Fixed-Bed Reactor Optimization for the Synthesisof Dimethyl Ether Using a 2D ModelHans-Jörg Zander1,*, Gregor Bloch1, and Thibaut Monnin1,2

DOI: 10.1002/cite.201300138

Dimethyl ether is typically produced from synthesis gas by conversion to methanol and subsequent etherification. Com-

pared to this classical approach, the direct synthesis has the advantage of higher equilibrium conversion and a simpler sin-

gle-stage process. The heat release of the strongly exothermic reactions requires a careful thermal reactor design. Pure axial

models are not sufficient, as they neglect the important radial heat transport resistance. Therefore, a 2D model including

axial and radial gradients was implemented and used for design optimization.

Keywords: Dimethyl ether, Fixed-bed reactor, Fixed-bed reactor model

Received: October 16, 2013; revised: February 04, 2014; accepted: February 08, 2014

1 Introduction

Dimethyl ether (DME) is a potential substitute for dieselfuel. Its clean combustion, moderate vapor pressure, higherenergy content compared to methanol, and its relativeindependence from crude oil are the main advantages ofDME [1]. Furthermore, DME is a base chemical for a varietyof substances, including olefins and polymers. A globalDME demand of 15 – 5 mio. t a–1 in the next few years is pre-dicted [2]. Extended use of DME as liquid petroleum gassubstitute may significantly exceed this figure.

Dimethyl ether is typically produced from synthesis gas.A steam reformer is followed by a methanol synthesis andan etherification. Adding CO2 to the reformer feed, the ratioC/H is shifted to the optimal value, reducing the energydemand of the overall process. Due to thermodynamicrestrictions in both reaction steps, the conventional processof sequential methanol and DME production suffers fromlimited conversion, and therefore, requires a higher processpressure. In this work, a direct reaction from synthesis gasto DME is investigated. Equilibrium constants and conver-sions are calculated from basic thermodynamic equations.The basic thermodynamic constants can be taken from [3].Fig. 1 shows the equilibrium conversion for the direct reac-tion

3CO � 3H2 � H3C � O � CH3 � CO2 (1)

compared to the equilibrium conversion of the methanolformation

CO � 2H2 � CH3OH (2)

The direct synthesis is clearly advantageous. If methanoland DME are formed simultaneously, and both substancesare considered to be products, the equilibrium conversionincreases further. This encourages a reactor design for thedirect synthesis.

Due to the significant heat release of the reaction and thelimited radial heat conductivity, temperature and concentra-tion gradients are expected in axial and radial direction. Topredict conversion and to prove thermal stability, a 2D mod-el is required. For economic reasons, the tube diameter ofany tube bundle reactor is extended as far as possible untilradial heat transfer limitations are reached. Therefore, a 2Dmodel is highly recommended in any case.

2 Fixed-Bed Reactor Modelling Approach

A fixed-bed is an object on multiple scales. As shown inFig. 2, a tube bundle can be divided in several tube in-stances, which are composed of single pellets, each of thembeing a porous system with gradients inside and kineticallycontrolled reactions at local conditions in every single pore.While all scales are implemented in a universal fixed-bed re-actor model, the focus of this work is the single-tube model.

The model was set up as a finite volume method. Thefixed-bed is axially and radially divided into annular con-centric control volumes. The species, heat and momentumbalances, which describe the quasi-homogeneous flow in-side the fixed-bed, are integrated over the control volumeand the Gauss divergence theorem is used to transformdivergence terms into flux terms which are balanced. Using

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–1Dr. Hans-Jörg Zander (hans-joerg.zander@linde-le.com), GregorBloch, Thibaut Monnin, Linde AG, Engineering Division, Dr.-Carl-von-Linde-Straße 6 – 14, 82049 Pullach, Germany; 2ThibautMonnin, TU München, Lehrstuhl I für Technische Chemie,Lichtenbergstraße 4, 85747 Garching, Germany.

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this method, discretization errors do not contribute to thebalance of the conserved variables mass, energy and mo-mentum. Such balance violations often lead to importantinconsistencies in finite difference models.

