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Ind. Eng. Chem. Res. 1996,34, 2889-2897
Behavior of Thermochemical Waves during Reactive Flow through Permeable Media Myra Ann Dria Western Atlas International, Houston, Texas 77042
R. S. Schechter and Larry W. Lake* The University of Texas, Austin, Texas 78712
The chromatographic theory and terminology of Helfferich carries over into the study of reactive flow in which temperature changes occur. In this paper, we show examples of how temperature alters wave velocities and mineral sequences during flow with precipitatioddissolution. We also give indications of the importance of heat of reaction on in-situ temperature changes. Finally, the added complication of heat losses to adjacent strata are incorporated into the theory.
Introduction The flow of a reactive fluid through a permeable medium, where reactions entail the dissolution of species comprising the matrix of the medium coupled with the subsequent precipitation of introduced components, results in a progression of zones that have a wavelike character. This behavior is similar to what is observed in chromatography, a subject whose fundamental wavelike character has been skillfully expounded upon by Fred Helfferich. Indeed, most discussions of chromatographic processes are in terms of the semantics imposed onto the subject by Hewerich; coherence, time-distance diagrams, chemical waves, profiles, and histories are all concepts important for understanding chromatographic behavior. In prior studies of precipitatioddissolution processes (Walsh, 1982; Bryant, 1986; Dria; 1988; Novak, 1990; Sevougian, 19921, such as occurs with the migration of pollutants in groundwater, the formation of ore bodies, or the propagation of caustic fluids within reservoirs to enhance oil production, we have found many of these concepts to be useful. Thus, it is appropriate to revisit these studies in a body of work honoring Professor Helfferich for his pioneering efforts. The previous investigations of precipitatioddissolution processes have not included thermal effects. It is, indeed, not clear just how the characteristic wave phenomena will change if thermal effects, specifically heat of reaction or heat transfer, cannot be ignored. This contribution addresses this question, using in so far as is possible the terminology already common in chromatography. In some cases it is proper to speak of thermochemical waves, although the processes may become so complex that the advantages engendered may be evanescent.
Background All of the cases presented in this paper are restricted to one-dimensional flow of a single aqueous phase, designated by a subscript 1, through a chemically reactive medium. The chemical reactions themselves are restricted to precipitatioddissolution (p/d) reactions in the manner first discussed by Walsh (1983). See also Helfferich (1989). We further assume that the solid and fluid phases are in equilibrium with each other a t all points throughout the medium. This is the familiar assumption of local thermodynamic equilibrium; see
Sevougian (1992) for the effects on reactive flow when this assumption is not satisfied. In this work, we also assume that the medium contains a nonreactive inert material that we designate by a subscript s. The final basic assumption is that dissipative effects-dispersion, diffusion, and thermal conductivity-within the medium are negligible. Basic Definitions. Helfferich and Klein (1970) define a waue t o be a change in composition that propagates as a result of bulk fluid flow. Waves do not, in general, have the same velocity as the bulk flow. Here the wave definition includes changes in temperature or thermal waves. If a change in chemical composition also occurs across the wave, it is a thermochemical wave. All of the waves treated in this paper are shock waves, discontinuous changes in composition or temperature, or indifferent waves. Indifferent waves, waves that neither spread nor sharpen on propagation, appear identical to shocks under the conditions considered here. These waves propagate as a result of a discontinuous change at the inlet of the medium from an initial condition I to an injected condition J. For these types of waves, material balances for each component (Bryant, 1986) reduce to
1
(1)
k
where Cj and c k are the concentrations of the flowing and solid species, respectively, Yik and 1 7 ~are the stochiometric coefficients that represent the number of moles of component i in the respective solid or aqueous component, and UAC, is the specific velocity of the change in Ci. The specific velocity is the velocity of the concentration change A C i divided by the interstitial velocity of the aqueous phase. The sums in eq 1 are over all possible components. The coherence condition introduced by Helfferich and Klein (1970) says that for these types of problems U A C , are equal for all components. Coherence reduces the problem of solving for each concentration at all positions and times t o one of solving for the concentrations on either side of the waves, a tremendous reduction in effort (Novak, 1990). It also means that the specific velocities now apply to the waves themselves; we will be using u1, u2, etc. for the wave specific velocities of waves 1,2, etc. in what follows instead of u ~ c , .The ratio
