1 Introduction

The water gas shift (WGS) reaction (see Eq. 1) is a proven method for increasing the hydrogen content in a product gas generated by gasification. Recent research has extensively investigated a WGS unit, employing a commercial Fe-/Cr-based catalyst, which processed product gas from dual fluidized bed (DFB) biomass steam gasification [8, 16, 17].

$$ CO+{\mathrm{H}}_2\mathrm{O}\leftrightarrow\ {\mathrm{H}}_2+\mathrm{C}{\mathrm{O}}_2\kern0.75em \Delta H=-41.1\kern0.3em \mathrm{kJ}\kern0.3em mo{\mathrm{l}}^{-1} $$
(1)

Since the first applications of the WGS reaction catalyzed by an Fe-/Cr-based catalyst (compare [23]), several kinetic models, based on power laws, have been proposed by different authors, for example, in [2, 3, 9, 13, 20] and [22]. However, most of the investigations of Fe-/Cr-based WGS catalysts were carried out for product gas derived from coal gasification [10, 11]. Zhu and Wachs [28] give a comprehensive summary about kinetic models of the WGS reaction catalyzed by Fe-/Cr-based catalysts.

In addition, Fail [7] proposed a kinetic model for the WGS reaction catalyzed by an Fe-/Cr-based catalyst. This kinetic model specifically takes the product gas from DFB biomass steam gasification into account. In the present paper, this model is referred to as the former kinetic model.

The main gas components of the product gas are H2, CO, CO2, and CH4 which makes the product gas a suitable source for different synthesis reactions, for example, see [19, 21] and [24]. In addition, it contains about 100 cm3 m−3 H2S. This H2S content was considered within the former model because H2S could lead to a performance decrease of the catalyst [26].

In [15], the former model was validated with experimental data from a WGS reactor which employed commercial Fe-/Cr-based catalyst disks (diameter about 6 mm and height about 3 mm, original size) and which processed real product gas from the industrial scale DFB biomass steam gasification plant in Oberwart, Austria. The validation showed significant inaccuracies of the temperature profile along the WGS reactor, which could be attributed to the fact that the former model was established with a milled Fe-/Cr-based catalyst, which did not have the original disk size. Therefore, a higher catalyst surface was available during the establishment of the former kinetic model compared to the validation experiments where the catalyst was used in its original size. However, the WGS reactor, which is located at the site of the gasification plant in Oberwart employed the catalyst in its original pellet size. Consequently, the differences between the measured and calculated temperature profile occurred.

The present paper improves the former kinetic model in order to obtain a better agreement between the experimental data and the model. Consequently, the improved kinetic model should be accurate enough to be used for basic design and engineering of WGS reactors employing Fe/Cr based catalysts which process product gas from DFB biomass steam gasification.

2 Materials and methods

First, this section presents the former kinetic model, which is based on the previous work carried out by Fail (see [7]) and which is improved in the present paper. Second, the experimental setup which generated the data material which was used to improve the former and to validate the improved kinetic model is presented. Third, the derivation of the numerical mass and energy balances of the WGS reactor is introduced, which enables the comparison of the measured data with the results of the model.

2.1 The former kinetic model

This paper is based on a kinetic model which was derived at the TU Wien from results obtained with a laboratory scale chemical kinetics test rig (see [7]). For that kinetic model, the Fe-/Cr-based catalyst was milled before the kinetic parameters were determined. Consequently, the model shows inaccuracies regarding the reaction rate and, therefore, the temperature profile (compare [15]) if the catalyst is used in its original size.

In this paper, the former kinetic model has been improved in order to meet the requirements for using this model for the basic design and engineering of a WGS reactor which processes product gas from DFB biomass steam gasification.

The starting point for both the former and the improved kinetic model was the power law in Eq. 2 which is based on [7] and [15]:

$$ r\left({\varphi}_i,T\right)={k}_0\cdot \mathit{\exp}\left(\frac{-{E}_a}{R\cdot T}\right)\cdot {p}_{CO}^a\cdot {p}_{\mathrm{H}2\mathrm{O}}^b\cdot {p}_{CO2}^c\cdot {p}_{\mathrm{H}2}^d\cdot \left(1-\frac{K_{MAL}}{K_{\mathrm{g}}}\right) $$
(2)

The reaction rate r(φ i , T) is a function of the reactive species (CO, H2O, CO2, and H2) and the temperature. Other gas components which are usually contained in small amounts in the product gas of the DFB biomass steam gasification are considered as inert.

