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On the effect of flow patterns on the rate of mass transfer with chemical reaction
Chemical Engineering Science, 1970, Vol. 25, pp. 1239-1242.
Pergamon Press.
Printed in Great Britain.
On the effect of ílow patterns on the rate of mass transfer with chemical reaction (First receioed 15 September 1969; in reuisedform IN A RECENT paper Porter and Roberts [ 1] presented a rather simple theoretical model for mass transfer with chemical reaction to show the effect of flow pattems on the rate of mass transfer. The authors analyzed the behaviour of the goveming differential equation by seeking a solution composed by two terms. The first of them represents the solution of the so called “steady state” equation where the rate of diffusion is exactly balanced by the rate of reaction while the second term is in fact the solution of the whole equations but with modified boundary conditions. Clearly, the fluid dynamic effects are only present in the second term of the solution. Moreover by introducing the contact time as the ratio of the spatial coordinate and the interfacial velocity, the second term of the solution can be interpreted as a contact time dependent solution, so that when the contact time approaches infinite the first term wil1 represent the actual solution of the problem. Under these conditions fluid dynamic effects wil1 become negligible. After analyzing some particular cases the authors concluded that laboratory apparatus for scale up studies could be chosen on grounds of simplicity and convenience due to the general smal1 effects on the mass transfer coefficient produced by fluid dynamic conditions. However proper limits of validity for Porter and Roberts’ criterium were not given. Moreover since the analysis was restricted to pseudo first or zero order kinetics the dependence of those limits on the concentration of the reacting component present on the absorbing fluid phase was not taken into account. The purpose of this communication is to show that the limits mentioned above can be established with a good degree of approximation using previous results of the penetration theory. Let US first assume that a pseudo first order irreversible reaction between components A and B is taking place in a moving liquid layer and that the velocity field is restricted to unidirectional flow. The goveming differential equation for this phenomenon can be written as:
This equation can be reduced to dimensionless taking a characteristic length (8) defined by: 6 = 9/kL0
form by (2)
where kLOis the average mass transfer coefficient in the absente of chemical reaction. Provided the penetration thickness is small, the velocity can be expanded in Taylor’s series from the interface and the final dimensionless equation can be written as:
18 February 1970)
(1+(o/y)rl+($y2)rlP+...)~=~-e~ (31 where a = ( LI;/ IJ,) 6; 7) = (Y/W
l
= +( U;l CJ,)8x; y = (k, cBOW’2/k,,o;
7 = (kz ce.x)/ U,;
CA = c,/cA,(4a,b,c,d,e,f)
s and ’ denoting interfacial values and differentiation with respect to y respectively. Equation (3) immediately suggests a solution of the form: c*A = C~to,(~,~)+~-‘C~,,,(r)>7)+Y-2CA*o(1).7+.
...
(3
It must be pointed oui that the present treatment is not uniformely valid when the interfacial velocity (U,) approaches zero. In such a case the Taylor series used above can become a poor representation of the actual velocity profile in the zone of thickness 6 where mass transfer is taking place. Due to this fact our approximate method of solution presented above breaks down in the proximity of zero interfacial velocity ( U,). By replacing (5) into differential Eq. (3) a series of differential equations is generated. Even if the solution for each czc”, is not known, from the form of the solution (5) it can be deduced that the first term of Eq. (51, czo,, wil1 be an . . approxtmatton to the actual solution with error of the order y-‘. Thus for large values of y the equation to be solved is just given by
a
apc*
_ CL = --c* A w
a7
a$
*KV
(6)
whose solution is very well known and coincides with that given by the penetration theory ifproper boundary conditions are used. The same approach could be used to consider the same problem but accompanied with a second order irreversible reaction. Again for large values of the parameter y two equations similar to (6) would have been obtained, except that the last term on the right hand side wil1 be now a product of cAto, and c&,,) The simultaneous solution of these two equations was given by Brian et af. [2] and their resuhs were correlated with very smal1 deviation by the approximate solution of Van Krevelen and HoBijzer[3] provided the ratio of diffusivities for components A and B is approximately one. Their approximate equation can be written in terms of the reaction factor as:
