xss = steady-state value of x, ( n x 1) y = xss + x
Greek Letters = lower bound on u, ( r x 1) fi = upper bound on u, ( r x 1) CP = transition matrix = exp (AT) = [JT0 exp (A(T - A))&] B Literature Cited N
+
Jaakola. T. H. I., Luus, R., Oper. Res.. 22, 415 (1974). Koepcke. R., Lapidus, L., Chem. Eng. Sci.. 16, 252 (1961). Lapidus, L., Luus, R., "Optimal Control of Engineering Processes," Blaisdell, Waltham. Mass., 1967. Luus, R., Can. J. Chem. Eng.. 52, 98 (1974a). Luus. R., Chem. Eng. Sci.. 29, 1013 (1974b). Luus. R . , Jaakola, T. H. J., A.l.Ch.E. J., 19, 760 (1973). Siebenthal, C. D.. Aris. R., Chem. Eng. Sci.. 19, 729 (1964).
Received for review October 29, 1973 Accepted May 28,1974
Edgar, T. F., Lapidus. L., A.l.Ch.E. J.. 18, 780 (1972)
Thermophoretic Deposition of Aerosol Particles in a Heat-Exchanger Pipe Gunji Nishio,* Susumu Kitani, and Kazuhiro Takahashi Nuclear Fuel Research Division, Japan Atomic Energy Research Institute, Tokai Research Establishment, Tokai-mura, Naka-gun, Ibaraki-ken, Japan
Aerosol particles s u s p e n d e d in hot fluids tend to deposit on the cold walls of piping in chemical plants, so that it is important to estimate t h e degree of scaling or fouling caused by the deposition of aerosol particles. T h e s t u d y was undertaken to obtain information on t h e thermophoretic deposition of aerosol particles in a heat-exchanger pipe. T h e experiments were m a d e to determine the deposition of aerosol on t h e pipe walls of a heat exchanger in a temperature gradient along its length, a s a function of velocity range from laminar to turbulent. T h e degree of thermophoretic deposition of aerosol agrees relatively well with its analysis, connecting thermal repulsive force acting on particles with parameters of t h e heat exchanger, such a s mean temperature difference, velocity, and temperature gradient between hot fluid and coolant.
Aerosol particles move toward the lower temperature under the influence of a temperature gradient. To elucidate this phenomenon called thermophoresis, experiments have been made by many workers with the modified Millikan apparatus. The theory of thermophoresis was first developed by Epstein (1929). He introduced the thermal force acting on aerosol particles for a continuum regime by solving the Navier-Stokes equations with the use of the boundary conditions considering the field of temperature gradient with thermobalance of heat conduction. I t is known that Epstein's equation can explain the experimental results for aerosol particles of low thermal conductivity, but the equation does not coincide with those of high thermal conductivity. To extend the Epstein theory, Brock (1962) derived a theory applying to the aerosol particles of high thermal conductivity, considering a slip flow condition. Deryagin and Yalamov (1965) reported a theory of thermophoresis of large aerosol particles by using the concept of a heat flux due to the temperature gradient. On the other hand, Schmitt (1959) studied a thermal force acting on particles in a transition regime of the Knudsen number, 0.1 < ( X / r ) < 10, and the equation of the thermal force was analytically derived by Brock (1967). A theoretical equation to predict the thermal force over the wide regime from the transition to the continuum was introduced by Jacobsen and Brock (1965), which agreed with the experiments of sodium chloride aerosol. Because of the scaling or fouling on the cold walls of piping in chemical plants, caused by deposition of aerosol particles suspended in hot fluid, information on the thermophoretic deposition of particles is important for the maintenance of such plants which usually use many pipes 408
Ind. Eng.
Chem., Process Des. Develop.,Vol. 13, No. 4, 1974
with temperature gradients, such as a heat exchanger, an evaporator, a steam boiler, etc. For the thermophoretic deposition of aerosol particles in the conduit tube, Postma (1961) analytically obtained a fractional loss of the particles in the tube under the condition of turbulent stream but did not make experiments to confirm the analytical results. Byers and Calvert (1969) have performed experiments and analysis for thermophoretic deposition of particles contained in the hot turbulent stream, from the aspect of air cleaning. These experiments have been performed to obtain information by varying the Knudsen number for which the particle size was controlled in the range of 0.3 to 1.3 pm. Recently, Singh and Byers (1972) have conducted the experiments of thermophoretic deposition of aerosol in the transition regime, under the turbulent flow through the externally cooled tube, using polydisperse sodium chloride aerosol. They evaluated the accommodation coefficients of a thermal force proposed by Brock in the regime as a function of fluid temperature and particle size. In a high temperature gas-cooled reactor (HTGR) using helium as the coolant, fine carbon dusts containing radioactive material may be suspended in the high-temperature stream. The dusts will deposit on the surface of inner pipes of a primary heat exchanger due to the thermophoresis, so that the dusts do not only reduce the heat transfer due to adherence of the particles but also carburize the pipe material by diffusion of carbon atoms into the wall. The life of heat-exchanger pipes may be governed by the brittleness due to carburization of the pipe. The purpose of the present study is to obtain basic information on the thermophoretic deposition of aerosol particles in the
hot gas stream through the heat exchanger pipe. The experiments were made to determine the deposition of aerosol on the wall of pipe in the temperature gradient along its length, as a function of the flow velocity from laminar to turbulent. The thermophoretic deposition of aerosol particles is studied by comparing the experimental results with analysis, connecting the thermal force acting on particles of thermophoresis with the heat transfer in the heat exchanger.