The parameters which describe the reaction rate are the rate constant k 0, the activation energy E a , and the reaction exponents (a, b, c, and d) of each component i which is a reactant of the WGS reaction (see Eq. 1). p i  is the partial pressure, R is the general gas constant, and T is the temperature along the reactor. K MAL is the mass action law and K g the equilibrium constant calculated from thermo-physical properties.

$$ {K}_{MAL}=\frac{\varphi_{\mathrm{H}2}\cdot {\varphi}_{CO2}}{\varphi_{CO}\cdot {\varphi}_{\mathrm{H}2\mathrm{O}}} $$
(3)
$$ {K}_{\mathrm{g}}=\mathit{\exp}\left(\frac{-\varDelta G(T)}{R\cdot T}\right) $$
(4)

The partial pressure of each component p i can be expressed by the overall absolute pressure p and the volumetric fraction φ i .

$$ {p}_i={\varphi}_i\cdot p $$
(5)

The work in [7] led to the following parameters for Eq. 2, resulting in the former model (see Table 1).

Table 1 Parameters of the former kinetic model (see [7] and [15])
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2.2 Setup for the generation of the experimental data

This section provides information about the plants and the measurement principles which were used for the data generation in order to improve the former and to validate the improved kinetic model.

2.2.1 The dual fluidized bed steam gasification plant

The product gas for the WGS reactor was extracted from the commercial DFB biomass steam gasification plant in Oberwart, Austria. The gasification plant is operated as combined heat and power (CHP) plant and generates electricity and heat for the district. Figure 1 shows a simplified flowchart of the gasification plant.

Fig. 1
figure 1

Simplified flowchart of the gasification plant in Oberwart, Austria

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For the improvement of the former and the validation of the improved kinetic model, the product gas was extracted from two extraction points (see also Fig. 1). The first extraction point was located before the rapeseed methyl ester (RME) gas scrubber, and the second extraction point was located after the RME gas scrubber. The product gas which was extracted before the RME gas scrubber showed a significantly higher amount of tar (about 10 g m −3) and steam (about 35 %) compared to the product gas which was extracted after the RME gas scrubber (about 2 g m −3 tar and about 7 % steam).

Further details of the plant can be found in [6, 15–17].

2.2.2 The water gas shift reactor

Figure 2 shows a simplified flowchart of the cylindrical WGS reactor located at the site of the gasification plant in Oberwart which was used for the generation of the experimental data.

Fig. 2
figure 2

Simplified flowchart of the WGS reactor which was used for the generation of the experimental data

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The WGS reactor successively processed a partial flow of the product gas of the gasification plant from the two extraction points (see Fig. 1). The product gas was mixed with additional steam which was provided by a steam generator. The gas inlet temperature was adjusted with a heating section to about 350 °C. At the bottom of the reactor, a screen plate was used for carrying the commercial Fe-/Cr-based catalyst which had been in operation for more than 2000 h at the time this research was conducted. The catalyst bed was a fixed bed with a diameter of 9 cm and a height of about 40 cm resulting in a catalyst bed volume of about 2.5 dm3. The bulk density of the catalyst bed was 1.24 kg dm−3. The wall thickness of the reactor was 3 mm and the insulation consisted of stone wool with a thickness of 10 cm and a thin layer of Al around the outer surface.

Seven thermocouples were used to measure and to record the temperature profile along the WGS reactor. The temperature profile allowed reaching some conclusion about the activity of the commercial Fe-/Cr-based catalyst due to the exothermic nature of the WGS reaction.

After the product gas was processed in the reactor, it was recycled back to the gasification plant. A sample flow of the processed gas extracted before the steam addition, before the inlet, and after the outlet of the WGS reactor was sent to the analytical line and, subsequently, to a gas chromatograph (GC) in order to measure the dry gas composition of the processed gas.

The WGS reactor operated at ambient pressure and the pressure drop along the catalyst bed was between 500 and 1000 Pa during the whole operating time.

2.2.3 Measurement of the temperature profile along the water gas shift reactor

Figure 2 shows the positions of the thermocouples (type J) along the WGS reactor. Thermocouple T0 was positioned before the fixed bed Fe-/Cr-based catalyst. Therefore, it was not in the reactive zone. T1 to T5 were positioned along the catalyst bed at a distance of 10 cm from each other. T1 was positioned right at the beginning of the catalyst bed, and T5 was positioned right at the end of the catalyst bed. T6 was outside the catalyst bed. T1 to T5 enabled the measurement and recording of the temperature profile along the fixed bed WGS catalyst.

2.2.4 Measurement of the gas composition at the inlet and the outlet of the water gas shift reactor

Figure 3 shows the setup of the gas conditioning before the gas chromatograph (GC) that was used for the determination of the gas composition.

Fig. 3
figure 3

Setup of the gas conditioning before the GC and the water content determination

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Before entering the GC, the gas stream passed through two gas washing bottles filled with glycol at a temperature of about −5 °C in order to condense and separate the steam. Therefore, a dry gas stream could be assumed after the gas washing bottles. The dry gas stream passed through another gas washing bottle filled with glass wool in order to prevent aerosols from entering the GC. After the glass wool bottle, a gas meter recorded the volumetric dry gas flow.