Q888-5885I95/2634-2889$09.QQlQ0 1995 American Chemical Society
2890 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
of concentration changes in eq 1 is a delay factor for the indicated wave. The larger the delay factor (that is, the larger the solid concentration change), the slower the wave. We use dimensionless position XD and time t~ in this work. XD is a position x within the medium divided by the total medium length; tD is the cumulative fluid injected divided by the medium's pore volume XD
= x/L, t, = ut/#L
= 0 is the inlet end of the medium. In these coordinates the specific velocity of a wave is the change in XD divided by the change in t ~ If. the wave has a constant velocity, the specific velocity is just XDltD. The specific velocity of the interstitial fluid is one in these coordinates. The Time-Distance Diagram. We represent the solutions here on time-distance diagrams. These are plots of time t~ on a horizontal axis versus distance XD on a vertical axis that show the progression of the various waves formed in the flow. Shock waves are represented by lines that separate regions of constant composition on XD-tD diagrams, and the slopes of these lines are the specific velocities of the indicated wave. In most cases, the lines on a time-distance diagram are straight, but at the end of this paper we show cases where heat losses cause waves with curved trajectories. GeochemicalFlow Modeling. We have applied the concepts of geochemical flow modeling to several problems over the past decade (Walsh, 1983; Bryant, 1986; Dria, 1988; Novak, 1990; Sevougian, 1992). There are a large number of potential chemical reactions, but the most important, and the only ones considered here, are those involving the precipitation andor dissolution of solid phases-pld reactions. We retain this restriction here but with the addition of temperature changes, either as a result of chemical reaction or imposed by the injection. All of the insights derived from these works were based on finite-difference simulations. However, in all cases, we present the results of these simulations as though they are solutions from the method of coherence. The ABCD Problem. Precipitationldissolution problems of practical interest involve aqueous solutions of many ions (H+ or C032-, for example), dozens of possible precipitates (CaC03 or BaS04, for example), oxidation and reduction reactions, intraaqueous speciation, and perhaps even heterogeneous reactions occurring at differing rates. These can lead to immense complexity even for apparently benign cases. See Sevougian (1992) for methods to efficiently treat these complexities. However, the basic features of the solution can be examined by ignoring multispecies solids and intraaqueous reactions. Thus, we deal only with hypothetical aqueous species, A, B, C, and D. These do not react in the aqueous phase and can form only binary combinations of solids (AC, CD, etc.). These solids cannot flow. The aqueous species are normally charged, and their equivalent concentrations should sum to zero for electroneutrality. We additionally hypothesize the existence of a chemically inert (and implicit) species that makes the charge balance unnecessary. This final step also means that the concentration units used for CA,CB, etc. are arbitrary. As we will see, the ABCD problem will also provide a good sampling of the complexities of thermal effects. Temperature Corrections. We now explore the equilibrium state of pld reaction changes with temperXD
ature. An application of the first and second law of thermodynamics and algebraic manipulations leads to the van't Hoff equation (Sandler, 1989) d(ln K,) -=+dT
AH,'
RP
where K, is the equilibrium constant for reaction r, AH/ is the standard enthalpy change of the reaction, and T is absolute temperature. Kr in this equation serves the same role as the solubility products to be discussed below. If AH" is negative, the reaction is exothermic, and K, decreases with temperature. A positive AH", an endothermic reaction, means that K, increases with temperature. If AH" is independent of temperature, the above equation integrates to
where Krl and Kj-2 are the equilibrium constants at temperatures TI and T2. This equation is valid only over small temperature differences, but it does not require heat capacity data and only requires enthalpy data at one temperature. Equation 2 will be used for the calculations in this paper.