Y
%+AX
‘x
-
Flow
\
V/A
do-
co-9
X
Flow-&
Flow
---
F d X
Figure 1. Scheme of heat balance in a countercurrent type heatexchanger pipe.
Theoretical Treatment Because of the low concentration of the aerosol used and the short duration of the experiment, the rate of heat transfer across the pipe wall is not affected by the adherence of particles, and so the effect of fouling factor due to the deposition of particles will not be considered. In the heat balance of the element (cross hatched) of heat exchanger as shown in Figure 1, the rate of heat transfer through a film of laminar sublayer of the fluid near the wall must be equal to the value of W,C,dT for the hot fluid, where Wg is the weight of flow ( = ( T i 4)DZup) and C, is the specific heat a t a constant pressure.
ZT(dT/dy) is the velocity of a particle to the cold wall due to thermophoresis. Epstein, Brock, and Jacobsen and Brock introduced the theoretical equations of 2.r for a single size particle with radius r. To apply the equations for the aerosol, r is substituted with the arithmetic mean radius 7 for simplification. These simplified values Z r are given as
2, =
i7
dQ = hi7D(Tg - T,)dx = - -D2vpCpdT (1) 4 where h is the film coefficient of heat transfer, D is the diameter of the inner pipe, u and p are the velocity and density of the fluid, and Tg and T, are the temperatures of hot fluid and cold surface of the pipe wall, respectively. The film coefficient ( h ) is obtained by dividing the thermal conductivity ( k g ) of fluid by the thickness (6) of film near the wall, i.e., h = k g / b . Since the film coefficient changes considerably with fluid velocity or with thickness of the laminar sublayer, the value of film coefficient in the pipe is obtained by the Nusselt-type equations
= 2.01
k,
( wc k,l ) g
p
1/3
(E)
0.14
at laminar flow
2J-B
=
(k$kp)
+ Ct(b’7) +
(4h/3)Cm(h/T){(k$kp) + C t h / $ - 1) 3p (8) g p T ( 1 + 3Cm(A/3)(1 + 2(k$kp) + 2 C t ( h m ) where ZE, ZB, and Z J - ~ are those in the equations by Epstein, Brock, and Jacobsen and Brock, respectively. is the average absolute temperature of the fluid, and h , is the thermal conductivity of particles. If the size distribution of particles is log-normal, the mean radius 7 is
-
Y = Y, exp(0.5
hD = 0.023 k,
(7)(
CE kR gE
)
at turbulent flow
where 1 is the length of the heat conduction path, G is the mass velocity of the fluid, p is the viscosity of the fluid, and p, is the viscosity of the fluid near the wall. If the temperature of the pipe wall is assumed to be constant, the temperature gradient through the laminar sublayer is given by solving eq 1for the steady heat flow.
where Tgois the temperature of the hot fluid at the inlet of the heat exchanger and Twois the temperature on the surface of wall, calculated from the measured temperature difference between the hot fluid and coolant at the inlet, considering the heat resistance of the heat exchanger. Since the concentration decrease of the airborne particles is equivalent to the increase of the deposited particles, the material balance in thermophoretic deposition in the element in Figure 1 is
where C is the concentration of airborne particles, and
In2 0,)
(9 ) where rg is the logarithmic mean radius of aerosol particles and ug is the logarithmic standard deviation. Substituting eq 4 into eq 5 , a fractional loss of the aerosol concentration a t a position x can be obtained by the integration of eq 5 using the boundary condition of C = COa t n = 0.
If the length of a heat exchanger is infinite, the fractional loss takes the minimum value, i.e., C,/Co, because the exponential term in eq 10 approaches zero. The fractional deposition in the elements tends to decrease in the temperature gradient along the pipe length. This fractional decrease due to thermophoresis, Le., the thermophoretic deposition of particles in the element ( A x ) a t a given point of the heat exchanger, is given as
=
c, - CX+& (c, + C,+,)/2
where the fractional deposition is experimentally related to (AF.r)x = (AFo), - (AF,),; ( A F o )is~ the apparent fractional deposition in Ax a t position n under the temperature gradient along the pipe length, (AF,), the fractional deposition due to other mechanisms such as diffusion of small particles, gravitational settling of large particles, Ind. Eng. Chem., Process Des. Develop.. Vol. 13, No. 4, 1974
409
and eddy diffusion due to fluid turbulence, in the absence of a temperature gradient. Since the logarithmic mean temperature difference (AT),, in a heat exchanger varies with length of the pipe, it is convenient to determine the relationship between (AT),, and the degree of thermophoretic deposition. In the heat exchanger, the rate of heat transfer from hot fluid to coolant over the length of 0-L can be expressed as QL
=sL
Thermo- Cwple
Thermo-Couple
I rHeat I To Pump
E
Reformer
Genefalor
3 T
h
e
Re
r
m
j
h
Exchonper
Z Filter TCoolant b
I
'
r
V p J
-
k a D ( T , - T,) dx = UaDL(AT),, =
0
where U is the overall coefficient of heat transfer. ATH is the temperature difference between hot fluid and coolant a t the inlet of the exchanger, and ATI, is the difference at the distance L. The mean temperature difference is thus obtained from eq 4 and 12.
