Energy efficient reactor design simplified by second law analysis

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Energy efficient reactor design simplified by second law analysis Øivind Wilhelmsen a,b,*, Eivind Johannessen c, Signe Kjelstrup a a

Department of Chemistry, Norwegian University of Science and Technology, N-7491 Trondheim, Norway SINTEF Energy Research, N-7465 Trondheim, Norway c Statoil Research Centre, N-7005 Trondheim, Norway b

article info

abstract

Article history:

Gas heated reformer configurations which all produce the same amount of hydrogen have

Received 9 June 2010

been investigated. By analysing various stationary states of operation formulated by

Received in revised form

optimal control theory, we find numerical support for the hypothesis of minimum entropy

20 August 2010

production, namely that the state of operation with constant entropy production, and also

Accepted 21 August 2010

in some cases constant thermal driving force, are good approximations to this most energy

Available online 8 October 2010

efficient state of operation. This result applies, also for non-linear transport equations, and conditions for which there exist no rigorous mathematical description of the most energy

Keywords:

efficient state. Based on the studies, we also formulate a set of guidelines to aid in an

Chemical reactors

energy efficient reactor design, which can be used once the best available heat transfer

Entropy production

coefficients have been obtained. The optimal reactor design depends on the relative size of

Energy efficiency

the heat transfer coefficient for heat transfer across the tubular reactor wall and typical

Heat transfer

heat transfer coefficients in heat exchangers. Very efficient heat transfer across the reactor

Optimization

tube wall favours a design consisting of an adiabatic pre-reactor followed by a tubular reactor section exchanging heat. Very poor heat transfer across the reactor tube wall favours a design consisting of one or more adiabatic reactor stages with interstage heating/ cooling in dedicated heat exchangers. The guidelines add to earlier proposals in the literature, and help define central optimization variables and boundary conditions in reactor design. ª 2010 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved.

1.

Introduction

The consumption of oil, coal, gas and other energy sources has more than doubled in the last fifty years on a worldwide basis [1]. During the same time, atmospheric CO2 levels have increased with over 20 percent [2]. More energy efficient processes in the industry are vital to change this situation. This work is motivated by the increased attention worldwide to improve energy efficiencies in process units and plants, to save valuable resources and reduce CO2-emission. We have chosen to study the energy efficiency of chemical reactors

with focus on production of hydrogen by steam reforming. Since the USA alone produces over nine million tons of hydrogen per year, mostly with steam reforming of natural gas, and the largest exergy loss has been located in the chemical reactor [3], steam reforming is a great place to start a discussion for more energy efficient reactor design. In the engineering literature, chemical reactors have traditionally been optimized with several objective functions [4], and the most common are those related to economy, materials or to energy. In the design of commercial reactor systems, all three are important. This work will only concern

* Corresponding author. SINTEF Energy Research, N-7465 Trondheim, Norway. Tel.: þ47 73594133. E-mail address: [email protected] (Ø. Wilhelmsen). 0360-3199/$ e see front matter ª 2010 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2010.08.118

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State variables of the optimal control problem, e Axial position in the reactor, m

Nomenclature

x z

Roman symbols c Constant, 1=K Constant associated with radiation,  crad Heat capacity, J=K mol Cp ðdS=dtÞirr Total entropy production, J=K s D Diameter of the reformer tube, m Molar flow of component i, mol=s Fi Molar Gibbs energy of reaction j, J=mol DrGj Wall heat transfer coefficient, W=K m2 hw H Hamiltonian, J=Kms Molar enthalpy of reaction j, J=mol DrHj Total heat flux, J=sm2 Jq kw Thermal conductivity of the wall, W=K m L Total length of the reactor, m Length of the reaction mode, m L1 Length of the heat transfer mode, m L2 Optimal length of a reactor, m Lopt P Pressure, ðN=m2 Þ Pr Prandtl number, e Rate of reaction j, mol=kg Cat:s rj Universal gas constant, J=K mol Rg R Radii of the reformer reactor tube, m Average radius of the pellets, m Rs Packed bed Reynolds number, e Rep S Extensive entropy, J=K s T Temperature, K Stoichiometric coefficient, e vj;i v Convective speed of the fluid, m=s

Greek symbols U Cross section area, m2 r Density, kg/m3 rB Density of catalyst, kg Cat:/m3 l Multiplier function, e s Local entropy production, J=K ms M Viscosity, ðPa$sÞ Degree of conversion for reaction j, e xj The Stefan Boltzmann constant, W=m2 K4 sr hj Effectiveness factor of reaction j, e e Void fraction of the pellet bed, e S Component of the total entropy production, J=K s D% Relative difference in percent, e

the energy efficiency. The energy efficiency of a steady state plug flow reactor shall be measured by its entropy production, and we shall compare reactors that all produce the same amount of hydrogen, to have a fair comparison. By energy efficiency we therefore mean the second law efficiency. A second law analysis or exergy analysis offers a systematic way to evaluate and compare technologies, here chemical reactor systems. At the centre of such an analysis is the total entropy production. In a reactor system, this has contributions from heat transfer, flow and chemical reactions. Maximum energy efficiency corresponds to minimum total entropy production from the three phenomena. We shall not compare the entropy production to any inlet exergy, but rather discuss its absolute value, as a measure of the efficiency. The state of reactor operation with minimum entropy production has been investigated from the perspective of nonequilibrium thermodynamics over quite some years [5e11]. This theory gives the same thermodynamic basis for all transport phenomena involved, and adds to exergy analysis in this manner [12]. It was first proposed that chemical reactors could be approximated by a system having parallel production paths [5,6]. The minimum is then that the driving forces for transport are constant (Equipartition of the thermodynamic Forces, EoF), when non-linearity in the fluxeforce relations is insignificant. For plug flow reactors, this does not properly take into account the balance equations, however. Later work has