The steam content in the product gas before the addition of steam, before the inlet of the reactor, and after the outlet of the reactor was determined with a gravimetrical method. The wet gas stream passed through the gas washing bottles for a certain time, where the steam was condensed. Subsequently, the volumetric dry gas flow was recorded and the gas washing bottles were weighed. Consequently, the steam content before the steam addition, before the inlet, and before the outlet of the reactor could be determined.

The volumetric dry gas flow rate at the inlet of the WGS reactor was calculated by the water balance of the steam generator according to the following equation and the known volumetric flow rate of the steam addition:

$$ {\dot{V}}_{\mathrm{Dry}}={\dot{V}}_{\mathrm{Wet},\mathrm{In}}\cdot \left(1-{\varphi}_{\mathrm{H}2\mathrm{O},\mathrm{In}}\right)=\frac{\left({\dot{V}}_{\mathrm{H}2\mathrm{O},\mathrm{Add}}\cdot {\varphi}_{\mathrm{H}2\mathrm{O},\mathrm{Out}}-{\dot{V}}_{\mathrm{H}2\mathrm{O},\mathrm{Add}}\right)}{\left({\varphi}_{\mathrm{H}2\mathrm{O},\mathrm{In}}-{\varphi}_{\mathrm{H}2\mathrm{O},\mathrm{Out}}\right)}\cdot \left(1-{\varphi}_{\mathrm{H}2\mathrm{O},\mathrm{In}.}\right) $$

In this paper, all volumetric gas flow rates and gas volumes are given at standard temperature and pressure (STP, 273.15 K and 101325 Pa).

A GC (Clarus 500™ from Perkin Elmer) was used to measure the gas composition at the inlet and at the outlet of the WGS reactor. A thermal conductivity detector (TCD) enabled the quantification of the CO, CO2, CH4, N2, and higher hydrocarbons in the gas stream with two different columns (7’ HayeSep N, 60/80 1/8“ SF and 9' molecular sieve 13× 45/60, 1/8” SF). The higher hydrocarbons were considered inert and, therefore, their amount was added to the N2 concentration. The GC was not able to measure the H2 concentration. Consequently, the H2 concentration was calculated by closing the overall mass balance. Furthermore, CH4 was also considered inert for the modeling process.

In addition, the measurements were validated by applying the least squares method on the elemental balances of C, H, O, and N describing the WGS reactor.

2.3 Balances and consideration of the heat losses of the water gas shift reactor

This section describes the derivation of the numerical molar and energy balances of the WGS reactor which was used for gathering the experimental data. Furthermore, the approach for the consideration of the heat losses along the reactor is shown.

For the derivation of the balances, the assumption of an ideal plug flow reactor was made.

2.3.1 Molar balance of the water gas shift reactor

Figure 4 shows the derivation of the molar balance of the WGS reactor.

Fig. 4
figure 4

Illustration for the calculation of the molar balance for the WGS reactor

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The molar balance of each reactive component i leads to

$$ {\dot{n}}_i\left(z+\varDelta z\right)-{\dot{n}}_i(z)=\pm r\left({\varphi}_i,T\right)\cdot {\rho}_s\cdot \Delta z\cdot A. $$
(6)

\( {\dot{n}}_i \)is the molar flow of component i and A is the cross section of the catalyst bed. ρ s  is the bulk density of the Fe-/Cr-based catalyst and Δz is the grid step, which was chosen with 1 mm. ±indicates whether component i is an educt or a product of the WGS reaction.

The limiting processlimΔz → 0 leads to the following differential equation:

$$ \frac{d{\dot{n}}_i}{dz}=\pm r\left({\varphi}_i,T\right)\cdot {\rho}_s\cdot A. $$
(7)

Replacing the molar flow rate \( {\dot{n}}_i \)of each component i with the overall molar flow rate \( \dot{n} \) and the volumetric fractions of each component i leads to

$$ \frac{d{\varphi}_i}{dz}=\pm r\left({\varphi}_i,T\right)\cdot {\rho}_s\cdot A\cdot \frac{1}{\dot{n}}. $$
(8)

This step is valid because of the equimolar character of the WGS reaction and the assumption of ideal gas behavior.

Using the finite difference approach (see [25]) leads to

$$ {\varphi}_i^{K+1}=\pm r\left({\varphi}_i^K,{T}^K\right)\cdot {\rho}_s\cdot A\cdot \frac{1}{\dot{n}}\cdot \Delta z+{\varphi}_i^K. $$
(9)

Equation 9 enables the calculation of the concentration profiles of CO, H2O, CO2, and H2 along the catalyst bed height of the WGS reactor.

2.3.2 Energy balance of the water gas shift reactor

Figure 5 shows a drawing for the derivation of the energy balance of the WGS reactor.

