Ind. Eng. Chern. Res. 1992, 31, 706-713
706
= electric conductivity [l/(Q.m)] = crtjL2
be
T
7,
cp
= dimensionless time constant for a first-order delay = scalar potential for an electric field [Wb/s = VI Registry No. Ga, 7440-55-3.
Literature Cited (1) Brown, R. A. Theory of transport processes in single crystal growth from the melt. AZChE J . 1988,34,881-911. (2)Chandrasekhar, S.Hydrodynamic and Hydromagnetic Stability; Oxford University Press: Oxford, 1961;pp 190-195. (3) Crespo del Arco, E.; Bontoux, P.; Sani, R. L.; Hardin, G.; Extremet, G. P. Steady and oscillatory convection in vertical cylinders heated from below. Numerical simulation of asymmetric flow regimes. Adu. Space Res. 1988,8, (12)281412) 292. (4) Cubberry, W. H. Metals Handbook ZZ, 9th ed.; ASM: Metals Park, OH, 1979;735-737. (5) Czochralski, J. Z . Phys. Chem. 1918,92,219. (6) Hoshi, K.; Izawa, N.; Suzuki, T. Single crystal growth of silicon in a magnetic field. Appl. Phys. (Jpn.) 1984,53,38. (7) Hurle, D.T. J. Temperature oscillations in molten metals and their relationehip to growth striae in melt-grown crystals. Philos. Mag. 1966,13,305-310. (8) Kesshou Kogaku (Crystal Engineering) Handbook; Kyoritsu Shuppan Inc.: Tokyo, 1971;p 504 (in Japanese). (9) Kobayashi, S. Effect of an external magnetic field on solute distribution in Czochralski grown crystals-a theoretical analysis. J. Cryst. Growth 1986,75,301-308. (10) Nacken, R. Neues Jahrb. Miner. Geb. 1915,2,133. (11) Nakagawa, Y.Experiments on the inhibition of thermal con-
vection by a magnetic field. Proc. R. SOC.London, A 1957,240, 108-113. (12) Nakamura, S.;Hibiya, T.; Yamamoto, F.; Yokota, T. Thermal conductivity measurement of mercury under a magnetic field. Znt. J. Heat Mass Transfer 1990,33,2609-2613. (13) Okada, K.;Ozoe, H. Experimental heat transfer rates of natural convection of molten gallium suppressed under an external magnetic field in either the x-, y- or z- directions. J. Heat Transfer, in press. (14) Ozoe, H.; Okada, K. The effect of the direction of the external magnetic field on the three-dimensional natural convection in a cubic enclosure. Znt. J . Heat Mass Transfer 1989,32,1939-1954. (15) Ozoe, H.; Matsuo, H. Transient numerical computations for natural convection of liquid metal in a tall enclosure heated from side and with a lateral external magnetic field. Numer. Methods Therm. Probl., Proc. 6th Znt. Conf., Swansea 1989,6, 440-450. (16) Ozoe, H.; Churchill, S. W. Hydrodynamic stability and natural convection in Newtonian and non-Newtonian fluids heated from below. AZChE Symp. Ser., Heat Transfer 1973,69 (No.131), 126-133. (17) Roux, B., Ed. Numerical simulation of oscillatory convection in low-Pr fluids. A GAMM Workshop;Vieweg: Wiesbaden, FRG, 1990;Vol. 27. (18) Utech, H. P.; Flemings, M. C. Elimination of solute banding in indium antimonide crystals by growth in a magnetic field. J. Appl. Phys. 1966,37,2021-2024. (19)Witt, A. F.; Herman, C. J.; Gatos, H. C. Czochralski-type crystal growth in transverse magnetic field. J. Mater. Sci. 1970, 5 , 822-824.
Received for review January 28, 1991 Revised manuscript received May 14,1991 Accepted May 31, 1991
Temperature Measurement in Condensing Jets M. L. Strum? and H. L. Toor* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
This paper focuses on three aspects in the measurement of the temperature distribution in condensing turbulent fog jets. These are the temperature increase due to condensation in the region of fog formation, the spatial distribution of the maximum condensation driving force, and the observed increase in temperature fluctuations in the fog jet over a dry jet. The fog jets were generated by issuing saturated water vapor-air mixtures at 63 and 85 O C into a room at ambient temperature and relative humidity. Temperature profiles a t various axial positions were obtained with small wetted and nonwetted thermocouples. Below 25 jet diameters, the magnitude of the temperature increase in the condensation regime was found to be in excellent agreement with that predicted by incorporating the familiar wet bulb equation into a continuum analysis of heat and mass transfer in the turbulent jet. The data are consistent with the spatial distribution of the maximum condensation driving force predicted by this analysis and thus delineate regions in which fog may form and evaporate. The turbulent temperature fluctuations in a fog jet are greater than those in a similar dry jet and are consistent with the large temperature fluctuations predicted by elementary mixing length arguments. Introduction The release of latent heat and subsequent temperature increase of the condensing species is inherent to the condensation process just the absorption of latent heat and temperatwe decrease of the evaporating species is inherent to the evaporation process. When fog forms in a turbulent jet due to condensation, the temperature of the disperse phase (Le. fog droplets) increases over that of the supersaturated vapol-air mixture in the noncondensing (frozen) jet, having the same effluent and ambient conditions as the fog jet. Several workers who have performed experi'Current address: Environmental Protection Agency, Research Triangle Park, NC 27711.