Sillco Gel
Figure 2. Schematic of experimental arrangement for thermophoretic deposition of aerosol particles.
Substituting eq 13 into 10, the fractional deposition of aerosol in the heat exchanger is obtained as " 0
In eq 14, it is seen that the larger the mean temperature difference, the higher becomes the deposition of aerosol due to thermophoresis; on the other hand, the higher the hot-stream velocity, the lower the deposition. Experimental Procedure Apparatus and Method. The apparatus consists of a sodium vapor generator, a reformer from sodium vapor to its oxide aerosol, a reservoir, a reheater, and a heat exchanger, as shown in Figure 2. The generator is heated in an electric furnace a t 550°C a t the fixed velocity of pure argon; sodium vapor is generated from the sodium metal in the boat. The vapor is converted to sodium oxide particles (NazO) in the reformer kept at 250°C, by air feed at a fixed flow through a silica gel column and particulate filter. The aerosol stream enters the reservoir, where large particles are removed by settling. The stream is then heated in a 4.5-kW electric furnace up to a specific temperature in the reheater filled with Ruschig rings. The adherence of particles onto the rings is small because of a thermal repulsive force to the particles by heating. The stream enters the heat exchanger which consists of a glass pipe (100 cm long and 0.5 cm in diameter, with 0.05 cm wall thickness) and a cylindrical glass jacket of 2.5 cm diameter. The stream after the heat exchanger is discharged through a particulate filter and a flow meter. Measurement of the Temperature. The temperature profile along the pipe length of the heat exchanger was measured by inserting a thermocouple in the pipe, as a function of the flow rate of the hot stream, the temperature of reheater, and the coolant temperature. The temperature of the hot stream at the inlet of the heat exchanger was continuously recorded. Figure 3 shows a typical temperature profile for the hot stream (T,) and for coolant in the jacket ( T c ) . Aerosol Concentration and Particle Size Distribution. A portion of the stream at a constant flow rate in front of the heat exchanger was taken through a particu410
Ind. Eng. Chem., Process Des. Develop.,Vol. 13, No. 4 , 1974
20
40 Distance
60 80 from inlef ( c m
100
Figure 3. Temperature profiles along distance from inlet for hot
stream (T,)and coolant (T,) in the heat exchanger. late filter as shown in Figure 2. The concentration of the aerosol was thus determined by atomic absorption spectrophotometry (Murata and Kitani, 1972). The size distribution of the aerosol was determined by sampling on grids in a thermal precipitator at the inlet of the test pipe; the numbers of particles collected on the grids were counted with an electron microscope. From the straight line in cumulative percentage of the particles plotted on a logarithmic-probability paper, the size distribution of aerosol was found to be log-normal as shown in Figure 4. Deposition Due to Thermophoresis. Experiments of the thermophoresis were made in the wide velocity range of laminar to turbulent. To obtain a sharp temperature profile along the pipe length, air was flowed a t 240 l./hr as the coolant for the laminar stream, and water a t 50 l./hr for the turbulent stream. The amount of aerosol deposited along the pipe length during a given period of experiment was obtained by cutting the regular intervals and measuring the deposited sodium atoms by atomic absorption spectrometry. Table I shows the apparent fractions ( A F O )deposited ~ on the pipe in each 5 cm along the pipe length, together with Tg,for various Re numbers. The effect of fluid disturbance due to the joints a t the inlet and outlet of the pipe might be in the amount of deposition, and therefore the deposition at the ends ( D / x = 0.1) was omitted in Table I. Deposition by Other T h a n Thermophoresis. To calculate the degree of thermophoretic deposition of aerosol in the heat exchanger, the contribution of the particle deposition (AF,)* other than by thermophoresis must be evaluated. With the apparatus, experiments were made to determine the amount of deposition for the contribution of other mechanisms as a function of fluid velocity at the room temperature. The curves by the deposition were
P
001
1 003
el
J
1
I 01
03
Re: 9 9 7 (Run 6 )
10
-
I pml
P w t i d e Diameter
Figure 4. Particle size distributions of sodium monoxide aerosol
-P
c3
8 r
f
0 10 20 30 40 50 60 70 80 90 1 0 0 Distance from inlet l c m I
0.15 0 Re 0 Re
oic
0
a
= 900
A Re
20
40
Distance
from
:
1400
80
60
inlet
Figure 7. Variations of fractional deposition of aerosol particles along the heat-exchanger pipe due to thermophoresis at Re = 997.