Sub- and superscripts 0 Initial conditions 1 Radial position at the reactor wall (see Fig. 2) 2 Radial position at the reactor wall (see Fig. 2) a Annular heating section c Constant f The value at either z ¼ 0 or z ¼ L i Component number in Into a process j Reaction number out Out of a process Abbreviations GHR Gas Heated Reformer EoF Equipartition of Forces EoEP Equipartition of Entropy Production

therefore taken advantage of optimal control theory in the formulation of the optimization problem, and included all balance equations as constraints in the optimization [7e11,13]. A typical control variable in reactor optimization has been the temperature profile at the outside of the reactor, from which heat has been transferred into the reactor. In 2005, Nummedal et al. used the optimization package in Matlab 6.0 to find this temperature profile for a plug flow model of a steam reformer [8]. Using optimal control theory [14,15], other reactor systems were studied, for example the SO2 oxidation and the sulphuric acid decomposition [9,11]. Optimal control theory gives a robust formulation of the optimization problem. Studying the properties of the state of minimum entropy production, Johannessen and Kjelstrup [10] formulated the hypothesis for the state of minimum entropy production. The hypothesis was recently put in a wider context [16]. It reads: Equipartition of entropy production, but also equipartition of forces, are good approximations to the state of minimum entropy production in the parts of an optimally controlled system that have sufficient freedom to equilibrate internally. The hypothesis was first postulated from studies of chemical reactors. It was found to also apply to distillation columns [17] and heat exchangers [18]. The meaning of the expression “sufficient freedom” shall be discussed later. Some

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intriguing circumstances were noted, but not fully resolved. The hypothesis was, for instance, found to apply to processes which are intrinsically non-linear, i.e. chemical reactions, but it was only possible to prove equipartition of entropy production (EoEP) in a strict mathematical sense for linear processes when the number of control variables were equal to the number of driving forces [10,19]. The mathematical result was thus rather limiting for the application of the hypothesis, as many transport phenomena are non-linear. Among them, chemical reactions and radiative heat transfer are central examples. A final answer to the surprising fact that EoEP is observed in chemical reactors being rather far from chemical equilibrium has thus not been found. The hypothesis for minimum entropy production can nevertheless be used to give guidelines for energy efficient design and operation [16]. We have earlier seen that the optimal reactor was characterized by what we called a reaction mode and a heat transfer mode. The name was taken from the dominating contribution to the entropy production. Knowing about the presence of these modes and the viscous contribution, the following guidelines were formulated [16]:

[16]. The present work can be seen in this context. It is an extension of the first studies of the hypothesis for minimum entropy production, on how the hypothesis can be applied. Given that the hypothesis can be used to describe the optimal control problem, what are the consequences for energy efficient reactor operation? Can more insight be added to guidelines [16,12] formulated already? We shall continue to study plug flow reactors using a stateof-the art gas heated reformer model [20] as test case to evaluate the hypothesis and the guidelines presented above [16]. Compared to the simple plug flow reformer [8], the gas heated reformer is more sophisticated. Additional details are thus added to the one-dimensional plug flow model, such as radial energy balances across the reactor wall. We explicitly model the inner heat transfer coefficient and transport properties such as the viscosity and the thermal conductivity. Our study is for a particular reactor, but we are seeking improved guidelines for reactors in general. We shall thus end by suggesting process schemes which illustrate such guidelines for energy efficient design.

1. A tubular chemical reactor of length L, operating in an energy efficient way, should have an inlet section, of length L1, that is (close to being) adiabatic. The heat of the reaction (s) will then move the reacting mixture toward chemical equilibrium, and the reactor operates in the reaction mode. 2. A tubular chemical reactor of length L, operating in an energy efficient way, should have a central section, L2, characterized by a fine balance between heat transfer and reaction rate(s), so that the entropy production is constant. This translates to a temperature of the reacting mixture being (approximately) at constant distance from the temperature at which the mixture is in equilibrium. This can in most cases be achieved by counter-current heat exchange. We say that the reactor should operate in a heat transfer mode in L2. 3. A tubular chemical reactor operating in an energy efficient way, should have a total reactor length L  L1 þ L2 , that gives the best trade-off between low entropy production of heat transfer and reactions (long reactors are favourable) and low entropy production due to pressure drop (short reactors are favourable).

2.

Given the length of the reactor, these guidelines tell that the most energy efficient reactor has an adiabatic reactor inlet part followed by a heat-exchange-part operated in most cases with counter-current heating. The length is limited by the trade-off of the entropy production by viscous flow and that from heat transfer and chemical reaction. Leites and coworkers [12] proposed twelve commandments for energy efficient reactor design and operation. Other guidelines have been proposed by us (see above). The main aim of this work is to further examine the hypothesis for the state of minimum entropy production, and also see if we can add to the guidelines published for energy efficient design. These subjects deserve further investigation, because information and criteria on their validity can help in energy efficient reactor design. With these tools available, complex modelling and optimization efforts can be kept to a minimum

The plug flow reactor model

We investigate the tubular reactor outlined in Fig. 1 with diameter D and length L. In this section, the model will be presented. By the word “model” we mean the equations necessary to describe a part of the chemical reactor. The boundary conditions of the balance equations will be given in Section 5. We shall use two main models. The inner reactor tube, shown in Fig. 1, is the target of optimization, and shall be referred to as the study case, or just Case. Several varieties of the Case shall be computed. The optimization results shall be compared to a state-of-art reformer reactor, called the Gas Heated Reformer (GHR). This provides a fixed reference for comparison, and discussion of potential improvements. The GHR, as described by Wesenberg [21], consists of an inner reactor tube surrounded by an annular heating section. The inner reactor tube is filled with catalyst pellets. A total of 3 reactions are modelled for a gas mixture of 6 components: CH4 þ H2 O#CO þ 3H2

(1)

CO þ H2 O#CO2 þ H2

(2)

CH4 þ 2H2 O#CO2 þ 4H2

(3)

The production of hydrogen based on these reactions is endothermic, and in addition to the five components in Equations 1e3, nitrogen is included as inert gas. Constant

Fig. 1 e A sketch of the tubular reactor, the inner tube.

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effectiveness factors for all reactions, hj, are used in order to account for diffusion limitations inside the catalyst pellets. This is a good approximation for the steam reformer [20]. Water participates in all reactions and is chosen as the reference component.

2.1.