mental studies on turbulent condensing jets have utilized this concept to identify spatial regions of condensation. Hidy and Friedlander (1964) first reported data on the temperature increase as part of their analysis of the structure of the mixing zone of a glycerol vapor air jet. They compared a radial temperature profile in the mixing zone of a condensing jet with that in a dry jet. This showed fog formed in the mixing zone along a well-defined region. They also showed that the magnitude of the temperature rise decreased with decreasing initial concentrations of glycerol vapor. Temperature measurements were also obtained by Soviet researchers investigating the possibility of controlling the condensation process in water vapor jets. Vatazhin et al. (1984) obtained axial temperature profiles in water
0888-5885/92/ 2631-0706$03.00/0 0 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 3,1992 707 vapor jets of varying jet effluent temperatures. They compared these profiles with those calculated for the case of the frozen jet to show the sharp transition into the condensation region. In a later study, Vatazhin et al. (1985) performed the same measurements on the jets with a greater variation of effluent and ambient conditions. In the above studies, condensation regions were characterized by a temperature increase over a dry jet resulting from the release of latent heat during fog formation. No theoretical analysis was given to determine the magnitude of this increase. The only previous work in which temperature measurements in condensing jets were compared to theoretical values was that of Bennett (1971). He used a local equilibrium model (Matvejev, 1964) of fog formation to predict the fog boundary in a 60 OC turbulent condensing airwater vapor jet. In the course of this work he measured the temperature distribution in condensing and dry 60 O C jets. He observed that the measurements compared well with the assumed theoretical value of the temperature which is in local thermodynamic equilibrium with the vapor-air mixture at that particular spatial location. He called this "the supersaturated analog of the usual wet bulb temperature". However, he never derived the theory. The objective of this work was to extend this result to more extreme conditions as well as show that it is the expected result for a gas mixture with a Lewis number of 1. The analysis, which incorporates the microscopic transfer of heat and mass around a condensing droplet into the large-scale time-averaged continuum equations of heat and mass transfer, served as the basis of several models developed as part of a larger study of the phenomenon of fog formation in a turbulent jet (Strum, 1990). We show the temperature in the condensing jet to be the equilibrium value which Bennett utilized. We obtain different results, however, in our approximation of the frozen jet since Bennett did not account for the density difference between the dry jet and the frozen jet, which is important in correctly predicting the temperature in jets with the large effluent water vapor concentrations considered here. In this paper we present the derivation of the temperature increase (decrease) of the disperse phase in the condensation (evaporation) region of a fog jet over a vapor-air mixture in a frozen jet. We use this relationship to determine the maximum condensation driving force and the temperature of the disperse phase. Results are compared with thermocouple profile measurements of fog jets generated by issuing saturated water vapor-air mixtures at 63 and 85 O C , respectively, into an ambient room. Observations of the temporal temperature fluctuation in the fog jet as compared to that in the dry jet are discussed in light of the Prandtl mixing length theory of turbulence.
Continuum Model The continuum transport equations for temperature, T , and water vapor mass fraction, w , in a vertical axisymmetric turbulent jet with axial coordinate z, radial coordinate r, and axial and radial velocities u and u are
where ii3 represents the time average rate of condensation and is a function of the fine-scale microscopic processes involved in droplet growth coupled with the large-scale
0.6
0.4
0
0.2
0.0
0
20
60
40
80
100
Temperature (C) Figure 1. Concentration-temperature diagram.
