Electrical and thermal analysis of an intermediate temperature IIR‐SOFC system fed by biogas

An Intermediate Temperature Solid Oxide Fuel Cells (IT‐SOFC) system fed by not conventional fuels such as biogas can produce electric energy with high conversion efficiency and thermal energy. Inside the energy system, the methane in the biogas is mixed with steam, and then it is converted into carbon monoxide, hydrogen and carbon dioxide through steam reforming and water gas shift chemical reactions in an indirect internal reformer (IIR) mostly and in the SOFC anode minimally. The chemical energy of the electro‐oxidation of hydrogen and carbon monoxide produced is directly converted into electric energy in the fuel cell anode. A part of the anode exhaust gas can be recirculated at the IIR inlet and the percentage of this recirculated anode exhaust gas together to the fuel utilization factor influence the performances (electric and thermal powers and efficiencies, primary energy saving and first law efficiency) of the SOFC system in a cogenerative arrangement. Through the simulation model of an IT‐SOFC system fed by biogas in cogenerative arrangement, which was formulated ad hoc and implemented in a Matlab environment, the influence of the above‐mentioned variables on the IT‐SOFC system performances was evaluated. The verification of carbon formation at the anode was made.


Introduction
generally contain a significant level of contaminants, is extremely high [13], although the SOFC intermediate operating temperature can be useful to achieve an internal reforming of organic fuels.
Commercial SOFCs fed by biogas with Ni-YSZ anode and YSZ electrolyte are subject to an irreversible degradation caused mainly by carbon deposit and by sulfur poisoning [14,15]. These drawbacks can be solved by adding a purification step for biogas [16] and a chemical process to produce a clean syngas from biogas through steam reforming (SR), partial oxidation, autothermal reforming (ATR) or dry reforming (DR) [17][18][19].
The addition of a considerable amount of steam (steam to carbon ratio equal to or greater than 2) may avert the carbon, promoting the SR of the fuel [20], which may take place either externally to the fuel cell in a dedicated reformer, which is far from the fuel cell with an external mode (ER), or is coupled to the fuel cell with an internal indirect mode (IIR) or internally to the fuel cell with a direct internal mode (DIR). The required heat for reforming can be obtained through the combustion of a part of fuel in a catalytic burner realizing an autothermal reformer.
In the numerical modeling field of SOFC systems fed by biogas, there are some articles in the literature [21][22][23][24][25]. In all these articles [21][22][23][24][25][26] SOFCs operate at high temperature. In some of these articles the performance of SOFC systems with steam and/or dry reformer reactors external to the anode with ER mode [21,24] and of an SOFC-gas turbine system with a steam and DR chemical processes inside the SOFC anode [23] were evaluated. In [25] a 2D steady-state mathematical model of a tubular solid oxide fuel cell with indirect internal reforming (IIRSOFC) has been developed to examine the chemical and electrochemical processes and the effect of different operating parameters on the cell performance.
In this article, the conversion of suitable purified biogas into a H 2 -rich gas mostly takes place outside the SOFC anode in a reformer with IIR mode through the SR chemical process favored by the heat generated by SOFC, that operates at intermediate temperature. Later the same conversion is completed inside the SOFC anode with DIR mode by the same SR chemical process that is favored by both the SOFC intermediate temperature and the direct H 2 and CO consumptions inside the same anode. In an SOFC a part of the anode exhaust gas, consisting mainly of steam and carbon dioxide (CO 2 ), can be recirculated at the anode inlet and the fraction of the recirculated anode exhaust gas influences the electric power produced by the fuel cell and the thermal power in output of the fuel cell in an energy system that includes it [2,[26][27][28].
In this article, a part of the anode exhaust gas, containing mainly steam and CO 2 , is recirculated to the reformer inlet, to reduce the amount of steam necessary to the reformer, which has to be produced externally to the SOFC, for increasing the thermal power produced by the IT-SOFC system in a cogenerative arrangement.
This configuration was chosen because it allows one to preserve the SOFC as much as possible and to increase the useful life of the SOFC and because the heat necessary to the SR of biogas is totally supplied by SOFC.
The fuel utilization factor and the recirculation factor of anode exhaust gas affect the IT-SOFC system performances, in terms of electric and thermal powers produced, of the electrical, thermal, and first law efficiencies and the primary energy saving (PES).
The influence of the above-mentioned variables on system performance was evaluated through the simulation model of an IT-SOFC system, fed by biogas and in cogenerative arrangement, which was formulated and implemented in a Matlab environment. The IT-SOFC system simulation model considers the chemical reactions of SR and water gas shift inside the reformer and the same chemical reactions with the electrochemical anodic reaction of CO and H 2 consumptions at the IT-SOFC anode.
Through numerical simulations the trends of the mass flow rates, temperatures and compositions of the working fluids in the IT-SOFC system analyzed were evaluated as a function of some of its characteristic parameters, such as the fuel utilization factor and the recirculation fraction of the anode exhaust gas at the reformer inlet for an equimolar biogas composition.
The IT-SOFC system performances, in terms of electrical and thermal powers produced, the electrical, thermal and first law efficiencies and the PESs, were evaluated as a function of the same above-mentioned characteristic parameters and the regions of variation in these characteristic parameters, in which the IT-SOFC system obtains the highest values of the performances, were identified.
Finally, a verification on formation of carbon, which may be produced inside the anode, was conducted.