1239
@ = y[~]“*~anh[y[$$)]“z]}-l
(71
Shorter Commumcations where @ = kLIkLa and
reducing exactly Eq. (3) to that used by the penetration theory. In fact for laminar flows, as it is the case, (UilU,) and (UI/ U,) are independent of Pe while S (see Eq. (2)) decreases as the Pe increases, approaching zero in the limiting case of Pe being infinite. So it can be concluded that at least for laminar flows, the penetration theory can be used to estimate the limits of validity for Porter and Roberts’ criterium when either the values of -y are large or liquid phases are involved. The five regions indicated in Fig. 1 are the usual ones: Region 1: where @ approximately equals 1, the rate of mass transfer is controlled by the diffusion of component A, so that the flow conditions wil1 have a latge effect on the mass transfer coefficient. The limits for this region can be obtained by solving y - tan gh (y), thus
@, = 1+ (Q$c~,)
k,, being the average mass transfer coefficient in the presence of the chemical reaction while the other variables were already detined. However Eq. (7) is not expected to give exact results when the values of y are very small. To show the deviation from Eq. (7) at low values of y, the results of Rutland and Pfeffer[4] and Johnson and Akehata[S] were used here. They studied the dissolution of a sphere accompanied with first order irreversible reaction at very low Reynolds numbers (less than 1). Their results are presented in Fig. 1 together with those given by Eq. (7) by plotting @ as a function of y defined as:
y = 2(/3/Sh0) = (k,c,,i30”*/k~”
0 < y i 0.3. (8)
Region 11: is a transition region where the rate of mass transfer changes from being diffusional controlled to take place on the interface itself as y increases. Thus, though the influence of the flow conditions are less important than in Region 1, they cannot be neglected. Region 111: it is just the zone where the mass transfer coefficient is truly independent of the flow conditions since the reaction takes place right at the interface. The limits for this zone can be obtained solving (7) for @ = (1 kO@t)y, thus, giving approximately
/3’ being the second Damköhler number: (kacB,/9)re; SIP the Sherwood number in the absente of chemical reaction: (2 kLor/ 9) and r the radius of the dissolving sphere.
The paramettic curves drawn in Fig. 1 for low values of y. (y less than 8) correspond to different values of the Peclet number (Pe = U,r/ 9% Since Pe = 0 wil1 be the case where the fluid is stagnant, the curve for Pe = 0, wil1 define the region in Fig. 1, where the fluid dynamic effects wil1 be negligible in this particular case. ft is also seen that as Pe increases for a fixed value of y, there is a trend to approach the curve described by the results of the penetration theory. This is also coherent with our analysis, since in the present situation it can be shown that as Pe -+ ~0, a and c + 0, thus
6 5 4
-
3
-
2
-
(9)
1.8 d y i (1 +0.1
(10)
For 0, less than 8, there is no Region 111 so that flow condi-
Y Fig. 1.
1240
Shorter Communications tions wil1 always affect the mass transfer coefficient when dealing with very dilute solutions of component B. Region IV: is similar to Region 11 and also here the mass transfer coefficient wil1 be aífected by the flow conditions. The rate of mass transfer is now controlled partially by the diffusion of component B and the limiting case wil1 be that where the reaction is taking place in a plane which is no longer located at the interface but inside the liquid layer. Thus in this region the fluid dynamic effect wil1 increase as y does. Region V: the reaction occurs on a plane where the concentration of both components is zero; thus the process is diffusional controlled as described in Region 1. Incidentally even if, as Porter and Roberts[l] stated, the reaction factor is independent of flow conditions in this zone, this does not mean that the mass transfer coefficient wil1 also be; on the contrary wil1 be as much affected as the physical one. The lower limit for this region can be obtained solving (7) for