The inner reactor tube

A pseudo-homogeneous one-dimensional plug flow model is chosen for the packed bed. The main assumptions are no radial gradients and a flat velocity profile. Details on the balance equations of a plug flow reactor can be found in [22]. The energy balance is: i P3 h  dT pDJq þ UrB j hj rj Dr Hj (4) ¼  P6  dz i Fi Cp;i Here, Jq is the measurable heat flux through the reactor wall, U is the cross section area of the reactor, rj is the reaction rate of reaction j, Dr Hj is the enthalpy of reaction j, Cp;i is the heat capacity of component i, rB denotes the density of the catalyst bed and Fi is the molar flow rate of component i. Information on the reaction kinetics was taken from Xu and Froment [23]. Ideal gas law is used as equation of state, giving an expression for the velocity: P  6 Rg T i Fi (5) v¼ UP Here, P is the pressure. Ergun’s equation has been popular to model the momentum balance of packed beds [9,8]. Ergun’s equation is only valid for Rep =ð1  eÞ500 where Rep is the Reynolds number of the packed bed and e is the catalyst bed void fraction. Larger Reynolds numbers where Ergun’s equation is invalid are typical for the steam reformer [24]. In correspondence with [24], Hicks equation is used to model the momentum balance: ð1  eÞ1:2 0:2 rv2 dP Rep ¼ 3:4 e3 Rs dz

(6)

Here, Rs denotes the radius of the catalyst pellets and m is the viscosity. The conversion of reaction j is defined as: xj ¼

moles of H2 O consumed by reaction j moles of H2 O at the inlet

inner reactor tube in an annular heating section. A flat temperature and velocity profile is assumed for the exhaust gas [22] pDJq dTa ¼ P6   dz i Fa;i Cp;i

This energy balance for the exhaust gas (Eq. (8)) is specific for the GHR, and does not apply when Ta is a free variable (see below). The energy balance connects the reactor tube and the annular section through the heat flux, Jq. The momentum balance of the annular section has an insignificant effect on the results [25] and was omitted.

2.3.

The heat flux

To calculate the heat flow across the reactor tube wall, we need the wall temperatures, Tw1 and Tw2 . The radii of the inner and the outer wall of the reactor tube are denoted R1 and R2 respectively. The different radii, along with the temperatures in the different sections are illustrated in a cross section of the reactor tube in Fig. 2. Convective as well as radiative heat transfer is taken into account. At R1 the heat flux is: Jq ¼

   R2  crad sr T4a  T4w2 þ hw2 ðTa  Tw2 Þ R1

The molar flow rate of component i becomes: 1 0 3 X   0 0 @ Fi ¼ Fi þ Fi¼H2 O nj;i xj Ai ¼ 1; .6

2.4.

The energy balance in the radial direction

To obtain consistent values for the wall temperatures Tw1 and Tw2, the energy balance has to be solved for every axial

(7)

(8)

(9)

j

Here, ni;j is the stoichiometric coefficient of component i in reaction j with water as the reference component.

2.2. The annular heating section of the gas heated reformer In the gas heated reformer, the reactor is heated by a hot exhaust gas flowing counter-currently at the outside of an

(11)

Heat is supplied to the reactor tube from the outside, at a temperature of Ta, with the help of a heating utility. The coefficient crad takes into account the radiative properties of the heating utility and hw2 is the convective heat transfer coefficient at the outside of the reformer tube. In the gas heated reformer the heating utility is the hot exhaust gas and Ta is given by Equation 10. In the optimization cases, the heating utility is left unspecified, but is assumed to have the same heat transfer properties as the exhaust gas.

Which gives the following component balances: dxj UrB ¼ h rj j ¼ 1; 2; 3 dz F0i¼H2 O j

(10)

Fig. 2 e Cross section of the inner reactor tube.

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position through the reactor unit. At the inner wall of the reactor tube, R1, an energy balance over the radial direction gives [26]: hw1 ðTw1  TÞ ¼

kw ðTw2  Tw1 Þ R1 lnðR2 =R1 Þ

(12)

Inside the reactor tube, at R1, we include the dependence of the heat transfer coefficient, hw1 , on the reactor tube conditions according to Peters [27]. He claims that this expression is good for Reynolds numbers up to 8000 [27]: hw1 ¼

9:8Rs 0:33 ð2Rs =DÞ0:26 Re0:45 p Pr kg

(13)

Here, Rs is the radius of the catalyst pellets. Pr and Nu denote the Prandtl and Nusselt number respectively. The dimensionless parameters used in this work are defined in [28]. At the outer radius of the reactor tube, R2, the energy balance for the radial direction gives: Jq ¼

kw ðTw2  Tw1 Þ R1 lnðR2 =R1 Þ

(14)

We did not consider radiative heat transfer from the inner tube wall to the packed bed, in accordance with Wesenberg and Svendsen [20], because the largest temperature gradient is at the outer wall, the wall temperature is considerably lower than Ta, and because convection due to turbulence is by far the most important heat transfer phenomenon in the packed bed.

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 ZL dS ¼ sðzÞdz ¼ ST þ Sp þ Sr dt irr

(16)

0

Here, the symbols ST ; Sp ; Sr denote the total entropy production from thermal, flow and reaction contributions.

3.1.

Consistency check of the total entropy production

The total entropy production can be obtained by Equation 16, or it can be calculated from an entropy balance over the whole reactor. For the case of a gas heated reformer with insulation on the outside as in the reference case, entropy is only transferred into or out from the process through the gasflows. The total entropy balance then gives:  dS ¼ Sout  Sin þ Sa;out  Sa;in dt irr

(17)

Here, subscript a denotes the annulus. Sout and Sin are the entropic contributions which follow the gas-flow out or into the reformer. In the case where only the reactor tube is investigated, and heat is transferred through the outer wall, the total entropy production as given by the entropy balance is:  ZL Jq ðzÞ dS dz ¼ Sout  Sin  pD Ta ðzÞ dt irr

(18)

0

2.5.