Numerical Simulation Model
The simplified lay-out of the IT-SOFC system in cogenerative arrangement considered is shown in Figure 1. It comprises an SOFC, which supplies a direct current user (U), an indirect internal reformer (IIR), a burner (B), and a heat exchanger (HE).

Chemical reactions
In IIR, which is fed by a gas mixture of biogas and steam and by a part of the anode exhaust gas, SR (1.1) and Water Gas Shift (WGS) (1.2) chemical reactions take place: The IIR operating temperature is equal to the operating one of the SOFC, which works at intermediate temperature (700°C). In the presence of steam and at intermediate temperature, the SR chemical reaction is much more favored than the DR chemical reaction, which can be neglected. The reformer exhaust gas consists mainly of CO, H 2 , and CO 2 and of a small CH 4 percentage.
The electrochemical reactions (2.1) and (2.2) and chemical reactions (1.1) and (1.2), albeit minimally, take place at the SOFC anode, which is fed by the reformer exhaust gas: The conversion of biogas into an H 2 -rich gas occurs mainly in the reformer with IIR mode by chemical reactions (1. (2.1)  The electrochemical reaction (3) takes place at the cathode: Electric energy is produced by the overall SOFC chemical reactions (4) and (5): A part of CO in reformed feeding gas is used by SOFC, to produce electric energy directly through the electrochemical reactions (2.2) and (3).

IT-SOFC system
The feeding biogas, before entering the reformer, is mixed with the recirculated anode exhaust gas and with steam (mixing point 1).
The gas mixture of biogas, steam and recirculated anode exhaust gas enters the reformer at the temperature T b-S-raeg,in,iir , such that reformer catalytic layer works properly, and exits transformed from the reformer at the SOFC average operating temperature T c .
The reformed gas and air enter in the anode and in the cathode, respectively, at the SOFC average operating temperature, T c , neglecting the thermal energy loss in the stretch linking IIR and anode, and at the temperature T air,in,ca , such that SOFC electrolyte works properly. At the SOFC anode electrochemical reactions (2.1) and (2.2) and the chemical reactions (1.1) and (1.2) encourage the further and almost complete conversion of CH 4 into H 2 . The anode and cathode exhaust temperatures are equal to the SOFC average operating temperature, T c .
A part of the anode exhaust gas is recirculated at the reformer inlet, until the mass flow rate of gas at the SOFC anode inlet reaches the maximum value imposed.
In the HE biogas and water preheat from their feeding temperatures, T bio and T H 2 O , up to the mixing temperature with the recirculated anode exhaust gas, T 1 , and the air preheat from its feeding temperature, T air , up to its inlet temperature into the SOFC cathode, T air,in,ca , thanks to the thermal energy supplied to it by burner B exhaust gas.
The cathode inlet air guarantees the thermal equilibrium of the IIR-SOFC group and is always in excess of that strictly necessary for the functioning of the SOFC.
In the combustor B the non-recirculated anode exhaust gas, which is high-temperature gas with combustible gases (CO and H 2 ), completely oxidizes thanks to the cathode exhaust gas, which contains oxygen (O 2 ). The auxiliary air enters the combustor, only in the case in which the cathode exhaust gas is not sufficient to ensure a complete oxidation of the non-recirculated anode exhaust gas at the IIR inlet, and in a quantity strictly necessary to ensure the full oxidation of the same gas.
The burned gas leaves the combustor B at the oxidation temperature, T gas,out,B and the HE at the temperature T gas,out,HE .
In the simulation model, the current collectors and the fuel cell gas distributors are considered equipotential surfaces and the amount of carbon that can be formed in the anode is calculated at chemical equilibrium condition.

