ANALYSIS OF T H E COMBUSTION OF GRAPHITE-URANIUM FUELS IN A FIXED BED OR IMOVING BED CHARLES D. SCOTT
Oak Ridge National Laboratory, Oak Ridge, Tenn.
A proposed first step in processing graphite-uranium nuclear fuels for the recovery of uranium is burning of the fuel by oxygen and/or air in a fixed or moving bed to remove the carbon and oxidize the uranium. To predict efects of operating variables and point out potential problems, an analysis was made of the reaction system which includes consideration of both reaction kinetics and mechanisms of heat transfer. A mathematicall model of the process was derived and numerical solutions were made. Results of the analysis compare favorably with experimental results, and several potential operational problems were revealed, such as high carbon monoxide content and excessive heating. Safe operating conditions were predicted for deep beds of fuel in tubular and annular reactors.
HE first step in a proposed processing method for recovery Tof uranium from graphite-uranium fuels consists of oxidation of the fuel by oxygen to volatilize the carbon (77). Residue ash from the combustion step can be treated in a variety of ways to recover and purify the uranium (3, 77). The combustion step may be carried out by bringing the solid fuel in a fixed or moving bed in contact with a stream of oxygen-bearing gas in a tubular or annular reactor. Oxidizing gas may be introduced to the reactor at several points up the reactor and there may be continuous or intermittent addition of fresh fuel and removal of residue ash (Figure 1). The small-scale combustion of graphite for recovering the included uranium has been a successful processing method in U. S. Atomic Energy Commission facilities for several years ( 7 ) , and the feasibility of engineering-scale graphite-uranium fuel combustion in fixtsd beds has been demonstrated a t Oak Ridge National Labor,itory, where the Ultra High Temperature Reactor Experimmt (UHTREX) type of fuels has been burned successfully (3,6). Initial results from the large-scale burning of graphiteuranium fuel in fixed beds indicate several potential problems resulting from scale-up. For example, such process variables as gas-phase carbon monoxide concentration and fuel temperature must be controlled within acceptable limits for safe operation. An analysis that allows the prediction of process variables is valuable as a tool for process design and as a guide for the experimental program. Such an analysis is presented in which reaction kinetics and heat transfer were considered, and a mathematical model of the process was derived. Numerical solutions of the mathematical model were the basis for the prediction of process variables. The result: of the analysis are directly applicable to UHTREX-type fuels, and the general approach to the problem is usable for other types of graphite-base fuels.
Rate-Controlling Mechanisms
It is desirable to be able to predict heat-general In rates, reaction rates, gas composition, and system temperature as functions of bed height and run time for the oxidation of graphite-uranium fuels in fixed or moving beds. This is accomplished by imposing possible mechanisms of reaction, heat generation, and heat transfer on the system and then
FUEL FEED
(BATCH OR CONTINUOUS)
7u 4
FUEL ~~
BED
4
0 0
U
ASH REMOVAL
OXIDIZING-GAS INLETS
Figure 1 . Fixed- or moving-bed graphitefuel oxidation process
deriving equations describing this assumed model of the system. If these assumed mechanisms are well based, the modelpredicted performance will adequately describe the real system. A favorable comparison of model-predicted and experimental results is necessary to establish the validity of the model. T h e major chemical reactions that occur in fixed beds of graphite fuels at the solid reaction site during burning with oxygen are: oxidation of the graphite to carbon monoxide or carbon dioxide
+ = 2co c + os = cos
2c
0 2
and oxidation of uranium compounds to U308 and other metal compounds to their oxides. The oxidation of C O to GO1 by VOL. 5
NO. 3 J U L Y 1 9 6 6
223
oxygen can also occur within the graphite fuel and throughout the reaction system: 2co
+
0 2
=
2c02
(3)