proceases and c is the eddy diffusivity of heat or mass. The boundary conditions are atr=O in the ambient room, at r =
a: P
= a,;
T = T,
in the jet effluent, at z = 0, r C d/2: P = wo; T = To The key assumptions/simplificationsare as follows: the vapor-air mixture is dilute, the molecular diffusivities are negligible compared with the turbulent diffusivities, the turbulent Lewis number is 1 (turbulent diffusivities are equal), and the difference in heat capacities between the species is negligible. If no condensation occurs in the above system, 4 is zero, and the equations describe the transport of a pansive scalar property in a jet. This situation has been denoted a frozen jet (Vatazhin et al., 1984). It may arise when there exists no nuclei upon which to condense the supersaturated vapor; however, in the atmosphere and in an ordinary room, there are abundant nuclei, and thus the frozen jet is normally fictitious. Defining a,(r,z) and T,(r,z) as the distribution of water vapor concentration and temperature in the frozen jet, it is seen that the dimensionless quantities
satisfy transport eq 1with F3 = 0 and boundary conditions at r = a: 8u,m = OT,m = 0 at
= 0, r C d/2:
Ow,, = OT,,
=1
Hence, Ow,, = BT,, = 8, = F(r,z), with the function F representing the spatial distribution of a mean passive scalar, such as temperature or vapor concentration, in a frozen jet. In terms of temperature and vapor mass fraction 0, - w , T, - T, -== F(r,z) (4) w0-0, To- T , This is the mixing line, reflecting the path of possible fluid conditions in a - T space resulting from turbulent diffusion in a noncondensing (frozen) jet whose effluent is at (To, wo) into an ambient environment at (T,, wJ. The mixing line is illustrated on a concentration-temperature diagram in Figure 1. The saturation curve,
708
Ind. Eng. Chem. Res., Vol. 31, No.3, 1992
o*(T),also plotted in Figure 1, represents the water vapor concentration corresponding to a saturated vapor at temperature T. The region in the figure which is bounded by the mixing line and the saturation curve, in which om> w*(Tm),is supersaturated. The dotted curve in the figure depicts a possible real path, associated with the actual condensation kinetics represented by ii3. For constant X and C, and arbitrary ii3, eqs 1 and 2 give the conservation of jet enthalpy: aH aH 1 a pii - + pb - = - - ( r p (5) az ar r ar
$)
with aH/ar = 0 at r = 0, H = H, in the ambient, and H = Hoin the jet effluent. The enthalpy in the frozen jet also satisfies (5) with identical boundary conditions. Thus, H = H,, or C,Tm + ha, = CPT + xa (6) The path taken in the limiting case in which the fog is in equilibrium with the surrounding vapor at temperature -
T* and vapor concentration o*(T*)is determined by intersecting the constant enthalpy line with the equilibrium curve or solving
CJF - ?,I
.-
= L[w, - 0 ( P j l
(7)
where we have neglected the error caused by assuming that (Strum, 1990)
-
O*(n= o*(fi
(8)
On the microscopic level, heat and mass transfer to a drop at temperature Td in an S i t e stagnant atmosphere at temperature T and vapor concentration w gives c,(Td - T ) = h[U - o * ( T ~ ) ] (9) This equation is the Lewis number 1form of the familiar wet bulb equation of psychometry. The key assumptions are the diffusion of heat and mass in the vicinity of the droplet is a quasi-stationary process, the interfacial resistance at the droplet surface is negligible, the vapor-air mixture is dilute, physical properties are constant, and the molecular Lewis number, Le = a / D , is unity. Time averaging (9), utilizing (8), and combining with (6) give c p ( T d - Tm) X[am - O * ( T d ) ] (10) Comparing (10) to (7), it is evident that the temperature of the fog water, p d is identical to the temperature of the continuous phase in the limit of thermodynamic equilibrium, the wet bulb temperature T*. Thus (10) may be rewritten as (7):
- -
C,(F
- T,) = nro,
- 0*(F)l
(11)
Geometrically, the disperse phase or wet bulb temperature is the intersection of the c_onstantenthalpy line (6) and the saturation curve, w*(7'). In Figure 1, the abcissa of point f is the disperse-phase temperature for a location ( r j ) in the jet whose constant-enthalpy line subtends points e and f. Thus, the disperse-phase temperature is indeed the supersaturated analogue of the usual wet bulb temperature.