Indirect internal reformer
The chemical equilibrium inside the internal indirect reformer (IIR) is governed by the equations (6.1) and (6.2): where K sr and K wgs are the equilibrium constants of chemical reactions (1.1) and (1.2), calculated at the IIR average operating temperature, which is assumed to be equal to the average SOFC operating temperature of the, T c ; F CO,wgs,iir and F CH 4 ,sr,iir are, respectively, the CO and CH 4 molar flows converted into H 2 and CO inside the IIR through chemical reactions (1.1) and (1.2).
The molar flows of the IIR exhaust gas constituents (H 2 O, CO, CO 2 , H 2 , N 2 , CH 4 ) are calculated using equations (6.3): where ν i,j is the stoichiometric coefficient of the ith constituent in the chemical reaction jth, which is positive or (6.1) 4 ,sr,iir , F 1.2 = F CO,wgs,iir and i = CH 4 , H 2 O, CO, CO 2 , H 2 , N 2 negative in cases in which the same constituent is, respectively, a product or a reactant in the chemical reaction jth.

Fuel cell
Main equations inside the anode are (7.1), (7.2), (7.3), and (7.4): where K sr and K wgs are the equilibrium constants of chemical reactions (1.1) and (1.2), calculated at the average SOFC operating temperature, T c ; F CO,wgs,a and F CH 4 ,sr,a are the molar flows of CO and CH 4 , which are, respectively, converted into H 2 and CO at the anode by chemical reactions (1.1) and (1.2); F CO,b and F H 2 ,b are the molar flows of CO and H 2 consumed by electrochemical anodic reactions (2.1) and (2.2). Equations (7.1) and (7.2) govern the equilibrium of the SR and WGS chemical reactions, equation (7.3) defines the parameter q, which represents the ratio of the CO and H 2 molar flows consumed by the electrochemical anodic reactions (2.1) and (2.2), F CO,b and F H 2 ,b , whereas equation (7.4) governs the electrochemical CO and H 2 consumption at the anode.
The parameter q was measured experimentally by Matsuzaki et al. [29] for an H 2 -CO-CO 2 -H 2 O system at the interface between the Ni-YSZ cermet electrode and YSZ electrolyte for two values of the temperature T c : 1023 and 1273 K.
The parameter q was evaluated by linear extrapolation at the fuel cell operating temperature investigated. The molar flows of the anode exhaust gas constituents (H 2 O, CO, CO 2 , H 2 , N 2 , CH 4 ), F i,out,a , are calculated through equations (7.5): The molar flows of H 2 O, CO, CO 2 , H 2 , N 2 , CH 4 recirculated at the IIR inlet, F i,rec,iir , are calculated using equation (8): where f rec is the recirculation factor.
At the cathode, the fundamental equation is equation (9.1): If technical air, which is composed only of O 2 and N 2 , is considered as cathode feeding gas, the molar flows of O 2 and N 2 at the cathode outlet, F O 2 ,out,ca and F N 2 ,out,ca , are calculated using equations (9.2):

