- Email: [email protected]
Dimensioning nomograms for the design of fixed-bed solid-gas thermochemical reactors with various geometrical configurations
ELSEVIER
Chemical Engineeringand Processing 36 (1997) 45-58
Chemical Enghyr mg Processing
Dimensioning nomograms for the design of tied-bed solid-gas thermochemical reactors with various geometrical configurations D. Stitou, G. Crozat CNRS, Instittrf de Science et de Ginnie des Matfriaux Scierzce et Inghierie des ~~at~riaux et Pro&d&
et Pro&d& (IMP), (SIMAP), UniversitP,
Lnboratoire Europeen Associd, 66860 Perpignan, France
Received30 March 1996;accepted21June1996
Abstract
The development of solid-gas thermochemicaltransformers requires the design and the construction of demonstration machineswhich could not exist without simulation models,allowing the dimensioningof fixed-bed solid-gas reactors.All of the modelsdevelopedso far are basedon a dynamic modelling of the processdescribingthe temporal evolution of the principal elementsof which the machine consists.This approach to dimensioningis certainly better adapted to the discontinuousand unstationary working mode of thesemachines.However, the dimensioningof the machine becomesa very complex and time consumingoperation. With the aim of simplifying this procedure,somedimensioningnomogramshave beendevelopedin this study. They will enable any designerto draft a rapid dimensioningof a fixed-bed solid-gas reactor, thus facilitating decisionmaking at the early stages of a project. These nomogramshave been set up from a simplified analytical model which considersthe averagefunctional characteristicsof a thermochemicalreactor over a given time step. Finally, several applied examplesare given to demonstratethe various possibilities,as well as the simple usageof these dimensioningnomograms.Copyright 0 1996Elsevier ScienceS.A. Keywords:
Average thermal power; Dimensioning nomograms; Fixed-bed solid-gas reactor; Thermochemical
1. Introduction
The development of sorption processes fits into the research field of new technologies for cold production or heat storage. These new processes are nowadays favourable for their development for reasons related to the use of inert fluids harmless to the ozone layer [1,2]. Industrialists are willing to promote these systems [3,4]. Thermochemical systems are based on the thermicity of reversible reactions between a salt and a gas. Typical reactants are alkaline-earth or transition metal halogenides that react with ammonia or one of its derivatives (methylamine):
The functioning principle of those sytems is based on the coupling of two elements, a tied-bed reactor in which a monovariant chemical reaction takes place allowing the synthesis of the salt S2 (or its decomposi0255-2701/95/$15.00 0 1996- ElsevierScienceS.A. All rightsreserved PII SO255-2701(96)04172-4
transformers
tion), and an evaporator that constitutes the source of the reactive gas (or a condenser collecting the gas produced). Recent research on thermochemical solid-gas-transformers has allowed a better understanding of the mechanisms involved in processes within fixed-bed reactors (heat and mass transfers coupled to chemical reaction kinetics). This has brought out a development of new reactive media [5] allowing an improvement in the heat and mass transfer conditions of such reactors. In order to realize these reactions, the reactor is submitted to a double constraint, a pressure P, and temperature T,, thus defining a functioning point out of the thermodynamic equilibrium of the reaction. These constraints are essential to overcome the various resistances to the heat and mass transfer taking place in fixed-bed reactors. In general, the rate of the liquid-to-vapor state change in the evaporator/condenser is higher than the
46
D. Stitou,
G, Crozat
/Chemical
Engineering
rate of the chemical reaction occurring in the fixed-bed reactor. Thus, the constraint pressure imposed on the reactor is the pressure that prevails within the evaporator/condenser, which is assumed to be oversized and non-limiting. The constraint in temperature is characterized by a temperature drop (AT,, = T, - T,, (I’,)) defined according to the thermodynamic equilibrium temperature of the reaction that corresponds to the constraint pressure tied by the evaporator/condenser. This equilibrium drop defines the type of reaction being produced in the reactor. When it is positive, a decomposition reaction occurs. Inversely, a synthesis reaction takes place when it is negative. This equilibrium drop constitutes an essential operational parameter and has a direct influence on the reactor’s performance. As it is a driving force for the global transformation rate of the thermochemical transformer, it acts directly on the thermal power extracted from (during the synthesis phase) or supplied (during the decomposition phase) to the reactor [6,7] (Fig. 1). The basic cycle of such thermochemical systems can be easily represented in the Clapeyron’s diagram (Fig. 2). It is a discontinuous and transient cycle functioning between three temperature levels. It consists of two main stages during which the synthesis and decomposition reactions are produced. Two intermediate stages are necessary in order to set up conditions for the reactor to react (pressurization and depressurization phases). The various models of transformation of Lxed-bed solid-gas reactors developed so far are based on the classical laws of energy and mass conservation coupled with a chemical kinetic law. The degree of complexity among these various models is characterized by the simplifying hypothesis and by the nature of the reference volume to which these laws of conservation are applied: Paver W) 4
25 ZOISlo5-
o!’ 0
\ 10
20
30
40 ATeq (“C)
Fig. 1. Influence of the equilibrium temperature drop on the average thermal power of the reactor. Experimental values obtained on a pilot plant with a power of 20-50 kW 171.
and Processing
36 (1997)
45-58
- The ‘local’ models consider local and uniform
variables on a small element of volume. They end in a numerical resolution of a set of governing equations with partial derivatives that need a discretization of the problem in space and time [8,9-j. - The ‘global’ models consider uniform variables, but in a reference volume in scale with the reactor. The quantities characterizing the global transformation of the reactor are averaged in the whole volume. This kind of model leads also to a numerical resolution, but of a set of differential equations that need only a discretization in time of the problem [lO,ll]. - The ‘analytical’ model, which has recently been developed (121, takes into consideration the average variables during the reaction time. By relying on the local laws of energy and mass conservation, this model results in the resolution of a set of differential equations where the characteristic variables are only related to the space variable. By considering the hypothesis of an absence of limitation by mass transfer in the reactive medium, the differential system is reduced to a single differential equation (heat transfer equation) whose resolution is analytic [12]. The obvious interest in this last model can be found in the dimensioning objective of a thermochemical reactor. Dimensioning consists of determining the size of the fixed-bed reactor able to carry out desired performances at a given time interval and under given working constraints. Actually, the thermal power of a reactor is mainly governed by several factors: - the working conditions (constraint pressure and thermal equilibrium drop) - the characteristic values of heat and mass transport within the solid reactive medium (effective conductivity, heat exchange coefficient between the wall of the reactor and the reactive medium, and permeability) - the geometry of the reactor (cylindrical, plane) - the structure of the heat exchange surface {external, internal or distributed to the core of the reactor). The actual procedure for the dimensioning of reactors requires dynamic simuIations of the process using either local or global models. From parameters initially fixed in an arbitrary way (geometry, characteristics of the medium, and constraints that are imposed on the reactor), one proceeds to the study of the dynamic response of the reactor (thermal power and global reaction extent). Usually, in order to achieve the desired dimensioning, it is then necessary to carry out several dynamic simulations by successive iterations of the parameters in order to reach the average thermal power wanted over a given time step. This methodology of dimensioning is more complex and sometimes leads to unnecessarily lengthy, time-consuming numerical calculations.
D. Stitou,
Lr
G. Crozat
/ Chemical
Engineering
and Processing
Equiiibrium S/G
36 (1997)
45-58
Decomoosition
phase
P cond
Pevap I/ t-l
W-&he&
ohase
Fig. 2. Representation in a Clapeyron’s diagram of the practical operating conditions and the thermodynamic trajectory followed by the fixed-bed reactor in its basic cycle in a cold production mode. The reactor is alternatively coupled to the evaporator during the synthesis phase and to the condenser during the decomposition phase. During the intermediate conditioning stages, the reactor is isolated. Heating (paths 3 and 4) or cooling (naths 1 and 2) of the isolated reactor induces the nressurization or the depressurization of the reactive medium, which follows its thermodynamic equilibrium line.
The latest model, developed in a previous publication [12], considerably simplifies the dimensioning procedure. Better adapted to this objective, it proposes a solution totally anaiytical to the problem and thus a rapid predimensioning of the reactor. In fact, thanks to its analytical form, the various parameters which influence the reactor’s performances are more easily linked (Fig. 3). Following with our previous study, the objective here is to develop dimensioning nomograms which are simple to use and well-adapted for fixed-bed solid-gas thermochemical reactors. These will be very helpful to any decision-maker or designer of such reactors. We propose, therefore, after a brief reminder of the basis of the simplified analytical model, to establish a general relationship between the average thermal power of the reactor and its geometry, thermal characteristics of the reactive medium, and working conditions for various types of heat exchanger structures (external, internal or distributed).