Thermodynamic data and transport properties

Ideal gas law is used as equation of state. This is reasonable considering the large temperatures encountered in the reformers. The viscosity of each specie is calculated by the use of Lucas’ method, and polarity effects are taken into account when the viscosity of the gas mixture is estimated by the method of Wilke [29]. Thermal conductivities and heat capacities of the components are modelled as polynomials [29]. The thermal conductivity of the gas mixture is calculated from the Wassilijewa equation, and the heat capacity of the gas mixture is found by a sum weighted with the mole-fractions [29,30]. The details of these models can be found in [21].

3.

The entropy production

Non-equilibrium thermodynamics gives the foundation for this work [31]. The local entropy production of the plug flow reactor is [16]:  s ¼ pDJq

1 1  T Ta



   m

X Dr Gj 1 dP þ Uv  hj rj  þ UrB T T dz j¼1

(15)

The expression is valid for all cases studied. The first term on the right hand side is called the thermal entropy production. The second term is entropy produced by pressure gradients or viscous flow. It can be shown that the third term covers the entropy production associated with both diffusion and reaction inside the catalyst pellets. The total entropy production, ðdS=dtÞirr , is obtained by integrating the local entropy production over the length of the reactor:

The last term in Equation 18 comes from the entropy production in the heating utility because of the heat transferred to or from it.

4. Optimal cooling/heating of plug flow reactors 4.1.

The optimization problem

The optimization problem was to find the optimal heating strategy with a fixed chemical conversion of hydrogen. The optimal heating strategy was defined as the temperature profile at the outside of the reactor tube wall, Ta ðzÞ, which gives minimum entropy production. Equation 16 was the objective function in the minimization. Equation 18 and 17 were used for consistency checks on its value.

4.2.

The state of minimum entropy production

The minimization problem was constrained by the onedimensional balance equations (Eq. 4, 6 and 8). This is a standard problem in optimal control theory. In correspondence with the standard formulation in optimal control theory we refer to the variables restricted by conservation equations as state variables, x ¼ fT; P; x1 ; x2 and x3 g. The temperature at the outside of the reformer tube, Ta ðzÞ, is referred to as the control variable and is allowed to take any positive value. The state variables can either have fixed or free values at the start of the reformer tube ðz ¼ 0Þ, or at the end ðz ¼ LÞ, and this is referred to as fixed or free boundary conditions. Optimal control theory will not be explained in detail here, but can be found in the

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literature [14,15]. The Hamiltonian for our optimal control problem is: H ¼ s þ lT ðzÞ

dT dP dx dx dx þ lP ðzÞ þ lx1 ðzÞ 1 þ lx2 ðzÞ 2 þ lx3 ðzÞ 3 dz dz dz dz dz

(19)

Here, l denotes the multiplier functions, which are functions of z. The necessary conditions for a minimum are given by Pontryagins’ minimum principle [14,15]: dT dH ¼ dz dlT ðzÞ

(20)

dP dH ¼ dz dlP ðzÞ

(21)

dxj dH ¼ j ¼ 1; 2; 3 dz dlxj ðzÞ

(22)

dlT ðzÞ dH ¼ dz dT

(23)

dlP ðzÞ dH ¼ dz dP

(24)

dlxj ðzÞ dz

dH ¼  j ¼ 1; 2; 3 dxj

(25)

We also have an algebraic restriction on the system: dH ¼0 dTa

(26)

This algebraic restriction provides an expression for the optimal temperature of the heating utility as function of the state variables and the multiplier functions. How Ta is extracted from Equation 26 is explained in Appendix A. If the Hamiltonian does not depend explicitly on z, such as in our optimal control problem, it is called autonomous. In such cases the Hamiltonian is constant. At the optimal length of the reactor, the Hamiltonian is also zero. These properties were used as a check of correct implementation of the equations. The boundary conditions used to solve the optimal control problem are summarised in Table 1.

5.

Calculations

5.1.

Description of the cases

We first established the entropy production in the gas heated reformer, using boundary conditions reported in the literature [21]. We proceeded to optimize a plug flow reactor with the same hydrogen production and the same heat transfer

mechanisms as the GHR. Finally, some studies were made of the effect of heat transfer mechanism. Table 2 summarizes the cases investigated. Columns 3 and 4 provide input parameters to the heat flux in Equation 11. Details on the feed gas properties or material and geometry related parameters can be found in Appendix B. These parameters are identical for all cases, unless something different is specified in Table 2. The hydrogen production of all the cases is fixed according to Table B 5 in Appendix B. All calculations were done with Matlab 7.9. The results reported have a relative accuracy of 5  106 . The total entropy balance was checked by comparing the results of Equation 16 to Equation 18, and is found to be the same within the given accuracy in all cases.

5.2.

The gas heated reformer

The Gas Heated Reformer (GHR) is state-of-art technology, and is therefore a good reference for evaluation and discussions of potential new designs. Representative temperature profiles for the GHR were obtained by solving the reactor model simultaneously with the energy balance of an annular heating section (Eq. (10)), including the local heat flux (Eq. (11)). The parameters of Equation 11, hw2 and crad , were chosen to give almost identical results for the convective and the radiative heat fluxes as obtained by Wesenberg et al. [21,24] in their case GHR-1 (see their Fig. 7.8). The inlet properties of the annular heating gas can be found in Appendix B. We used the Matlab 7.9 boundary value solver bvp4c to solve the GHR.

5.3.

The state of minimum entropy production

The optimization was conducted using a routine first proposed by Johannessen and Kjelstrup [9]. A brief description is as follows:  A coarse grid (30e40 points) was used in a numerical optimization with the Matlab 7.9-routine fmincon to create an initial guess for the optimal control problem. The hydrogen production was specified by the reference case, and Ta was the control variable.  The boundary value solver, bvp4c, was used to solve the optimal control problem outlined in Section 4. The boundary values of the optimal control problem were taken into account according to Table 1.  After each simulation, the Hamiltonian was checked and confirmed to be constant, as predicted in Section 4. For the

Table 2 e Description of the cases. Table 1 e Possible boundary conditions for the optimal control problem. Here, x denotes the state variables and subscript f denotes the value at either z[0 or z[L. Description Fixed L, fixed end state Fixed L, free end state Free L, fixed end state Free L, free end state

HðzÞ

xðzf Þ

l(zf)

constant constant 0 0

xf e xf e

e 0 e 0

Case:

Heat flux:

hw2

crad

T0

Ta

L

GHR 1a 1b 1c 2a 2c

mixed mixed mixed mixed convective convective

100 100 100 100 100 100

0.45 0.45 0.45 0.45 0.00 0.00

fixed fixed free free fixed free

fixeda free free free free free

fixed fixed fixed free fixed free

a fixed by Eq. (10)

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Cases 1c and 2c, the Hamiltonian was also confirmed to be zero.