Prediction of the Temperature Distribution in the Frozen Jet Recall that we have denoted the frozen jet as a noncondensing jet, of identical Reynolds number, and temperature and vapor concentration boundary conditions as
a condensing jet. That is, the scalar properties of temperature and vapor concentration, pm(r,z) and am(r,z) satisfy the turbulent time-averaged conservation equations of mass and heat, eqs 1and 2,with source or sink term, ii3, identically zero, and may be denoted passive scalars. In dimensionless form they are represented by the function F(r,z). We cannot directly measure the temperature profile in a frozen jet, since a jet experimentally generated with a hot saturated effluent condition experiences fog formation, and Y3 # 0. Even if we attempted to prevent condensation by producing a vapor-air jet in a room perfectly free of condensation nuclei, we still could not measure a frozen jet temperature profile. The thermocouple would provide a site for condensation and, once again, ii3 would no longer be zero. Because the thermocouple becomes wet, it measures the wet bulb temperature, T*. We can approximate the frozen jet by a dry jet of the same Reynolds number and jet effluent temperature. The main difference between the frozen jet and the dry jet is the concentration of water vapor in the jet effluent. In the frozen jet, the effluent is a saturated vapor-air mixture, while in the dry jet it is practically devoid of water vapor. The form of the dimensionless temperature distribution, i.e., F(r,z) in the frozen and dry jets will be equal as long as this difference does not alter the flow field (velocity profile and turbulent diffusivities) between the dry and frozen cases. This assumes that the effect of buoyancy is the same in the two cases. Since the magnitude of the densimetric Froude number, Fr = uo/[gD(po- p m ) / p 0 ] 1 / 2 , characterizes the effect of buoyancy (Chen and Rodi, 1980), we compare the densimetric Froude numbers of the dry jet and fog jet experiments. For the 63 "C jets, a dry effluent has Fr = 900 as compared with Fr = 650, in the frozen jet, and for the 85 "C jets, dry and frozen effluents have Fr = 650 and 400, respectively. Based on the correlation of experimental buoyant jet data with densimetric Froude number given in Chen and Rodi (1980), it is seen that the assumption of identical flow fields is valid near the jet mouth. Since our measurements, z / d < 40, do not go far enough into the transition region to deviate significantly from the nonbuoyant behavior, this assumption is reasonable throughout our measurement region. In the self-similar region, z / d > 8 jet diameters, the function F(r,z), is computed by fitting the dimensionless temperature profile in the dry jet to a Gaussian profile of the form (Hinze, 1975)
where A , b, and X are fit parameters, and the factor (pol p a ) l i 2 accounts for the difference between the density of the jet effluent, po, and the environment, pa. In the near field region of the jet, the density ratio does not play a role in diffusion of a passive scalar. Thus, in the near field region of the frozen jet, the distribution of the dimensionless passive scalar is given by the dimensionless temperature distribution in the dry jet: (13)
Prediction of the Wet Bulb Temperature and Saturation Border The theoretical diaperse-phasetemperature or wet bulb temperature is obtained by solving the equilibrium relationship together with eqs 11 and 4 with F(r,z) given by (121, for a spatial location in the self-similar region or (13)
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 709 in the near field region. Alternatively, once F(r,z) is determined, the disperse-phase temperature is graphically obtained from the intersection between the constant enthalpy line which passes through the point (a,,T',) and the saturation curve. Since the sign of the actual condensation driving force, a - LO*(-), demarcates unsaturated, saturated, and supersaturated regions in the jet, the actual saturation border requires knowledge of the actual P - T path. However, the maximum driving force, a, - w*(-), which gives the saturation border for the frozen jet, is easily obtained from the disperse-phase temperature, as illustrated in Figure 1. In some situations, the maximum driving force provides information on the saturation condition of the real jet. For example, a region of the jet in which T* < T",, is unsaturated regardless of the actual P - 'I' path, since the maximum condensation driving force and thus the actual condensation driving force, are less than zero.
00000 r -66 rrd- 1.7 o r n o fa'rrd. 1 .a 1.20
1.00
0.80 @ 0.60 0.40
0 0 0 ~ 0T&:
***** 7-07
,
Ud- 2.7
rrd.37
+++++ r 3 4 : r r d . 3 7 1
1
1 1 1
o.20
0.00 -2.0 -1.5-1.0 -0.5 0.0 0.5 1.0 1.5 2.0
Radial Distance (r / d) Figure 2. Dimensionless profile in the near field region. 1.0 [
Experiment A saturated vapor-air flow was produced by bubbling fitered air through a temperature-controlledtank partially filled with distilled water. The exit stream was filtered to remove water droplets and then sent to a straight vertical tube of 1.1-cm diameter, where it issued into a room and formed fog. Fog jets were generated with effluent temperatures of 63 and 85 "C, corresponding to a vapor mass fraction, coo, of 0.14 and 0.47, respectively. Dry jets were also produced with these exit temperatures. In both the fog and dry jets, the Reynolds number was approximately 2000. Details of the apparatus are given in Strum (1990). Temperature profie measurements in the jet were taken with chromel-alumel (type K)bare-wire thermocouples of 0.13-mm diameter and an OMEGA "-52 digital thermometer. In some fog jet experiments a wetted cloth wick was tied to the bead of the thermocouple (wet bulb thermocouple). The system was calibrated with an ASTM total immersion mercury thermometer accurate to 0.2 "C. Fluctuations were monitored by recording the average temperature over 10-s intervals. The mean temperature was obtained by averaging over a period of approximately 5 min. The precision of the measurement was obtained by taking measurements over long periods of time and comparing them to averages over 5-min time intervals. In the main section (close to the center line) it was found that the precision of the temperature measurement is approximately 0.2 "C. Away from the center line, the precision is approximately 1 "C. The dry jets showed lower fluctuations in temperature, and a shorter averaging time interval was used. Temperature measurements in the fog jet were taken with dry bulb and, in some cases, wet bulb (wicked) thermocouples. Since dry bulb thermocouples may be wetted due to either impaction by fog droplets or condensation of supersaturated water vapor, at each spatial location, the temperature measured reflected one of three conditions of the thermocouple: fully wet, partially wet, or dry. A fully wetted thermocouple (wet bulb thermocouple or wetted dry bulb thermocouple) measures the wet bulb temperature, or the temperature of the disperse phase; the measurement may be compared with the predicted wet bulb temperature, T*. A dry thermocouple reading corresponds to the temperature of an unsaturated or saturated vapor-air mixture; the measur3mer.t may be compared with the frozen jet temperature, T,. A partially wetted thermocouple is due to dry regions on a wetted thermocouple tip. It will occur in an unsaturated region
I
&
&
t 0.6 0.4
0.0
'
\ 0
10
20
30
40
Axial Distance (z I d )
Figure 3. Axial dimensionless temperature divided by the density ratio along the jet center line.