Energy Analysis of Cogenerative Plant
When the fuel cell is fed by the indirect internal reformer exhaust gas, which is a gas mixture mainly consisting of CO and H 2 , similar to a syngas, the real fuel cell voltage, V c , is calculated by the equation (10) (4) and (5), respectively; ΔV act,a,CO , ΔV act,a,H 2 and ΔV act,ca,O 2 are the activation voltage losses in the anode and at the cathode. In this article, the SOFC in cogenerative arrangement is considered to operate in the low to medium current density region, where the concentration voltage losses are expected to be limited [25,32] in order to extend its working life. Therefore, in equation (10)   where j CO,a is the current density produced by the electrochemical reaction (2.2); K CO∕O 2 is the equilibrium constant of the chemical reaction (5).
The parameter CO is constant and its value is calculated in such a way as the condition expressed by the equation (14) occurs [2]: therefore its value depends on the value of parameter q.
The current density produced by the fuel cell, j c , is calculated by equation (15): The electric power produced by the IT-SOFC system, P el,SS , is calculated using equation (16): The IT-SOFC system electrical efficiency, η el,SS , is calculated through equation (17): where G bio and LHV bio are the mass flows and the low heating value of feeding biogas at the inlet of the IT-SOFC system considered.
The thermal power surplus produced by the fuel cell, P th,c , is calculated by equation (18) [26]: where P th,wgs,a and P th,sr,a are the thermal powers produced and consumed by the WGS and SR chemical reactions in the anode. P th,c must be removed to avoid the fuel cell overheating.
The thermal power needed for preheating the feeding gases of the fuel cell, P th,tot,ph , is calculated through an equation formulated by the authors ([2] (eq. 21)).
The thermal power needed for heating the IIR, P th,iir , is calculated by equation (19): The fuel cell is in thermal equilibrium condition if equation (20) is verified: Equation (20) expresses the equality of the thermal power P th,c and the sum of the thermal powers P th,tot,ph and P th,iir .
The burner B thermal balance is expressed by equation (21): In equation (21) the two members represent the net thermal energy of the reactants (not-recirculated anode exhaust gas, the cathode exhaust gas, and auxiliary air) and the thermal energy of the combustion products.
The thermal balance of HE is expressed by an equation similar to that given by the authors ([2] (eq. 24)). The latter equation considers that in the HE of the analyzed system biogas, water and air are preheated. In mixing point 1 in Figure 1 the thermal balance equation is the equation (22): The net thermal power produced by the IT-SOFC system, P th,net,SS , is calculated through equation (23): where T ref is the reference temperature used to calculate this thermal power.
The IT-SOFC system thermal efficiency, th,SS , is calculated using equation (24): j c = j H 2 ,a + j CO,a = j O 2 ,ca .

Verification of Anodic Carbonation
The well-known chemical reactions of methane cracking, Boudouard and of steam production [26] take place in the anode with the electrochemical reactions (2.1) and (2.2) and the chemical reactions (1.1) and (1.2) and produce carbon. The product carbon might be deposited on the anode, causing the progressive deactivation of the entire fuel cell.
The gases pass through the anode quite quickly and the above chemical reactions could not have time to reach the chemical equilibrium condition. Therefore, the amount of carbon produced by these three chemical reactions in chemical equilibrium condition is definitely larger than or equal to the real amount of carbon produced in the fuel cell anode.
It is possible to refer to the composition of the anode inlet gas or to the composition of the anode exhaust gas or to the average gas composition between the anode inlet and outlet to calculate the amount of carbon produced by the above chemical reactions in equilibrium condition.
The triple chemical equilibrium was modeled numerically through three not-linear equations formulated by the authors ( [26] (eq. 24)). In the latter system of equations constants k 7 , k 8 , and k 9 assume unit value in the chemical equilibrium condition.
Three parameters, which represent the carbon masses produced in chemical equilibrium per unit mass of feeding biogas at the inlet or at the outlet of the anode or average in the same anode condition are defined by equations (27): when parameter χ C is negative or zero it means there is no carbon formation inside the anode in chemical equilibrium condition.

Validation of Numerical Simulation Model
The reformed gas at the IIR outlet and at the inlet of the fuel cell contains a very small percentage of CH 4 (below 0.1%), which can be neglected, and its composition is similar to that of a syngas.
In Ref. [33] a planar SOFC operating at a temperature of 800°C was fed by a simulated reformed gas (syngas) at different compositions and by air and it has been electrically tested. The compositions of the simulated reformed gas (syngas) used in experiments [33] and of particular interest in this work are shown in Table 1.
The input data to the SOFC calculation code were chosen to make the comparison between the results of the calculation code and the experimental data found in Ref. [33] as homogeneous as possible. Figure 2 shows the comparison of the two polarization curves obtained by the calculation code, considering two different types of feeding reformed gas, and the experimental data found in Ref. [33], when the operating SOFC temperature is at 800°C.
The mean absolute percentage error of the voltage, |(err)|(%), is defined in equation (28): Figure 2 shows that the percentage average absolute errors are, respectively, at 1.37 and 0.84 for the two reformed gases, therefore, for both the reformed gases there is a good agreement between the theoretical polarization  curves produced by SOFC calculation code and the experimental data.