Fig. 5
figure 5

Illustration for the calculation of the energy balance for the WGS reactor

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The energy balance of the differential height element leads to

$$ \dot{h}\left(z+\varDelta z\right)-\dot{h}(z)=\pm \Delta {h}_R\left({\varphi}_i,T\right).r\left({\varphi}_i,T\right).{\rho}_s.A.\Delta z-\Delta \dot{Q}(z). $$
(10)

Here, Δh R (φ i , T) is the formation enthalpy of the WGS reaction, \( \dot{h} \) is the overall enthalpy flow, and \( \Delta \dot{Q}(z) \) is the term which describes the heat losses along each height element Δz. The limiting process limΔz → 0 and the overall molar heat capacity c P (φ i , T) of the gas stream lead to

$$ \frac{dT}{dz}=\pm \kern0.5em \Delta {h}_R\left({\varphi}_i,T\right)\cdot r\left({\varphi}_i,T\right)\cdot \frac{\rho_s\cdot A}{c_P\left({\varphi}_i,T\right)\cdot \dot{n}}-\frac{d\dot{Q}(z)}{c_P\left({\varphi}_i,T\right)\cdot \dot{n}\cdot dz}. $$
(11)

Applying the finite difference approach to Eq. 11 leads to

$$ {T}^{K+1}=\pm \kern0.5em \Delta {h}_R\left({\varphi}_i^K,{T}^K\right)\cdot r\left({\varphi}_i^K,{T}^K\right)\cdot \frac{\rho_s\cdot A}{c_P\left({\varphi}_i,T\right)\cdot \dot{n}}\cdot \Delta z-\frac{\Delta {\dot{Q}}^K}{c_P\left({\varphi}_i^K,{T}^K\right)\cdot \dot{n}}+{T}^K. $$
(12)

Equation 12 enables the calculation of the temperature profile along the catalyst bed height of the WGS reactor. It also considers the heat losses which occur along the WGS reactor.

Equations 9 and 12 form a system of equations which describes the concentration and temperature profiles along the WGS reactor. This system of equations was solved using an algorithm which was written with the numerical software Scilab™ [27].

The thermo-physical properties of the product gas components were calculated by NASA polynomials [18].

The input values for the numerical solution are the volumetric steam flow rate\( {\dot{V}}_{\mathrm{H}2\mathrm{O}} \), the volumetric dry gas flow rate\( {\dot{V}}_{\mathrm{Dry}} \), the kinetic model coefficients, the reactor geometry, the gas composition at the reactor inlet\( {\varphi}_i^0 \), the reactor inlet temperature T 0, and the ambient temperature T 0.

With the numerical calculation of the mass and energy balance of the WGS reactor, the former kinetic model (see Table 1) was improved and the resulting improved kinetic model (see Table 3) was validated with experimental data. To acquire the experimental data, the WGS reactor was operated with real product gas from the gasification plant.

2.3.3 Consideration of the heat losses along the water gas shift reactor

The heat losses along the WGS reactor were considered because they have a significant influence on the energy balance of the WGS reactor.

The heat losses \( \Delta \dot{Q}(z) \)are calculated for each Δz along the WGS reactor according to

$$ \Delta \dot{Q}(z)=\left(T(z)-{T}_0\right).{R}_Q(z). $$
(13)

Using the finite difference approach leads to

$$ \Delta {\dot{Q}}^K=\left({T}^K-{T}_0\right)\cdot {R}_Q^K. $$
(14)

Here, T(z) is the actual reactor temperature for each z, T 0 is the ambient temperature, and R Q (z) is the overall heat transfer conductivity for each z; it can be described by four different single heat transfer conductivity coefficients (ambient, insulation, reactor wall, and inner reactor) according to Eq. 15.

$$ \begin{array}{c}\hfill \frac{1}{R_{\mathrm{Q}}}=\frac{1}{\alpha_0\cdot {A}_0}+\frac{\delta_I}{\lambda_{\mathrm{I}}\cdot {A}_{\mathrm{I}}}+\frac{\delta_S}{\lambda_{\mathrm{S}}\cdot {A}_{\mathrm{S}}}+\frac{1}{\alpha_{\mathrm{R}}\cdot {A}_{\mathrm{R}}}\hfill \\ {}\hfill =\frac{1}{\pi \cdot \Delta z}\cdot \left(\frac{1}{\alpha_0\cdot {d}_0}+\frac{\delta_{\mathrm{I}}}{\lambda_I\cdot \frac{d_{\mathrm{I}}-{d}_{\mathrm{S}}}{\mathit{\ln}\left(\frac{d_{\mathrm{I}}}{d_{\mathrm{S}}}\right)}}+\frac{\delta_{\mathrm{S}}}{\lambda_S\cdot \frac{d_{\mathrm{S}}-{d}_{\mathrm{R}}}{\mathit{\ln}\left(\frac{d_{\mathrm{S}}}{d_{\mathrm{R}}}\right)}}+\frac{1}{\alpha_{\mathrm{R}}\cdot {d}_{\mathrm{R}}}\right)\hfill \end{array} $$
(15)

These four terms describe the heat transfer conductivity of the shell of the reactor (compare Fig. 6). The heat losses of the bottom surface and the top surface of the reactor are neglected because the surfaces are significantly smaller than the shell surface.