carbon dioxide to sodium hydroxide solutions on an unstable film falling down on the external surface of a wire. Their measurements clearly indicate the influence of the Reynolds’ number on the mass transfer coefficient. The experimental results presented graphically kJ 49 as a function of Reynolds’ number in a double log. paper, showed that the slope changes from 1 to 0 as the sodium hydroxide concentration is increased from 0 to 1 mol/1 approximately. It has been shown that the penetration theory can be used as a first approximation to determine the limits on y(Eq. (10)) where the fluid dynamic effects are expected to be negligible. The prediction given by the penetration theory wil1 be better as y increases to large values or when a and Qwil1 be smal1 this being the case for liquid laminar flows. However it is also shown that this field is quite open to more theoretical and experimental work. J. C. GOTTIFREDIt A. A. YERAMIAN J. J. RONCOt
(11)
This simple model is valid when the intensity of turbulente in the vicinity of the interface is very low but when the interfacial tension is smal1 so as to allow for some sart of instabilities to be present other concepts should be used to validate our conclusions since under these conditions velocities profiles are not very wel1 known. However Banetjee er al.[6] succeeded in predicting mass transfer coefficient for physical absorption to an unstable falling film by using the result of the penetration theory, namely k,O being inversely proportional to the square root of the time of renewal, and by calculating this time through the applications of the similarity hypothesis of Kolmogorov. Thus Fig. 1 could stil1 be used if kLo is calculated in a similar fashion to deterrnine the magnitude of y. We recall that the limits given above are not valid for every system to be dealt with. However as shown in the case of the dissolution of a sphere they can be used with a great deal of confidence provided the Peclet number is large (-- 104). In a such a case the net effect of Q and l becomes negligible so as to render valid our treatment even for values of y of the order of 1. Thus depending on the system to be analyzed the solution of infinite terms proposed by Eq. (5) can be very wel1 approximated by the first term (cAt,,,*) even for smal1 values of y. In any case, keeping in mind that the principal scope here is to determine the zone where the fluid dynamic effects are negligible, when dealing with fluidsolid or some liquid-liquid systems characterized by a smal1 Pe number the lower limit of y given by Eq. (10) should only be considered as a rough approximation. Ronco and Coeuret [7] have used some results obtained by Marangozis et al.[8], who presented a model for mass transfer with and without chemical reaction in a turbulent boundary layer where the fluid dynamic effect was only included through the turbulent diffusivity but otherwise considering the fluid velocity to be negligible, to show that Fig. 1 wil1 give correct results provided the Schimdt number was of the order of 1000 though for Sc of the order of 1 the lower limit for y defined by Eq. (10) is modified to 5.5 approximately. Our conclusions are also supported by recent experiments reported by Ronco et al. [9] who studied the absorption of
Departamento de Tecnologia Quimica Universidad Nacional de La Plata La Plata, Argentina
tResearch
Fellow C.N.1.C.y.T.
NOTATION radius of the sphere concentration C: dimensionless concentration 9 diffusion coefficient k kinetic constant for a second order reaction k, average mass transfer coefficient Sc Schmidt number Sh” Sherwood number 2 (k,,Or/ 9) velocity u X spatial coordinate Y spatial coordinate r
Greek symbols a
dimensionless parameter defined by (4a) Damkohler number (k2cs,/9)r* dimensionless variable defhied by (4~) dimensionless variable delïned by (4b) characteristic length defined by (2) dimensionless distance defined by (4d) dimensionless time defined by (4e) reaction factor (kJkLo) reaction factor for instantaneous reaction (csJc4, ))
Subscripts
A,B
1241
: 0
(1 +
and superscripts
components A and B interfacial value bulk value differentiation with respect to y refers to the correspondent case in the absente chemical reaction
of
Shorter Communications
[1] [2] [3] [4] [5] [6] [7] [8] [9]
REFERENCES PORTER K. E. and ROBERTS D., Chem. Engng Sci. 1969 24 695. BRIAN P. T. L., HURLEY J. F. and HASSELTINE E. H.,A.I.Ch.E. Jll9617 226. VAN KREVELEN D. and HOFTIJZER R., Rec. Trau. Chic. 1948 67 563. RUTLAND L. and PFEFFER R.,A. Z. Ch. E. Jll967 13 182. JOHNSON A. 1. and AKEHATA T., Can. J. chem. Engng 1967 43 10. BANERJEE S., SCO’M D. S. and RHODES E., Znd. Engng Chem. Fundls 1968 7 22. RONCO J. J. and COEURET F., Génie,chim. 1968 100 189. MARANGOZIS J.,TRASS 0. andJOHNSON A. I., Can.J. chem. Engng 1963 41 195. COEURET F., JAMET B. and RONCO J. J., Chem. Engng Sci. 197025 17.
Chemical Engineering Science, 1970, Vol. 25, pp. 1242-1244.
Pergamon Pres%
Printed in Great Britain.