5.4.

Defining the solution with EoEP

In a chemical reactor with only one control variable, it may not be possible to achieve EoEP at every point. To investigate properties of the optimum solution, and to see how far the GHR is from EoEP, we defined a practical way to find profiles giving the property of EoEP. We thus defined the heat flux to be zero at points where equipartition is impossible, corresponding to adiabatic conditions: Ta ðzÞ ¼

gives sðzÞ ¼ sc Ta ðzÞ ¼ TðzÞ

if sðTa ðzÞ ¼ TðzÞÞhsc if sðTa ðzÞ ¼ TðzÞÞ  sc

The constant sc was always adjusted to give the same hydrogen production as in the GHR.

5.5.

Defining the solution with EoF

Equipartition of forces means that the thermodynamic driving forces are constant over the reactor length. With Ta as the only control variable, the only force to equipartition is the thermal force. We have: Ta;EoF ðzÞ gives

1 1 ¼c  TðzÞ Ta ðzÞ

(27)

where c is a constant that is adjusted to give the same hydrogen production as in the GHR.

6.

Results and discussion

6.1. The gas heated reformer e comparison to earlier results We first confirmed that our set of equations reproduced case GHR-1 of Wesenberg [21]. This was done by examining the heat flux through the wall from the annulus to the reactor. The heat flux at R2, shown in Fig. 3, is very close to that reported in the Fig. 7.8 by [21]. Outlet conditions such as pressures, temperatures and compositions shown in Appendix B were also similar to [21]. Wesenberg used a more sophisticated two-dimensional model with the discrete ordinate method to model the radiation. We explain a small discrepancy between the results by this difference. We therefore consider the solution presented for the GHR as being validated. On this background we proceeded to determine the entropy production of the 12.93 m long GHR. Table 3 gives a total entropy production of 62.47 W/K for the GHR. It is completely dominated by the thermal contribution (55.15 W/K), the viscous and reaction contributions being only 4.33 and 2.99 W/K, respectively.

6.2.

The total entropy production

Table 3 gives the total entropy production for all the cases investigated along with the different contributions. The different contributions were explained in Section 3 and are

Fig. 3 e The different heat flux contributions in the reference GHR case. Radiative (solid line), convective (dashed line) and the total heat flux (dash-dot line).

found in Equation 15. In addition to the table values, local information about the cases are presented in the Figs. 4e7. The Cases 1a, b and c have the same reactor and wall conditions as the GHR, but here the variable Ta is set free in the optimization. When the inlet temperature and reactor length are fixed to the same value as in the GHR (Case 1a), a minor ð1:6%Þ reduction is observed in the optimum total entropy production, from 62.47 to 61.46 W/K. The immediate conclusion is that the state-of-the art GHR-technology is excellent, given the boundary conditions. A drastic reduction in the entropy production to 29.33 W/K is possible by setting also T0 free. This however, leads to a rise in the temperature at the inlet to about 1060 K, and excess entropy production by preheating the mixture (not accounted for here). A further reduction in the optimum value to 29.28 W/K, by increasing the reactor length beyond 12.93 m to the optimal value 15.06 m (Case 1c), does not seem to improve the situation much beyond the results of Case 1b.

Table 3 e The total entropy production and its contributions in W/K. Symbols were explained in connection with Eq.(16). Solutions with EoEP and EoF are also shown. All reactors produce the same amount of hydrogen. Sp

Sr

ðdS=dtÞirr

4.33

2.99

The cases with mixed heat flux Case 1a 53.94 4.48 3.02 Case 1b 7.41 5.68 16.24 Case 1c 7.80 6.63 14.85

ST

EoEP

EoF

62.47

e

e

61.46 29.33 29.28

61.74 29.45 29.41

63.24 29.37 29.31

The cases with purely convective heat flux Case 2a 81.93 4.54 3.05 89.52 Case 2c 4.36 4.16 24.16 32.68

89.89 32.77

90.91 32.71

The reference case GHR 55.15

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Fig. 4 e The T and Ta-profiles of the GHR (dash-dot lines) and the corresponding optimal solution 1a (solid lines). The Ta-profile from the EoEP solution is given by the dashed line. All the cases have a mixed heat flux.

By artificially allowing the system to have only one type of heat transfer mechanism, we found large effects on the entropy production. The Cases 2a and 2c in Table 3 have the same conditions as the Cases 1a and 1c, but now the heat flux is purely convective. The optimal entropy production rises from 61.46 to 89.52 W/K when the radiative heat flux contribution is removed. The relative contribution from the thermal part of the entropy production stays very high (81.93 W/K). By allowing the inlet temperature and the reactor length to vary (see Table 3), one can again obtain a large reduction in the total entropy production, to 32.68 W/K. The optimal reactor length is then considerably shorter than the length of the GHR, 9.15 m versus 12.93 m. The cases are included to illustrate that there are large variations possible in the state of minimum

Fig. 5 e The temperature of the gas mixture, T (lower lines), and the heating utility, Ta (upper lines) for the optimal solution 1b (solid lines). The Ta-profile from EoEP solution is given by the dashed line. All the cases have a mixed heat flux.

Fig. 6 e The local entropy production for the reference case (GHR, solid line), the optimal cases with fixed T0 and L (1a, upper dashed line), free T0 and fixed L (1b, dash-dot line) and free T0 and L (1c, lower dashed line). All the cases have a mixed heat flux. entropy production. It is thus important to use a realistic heat transfer model for a given reactor design in order to obtain the relevant optimum.

6.3.

