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Ammonia synthesis reactors operate in conditions of high pressure and high temperature. Consequently, the flow inside these reactors always presents interaction between components in the feed mixture. A modeling accounts these interactions with pressure, temperature and the molar fraction is essential to converter simulation more realistic. The compositional approach based on cubic equations of state provides the influences of the component of a gas mixture using mixing rules and binary interaction parameters. This multicomponent description makes the model more robust and reliable for properties mixture prediction. In this work, two models of ammonia synthesis reactors were simulated: adiabatic and autothermal. The fitted expression of Singh and Saraf was used. The adiabatic reactor model presented a maximum relative error of 1.6% in temperature and 11.4% in conversion while the autothermal reactor model presents a maximum error of 2.7% in temperature, when compared to plant data. Furthermore, a sensitivity analysis in input variables of both converter models was performed to predict operational limits and performance of the Models for Ammonia Reactor Simulation (MARS).

In ammonia synthesis, fixed bed reactors are always used. Only one reaction occurs in the gas phase using iron catalyst particles [

N 2 + 3H 2 ↔ 2NH 3 ,Δ H r o = − 45 .94 kJ / mol of NH 3 (1)

The reaction is exothermic in pressures from 150 to 300 atm, due to the decrease in mole numbers. Moreover, the temperature range is 600 to 800 K. After all, even with an exothermic reaction, the reaction rate in converters must be high. The High Pressure and High Temperature (HPHT) conditions in gas phase make properties in reactive fluid temperature vary too much along the converter. Some examples are density, fugacity, chemical activity and heat capacity.

The mathematical modeling of ammonia converters uses experimental rate expressions. They generally contain partial pressures [

After a few years, many researchers continued to use Temkin and Pyzhev rate in their calculations [

Dyson and Simon [

However, in previous publications, the fugacity coefficients were derived from a correlation that they varied with temperature and pressure, but not composition [

Therefore, a more sophisticated model can be used to provide a multicomponent description. This method is the principle of the compositional approach: the reactant fluid present changes of properties influenced not only by pressure and temperature but also by composition and intermolecular forces. The last and more important part is computed due to mixing rules in cubic EoS calculations. Furthermore, two facts reinforce the use of a compositional approach in ammonia synthesis: 1) the molecules in the gas phase are smaller and 2) the high-pressure deviates gas from ideal treatment, approximating gas molecules.

As most of substances are non-polar (except for NH_{3}) and gases are at HPHT conditions, the Peng-Robinson (PR) and Soave-Redlich-Kwong (SRK) cubic EoS are chosen to model the system. Therefore, the EoS approach will replace the previous activities used in reaction rate. As discussed above, these models present a more reliable theory and a molecular formulation.

The main objectives of this work are: validate of two ammonia reactor models with plant data (adiabatic and autothermal); use a fitted reaction rate using a compositional approach; solve mass, energy and momentum balances in PFR model in steady-state and present a study of parametric sensitivity in input variables of ammonia converters.

The SRK [

p = R T v − b − a ( v + δ 1 b ) ( v + δ 2 b ) (2)

For the results of this work, only PR-EoS was used, with the parameters δ 1 = 1 + 2 0.5 and δ 2 = 1 + 2 0.5 . Defining the following Equations (3) to (5) below:

A = a p ( R T ) 2 (3)

b = b p R T (4)

Z = p v R T (5)

and substituting into Equation (2), as described at Barbosa Neto [

Z 3 + [ ( δ 1 + δ 2 − 1 ) B − 1 ] Z 2 + [ A + δ 1 δ 2 B 2 − ( δ 1 + δ 2 ) B ( B + 1 ) ] Z − [ A B + δ 1 δ 2 B 2 ( B + 1 ) ] = 0 (6)

The van der Waals one-fluid mixing rules are used for the energy A, and for the volume B, parameters of the EoS, are expressed by Equations (7) and (8), respectively.

A = ∑ i = 1 n c ∑ j = 1 n c y i y j A i j (7)

B = ∑ i = 1 n c y i B i (8)

The parameter A_{ij}, in the Equation (7), is calculated from Equation (9).

A i j = A i A j ( 1 − k i j ) (9)

The terms A_{i} and B_{i} are defined by Equations (10) and (11), respectively. The factor m(ω_{i}) is given in the literature [_{a} and Ω_{b} changes with EOS.

