CACHE Modules on Energy in the Curriculum
Fuel Cells
Module Title: Heats of Reaction and Energy Balances in an SOFC
Module Author: Michael D. Gross
Module Affiliation: Department of Chemical Engineering
Bucknell University, Lewisburg, PA 17837
Course: Material and Energy Balances (Introduction to Chemical Engineering)
Text Reference: Felder and Rousseau 3rd edition, Sections 9.1-9.3 and 9.5, Tables B.1-B.2
Concept Illustrated: Heats of reaction and reactive energy balances
Problem Motivation: Fuel cells are a promising alternative energy conversion technology. There are numerous types of fuel cells, as described in Module 0, which are typically distinguished (and named) by either 1) the ion conducted across the electrolyte or 2) the electrolyte material. The solid oxide fuel cell has a dense metal oxide electrolyte and operates in the temperature range of 700-1000°C. In this module, the SOFC will be explored with focus on the heat generated by the reactions in the SOFC.
A general schematic of an SOFC operating on H2 fuel is shown in Figure 1. Oxygen from air supplied to the cathode is reduced to O2-, then the O2- is transported across the electrolyte from the cathode to the anode, and finally the O2- is reacted with H2 fuel at the anode releasing electrons. The electrons released at the anode are transported through an external circuit where electrical power can be drawn. The layered cathode-electrolyte-anode structure is often referred to as the membrane electrode assembly.
SOFC reactions:
Anode: H2 + O-2 ® H2O + 2 e-
Cathode: ½ O2 + 2 e- ® O-2
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Overall: H2 + ½ O2 ® H2O
Figure 1. General schematic of SOFC.
Besides the membrane electrode assembly where the reaction takes place, there is a balance of system, where gas flow, temperature, pressure, and a variety of other processes are managed. These systems must be operating correctly for the fuel cell to work properly. Temperature management is important for all types of fuel cells. SOFCs operate isothermally somewhere in the range of 700-1000°C. Changes in temperature may occur because the heat generated by the fuel cell reaction and the heat transferred away from the fuel cell are not balanced. The fuel cell reaction generates heat because the oxidation reaction is exothermic. At the same time, heat is being lost to the surroundings because the temperature of the cell is greater than the temperature of the surroundings.
The energy released in a reaction is dependent on temperature because the enthalpies of chemical species change with temperature. These changes must be accounted for when calculating the amount of heat released from the exothermic reaction.
Calculating heat of reaction:
The oxidation reaction of methane at 800°C will be used as an example for calculating heat of reaction.
CH4(g) + 2O2(g) à CO2(g) + 2H2O(g)
Note that all chemical species are in the gas phase at 800°C. The heat of reaction is calculated with the heats of formation for each chemical species involved in the reaction.
here, is the heat of formation of species i at the fuel cell temperature, TFC, and νi is the stoichiometric coefficient for each species for the given reaction. For the oxidation of methane, the equation is as follows:
The heat of formation at TFC can be calculated for each species as follows:
where is the heat of formation for species i at standard temperature (25°C) and Cp is the heat capacity at a constant pressure. The integrated CP term accounts for changes in enthalpy due to temperature. can be found in Table B.1 of Felder and Rousseau.
CP values are a function of temperature and can be calculated with Table B.2 of Felder and Rousseau. In Table B.2, CP as a function of temperature is reported in the form a+bT+cT2+dT3. Therefore,
Note in Table B.2 that CP values for H2, O2, and H2O are to be calculated with temperature units of °C.
Problem Information
Example Problem Statement:
A SOFC is being fed 0.4 mol/s H2 and 0.2 mol/s O2 and the reaction goes to 80% completion. The fuel cell generates heat due to the exothermic reaction and heat is lost to the surroundings. Both the rate of heat generation and rate of heat loss to the surroundings are a function of temperature. Calculate the temperature at which the rate of heat loss is equal to the rate of heat generation. Consider a temperature range of 500°C to 1000°C.
Necessary Information:
Overall fuel cell reaction: H2(g) + ½ O2(g) → H2O(g)
For H2, O2, and H2O, standard heats of formation can be found in Table B.1 and formulas for Cp can be found in Table B.2 of Felder and Rousseau.