The state of minimum entropy production and EoEP

According to the hypothesis for the state of minimum entropy production, EoEP is a good approximation to this state. This can now be illustrated, for the realistic as well as the less realistic cases above. The total entropy production of each reactor is given in the sixth column of Table 3 for all cases. This column can now be compared to the fifth column, to test the statement in the hypothesis. Clearly, the total entropy production in the sixth column is a very good approximation to

Fig. 7 e Optimal T and Ta-profiles for a mixed heat flux (1a, solid lines, lower/upper) and a purely convective heat flux (2a, dashed lines lower/upper).

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 1 3 2 1 9 e1 3 2 3 1

the value in the fifth column. The numbers agree within 0.5%. Agreement of the total values does not necessarily mean that the local profiles are the same, however. We therefore plotted the local temperature profiles of the GHR, Case 1a and the EoEP solution in Fig. 4, the solution of Case 1b plus the EoEP solution in Fig. 5, and the local entropy production arising from these profiles for the GHR, and the Cases 1a, b, and c in Fig. 6. The profiles of T and Ta for the GHR are slightly curved in Fig. 4, Ta more than T (dash-dot lines). In the optimal Case 1a, Ta is a variable (solid lines). The solid lines show that there is some difference between the optimal reactor 1a and the GHR reactor, but the temperature profiles from the EoEP solution nearly coincide with that of solution 1a. It is surprising, given the highly non-linear mechanism for heat transfer and highly nonlinear chemical reactions, that the EoEP solution is such a good approximation to the state of minimum entropy production. Fig. 5 shows a similar picture for Case 1b, which has a very different entropy production from 1a, and also very different relative contributions to the entropy production (compare the Cases 1a and 1b in Table 3). By setting the inlet temperature free, we see that it rises to a very high value (1060 K). This is favourable for the chemical reaction, because the figure immediately continues with a large drop in both optimal temperature profiles (solid lines). The same was observed earlier [8]. At the inlet, it is impossible to construct an EoEP solution, because of the large entropy produced by the reactions and the varying reaction Gibbs energies. Away from the inlet and outlet, we see that the solution constructed from EoEP is very close to the optimal solution. Again we conclude that EoEP is a good approximation to the state of minimum entropy production, when the system is not bounded by the inlet or the outlet. The drop in Ta near the outlet is a consequence of the boundary conditions, as was explained earlier [9]. Fig. 6 summarizes how the entropy production varies through the reformer in all four cases with mixed heat flux (GHR, 1a, b and c). The entropy production of the GHR is less constant inside the reactor than the other cases. The Cases 1a, b and c have fairly constant entropy production inside the reactor, 1a more than 1b and c, except for the end sections as explained by the temperature profiles. The figure confirms that the entropy production is reduced significantly from 1a to the Cases 1b and 1c. Clearly EoEP is not exactly obeyed, but we can conclude that it gives a good approximation to the state of minimum entropy production for all the Cases 1a, b and c. The entropy production seems to be more constant with a fixed than with a free inlet temperature. We explain this as a consequence of the change in the relative importance of the various contributions to the entropy production between the Cases 1a and b. It is more difficult to maintain a constant level, with the temperature Ta as the only control variable, when the entropy production is more dominated by chemical reactions. Next, we examine the case with a simpler expression for the heat flux, the one having convection only. Fig. 7 shows that the shape of the optimal temperature profile outside the reactor varies largely from Case 1a to 2a. A purely convective heat flux (dashed line) is less effective in heating the reactor than one which includes radiation (Case 1a). Better heat transfer mechanisms allow for the same amount of hydrogen to be produced with a smaller temperature gradient across the wall and

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a smaller thermal entropy production. Again, EoEP is a good approximation to any of the two optimal solutions (Table 3). To summarize so far, we have seen that the hypothesis for the state of minimum entropy production applies to widely different reactor solutions, for various boundary conditions and for various heat transfer mechanisms. Except from the inlet section of the reactor, EoEP is a very good approximation to the state of minimum entropy production for all the cases studied above, also the highly non-linear cases involving radiation! The reactors, which all produce the same amount of hydrogen, have an entropy production in the state of EoEP which is less than 0.5% away from the numerically optimal solution. Spirkl and Ries [19] and Johannessen and Kjelstrup [10] investigated processes with minimum entropy production and radiation, and found that the local entropy production was generally not constant. Formally speaking their theoretical finding is true and agrees with our results, but in practice, we find that EoEP nevertheless is a surprisingly good practical approximation to the optimal solution. The more practical condition, EoF, can also give a good approximation to the state of minimum entropy production according to the last column of Table 3. EoF gives a practical result, as it is directly related to the temperature profiles of the optimal solutions, cf. the Figs. 4e7. The result is more casespecific and more dependent on boundary conditions, however. The hypothesis, which gives a priority to EoEP over EoF, is thus confirmed. These observations have some support from the findings of Sorin et al. [32], who found that equipartition of power of separation in a mass exchanger, gave a better fit to the numerical optimum than a fit defined by the chemical driving force.

6.4.

Changing the reactor length

Fig. 8 shows how the total entropy production changes with reactor length for the Cases 1b and 1c. There is an optimal reactor length for both cases, but the minima are rather broad, so only small improvements can be obtained by adjusting the reformer length. To find the optimal reactor length, is thus not a main issue with respect to more energy efficient reformer design. But, according to the figure, the optimal length depends very much on the mechanism of heat transfer. This confirms that it is important to use a realistic heat transfer model in the optimization in order to get the relevant optimum. In order to demonstrate the effect of non-linearity on the hypothesis for minimum entropy production, Fig. 9 shows the relative difference between the total entropy production of the EoEP solution and the optimal solution as function of reactor length for mixed and purely convective heat fluxes. Fig. 9 shows that EoEP is a better approximation when the heat flux is purely convective than when we have radiation. This applies to all reactor lengths investigated. The first heat mechanism has a linear fluxeforce relationship, while the second is highly non-linear. The difference is relatively small, however, only 0.03e0.06%. So, the hypothesis obtains support for validity, also for non-linear heat transfer mechanisms. The distance of the reactor operation from equilibrium is important for how well EoEP fits the optimal solution. This distance can, to some degree, be measured by the average conversion in the reactor as quantified by the process intensity,

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Table 3 and the Figs. 4e10, we can discuss other aspects of the hypothesis. The hypothesis refers to degrees of freedom in the system. In the mathematical proof leading to EoEP [10,19], the number of control variables is equal to the number of driving forces; in the present case this number is the number of reactions plus 2, j þ 2. Clearly, we do not have the freedom to control this many forces, having only one control variable, Ta. The lack of possibilities for control is most visible near the start and end of the reactor. The freedom is here rather limited, due to inlet and outlet restrictions set by other process units. In these parts, EoEP is impossible with Ta as the only control variable. It is in this sense that the degrees of freedom are insufficient for EoEP. What is surprising to see, is that the system can adapt itself to EoEP as soon as it becomes more free away from the inlet and outlet.

