International journal of hydrogen energy

international journal of hydrogen energy 35 (2010) 308–312

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journal homepage: www.elsevier.com/locate/he

Technical Communication

A simple equation for temperature gradient in a planar SOFC stack
A.A. Kulikovsky a,b,*
a b

Institute for Energy Research – Fuel Cells (IEF–3), Research Centre ‘‘Julich’’, Leo-Brandt Str, D-52425 Julich, Germany ¨ ¨Moscow State University, Research Computing Center, 119991 Moscow, Russia

article info
Article history: Received 8 October 2009 Received in revised form 22 October 2009 Accepted 23 October 2009 Available online 13 November 2009 Keywords: SOFC stack Heat transport Temperature gradient Modeling

abstract
Our recent model of heat transport in a planar SOFC stack is extended to take into accountfinite hydrogen utilization. The extended model includes the heat balance equations in the interconnect and air flow, and the hydrogen mass balance equation in the anode channel. An approximate analytical expression for the gradient of stack temperature along the air channel is derived. The analytical result is in excellent agreement with the exact numerical solution. The resulting expression can beused for rapid estimate of the temperature gradient in a planar SOFC stack under real operating conditions. ª 2009 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved.

1.

Introduction

The operating temperature is one of the key parameters determining the efficiency and lifetime of a solid oxide fuel cell (SOFC) stack. Important issue in stack design is minimizationof temperature gradients over the stack volume [1]. These gradients induce thermal stress on the functional layers and sealant, which reduces stack lifetime. Direct measurements of the temperature distribution over the surface of individual fuel cells in a stack is a difficult and expensive task. To our best knowledge, so far such measurements have not been published. In this situation, modeling isthe only way to rationalize the temperature field in stacks. Numerical CFD models give the temperature distribution over the surface of individual cells [2,3], or along the single air channel [4,5] (recent CFD models of SOFC are reviewed in refs.

[6,7]). However, CFD calculations are time-consuming. An analytical expression suitable for fast engineering calculations of the temperature gradientin the stack would be very desirable. In this work, our recent model for the temperature shape in a planar SOFC stack is employed to derive a simple analytical expression for the temperature gradient along the air channel. The physics which stands behind the resulting expression is discussed.

2.
2.1.

Model
Heat transport: general assumptions and equations

Consider an element of a planarSOFC stack, a piece of interconnect with the single straight air channel (Fig. 1). From the

* Institute for Energy Research – Fuel Cells (IEF–3), Research Centre ‘‘Julich’’, D-52425 Julich, Germany. ¨ ¨ E-mail address: A.Kulikovsky@fz-juelich.de 0360-3199/$ – see front matter ª 2009 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2009.10.066international journal of hydrogen energy 35 (2010) 308–312

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Table 1 – The physical parameters.
Air temperature at the inlet T0 , (K) a Standard temperature T298 (K) Interconnect thickness hp (m) Characteristic length for heat exchange hc (m) Channel length L (m) Interconnect thermal conductivity lp (W mÀ1 KÀ1) Air thermal conductivity la (W mÀ1 KÀ1) Entropy change in hydrogen–oxygenreaction at 1000 K DS (J molÀ1 KÀ1) Specific heat of air at 700  C cPa (J kgÀ1 KÀ1) Air density at 700  C Pa (kg mÀ3) Flow velocity in the cathode channel v0 (m sÀ1) a Characteristic current density jref (A mÀ2) Tafel slope b (V) Nusselt number in the air channel 600 þ 273 298 10À3 10À3 0.1 12 0.073 54.8 [11] 1160 0.32 10 104 0.15 4

Fig. 1 – Sketch of the stack element: the single linear...