Fig. 6
figure 6

Drawing for the consideration of the heat losses along the reactor shell

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In the following, the derivation of the four heat transfer conductivity terms (ambient, insulation, reactor wall, and inner reactor) is shown.

Ambient

This term consists of the heat transfer coefficients α 0 and the outer surface of the reactor A 0.

α 0 can be calculated with the heat transfer coefficient caused by the radiation emitted by the insulation surface (α Rad, thin Al layer) and the heat transfer coefficient caused by natural convection (α Conv):

$$ {\alpha}_0={\alpha}_{\mathrm{Rad}}+{\alpha}_{\mathrm{Conv}} $$
(16)

According to [1], α Rad can be calculated with Eq. 17.

$$ \begin{array}{c}\hfill {\alpha}_{\mathrm{Rad}}={\epsilon}_0\cdot \sigma \cdot \frac{T_{\mathrm{S}}^4-{T}_0^4}{T_{\mathrm{S}}-{T}_0}=\hfill \\ {}\hfill =0.09\cdot 5.67\cdot {10}^8\frac{W}{{\mathrm{m}}^2\cdot {K}^4}\cdot \frac{{\left(\left(50+273.15\right)K\right)}^4-{\left(\left(25+273.15\right)K\right)}^4}{\left(50+273.15\right)K-\left(25+273.15\right)K}\hfill \\ {}\hfill =0.61\frac{W}{{\mathrm{m}}^2\cdot K}\hfill \end{array} $$
(17)

The convective heat transfer coefficient α Conv can be calculated by the Nusselt number Nu with the correlation in Eq. 18 (see [5]).

$$ \begin{array}{c}\hfill Nu=\frac{\alpha_{\mathrm{Conv}}\cdot {d}_0}{\lambda}\hfill \\ {}\hfill ={\left\{0.6+\frac{0.387\cdot {\left(Gr\cdot \mathit{\Pr}\right)}^{\frac{1}{6}}}{{\left[1+{\left(0.559\cdot P{r}^{-1}\right)}^{\frac{9}{16}}\right]}^{\frac{8}{27}}}\right\}}^2\hfill \end{array} $$
(18)

In this equation, d 0 is the outer diameter of the reactor and λ is the heat conductivity of air.

The Grashof number Gr with the gravitational constant g, the thermal expansion coefficient of air β, the outer reactor diameter d 0, and the kinematic viscosity of air ν

$$ \begin{array}{c}\hfill Gr=\frac{g\cdot \beta \cdot \left({T}_{\mathrm{S}}-{T}_0\right)\cdot {d}_0^3}{\nu^2}\hfill \\ {}\hfill =\frac{9.81\frac{m}{s^2}\cdot \frac{1}{\left(25+273.15\right)}\ \frac{1}{K}\cdot \left(25\ K\right)\cdot {0.296}^3\ {\mathrm{m}}^3}{{\left(1.6\cdot {10}^{-5}\right)}^2\ {\left(\frac{{\mathrm{m}}^2}{s}\right)}^2}\hfill \\ {}\hfill =8.33\cdot {10}^7\hfill \end{array}= $$
(19)

and the Prandtl number of air

Pr = 0.7

lead to

Nu = 47.89

and, consequently, to

$$ {\alpha}_{\mathrm{Conv}}=\frac{Nu\cdot \lambda }{d_0}=\frac{47.89\cdot 2.6\cdot {10}^{-2}\ \frac{W}{m\cdot K}}{0.296\ \mathrm{m}}=4.21\ \frac{W}{{\mathrm{m}}^2\cdot K}. $$
(20)

With Eqs. 17 and 20, the overall heat transfer coefficient for the ambient can be calculated with

$$ \begin{array}{c}\hfill {\alpha}_0={\alpha}_{\mathrm{Rad}}+{\alpha}_{\mathrm{Conv}}=\hfill \\ {}\hfill =0.61\ \frac{W}{{\mathrm{m}}^2\cdot K}+4.21\ \frac{W}{{\mathrm{m}}^2\cdot K}=4.82\ \frac{W}{{\mathrm{m}}^2\cdot K}\hfill \end{array} $$
(21)

It can be seen that the heat transfer coefficient which is caused by the natural convection is about seven times higher than the heat transfer coefficient of the radiation at this low ambient temperature.

Furthermore, the outer surface of the reactor can be calculated with

$$ {A}_0={d}_0\cdot \pi \cdot \Delta z $$
(22)

Insulation

This term consists of the thickness of the insulation δ I , the heat conductivity of the insulation,  λ I  and the mean logarithmic surface of the insulation A I. In general, the mean logarithmic surface (A m) can be calculated using the outer (A o) and inner (A i) surface of a cylindrical object according to Eq. 23.

$$ {A}_{\mathrm{m}}=\frac{A_{\mathrm{o}}-{A}_{\mathrm{i}}}{\mathit{\ln}\frac{A_{\mathrm{o}}}{A_{\mathrm{i}}}} $$
(23)

Reactor wall

This term consists of the thickness of the reactor wall δ S, the heat conductivity of steel  λ S, and the mean logarithmic surface of the inner reactor A R and the outer surface of the reactor wall A S.