Rate of heat loss to the environment:
where TFC is the temperature of the fuel cell and To is the temperature of the surroundings. The equation applies for temperature in °C and To = 25°C.
Assume a continuous system at steady state.
Example Problem Solution:
The problem can be broken into three parts: 1) determine the rate of heat generation due to the reaction, 2) determine the rate of heat loss to the environment, and 3) determine the temperature at which the rate of heat generation and heat loss are equal. For each part, example calculations are shown for a fuel cell temperature of 500°C.
Part 1.
Determine the rate of heat generation due to the reaction. (Example calculations are shown for 500°C.)
Step 1.
The heat generated per mole of reaction can be calculated as follows:
where, for example, is the heat of formation of H2O at the fuel cell temperature. Note that stoichiometric coefficients are used for the values.
Step 2.
The heat of formation at the fuel cell temperature can be calculated for each species as follows:
where is the heat of formation for species i at standard temperature (25°C) and Cp is the heat capacity at a constant pressure. The integrated CP term accounts for changes in enthalpy due to temperature.
Step 3.
CP values are a function of temperature and can be calculated with Table B.2 of Felder and Rousseau. In Table B.2, CP as a function of temperature is reported in the form a+bT+cT2+dT3. Therefore,
Note in Table B.2 that CP values for H2, O2, and H2O are to be calculated with temperature units of °C.
Step 4.
Calculate for H2, O2, and H2O.
Step 5.
Calculate
The negative value indicates that heat is being released during the reaction, i.e. the reaction is exothermic. In terms of heat generated in the fuel cell, we will use a positive value of 246.16 kJ per mole of reaction.
Step 6.
Convert the heat released per mole of reaction to a rate of heat generation with units kJ/s.
The heat added to the fuel cell system by the reaction should be converted to a rate of heat transfer (kJ/s).
Rate of heat added to fuel cell by reaction =
Part 2.
Determine the rate of heat loss to the surroundings. (Example calculations are shown for 500°C.)
Step 1.
The rate of heat loss is described by the following equation, which was given in the problem statement.
For TFC = 500°C and To = 25°C
Part 3.
Determine the temperature at which the rate of heat generation and heat loss are equal; i.e. the net heat generation is zero. (Example calculations are shown for 500°C)
Step 1.
The net heat generation in the fuel cell can be calculated as follows:
Net heat generation = Heat added to fuel cell by reaction – heat lost to surroundings
The value of was negative in Part 1, which corresponds to an exothermic reaction. In this case, heat was generated in the fuel cell. For calculating net heat generation, the heat generated by the reaction will be taken as a positive value, which corresponds to adding heat to the fuel cell system. Heat lost to the surroundings will be taken as a negative value, which corresponds to the fuel cell system losing heat.
Step 2.
Calculate the net heat generation
Net heat transfer =
Step 3.
Repeat calculations for temperatures from 500°C to 1000°C in 50°C increments. The following table and figure shows values for heat generation from the reaction, heat loss to the surroundings, and net heat generation.
Based on the graph, the net heat generation is zero at approximately 880°C.
Step 3 alternative.
An alternative way to solve for the temperature that results in a net heat generation of zero would be to use a computer solver, such as Goal Seek in Excel.
Note: The heat generated by the reaction only changes by 1% over a temperature range of 500-1000°C, whereas the rate of heat loss increased by a factor of 6 over the same temperature range.
Home Problem Statement:
A co-worker has collected the following experimental data on a fuel cell system you have developed.
With the collected data, calculate the rate of heat loss to the environment, rate of heat generated by reaction, and the H2 reaction rate (in mol/s).
Necessary Information:
Overall fuel cell reaction: H2(g) + ½ O2(g) → H2O(g)
For H2, O2, and H2O, standard heats of formation can be found in Table B.1 and formulas for Cp can be found in Table B.2 of Felder and Rousseau.
Rate of heat loss to the surroundings:
where TFC is the temperature of the fuel cell and To is the temperature of the surroundings. The equation applies for temperature in °C and To = 25°C.
Assume a continuous system at steady state.
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