2. The basis of the analytical model
A solid-gas reactor may be considered globally as a double exchanger of heat and mass. The supply or extraction of heat is carried out by the use of an external thermal fluid. Considering that the wall of the heat exchanger is isothermal, it is possible to uncouple the reactor part from’the thermal fluid part. Only one half of the exchanger will be taken into account here: it occurs only in the reactor whose isothermal wall is at a constraint temperature T,. The study of the other half of the exchanger concerning the thermal fluid is assimilated to the classical case of a condenser/evaporator type exchanger and will not be developed in the present
work. This step will free the designer from the difficulties inherent to the choice of the structure of the fluid/reactor heat exchanger, the nature of the external thermal fluid, its mass flow, and its temperature level. The analytical model which has thus been developed takes into account the following hypothesis: - sources or sinks of heat and gaseous matter are supposed to be i&mite. From there, the working conditions (constraints in pressure PC and temperature TJ applied to the reactor will be considered constant; - the porosity of the reactive medium is sufficiently high to assume a high permeability to the reactive gas and, therefore, to consider that there is a uniform pressure in the reactor, as well as no limiting mass transfer; - the effective conductivity 2, in the reactive medium and the heat exchange coefficient H,, at the wall are assumed to be independent of the global extent of the reaction and temperature. In the context of this study, only the case of Axed-bed reactors whose geometry is cylindrical and within which is confined the reactive medium will be considered. An approximation, subsequently brought in, will show how this model can be adapted to reactors with a plane geometry. The objective here, is not to describe the development of the analytical model, which has already been presented in a recent study, but to show the essential points that brought about its formulation. The basis of this model relies on the resolution of the equation of heat (1) whose characteristic variables are local and averaged over a given period of time (reaction time t, corresponding to a global reaction extent AX& Boundary conditions, which are also averaged on the reaction time, are associated with this equation:
D. Stitou,
G. Crozat
/Chemical
Engineering
and Processing TherFal P
‘DYNAMIC MODEL
0
a
NUMERICAL RESOLUTION
36 (1997)
power
aver
’ tr q
Global reaction
I
extent
,n P aver
time *
A% tr
8
tr
time
r
thermal power Pavei *Geometry of the reactor *Characteristic of the reactive medium
45-58
Thermal equilibrium drop
ANALYTICAL
AT,
Fig. 3. Comparison of the dimensioning metholodologies of thermochemical reactors using either a dynamic model (a) or the analytical model (b).
(WL)) where P,,,, is the average linear thermal power (per meter of reactor) that is extracted from, or supplied to, the reactor during the reaction phase. This power takes into account both the thermal power absorbed or freed by the medium under a sensible form (P,,,,) and under a reactive form (P,,). The comparison between the molar heat capacity of the salt and the molar enthalpy of the reaction gives an estimation of the number {, defined by the ratio (P,,,,/P,,,,), of about 5%. The profile of local extents AX(r, tn) that corresponds to the global reaction extent AX, at the reaction time t, is calculated by the integration of Eq. (2) over the whole volume,and by considering a linear profile in the reactional zone whose slope is a function of the thermal characteristics of the reactive medium. A,=1
AX@, tR)d V VYs The averaged boundary conditions are adapted according to the form of the local extents profile obtained and according to the geometric configuration of the reactor. Thus the flux condition at the heat exchanger wall is: _
3 s ‘e exch
d(W))
(3) Far from the heat exchanger wall, the averaged temperature condition is a function of the equilibrium temperature of the salt at the constraint pressure imposed on the reactor:
=fKpcN
(4)
The geometric configuration usually adopted for fixed-bed reactors is cylindrical and can be mono-tubular or multi-tubular (Fig, 4). In a mono-tubular configuration, the heat exchange occurs on the outer periphery (external configuration) or the inner periphery (internal configuration), whereas for a multi-tubular configuration, the thermal exchange is distributed to the core. In order to model the multi-tubular reactor, each inner tube is considered as a micro-reactor in itself with a unitary thermal power Pavcr/Ntub.The study of this multi-tubular exchanger is then reduced to the study of N monotubular exchangers, such that their heat exchange takes place on the inner periphery. Having chosen the geometric configuration of the reactor, the integration of the heat Eq. (1) associated with the averaged boundary condition leads to an analytical relation linking: - the average thermal power of the reactor (P& per meter of reactor for a given global extent of the reaction (AX& - the temperature drop AT,, in relation to the thermodynamic equilibrium of the reaction under a given constraint pressure P,; - the characteristic values of the heat transfer within the reactive medium (effective conductivity 2, and heat exchange coefficient at the wall H,,); - the reactor’s geometry (external radius R, and internal radius Rtub, number of inner tubes N&L Thus, for a mono-tubular reactor where the heat exchange is external, the analytical model gives the following general formulation:
D. Stitou,
G. Crozar
/ Chemical
P aYer= @extWeq> A, Ax., ~extt pex3
Engineering
(5)
For a mono- or multi-tubular reactor with an internal heat exchange surface, the formulation is: paver = Ntub@int(ATeqs A, Axg, Yint, Pint) (6) The variables pext and Pint characterize the geometric configuration of the reactor and are defined according to the type of exchange by: (7)
The variables yext and Yint, characterizing the spread of the reactional zone into the reactive medium, are defined as:
Yext =
? H,,R,(;E-
p,.,)
and Yint =
k/K H,JU1
- Pin3
(8)
and Processing
36 (1997)
49
45-58
These last variables clearly determine the influence of the geometric and thermal factors of the reactor. When y tends to 0, the transformation of the reactor is bound to a radial movement of a narrow reactional zone denoting, typically, a limitation by conductive transfer in the reactive medium (influence of 1,). Conversely, when the form factor y increases to above unity, the transformation of the reactor extends over the whole reactor and, therefore, becomes more and more uniform. In this case, the absence of a typical reaction front is characteristic of a limitation by heat transfer at the wall (influence of Hsw). The analytical relations established by the model allow the performances of fixed-bed reactors to be anticipated satisfactorily. A comparison between the results of this model and those of a local and dynamic model [EL] was carried out under the same conditions and considering the same hypothesis for both models. That comparison .showed that the reactor’s average performance obtained with the dynamic model agreed well with that obtained with the analytical model. Furthermore, the analytical model was compared to the experimental results obtained in a pilot plant. It showed that the deviation between the simplified model and experience remains broadly acceptable for global reaction extents AX, from 0.2 to 0.8. The interest of the analytical formulation proposed by this model will become apparent. The relations determined for various geometric configurations lead themselves easily to establishing nomograms (or abaci) that allow for a realistic and rapid pre-dimensioning of faxed-bed solid-gas reactors. This will be a very useful tool for designers in the task of making choices during early projects.
fm3-nal heat exchange
internal heat exchange 1 inner tube
distributed heat exchange Ntub
inner tubes
Fig. 4. Reactors’ geometric configurations. The extraction or input of heat, which is necessary to the transformation of the reactive medium, can be achieved with an external thermal fluid by a heat exchange either externally, internally or distributed to the core.
3. Establishing nomograms for the dimensioning of reactors
The construction of dimensioning abaci for reactors is based on the analytical expressions (5) and (6) that result from the simplified model. By successively fixing values to the different variables, these expressions may be formulated in a new form which is better adapted for establishing nomograms. Three analytical functions are used in this new formulation which ensure the uncoupling of the constraints .imposed by the load specifications of the dimensioning: Paver= @We,,
Wea dA&
Y, PI>>
(9)
By varying the parameters, the bundles of curves obtained by these functions are then regrouped to form a characteristic nomogram for a given global extent of the reaction AX, (Fig. 5). These, dimensioning nomograms enable the determination not only of the reactor’s characteristics (average performance, geome-
D. Stitou,
G. Crazat
/ Chemical
Engineering
Paver
ATeq
Fig. 5. Principle diagram of the dimensioning nomogram of tied-bed solid-gas thermochemical reactors.
try, reactive medium), but also the evolution of these characteristics when one of the parameters is varied. They have been established for several global extents (0.4, 0.6, and 0.8) and in a coordinates system with the form factor g of the reactional front and the average linear thermal power per meter of reactor, either unitary or totally (Appendices A, B and C). Here there are three variables carried as parameters: - the geometric form factor y of the reactor - the effective conductivity /2, of the reactive medium - the operational constraint in temperature (equilibrium drop ATeg at a considered constraint pressure PC) The parameters used in the construction of these nomograms vary according to a scale of values which cover a large range of reactor configurations. They even offer the possibilty of extrapolating these results from reactors whose geometric configuration is plane (plate exchanger); this will be subsequently undertaken. Several applied examples will show the extreme simplicity of the use of these nomograms.