Inner reactor

The inner reactor is considered as an ideal pipe reactor with the inner surface area A R. Therefore, [1] (page 418) gives a Nusselt number of Nu = 3.657. With Nu, the inner reactor diameter d R, and the estimated heat conductivity of the product gas  λ R, the heat transfer coefficient  α R can be calculated.

Hence, the overall heat transfer conductivity R Q for each Δz is 2.46 ⋅ 10−4 W ⋅ K −1 with the chosen parameters. With this information, the heat losses along the reactor can be calculated according to Eqs. 13 and 14, respectively.

3 Result and discussion

First, this section presents the determination of the parameters for the improved kinetic model. Second, the validation results of the improved kinetic model are discussed.

Table 2 gives an overview of the four different operating points of the WGS reactor (OP1 to OP4). OP1 was used to improve the former kinetic model, and OP2 to OP4 were used for the validation of the improved kinetic model with experimental data from the WGS reactor.

Table 2 Overview of the operating points for the improvement of the kinetic model and its subsequent validation
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The operating conditions of the WGS reactor were described by the gas hourly space velocity (GHSV) and the steam to dry gas ratio (STDGR) in Eqs. 24 and 25.

$$ \mathrm{GHSV}=\frac{{\dot{V}}_{\mathrm{Dry}}}{V_{\mathrm{Cat}}} $$
(24)
$$ \mathrm{STDGR}=\frac{{\dot{V}}_{\mathrm{H}2\mathrm{O}}}{{\dot{V}}_{\mathrm{Dry}}} $$
(25)

OP1, OP2, and OP3 processed product gas which was extracted before the RME gas scrubber of the gasification plant and OP4 processed gas which was extracted after the RME gas scrubber.

3.1 The improved kinetic model

In order to improve the former kinetic model, a new parameter for the activation energy E a was sought, one that should consider the fact that the catalyst was used in its original pellet size during this experimental approach, which was in contrast to the determination of the former kinetic model, where the catalyst was milled and, therefore showed a higher specific surface. E a was chosen as the parameter to improve because it significantly affects the temperature profile (compare Eq. 2). To do so, a variation of E a with subsequent error calculation regarding the temperature profile was done.

For each E a, the temperature profile along the WGS reactor (thermocouples 1 to 5) was calculated and compared with the measured values of OP1. Subsequently, the overall error for each E a was calculated according to Eq. 26.

$$ \mathrm{Err}={\displaystyle \sum_{i=1}^5}\sqrt{{\left({T}_{c,i}-{T}_{m,i}\right)}^2} $$
(26)

The result can be seen in Fig. 7.

Fig. 7
figure 7

Variation of E a with corresponding error for OP1

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The new value for E a , j was chosen to be that for which the error Err was a minimum.

The new value of E a and, consequently, the other parameters for Eq. 2 which were not changed can be seen in Table 3.

Table 3 Parameters of the improved kinetic model
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Compared to the models in [2, 7, 13, 15] and [22], the activation energy E a is higher. In these approaches, the activation energy values varied between 95 and 118 kJ mol−1. However, Chinchen et al. [4] give a value of 129.4 kJ mol−1 for the activation energy for WGS catalysts used at industrial scale and at pressures of up to 3.0 MPa, which is in good agreement with the value found in this work.

In addition, the presented value of factor a shows that the influence of the CO partial pressure is also higher (1.77 in this work versus about 1.0 in [2, 13], and [22]). This could be explained by the low operating pressure in this work and, therefore, an even lower adsorption of CO on the catalyst surface.

In contrast, the reaction exponents b, c, and d are in the same order of magnitude, which also indicates that the supply of H2O, CO2, and H2 to the catalyst surface is not limiting the reaction.

3.2 Validation of the improved kinetic model with experimental data

The improved kinetic model was validated with temperature measurements and concentration measurements of all four operating points where real product gas from the gasification plant was processed in the WGS reactor.

Figure 8 shows the validation of the kinetic model with the measured temperature profiles along the WGS reactor from all four operating points (OP1 to OP4). The temperatures at T1 were the boundary conditions; therefore, the measurements and the calculations have the same value.

Fig. 8
figure 8

Temperature profiles along the water gas shift reactor for all four operating points

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It can be seen that the calculated and measured temperature profiles have nearly no difference at OP1 because the improved model is based on this operating point.

The temperature profiles of OP2 also have a higher difference. However, the calculated profile was slightly lower than the measured profile. Consequently, the chosen activation energy of E a = 126.6 kJ mol−1 seems to be too high for OP2.

For OP3, the temperature profiles show good agreement for the first three measurement points. However, it seems that the calculated heat losses in the improved model are too low compared to the measured values, which explains the higher calculated temperature profile at measurement points 4 and 5. Consequently, the highest error occurred at the last measurement point.