A i = Ω a ( p r i T r i 2 ) [ 1 + m ( ω i ) ( 1 − T r i ) ] 2 (10)

B i = Ω b ( p r i T r i ) (11)

The expression shown at Equation (12) for fugacity coefficients is obtained.

ln ( φ i ) = ( Z − 1 ) ( B i B ) − ln ( Z − B ) − ( A Δ B ) ( 2 ψ i A − B i B ) ln [ Z + δ 1 B Z + δ 2 B ] (12)

With Δ = δ 1 − δ 2 and

ψ i = ∑ i = 1 n c A i j y j (13)

The fugacity and activity are also important for reaction rate used. They are given in Equations (14) and (15).

f i = φ i y i P (14)

a i = φ i y i P P r e f = f i P r e f (15)

Other important properties are the heat capacities, which are given in Equations (16) and (17). More details are found in Tosun [

C p = C p i g + C p r e s (16)

C v = C v i g + C v r e s (17)

The rate used in ammonia reactors of this research was made by Singh and Saraf [

r NH 3 = 4.11 × 10 10 exp ( − 163422 R T ) [ K e q 2 a N 2 ( a H 2 3 a NH 3 2 ) α − ( a NH 3 2 a H 2 3 ) 1 − α ] (18)

The rate given in Equation (18) is pseudo-homogeneous. Nielsen and collaborators provided a suitable range for ammonia rate expressions: 640 to 770 K and 150 to 310 atm [

η = b 0 + b 1 T + b 2 x N 2 + b 3 T 2 + b 4 x N 2 + b 5 T 3 + b 6 x N 2 3 (19)

To summarize this section, the equilibrium constant (K_{eq}) in Equation (19) is found in correlation given by Gillespie and Beattie [

P (atm) | b_{0} | b_{1} | b_{2} | b_{3} | b_{4} | b_{5} | b_{6} |
---|---|---|---|---|---|---|---|

150 | −17.539 | 0.0769 | 6.901 | −1.083 × 10^{−4} | −26.425 | 4.928 × 10^{−8} | 38.937 |

225 | −8.213 | 0.0377 | 6.190 | −5.355 × 10^{−5} | −20.869 | 2.379 × 10^{−8} | 27.880 |

300 | −4.676 | 0.0235 | 4.687 | −3.463 × 10^{−5} | −11.280 | 1.541 × 10^{−8} | 10.460 |

In the adiabatic operation of ammonia reactors, the volume of control increases its temperature using only the heat of reaction. However, this operation is not possible in one converter. The reactor volume must be separated into several reactors, as given in

The mass balance in one reactor is expressed only regarding nitrogen conversion (Equations (20) and (21)) because only one reaction takes place in converter [

d x N 2 d L = A r NH 3 η 2 F N 2 0 (20)

d x N 2 d V = r NH 3 η 2 F N 2 0 (21)

Equations (22) and (23) express the energy balances. As the reactor is at high pressure, the estimation of C_{p} is essential. Moreover, the heat of reaction is computed according to correlation [

d T d L = A r NH 3 η ( − Δ H r ) m ˙ C p m i x (22)

d T d V = r NH 3 η ( − Δ H r ) m ˙ C p m i x (23)

Besides, as reactor contains catalyst particles, there is a pressure loss along the converter. It is estimated using the Ergun Equation [

d P d L = 150 ( 1 − ε ) 2 μ u ε 3 d p 2 − 1.75 ( 1 − ε ) ρ u 2 ε 3 d p (24)

The autothermal reactor uses the energy released by a reaction to heat the reactant gas. It operates with countercurrent flow, and it is divided into several tubes, as given in

The mass balance and pressure loss in the autothermal converter do not change compared to the adiabatic model. The difference is in the temperature: now we have the reactant gas (that increases its temperature by the reaction and is cooling by cooling gas) and the cooling gas (that always increases its temperature). The energy balances are given in Equations (25) and (26) (for length variations) and Equations (27) and (28) (for volume variations). The minus sign in Equations (26) and (28) is related to countercurrent operation.

d T d L = A r NH 3 η ( − Δ H r ) m ˙ C p m i x − U A ′ ( T − T g ) m ˙ C p m i x (25)

d T g d L = − U A ′ ( T − T g ) m ˙ C p m i x (26)

d T d V = r NH 3 η ( − Δ H r ) m ˙ C p m i x − U a ′ ( T − T g ) m ˙ C p m i x (27)

d T g d V = − U a ′ ( T − T g ) m ˙ C p m i x (28)

The adiabatic and autothermal reactors must be solved using a numerical method, once an analytical solution is complex. The method used was the Runge-Kutta-Fehlberg(RKF) using an error control strategy, described by Chapra and Canale [

Another challenge is the interdependence of variables in differential equations. Therefore, the code was made using a modular structure in Wolfram Mathematica® Programming Language. Moreover, our algorithm is called MARS (Models for Ammonia Reactor Simulation).