The equation system is closed with a density equation, anenthalpy equation, kinetics equations and empirical correla-tions for effective mass and energy transport, heat transferand pressure drop. As these correlations require furtherproperty data, such as heat conductivity, viscosity, etc., thisdata also has to be provided. At last, a number of coeffi-cients have to be provided such as fixed bed density, particlesize, bed void fraction, etc.

The kinetics equations are provided as a function. Thiscan either be an explicit function of process conditions or apellet model which is solved for every fixed-bed grid point tocalculate effective reaction rates including pore diffusionand external heat and mass transfer. Although radial con-vective flow is very small compared to conductive transportas well as to axial convection, it has to be considered in themodel equations to obtain a mathematically consistent sys-tem. Using this approach, the finite mass balance over anycontrol volume derived from the standard mass balanceequation [4] then reads

�k

q u wi� �k dAk � ��

kDeff q ∇wi� �k dAk

� Mi

�jvij rj�eff dV

(3)

with density q, velocity u, mass fractionwi, effective conductive transport coeffi-cient Deff , molecular weight Mi of spe-cies i and the control volume dV with itsk faces dAk.

The heat balance equation denotes

�k

que � pu� �k dAk ���

kkeff ∇T� �k dAk � qgu dV � qdV

(4)

The source term q includes heat trans-fer from the catalytic pellet to the gasphase in case the reaction is consideredheterogeneous. As the exothermicity of achemical reaction is the conversion ofchemical energy into sensible heat, andthe inner energy e is defined includingall forms of energy, it is conserved andno explicit heat of reaction occurs.

The momentum equation is mostlydominated by the irreversible pressuredrop.

�k

quui� p� �kdAk � qgidV � ∂p∂xi

� �irrev�

(5)

In the finite volume form, boundary conditions are easy.In the inlet, axial mass flow and inlet temperature are given.At the wall, the radial mass flow is zero, while the heat fluxis

qr � a T R� � � Tw� � (6)

with heat transfer coefficient a and wall temperature Tw. Noboundary conditions are required for the outlet crosssec-tional area.

According to fluid dynamics, it is not possible to deter-mine flow and pressure simultaneously on the same bound-ary. If the flow rate to the reactor is fixed, a pressure set atthe outlet boundary is propagated against flow direction, re-sulting in an unknown inlet pressure. In an algebraic sense,this can be reverted. Flow and pressure are algebraically setat the inlet boundary, the equations result in the physicallyindependent outlet pressure. In a physical sense, the outletpressure is the independent pressure that leads to theselected inlet pressure. The advantage is that the equationsystem now decouples completely in flow direction. Fromany state, the next downstream point can be calculated.Therefore, the reactor can be calculated sequentially frominlet to outlet cross sections. This strongly reduces computa-tion time, but requires neglecting axial dispersion. Due tothe high axial Péclet number this is no practical constraint.

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a) b)

Figure 1. Equilibrium conversion of syngas to dimethyl ether (a) and to methanol (b) asfunction of temperature and pressure.

Figure 2. Modelling levels of fixed-bed reactors.

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The equations describing each cross section are solvedsimultaneously using a Newton-based nonlinear algebraicsolver. To overcome convergence problems due to stronglynonlinear terms, especially of fast chemical reactions, ahomotopy method is implemented. The reaction rate con-stant is increased step by step to the final value, each stepdelivering the initial guess for the next solver call. Usingthis approach, a proprietary C/C++ implementation of thisreactor model converged for the given system using 50 – 100axial and 5 – 20 radial points within a few seconds on a stan-dard PC. As a validation, a grid independency was shownand a comparison was done against a number of cases forwhich analytical solutions are available.

3 Dimethyl Ether Reaction System

First, the species and the reaction system have to be defined.The model includes the species H2, CO, CO2, H2O, CH3OHand DME. The considered reaction system is

CO2 + 3 H2 � CH3OH + H2O (7)

CO + 2 H2 � CH3OH (8)

2 CH3OH � DME + H2O (9)

CO2 + H2 � CO + H2O (10)

The reaction rates have been adapted to experimentaldata.

r1 � k1 exp � E1

RT

� � pCOp3H2

� pCH3OHpH2O

K1

� �m1

1 ��i Ki�1pi

(11)

r2 � k2 exp � E2

RT

� � pCOp2H2

� pCH3OH

K2

� �m2

1 ��i Ki�2pi

(12)

r3 � k3 exp � E3

RT

� � p2CH3OH � pDMEpH2O

K3

� �m3

1 ��i Ki�3pi

(13)

r4 � k4 exp � E4

RT

� � pCO2pH2

� pCOpH2O

K4

� �m4

1 ��i Ki�4pi

(14)

At the given conditions, the gas phase can be simplifiedto be an ideal gas. The thermal state equations and equili-bria result from temperature integration of cp and cp/T.