4. Performance comparison with an existing laboratory pilot reactor The comparison of average thermal powers, determined by the dimensioning nomograms developed in this study, has been carried out in several experiments on a laboratory pilot reactor [13]. The particularity of this experimental reactor resides in its way of supplying and extracting the heat from the reactor during reaction phases. The heat exchange is achieved by two toluene heat pipes. The cylindrical reactor, whose characteristics are presented in Fig. 6, is made up of internal and external heat pipes. In the synthesis phase, the extraction of heat from the exothermic reaction is carried out by the evaporator of
and Processing
36 (1997)
45-58
the external heat pipe, whereas the input of the necessary heat to the endothermic decomposition reaction is carried out by the condenser of the internal heat pipe. With the characteristics of this reactor, mentioned in Fig. 6, it is possible to calculate the various quantities intervening in the graphic determination of the average linear power of the reactor on the nomogram. In this configuration of a mono-tubular reactor, the geometric form factors, Pint and pext, are identical for both types of exchange (internal in decomposition and external in synthesis). It is similar for the form factors of the reaction front, l/int and yext. The calculation of these quantities p and y gives the values 0.33 and 0.55, respectively. Proceeding as indicated on the principle diagram of nomograms, one can read the average linear power of the reactor (on the nomogram established for a global reaction extent of 0.8), that is to say 3300 W m-i in synthesis and 2200 W m - ’ in decomposition. That is in a good concordance with the average power actually obtained on the experimental reactor (deviation lower than 10%). 5. Applied examples of nomograms Any technical decision-maker who conceives and creates a thermochemical transformer in order to fulftll a given objective has to make a choice while following technological constraints iixed by the load specifications. The dimensioning nomograms of fixed-bed solid-gas reactors renders the conceptor’s decisionmaking easier. In the conception of a thermochemical reactor, there are four groups of parameters: - specific parameters for the geometric conE.guration of the reactor, - parameters characterizing the reactional medium, - operational functioning constraints, - the reactor’s average performances. In general, the designer chases one objective to be obtained from these four groups. While following the conditions of two imposed groups, he will choose the fourth group of parameters to full?ll the fixed objective. Fig. 7 summarizes the different dimensioning cases with which a reactor designer may be confronted. According to the situation, for example, the designer will have to determine the temperature levels to be imposed in the reactor in order to fulfill an average power objective, with regard to a reactor with a fixed geometry and given thermal characteristics of a known reactive medium. Other problems may also appear and the use of dimensioning nomograms ensures a response to a certain number of cases. In this work and through several examples, it will be envisaged only all the cases of dimensioning relative to
D. Stitou,
G. Crozat
/ Chemical
Engineering
and Processing
36 (1997)
Geometric characteristics inner radius (mm) outer radius (mm) usable lenght (m) Thermal
characteristics of the reactive medium medium’s effective conductivity (W/m.K) heat exchange coefficient at the wall (W/m*.K)
Working l
51
45-58
16,8 50,2 0.62
13 700
constraints
In the synthesisphase (external exchange): equilibrium temperature of the salt (“C) constraint temperature(%)
120 90
* In the decompositionphase(infernal exchange): equilibrium temperature of the salt (“C) constraint temperature(%) Energetic performances (W ) average thermal power extracted from the reactor in synthesis average thermal power supplied to the reactor in decomposition
135 180
1700 1200
Fig. 6. Characteristics and performances (for AXp = 02) of the experimental reactor [13].
a performance objective (average power over a given time interval that gives a desired global extent of the reaction) will be used. The aim of the dimensioning will be to determine either the functionning conditions, or the geometric configuration of the reactor, or even the nature of the reactive medium to be used. 5.1. Determination of the working conditions of a thermochemical reactor
With this type of dimensioning
Average level of thermal power over a given reaction
problem,
the aim is to
Constraint in pressure and
CONTRAINTS UNKNOWN
T O BE
OBJECTIVE
Fig. 7. Descriptive diagram of the different cases of dimensioning of a fixed-bed solid-gas thermochemical reactor.
determine the constraint temperature level that must be imposed on a reactor with a given geometric configuration, and within which is confined a perfectly characterized reactive medium, in order to fulfill a desired average level of thermal power. To illustrate this, let us consider a reactor with a cylindrical geometry with a 15 cm outer radius. The input or the extraction of heat is realized on the external periphery of the reactor. The arrival or departure of the reactive gas occurs through a 2 cm radius central hole whose role is to facilitate the diffusion of the reactive gas into the reactive medium. The chosen reactive medium is characterized by a large gas permeability, an effective conductivity 1, of 10 W m - l K - l and a thermal contact coefficient I&, of 300 W me2 K-’ at the exchanger wall. The problem is the following: what is the temperature difference ATeg of the thermodynamic equilibrium of the reaction enabling the reactor to ensure an average linear power of 5 kW m-l during a reaction time corresponding to a global reaction extent AX, of 0.8? The use of a nomogram established for a global extent of 0.8 gives an instantaneous response to the question by a simple graph reading. However, in order to make this reading possible, the various parameters intervening in these nomograms must be determined, notably, form factors p and y. According to their definitions in Eqs. (7) and (8), the values of p and y are, respectively, 0.13 and 0.25 for this configuration of the reactor.
52
D. Stitolr,
External
G. Crozat
/ Chemical
Engineering
Exchanae
Fig. 8. Principle diagram for the determination conditions.
of the working
The method of graphic reading illustrated in Fig. 8, determines the equilibrium temperature drop ATes to which the reactor must be submitted. This value 45°C in this case. The constraint temperature T, will then be determined if the constraint pressure submitted to the reactor is known. This pressure is iixed by the functioning conditions of the evaporator or the condenser connected to the reactor both in the synthesis and decomposition phases. 5.2. Determination of the geometrical configuration of a reactor
The problem here consists of the determination of the best adapted reactor’s geometry to fulfill a fixed power objective for some given functioning conditions (T, and P,) and given thermal characteristics of the reactive medium. For example, the constraints in temperature and pressure (T, and P,) are such that they induce a temperature drop relative to the thermodynamic equilibrium of the reaction (AT,, = T, - T,,(P,)) of 40°C. The reactive medium is supposed to have a high permeability to the reactive gas and is characterized by an effective conductivity 2, of 10 W m - 1 K-r and a contact coefficient H,, at the exchanger wall of 300 W m-2 K-r. Considering several types of exchange with the external thermal fluid (external or internal heat exchange or distributed heat exchange to the core with five inner tubes), the aim is to determine the reactor’s geometry, which enables an average linear thermal power of 5000 W m-l to be obtained. The method of ,graphic reading explained in Fig. 9, determines the couples (p, y) which will satisfy all the constraints inherent to the reactor. These couples are,
and Processing
36 (1997)
45-58
in fact, for a known reactive medium, characteristics of the reactor’s geometric configuration. On refering to the Eqs. (7) and (8) for these variables, it is possible to calculate the outer radius of the reactor and that of the inner tube or tubes. The designer then chases the geometry which is the most suitable to the possible supplementary technical constraints (reaction time, cycling time, external thermal fluid, cost, . ..) which have not been taken into consideration in this study. For example, if the supplementary technical constraints of having an external radius of 15 cm is imposed on the reactor, then the project designer will have to choose in order to fulhll the objective of an average power level and according to the type of exchange retained: - a 5 cm inner tube for an external exchange - a 11 cm inner tube for an internal exchange - 5 x 2 cm inner tubes for an exchange distributed to the core The reactor’s geometry is thus totally determined. 5.3. Characterization of the reactive medium to be used
Decision-makers may often be faced with another type of problem: the choice of the reactive medium to be used in a reactor with a defined geometry and given working conditions. The difficulty then resides in the determination of the thermal characteristics of the reactive medium that enables the obtention of a desired average level of thermal power. Consider, for example, a cylindrical reactor with a 10 cm outer radius and a 1 cm inner radius, This reactor will exchange heat with the outer environment through a thermal fluid at its external periphery. The working conditions (T, and P,) imposed on the reactor bring about a temperature drop AT,, to the thermodynamic equilibrium of 40°C. Through this example, we propose to characterize, thermally, the reactive medium which ensures an average linear thermal power of 3000 W m -1 . This characterization is carried out through a graphic reading whose principle is detailed in Fig. 10. This enables a research of couples (A,, r) satisfying the constraints of the given problem. These couples thus characterize the thermal parameters of the reactive medium to be used for a known reactor’s geometry and for the given constraints. From the equation ‘for the form factor Eq. (8) y, the contact coefficient H,, at the exchanger wall can be calculated for each effective conductivity 2, considered. The designer in charge of the reactor’s conception is then able to choose a reactive medium with a couple of appropriate thermal parameters (I,, H,,J that will enable him, on addition, to be free of limitations by mass transfer (that is to say a medium with a high permeability to reactive gas).
il.
Stitou,
G. Crozat
/ Chemical
Engineering
36 (1997)
53
45-58
Internal or Distributed
External ___.._ _..-_ - .___... -_. -Exchanae 1
hXq=O,B
and Processing
Exchange ] AX$l=O,8
L
Outer radius of the reactor (cm)
Y 10
j
/
A Exchange distributed
I
to
1
OrI 0,O
0,l
0,2
0,3
0,4
0,5
O,6
0,7
0,8
0,9
0
2
4
6
8
10
12
14
16
18
20
Radius of the internal tubes (cm)
P
Fig. 9. Principle diagrams of the nomogram in order to determine the geometric configuration of a reactor. The couples (p and 71; thus determined for each type of heat exchange, enable the calculation of radii (reactor’s external radius and the radius of the inner tube or tubes).
6. Extrapolation to reactors with a plane geometry
For several reasons, linked especially to the formation of the reactive medium or to the search of compact reactors, a designer could be interested in reactors with a plane geometry whose principle resembles that of plate exchangers. With a modular structure, this type of reactor is a result of the adaptation of a fluid/fluid plate exchanger where one of the fluids is replaced by a layer of reactive solid. It consists of modules piled one upon each other, and composed of: - a layer of solid reactive of e thickness, - a heat exchanger plate in which flows the external thermal fluid, - a layer of solid reactive of’e thickness. Between each ,elementary module, a diffuser plate is installed in order to facilitate the diffusion of the reactive gas into the reactive medium (Fig. 11).