The variables of interest in both models are the outlet temperature, pressure, and composition. The thermodynamic module computes the thermodynamic and transport properties; the kinetic module calculates the reaction rate and another module joins all the previous in a numerical method to solve the balances. As the problem is solved in one dimension, the stopping-criteria is the end of the reactor, as expressed in

As discussed before, the chemical activities originally are computed using a correlation―Singh and Saraf Rate (Equation (18)). However, when using the multicomponent approach, it is expected that chemical activity decreases, due to compositional interactions. Therefore, the reaction rate computed also decreases its value. So, the kinetic factor α is fitted in MARS. The fit is made according to an adiabatic reactor in the literature [

_{2} conversion variation in an adiabatic converter with α alterations.

Parameter | Value | Parameter | Value | Parameter | Value | Exp. Data | Value |
---|---|---|---|---|---|---|---|

y N 2 | 0.2219 | y NH 3 | 0.0546 | T_{in} (K) | 658.15 | T_{out1} (K) | 780.15 |

y H 2 | 0.6703 | y_{Ar} | 0.0256 | P_{in} (atm) | 226 | x N 2 out1 (%) | 15.78 |

y NH 3 | 0.0276 | m ˙ (kg/s) | 29.821 | V_{1} (m³) | 4.75 |

The outlet conversion increases when α rises because the reaction rate is higher. Moreover, at α = 0.570, the model does well in predicting the outlet conversion. Therefore, the original value of α = 0.550 in Singh and Saraf rate [

For validations in adiabatic and autothermal models, the relative error was calculated as expressed in Equation (29). In the case of multiple reactors, the maximum relative error will be taken.

rel error = 100 | plant data − simulation data | plant data (29)

Even with a suitable numerical method and a robust code, validations of reactors models are necessary. The RKF method and α = 0.570 are selected for Singh and Saraf modified rate. Furthermore, calculates an adiabatic reactor containing three fixed beds in series and an autothermal converter. Both models are reliable compared to plant data.

In adiabatic case, we have three reactors in series. The first bed is computed in a variation of α. Therefore, all inlets parameters remain the same. The only additional information for simulation is the inlet temperature of the second and third reactors and their respective volume, as given in

Bed | V (m³) | T_{in} (K) | T_{out} (K) | Outlet x N 2 (%) |
---|---|---|---|---|

1 | 4.75 | 658.15 | 780.15 | 15.78 |

2 | 7.2 | 706.15 | 775.15 | 25.55 |

3 | 7.8 | 688.15 | 728.15 | 30.91 |

place where the rate has its highest values. Therefore, it can provide more errors compared to the others. However, in the second and third converters, the simulated temperature gives good results compared to plant data. In all simulations of the adiabatic arrangement, 74 iterations are required with RKF. It is a good result compared if the 4^{th} Order Runge-Kutta with fixed step size was used (with 50 iterations at each reactor, for example).

In addition,

Even with good results in temperature predictions, the conversion is another crucial variable. The composition of the reactor depends on conversion, after all. Furthermore, it is more sensitive to variations than temperature.

In ^{rd} reactor, the MARS model is reliable for adiabatic reactor

Bed | T_{out} _{Plant} (K) | T_{out} _{MARS} (K) | Rel Error (%) |
---|---|---|---|

1 | 780.15 | 768.04 | 1.55 |

2 | 775.15 | 773.14 | 0.26 |

3 | 728.15 | 730.46 | 0.32 |

Bed | x N 2 out Plant (%) | x N 2 out MARS (%) | Rel Error (%) |
---|---|---|---|

1 | 15.78 | 15.64 | 0.88 |

2 | 25.55 | 26.64 | 4.27 |

3 | 30.91 | 34.42 | 11.36 |

simulation, because the temperature is usually the control variable in ammonia reactors. Errors in literature reached less than 0.5% [

The autothermal converter has the reactor parameters given in

In

However, the maximum temperature point is well predicted, which reinforces the method’s effectiveness. The maximum relative error was 2.7% in the end of reactor, due to high nonlinearity of equations.

Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|

y N 2 | 0.2190 | y_{Ar} | 0.0360 | V (m^{3}) | 4.07 |

y H 2 | 0.6500 | m ˙ (kg/s) | 6.038 | a' (m^{2}/m^{3}) | 10.29 |

y NH 3 | 0.0520 | T_{in} (K) | 694.15 | U (W/m^{2}∙K) | 465.2 |

y CH 4 | 0.0430 | P_{in} (atm) | 279 |

V (m^{3}) | T_{plant} (K) | Rel. Error (%) | V (m^{3}) | T_{plant} (K) | Rel. Error (%) |
---|---|---|---|---|---|

0 | 694.15 | 0.00 | 2.21 | 781.15 | 0.01 |

0.17 | 716.15 | 1.47 | 2.54 | 771.15 | 0.17 |

0.51 | 759.15 | 1.78 | 2.88 | 756.15 | 1.27 |

0.85 | 789.15 | 1.05 | 3.22 | 748.15 | 1.28 |

1.19 | 799.15 | 0.07 | 3.56 | 733.15 | 1.68 |

1.53 | 796.15 | 0.46 | 3.90 | 719.15 | 1.84 |

1.87 | 787.15 | 0.56 | 4.07 | 711.15 | 2.66 |

proves the reliability of the MARS model for the autothermal reactor. Other authors reached 3.2% of maximum relative difference in temperature, which was an acceptable value [

In the sensitivity analysis the same data set of validation was used to fit α. In the adiabatic model, the analysis is realized in three reactors in series. The inlet temperature was varied. Then, the effects on temperature, conversion and effectiveness factor profiles were detected in each case. While, in the autothermal model, the heat exchange coefficient was varied, with the same detections on output of reactor. The variations were computed in length L.

In adiabatic operation, only the inlet temperature of the 1^{st} reactor is varied. The temperatures T_{in}_{2} and T_{in}_{3} maintain the same values (as given in

The inlet temperature effect along converter temperature is given in _{in} = 643.15 K) gave the highest profiles along second and third reactors. After all, smaller temperatures provide smaller rates, which give minor conversions. Therefore, after first reactor, more N_{2} can be converted, giving a more accentuated profile (T_{in} = 643.15 K).

On the other hand, high inlet temperatures present high reaction rates, converting more N_{2} in the first bed. However, it becomes more difficult to react in the second and third converters. Therefore, the temperature rise is not so significant as in smaller T_{in} values, even with high rates. In the three cases, the number of iterations during the converging process is similar.

Constraints for adiabatic simulation T_{in}_{1} | |||
---|---|---|---|

P_{in} (atm)―1^{st} bed | 150 | y N 2 | 0.22 |

T_{in}_{2} (K)―2^{nd} bed | 693.15 | y H 2 | 0.66 |

T_{in}_{3} (K)―3^{rd} bed | 683.15 | y NH 3 | 0.03 |

D_{r} (m) | 1.7 | y CH 4 | 0.045 |

d_{part} (m) | 0.006 | y_{Ar} | 0.045 |

ε_{bed} (-) | 0.4 | m ˙ (kg/s) | 30.0 |

Another essential measure inside our reactor is the composition. As there is only one reaction, the conversion profile gives the other molar fractions indirectly. For the inlet temperature variation case,

In _{2} conversions. However, a high Tin does not guarantee to elevate conversions, because the reaction is exothermic. After all, the reaction is reversible, and high temperatures provide low equilibrium conversion. Moreover, the highest rise in conversions occurs in the first bed. It can be seen that the growth of conversion between interval of 683.15 K and 663.15 K is smaller than 663.15 K and 643.15 K. Therefore, there is a limit to the T_{in} value.

The parameters for simulation are given in

^{2}∙K is 753 K, while for U = 650 W/m^{2}∙K, is 739 K. Moreover, another important value is the final value of temperature for the reactant gas at the end of the reactor. For U = 450 W/m^{2}∙K, the final T value is 724 K and U = 650 W/m^{2}∙K is 687 K. It takes place because a large U removes more energy in the reactor, decreasing temperature more rapidly.

Constraints for countercurrent simulation | |||
---|---|---|---|

y N 2 | 0.22 | nt | 250 |

y H 2 | 0.66 | D_{r} (m) | 1.5 |

y NH 3 | 0.03 | L_{r} (m) | 2.5 |

y CH 4 | 0.045 | a' (m^{2}/m) | 11.78 |

y_{Ar} | 0.045 | m ˙ (kg/s) | 6.0 |

P_{in} (atm) | 225 | T_{in} (K) | 613.15 |

The effectiveness factor η profiles are given in ^{2}∙K, there are errors computing η (higher than 1).