Since carbon is the major chemical species in the fuel (more than 90 mole yo),only the oxidation of carbon and carbon monoxide will be considered significant for determining bulk gas-phase composition and heat generation. Carbon Oxidation. The reaction of oxygen with carbon a t the carbon reaction site will produce only carbon dioxide at low temperature (below 500’ C.), and only carbon monoxide at high temperature (above 1000° C.). However, carbon monoxide is further oxidized to the dioxide in the presence of oxygen and is catalyzed by available surface such as metal oxide ash. Many kinetic mechanisms can control heterogeneous reaction systems. For irreversible reactions, these can include : Mass transport of the reacting gas from the bulk gas to the solid surface Mass transport of the reacting gas through the porous solid to a reaction site Reaction of the oxygen with graphite a t the graphite-oxygen reaction site (assumed to be controlled by a first-order reaction) FIRST-ORDERKINETICS.T h e rate of reaction of oxygen with graphite at the graphite reaction site can be approximated by a first-order relation dependent upon the oxygen concentrations in the temperature range of 500’ to 700’ C. (6) :
RC = klaCo,,
The correlation by Hobson and Thodos (5) was used in this study : J D
= 1.30N~,-O.~‘
(6)
where NRe is the modified Reynolds number. INTERNAL MASS TRANSPORT. During oxidation of UHTREX-type fuels, some of the resulting metal oxide ash that is formed tends to adhere to the graphite surface in a layer. The ash remains either as a sintered mass, closely approximating the shape of the original fuel rod, or as a thin layer on the unburned fuel, depending upon the type of fuel. Thus, in the reaction between partially reacted fuel and oxygen, the oxygen must diffuse through the porous ash to the graphite reaction site. This “internal” mass transport of the oxygen represents another resistance to the reaction, and may also contribute to the control of the reaction rate. Considering Fickian diffusion in the partially reacted fuel, the internal diffusional relationship can be expressed by :
(7) where Co,s = 0 2 concentration in the gas phase within the solid, and DE = effective diffusivity. For any small increment of time, the diffusional process can be considered to be at steady state, and, for fuel particles that are approximated by an equivalent sphere, Equation 7 becomes :
(4)
where
Rc = rate of oxygen reaction with carbon to form carbon monoxide kl = 20,000 exp (-21,4OO/T) gram-mole/sq. cm. sec. atm. a = area available for reaction CO, = bulk gas oxygen concentration EXTERNALMASS TRANSPORT. Resistance due to mass transport of the oxygen from the bulk gas to the external surface of the fuel can contribute to the control of the C-O2 reaction. I n heterogeneous flow systems such as the graphitefuel-oxygen system, the resistance to “external” mass transport can be approximated by a stagnant gas film surrounding the solid phase through which the oxygen must diffuse. The rate of mass transport can be expressed by:
R
=
k,a(C02
- CoZE)
(5)
where
R
where r represents radial distance within the sphere. The general solution to Equation 8 is: C O , ~= A
+ B/r
(9)
where A and B are constants of integration. If first-order kinetics are not important for reaction rate control a t the same time that internal mass transport contributes to reaction rate control, and if the reaction a t the reaction site is essentially irreversible, the following boundary conditions apply:
r = rl
Co: = CoF
r
CO,’ = 0
=
rl
(10)
where rE = external radius of the fuel rod and rl = radius of the reaction interface. After these boundary conditions are applied, internal concentration can be expressed by:
= total
0 2 reaction rate external mass transport rate constant k, Cop = 0 2 concentration on the surface of the solid phase
=
The rate constant, k,, is a function of the physical properties of the gas and the mass flow rate. The usual correlation for the mass transfer coefficient is the mass transfer factor, J D , as a function of the modified Reynolds number,
where a = effective solid-phase porosity, and D = molecular diffusivity of oxygen. The partial derivative of Equation 11 evaluated a t the reaction interface is:
where
gas density
p
=
G M
= gas mass flow rate
hr8,
= Schmidt number
224
The rate of reaction of oxygen with carbon must be equal to the rate of oxygen diffusing to the reaction interface :
=
average molecular weight of gas
I&EC PROCESS DESIGN A N D DEVELOPMENT
therefore,
Carbon Monoxide Oxidation. If CO is the product of the graphite-oxygen reaction, the kinetics of the oxidation of the CO must also be considered because this represents additional oxygen usage and will influence the extent of carbon oxidation and the C O content in the bulk gas phase. The CO-02 reaction is surface-catalyzed and dependent on both the 0 2 and CO gas-phase concentration (2, 70). Although the order of dependence on the CO or 0 2 concentration has not been established for the conditions of this system, it was assumed that the rate could be expressed by:
RCO= kcaCoISCcoS
-7
4 I
t
(15 )
where k, = the rate constant dependent on temperature and Rco = the rate of oxygen reacting with CO. Figure 2.
