An assessment on convective and radiative heat transfer modelling in tubular solid oxide fuel cells☆
Introduction
SOFCs are devices operating at temperatures ranging from 800 to 1050 °C for state of the art materials. Below this range, voltage losses due to ionic/electronic resistivity of materials increase noticeably as conductivity grows exponentially with temperature [1], [2]. On the other hand, SOFCs cannot be operated continuously at a very high temperature, say 1100 °C, as this would lead to a considerable decrease in performance, probably caused by a thermal expansion mismatch between electrodes and electrolyte [3]. Therefore, the management of heat transfer inside a solid oxide fuel cell, either with tubular or planar technology, is essential in order to guarantee the reliability and long life demanded by the market to this sort of power generation devices. Fig. 1 shows the amount of energy released and/or consumed inside an SOFC fed with natural gas as a function of operating voltage and for different pressures.
Three reactions are considered to take place: hydrogen oxidation, Eq. (1), methane reforming, Eq. (2), and carbon monoxide shifting, Eq. (3).CH4 + H2O → 3H2 + COCO + H2O →H2 + CO2
The net amount of heat released according to Fig. 1 must be evacuated from inside the cell by the air mass flow, which is supplied well in excess with respect to the stoichiometry of Eq. (1). Thus, under normal operating conditions, only 15–20% of the air is used to oxidize the fuel.
This work deals with heat transfer characterization and modelling inside tubular SOFCs, particularly applied to a 1.5 m long Siemens Westinghouse cell with 100 W rated power for ambient pressure operation, Fig. 2. More precise geometric data of this technology can be found in reference [4] and is shown in Table 1.
As said before, heat released in Eq. (1) is evacuated from the electrodes/electrolyte solid structure, also known as PEN from Positive Electrolyte Negative, mainly by convection but, in the case of the tubular technology shown in Fig. 2, radiation between PEN and air supply pipe also plays an important role. Fig. 3 shows the proportion of total heat transfer which takes place by convection and radiation. It can be concluded that each one of these heat transfer mechanisms is dominant at a different part of the cell: convection for the first third of it and radiation from that point to the exhaust section. Although this distribution is not constant and may vary according to operating conditions, it is clear that these two phenomena must be very well described when developing a model of performance suitable for tubular SOFCs.
The work is divided in two parts. First, a model for describing convective heat transfer, based on a local evaluation of transfer coefficients, is proposed. This model is later simplified and the loss of accuracy evaluated. Secondly, two models of radiative heat transfer are proposed. One of them is a simple model, extensively used in previous works, for radiative exchange in the radial direction which is based on the hypothesis of infinite walls. Then, a complete model of radiation is described. This second model considers radiative exchange in all directions, radially and obliquely, and introduces additional complexity to heat balance equations.
Fig. 4 shows the discretization of the cell which is used for the heat transfer models. The cell is divided axially into a number of slices which are again divided radially into five annular volumes, cylindrical for the inner one, called elements.
It is commonly agreed that multidimensional models of performance of an SOFC are based on a decoupled solving strategy for temperatures and composition. In other words, an iterative method is used that solves thermal and electrochemical models consecutively [5]. Firstly, temperatures are calculated by solving the system of linear equations resulting from local heat balance equations. Compositions and current density remain constant at all the slices and elements. This system of equations is shown in Eq. (4) where T stands for local temperatures, CT for heat transfer coefficients and G for generation terms:
This temperature field is then used as an input to solve the electrochemical model and obtain compositions, current density, voltage losses, reaction rates, etc.
As stated above, this work is aimed at describing radiative and convective heat transfer equations involved in heat balance local equations. In other words, models will be presented to calculate coefficients included in CT, Eq. (4).
Section snippets
Convective heat transfer: model description
Convective heat transfer is described by Newton's law of cooling:where hcv is the convective heat transfer or film coefficient. Calculating this coefficient accurately is the key task to obtaining a precise heat transfer model. The following lines describe a step-by-step procedure to obtain hcv:
- 1.
Calculation of fluid properties: viscosity and thermal conductivity.
- 2.
Calculation of Reynolds number from fluid properties and duct geometry.
- 3.
Calculation of flow regime from Reynolds
Radiative heat transfer: model description
Previous works by the authors have shown that radiation involves not only heat exchanged between solid walls but between a solid wall and certain gases. However, the latter is out of the scope of this work as it is only relevant under abnormal operation inside the cell, i.e. high current density [5], [9].
Two models are to be presented in the following lines. One of them is a so-called radial model and is based on the hypothesis of infinite coaxial cylinders. It will be called the simple model.
Convective heat transfer: model results
As said in the introductory section of this work, convective heat transfer coefficients cannot be taken as constant along the cell tube. In fact, as shown in Fig. 8, Re varies significantly from the entrance to the exhaust section of any of the three ducts inside the stack, especially at the anode as a consequence of the rapid increase in temperature and mass flow. Convective heat transfer coefficients are depicted in Fig. 9. It can be seen that, for the operating conditions considered, 0.45 V
Radiative heat transfer: model results
Results obtained when applying the simple model have been published in previous works by the authors [9] and other researchers [4] and will not be repeated here. However, it must be said that they have been considered satisfactory as the impact over global performance of using the simple or complex models is not dramatic. This will be shown later.
Fig. 13 shows the complexity of the radiative heat transfer when applying the complex model. A is the inner wall of the cathode of slice n and is
Conclusions
The work presented here is based on another work previously published by the authors [5]. It focuses on heat transfer modelling inside tubular fuel cells and tries to improve the weakest aspects of those models being used by other authors currently [2], [12], [13]. The following particular conclusions with respect to the models presented can be drawn:
- 1.
If the model is intended to predict the global performance of the fuel cell and no internal information is needed, complex models are not
References (13)
- et al.
J. Power Sources
(2005) - et al.
J. Power Sources
(2004) - et al.
J. Power Sources
(2002) - et al.
J. Power Sources
(2006) - et al.
J. Power Sources
(2005) - U.G. Bossel, Final Report on SOFC Data Facts and Figures, Swiss Federal Office of Energy, Berne,...
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This paper presented at the 2nd National Congress on Fuel Cells, CONAPPICE 2006.