The compositional modeling using PR cubic EoS was essential to calculate the properties in reactive streams at HPHT conditions during the ammonia synthesis reactors simulation. The adiabatic reactor model presented a maximum relative error of 1.6% in temperature and 11.4% in conversion when compared to plant data. In sensitivity study, a value of T_{in} = 683.15 K gave the highest conversion of 25.16%. The autothermal reactor model presented a maximum error of 2.7% in temperature when compared to experimental points. In parametrical sensitivity, the highest conversion of 37.22% was provided by a value of U = 850 W/m^{2}∙K. Therefore, both models proved reliable in simulating ammonia reactors for the set of data used.

Another improvement of the ammonia synthesis reactor models can be achieved using intraparticle diffusional approach. It computes the effectiveness factor without using an experimental correlation.

The authors acknowledge CNPq (Process Number 131744/2016-0) and Unicamp for providing financial support.

Jorqueira, D.S.S., Neto, A.M.B. and Rodrigues, M.T.M. (2018) Modeling and Numerical Simulation of Ammonia Synthesis Reactors Using Compositional Approach. Advances in Chemical Engineering and Science, 8, 124-143. https://doi.org/10.4236/aces.2018.83009

HPHT High Pressure and High Temperature

PR Peng Robinson

SRK Soave Redlich Kwong

BWR Benedict Webb Rubin

EoS Equation of State

PFR Plug Flow Reactor

i Substance i

List of SymbolsN_{2} Nitrogen

H_{2} Hydrogen

NH_{3} Ammonia

CH_{4} Methane

Ar Argon

List of Roman Lettersr NH 3 kmol / ( m 3 ⋅ s ) Ammonia production rate

P (Pa) System pressure in reactor

T (K) System temperature

R J / ( mol ⋅ K ) Ideal gas constant

K e q Equilibrium constant for ammonia rate

y i Molar fraction of substance i in reactor

v m 3 / mol Molar volume of gaseous system

b m 3 / mol Covolume term for cubic EoS

a ( Pa ⋅ m 6 ) / mol 2 Function of temperature in cubic EoS

A Mixture term for cubic EoS computing interactions

B Mixture term for cubic EoS computing interactions

A i j Mixture term for cubic EoS computing interactions

Z Mixture compressibility factor

k i j Binary interaction parameter

T r i Reduced temperature of component i

P r i Reduced pressure of component i

P r e f Pa Reference pressure

f i Pa Fugacity of i component in gaseous mixture

a i Chemical activity of i component in gaseous mixture

g e J / mol Gibbs excess energy for mixture

C p m i x J / ( kg ⋅ K ) Mixture heat capacity in mass units

C p m i x J / ( mol ⋅ K ) Real heat capacity at constant pressure of mixture

C p r e s J / ( mol ⋅ K ) Residual heat capacity at constant pressure of mixture

C p i g J / ( mol ⋅ K ) Ideal gas heat capacity at constant pressure of mixture

C v J / ( mol ⋅ K ) Real heat capacity at constant volume of mixture

C v r e s J / ( mol ⋅ K ) Residual heat capacity at constant volume of mixture

C v i g J / ( mol ⋅ K ) Ideal gas heat capacity at constant volume of mixture

b 0 - b 6 Experimental coefficients for η calculation

x N 2 Nitrogen conversion

L m Reactor length

V m 3 Reactor volume

y i Molar fraction of substance i in reactor

Δ H r J / mol Heat of reaction for ammonia synthesis

F N 2 o mol / s Initial molar flow rate of nitrogen in reactor

m ˙ kg / s Mass flow inside the reactor

A m 2 Sectional area of the reactor

T K Reactant gas temperature in reactor models

T g K Cooling gas temperature in autothermal reactor

U W / ( m 2 ⋅ K ) Overall heat transfer coefficient in autothermal converter

A ′ m 2 / m Specific heat exchange area in length

a ′ m 2 / m 3 Specific heat exchange area in volume

u m / s Superficial velocity of gas in reactor

d p m Particle diameter

h m or m^{3} Step size along the reactor in numerical method

ρ mol / m 3 Molar density of gaseous system

Δ Constant for cubic EoS (PR or SRK)

δ 1 Constant for cubic EoS (PR or SRK)

δ 2 Constant for cubic EoS (PR or SRK)

Ω a Constant for cubic EoS (PR or SRK)

Ω b Constant for cubic EoS (PR or SRK)

ψ i Mixture factor in cubic EoS for i component

ω i Acentric factor for i component

m ( ω i ) Acentric factor function

φ i Fugacity coefficient in gas phase

η Effectiveness factor inside the catalyst pellet

ε Bed porosity

μ (Pa∙s) Gas viscosity