Heat transfer and the gaseous reduction of haematite (First received 11 August 1969; in revised form 22 January 1970) INTRODUCTION reduction of iron oxides has frequently been investigated, and a large number of contributions to this field are to be found in the literature. The temperature of samples undergoing reduction is usually assumed[l] to be the same as that indicated by an adjacent thermocouple, on the ground that the heats of reactions (Table 1) are too smal1 to have much effect. However, Strangway, Toppi and Ross THE GASEOUS
Table 1. Heats of reactiont (K. cal./atom of Fe) for the reduction of iron oxides by hydrogen and carbon monoxide at 1000°C Reduction step Fe,O, + Fe,O, Fe304 + Fe,.,0 Fe,.,0 + Fe tcalculated
CO AH” 1000°C
H* AH” 1000°C
-1.92 +1.55 -4.34
-064 +3.71 +3.75
the samples made from BDH ferric oxide approached 95 per cent of the theoretical density, while those made from the magnetite concentrate were fully converted to porous haematite (- 30 per cent porosity). The reduction tests were carried out in a Stanton Massflow thermobalance, and the sample temperature during reduction was measured by a thermocouple inserted in the central hole as shown in Fig. 1. The output of thermocouple A was opposed by an emf obtained from a Cambridge potentiometer, and the resultant potential fed to a Speedomax W 1 mV recorder. Prior to the introduction of reducing gas, the recorder indicated a sinusoidal temperature variation of
from tabulated data[2].
[3] have shown that this assumption is not justified in the case of hydrogen reduction. In view of the fact that the present concept of the reduction mechanism is largely based upon the results of experiments with hydrogen[4,5], while in practice carbon monoxide is the more common-reducing agent, a comparison of the heat flux in haematite during reduction by both gases was considered to be appropriate. It is this work that is reported here. EXPERIMENTAL PROCEDURE Cylindrical samples (approx. 1 cm X 1 cm) were manufactured from both BDH calcined ferric oxide powder and a magnetite concentrate. In each case, approximately 3 g of powder was placed in a single-acting die and compacted using a plunger that also produced a centra1 hole (0.25 cm dia.) extending to the mid-height of the sample. Previous experience with these materials indicated that after firing in oxygen,
Fig. 1. Arrangement
1242
of sample and thermocouples tube.
in fumace
Pergamon Press.
Printed in Great Britain.
On the effect of ílow patterns on the rate of mass transfer with chemical reaction (First receioed 15 September 1969; in reuisedform IN A RECENT paper Porter and Roberts [ 1] presented a rather simple theoretical model for mass transfer with chemical reaction to show the effect of flow pattems on the rate of mass transfer. The authors analyzed the behaviour of the goveming differential equation by seeking a solution composed by two terms. The first of them represents the solution of the so called “steady state” equation where the rate of diffusion is exactly balanced by the rate of reaction while the second term is in fact the solution of the whole equations but with modified boundary conditions. Clearly, the fluid dynamic effects are only present in the second term of the solution. Moreover by introducing the contact time as the ratio of the spatial coordinate and the interfacial velocity, the second term of the solution can be interpreted as a contact time dependent solution, so that when the contact time approaches infinite the first term wil1 represent the actual solution of the problem. Under these conditions fluid dynamic effects wil1 become negligible. After analyzing some particular cases the authors concluded that laboratory apparatus for scale up studies could be chosen on grounds of simplicity and convenience due to the general smal1 effects on the mass transfer coefficient produced by fluid dynamic conditions. However proper limits of validity for Porter and Roberts’ criterium were not given. Moreover since the analysis was restricted to pseudo first or zero order kinetics the dependence of those limits on the concentration of the reacting component present on the absorbing fluid phase was not taken into account. The purpose of this communication is to show that the limits mentioned above can be established with a good degree of approximation using previous results of the penetration theory. Let US first assume that a pseudo first order irreversible reaction between components A and B is taking place in a moving liquid layer and that the velocity field is restricted to unidirectional flow. The goveming differential equation for this phenomenon can be written as:
This equation can be reduced to dimensionless taking a characteristic length (8) defined by: 6 = 9/kL0
form by (2)
where kLOis the average mass transfer coefficient in the absente of chemical reaction. Provided the penetration thickness is small, the velocity can be expanded in Taylor’s series from the interface and the final dimensionless equation can be written as:
18 February 1970)
(1+(o/y)rl+($y2)rlP+...)~=~-e~ (31 where a = ( LI;/ IJ,) 6; 7) = (Y/W
l
= +( U;l CJ,)8x; y = (k, cBOW’2/k,,o;
7 = (kz ce.x)/ U,;
CA = c,/cA,(4a,b,c,d,e,f)
s and ’ denoting interfacial values and differentiation with respect to y respectively. Equation (3) immediately suggests a solution of the form: c*A = C~to,(~,~)+~-‘C~,,,(r)>7)+Y-2CA*o(1).7+.