= 350
26 v) ._ v)
; n
0
24
6
4
22
2
20
B
f
'0
50 40 50 60 70 80 90
10
100
16
I cm I
Figure 5 . Amounts of particle deposited in each 10-cm length along distance from inlet, in the absence of temperature gradient.
18
14
L-
I7
RB_=3 2 4 0 (Run I O )
12 10
8
8
6
6
4
4
2
2
0
0
10 20 30 40 x) 60 70 80 90
Distance from inlet
I cm
0
0 10 20 30 40 50 60 70 80 90 Distance from inlet
I cm
)
Figure 8. Variations of fractional deposition of aerosol particles along the heat-exchanger pipe due to thermophoresis at Re = 3240.
25
h
0.0001
IO
100 Re:
1000
iOcQ0
DijP p
Figure 6. Dependence of fractional deposition in each 5-cm length on Reynolds number, in the absence of temperature gradient. nearly horizontal along the pipe length, as shown in Figure 5. Figure 6 shows the fractional deposition ( A F , ) us. the Reynolds number; that is, the deposition may be caused by in the absence of a temperature gradient. Because of the polydisperse aerosols used, the fraction in the laminar range decreases with increase of Reynolds number; that is, the deposition may be controlled by diffusion for small particles and settling of large particles. The fraction in the turbulent range, on the other hand, increases with Reynolds number; that is, the deposition may be caused by inertial effects of the fluid eddies. Thermal Conductivity of Aerosol Particle. In calculations of the thermophoretic deposition of particles, it is necessary to know the thermal conductivity of the aerosol particles. Its conductivity was determined as follows. The pellet of sodium monoxide was sandwiched between two
silver plates. Acetone and benzene were put in the cylindrical container below and above the sandwich, respectively, and both liquids were set boiling. The desired thermal conductivity was thus derived from the rate of evaporation of acetone and the temperature difference between the upper and lower silver plates. The measurements were made carefully, considering the influence of the humidity. The value of the conductivity was corrected by means of that of a standard quartz plate; it was taken at k , = 0.650 kcal/m hr "C a t the 83.3% theoretical density ( = 2.27 g/ cm3) of NapO. Results The apparent fractional depositions ( A F O ) in ~ Table I were corrected by substracting the deposition ( S F , ) in Figure 6 other than thermophoresis, L.e., ( A F , i x = ( A F o ) ~ - ( A F , ) . Figure 7 shows the value of ( A F r ) x deposited in the 5-cm intervals along pipe length in a laminar region (Re = 997) as a function of the temperature difference between stream and pipe wall at distance 5 cm from the inlet of the heat exchanger. Figure 8 also shows the same relation in a turbulent stream (Re = 3240). It is noted that the fractional deposition ( A F r ) x increases with temperature of the reheater. The relation between the fractional deposition ( A F r ) x along the pipe length and the Reynolds number is shown Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 4, 1974
411
Table I. Experimental Data on Apparent Fractional Deposition of Aerosol Particles in Each 5-cm Length with Hot Stream Temperature in the Pipe, Shown with Reynolds Number and Aerosol Concentration Run no. 1 Re = 407 11 = 1.45 l./min C, = 1.390 mg/l.
Run no. 2 Re = 407 u = 1.45 l./min C, = 0.374 mg/l.
s, em
T,, 'C
(AFo),
T,, "C
(AF,),
T,. 'C
5- 10 10- 15 15-20 20-30 30-40 40-50 50- 65 65- 80 80- 95
75-56 56-45 45-38 38-31 31-27 27-26 26-24 24-23 23-22
0.00540 0.00426 0.00397 0.00364 0.00240 0.00240 0.00201 0.00195 0.00190
114-90 90-71 71-58 58-41 41-32 32-26 26-23 23-22 22-22
0.01070
128-103 103-84 84- 69 69-51 51-40 40-32 32-27 27-23 23-22
Run no. 5 Re = 997 = 3.85 l./min C, = 0.403 mg/l.
0.00817 0.00653 0.00500 0.00385 0.00334 0.00309 0.0026 1 0.00223
Run no. 6 Re = 997 zi = 3.85 l./min C, = 0.170 mg/l.
Run no. 3 Re = 407 21 = 1.45 l./min C, = 0.188 mg/l.
0.01320 0.00880 0.00770
0.00385 0.00238 0.00183 0.00167 0.00134 0.00154
Run no. 7 Re = 1657 u = 6.40 l./min C, = 0.396 mg/l.
Run no. 4 Re = 997 u = 3.85 l./min C, = 1.335 mg/L TR,'C
(AFo),
85-73 73-68 68-60 60-53 53-46 46-41 41-34 34-29 29-26
0.00580 0.00440 0.00270 0.00255 0.00206 0.00205 0.00134 0.00116 0.00072
Run no. 8 Re = 1657 it = 6.40 l./min C, = 0.447 mg/l.