Temperature Dependence in Isothermal Flow Two examples demonstrate the importance of temperature in isothermal flow. Temperature changes caused by p/d reactions are neglected until the next section. We show the results of two cases at high and low temperature, respectively. In the first case, the sequence of solids is the same at both temperatures. However, the temperature-altered solubilities cause changes in the wave velocities. In the second case, the solubilities change in opposite directions from those of the first example, resulting in a different solid sequence. Consider the possible precipitationldissolution of four solids whose solubility products at a low temperature are Ksp(AB) = 0.5, Ksp(AC) = 2.0, Ksp(DC) = 1.0, and Ksp(EC) = 0.2. These solubility products were selected t o facilitate presentation of the solution; only the most soluble salts have solubility products this large. The fluid is initially in equilibrium with the solids AB,AC, and DC; EC is a potential precipitate. The fluid injected into the medium is undersaturated with respect to all the solids. Figure l a shows the XD-tD diagram resulting from this injection. Since this result is typical of several to be shown later, we explain Figure l a in some detail. The conversion of a medium with three solids initially to one with no soluble solids takes place across 5 waves. The locations of these waves are indicated by the straight lines radiating from the origin in Figure la. The concentrations (both aqueous and solid) are constant on either side of the wave and are shown in Table 1. Furthermost downstream (XD large) is a salinity wave across which only the aqueous concentrations change. Such waves, shown as dotted lines in the figures, travel with the interstitial velocity (specificvelocity = 1). The region between the salinity wave and the next most upstream pld wave carries the reaction products from all of the upstream reactions. Salinity waves are ubiquitous features of pld problems.
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2891 Table 2. Composition of Regions, Wave Velocities, and Solubility Products of Displacement in Figure IC flowing phase
X
D
region
A
B
C
D
Ria
0.82 0.36 1 x lo-' 1 10-7
1.10 0.25 0.25 0.25
1.23 0.28 0.13 I x 10-7
0.41 0.18 0.38 0.25
R2 R3
J
b 1
X
I
R.
D
tD
Figure 1. (a)The low-temperature sequence whose composition is listed in Table 1. (b) The resulting sequence from a simulation of the same conditions of part a at an arbitrary high temperature. Sequence is identical but wave velocities are modified. (c) The resulting sequence from a simulation of the same initial conditions as part a at a higher temperature. Sequence is modified because of the changes in solubility products. Composition of each region is listed in Table 2.
Table 1. Composition of Regions, Wave Velocities, and Solubility Products of Displacement in Figure l a flowing phase region
A
B
C
Ria Rz Rs R* J
1.60 0.59 0.59 1x 1 x 10-7
0.31 0.84 0.84 0.25 0.25
1.25 0.64 0.17 0.17 1 x 10-7
solids
D
E
AB AC DC EC 0.80 1.60 4.0 2.0 4.0 1.57 0.25 0.25 0.25
0.31 4.7 1.17 4.7 1.17 1.0
5.0 0.2 3.4 3.4
Species AC dissolves across the first pld wave (from region R1, to R2) in Figure la. A fast rate of propagation is reasonable since AC is the most soluble (largest Ksp) of the species in the problem; however, reaction products from more upstream dissolutions cause a new species EC to form and species AB and DC to increase in concentration across this wave (Table 1). DC dissolves across the second pld wave (R2 to R3) and EC increases; the second pld wave is slower than the first pld wave because the concentration changes across it are larger
solids
E
AB AC DC EC
1.00 4.0 2.0 4.0 1.00 3.8 2.9 1.00 5.1 1.00
(eq 1). AB disappears across the third pld wave and, finally, EC across the fourth pld wave, each still slower than the preceding wave. The result of this low temperature flow is the sequence {AB-AC-DC}l{AB-DCEC}/{AB-EC}/{EC}/{ } from downstream to upstream. The specific velocity of wave 1, separating R1a and Rz, is u 1 = 0.44. Subsequent velocities are u2 = 0.21, u3 = 0.11, and u4 = 0.05, for the DC, AB, and EC dissolution, respectively. Now consider the same flow a t a high temperature such that the new solubility products are Ksp(AB) = 0.29, Ksp(AC) = 40.0, Ksp(DC) = 2.25, and Ksp(EC) = 0.1. These changes in solubility products indicate a decrease in the solubility for AB and EC and increases in solubility for AC, DC, and DB. The resulting wave response is in Figure lb. The sequence is unchanged from that shown in Figure la; however, the pld waves that primarily involve the dissolution of a species whose solubility product is increased show an increase in velocity. These pld waves are wave 1 (dissolution of AC), u 1 = 0.78, and wave 2 (dissolution of DC), uz = 0.30. Changes across pld waves whose solids are now less soluble show decreased velocity. These waves are wave 3 (dissolution of AB), u3 = 0.10, and wave 4 (dissolution of EC), ~4 = 0.04. We also see from Figure l b why we picked the solubility products to be so large. Typical solubility products will result in wave velocities that are so small as to make their representation difficult on a X D - ~ D diagram. If we include the salinity wave, the pld waves will all be compressed against the XD axis. Once we know the wave sequence for a given problem, the velocities can be constructed, regardless of the solubility products (Walsh, 1983; Novak, 1990). In the next example, the changes in solubility products result in the formation of an entirely different sequence. We now consider the solubility products to be altered a t high temperature such that the direction of change for each solid is opposite from that in the Figure l b example; the solubility products for Ksp(AB) = 0.9 and Ksp(EC) = 0.4 increase and decrease for KSP(AC) = 0.1. Ksp(DC) is unchanged. The sequence that results, Figure ICand Table 2, is {AB-AC-DC}l(AC-DC}/(DC}/( }. Now, EC is so soluble that it never precipitates. The increase in solubility of AB makes it the first solid to dissolve from R1a to R2. From these two examples, we see that the use of the correct temperature can be important for the identification of the precipitating minerals. For a well treatment application, for example, significant change in well treatment design or change in injection procedures might be required if damage, for example, is caused by Fen03 (rust from well casing) rather than CaC03 (calcite).