Numerical Simulations and Results Analysis
Countless test campaigns were carried out by means of the calculation code implemented in Matlab language. The simplified flowchart of IT-SOFC system calculation code is shown in Figure 3.
The main input data are the variation ranges of the fuel utilization factor and of the anode exhaust gas recirculation factor at the IIR inlet, → U f,conv and ⃗ f rec , the fuel cell active surface, A c , the biogas and air compositions, ⃗ x bio and ⃗ x air , the maximum mass flows of biogas and gas at the system and anode inlets, G max,bio and G max,gas,in,a , and the steam to carbon ratio, S/C.
When the input data are loaded, the IT-SOFC system calculation code: • evaluates the molar flows of all chemical species at the reformer inlet and outlet, F i,in,iir and F i,out,iir , and at the anode outlet, F i,out,a , by an iterative process; • verifies whether the anode inlet mass flow, G gas,in,a , is greater than the maximum value imposed, G max,gas,in,a , and only in this last case recalculates the biogas mass flow at the system inlet, G bio , and the molar flows F i,in,iir , F i,out,iir , and F i,out,a ; • evaluates the molar flows of all the chemical species at the cathode inlet and outlet, F i,in,ca and F i,out,ca , the fuel cell voltage, V c , and the system inlet air mass flow, G air , by another different iterative process; • evaluates the burner and HE exhaust gases temperatures, T out,gas,B , T out,gas,HE , the SOFC system electric and thermal powers, P el,SS , P th,net,SS , the anode carbonation parameters, χ C,in,a , χ C,out,a , χ C,m,a , and IT-SOFC system electrical, thermal, first law efficiencies, η el,SS , η th,SS , and η I,SS , and PES percentage.
In the most significant test campaign, the results of which are shown in this article, the calculation code was used to simulate the IT-SOFC system fed by an equimolar biogas (50% CH 4 and 50% CO 2 ) varying the fuel utilization factor and the anode exhaust gas recirculation factor at the IIR inlet. In this test campaign, the main input data of calculation code are given in Table 2.
The maximum mass flow of the anode inlet gas was defined through appropriate numerical simulations. When the recirculation factor, f rec , increases, the mass flow of Figure 2. Comparison of the theoretical polarization curves of a fuel cell fed by two different feeding reformed gases (syngases) and the experimental data found in Ref. [33]. the mixture gas of biogas and steam at the anode inlet has been suitably reduced, so that the overall mass flow in the anode gas inlet, G gas,in,a does not exceed the maximum set value, G max,gas,in,a .
For anodes already developed and at the SOFC operating temperature considered (973.15 K) the value of the parameter q is equal to 0.497. From the analysis of numerous test campaigns the influence of the fuel utilization factor and of the recirculation factor on the main parameters characterizing the IT-SOFC system fed by biogas considered was evaluated. Subsequently, from the analysis of the above described test campaigns the influences of the fuel utilization factor and the anode exhaust gas recirculation factor on the IT-SOFC system performances, in terms of electric and thermal power produced, electrical, thermal, and first law efficiencies and PES, and on the anodic carbonation parameters were evaluated. Figure 4 shows the trend of the electric power generated by the IT-SOFC system, P el,SS , varying the anode exhaust gas recirculation factor at the IIR inlet, f rec , and the conventional fuel utilization factor, U f,conv .
When G gas,in,a is equal to G max,gas,in,a , f rec assumes the value f * rec . The f * rec value decreases, if U f,conv increases, and it is between 0.30 and 0.32.
The increasing of f rec and for each value of U f,conv before P el,SS slowly increases, when f rec is lower than f * rec , since at constant SOFC operating current density, j c , operating SOFC voltage, V c , slowly increases. In the same conditions then P el,SS decreases rapidly, when f rec is greater than f * rec , since j c decreases rapidly, despite V c increases.
For each value of f rec P el,SS always increases when U f,conv increases, since j c increases despite V c being reduced. The maximum value of electric power is greater than 68 W obtained at the maximum value of U f,conv investigated (0.65) and for f rec equal to f * rec . In Figure 5 the trend of the net thermal power produced by the IT-SOFC system, P th,net,SS , varying f rec and U f,conv , is shown.