Run no. 9 Re = 3240 z( = 13.0 l./min C, = 0.010 mg/l.
x. em
Tgr"C
(AFo)*
T,, "C
(AF,),
T,, "C
(AFO),
T,, "C
(AFO),
T,, "C
(AFo),
5- 10 10- 15 15-20 20-30 30-40 40-50 50-65 65-80 80-95
130-117 117- 104 104-93 93-76 76-61 61-51 51-41 41-35 35-31
0.00700 0.00558 0.00455 0.00319 0.00244 0.00206 0.00177 0.00139 0.00131
145-129 129-119 119-109 109-91 91-78 78-66 66-53 53-41 41-30
0.0129 0.0100 0.00788 0.00558 0.00450 0.00372 0.00275 0.00210 0.00198
150-138 138-128 128-121 121-109 109-99 99-89 89-76 76-64 64-53
0.0064 1 0.00474 0.00368 0.00284 0.00229 0.00199 0.00157 0.00140 0.00129
102-97 97-94 94-90 90-83 83-75 75-67 67-57 57-47 47-37
0.00380 0.00268 0.00220 0.00192 0.00182 0.00149 0.00134 0.00128 0.00116
104-93 93-88 88-80 80- 66 66-58 58-52 52-47 47-45 45-40
0.00642 0.00497 0.00458 0.00369 0.00273 0.00209 0.00168 0.001 15 0.00108
Run no. 10 Re = 3240 ZI = 13.0 l./min C, = 0.0215 mg/l.
Run no. 11 Re = 3240 z i = 13.0 l./min C o = 0.00653 mg/l.
Run no. 12 Re = 5770 u = 24.0 l./min C, = 0.0212 mg/l.
Run no. 13 Re = 8450 I I = 34.0 l./min Co = 0.0173 mg/l.
Run no. 14 Re = 10,3000 = 41.5 l./min C, = 0.00685 mg/l.
s. em
TB'"C
(AF,),
T,, "C
(AFO)*
T,, "C
(AFO),
T,, "C
(AFOL
Tg. "C
(AFo),
5- 10 10- 15 15-20 20-30 30-40 40-50 50-65 6 5- 80 80-95
152- 143 143- 127 127- 106 106-94 94- 85 85-71 7 1- 50 50-44 44-39
0.0123 0.00957 0.00840 0.00606 0.00515 0.00405 0.00225 0.00225 0.00105
212-197 197- 182 182-167 167-127 127-102 102-83 83-66 66-55 55-50
0.0269 0.0168 0.0148 0.0126 0.00974 0.00732 0.00585 0.00461 0.00378
181-157 157-141 141- 124 124- 104 104-85 85-73 73-58 58-50 50-45
0.0123 0.0105 0.00883 0.00813 0.00665 0.00526 0.00368 0.00303 0.00236
188-168 168- 152 152-135 135-122 122-111 111-91 91-64 64- 54 54-49
0.0138 0.0128 0.0125 0.0112 0.00905 0.00704 0.00493 0.004 12 0.00273
188- 172 172- 157 157-139 139-116 116-98 98- 83 83-69 69-61 61-51
0.0185 0.0164 0.0148 0.0131 0.0113 0.00986 0.00785 0.00599 0.00446
in Figure 9. Since the coolant is air for the laminar region, the fractional deposition for Re = 1657 decreases gradually along the pipe length. The fraction for turbulence decreases sharply, however, because of water as the coolant. The thermophoretic deposition is related to Z r. Therefore, the fractional deposition is determined by eq 10 and 11 using Z E , ZB, and ZJ-Hshown in Figures 7-9. In the figures, A corresponds to the curves obtained by ZJ.B, B by ZB and C by ZE. For the curves A and B, the following values are used in the calculation; (A/?) = 0.54, C, = 1.0, Ct = 3.32, and b = 2.4. The value of ( k g / k p ) ,which is the ratio of thermal conductivity of NazO to that of air, is about 0.0406. It is seen that the calculated curves A and B agree relatively well with experimental results, except C. Table I1 gives the experimental relation between the 412
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 4 , 1974
mean temperature difference and the fraction of aerosol deposited over the specific lengths of L for the heat-exchanger pipe, together with the Reynolds number (Re), flow rate of hot stream ( u ) , and overall heat transfer coefficient (U).T o elucidate the relation between the mean temperature difference and the degree of thermophoretic deposition of particles in the heat exchanger, Figures 10 and 11 show the depositions (1 - CI,/C,) over the lengths of L = 15 and = 45 cm, us. the term of 4LU(AT),,/Dukg in eq 14, which is given as a function of (AT),,, U, and u. The number quoted in the figures corresponds to that in Table I. The curves are calculated from eq 14, using the values in Table I1 and Z,r of eq 6-8. The experimental values lie between the lines A and B, so the experimental results appear to agree with those obtained by the calcu-
Table 11. Experimental Relation between Logarithmic Mean Temperature Difference and Fractional Deposition of Aerosol Particles Due to Thermophoresis in Two Lengths of Heat-Exchanger Pipe L = 15 c m (5-20 c m ) Run no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 101
z',
u,
Re
m/hr
k c a l / m 2 h r "C
(AT)av, "C
407 407 407 997 997 997 1,657 1,657 3,240 3,240 3,240 5,770 8,450 10,300
4,420 4,420 4,420 11,730 11,730 11,730 19,500 19,500 39,600 39,600 39,600 73,200 104,000 127,000
4.48 4.48 4.48 5.09 5.09 5.09 5.34 5.34 39.40 39.40 39.40 48.8 56.0 62.5
27.8 57.6 71.7 48.3 86.9 99.3 110.7 72.3 66.7 100.7 161.9 124.5 132.4 137.3
,
,
141
,
1
1 - C,/Co 0.00763 0.0194 0.0237 0.0103 0.0141 0.0281 0.0129 0.00679 0.0130 0.0269 0.0555 0.0269 0.0310 0.0377
L = 45 c m (5-50 c m ) ( A T ) , , "C