Heat of Reaction Effects without Heat Loss This section discusses the interaction between the heat generated by pld reactions and the formation of
2892 Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995
an additional temperature wave because of this enthalpy change. These thermochemical waves, across which temperature changes, also mark a change in chemical composition. We discover the importance of heat effects that,. accompany chemical reactions and define parameters that can determine when such effects are significant. Problem Statement. For simplicity of presentation, consider the dissolution of a mineral A to its solution forms, A and B.
I
/
XD
' D
AB-A+B
a. Dissolution only.
We inject a fluid at temperature P into a medium that is initially at temperature TI. The J fluid is free of A ( C A ~= O), meaning that dissolution will occur. Accompanying this dissolution will be a temperature increase, if the reaction is exothermic, or a decrease, if the reaction is endothermic. Of interest is the magnitude of temperature and solid concentration change. We neglect changes in CB in the following by assuming it is present in excess. We present this case accompanied by its end members in Figure 2. Figure 2a shows the dissolution-only, isothermal case, Figure 2b the propagation of the temperature change in the absence of chemical reaction, and Figure 2c their combination. The specific velocity of the p/d waves in Figure 2 are of the form (1 where Dfis a delay factor. From eq 1,the delay factor for the dissolution wave in Figure 2a is Df= A C ~ A C A , the retardation being brought about by the chemical reaction. Dffor the thermal wave in Figure 2b is
XD
tD
b. Temperature change only.
+
where C, is the concentration (density) of the inert material s, Cpsis its specific heat capacity per unit mass; C1 is the water concentration in the aqueous phase, and Cpl is its specific heat capacity per unit mass. The delay factor in eq 3 follows from an energy balance across a propogating step change in temperature (Lake, 1989). Equation 3 shows that the retardation of a thermal ,wave is caused by the need to heat up the inert material. The product of heat capacity and concentration (or density) is the volumetric heat capacity. Such temperature changes propagate as indifferent waves when, as is the case here, the energy content on the fluid and solid linearly depend on temperature. The case with dissolution and an imposed temperature change, Figure 2c, shows the formation of a new region whose concentration and temperature is marked with an asterisk. This region occurs just downstream of the dissolution wave. We calculate F and Cg in the following development. Coherence Condition for Thermochemical Waves. One of the contributions of this work is the expansion of the coherence condition to include propogating temperature changes. Coherence is a property of the form of the conservation law-in this case, firstorder, reducible hyperbolic partial differential equations with step change inlet conditions (Walsh and Lake, 1989). Thus, both mass and energy balances can exhibit coherent solutions as long as their conservation laws satisfy these conditions. As in Figure 2c, we denote the unknown concentrations and temperature as C h , CZ, and T*. We let the heat capacities be independent of temperature and the
"TI = "Cl
tD
c. Dissolution and temperature change. Figure 2. Three cases on heat effects in adiabatic precipitation/ dissolution example. The shading represents zones of differing temperature.
furthest downstream region be the reference enthalpy; i.e., HT= H: c,~