in which a wetted thermocouple has partidy dried. Such a reading cannot be compared with any theoretical value; however, it does indicate that the fog jet is unsaturated at the measurement location.
Results and Discussion Temperature Distribution in Heated Dry Jets. Temperature profiles in dry heated jets, approximating frozen jets, were taken at various axial positions up to 37 jet diameters. Figure 2 shows the results of profiles in the near field region of the jet, z / d < 4. These positions subtend the potential core. As expected, the temperature profile in the vicinity of the tube ( r / d < 0.5) is constant and identical to that of the effluent. The core region narrows with increasing axial distance, as shown by the profiles taken at 4 jet diameters from the mouth. In the dry jets, the dimensionless temperature profiles showed self-similarity a t 8 jet diameters. The following Gaussian function was fit to the data: F(i',z^,(Po/P,)'/2) =
The goodness of fit is illustxated in Figurea 3 and 4. The functional form of the profile, hyperbolic in center-line axial distance, and Gaussian in the radial coordinate, is well documented in the literature for the transport of a scalar in turbulent round jets (Hinze, 1975). The best fit constants to this form are consistent with the table of experimental values compiled by Chen and Rodi (1980). Equation 14 represents the distribution of the dimensionless passive scalar in the frozen jet, F ( t , i , ( p o / p a ) 1 ~ 2 ) .
710 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 Table I. Experimental Conditions for Temperature Measurements in Fog Jets z 8.1 8.1 15.4 15.4 25.1 25.1 3.66 8.1 8.1 8.1 15.4 23.6
name o318a o318b o2415a o2415b o3025a o3025b j174 n88ad n88bd n88aw d2415 d1825w 1,2 1.0
TOIOC 64.2 64.2 63.2 63.2 64.0 64.0 85.6 86.4 86.4 86.4 85.8 84.8
WO
0.167 0.167 0.159 0.159 0.166 0.166 0.483 0.501 0.501 0.501 0.486 0.462
T./OC 23.4 23.4 21.4 21.4 22.5 22.5 25.3 22.3 22.3 22.3 20.2 21.1
w,
Re
Fr
0.006 0.006 0.010 0.010 0.009 0.009 0.013 0.0011 0.0011 0.0011 0.004 0.003
2042 2042 2117 2117 2052 2052 1840 1990 1990 1990 1878 1783
625 625 669 669 628 628 397 448 448 448 389 356
r
ud(cmls) 408 408 419 419 409 409 480 526 526 526 492 460
PolPa
.80 .80 .80 .a0 -80 .80 .65 .63 .63 .63 .63 .64
--C 00000031 Bb. i-8,64‘ C -****e 031&,i=8,64’C
t
----- theorelical
T 01 Imzen jeI
0.8 -
0 -
0.6 -
@I, 0.4
-
50
40
T (“C) 0.2
-
30 20
10 -5 -4 -3 -2 -1
0 1 2 3
4
5
Radial Distance (r/d) Figure 5. Fog and frozen jet temperature profiles: z l d = 8; 64 O C jet.
10
-6
-4
-2
0
2
4
6
Radial Distance (r/d) Figure 6. Fog and frozen jet temperature profiles: z l d = 15; 63 O C jet.