6.6. Fig. 8 e The variation in total entropy production, ðdS=dtÞirr , with the length of the reactor, L. Mixed heat flux (solid lines) and purely convective heat flux (dashed lines). The vertical lines represent the cases with optimal length, Lopt : Lopt (1c) [ 15.06 and Lopt (2c) [ 9.15. Dxj =L, where Dxj ¼ xj;out  xj;in and L is the length of the reactor. The nearness of the EoEP solution to the optimal solution becomes worse, as the process intensity increases, but again the deviations are rather small. This is illustrated for the total entropy production in Fig. 9 and the temperature profiles in Fig. 10. The same phenomenon was observed for the SO2reactor and for distillation columns [17]. Johannessen and Røsjordet noticed that EoEP performed worse as the available heat transfer area in the distillation column decreased.

6.5.

The degrees of freedom in the system

Having established that EoEP, and also EoF, are good approximations to the state of minimum entropy production, by

Fig. 9 e The relative difference, D%ðdS=dtÞirr , between the total entropy production of the EoEP solution and the optimal solution as function of reactor length for mixed (solid line) and purely convective heat flux (dashed line).

Guidelines for energy efficient design

The above analysis has shown that the present state-of-art reformer technology, the GHR, operates with an entropy production very close to the optimal obtainable one, for given feed and product conditions. The GHR and the optimal entropy production differ by only 1.6%. The result is not surprising, considering that the reformer reactor for hydrogen production represents mature technology, developed over several decades [33]. The fact that the results make sense from a perspective of being developed, gives credence to the ability of optimal control theory to find the minimum of the system’s total entropy production. We can trust the tool when applied to new reactors, or other technologies less mature. Also, we may trust that the optimal solution that we are seeking has the same characteristic properties as the optimal reformers have. The tool was earlier used to study exothermic reactions like the ammonia reaction, the methanol reaction and the sulphur-dioxide oxidation [8e11]. The accumulated experience can now be taken advantage of.

Fig. 10 e Three different reactor lengths and optimal Taprofiles (solid lines) and Ta-profiles of the EoEP solutions (dashed lines). Upper solution has L [ 5.93 m, middle has L [ 7.93 m and lower solution has L [ 15.93 m.

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The amount of heat to be transferred and the heat transfer coefficient in an endothermic or an exothermic reactor are vital for its entropy production. We saw the importance of the mechanism of heat transfer by comparing the Cases 1a and 2a. Heat transfer coefficients are therefore central when it comes to reducing the system’s entropy production. In some respects this fact is similar to the statement made by Leites et al. [12] in their first commandment: The driving force of a process must approach zero at all points in a reactor, at all times. A thermal driving force can be made small, by increasing the heat transfer coefficient. The interesting question beyond that becomes; what can be done, once the heat transfer coefficients have been maximized? In all optimal reactor solutions presented here, and earlier [8e11], we have seen a picture emerge; the optimal solution enters first a reaction mode at the inlet, before it proceeds into a heat transfer mode of operation in the central part (see Fig. 5). It follows for single tubular reactors of length L, that a (close to) adiabatic inlet section L1, is an advantage for the total entropy production. Furthermore, the next part, L2, can best be characterized by equipartition of the entropy production, in some cases also by equipartition of the forces. In other words, to find the optimal solution for a system translates into a procedure where one considers a scheme with separate units, like that illustrated in Fig. 11. The reactor part of the system consists of two subunits, an adiabatic pre-reactor and a tubular reactor with heat transfer. To complete the system analysis, a heat exchanger is added in front of the adiabatic reactor as in Scheme 1. This system can now be used to account for the trade-off exemplified by the Cases 1a and 1b, including also the contribution to the entropy production from heat exchange upfront of the reactor system. To be more specific, consider also Scheme 1 in Fig. 11 in connection with Fig. 5. The purpose of the heat exchanger (the first item in Scheme 1) is to bring the reacting mixture to the initial temperatures set on the y-axis of Fig. 5. The purpose of the adiabatic reactor, the next unit, is to bring the reaction to the point where the temperature profiles start to become parallel. The tubular reactor should have these temperatures as inlet conditions. Whether it pays, in terms of entropy production, to use Scheme 1 with the reactor in the heat transfer mode of operation, or to transfer to more discrete units as illustrated in Scheme 2 in Fig. 11 depends on the relative values of the heat transfer coefficients. When the heat transfer coefficients across the reactor tube wall are very low, it is better to use dedicated heat exchangers for heat transfer. It will then pay to split the operation in heat

Fig. 11 e Process set-ups for energy efficient reactor design. H.ex denotes the heat exchanger while Rct. is the chemical reactor. A denotes adiabatic.