X
Oxygen material balance in bed
Mathematical Model
The reaction system (can be represented by a set of differential and difference equations which describe the reaction-ratecontrolling processes and material balance relationships. These equations constitute the mathematical model of the reaction system, and a solution of the set of equations results in a complete description of the system if the original assumptions are correct. Simple heat balance equations also allow determination of the system temperature. Bed Material Balance. Several additional assumptions were made to simplify the mathematical problem: T h e graphite fuel is composed of a bed of graphite pieces that can be represented by fuel spheres of equivalent volume and surface area. All physical and chemical properties in the external gas phase and the specific reaction rates are independent of radial position in the bed. Diffusion of gas components in the direction of flow is negligible. Changes in gas properties for any small increment of bed height are negligible. The gas holdup in the graphite fuel pores is negligible. Changes in the chemical or physical properties of the solids in a small increment of bed height occur as a step change a t the end of a small increment of time. Changes in the external gas stream as a result of reaction or heat transfer occur in the bed as a step change a t the end of an increment of bed height but before the gas enters the next increment of bed height. The system pressure is constant. A rigorous derivation of differential equations from the differential material balance of each gas component would be cumbersome since several of the properties of the gas phase, such as volumetric flow rate and temperature, do not remain constant with bed position and reaction time; therefore, the bed material balance relationships will be expressed directly in a form suitable for numerical solution Consider an increment of bed volume in the fixed bed that results from an increment of bed height Az at a distance t from the gas entrance (Figure 2). The gas enters the volume increment at a volumetric flow rate of F and an oxygen concentration of ( C O ? ) ~Gas . leaves the differential section at a flow rate of (F A F ) and at an oxygen concentration of ( C O ~ ) ~ + *Oxy~. gen is removed from the gas stream in the increment of bed volume by its reaction with graphite to form CO, or with carbon monoxide to form COZ. The material balance relationship will be :
+
F(Cdz
-
( F f - W ( C o d z + ~ z= (Rc
+ Rco)AAz
(16)
where reaction rates Rc and Rco are expressed in rate of oxygen reaction per unit volume of bed, and A is the cross-sectional area of the reactor. Dividing all gas concentrations by the gas density, p , to give mole fractions, and rearranging, Equation 16 gives :
The change in the volumetric flow rate, AF, must also be evaluated. Since the pressure is constant, the change in flow rate will result from the generation of C O (Reaction 1). Therefore, AF can be expressed by: AF =
(Rc - Rco)
AAz
P
Similar material balance expressions can be determined for both CO and COZ: pF(xco)z (xCO)r+A.e
=
(XCO?)r+Az
=
+
-
2(Rc Rco)AAz P(F AF)
+ PF(XCOJZ+ 2RcoAA.t P(F + A F )
(19)
(20)
Specific Reaction Rates. T o use the material balance equations, it is necessary to have expressions for the specific reaction rates, Rc and Reo. REACTION RATECONTROLBY FIRST-ORDER KINETICSA N D DIFFUSION.For the case of reaction rate control by first-order kinetics and film diffusion (Equations 4 and j), the two rate resistances represented by the rate constants, kl and k,, can be combined and the specific reaction rate per unit volume of bed will be:
where R is the over-all rate of reaction of oxygen with graphite and r is the specific fuel-particle population in the bed. REACTION RATE CONTROL BY EXTERNAL-FILM DIFFU5ION AND INTERNAL DIFFUSION.I n the region where first-order kinetics does not contribute to control, only external diffusion and internal diffusion resistances need be considered. This will occur a t high temperatures, where C O will be the primary reaction product a t the 0 2 - C reaction site. The 0 2 - C 0 reaction (Equation 3) will also be occurring throughout the VOL. 5
NO. 3
JULY 1966
225
zone of internal oxygen diffusion (in the oxidation ash). This system can be approximated by separating the diffusion problem from the kinetics problem and considering only the diffusion of one species, 0 2 . This can be done by assuming that the CO-02 reaction occurs in a hypothetical reaction shell of differential thickness at the outer surface of the fuel
piece. In this reaction shell, the 0 2 concentration will be that of the 0 2 after diffusing through the external gas film, and the C O concentration will be that of CO a t the 02-C reaction site. Further, the surface area available for the surfacecatalyzed 02-CO reaction will be the active external surface of the fuel piece plus the surface area in the oxidation ash. This approximation should be good when the ash layer is small compared with the size of the total fuel piece. The physical model that results from these assumptions (Figure 3) gives the fuel piece of radius rE, surrounded by a stagnant gas film of radius TF, and immediately surrounding the outer surface is the hypothetical CO reaction zone of radius TH with P H = Y E . The reaction interface or carbon-ash interface is at a radius of rl, and the oxidation ash is in the space between r I and TE. The gas composition in the porous fuel particle of the assumed model can also be qualitatively shown. Oxygen concentration will decrease from the bulk gas value, Co,, to Co?, because of the external gas film resistance, T F - Y H (Figure 4). I t remains constant in the hypothetical reaction zone ( r H - Y E ) , and there will be a predictable concentration gradient from r E to rl due to the resistance caused by oxygen diffusing in the porous ash. The oxygen concentration will go to zero a t the reaction site. Carbon monoxide concentration will be constant from the r e ) , where reaction site to the hypothetical reaction zone (rr it decreases to its bulk gas-phase value. Carbon dioxide and inert-gas concentrations will be then obtained by material balance and stoichiometry considerations. The reaction rate of oxygen with CO in the hypothetical reaction zone can be expressed by:
=
-
i 4 r aTE D r I r E irl
r[(kcllE2)
+ + kc,(rE3 - r1”1 (p - cco, - cr)}cot
(23)
Equations 5 and 23 can be combined to obtain an expression for Co::
Equations 23 and 24 give the specific reaction rate of oxygen for the case of external and internal diffusion of oxygen controlling the rate of its reaction with graphite. Equations 22 and 14 with Equation 24 give the specific reaction rates for the 0 2 4 and 02-CO reactions. The specific reaction rates are expressed in terms of the position of the reaction interface, r I . Because this interface position will change with time because of the 02-C reaction, it is nec-
r
GAS F’uI
-
Figure 3. piece
Physical model of oxidation of a graphite fuel This model results from mathematical model
where reaction rate constant dependent on external surface of fuel = reaction rate constant dependent on internal surface k,, in the ash Cco’ = CO concentration at reaction interface kc,
=
The rate of reaction of oxygen with graphite, Rc,is dependent on the diffusion of oxygen from the bulk gas phase to the reaction site at 71. The total oxygen reaction rate must be equal to the combined for graphite oxidation, Rc, and CO oxidation, Reo. After combination of Equations 14 and 22 and because the CO concentration at the reaction interface will be the difference between the total gas concentration or gas molar density and the sum of the COZ and inert gas concentrations, the total reaction rate is: 226
I & E C PROCESS DESIGN AND DEVELOPMENT
r, DISTANCE FROM CENTER OF FUEL ROD
Figure 4. Concentration profiles in graphite-fuel oxidation system as predicted by mathematical model
essary to derive an expression for its movement. This can be done by equating the ‘carbon reaction rate to the movement of the interface which satisfies the stoichiometry of Equation 1 : 2Rcl = -44sr12b
dr1
(25)
-
dt
where b is molar density of graphite in the fuel phase. rearrangement, Equation 25 becomes :
Upon
Mathematical Solution. The necessary partial differential equations and difference equations are now available for a mathematical solution of the proposed mathematical model. Simultaneous solution of Equations 17, 19, 2 0 , 2 1 , and 26, or 17. 19, 23, 24, and 26 I-esults in a mathematical solution of the system and allows prediction of the 02-C and OYCO reaction rates at any point in the reactor as a function of time. This will also allow prediction of the bulk gas-phase composition as a function of time and bed position. Heat Transfer Mechanisms. Heat is transferred from the fuel rods by radiation, convection, and conduction. For large fuel pieces at hi:h temperature, the main heat transfer mechanisms will be radiation and convection. The equation for net radiative heat transfer between two bodies with emissivity and absorptivity of unit can be expressed as: q12
=
RidT14 - T24)
(27)
here
h=-
where rw is the radius of the reactor wall. Heat Balances. If reactor wall temperatures T I , and Tow, inlet gas temperature TG,reaction rates Ro and Reo, past temperature history of fuel particles, and shapes of the fuel particles are known for an incremental section of the bed, then heat transfer to the wall, the fuel rod temperature, and the change in gas temperature can be determined from heat balance equations that utilize the heat generation rate resulting from chemical reaction. The heats of reaction for Equations 1 , 2 , and 3 were considered to be constant with temperature and expressed by (7) : AH1 = -136 kcal./gram-mole
AH2 = -91 kcal./gram-mole AH3 = -46 kcal./gram-mole
The heat balance for an increment of bed height per unit time for the center radial shell will be: = Kc(Tc*- To4)
QC
0, I, and C
=
Q
=
q12
n
where
=
Several correlations for convective heat transfer coefficients are available for systems similar to the fixed-bed graphite oxidation. An expression developed by Satterfield and Resnick ( 9 ) was used for heat transfer from a fuel particle to the gas h = 0.992CpG(.VRe)-0.34(.l’pr)
-0.667
+
The three radial heat balance equations will give values of
Tc, To, and TI. A heat balance around the gas in the differential section of the reactor results in the following expression for change in the gas temperature as it passes through the incremental bed section :
where Hw is the specific convective heat transfer coefficient a t the walls. Finally, the heat flux to the reactor walls can be determined by the amount of heat transferred by radiation and convection and the wall heat balance equations will be of the form
- Tv)
+ Kv(Ti4- Tw4)
(33)
Gas Properties. Each of the gas-phase physical properties was assumed to be a linear combination of the properties of the individual gases in the mixture. All gases and the gas mixture were assumed to be ideal.