The radial profile at 8 jet diameters is shown in Figure 5. In case o318b, (the circles in Figure 5)) the thermocouple traversed the jet from the room into the fog, as indicated by the arrow from left to right. In the initial stage of the traverse, the data points taken by the dry thermocouple (unsaturated or saturated region) lie on the frozen jet curve. This indicates that no significant concentration of fog exists in the unsaturated region. If it had existed, the dry bulb thermocouple would have become wet by impaction, and the thermocouple would have given temperatures below the frozen jet curve. The profile shifts from the frozen jet curve to the wet bulb curve at around 1.5 jet diameters, in agreement with the theoretical border between unsaturated and supersaturated regions of the frozen jet. Movement of the measured data of case o318a off of the wet bulb curve at r / d = 2 may have been due to the drying off of the thermocouple as it entered the unsaturated region of the jet. It is seen that the experi-
Ind. Eng. Chem. Res., Vol. 31,No. 3, 1992 711
---
00300 n88ad t I 8 86'C
***** n 8 8 W . i - 5 86'C theoretical
----.T o l l m e n jet +++++ n88aw.i-8 85'C
70 60
-
10 -16 -12 -8
-4
0
4
8
50
T ('C) 40
30 20
1 1 10
12 16
Radial Distance (r/d) Figure 7. Fog and frozen jet temperature profiles: z / d = 25; 64 "C jet.
"
I
'
-5 -4 -3 -2 -1
'
I
I
I
0
1
2
3
4
-
-
00300 42415,2=15,86'C
I
----_theorelica T 01 frozen je1
60
10 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
Radial Distance (r/d)
5
Radial Distance (r/d) Figure 9. Fog and frozen jet temperature profiles: z / d = 8;86 O C jet; n88aw (wicked thermocouple experiment, +).
& & A 4 frozen (mza4) lheormka c 4+++4 j174,i.4,BB
an
1 1 I
-
I
Wl
0 -8
-6
-4
-2
0
2
4
6
8
Figure 8. Fog and frozen jet temperature profiles: z / d = 4; 86 O C jet.
Radial Distance (r/d) Figure 10. Fog and frozen jet temperature profiles: z / d = 15; 86 O C jet.
mental profiles from the wetted thermocouples are in excellent agreement with the calculation of T* from the wet bulb type relation, (11). Error bars in the predicted wet bulb and frozen jet curves reflect the uncertainty in the parameters used in the calculation. At 15 jet diameters downstream (Figure 6),the disperse-phase temperature profile measured by the wetted thermocouples is also in excellent agreement with that predicted from eq 11. Both predicted and measured increasesof the wet bulb temperature at the jet center line, T, - P,at 25 jet diameters (Figure 7) dropped sharply from the 7 O C difference obtained at 8 and 15 jet diameters. In contrast with the very good agreement between measured and predicted wet bulb temperatures at 8 and 15jet diameters, the disperse-phase profile at 25 jet diameters lies barely within the error bands. In Figure 7,the center line predicted temperature difference is 3.7 "C, and the measured difference is about half of that value. However, excellent agreement exists between the fog jet experimental data in the clear region and the frozen jet profile. This supports the assumption of identical flow fields in the frozen and dry jets, and it also indicates that no significant concentrations of fog exist in the unsaturated region. Fog Jet Temperature Profiles at 85 "C. The disperse-phase temperature and condensation temperature increase in the 85 OC fog jet experiments were considerably greater than in the 63 "C experiments. This effect is directad by the wet bulb equation, considering that mixing brings about a linear dependence of water vapor concentration on temperature while the saturation concentration is exponentially dependent on temperature. The data at 4 jet diameters, corresponding to the near field region in
the jet, are shown in Figure 8. The measurement was taken with a wetted thermocouple, since it was initially traversed through the fog region. In both the measured and theoretical fog temperature - profiles, the greatest temperature increase, Tm- P ,lies off of the center line. This is due to the presence of the potential core at the jet center line. Visual measurements made by passing a laser through the jet have shown that, in this region, fog does not form. The fact that the frozen- and disperse-phase temperatures are nearly identical at the center line reflects the saturated condition of the vapor-air mixture at this location. Temperature profiles measured at 8 jet diameters in the 85 O C jet are shown in Figure 9. As in the 63 O C jet, the measured wet bulb temperature profile at 8 jet diameters is in excellent agreement with the theory. The cases shown in Figure 9 encompass all conditions of the thermocouple. The plus signs represent data from the wet bulb thermocouple; this data is approximately the same as that taken from the wetted dry bulb thermocouple (circles and stars in the fog region). This fiiding is in contrast to Bennett's (1971)data in which wet bulb thermocouple readings were consistently lower than the wetted dry bulb readings. The stars represent the data taken from an initially dry thermocouple traversing from the room to the fog; thus in the initial part of the traverse, the data fall upon the frozen jet curve. They move to the theoretical wet bulb curve once the thermocouple becomes wetted. The circles represent measurements from a dry thermocouple which was originally wetted by the fog and allowed to dry (-4 < r / d < 2) before traversing through the fog from the negative r direction. The measured profiles at 8 jet diameters
712 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
--
o3300 a1825a. i - 2 4 . 8 5 ' C
-+++++
-----
41825*,5-24 8 5 ' C theoretical T of frozen ]e1
I
10
ML
I
1
I
I
I
0 -8
-6 -4
-2 0
2
4
6
0
Radial Distance (r/d) Figure 11. Fog and frozen jet temperature profiles: z / d = 25; 85 "C jet; d1825w (wicked thermocouple experiment, +).
