, the current is collected through the cathode surface. On the anode side, the current is collected through the anode current collector(s) at one or both sides of the anode tube. Due to the axial symmetry of mtSOFC, it is necessary only to apply a twodimensional (2D) geometric model illustrated in FIG. 1(b) for the numerical simulation. The actual 3D structure of the mtSOFC is obtained by revolving the 2D computational domain around the symmetry axis.
The goal of multiphysics simulations is to find the geometric parameters that maximize VPD=P_{m}/V_{cell}, where P_{m} and V_{cell} are respectively the maximum electrical power output and the overall volume of the mtSOFC cell. V_{cell} is calculated as,
$\begin{eqnarray} V_{\textrm{cell}} = l_{\textrm{cell}} \cdot \pi r_{\textrm{out}}^2 = \pi l_{\textrm{cell}} (r_{\textrm{in}} + t_{\textrm{an}} + t_{\textrm{el}} + t_{\textrm{ca}} )^2 \end{eqnarray}$  (1) 
where l_{cell}, r_{out}, and r_{in} are respectively the length, the outer radius and inner radius of the mtSOFC cell, tan,
It is reasonable to assume that the current output is roughly proportional to the area of electrochemically active region,
$\begin{eqnarray} A_{\textrm{EC}}= 2\pi (r_{\textrm{in}} + t_{\textrm{an}} )l_{\textrm{ca}} \end{eqnarray}$  (2) 
where
Note that the cell volume increases quadratically with
Unlike the cases with
A related but different consideration is required for the geometric parameter
Based on the above analysis, there are basically two optimizing parameters,
A standard set of governing equations for the currentvoltage (IV) relation, mass and momentum transports are applied. The multiphysics equations and the associated source terms used are shown as follows.
For IV relation:
$\begin{eqnarray} V_{\textrm{cell}} = E_{\textrm{Nernst}}  \eta _{\textrm{ohm}}  \eta _{\textrm{con}}  \eta _{\textrm{act}} \end{eqnarray}$  (3) 
For charge transport:
$\begin{array}{l} \nabla \cdot {{\bf{i}}_{{\rm{el}}}} = \nabla \cdot (  {\sigma _{{\rm{el}}}}\nabla {\varphi _{{\rm{el}}}})\\ \quad \quad = \pm {j_{{\rm{TPB}}}}{\lambda _{{\rm{TPB}}}}\quad \quad \quad \quad ({\rm{cathode/anode}}) \end{array}$  (4) 
$\begin{array}{l} \nabla \cdot {{\bf{i}}_{{\rm{io}}}} = \nabla \cdot (  {\sigma _{{\rm{io}}}}\nabla {\varphi _{{\rm{io}}}})\\ = \left\{ \begin{array}{l}  {j_{{\rm{TPB}}}}{\lambda _{{\rm{TPB}}}}\quad \quad \;\;{\rm{in}}\quad {\rm{cathode}}\\ {\rm{0}}\quad \quad \quad \quad \quad \quad {\rm{in}}\quad {\rm{electrolyte}}\\ {j_{{\rm{TPB}}}}{\lambda _{{\rm{TPB}}}}\quad \quad \quad \,{\rm{in}}\quad {\rm{anode}} \end{array} \right. \end{array}$  (5) 
${j_{{\rm{TPB}}}} = {j_0}\left[ {\exp \left( {\frac{{2{\alpha _f}F}}{{RT}}{\eta _{{\rm{act}}}}} \right)  \exp \left( {  \frac{{2{\beta _f}F}}{{RT}}{\eta _{{\rm{act}}}}} \right)} \right]$  (6) 
For mass transport [20]:
$\begin{eqnarray} \nabla N_i \hspace{0.15cm}&=&\hspace{0.15cm} \nabla \cdot (  D_i \nabla C_i + C_i \textbf{u})=R_i \end{eqnarray}$  (7) 
$\begin{eqnarray} N_i &= &N_i^{\textrm{diffusioin}} + N_i^{\textrm{convection}} \nonumber\\ &=&  D_i \nabla C{}_i  C_i \frac{{\bar B_2 }}{\mu }\nabla p \end{eqnarray}$  (8) 
$\begin{eqnarray} &&{\rm{Anode: }} \hspace{0.3cm}R_{\textrm{H}_2 } =  R_{\textrm{H}_2 \textrm{O}} =  \frac{{i_{\textrm{el}} }}{{2F}} \end{eqnarray}$  (9) 
$\begin{eqnarray} &&{\rm{Cathode: }}\hspace{0.3cm}R_{\textrm{O}_2 } =  \frac{{i_{\textrm{el}} }}{{4F}} \end{eqnarray}$  (10) 
$\begin{eqnarray} &&{\rm{All \hspace{0.3cm}others: }}\hspace{0.3cm}R_i = 0{\rm{ }} \end{eqnarray}$  (11) 
For momentum transport:
(ⅰ) fuel channel,
$\nabla \cdot \left\{ {\mu \left. {\left[ {\nabla {\bf{u}} + {{\left( {\nabla {\bf{u}}} \right)}^T}} \right]} \right\}  \nabla p = \rho ({\bf{u}} \cdot \nabla ){\bf{u}}} \right.$  (12) 
(ⅱ) porous electrode,
$\begin{array}{l} \frac{\mu }{{{B_0}}}{\bf{u}} =  \nabla p + \nabla \cdot \left\{ {\frac{\mu }{{{\phi _{\rm{g}}}}}\left[ {\nabla {\bf{u}} + {{\left( {\nabla {\bf{u}}} \right)}^T}} \right]} \right\}  \\ \nabla \cdot \left( {\frac{{2\mu }}{{3{\phi _{\rm{g}}}}}\nabla \cdot {\bf{u}}I} \right) \end{array}$  (13) 
Most variables and parameters mentioned in Eq.(3)Eq.(13) are self explanatory. Details about the governing equations and their source terms, boundary conditions, numerical grids and solver, basic parameters for physical properties of materials and cell operating conditions, etc., are referred to Ref.[7]. The multiphysics model has been shown to provide IV curves that are in very good agreement with the experimental results for both pSOFC and mtSOFC consisting of NiYSZ anode/YSZ electrolyte/LSMYSZ cathode [7, 10, 21, 22].
Although the multiphysics model employs a set of governing equations that are quite general, the numerical results are dependent on the values of model parameters. To avoid using a large number of variable material parameters, only the material combination of NiYSZ anode/YSZ electrolyte/LSMYSZ cathode with parameters described in Ref.[7] is considered here. Notice that this is not a limitation as it appears to be. Instead, the optimization results are in fact quite general. This is because that, as discussed above, the thicknesses of electrolyte and cathode are not the true geometric optimization targets. The optimal cell and electrolyte/cathode lengths are closely related to the electronic conductivity of the anode. Considering the fact that Ni is currently a universal material choice for SOFC anodes, the anode electronic conductivity is determined by the Ni content. Consequently, the NiYSZ based optimization results are of broad implications as they are valid also for other Ni based anode materials, e.g., NiGDC, NiSDC, NiCGO, etc.
Ⅲ. RESULTS AND DISCUSSION A. Dependence of cell performance on the cathode locationThe influence of the cathode location on the cell performance has been examined experimentally by comparing the electrochemical performances of four single cells [11]. The four single cells composed of NiYSZ anode/YSZ electrolyte/LSMYSZ cathode were essentially identical, but differed in the distance, d, between the cathode and the anode current collector. The four single cells, cell A, cell B, cell C and cell D, correspond to d=2, 5, 8 and 14 cm, respectively. Geometric models were built to correspond to the specifications of the four single cells and the same set of property parameters as described in Ref.[7] was applied for the multiphysics simulations. The parameter set of Ref.[7] have been shown to reproduce the experimental results of Ref.[10] very well. With this set of parameters, the theoretical IV curves for the four single cells are shown together with the experimental data in FIG. 2. As seen in FIG. 2, the theoretical and experimental results are in very good agreement. The result is remarkable as the values of all the model parameters are exactly the same as that in Ref.[7] and there is no fitting parameter used in this study. The ability to reproduce two independent experiments [10, 11] with the same set of parameters demonstrates convincingly the predictive power of the multiphysics model employed here.