The increasing of f rec and for each value of U f,conv before P th,net,SS increases rapidly, when f rec is lower than f * rec , since the mass flow and the temperature of the hot gases at the outlet of the HE, G out,gas,HE and T out,gas,HE , increase rapidly. Under the same conditions then P th,net,SS decreases rapidly, when f rec is greater than f * rec , since G out,gas,HE decreases rapidly, despite T out,gas,HE continues to increase.
When U f,conv increases and for each value of f rec before P th,net,SS increases, since G out,gas,HE increases, despite T out,gas,HE is reduced. When U f,conv increases and for each value of f rec P th,net,SS decreases because T out,gas,HE decreases much faster, despite G out,gas,HE continues to increase.
The maximum value of thermal power is greater than 129 W and this value is reached for U f,conv equal to 0.52 and for f rec equal to f * rec .  When f rec is lower than f * rec the mass flow of biogas at the IT-SOFC system inlet, G bio , is constant and equal to the maximum value imposed, G max,bio , while when f rec is greater than f * rec , G bio is progressively reduced to allow the increase in the recirculation gas mass flow at a fixed value of anode inlet gas maximum mass flow.
When f rec is lower than f * rec the trends of η el,SS and η th,SS follow the trends of P el,SS and P th,net,SS . When f rec is greater than f * rec the trends of η el,SS and η th,SS deviate from the trends of P el,SS and P th,net,SS since G bio decreases.
In nominal operating range (U f,conv,min ≤ U f,conv ≤ U f ,conv,max ; 0 ≤ f rec ≤ f * rec ) the maximum values of η el,SS and η th,SS are, respectively, greater than 0.31 for f rec equal to f * rec and U f,conv equal to U f,conv,max and greater than 0.57 for f rec equal to f * rec and U f,conv equal to 0.52. Figure 8 shows the IT-SOFC system first law efficiency, η I,SS , varying f rec , and U f,conv . Obviously its trend is derived from the trends of η el,SS and η th,SS .
In nominal operating range (U f,conv,min ≤ U f,conv ≤ U f,con v,max ; 0 ≤ f rec ≤ f * rec ) the maximum value of η I,SS is greater than 0.88 for f rec equal to f * rec and U f,conv equal to 0.52. Figure 9 shows the isolevel curves of PESs, PES, trend, varying the parameters f rec and U f,conv .
In nominal operating range (U f,conv,min ≤ U f,conv ≤ U f,conv,max ; 0 ≤ f rec ≤ f * rec ) the maximum value of PES is higher than 30% for f rec equal to f * rec and U f,conv equal to 0.52.
For f rec equal to f * rec and U f,conv equal to 0.52 graphs shown up to now show that the IT-SOFC system is certainly considered a cogenerative system, according to current Italian and European legislation, at high efficiency and obtains the best values of PES and η I,SS . Finally, the    calculation code was used to verify the formation of carbon at the SOFC anode. Figure 10 shows the trends of the anodic carbonation parameters, χ C,in,a , χ C,out,a , and χ C,m,a , varying f rec and U f,conv . As expected, the resulting graphs confirm that there is no carbon formation either at the anode inlet and outlet or on average in the anode, because S/C ratio is largely precautionary.
Regarding the carbon formation, the most critical area is the anode inlet section of the fuel cell, since in it the molar flows of CO and H 2 are significant, but in the same section the presence of equally significant CO 2 and steam molar flow rates averts the carbon formation.

Conclusions
The calculation tool set up in this work allows the IT-SOFC energy system fed by biogas in cogenerative arrangement to be described completely.
The energy-cogenerative analysis conducted has permitted output parameters to be obtained such as electrical and thermal powers produced, PES and electrical, thermal and first law efficiencies, as a function of two evaluation parameters, such as the fuel utilization factor and the recirculation fraction of the anode exhaust gas at the internal indirect reformer inlet.
The same calculation tool has made possible the identification of the regions of variation in the above evaluation parameters, in which the IT-SOFC system obtains the highest values of the output parameters chosen.
An analysis of the results showed that the IT-SOFC system is a high-performance cogenerative system, in accordance with Italian and European legislation, and the same system obtains values of electrical, thermal and first law efficiencies greater than 0.31, 0.57, and 0.88 and a PES greater than 30% of the parameters variation in the regions identified.
This calculation tool is very flexible and can also be used to evaluate other output parameters as a function of different evaluation parameters.