...
(3
It must be pointed oui that the present treatment is not uniformely valid when the interfacial velocity (U,) approaches zero. In such a case the Taylor series used above can become a poor representation of the actual velocity profile in the zone of thickness 6 where mass transfer is taking place. Due to this fact our approximate method of solution presented above breaks down in the proximity of zero interfacial velocity ( U,). By replacing (5) into differential Eq. (3) a series of differential equations is generated. Even if the solution for each czc”, is not known, from the form of the solution (5) it can be deduced that the first term of Eq. (51, czo,, wil1 be an . . approxtmatton to the actual solution with error of the order y-‘. Thus for large values of y the equation to be solved is just given by
a
apc*
_ CL = --c* A w
a7
a$
*KV
(6)
whose solution is very well known and coincides with that given by the penetration theory ifproper boundary conditions are used. The same approach could be used to consider the same problem but accompanied with a second order irreversible reaction. Again for large values of the parameter y two equations similar to (6) would have been obtained, except that the last term on the right hand side wil1 be now a product of cAto, and c&,,) The simultaneous solution of these two equations was given by Brian et af. [2] and their resuhs were correlated with very smal1 deviation by the approximate solution of Van Krevelen and HoBijzer[3] provided the ratio of diffusivities for components A and B is approximately one. Their approximate equation can be written in terms of the reaction factor as:
1239
@ = y[~]“*~anh[y[$$)]“z]}-l
(71
Shorter Commumcations where @ = kLIkLa and
reducing exactly Eq. (3) to that used by the penetration theory. In fact for laminar flows, as it is the case, (UilU,) and (UI/ U,) are independent of Pe while S (see Eq. (2)) decreases as the Pe increases, approaching zero in the limiting case of Pe being infinite. So it can be concluded that at least for laminar flows, the penetration theory can be used to estimate the limits of validity for Porter and Roberts’ criterium when either the values of -y are large or liquid phases are involved. The five regions indicated in Fig. 1 are the usual ones: Region 1: where @ approximately equals 1, the rate of mass transfer is controlled by the diffusion of component A, so that the flow conditions wil1 have a latge effect on the mass transfer coefficient. The limits for this region can be obtained by solving y - tan gh (y), thus
@, = 1+ (Q$c~,)
k,, being the average mass transfer coefficient in the presence of the chemical reaction while the other variables were already detined. However Eq. (7) is not expected to give exact results when the values of y are very small. To show the deviation from Eq. (7) at low values of y, the results of Rutland and Pfeffer[4] and Johnson and Akehata[S] were used here. They studied the dissolution of a sphere accompanied with first order irreversible reaction at very low Reynolds numbers (less than 1). Their results are presented in Fig. 1 together with those given by Eq. (7) by plotting @ as a function of y defined as:
y = 2(/3/Sh0) = (k,c,,i30”*/k~”
0 < y i 0.3. (8)
Region 11: is a transition region where the rate of mass transfer changes from being diffusional controlled to take place on the interface itself as y increases. Thus, though the influence of the flow conditions are less important than in Region 1, they cannot be neglected. Region 111: it is just the zone where the mass transfer coefficient is truly independent of the flow conditions since the reaction takes place right at the interface. The limits for this zone can be obtained solving (7) for @ = (1 kO@t)y, thus, giving approximately
/3’ being the second Damköhler number: (kacB,/9)re; SIP the Sherwood number in the absente of chemical reaction: (2 kLor/ 9) and r the radius of the dissolving sphere.