heat transfer from solid to gas temperature of the surface and gas = convective heat transfer coefficient =
T s , TG
TI^)
outside, inside, and center of the reactor heat generated by the chemical reaction = heat capacity of the graphite fuel = specific convective heat transfer coefficient from solid to gas = time increment designation
Q w = Hw(To
q h
+ K I ( T c -~
There will be similar expressions for the outside and inside radial shells. Here
CP
Three equidistant radial shells in the fixed bed of fuel were considered for the heat transfer problem and radiative heat transfer was assumed to be radial only. Each zone was assumed to be a constant radial temperature and it radiated heat as a single surface. These assumptions allow the radiative heat transfer phenomena to be described by three equations and three temperatures. Heat generated from the graphite oxidation will also be transferred from the fuel particles by convective heat transfer to the flowing gas stream and from the gas to the reactor wall. The rate of convective heat transfer from a solid surface to a flowing gas can be drscribed by an equation of the following form :
(30)
(AVRe)O.yNp,)OJ TW
HS
= net heat transfer from surface 1 to surface 2 TI, T2 = absolute temperature of surfaces 1 and 2 K12 = radiative heat transfer coefficient
0.55kT
(29)
and an expression developed by Hanrattry ( 4 ) was used for the convective heat transfer from the gas to the reactor wall,
Computer Solution
The mathematical model was solved by a digital computer, utilizing a finite-difference technique in which the difference and differential equations (in difference form) were solved for successive increments of bed, which allowed establishment of the gas composition and reaction rate profiles in the bed. Heat balance equations allowed determination of system temperatures for each bed volume increment. The numerical VOL 5
NO. 3
JULY 1 9 6 6
227
or less constant layer of one or two uranium compound particles in thickness (4). Single-Particle Tests. Experimental information that can be obtained from single-particle tests are : the rate constants for carbon monoxide oxidation within the fuel ash and internal mass transport property. Some single-particle tests were made to establish rate constants for CO oxidation and internal mass transport properties. The tests were made by suspending fuel specimens from an analytical balance into a heated tube furnace through which oxygen or air could be metered. The temperature, fuelelement weight, and effluent-gas composition were then monitored for various operational conditions to establish rates of reaction of 0 2 with graphite and CO. Equations 14 and 22 were used with observed reaction rates to determine average values of kc,, kc,, and a in the temperature range of 700" to 900" C. ( 6 ) . The average values are:
solution was started at time equal to At a t the entrance point of the gas where initial conditions were known. This allowed computation of values of interest at a point Az further in the bed. This procedure was then repeated for succeeding bed inerements until the end of the bed was reached. All the gas composition and temperature profiles in the bed were stored for future use. Another time increment, t At, was then taken and, with initial conditions and previous bed profile conditions, another transit of the bed was made. Bed profile values were periodically recorded. The computer solution was stable throughout the range of parameters tested, with bed height increments, Az, of 0.5 to 2.0 cm. and time increments, At, of 10 to 30 seconds. The computer program was run on a digital computer a t Oak Ridge National Laboratory. With Az = 1 cm. and At = 10 seconds, for a lOO-cm.-deep bed, machine time was approximately 5% of real time.
+
kc, = 1.8 X lo7 ~ m . ~ / s e .gram-mole c. k,, = 2.8 X 106 ~ m . ~ / s e .gram-mole c. = 0.085 effective open area/total area CY
Experimental Tests
T o use the computer solution, it was necessary to determine some properties of the graphite fuel experimentally by differential bed (single-particle) tests. I t was also desirable to compare the results from experimental deep-bed tests with predicted results from the computer solution to validate the reliability of the predictions from the mathematical model. Experimental tests were made with two different types of graphite-uranium fuels. Fuel Characteristics. Fuels tested were typical of the fuel used for the U H T R E X in which UC2 spheres, either uncoated or coated with pyrolytic carbon, are dispersed in a graphite extrusion in the form of rods ( 8 ) . The fuel rods were charged to the oxidation reactor either as long pieces or as small pieces resulting from breaking the fuel rods. In either case, the fuel charge could be approximated by a bed of equivalent fuel spheres. The graphite-base fuels in which the UC2 spheres were uncoated gave upon oxidation a residue ash that tended to adhere and sinter in the approximate form of the original fuel piece; however, the fuel with coated UC2 spheres gave an ash that tended to slough off as oxidation progressed, with a more
OPERATING 5s E CONDITIONS
$