The decreased cell performance with the increased d shown in FIG. 2 is simply due to the associated increase of the ohmic loss of current collection. The current is collected by traveling a distance of d and passing through a cross section area:
$\begin{eqnarray} A_{\textrm{an}} &=& \pi [(r_{\textrm{in}} + t_{\textrm{an}} )^2r_{\textrm{in}}^2] \nonumber\\ &=& \pi t_{\textrm{an}} (2r_{\textrm{in}} + t_{\textrm{an}} ) \end{eqnarray}$  (14) 
For the above four single cells,
Multiphysics simulations were performed for 20 combinations of (
As seen in Table Ⅰ, the optimal cell length, or the corresponding
$\begin{eqnarray} \frac{{\Delta A_{\textrm{an}} }}{{A_{\textrm{an}} }} = \frac{{\Delta t_{\textrm{an}} }}{{t_{\textrm{an}} }}\frac{{r_{\textrm{in}} + t_{\textrm{an}} + \Delta t_{\textrm{an}} /2}}{{r_{\textrm{in}} + t_{\textrm{an}} /2}} > \frac{{\Delta t_{\textrm{an}} }}{{t_{\textrm{an}} }} \end{eqnarray}$  (15) 
and a relative increase of
$\begin{eqnarray} \frac{{\Delta A_{\textrm{EC}} }}{{A_{\textrm{EC}} }} = \frac{{\Delta t_{\textrm{an}} }}{{r_{\textrm{in}} + t_{\textrm{an}} }} < \frac{{\Delta t_{\textrm{an}} }}{{t_{\textrm{an}} }} \end{eqnarray}$  (16) 
Due to the extra capacity of current conduction provided by the larger increase of
$\begin{eqnarray} \frac{{\Delta A_{\textrm{an}} }}{{A_{\textrm{an}} }} \hspace{0.15cm}&=& \hspace{0.15cm} \frac{{\Delta r_{\textrm{in}} }}{{r_{\textrm{in}} + t_{\textrm{an}} /2}}\end{eqnarray}$  (17) 
$\begin{eqnarray} \frac{{\Delta A_{\textrm{EC}} }}{{A_{\textrm{EC}} }} \hspace{0.15cm}&= &\hspace{0.15cm}\frac{{\Delta r_{\textrm{in}} }}{{r_{\textrm{in}} + t_{\textrm{an}} }} \end{eqnarray}$  (18) 
That is,
The optimal
It should be noticed that the data in Table Ⅰ are obtained with a set of conventional and mature materials, i.e., NiYSZ anode/YSZ electrolyte/LSMYSZ cathode. A thickness of 5 μm assumed for the electrolyte layer should also impose no significant challenge on fabrication technique [4, 17, 18]. The tube outer diameter for practical mtSOFCs is often under 2 mm [4]. Such a tube is about the size of the tube with
As the practical
Similar to the cases with singleterminal anode current collection (STACC), multiphysics simulations were performed for mtSOFCs with doubleterminal anode current collection (DTACC). The results for the optimal
To provide more information about the performance of mtSOFC versus the tube size, FIG. 4 shows the dependence of
It is a common perception that mtSOFC is advantageous on thermal shock resistance and fast startup as well as thermal cycling, but suffers from the drawback of much lower current output than that of pSOFC. As shown above, however, the cell current production can be substantially improved by the geometric optimization and by using DTACC instead of the conventional STACC. In fact, FIG. 4 indicates that the performance of mtSOFC can be comparable with that of the state of the art pSOFC [21]. It should be interesting to compare the performances of mtSOFC and pSOFC. However, a quality comparison should examine the effect of a number of key design parameters and require a dedicated effort of study. Consequently, only a preliminary comparison is made here.
As the experimental results reported in Ref.[21] are representative of the best performing pSOFC, the same set of materials and relevant geometric parameters are used for mtSOFC. In addition, the practical parameters of (
As shown in FIG. 5, the power output of mtSOFC is in fact quite comparable to its pSOFC counterpart. This is understandable as the material properties of the two cells are similar and the extra ohmic loss in mtSOFC with
Based on this study, the following results are obtained. (ⅰ)
The numerical results show that: (ⅰ) for (
This work was supported by the National Natural Science Foundation of China (No.11374272 and No.11574284) and the Collaborative Innovation Center of Suzhou Nano Science and Technology.
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