Here again, the use of dimensioning nomograms is of great use to the designer in order to predict satisfactorily the performances of a reactor with such a geometric configuration. In order to adapt the nomograms to this type of reactor, we are only interested in half of an elementary module, consisting of the exchanger wall and the layer of reactive solid. This may then be considered as the result of a cylindrical reactor, the interior of which is a thin layer of reactive solid compared with the outer radius (Fig. 12). This geometric approximation, which is currently accepted in heat transfer, is very satisfactory when the geometric form factor p is close to unity. In order to remain within the context of this approximation, a value of 0.95 will be chosen for ~this factor. The outer radius of the equivalent cylindrical reactor is then defined by Eq. (10):
Req=L 1-P
D. Stirou,
G. Crozat
/Chemical
Engineering
Form factor y
and Processing
He:oychange i
36 (1997)
45-58
coefficient
Hsw at the wall (W.m2 K-l )
t
500 -
100 ,)
0
5
10
15
20
25
Effective conductivity he (W.m-’ K-‘)
0
5
10
15
20
25
Effective conductivity he (W.m-’ K*’ )
Fig. 10. Principle diagram of the nomogram used for the determination of the thermal characteristics of the reactive medium to be used. The couples (A=,,y) determined on the graph ensure access to the appropriate reactive medium characterized thermally by a couple (A,, I&,).
If the thermal characteristics of the layer of reactive solid are known, then the equivalent outer radius (Eq. (10)) authorizes the calculation of the factor y and thus, through the nomogram, enables the access to the average linear power of the equivalent cylindrical reactor. A simple relation (Eq. (11)) then determines the average power per
Fig. 11. Geometric configuration of a plate reactor.
square meter of heat exchanger characterizing reactor:
the plate
Dlin
(11) In order to illustrate this point, consider a plate reactor with an average reaction power of 25 kW for a global reaction extent of 0.6, and try to determine the necessary heat exchanger area. The working conditions of this reactor lead to a temperature drop AT,, = 45°C. The reactive medium is thinly layered (8.5 mm) and is characterized by an effective conductivity of 1 W m - * K - i and a heat exchange coefficient at the wall of 70 W m - 2 K - l. The value of p being fixed at 0.95, the equivalent outer radius of the cylindrical reactor is equal to 0.17 m and the calculation of the y factor gives a value of 1.68. By a direct reading on the nomogram established for an extent of the reaction of 0.6, the average linear power of the equivalent cylindrical reactor is equal to 2600 W
D. Stitou,
G. Crozat
/ Chemical
Engineering
and Processing
36 (1997)
45-58
55
8. Nomenclature e
thickness of the layer of reactive solid in a plate reactor (m) heat exchange coefficient at the wall (W m-2
N tub PC
Paver
8, Rexch Fig. 12. The transformation of a plane geometry reactor into an equivalent cylindrical geometry reactor. This approximation is only possible when e CCR,,.
m - ‘. This linear power corresponds then to an average thermal power density for the plate reactor of 2430 W m - 2. At the end, the necessary exchange surface for this plate reactor in order to develop a global power of 25 kW should be 10.3 m2. The dimensioning of this plate reactor, obtained by the nomogram, is in fact very close to that of an industrial pilot plant already built and whose dimensioning has been carried out by dynamic simulations in identical conditions (Fig. 13).
RCO
R,,
R
Rx
Rtub
Sexch 43
Tb,
Trm
Th
TC
Teq ht
7. Conclusion
V
The dimensioning nomograms established and presented in the current study enable us fInally to consider every case of dimensioning of fixed-bed solid-gas thermochemical reactors. They can, in fact, be applied to reactors having very different geometric configurations, working with a large scale of operating conditions and within which a reactive medium is confined. This reactive medium can have a large variety of thermal characteristics, as long as, however, it does not lead to limitations by mass transfer. Through several examples we have shown the different possibilities of application of these dimensioning nomograms, as well as their simplicity and their flexibility of use. The rapidity of dimensioning of a solid-gas reactor constitutes one of the main interests of these nomograms. At the end, they represent a new helpful tool for assisting desision-making that will enable any technical designer to draft out quickly a realistic dimensioning of tixed-bed solid-gas thermochemical reactors.
Greek letters
K-1
>
number of inner tubes constraint pressure applied to the reactor (pa) average power supplied or extracted from the reactor at the level of the heat exchanger during the reaction phase (w) thermal power developed respectively under reactive and sensible form by the medium during the reaction phase (I+‘) radius (m) equivalent outer radius of a plate reactor (m) radius of the heat exchanger wall of the reactor radius of the wall far from the heat exchanger wall (m) reactor’s outer radius (m) reactor’s inner radius or that of the inner tubes (m) heat exchange surface (m”) reacton time corresponding to a given extent of the reaction (s) heat sources or sinks temperatures (K) constraint temperature applied to the reactor (K) equilibrium temperature of the reaction to the considered constraint pressure reaction time corresponding to a given extent of the reaction (s) Volume of the reactive medium (m3) Analytical functions that link the dimensioning constraints Number
defining the power ratio (Psens/
Preac)
Stoichiometric ratio of the chemical reaction Effective conductivity of the reactive medium (W m-l K-r) Geometric form factor of the reactor Form factor characterizing the reactional front protie within the reactor Temperature drop with regard to the thermodynamic equilibrium of the reaction at a considered pressure (K) Global extent of the reaction Local extent of the reaction at a considered time t,
D. Stitou,
G. Crozat
/ Chemical
Engineering
and Processing
36 (1997)
45-58 Positioning
wall -.
-..
,-..----,-/C,
Reactor’s ---\
CONDENSATION
10
--\_
20
-\---
t (ml
characteristics
of the pilot plant
Number of heat exchanger plates Number of gas diffuser plates Number of layer of solid reactive Thickness of the layers (mm) Plates diameter (mm) Total heat exchange surface (m*) Salt mass of CaCl2 (kg) Expanded graphite mass (kg) Mixture density (kg of mixture / m3)
20 21 40 815 565 10 14,2 4,7 230
Fig. 13. Diagram and characteristics of the industrial pilot plant reactor of 20-50 kW. Experimental powersextractedfrom the reactorandthe condenser during a decomposition phase obtained with a ATes= 45°C (according to [7]). Appendix A Pre-dimensioning nomogram for thermochemical reactors established for three types of heat exchanger s~ruc~ure
and a global extent of the-reaction
External
heat exchanae
AX, of 0.8.
mono-
or multi-tubular
D. Stitou,
Appendix
G. Crozat
/ Chemical
Engineering
and Processing
36 (1997)
45-58
57
3
Pre-dimensioning nomogram for thermochemical and a global extent of the reaction AX, of 0.6.
reactors established for three types of heat exchanger structure
mono- or multi-tubular Internal heat exchanae
Appendix C
Pre-dimensioning nomogram for thermochemical and a global extent of the reaction AX, of 0.4.
ial heat exchanae
reactors established for three types of heat exchanger structure
58
D. Stitou,
G. Crozat
/Chemical
Engineering
References [l] E. Merlin, Environmental protection: New challenge for technical competition and innovation, Proc. Symp. Solid Sorption Refrigeration, 18j20 November, Paris, France, 1992, pp. 21-23. [Z] F. Meunier, Solid sorption: an alternative to CFCs, J. Heat Recovery Syst. and CHP 13, 4 (1993) 289-295. [3] MSuzuki, Application of adsorption cooling systemsto automobiles, J. Heat Recovery Systems and CHP 13, 4 (1993) 335-340. [4] P. Picard, Sorption systems:the point of view of gas utility, J. Heat Recovery Systems and CHP 13, 4 (1993) 329-334. [5] S. Mauran, M. Lebrun, P. Prades, M. Moreau, B. Spinner, C. Drapier, Composite actif et pro&de de mise en oeuvre de processus physicochimiques gaz-solide ou liquide-gaz utilisant comme milieu reactionnel un tel composite, FR Patent 910303(April1991). [6] J. Vila, Influence du protocole d’ttablissement des contraintes thermodynamique sur les performances dun rkacteur solide-gaz de pompe g chaleur chimique, Ph.D. Thesis, University of Perpignan, France, 1992. [7] M. Lebrun, P. Neveu, Conception, simulation, dimensioning and testing of an experimental chemical heat pump, ASHRAE Trans., 98 (1992)
420-429.
and Processing
36 (1997)
45-58
[S] V. Goetz, A. Marty, A model for reversible solid-gas reactions submitted to temperature and pressure constraints, simulation of the rate of reaction in solid-gas reactors used in chemical heat pumps, C/rem. Eng. Sci., 47 (1992) 479-498. [9] N. Mazet, M. Amouroux, B. Spinner, Analysis and experimental study of the transformation of a non isothermal solid-gas reacting medium, Chem. Eng. Comm., 99 (1991) 155-174. [lo] M. Lebrun, B. Spinner, Simulation for the development of solid-gas chemical heat pump pilot plants Part 1: Simulation and dimensioning, Chem. Eng. Process., 28 (1990) 55-66. [I l] P. Neveu, J. Castaing, From materials to process: development of a numerical sizing tool applied to solid-gas thermochemical transformers-parts I and II, J. Heat Recovery Systems and CHP., (1996) submitted. [12] D. Stitou, V. Goetz, B. Spinner, A new analytical model for solid-gas thermochemical reactors based on thermophysical properties ofthefixed-bed reactivemedium, Cijem. Eng. Process., (1996) accepted. [13] A Wagner, Machine a froid h sorption solide-gaz a multiple effets g&r6 par caloduc: Mod&ation et validation experimental sur un pilote de 2 kW de production de froid, Ph.D. Thesis, University of Perpignan, France.