The model calculation result in concentration, tempera-ture and pressure profiles for any given operating point.Fig. 3 shows one temperature profile obtained in an arbi-trary case. As expected, the highest temperature occurs inthe tube center near the inlet. For comparison, a 1D modelcompletely neglecting the radial profile also strongly under-estimates the axial profile. This can result in a serious un-derestimation of the catalyst temperature, and therefore, inan insufficient, and in the worst case unsafe reactor design.

The 2D model calculation is able to predict the thermalstability limits of the reactor. The system becomes thermallymore stable with– a smaller bundle tube diameter,– a smaller temperature difference between process and

cooling medium,– higher gas velocity, and– operation at smaller space-time yields.

Although this is expected by theoretical considerations,the model quantitatively shows the stability limits and pre-dicts the available conversion and selectivity.

4 Reactor Operating Point Optimization

The reactor is optimized towards a desired operation point,which is characterized by– a temperature profile within the allowed catalyst tempera-

ture range,– safe operation with sufficient distance to thermal run-

aways,– a small reactor volume,– high DME and methanol conversion rates,– high space-time yield, and– optimal DME and methanol product ratios.

While safe operation in a defined temperature range is anunconditional constraint, the other items are subject to opti-mization. The prediction of thermal stability is the most

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0

0.5

1

0

0.5

1+0

+5

+10

+15

+20

z/z0r/r0

∆ T

[°C

]

Figure 3. Exemplary temperature profile.

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important ability of the model calculation, since a lowerbundle tube diameter enhances thermal stability, but alsosignificantly increases reactor costs. Therefore, the goal is tominimize reactor costs by increasing the tube diameter asfar as possible, while avoiding runaway regions. Addition-ally, the less significant variables also have an impact on thethermal stability as well as on the investment and operatingcosts.

The desired operating point can now be selected by a com-parison of different operating points, or the model is usedas subject function for a numerical constraint optimizationroutine. In the latter case, a scalar cost function derivedfrom the calculated profiles including project details is re-quired as well as a numerical definition of “runaway“ and avariable describing the distance to the instable region. As noderivatives of the profiles with respect to the process condi-tions are available, and because of the low dimension ofindependent process variables, a gradient-free optimizationis the method of choice.

5 Conclusion

The dimethyl ether production from synthesis gas in afixed-bed reactor was described using a 2D model. The reac-tor simulation is able to predict the reactor operating pointincluding conversion, selectivity and its thermal stability.The model helps determining the economically optimaloperating point under the constraint of safe operation. Thisrequires the use of a 2D model including axial and radial

profiles. Often-used 1D models are insufficient to predicthot spots and runaways. As tube bundle reactors are gener-ally designed minimizing the tube number and maximizingits diameter, radial gradients are an important issue. Theneed to use 2D models for fixed-bed reactor design is em-phasized.

This paper is based on a project funded by the GermanBundesministerium für Bildung und Forschung.Financial support 033RC1108F is gratefully acknowl-edged. The authors are responsible for the publication.The project team “Integrierte Dimethylethersyntheseaus Methan und CO2 (DMEEXCO2)“ consists of BASFSE, Linde AG, hte AG, Technische Universität Münch-en, MPI für Kohlenforschung and FraunhoferUMSICHT.

References

[1] G. A. Olah, A. Goeppert, G. K. Surya Prakash, Beyond Oil andGas – The Methanol Economy, Wiley-VCH, Weinheim 2009.

[2] Dimethyl Ether Technology and Markets, Report 07/08S3, NexantInc., New York 2008.

[3] B. E. Poling, J. M. Prausnitz, J. P. O’Connell, The Properties ofGases and Liquids, 5th Edition, McGraw-Hill, New York 2001.

[4] M. Schäfer, Numerik im Maschinenbau, Springer, Berlin 1998.

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