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transfer mode (taken care of by the tubular reactor in Scheme 1) by separate sets of one or more adiabatic reactor stages with interstage heating/cooling, as illustrated in the lower part of Fig. 11. Two or more heat exchanger-adiabatic reactor pairs may be cost effective and energy effective as well. An optimization of this kind of system was carried out by de Koeijer et al. [7]. Their SO2 converter had four catalytic beds (lengths of type L1) and five external heat exchangers (lengths of type L2). The optimization yielded the lengths L1 of the catalytic beds and boundary conditions for the heat exchangers. Also in this case, the optimization becomes more straight forward, when the process units can be identified. This shows that a complex optimal control problem can be reduced, if not avoided, and that the process of finding an energy efficient reactor design can be simplified significantly. We have seen that a practical approximation of the optimal design can be found as a combination of heat exchangers, adiabatic reactor sections and heat exchanging tubular reactor sections. In such a combination of process units, the decision variables in the optimization problem are significantly fewer than in the complete system. The variables amount to inlet temperatures/pressures, the size of heat exchangers, reactor stages, and similar variables that easily can be changed in a design phase. Moreover, these variables can be related more easily to other objective functions than the one given by the Ta-profile. This means that a combined assessment of energy efficiency, economics, increased production (see [4]), and other criteria becomes more feasible. In their first commandment, Leites and coworkers [12] stated that equipartition of forces should be preferred at all points in a chemical reactor from an energy efficiency point of view. From the above analysis, we now see that this applies only to reactors that already operate in the heat transfer mode. Since our model includes rates, we have been able to obtain a more detailed picture. Unlike Leites et al. [12] we have found that the optimal solution is characterized by an adiabatic inlet section. Their third and forth commandment on exothermic and endothermic reactions should therefore be adjusted accordingly, to include adiabatic inlet sections in a search for systems with a smaller total entropy production.

7.

Conclusion

We have investigated the state of minimum entropy production for energy efficient operation of tubular reactors, and arrived at several conclusions, using a gas heated reformer reactor as a reference case.  The gas heated reformer can be said to operate with an energy efficiency as near to the optimal one as it is possible to come in practice given the boundary conditions.  The results support the hypothesis for the state of minimum entropy production, meaning that Equipartition of Entropy Production (EoEP) and also in some cases Equipartition of Forces (EoF) are good approximations to the optimal solution in several plug flow reformer cases.  The EoEP approximation becomes better, the more linear the transport processes become, as tested by varying radiative heat transfer, and varying the process intensity. EoEP

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however, is still a good approximation in the presence of non-linear phenomena, again in agreement with earlier findings.  It follows that energy efficient reactor design can be obtained by proper combinations of adiabatic pre-reactor(s) with dedicated heat exchangers and/or tubular reactors operating in the heat transfer mode. Heat transfer should in both cases obey equipartition of the entropy production.  This set of process units with their boundary conditions provide a good basis for further optimization of process variables, without the need to elaborate tools like control theory.

Acknowledgements Øivind Wilhelmsen would like to thank Margrete H. Wesenberg for valuable input on the GHR-technology.

Appendix A. The constraint of the optimal control problem

1

All the cases in this work have a feed gas composition according to the top second column of Table B 5, hydrogen outlet flow according to the top third column and geometry and material related parameters according to Table B 4 unless some variable is free according to Table 2. Results from the reference case (the GHR) are shown in Table B 4. They show that the outlet pressure, temperature and hydrogen conversion of the reference case is similar to that in a gas heated reformer [21]. The bottom columns of Table B 4 show the inlet and outlet conditions of the heating gas in the reference case.

Description

Value

Tube length, L Inner tube radius, R1 Outer tube radius, R2 Catalyst pellet radius, Rs Wall thermal conductivity, kwall Effectiveness factors, hj

A2

Table B 5 Inlet gas composition, temperatures and pressures used in all cases (top column 2). Packed bed outlet of the Case GHR (top column 3). Conditions of the GHR heating gas inlet and outlet (bottom columns 2 and 3).

The algebraic constraint of the optimal control problem gives:   vJq vH 1 1 lT , ¼ 00  þP vTa vTa T Ta i Fi Cp;i 1  þ 2 Jq ¼ 0 Ta



A3

Here, Jq is given by Equation 11, and the derivative with respect to Ta is:    vJq R2 vTw2 vTw2 þ hw2 1  ¼ , crad sr 4T3a  4T3w2 , vTa R1 vTa vTa

A4

Temperature, [ C] Pressure, [bar] Mole fraction CH4 Mole fraction CO2 Mole fraction CO Mole fraction H2O Mole fraction H2 Mole fraction N2 Molar flow, [kmole/h]

The derivative vTw2 =vTa is found by an implicit derivation of Equation A 1. The result is: vTw2 hw2 þ 4crad sr T3a ¼ 1 vTa hw2 þ 4crad sr T3w2 þ R1 hw1 aðR2 ða þ hw1 ÞÞ

12.93 m 0.045 m 0.057 m 3.63 103 m 100 W/mK 0.09

A1

Here, the constant a is given by: a ¼ kw ðR1 lnðR2 =R1 ÞÞ

Appendix B. The simulations

Table B.4 Geometrical, and material related parameters used in all the cases [21].

With the heat flux displayed in Equation 11, Equation 26 only gives Ta implicitly as function of the state variables and the multiplier functions. An analytic expression for the implicit relation is derived in this appendix. Tw2 is neither a state variable as T, nor a control variable like Ta. Tw2 is however, a function of both T and Ta. The relation between T, Ta and Tw2 is found by combining Equation 12 and 14, which are the radial energy balances at R1 and R2. This gives:    R1 hw1 aTw2 þ hw1 T hw2 ðTa  Tw2 Þ þ crad sr T4a  T4w2 ¼ , T R2 a þ hw1

The analytical and numerical derivations were found to give the same value to a relative accuracy of approximately 106, which was the accuracy of the numerical derivation.



A5

The Equations (11), and A 3eA 5 gave an implicit relation for Ta which was solved for every iteration of the boundary value solver bvp4c. Ta was extracted from the implicit relation with the root-solver in Matlab 7.9 together with the script fsolve. The sceptical reader should note that the performance of Equations A 3eA 5 was compared to numerical derivations.

Temperature, [ C] Pressure, [bar] Mole fraction CH4 Mole fraction CO2 Mole fraction CO Mole fraction H2O Mole fraction H2 Mole fraction N2 Molar flow, [kmole/h]

Packed bed (inlet)

Packed bed (GHR, outlet)

400 40.0 0.290 0.041 0.001 0.657 0.005 0.007 23.0

693 37.1 0.178 0.078 0.028 0.451 0.259 0.006 26.8

Heating section (GHR, inlet)

Heating section (GHR, outlet)

1050 38.7 0.0008 0.072 0.113 0.451 0.359 0.004 40.9

650 38.7 0.0008 0.072 0.113 0.451 0.359 0.004 40.9

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