Chemical Engineeringand Processing 36 (1997) 45-58
Chemical Enghyr mg Processing
Dimensioning nomograms for the design of tied-bed solid-gas thermochemical reactors with various geometrical configurations D. Stitou, G. Crozat CNRS, Instittrf de Science et de Ginnie des Matfriaux Scierzce et Inghierie des ~~at~riaux et Pro&d&
et Pro&d& (IMP), (SIMAP), UniversitP,
Lnboratoire Europeen Associd, 66860 Perpignan, France
Received30 March 1996;accepted21June1996
Abstract
The development of solid-gas thermochemicaltransformers requires the design and the construction of demonstration machineswhich could not exist without simulation models,allowing the dimensioningof fixed-bed solid-gas reactors.All of the modelsdevelopedso far are basedon a dynamic modelling of the processdescribingthe temporal evolution of the principal elementsof which the machine consists.This approach to dimensioningis certainly better adapted to the discontinuousand unstationary working mode of thesemachines.However, the dimensioningof the machine becomesa very complex and time consumingoperation. With the aim of simplifying this procedure,somedimensioningnomogramshave beendevelopedin this study. They will enable any designerto draft a rapid dimensioningof a fixed-bed solid-gas reactor, thus facilitating decisionmaking at the early stages of a project. These nomogramshave been set up from a simplified analytical model which considersthe averagefunctional characteristicsof a thermochemicalreactor over a given time step. Finally, several applied examplesare given to demonstratethe various possibilities,as well as the simple usageof these dimensioningnomograms.Copyright 0 1996Elsevier ScienceS.A. Keywords:
Average thermal power; Dimensioning nomograms; Fixed-bed solid-gas reactor; Thermochemical
1. Introduction
The development of sorption processes fits into the research field of new technologies for cold production or heat storage. These new processes are nowadays favourable for their development for reasons related to the use of inert fluids harmless to the ozone layer [1,2]. Industrialists are willing to promote these systems [3,4]. Thermochemical systems are based on the thermicity of reversible reactions between a salt and a gas. Typical reactants are alkaline-earth or transition metal halogenides that react with ammonia or one of its derivatives (methylamine):
The functioning principle of those sytems is based on the coupling of two elements, a tied-bed reactor in which a monovariant chemical reaction takes place allowing the synthesis of the salt S2 (or its decomposi0255-2701/95/$15.00 0 1996- ElsevierScienceS.A. All rightsreserved PII SO255-2701(96)04172-4
transformers
tion), and an evaporator that constitutes the source of the reactive gas (or a condenser collecting the gas produced). Recent research on thermochemical solid-gas-transformers has allowed a better understanding of the mechanisms involved in processes within fixed-bed reactors (heat and mass transfers coupled to chemical reaction kinetics). This has brought out a development of new reactive media [5] allowing an improvement in the heat and mass transfer conditions of such reactors. In order to realize these reactions, the reactor is submitted to a double constraint, a pressure P, and temperature T,, thus defining a functioning point out of the thermodynamic equilibrium of the reaction. These constraints are essential to overcome the various resistances to the heat and mass transfer taking place in fixed-bed reactors. In general, the rate of the liquid-to-vapor state change in the evaporator/condenser is higher than the
46
D. Stitou,
G, Crozat
/Chemical
Engineering
rate of the chemical reaction occurring in the fixed-bed reactor. Thus, the constraint pressure imposed on the reactor is the pressure that prevails within the evaporator/condenser, which is assumed to be oversized and non-limiting. The constraint in temperature is characterized by a temperature drop (AT,, = T, - T,, (I’,)) defined according to the thermodynamic equilibrium temperature of the reaction that corresponds to the constraint pressure tied by the evaporator/condenser. This equilibrium drop defines the type of reaction being produced in the reactor. When it is positive, a decomposition reaction occurs. Inversely, a synthesis reaction takes place when it is negative. This equilibrium drop constitutes an essential operational parameter and has a direct influence on the reactor’s performance. As it is a driving force for the global transformation rate of the thermochemical transformer, it acts directly on the thermal power extracted from (during the synthesis phase) or supplied (during the decomposition phase) to the reactor [6,7] (Fig. 1). The basic cycle of such thermochemical systems can be easily represented in the Clapeyron’s diagram (Fig. 2). It is a discontinuous and transient cycle functioning between three temperature levels. It consists of two main stages during which the synthesis and decomposition reactions are produced. Two intermediate stages are necessary in order to set up conditions for the reactor to react (pressurization and depressurization phases). The various models of transformation of Lxed-bed solid-gas reactors developed so far are based on the classical laws of energy and mass conservation coupled with a chemical kinetic law. The degree of complexity among these various models is characterized by the simplifying hypothesis and by the nature of the reference volume to which these laws of conservation are applied: Paver W) 4
25 ZOISlo5-
o!’ 0
\ 10
20
30
40 ATeq (“C)
Fig. 1. Influence of the equilibrium temperature drop on the average thermal power of the reactor. Experimental values obtained on a pilot plant with a power of 20-50 kW 171.
and Processing
36 (1997)
45-58
- The ‘local’ models consider local and uniform
variables on a small element of volume. They end in a numerical resolution of a set of governing equations with partial derivatives that need a discretization of the problem in space and time [8,9-j. - The ‘global’ models consider uniform variables, but in a reference volume in scale with the reactor. The quantities characterizing the global transformation of the reactor are averaged in the whole volume. This kind of model leads also to a numerical resolution, but of a set of differential equations that need only a discretization in time of the problem [lO,ll]. - The ‘analytical’ model, which has recently been developed (121, takes into consideration the average variables during the reaction time. By relying on the local laws of energy and mass conservation, this model results in the resolution of a set of differential equations where the characteristic variables are only related to the space variable. By considering the hypothesis of an absence of limitation by mass transfer in the reactive medium, the differential system is reduced to a single differential equation (heat transfer equation) whose resolution is analytic [12]. The obvious interest in this last model can be found in the dimensioning objective of a thermochemical reactor. Dimensioning consists of determining the size of the fixed-bed reactor able to carry out desired performances at a given time interval and under given working constraints. Actually, the thermal power of a reactor is mainly governed by several factors: - the working conditions (constraint pressure and thermal equilibrium drop) - the characteristic values of heat and mass transport within the solid reactive medium (effective conductivity, heat exchange coefficient between the wall of the reactor and the reactive medium, and permeability) - the geometry of the reactor (cylindrical, plane) - the structure of the heat exchange surface {external, internal or distributed to the core of the reactor). The actual procedure for the dimensioning of reactors requires dynamic simuIations of the process using either local or global models. From parameters initially fixed in an arbitrary way (geometry, characteristics of the medium, and constraints that are imposed on the reactor), one proceeds to the study of the dynamic response of the reactor (thermal power and global reaction extent). Usually, in order to achieve the desired dimensioning, it is then necessary to carry out several dynamic simulations by successive iterations of the parameters in order to reach the average thermal power wanted over a given time step. This methodology of dimensioning is more complex and sometimes leads to unnecessarily lengthy, time-consuming numerical calculations.
D. Stitou,
Lr
G. Crozat
/ Chemical
Engineering
and Processing
Equiiibrium S/G
36 (1997)
45-58
Decomoosition
phase
P cond
Pevap I/ t-l
W-&he&
ohase
Fig. 2. Representation in a Clapeyron’s diagram of the practical operating conditions and the thermodynamic trajectory followed by the fixed-bed reactor in its basic cycle in a cold production mode. The reactor is alternatively coupled to the evaporator during the synthesis phase and to the condenser during the decomposition phase. During the intermediate conditioning stages, the reactor is isolated. Heating (paths 3 and 4) or cooling (naths 1 and 2) of the isolated reactor induces the nressurization or the depressurization of the reactive medium, which follows its thermodynamic equilibrium line.
The latest model, developed in a previous publication [12], considerably simplifies the dimensioning procedure. Better adapted to this objective, it proposes a solution totally anaiytical to the problem and thus a rapid predimensioning of the reactor. In fact, thanks to its analytical form, the various parameters which influence the reactor’s performances are more easily linked (Fig. 3). Following with our previous study, the objective here is to develop dimensioning nomograms which are simple to use and well-adapted for fixed-bed solid-gas thermochemical reactors. These will be very helpful to any decision-maker or designer of such reactors. We propose, therefore, after a brief reminder of the basis of the simplified analytical model, to establish a general relationship between the average thermal power of the reactor and its geometry, thermal characteristics of the reactive medium, and working conditions for various types of heat exchanger structures (external, internal or distributed).