The paramettic curves drawn in Fig. 1 for low values of y. (y less than 8) correspond to different values of the Peclet number (Pe = U,r/ 9% Since Pe = 0 wil1 be the case where the fluid is stagnant, the curve for Pe = 0, wil1 define the region in Fig. 1, where the fluid dynamic effects wil1 be negligible in this particular case. ft is also seen that as Pe increases for a fixed value of y, there is a trend to approach the curve described by the results of the penetration theory. This is also coherent with our analysis, since in the present situation it can be shown that as Pe -+ ~0, a and c + 0, thus
6 5 4
-
3
-
2
-
(9)
1.8 d y i (1 +0.1
(10)
For 0, less than 8, there is no Region 111 so that flow condi-
Y Fig. 1.
1240
Shorter Communications tions wil1 always affect the mass transfer coefficient when dealing with very dilute solutions of component B. Region IV: is similar to Region 11 and also here the mass transfer coefficient wil1 be aífected by the flow conditions. The rate of mass transfer is now controlled partially by the diffusion of component B and the limiting case wil1 be that where the reaction is taking place in a plane which is no longer located at the interface but inside the liquid layer. Thus in this region the fluid dynamic effect wil1 increase as y does. Region V: the reaction occurs on a plane where the concentration of both components is zero; thus the process is diffusional controlled as described in Region 1. Incidentally even if, as Porter and Roberts[l] stated, the reaction factor is independent of flow conditions in this zone, this does not mean that the mass transfer coefficient wil1 also be; on the contrary wil1 be as much affected as the physical one. The lower limit for this region can be obtained solving (7) for
carbon dioxide to sodium hydroxide solutions on an unstable film falling down on the external surface of a wire. Their measurements clearly indicate the influence of the Reynolds’ number on the mass transfer coefficient. The experimental results presented graphically kJ 49 as a function of Reynolds’ number in a double log. paper, showed that the slope changes from 1 to 0 as the sodium hydroxide concentration is increased from 0 to 1 mol/1 approximately. It has been shown that the penetration theory can be used as a first approximation to determine the limits on y(Eq. (10)) where the fluid dynamic effects are expected to be negligible. The prediction given by the penetration theory wil1 be better as y increases to large values or when a and Qwil1 be smal1 this being the case for liquid laminar flows. However it is also shown that this field is quite open to more theoretical and experimental work. J. C. GOTTIFREDIt A. A. YERAMIAN J. J. RONCOt
(11)
This simple model is valid when the intensity of turbulente in the vicinity of the interface is very low but when the interfacial tension is smal1 so as to allow for some sart of instabilities to be present other concepts should be used to validate our conclusions since under these conditions velocities profiles are not very wel1 known. However Banetjee er al.[6] succeeded in predicting mass transfer coefficient for physical absorption to an unstable falling film by using the result of the penetration theory, namely k,O being inversely proportional to the square root of the time of renewal, and by calculating this time through the applications of the similarity hypothesis of Kolmogorov. Thus Fig. 1 could stil1 be used if kLo is calculated in a similar fashion to deterrnine the magnitude of y. We recall that the limits given above are not valid for every system to be dealt with. However as shown in the case of the dissolution of a sphere they can be used with a great deal of confidence provided the Peclet number is large (-- 104). In a such a case the net effect of Q and l becomes negligible so as to render valid our treatment even for values of y of the order of 1. Thus depending on the system to be analyzed the solution of infinite terms proposed by Eq. (5) can be very wel1 approximated by the first term (cAt,,,*) even for smal1 values of y. In any case, keeping in mind that the principal scope here is to determine the zone where the fluid dynamic effects are negligible, when dealing with fluidsolid or some liquid-liquid systems characterized by a smal1 Pe number the lower limit of y given by Eq. (10) should only be considered as a rough approximation. Ronco and Coeuret [7] have used some results obtained by Marangozis et al.[8], who presented a model for mass transfer with and without chemical reaction in a turbulent boundary layer where the fluid dynamic effect was only included through the turbulent diffusivity but otherwise considering the fluid velocity to be negligible, to show that Fig. 1 wil1 give correct results provided the Schimdt number was of the order of 1000 though for Sc of the order of 1 the lower limit for y defined by Eq. (10) is modified to 5.5 approximately. Our conclusions are also supported by recent experiments reported by Ronco et al. [9] who studied the absorption of
Departamento de Tecnologia Quimica Universidad Nacional de La Plata La Plata, Argentina
tResearch
Fellow C.N.1.C.y.T.