OP4 shows the highest gap regarding the calculated and the measured temperature profile which could be explained by the fact, that this operating point processed product gas with a lower STDGR of 1.2 compared to the other operating points. For this operating point, the chosen value of E a was too low, which indicates that a lower STDGR ratio has a negative effect on the kinetics. This is in agreement with Hla et al. [12], where the authors observed that higher steam content increased the reaction rate of the WGS reaction. However, the measurement points 3 to 5 indicate good agreement between the calculated and measured heat losses.

Table 4 shows the calculated and measured values of the gas compositions at the inlet and outlet of the WGS reactor. The measured inlet concentrations were the boundary conditions for the calculation.

Table 4 Concentrations at the inlet and outlet of the WGS reactor for all operating points. The measured reactor inlet concentrations as well as the measured and calculated reactor outlet concentrations are shown. The H2 concentration of the measurements was calculated by closing the mass balance as H2 could not be detected by the GC. The measurements were single sample measurements; therefore, no standard deviation can be given
Full size table

Overall, it can be seen that the error of the CO concentration is the highest of all measured and calculated components. However, the absolute error is quite low, which indicates a good agreement of the measured and calculated values.

Taking a look at OP1, it can be seen that the measured and calculated values show good agreement regarding the concentrations of H2, CO2, CH4, and N2. The relative error of CO is higher than the relative errors of the other gas components. However, the absolute error is about 1 %.

Looking at OP2, it can be seen that the absolute and relative errors of the CO concentration are slightly higher than for OP1. In addition, OP2 has a higher CO2 concentration than the CO concentration at the inlet of the WGS reactor which can be attributed to a partial load operation of the gasification plant during this measurement. At partial load operation, a higher steam to fuel ratio leads to higher CO2 concentrations and, consequently, to lower CO concentrations (compare [14]).

OP3 shows similar results as for OP1. However, the overall errors are higher; especially, the CO concentration shows a higher relative error and a higher absolute error compared to OP1.

OP4 shows, again, good agreement of the measured and calculated concentrations at the outlet of the WGS reactor. It shows again the highest relative error for the CO concentrations. The absolute error of 0.9 % is also in the same order of magnitude as for the other operating points.

3.3 Comparison of the improved with the former model

In this section, the improved and the former kinetic model are compared by using the experimental data from OP1. Figure 9 shows the comparison of the temperature profile between the measured data, the improved kinetic model, and the former kinetic model.

Fig. 9
figure 9

Comparison of the temperature profile along the WGS reactor between the measured data, the improved kinetic model, and the former kinetic model using data from OP1

Full size image

It can be seen that the temperature profile, which was calculated with the former kinetic model, shows significant deviation compared with the measured data and, therefore, also with the improved kinetic model.

Table 5 shows the comparison of the volumetric fractions of the gas components between the measured data, the improved kinetic model, and the former model.

Table 5 Comparison of the volumetric fractions of the gas components between the measured data, the improved kinetic model, and the former model using data from OP1
Full size table

It can be seen that the former kinetic model shows a higher H2 and CO2 concentration and, consequently, a lower CO content. This could be explained by the higher reaction rate of the former kinetic model. In addition, the volumetric fractions of the former kinetic model are very close to the equilibrium composition at the given parameters.

In contrast, the CH4 and N2 concentrations of the former and improved are at the same level, which indicates, that CH4 and N2 did not take part in a reaction and the error between the measured and calculated values was caused by a measurement error. The small deviation can be explained by the slightly higher volumetric dry gas flow rate if the former kinetic model is employed for the calculation.

4 Conclusion and outlook

In this paper, a former kinetic model for the water gas shift reaction was improved with experimental data from a water gas shift reactor which processed real product gas from dual fluidized bed biomass steam gasification and which employed a commercial Fe-/Cr-based catalyst. Both kinetic models, the former one and the improved one, considered a H2S amount of about 100 cm3 m−3 which is usually contained in the product gas generated from dual fluidized bed biomass steam gasification.

\( \begin{array}{c}\hfill r\left({\varphi}_i,T\right)=117.8\frac{\mathrm{mol}}{g\kern0.5em P{a}^{1.71}s}\cdot \mathit{\exp}\left(\frac{-126.6\frac{\mathrm{kJ}}{\mathrm{mol}}}{R\cdot T}\right)\cdot {p}_{CO}^{1.77}\cdot {p}_{\mathrm{H}2\mathrm{O}}^{0.23}\cdot {p}_{CO2}^{-0.17}\cdot {p}_{\mathrm{H}2}^{-0.12}\hfill \\ {}\hfill \cdot \left(1-\frac{K_{MAL}}{K_g}\right)\hfill \end{array} \)

Furthermore, this improved kinetic model was validated with the experimental data from four different operating points of the water gas shift reactor.

In order to enhance the accuracy of the validation, the heat losses of the water gas shift reactor, which play a significant part in the energy balance, were also considered.