1 - C,/Co 0.0125 0.0326 0.0290 0.0192 0.0235 0.0503 0.0234 0.0233 0.0240 0.0508 0.1088 0.0556 0.0694 0.0822
14.8 25.8 39.2 35.7 65.3 65.5 93.9 59.7 47.4 76.8 109.0 91.0 106.0 101.0
0.06
8
0
6 4
2 0 8 6 4
2 0 12
10 8 6
4
2
0
0 10 20 30 40 50 60 70 80 90 100
Distance from inlet
lcm )
0 10 20 30 40 M 60 70 80 90
Distance frm inlet ( c m
4LU ( ATlov D v Kq
I
Figure 9. Fractional deposition of aerosol particles due to thermophoresis along the pipe length us. Reynolds number. lation of ZJ-H and ZB. The scattering in the figures is probably caused by the inaccuracy of measurements of the temperature and velocity of the hot and cold fluids, or alternatively, it may also be due to fluctuation of the aerosol concentration from the sodium metal and variation of the heat transfer caused by adherence of aerosol to the pipe wall.
Figure 10. Fraction of particles deposition in heat-exchanger pipe of 15 cm length as a function of the parameter given in eq 14. 0.12
Discussion In the absence of a temperature gradient, aerosol particles are deposited on the inner wall of the pipe by Brownian diffusion of particles less than submicron size (Fuchs, 1964) and gravitational settling of micron size particles (Thomas, 1958) in the laminar stream. These depositions tend to decrease steadily with Reynolds number, as shown in Figure 6, while in the turbulent stream the deposition is due to the eddy diffusion, which decreases with Reynolds number (Friedlander, 1957; Beal 1970).
The apparent fractional deposition ( A F o ) ~under the condition of no temperature gradient was only from 0.1 to 0.5% per 5 cm of pipe length, depending on the Reynolds number. This phenomenon occurred only in the end of the test pipe in which the temperature difference was small
4Lu Dv Kq
Figure 11. Fraction of particles deposited in heat-exchanger pipe of 45 cm length as a function of the parameter given in eq 14.
between the gas stream and the wall, as seen in Table I (cf. Figure 6). Equation 4 was introduced from eq 1 by assuming the Ind. Eng. Chern..
Process Des. Develop.. Vol.
13, No. 4, 1974
413
wall temperature T , to be constant. Strictly, this assumption is not justified, because the temperature T , also has a gradient in the flow direction. The value of T , along the pipe length, however, is not changed much because of extensive cooling by the coolant to the pipe wall. Therefore, the analytical treatment described may be applicable to the experiments. With the film coefficient obtained from Nusselt number, the temperature gradient governing the thermophoretic deposition of particles in the heat-exchanger pipe may be determined in the whole region of laminar to turbulent flows (Re = 200-10,500). However, eddy currents appeared a t extremely high Reynolds numbers (Re = 20,000), so that these aerosol particles deposited on the pipe wall rolled off the wall surface. Since the experiments were made with very low concentrations of the aerosol, the heat transfer due to migration of the heated particles and the effect of the fouling factor were not taken into account for the analysis. If the aerosol concentration were dense, the film coefficients of eq 2 and 3 would not be adoptable. In this case, it is necessary to modify these equations by considering the heat transfer of aerosol particles. Since aerosols of the experiment were situated in the wide distribution of submicron particles (see Figure 4 ) , the Knudsen number (A/?) obtained from the size distribution extended over the wide range from the transition regime to the slip-flow regime. It is well known that the thermophoretic deposition of particles in the transition regime depends strongly upon the Knudsen number. The value of the thermal force calculated from ZJ-g was large compared with that obtained from ZB in the transition regime. In fact, the degree of thermophoretic deposition obtained from the present experiment was larger than the calculated value predicted from ZB and agreed relatively well with that obtained from ZJ-B. The values of ZE, ZB, and Zi-B in eq 6-8 are altered with the ratio of thermal conductivity of particle material to that of fluid, i.e., (kg/kp).The value of ZE approached zero for large value of k , ; consequently, the experimental results were not in agreement at all with those calculated from ZE. The thermal conductivity of carbon dust particles in the helium coolant of an HTGR is much larger than that of NazO particles; the conductivity ratio of heliKeng and Orr um to carbon (kg/kp)is about 5 x (1966) reported that the thermal force