NOTATION radius of the sphere concentration C: dimensionless concentration 9 diffusion coefficient k kinetic constant for a second order reaction k, average mass transfer coefficient Sc Schmidt number Sh” Sherwood number 2 (k,,Or/ 9) velocity u X spatial coordinate Y spatial coordinate r
Greek symbols a
dimensionless parameter defined by (4a) Damkohler number (k2cs,/9)r* dimensionless variable defhied by (4~) dimensionless variable delïned by (4b) characteristic length defined by (2) dimensionless distance defined by (4d) dimensionless time defined by (4e) reaction factor (kJkLo) reaction factor for instantaneous reaction (csJc4, ))
Subscripts
A,B
1241
: 0
(1 +
and superscripts
components A and B interfacial value bulk value differentiation with respect to y refers to the correspondent case in the absente chemical reaction
of
Shorter Communications
[1] [2] [3] [4] [5] [6] [7] [8] [9]
REFERENCES PORTER K. E. and ROBERTS D., Chem. Engng Sci. 1969 24 695. BRIAN P. T. L., HURLEY J. F. and HASSELTINE E. H.,A.I.Ch.E. Jll9617 226. VAN KREVELEN D. and HOFTIJZER R., Rec. Trau. Chic. 1948 67 563. RUTLAND L. and PFEFFER R.,A. Z. Ch. E. Jll967 13 182. JOHNSON A. 1. and AKEHATA T., Can. J. chem. Engng 1967 43 10. BANERJEE S., SCO’M D. S. and RHODES E., Znd. Engng Chem. Fundls 1968 7 22. RONCO J. J. and COEURET F., Génie,chim. 1968 100 189. MARANGOZIS J.,TRASS 0. andJOHNSON A. I., Can.J. chem. Engng 1963 41 195. COEURET F., JAMET B. and RONCO J. J., Chem. Engng Sci. 197025 17.
Chemical Engineering Science, 1970, Vol. 25, pp. 1242-1244.
Pergamon Pres%
Printed in Great Britain.
Heat transfer and the gaseous reduction of haematite (First received 11 August 1969; in revised form 22 January 1970) INTRODUCTION reduction of iron oxides has frequently been investigated, and a large number of contributions to this field are to be found in the literature. The temperature of samples undergoing reduction is usually assumed[l] to be the same as that indicated by an adjacent thermocouple, on the ground that the heats of reactions (Table 1) are too smal1 to have much effect. However, Strangway, Toppi and Ross THE GASEOUS
Table 1. Heats of reactiont (K. cal./atom of Fe) for the reduction of iron oxides by hydrogen and carbon monoxide at 1000°C Reduction step Fe,O, + Fe,O, Fe304 + Fe,.,0 Fe,.,0 + Fe tcalculated
CO AH” 1000°C
H* AH” 1000°C
-1.92 +1.55 -4.34
-064 +3.71 +3.75
the samples made from BDH ferric oxide approached 95 per cent of the theoretical density, while those made from the magnetite concentrate were fully converted to porous haematite (- 30 per cent porosity). The reduction tests were carried out in a Stanton Massflow thermobalance, and the sample temperature during reduction was measured by a thermocouple inserted in the central hole as shown in Fig. 1. The output of thermocouple A was opposed by an emf obtained from a Cambridge potentiometer, and the resultant potential fed to a Speedomax W 1 mV recorder. Prior to the introduction of reducing gas, the recorder indicated a sinusoidal temperature variation of
from tabulated data[2].
[3] have shown that this assumption is not justified in the case of hydrogen reduction. In view of the fact that the present concept of the reduction mechanism is largely based upon the results of experiments with hydrogen[4,5], while in practice carbon monoxide is the more common-reducing agent, a comparison of the heat flux in haematite during reduction by both gases was considered to be appropriate. It is this work that is reported here. EXPERIMENTAL PROCEDURE Cylindrical samples (approx. 1 cm X 1 cm) were manufactured from both BDH calcined ferric oxide powder and a magnetite concentrate. In each case, approximately 3 g of powder was placed in a single-acting die and compacted using a plunger that also produced a centra1 hole (0.25 cm dia.) extending to the mid-height of the sample. Previous experience with these materials indicated that after firing in oxygen,
Fig. 1. Arrangement
1242
of sample and thermocouples tube.
in fumace