sharply delineate the unsaturated and supersaturated regions of the frozen jet at 2 jet diameters, in accordance with the theory. The wet bulb temperature profiles in the fog jet are also predicted very well by the theory at 15 jet diameters, as shown in Figure 10. The agreement of the dry data with the frozen jet curve supports the assumption of identical flow fields in the frozen and dry jets and indicates that no significant concentrations of fog exist in the unsaturated region. As in the 63 "C cases, the comparison between theoretical and experimental disperse-phase temperature profiles worsens slightly at 24 jet diameters (Figure 11). The experimental points lie mostly below, but near, the theory. Fog Temperature Fluctuations. Observations that the temperature fluctuations in fog jets were much greater than in the equivalent dry jets led us to examine this effect more closely, even though the system was not designed for this. To do this we approximated the root mean square temperature fluctuation using the 10-5 averages:
where N represents the number of 10-5 measurements at the same position (N = 30). The difference in this estimated root mean square value between fog and dry jets at the same position can be obtained from Prandtl's mixing length theory. The view (Schlichting, 1968) is that a fluid eddy originating at the spatial location (I+), with a mean temperature of T ( ~ 1 , z ) is displaced by a mixing length 1, to the position ( r j ) ,and experiences a new mean temperature T(r,z). The temperature difference caused by the transverse motion over the mixing length may be regarded as the turbulent temperature fluctuation at (r,z). That is,
T'(r,z) = T(r,z) - T(r-1,z) = 1
ar
(16)
If the turbulent structures of the fog and dry jets are equivalent so that the mixing length is the same, the magnitude of the fluctuations are related by:
-6
-4
-2
0
2
4
6
Radial Distance (r/d) Figure 12. Fluctuations in fog temperature.
crease is proportionately greater than the increased temperatures themselves. We used eq 17 to predict the behavior of the root mean square fluctuation data of a set of 85 "C fog jet temperature profilw measured at 25 jet diameters, cam d1825 and d1825d from the fluctuation data of the analogous dry experiment. The results are plotted in Figure 12. Considering the crude measurement, the behavior of the calculated profile of the temporal fluctuations matches the predicted behavior reasonably well. The maximum in both the predicted and measured (using both dry bulb and wet bulb thermocouples) fluctuation profile lies off the center line. At the center line, the fluctuation is a relative minimum. These qualitative results provide evidence that the turbulent structure in condensing and noncondensing jets is similar. Conclusion The macroscopic continuum theory of heat and mass transfer in a turbulent jet and microscopic theory of heat and mass transfer around a condensing droplet enable calculation of the fog droplet temperature profile and maximum condensation driving force profile independent of the rate of condensation. These theories predict the profiles very well in the near field region and at 8 and 15 jet diameters in the self-similar region in both 63 and 85 "C jets. At 25 jet diameters, in the 63 "C case, the measured temperature increase is less than the predicted value, and the spatial region in which the temperature increase is positive is smaller. The comparison between the 85 "C data and theoretical predictions shows the same trend at these distances from the jet mouth, although the 85 "C measurements agree much better with the theory than the 63 "C measurements. The disagreement between theory and experiment in these regions might be caused by the difficulty in measuring the mean temperature in a fogladen mixture when the fluid conditions fluctuate between unsaturated and supersaturated stabs. The fluctuations in the fog jet temperatures were significantly greater than those in dry jets, consistent with mixing length theory. Nomenclature
C, = heat capacity of total gaseous species d, do = diameter of nozzle
From this viewpoint, the steeper profiles in the fog jets cause the increased temperature fluctuations and the in-
D = water vapor-air diffusivity F(p,2,(po/p31/2) = profile of mean dimensionless scalar quantity in the self-similar region of the turbulent jet
Ind. Eng. Chem. Res. 1992, 31, 713-721
Fr = uo/[Bd(po- ~ ~ ) / p , , ] ' densimetric /~, Froude number H = enthalpy = Xu + C,T