acting on several aerosol materials agreed well with Brock's theory corresponding to eq 7, though metallic aerosols had high thermal conductivities. In the slip-flow regime, the value of ZB was nearly equal to that of ZJ-B.Therefore, in the calculation of thermophoretic deposition in the heat exchanger pipe of HTGR, the ZB may be fitted in the case of large carbon particles. However, if very fine carbon particles are suspended in the coolant stream (the transition regime), ZJ-B proposed by Jacobsen and Brock may be suitable for estimation of the degree of thermophoretic deposition. Rowland, et al. (1969), performed the measurement of radioactive fine dust particles deposited along the inner wall of heat exchanger pipe in the Doragon-HTGR. The airborne dusts consisted of carbon (26-50 wt %) containing metallic oxides of Fe and Cr. For the heat exchanger pipes of HTGR, the degree of thermophoretic deposition of the carbon dusts is estimated by using eq 14 with Z.I-Bin the transition regime, Le., (A/r) = 0.5. Data of the heat exchanger applied to the calculation are tentatively assumed: L = 5 m, D = 0.02 m, (AT)av = 300"C, u = 25 m/sec, and U = 50 kcal/m2 hr"C, respectively. The percentage of deposition, (1 - C/Co) is found to be about 414
Ind. Eng. Chem., Process Des. Develop., Vol. 13,No. 4 , 1974
1.6%. For a long period of the operation, the fouling by the deposition of carbon dusts on the wall of the heat exchanger pipe may cause the need for maintenance of the pipe material, even though the dust concentration in helium coolant is very low.
Acknowledgment The authors wish to thank Mr. S. Uno, Mr. J. Takada, and Mr. T. Shiratori in JAERI for their assistance in performance of the work. Nomenclature b = numerical constant of second order in the thermal force derived by Jacobsen and Brock ( = 2 . 4 ) C = aerosol concentration a t any point, mg/l. C, = numerical constant in thermal force derived by Brock, ( = 1.0) C, = specific heat capacity a t a constant pressure, kcal/g Ct = numerical constant by Brock ( = 3.32) D = diameter of test pipe, m F, = fractional deposition of aerosol in the absence of temperature gradient FO = apparent fractional deposition of aerosol under the temperature gradient F,r = fractional deposition by the thermophoresis G = mass velocity of fluid, g/m2 hr h = heat transfer coefficient, kcal/m2 hr "C k , = thermal conductivity of gas, kcal/m hr "C k , = thermal conductivity of aerosol particles, kcal/m hr
"C L = length of test pipe, m 1 = length of heat conduction, m Q = quantity of heat, kcal/hr r = radius of aerosol particle, m ? = arithmetic mean radius of particles, m rg = logarithmic mean radius, m = average absolute temperature of fluid, "K T, = temperature of coolant, "C Tg = temperature of fluid, "C Tgo = temperature of fluid a t inlet, "C T , = temperature of wall, "C T,, = temperature of wall a t inlet, "C (AT),, = mean temperature difference of heat exchanger, "C A TH = temperature difference between hot and cold fluids a t inlet, "C ATI, = temperature difference a t outlet, "C L' = overall heat transfer coefficient, kcal/m2 hr "C u = flow rate of hot fluid, l./min v = velocity of hot fluid, m/hr W, = weight of flow ( = ( x / 4 ) D 2 u p ) ,g/hr x = length from inlet of pipe, m y = distance toward cold wall, m Zg = a term in thermal force of thermophoresis derived by Brock, m2/hr "C ZE = a term in thermal force of thermophoresis by Epstein, m2/hr "C ZJ-B = a term in thermal force of thermophoresis by Jacobsen and Brock, m2/hr "C Z,r(dT/dy) = velocity of particle, m/hr, (dT/dy) = the temperature gradient to wall Greek Letters p = density of fluid, g/m3 6 = thickness of boundary sublayer, m
X = mean free path of fluid, m p = viscosity of fluid, g/m hr pw = uR =
viscosity of fluid near cold wall, g/m hr logarithmic standard deviation
Literature Cited Beal, S. K.. Nucl. Sci. Eng.. 40, 1 (1970). Brock, J. R . . J. ColloidSci., 17, 768 (1962). Brock, J. R., J. Colloid Interface Sci., 23,448 (1967).
Byers, R . L., Calvert, S., Ind. Eng. Chem., Fundam., 8, 646 (1969). Deryagin. B. V.. Yalarnov, Yu., J. ColioidSci., 20, 555 (1965). Z. Phys., 5 4 , 537 (1929). Epstein, P. S.. Friedlander, S. K.. Johnstone, H. F., Ind. Eng. Chem., 49, 1151 (1957). Fuchs, N. A,, "The Mechanics of Aerosols," pp 204-212, Pergamon Press, Elmsford, N. Y., 1964. Jacobsen, S., Brock. J. R . , J . CoIloidSci., 20, 544 (1965). Keng, E. Y. H., Orr, C. Jr., J. Colloid lnterface Sci., 22, 107 (1966). Murata, M.. Kitani. S..J. Nucl. Sci. Techno/., 9, 622 (1972).