2. The basis of the analytical model
A solid-gas reactor may be considered globally as a double exchanger of heat and mass. The supply or extraction of heat is carried out by the use of an external thermal fluid. Considering that the wall of the heat exchanger is isothermal, it is possible to uncouple the reactor part from’the thermal fluid part. Only one half of the exchanger will be taken into account here: it occurs only in the reactor whose isothermal wall is at a constraint temperature T,. The study of the other half of the exchanger concerning the thermal fluid is assimilated to the classical case of a condenser/evaporator type exchanger and will not be developed in the present
work. This step will free the designer from the difficulties inherent to the choice of the structure of the fluid/reactor heat exchanger, the nature of the external thermal fluid, its mass flow, and its temperature level. The analytical model which has thus been developed takes into account the following hypothesis: - sources or sinks of heat and gaseous matter are supposed to be i&mite. From there, the working conditions (constraints in pressure PC and temperature TJ applied to the reactor will be considered constant; - the porosity of the reactive medium is sufficiently high to assume a high permeability to the reactive gas and, therefore, to consider that there is a uniform pressure in the reactor, as well as no limiting mass transfer; - the effective conductivity 2, in the reactive medium and the heat exchange coefficient H,, at the wall are assumed to be independent of the global extent of the reaction and temperature. In the context of this study, only the case of Axed-bed reactors whose geometry is cylindrical and within which is confined the reactive medium will be considered. An approximation, subsequently brought in, will show how this model can be adapted to reactors with a plane geometry. The objective here, is not to describe the development of the analytical model, which has already been presented in a recent study, but to show the essential points that brought about its formulation. The basis of this model relies on the resolution of the equation of heat (1) whose characteristic variables are local and averaged over a given period of time (reaction time t, corresponding to a global reaction extent AX& Boundary conditions, which are also averaged on the reaction time, are associated with this equation:
D. Stitou,
G. Crozat
/Chemical
Engineering
and Processing TherFal P
‘DYNAMIC MODEL
0
a
NUMERICAL RESOLUTION
36 (1997)
power
aver
’ tr q
Global reaction
I
extent
,n P aver
time *
A% tr
8
tr
time
r
thermal power Pavei *Geometry of the reactor *Characteristic of the reactive medium
45-58
Thermal equilibrium drop
ANALYTICAL
AT,
Fig. 3. Comparison of the dimensioning metholodologies of thermochemical reactors using either a dynamic model (a) or the analytical model (b).
(WL)) where P,,,, is the average linear thermal power (per meter of reactor) that is extracted from, or supplied to, the reactor during the reaction phase. This power takes into account both the thermal power absorbed or freed by the medium under a sensible form (P,,,,) and under a reactive form (P,,). The comparison between the molar heat capacity of the salt and the molar enthalpy of the reaction gives an estimation of the number {, defined by the ratio (P,,,,/P,,,,), of about 5%. The profile of local extents AX(r, tn) that corresponds to the global reaction extent AX, at the reaction time t, is calculated by the integration of Eq. (2) over the whole volume,and by considering a linear profile in the reactional zone whose slope is a function of the thermal characteristics of the reactive medium. A,=1
AX@, tR)d V VYs The averaged boundary conditions are adapted according to the form of the local extents profile obtained and according to the geometric configuration of the reactor. Thus the flux condition at the heat exchanger wall is: _
3 s ‘e exch
d(W))
(3) Far from the heat exchanger wall, the averaged temperature condition is a function of the equilibrium temperature of the salt at the constraint pressure imposed on the reactor:
=fKpcN
(4)
The geometric configuration usually adopted for fixed-bed reactors is cylindrical and can be mono-tubular or multi-tubular (Fig, 4). In a mono-tubular configuration, the heat exchange occurs on the outer periphery (external configuration) or the inner periphery (internal configuration), whereas for a multi-tubular configuration, the thermal exchange is distributed to the core. In order to model the multi-tubular reactor, each inner tube is considered as a micro-reactor in itself with a unitary thermal power Pavcr/Ntub.The study of this multi-tubular exchanger is then reduced to the study of N monotubular exchangers, such that their heat exchange takes place on the inner periphery. Having chosen the geometric configuration of the reactor, the integration of the heat Eq. (1) associated with the averaged boundary condition leads to an analytical relation linking: - the average thermal power of the reactor (P& per meter of reactor for a given global extent of the reaction (AX& - the temperature drop AT,, in relation to the thermodynamic equilibrium of the reaction under a given constraint pressure P,; - the characteristic values of the heat transfer within the reactive medium (effective conductivity 2, and heat exchange coefficient at the wall H,,); - the reactor’s geometry (external radius R, and internal radius Rtub, number of inner tubes N&L Thus, for a mono-tubular reactor where the heat exchange is external, the analytical model gives the following general formulation:
D. Stitou,
G. Crozar
/ Chemical
P aYer= @extWeq> A, Ax., ~extt pex3
Engineering
(5)
For a mono- or multi-tubular reactor with an internal heat exchange surface, the formulation is: paver = Ntub@int(ATeqs A, Axg, Yint, Pint) (6) The variables pext and Pint characterize the geometric configuration of the reactor and are defined according to the type of exchange by: (7)
The variables yext and Yint, characterizing the spread of the reactional zone into the reactive medium, are defined as:
Yext =
? H,,R,(;E-
p,.,)
and Yint =
k/K H,JU1
- Pin3
(8)
and Processing
36 (1997)
49
45-58
These last variables clearly determine the influence of the geometric and thermal factors of the reactor. When y tends to 0, the transformation of the reactor is bound to a radial movement of a narrow reactional zone denoting, typically, a limitation by conductive transfer in the reactive medium (influence of 1,). Conversely, when the form factor y increases to above unity, the transformation of the reactor extends over the whole reactor and, therefore, becomes more and more uniform. In this case, the absence of a typical reaction front is characteristic of a limitation by heat transfer at the wall (influence of Hsw). The analytical relations established by the model allow the performances of fixed-bed reactors to be anticipated satisfactorily. A comparison between the results of this model and those of a local and dynamic model [EL] was carried out under the same conditions and considering the same hypothesis for both models. That comparison .showed that the reactor’s average performance obtained with the dynamic model agreed well with that obtained with the analytical model. Furthermore, the analytical model was compared to the experimental results obtained in a pilot plant. It showed that the deviation between the simplified model and experience remains broadly acceptable for global reaction extents AX, from 0.2 to 0.8. The interest of the analytical formulation proposed by this model will become apparent. The relations determined for various geometric configurations lead themselves easily to establishing nomograms (or abaci) that allow for a realistic and rapid pre-dimensioning of faxed-bed solid-gas reactors. This will be a very useful tool for designers in the task of making choices during early projects.
fm3-nal heat exchange
internal heat exchange 1 inner tube
distributed heat exchange Ntub
inner tubes
Fig. 4. Reactors’ geometric configurations. The extraction or input of heat, which is necessary to the transformation of the reactive medium, can be achieved with an external thermal fluid by a heat exchange either externally, internally or distributed to the core.
3. Establishing nomograms for the dimensioning of reactors
The construction of dimensioning abaci for reactors is based on the analytical expressions (5) and (6) that result from the simplified model. By successively fixing values to the different variables, these expressions may be formulated in a new form which is better adapted for establishing nomograms. Three analytical functions are used in this new formulation which ensure the uncoupling of the constraints .imposed by the load specifications of the dimensioning: Paver= @We,,
Wea dA&
Y, PI>>
(9)
By varying the parameters, the bundles of curves obtained by these functions are then regrouped to form a characteristic nomogram for a given global extent of the reaction AX, (Fig. 5). These, dimensioning nomograms enable the determination not only of the reactor’s characteristics (average performance, geome-
D. Stitou,
G. Crazat
/ Chemical
Engineering
Paver
ATeq
Fig. 5. Principle diagram of the dimensioning nomogram of tied-bed solid-gas thermochemical reactors.
try, reactive medium), but also the evolution of these characteristics when one of the parameters is varied. They have been established for several global extents (0.4, 0.6, and 0.8) and in a coordinates system with the form factor g of the reactional front and the average linear thermal power per meter of reactor, either unitary or totally (Appendices A, B and C). Here there are three variables carried as parameters: - the geometric form factor y of the reactor - the effective conductivity /2, of the reactive medium - the operational constraint in temperature (equilibrium drop ATeg at a considered constraint pressure PC) The parameters used in the construction of these nomograms vary according to a scale of values which cover a large range of reactor configurations. They even offer the possibilty of extrapolating these results from reactors whose geometric configuration is plane (plate exchanger); this will be subsequently undertaken. Several applied examples will show the extreme simplicity of the use of these nomograms.