r = radial coordinate r3= rate of formation of maas of liquid water per unit volume due to condensation T = temperature of vapor-air mixture Td = droplet temperature u = axial velocity u = radial velocity z = axial coordinate
Greek Symbols c = eddy diffusivity of scalar (heat or water vapor) h = heat of vaporization of water p = density of air + water vapor 0 = dimensionless passive scalar quantity w = mass fraction of water vapor in total gaseous species Subscripts = value in the ambient environment cl = value at the jet center line m = value along the mixing line (frozen jet case) meas = a measured quantity 0 = value at nozzle effluent Superscripts - = time smoothed quantity * = corresponding to the value at local thermodynamic equilibrium or conditions at the surface of the droplet ' = the turbulent fluctuation of a quantity
713
= dimensionless quantity
Literature Cited Bennett, D. Fog Formation and Vaporization in Turbulent Jets. M.S. Thesis, Carnegie Mellon University, 1971. Chen, C. J.; Rodi, W. Vertical Turbulent Buoyant Jets; Pergamon Press: New York, 1980. Hidy, G. M.; Friedlander, S. K. Vapor Condensation in the Mixing Zone of a Jet. AZChE J. 1964, I, 115-123. Hinze, J. 0. Turbulence; McGraw-Hill: New York, 1975. Matvejev, L. T. On the Formation and Development of Layer Clouds. Tellus 1964, 16 (2), 139-146. Schlichting, H. Boundary-Layer Theory; McGraw-Hik New York, 1968. Strum, M. L. Fog Formation in a Turbulent Jet. Ph.D. Thesis, Carnegie Mellon University, 1990. Sunavala, P. D.; Hulse, C.; Thring, M. W. Mixing and Combustion in Free and Enclosed Turbulent Jet Diffusion Flames. Combust. Flame 1957, I, 179-193. Vatazhin, A. B.; Valeev, V. A.; Likhter, V. A.; Shulgin, V. I.; Yagodkin, V. I. Investigation of Turbulent Vapor-air Jets in the Presence of Condensation and the Injection of Foreign Particles. Fluid Dyn. 1984,3, 385-392. Vatazhin, A. B.; Valeev, V. A.; Likhter, V. A.; Shuigin, V. I. Turbulent Condensation Jets and the Possibilities of Their Control by Means of an Electric Field. Fluid Mech.-Sou. Res. 1985, 14, 10-22. Received for review December 10, 1990 Revised manuscript received July 19, 1991 Accepted August 13, 1991
History Effects in Transient Diffusion through Heterogeneous Media Joe D.Goddardt Department of Chemical Engineering, University of Southern California, Los Angeles, California 90089-1211
This article shows theoretically how scalar transport in a locally inhomogeneous medium characterized by variable capacity C(X) and conductivity K(X) in the diffusion equation c(x) a$/& = V*K(X)V$ y(x,t) can be represented by the homogenized form E*d$/dt = n*Vv l*y. Here E(t) and ~ ( tare ) memory kernels and asterisks denote convolution integrals in t, which serve to represent the history effects arising from local diffusional relaxation. The analysis rests mainly on the effective medium approximation (EMA) applied to the lattice analogue or discretized version of the above equations. New EMA formulas are derived to account for the heterogeneous capacity c(x) and the resulting history effect represented by E*d$/dt. Also, based on symmetry arguments, new representations are given for the associated lattice Green's functions of general, non-Bravais lattices. In the shortrtime or high-frequency limit the discrete EMA predicts neither the diffusive response one expects of heterogeneous continua nor the wave-propagation effecta that often accompany microscopic relaxation. Underlying reasons and the possible need for a nonlocal model involving higher order spatial gradients are discussed.
+
+
1. Introduction 1.1. Transport Model. We deal here with diffusive
scalar transport in a locally isotropic heterogeneous medium which, with boldface symbols denoting physical vectors, is assumed to be governed by the well-known balance and flux relations (1)
and
J = -KV# (2) where, p(x,t), J(x,t), and y ( x , t ) denote, respectively, a spatial density, associated vector flux, and volumetric
source, while #(x,t) is a potential and K = K(X) a scalar conductivity. We assume continuity of # at the interface between distinct phases, and we restrict ourselves to mathematically linear systems, on the basis of (2) and the further assumption P = c# (3) where c = c(x) is a volumetric capacity. In any materially uniform region of space K and c are constant, and (1)-(3) reduce to the diffusion equation:
a#
Y
at
C
- DV2$ + where
Present address: Department AMES-0310, University of California, San Diego, La Jolla, CA 92093-0310.
0888-588519212631-0713$03.00/00 1992 American Chemical Society
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