An assessment on convective and radiative heat transfer modelling in tubular solid oxide fuel cells

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Abstract

Four models of convective and radiative heat transfer inside tubular solid oxide fuel cells are presented in this paper, all of them applicable to multidimensional simulations. The work is aimed at assessing if it is necessary to use a very detailed and complicated model to simulate heat transfer inside this kind of device and, for those cases when simple models can be used, the errors are estimated and compared to those of the more complex models.

For the convective heat transfer, two models are presented. One of them accounts for the variation of film coefficient as a function of local temperature and composition. This model gives a local value for the heat transfer coefficients and establishes the thermal entry length. The second model employs an average value of the transfer coefficient, which is applied to the whole length of the duct being studied. It is concluded that, unless there is a need to calculate local temperatures, a simple model can be used to evaluate the global performance of the cell with satisfactory accuracy.

For the radiation heat transfer, two models are presented again. One of them considers radial radiation exclusively and, thus, radiative exchange between adjacent cells is neglected. On the other hand, the second model accounts for radiation in all directions but increases substantially the complexity of the problem. For this case, it is concluded that deviations between both models are higher than for convection. Actually, using a simple model can lead to a not negligible underestimation of the temperature of the cell.

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).H2+12O2H2OCH4 + 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:CTT=G

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:qconv=hcv(TwallTgas)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)

  • A. Selimovic et al.

    J. Power Sources

    (2005)
  • S. Campanari et al.

    J. Power Sources

    (2004)
  • B. Todd et al.

    J. Power Sources

    (2002)
  • D. Sánchez et al.

    J. Power Sources

    (2006)
  • C. Stiller et al.

    J. Power Sources

    (2005)
  • U.G. Bossel, Final Report on SOFC Data Facts and Figures, Swiss Federal Office of Energy, Berne,...
There are more references available in the full text version of this article.

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This paper presented at the 2nd National Congress on Fuel Cells, CONAPPICE 2006.

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