4. Performance comparison with an existing laboratory pilot reactor The comparison of average thermal powers, determined by the dimensioning nomograms developed in this study, has been carried out in several experiments on a laboratory pilot reactor [13]. The particularity of this experimental reactor resides in its way of supplying and extracting the heat from the reactor during reaction phases. The heat exchange is achieved by two toluene heat pipes. The cylindrical reactor, whose characteristics are presented in Fig. 6, is made up of internal and external heat pipes. In the synthesis phase, the extraction of heat from the exothermic reaction is carried out by the evaporator of
and Processing
36 (1997)
45-58
the external heat pipe, whereas the input of the necessary heat to the endothermic decomposition reaction is carried out by the condenser of the internal heat pipe. With the characteristics of this reactor, mentioned in Fig. 6, it is possible to calculate the various quantities intervening in the graphic determination of the average linear power of the reactor on the nomogram. In this configuration of a mono-tubular reactor, the geometric form factors, Pint and pext, are identical for both types of exchange (internal in decomposition and external in synthesis). It is similar for the form factors of the reaction front, l/int and yext. The calculation of these quantities p and y gives the values 0.33 and 0.55, respectively. Proceeding as indicated on the principle diagram of nomograms, one can read the average linear power of the reactor (on the nomogram established for a global reaction extent of 0.8), that is to say 3300 W m-i in synthesis and 2200 W m - ’ in decomposition. That is in a good concordance with the average power actually obtained on the experimental reactor (deviation lower than 10%). 5. Applied examples of nomograms Any technical decision-maker who conceives and creates a thermochemical transformer in order to fulftll a given objective has to make a choice while following technological constraints iixed by the load specifications. The dimensioning nomograms of fixed-bed solid-gas reactors renders the conceptor’s decisionmaking easier. In the conception of a thermochemical reactor, there are four groups of parameters: - specific parameters for the geometric conE.guration of the reactor, - parameters characterizing the reactional medium, - operational functioning constraints, - the reactor’s average performances. In general, the designer chases one objective to be obtained from these four groups. While following the conditions of two imposed groups, he will choose the fourth group of parameters to full?ll the fixed objective. Fig. 7 summarizes the different dimensioning cases with which a reactor designer may be confronted. According to the situation, for example, the designer will have to determine the temperature levels to be imposed in the reactor in order to fulfill an average power objective, with regard to a reactor with a fixed geometry and given thermal characteristics of a known reactive medium. Other problems may also appear and the use of dimensioning nomograms ensures a response to a certain number of cases. In this work and through several examples, it will be envisaged only all the cases of dimensioning relative to
D. Stitou,
G. Crozat
/ Chemical
Engineering
and Processing
36 (1997)
Geometric characteristics inner radius (mm) outer radius (mm) usable lenght (m) Thermal
characteristics of the reactive medium medium’s effective conductivity (W/m.K) heat exchange coefficient at the wall (W/m*.K)
Working l
51
45-58
16,8 50,2 0.62
13 700
constraints
In the synthesisphase (external exchange): equilibrium temperature of the salt (“C) constraint temperature(%)
120 90
* In the decompositionphase(infernal exchange): equilibrium temperature of the salt (“C) constraint temperature(%) Energetic performances (W ) average thermal power extracted from the reactor in synthesis average thermal power supplied to the reactor in decomposition
135 180
1700 1200
Fig. 6. Characteristics and performances (for AXp = 02) of the experimental reactor [13].
a performance objective (average power over a given time interval that gives a desired global extent of the reaction) will be used. The aim of the dimensioning will be to determine either the functionning conditions, or the geometric configuration of the reactor, or even the nature of the reactive medium to be used. 5.1. Determination of the working conditions of a thermochemical reactor
With this type of dimensioning
Average level of thermal power over a given reaction
problem,
the aim is to
Constraint in pressure and
CONTRAINTS UNKNOWN
T O BE
OBJECTIVE
Fig. 7. Descriptive diagram of the different cases of dimensioning of a fixed-bed solid-gas thermochemical reactor.
determine the constraint temperature level that must be imposed on a reactor with a given geometric configuration, and within which is confined a perfectly characterized reactive medium, in order to fulfill a desired average level of thermal power. To illustrate this, let us consider a reactor with a cylindrical geometry with a 15 cm outer radius. The input or the extraction of heat is realized on the external periphery of the reactor. The arrival or departure of the reactive gas occurs through a 2 cm radius central hole whose role is to facilitate the diffusion of the reactive gas into the reactive medium. The chosen reactive medium is characterized by a large gas permeability, an effective conductivity 1, of 10 W m - l K - l and a thermal contact coefficient I&, of 300 W me2 K-’ at the exchanger wall. The problem is the following: what is the temperature difference ATeg of the thermodynamic equilibrium of the reaction enabling the reactor to ensure an average linear power of 5 kW m-l during a reaction time corresponding to a global reaction extent AX, of 0.8? The use of a nomogram established for a global extent of 0.8 gives an instantaneous response to the question by a simple graph reading. However, in order to make this reading possible, the various parameters intervening in these nomograms must be determined, notably, form factors p and y. According to their definitions in Eqs. (7) and (8), the values of p and y are, respectively, 0.13 and 0.25 for this configuration of the reactor.
52
D. Stitolr,
External
G. Crozat
/ Chemical
Engineering
Exchanae
Fig. 8. Principle diagram for the determination conditions.
of the working
The method of graphic reading illustrated in Fig. 8, determines the equilibrium temperature drop ATes to which the reactor must be submitted. This value 45°C in this case. The constraint temperature T, will then be determined if the constraint pressure submitted to the reactor is known. This pressure is iixed by the functioning conditions of the evaporator or the condenser connected to the reactor both in the synthesis and decomposition phases. 5.2. Determination of the geometrical configuration of a reactor
The problem here consists of the determination of the best adapted reactor’s geometry to fulfill a fixed power objective for some given functioning conditions (T, and P,) and given thermal characteristics of the reactive medium. For example, the constraints in temperature and pressure (T, and P,) are such that they induce a temperature drop relative to the thermodynamic equilibrium of the reaction (AT,, = T, - T,,(P,)) of 40°C. The reactive medium is supposed to have a high permeability to the reactive gas and is characterized by an effective conductivity 2, of 10 W m - 1 K-r and a contact coefficient H,, at the exchanger wall of 300 W m-2 K-r. Considering several types of exchange with the external thermal fluid (external or internal heat exchange or distributed heat exchange to the core with five inner tubes), the aim is to determine the reactor’s geometry, which enables an average linear thermal power of 5000 W m-l to be obtained. The method of ,graphic reading explained in Fig. 9, determines the couples (p, y) which will satisfy all the constraints inherent to the reactor. These couples are,
and Processing
36 (1997)
45-58
in fact, for a known reactive medium, characteristics of the reactor’s geometric configuration. On refering to the Eqs. (7) and (8) for these variables, it is possible to calculate the outer radius of the reactor and that of the inner tube or tubes. The designer then chases the geometry which is the most suitable to the possible supplementary technical constraints (reaction time, cycling time, external thermal fluid, cost, . ..) which have not been taken into consideration in this study. For example, if the supplementary technical constraints of having an external radius of 15 cm is imposed on the reactor, then the project designer will have to choose in order to fulhll the objective of an average power level and according to the type of exchange retained: - a 5 cm inner tube for an external exchange - a 11 cm inner tube for an internal exchange - 5 x 2 cm inner tubes for an exchange distributed to the core The reactor’s geometry is thus totally determined. 5.3. Characterization of the reactive medium to be used
Decision-makers may often be faced with another type of problem: the choice of the reactive medium to be used in a reactor with a defined geometry and given working conditions. The difficulty then resides in the determination of the thermal characteristics of the reactive medium that enables the obtention of a desired average level of thermal power. Consider, for example, a cylindrical reactor with a 10 cm outer radius and a 1 cm inner radius, This reactor will exchange heat with the outer environment through a thermal fluid at its external periphery. The working conditions (T, and P,) imposed on the reactor bring about a temperature drop AT,, to the thermodynamic equilibrium of 40°C. Through this example, we propose to characterize, thermally, the reactive medium which ensures an average linear thermal power of 3000 W m -1 . This characterization is carried out through a graphic reading whose principle is detailed in Fig. 10. This enables a research of couples (A,, r) satisfying the constraints of the given problem. These couples thus characterize the thermal parameters of the reactive medium to be used for a known reactor’s geometry and for the given constraints. From the equation ‘for the form factor Eq. (8) y, the contact coefficient H,, at the exchanger wall can be calculated for each effective conductivity 2, considered. The designer in charge of the reactor’s conception is then able to choose a reactive medium with a couple of appropriate thermal parameters (I,, H,,J that will enable him, on addition, to be free of limitations by mass transfer (that is to say a medium with a high permeability to reactive gas).
il.
Stitou,
G. Crozat
/ Chemical
Engineering
36 (1997)
53
45-58
Internal or Distributed
External ___.._ _..-_ - .___... -_. -Exchanae 1
hXq=O,B
and Processing
Exchange ] AX$l=O,8
L
Outer radius of the reactor (cm)
Y 10
j
/
A Exchange distributed
I
to
1
OrI 0,O
0,l
0,2
0,3
0,4
0,5
O,6
0,7
0,8
0,9
0
2
4
6
8
10
12
14
16
18
20
Radius of the internal tubes (cm)
P
Fig. 9. Principle diagrams of the nomogram in order to determine the geometric configuration of a reactor. The couples (p and 71; thus determined for each type of heat exchange, enable the calculation of radii (reactor’s external radius and the radius of the inner tube or tubes).
6. Extrapolation to reactors with a plane geometry
For several reasons, linked especially to the formation of the reactive medium or to the search of compact reactors, a designer could be interested in reactors with a plane geometry whose principle resembles that of plate exchangers. With a modular structure, this type of reactor is a result of the adaptation of a fluid/fluid plate exchanger where one of the fluids is replaced by a layer of reactive solid. It consists of modules piled one upon each other, and composed of: - a layer of solid reactive of e thickness, - a heat exchanger plate in which flows the external thermal fluid, - a layer of solid reactive of’e thickness. Between each ,elementary module, a diffuser plate is installed in order to facilitate the diffusion of the reactive gas into the reactive medium (Fig. 11).