For the improvement of the former kinetic model and the validation of the improved kinetic model, the gas composition at the inlet and the outlet of the water gas shift reactor was measured. In addition, the temperature profile along the catalyst bed of the reactor was recorded.

Overall, the validation showed good agreement of the measured and calculated values for the gas compositions and the temperature profiles of the water gas shift reactor. Of all considered gas components, the CO concentration showed the highest error. However, the highest absolute error was about 1.5 % (relative error of 30.0 %). The highest absolute error of the temperature profile was 21 °C (relative error of 4.8 %). In addition, a low steam to dry gas ratio at the reactor inlet (from 1.2 and below), reduced the accuracy of the model.

Hence, these results qualify the presented improved kinetic model for basic design and engineering of a water gas shift reactor which employs a commercial Fe-/Cr-based catalyst and which processes product gas derived from dual fluidized biomass steam gasification if the steam to dry gas ratio at the reactor inlet is set to about 1.5.

Furthermore, future work should focus on an additional kinetic model which considers a product gas which is derived from the gasification of alternative fuels like waste or, for example, plastic residues.

Abs, absolute; BDL, below detection limit; Cat, catalyst; CHP, combined heat and power; d.b., dry basis; DFB, dual fluidized bed; DL, detection limit; FR, flow record; GC, gas chromatograph; OP, operating point; ORC, organic Rankine cycle; Rel, relative; RME, rapeseed methyl ester; STP, standard temperature and pressure (273.15 K and 101325 Pa); TX, thermocouple X along the WGS reactor; TCD, thermal conductivity detector; WGS, water gas shift.

Greek Symbols

α 0 Heat transfer coefficient of the outer reactor wall to the ambient in W m−2 K−1

α Conv Heat transfer coefficient of the outer reactor wall caused by natural convection in W m−2 K−1

α R Heat transfer coefficient of the inner reactor wall in W m−2 K−1

α Rad Heat transfer coefficient of the outer reactor wall caused by radiation in W m−2 K−1

β Thermal expansion coefficient of air in K−1

φ i Volumetric fraction of component i in −

δ I Insulation thickness in m

δ S Reactor wall thickness in m

∈0 Emission coefficient of the thin aluminum layer in −

ΔG(T) Gibbs enthalpy as function of temperature in kJ mol−1

Δh R Enthalpy of formation for a certain temperature in kJ mol−1

ΔH Enthalpy of formation at 298.15 K and 101,325 Pa in kJ mol−1

\( \Delta \dot{Q}(z) \) Heat losses along the reactor in W

λ Heat conductivity of air in W m−1 K−1

λ I Heat conductivity of the insulation in W m−1 K−1

λ S Heat conductivity of the reactor wall in W m−1 K−1

v Kinematic viscosity of air in m2 s−1

ρ S Catalyst bulk density

σ Stefan-Boltzmann constant in W m−2 K−4

Δz Differential height element of the reactor in m

Latin Symbols

a Reaction exponent for CO in -

A Cross section of the reactor in m2

A 0 Outer reactor surface in m2

A i General inner cylindrical surface in m2

A I Mean logarithmic surface of the insulation in m2

A m General mean logarithmic surface in m2

A o General outer cylindrical surface in m2

A R Inner reactor surface in m2

A S Mean logarithmic surface of the reactor in m2

b Reaction exponent for H2O in −

c Reaction exponent for CO2 in −

d Reaction exponent for H2 in −

d 0 Outer reactor diameter in m

d I Inner reactor diameter including reactor wall and insulation in m

d R Inner reactor diameter in m

d S Inner reactor diameter including reactor wall in m

E a Activation energy in kJ mol−1

Err Error in K

g Gravitational constant in m2 s−1

GHSV Gas hourly space velocity in h−1

Gr Grashof number in −

k 0 Rate constant in mol g−1 Pa−(a + b + c + d) s−1

K Control variable for the finite difference approach in −

K g Equilibrium constant calculated by thermo-physical properties in −

K MAL Equilibrium constant calculated by the mass action law in −

\( {\dot{n}}_i \) Molar flow rate of component i in mol s−1

Nu Nusselt number in −

p Absolute pressure in Pa

p i Partial pressure of component i in Pa

Pr Prandtl number in −

r Reaction rate in mol g−1 s−1

R General gas constant in J mol−1 K−1

R Q Overall heat transfer conductivity in W K−1

STDGR Steam to dry gas ratio in −

T Temperature in K

T 0 Ambient temperature in K

T S Surface temperature of the thin Al layer of the insulation in K

T c , i Calculated temperature at thermocouple i in K

T m , i Measured temperature at thermocouple i in K

V Cat Catalyst bulk volume in m3

\( {\dot{V}}_{\mathrm{Dry}} \) Volumetric dry gas flow rate at STP in m3 h−1

\( {\dot{V}}_{\mathrm{H}2\mathrm{O}} \) Volumetric steam flow rate at STP in m3 h−1

X CO CO conversion in −