Postma, A. K., HW-70791, (Hanford), (1961). Rowland, P. R . , Corlyle, M., Sonbelet, G . C., "Diffusion des Produits de Fission," No. 16, 191. Saclay, 1969. Schmitt. K. H., Z. Naturtorsch., 14a, 870 (1959). Singh, E., Eyers. R . L., lnd. Eng. Chem., Fundam., 11, 127 (1972). Thomas, J. M., J . Air Pollut. Contr. Ass., 8, 32 (1958).
Received for reuieu November 5, 1973 Accepted May 10, 1974
Modeling the Water-Gas Shift Reaction Walter F. Podolski' and Young G. Kim* Department of Chemical Engineering, Northwestern University, Evanston, lllinois 6020I
The kinetics of the water-gas shift reaction over an iron oxide-based catalyst has been studied. The experiments were performed in a glass recycle reactor at approximately atmospheric pressure. The temperature dependence of the reaction and the activation energy were determined in conjunction with a study of the diffusional limitations of different size catalyst. The statistical procedure proposed by Box for experimental design and analysis of data in order to discriminate among the rival models was used in the experimental program. A number of representative models were examined, and it was found that only the Langmuir-Hinshelwood and the power-law models could adequately describe the reaction behavior over the temperature and concentration ranges investigated. In addition, some existing data from the literature were reexamined in the same manner as used here.
Introduction Numerous kinetic studies (Barkley, et al., 1952; Bohlbro, 1961, 1962, 1963, 1964, 1966; Giona et al., 1961; Hulburt and Vasan, 1961; Kodama, et al., 1953; Kul'kova and Temkin, 1949; Paratella, 1965; Shchibrya, et al., 1965) on the water-gas shift reaction have been conducted using iron oxide catalysts, and there is general agreement among most of the results reported which can be summarized as follows: the rate of reaction is approximately proportional to the carbon monoxide concentration; the rate is usually retarded by increasing carbon dioxide concentration and almost independent of hydrogen concentration and independent of steam concentration when it is in excess of the stoichiometric amount. Temkin was the first to publish a systematic attempt to combine the reaction mechanism with a rate equation. The mechanism assumes an alternate oxidation and reduction of the catalyst surface; carbon monoxide reacts with the catalyst surface creating oxygen ion vacancies which are immediately filled by oxygen from the water molecules. Both steps proceed a t the same rate on the catalyst surface which is assumed to be nonuniform. The oxidation-reduction models in Table I are the result of this treatment. Later, Boreskov, et al. (19701, measured the rates for the oxidation and reduction steps of the reaction over an iron oxide catalyst to determine if the rates for the individual stages and the overall rate of the reaction are in agreement. They concluded that within the experimental error the rates of the individual steps agreed with the overall rate, thus lending support to Temkin's model. The same type of experiments on copper chromite and chromium oxide catalyst (Yur'eva, et al., 1969) yielded different
' Argonne National Laboratory, Argonne, 111.
60439
results. Over copper chromite, they concluded that the reaction proceeded through the formation of an active complex including both a CO and an HzO molecule according to the Langmuir-Hinshelwood scheme. Over chromium oxide, carbon monoxide was not adsorbed in noticeable amounts, and therefore they concluded that the shift reaction proceeded by reaction of an adsorbed water molecule with gaseous carbon monoxide. Kaneko and Oki (1965a,b) measured the stoichiometric number of the rate-determining step of the reaction using deuterium and 14C. These experiments gave values of the stoichiometric number close to unity. They concluded that the rate determining step, if it existed, must involve both carbon monoxide and water molecules. These results supported the Langmuir-Hinshelwood and Eley-Rideal models but not Temkin's model. Reexamination of this data and subsequent experimentation with 1 8 0 (Oki, et al., 1972; Oki and Mezaki, 1973a,b) has led to a value of 2 for the stoichiometric number, which is compatible with a mechanism with two rate- determining steps, namely, the adsorption of CO and the associative desorption of hydrogen. In the initial stages of the reaction only the adsorption of CO is rate determining, while near equilibrium both steps are rate determining. These latter data were published after this work was completed and were not considered. The models considered, shown in Table I, fall into one of the following general classes: (1) Langmuir-Hinshelwood model, (2) Eley-Rideal model, (3) oxidation-reduction model, (4) Hulburt-Vasan model, (5) Kodama model, or (6) empirical (power law) model. Experimental Strategy In this study, we have made use of some of the techniques developed by Box and his coworkers (Box and Hill, 1967; Box and Henson, 1969) for discrimination among Ind. Eng. Chem.,
Process Des. Develop., Vol.
13, No. 4 , 1974
415