Here again, the use of dimensioning nomograms is of great use to the designer in order to predict satisfactorily the performances of a reactor with such a geometric configuration. In order to adapt the nomograms to this type of reactor, we are only interested in half of an elementary module, consisting of the exchanger wall and the layer of reactive solid. This may then be considered as the result of a cylindrical reactor, the interior of which is a thin layer of reactive solid compared with the outer radius (Fig. 12). This geometric approximation, which is currently accepted in heat transfer, is very satisfactory when the geometric form factor p is close to unity. In order to remain within the context of this approximation, a value of 0.95 will be chosen for ~this factor. The outer radius of the equivalent cylindrical reactor is then defined by Eq. (10):
Req=L 1-P
D. Stirou,
G. Crozat
/Chemical
Engineering
Form factor y
and Processing
He:oychange i
36 (1997)
45-58
coefficient
Hsw at the wall (W.m2 K-l )
t
500 -
100 ,)
0
5
10
15
20
25
Effective conductivity he (W.m-’ K-‘)
0
5
10
15
20
25
Effective conductivity he (W.m-’ K*’ )
Fig. 10. Principle diagram of the nomogram used for the determination of the thermal characteristics of the reactive medium to be used. The couples (A=,,y) determined on the graph ensure access to the appropriate reactive medium characterized thermally by a couple (A,, I&,).
If the thermal characteristics of the layer of reactive solid are known, then the equivalent outer radius (Eq. (10)) authorizes the calculation of the factor y and thus, through the nomogram, enables the access to the average linear power of the equivalent cylindrical reactor. A simple relation (Eq. (11)) then determines the average power per
Fig. 11. Geometric configuration of a plate reactor.
square meter of heat exchanger characterizing reactor:
the plate
Dlin
(11) In order to illustrate this point, consider a plate reactor with an average reaction power of 25 kW for a global reaction extent of 0.6, and try to determine the necessary heat exchanger area. The working conditions of this reactor lead to a temperature drop AT,, = 45°C. The reactive medium is thinly layered (8.5 mm) and is characterized by an effective conductivity of 1 W m - * K - i and a heat exchange coefficient at the wall of 70 W m - 2 K - l. The value of p being fixed at 0.95, the equivalent outer radius of the cylindrical reactor is equal to 0.17 m and the calculation of the y factor gives a value of 1.68. By a direct reading on the nomogram established for an extent of the reaction of 0.6, the average linear power of the equivalent cylindrical reactor is equal to 2600 W
D. Stitou,
G. Crozat
/ Chemical
Engineering
and Processing
36 (1997)
45-58
55
8. Nomenclature e
thickness of the layer of reactive solid in a plate reactor (m) heat exchange coefficient at the wall (W m-2
N tub PC
Paver
8, Rexch Fig. 12. The transformation of a plane geometry reactor into an equivalent cylindrical geometry reactor. This approximation is only possible when e CCR,,.
m - ‘. This linear power corresponds then to an average thermal power density for the plate reactor of 2430 W m - 2. At the end, the necessary exchange surface for this plate reactor in order to develop a global power of 25 kW should be 10.3 m2. The dimensioning of this plate reactor, obtained by the nomogram, is in fact very close to that of an industrial pilot plant already built and whose dimensioning has been carried out by dynamic simulations in identical conditions (Fig. 13).
RCO
R,,
R
Rx
Rtub
Sexch 43
Tb,
Trm
Th
TC
Teq ht
7. Conclusion
V
The dimensioning nomograms established and presented in the current study enable us fInally to consider every case of dimensioning of fixed-bed solid-gas thermochemical reactors. They can, in fact, be applied to reactors having very different geometric configurations, working with a large scale of operating conditions and within which a reactive medium is confined. This reactive medium can have a large variety of thermal characteristics, as long as, however, it does not lead to limitations by mass transfer. Through several examples we have shown the different possibilities of application of these dimensioning nomograms, as well as their simplicity and their flexibility of use. The rapidity of dimensioning of a solid-gas reactor constitutes one of the main interests of these nomograms. At the end, they represent a new helpful tool for assisting desision-making that will enable any technical designer to draft out quickly a realistic dimensioning of tixed-bed solid-gas thermochemical reactors.
Greek letters
K-1
>
number of inner tubes constraint pressure applied to the reactor (pa) average power supplied or extracted from the reactor at the level of the heat exchanger during the reaction phase (w) thermal power developed respectively under reactive and sensible form by the medium during the reaction phase (I+‘) radius (m) equivalent outer radius of a plate reactor (m) radius of the heat exchanger wall of the reactor radius of the wall far from the heat exchanger wall (m) reactor’s outer radius (m) reactor’s inner radius or that of the inner tubes (m) heat exchange surface (m”) reacton time corresponding to a given extent of the reaction (s) heat sources or sinks temperatures (K) constraint temperature applied to the reactor (K) equilibrium temperature of the reaction to the considered constraint pressure reaction time corresponding to a given extent of the reaction (s) Volume of the reactive medium (m3) Analytical functions that link the dimensioning constraints Number
defining the power ratio (Psens/
Preac)
Stoichiometric ratio of the chemical reaction Effective conductivity of the reactive medium (W m-l K-r) Geometric form factor of the reactor Form factor characterizing the reactional front protie within the reactor Temperature drop with regard to the thermodynamic equilibrium of the reaction at a considered pressure (K) Global extent of the reaction Local extent of the reaction at a considered time t,
D. Stitou,
G. Crozat
/ Chemical
Engineering
and Processing
36 (1997)
45-58 Positioning
wall -.
-..
,-..----,-/C,
Reactor’s ---\
CONDENSATION
10
--\_
20
-\---
t (ml
characteristics
of the pilot plant
Number of heat exchanger plates Number of gas diffuser plates Number of layer of solid reactive Thickness of the layers (mm) Plates diameter (mm) Total heat exchange surface (m*) Salt mass of CaCl2 (kg) Expanded graphite mass (kg) Mixture density (kg of mixture / m3)
20 21 40 815 565 10 14,2 4,7 230
Fig. 13. Diagram and characteristics of the industrial pilot plant reactor of 20-50 kW. Experimental powersextractedfrom the reactorandthe condenser during a decomposition phase obtained with a ATes= 45°C (according to [7]). Appendix A Pre-dimensioning nomogram for thermochemical reactors established for three types of heat exchanger s~ruc~ure
and a global extent of the-reaction
External
heat exchanae
AX, of 0.8.
mono-
or multi-tubular
D. Stitou,
Appendix
G. Crozat
/ Chemical
Engineering
and Processing
36 (1997)
45-58
57
3
Pre-dimensioning nomogram for thermochemical and a global extent of the reaction AX, of 0.6.
reactors established for three types of heat exchanger structure
mono- or multi-tubular Internal heat exchanae
Appendix C
Pre-dimensioning nomogram for thermochemical and a global extent of the reaction AX, of 0.4.
ial heat exchanae
reactors established for three types of heat exchanger structure
58
D. Stitou,
G. Crozat
/Chemical
Engineering
References [l] E. Merlin, Environmental protection: New challenge for technical competition and innovation, Proc. Symp. Solid Sorption Refrigeration, 18j20 November, Paris, France, 1992, pp. 21-23. [Z] F. Meunier, Solid sorption: an alternative to CFCs, J. Heat Recovery Syst. and CHP 13, 4 (1993) 289-295. [3] MSuzuki, Application of adsorption cooling systemsto automobiles, J. Heat Recovery Systems and CHP 13, 4 (1993) 335-340. [4] P. Picard, Sorption systems:the point of view of gas utility, J. Heat Recovery Systems and CHP 13, 4 (1993) 329-334. [5] S. Mauran, M. Lebrun, P. Prades, M. Moreau, B. Spinner, C. Drapier, Composite actif et pro&de de mise en oeuvre de processus physicochimiques gaz-solide ou liquide-gaz utilisant comme milieu reactionnel un tel composite, FR Patent 910303(April1991). [6] J. Vila, Influence du protocole d’ttablissement des contraintes thermodynamique sur les performances dun rkacteur solide-gaz de pompe g chaleur chimique, Ph.D. Thesis, University of Perpignan, France, 1992. [7] M. Lebrun, P. Neveu, Conception, simulation, dimensioning and testing of an experimental chemical heat pump, ASHRAE Trans., 98 (1992)
420-429.
and Processing
36 (1997)
45-58
[S] V. Goetz, A. Marty, A model for reversible solid-gas reactions submitted to temperature and pressure constraints, simulation of the rate of reaction in solid-gas reactors used in chemical heat pumps, C/rem. Eng. Sci., 47 (1992) 479-498. [9] N. Mazet, M. Amouroux, B. Spinner, Analysis and experimental study of the transformation of a non isothermal solid-gas reacting medium, Chem. Eng. Comm., 99 (1991) 155-174. [lo] M. Lebrun, B. Spinner, Simulation for the development of solid-gas chemical heat pump pilot plants Part 1: Simulation and dimensioning, Chem. Eng. Process., 28 (1990) 55-66. [I l] P. Neveu, J. Castaing, From materials to process: development of a numerical sizing tool applied to solid-gas thermochemical transformers-parts I and II, J. Heat Recovery Systems and CHP., (1996) submitted. [12] D. Stitou, V. Goetz, B. Spinner, A new analytical model for solid-gas thermochemical reactors based on thermophysical properties ofthefixed-bed reactivemedium, Cijem. Eng. Process., (1996) accepted. [13] A Wagner, Machine a froid h sorption solide-gaz a multiple effets g&r6 par caloduc: Mod&ation et validation experimental sur un pilote de 2 kW de production de froid, Ph.D. Thesis, University of Perpignan, France.