210
Ind. Eng. Chem. Fundam.,Vol. 17, No. 3, 1978
Binary Diffusivities of Nitrogen-Methane and Nitrogen-Methyl Chloride Systems Ming-Whe Hsieh'' and Huk Y. Cheh Department of Chemical Engineering and Applied Chemistry, Columbia University, New York, New York 10027
Binary diffusivities of nitrogen-methane and nitrogen-methyl chloride systems were measured by using a modified Stefan diffusion cell, at temperatures between 25 and 110 O C . The results agree well with those calculated by using the Kihara potential model for the nitrogen-methane system and the effective potential model for the nitrogen-methyl chloride system.
Introduction Various unsteady-state methods have been used to measure diffusivities in gaseous systems. The two earliest developments were those by Loschmidt (1870) and Stefan (1874). Loschmidt used a long glass tube which was divided into two sections. Each section was filled with a different gas and the two gases were then permitted to diffuse into each other. The diffusivity was calculated from the concentration history and the dimension of the two sections. On the other hand, Stefan used a tube which was closed at one end and initially filled with a pure sample gas. A purge gas was then swept over the open end of the tube and equimolar counterdiffusion was allowed to take place. The diffusivity was then calculated from the mean concentration of the gas in the tube a t the end of a run. In order to simplify the technique developed by Stefan, Rhodes (1962) measured diffusivities from the continuous concentration history of the effluent purge gas stream. However, the measured diffusivity was found to depend on the flow rate of the purge gas. Further modification by Frost and Amick (1973) permitted a relatively flow rate invariant diffusivity determined directly from the concentration of the purge gas effluent. In this investigation, a mathematical model was established to account for the flow rate-diffusivity dependence. In addition, the analysis of the effluent stream was simplified considerably by the use of a gas density balance. Experimental measurements were made for a nonpolarnonpolar (NrCH4) system and a nonpolar-polar (NpCHSCl) system. The results were then compared with predictions made by intermolecular potential models. Experimental Section Apparatus. Schematic diagrams of the diffusion cell and the experimental setup are shown in Figures 1and 2. The four essential parts of the experimental apparatus are the manifold system, diffusion cell, gas density balance, and recording system. The manifold system allows the diffusion cell to be alternately filled with the sample gas and purged by the second component. A provision was also made in the manifold to supply a calibrated amount of the purge gas to the gas density balance. The diffusion cell used in this work was similar to that used by Rhodes (1962). It was made of stainless steel and had a diffusion path of 9.756 cm. The cell was filled with capillary tubes of 0.0696-cm i.d. in order to prevent the purge gas from disturbing the gas in the cell. The diffusion volume of the cell was 21.951 cm3. The cell consisted of three interconnecting
parts which were bolted together and sealed by copper gaskets. The effluent gas from the diffusion cell was continuously analyzed by a gas density balance (Model 11-140 Gow Mac) which was used mainly by other investigators for quantitative work in gas chromatography (Martin and James, 1956; Guillemin et al., 1966, 1967; Walsh, 1967). However, the linear relationship between the output voltage and the concentration difference of the reference and purge gases makes the gas density balance well suited for the measurement of gaseous diffusivity (Nerheim, 1963; Schefflan, 1971). Possible error for the determination of concentration is f1.6%. Flow to the diffusion cell and the gas density balance was controlled by needle valves and monitored by a calibrated bubble flow meter. The diffusion cell and the gas density balance were placed in separate constant temperature baths with temperature fluctuation less than f 0 . 2 "C. The power supply for the gas density balance was supplied by Gow Mac (Model 9999D1). The output voltage was recorded on a strip-chart recorder (Model 1101S, Esterline Angus). The purity for nitrogen, methane, and methyl chloride used in this work was 99.997%,99.0%,and 99.5%, respectively. Procedure. The experimental procedure consisted of two modes of operation, the preliminary preparation and the actual run. When the system was operated in the preliminary preparation mode, the cell was filled with the sample gas. Flow rates of purge and reference gas for the gas density balance were set and measured. The gas density balance was zeroed. When the preliminary preparation was completed, the system was switched to the actual run. This was accomplished by changing the flow path of the purge gas so that it swept over the top of the diffusion cell prior to the gas density balance.
Diffusion Model The differential equation for the modified Stefan cell is described by Fick's second law of diffusion
acl - D- a w l --
at ax2 where C1 is the concentration of the sample gas in the diffusion cell, t is time, x is the distance along the axis of the cell measured from its bottom, and D is the binary diffusivity to be measured. The boundary conditions are: (1)a t t = 0, x 2 0 (2) a t t
> 0, x
=0
Address correspondence to this author at the Engineering Technology Center, Allied Chemical, Morristown, N.J. 07960.
0019-7874/78/1017-0210$01.00/0
(3) 0 1978 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 17, No. 3, 1978 211
..-A
Vent
Figure 2. Schematic diagram of equipment layout. Side View
diffusivity calculated by eq 9 the apparent diffusivity and denote it by D’. In addition, let us refine the boundary condition a t x = L by using the equation Figure 1. Schematic diagram of the diffusion cell.
( 3 ) at t
> 0, x
at t
(4)
0
where Cl,o is the initial concentration of the sample gas in the cell and L is the diffusion or cell length. Equation 4 is an approximation suggested originally by Rhodes (1962). The solution of eq 1 subject to boundary conditions eq 2 to 4 can be found in the work of Carslaw and Jaeger (1959)
c = -4 1 c1,0
-
P n=O
(-1)n e-(2n+1)2r2Dt/4Lz (2n
+ 1)
x cos
(2n
+ 1)xx
(5) 2L A relation between the diffusivity, the concentration of the effluent gas, Cl,out,and the concentration of the inlet gas, Cl,in, can be obtained by a material balance of the sample gas over the entire cell
where V is the free volume of the cell and F is the volumetric flow rate. Since pure nitrogen is used as the purge gas, C l , i n is equal to zero. Substituting eq 5 into eq 6 gives
2
Cl,out = 2c1vovD e-(2n+1)%2Dt/4L2 FL2 The asymptotic form for eq 7 a t large times is
(7)
> 0, x
= L, C1=
V D acl - Y-~ FL ax
Equation 1 subject to boundary conditions eq 2,3, and 11 can be solved by the standard separation-of-variable technique. The solution is
-C1
m - 2 E---
(k,x/L) e-kn2Dt/LZ (12) C1,o n=l k n [ l + ( Y V D ~ ~ / F L ~ ) ~ ] ~ ’ ~ where kn is the root of the equation Y E k , tan k n = I FL Substituting eq 12 into eq 6 yields
In analogy to the derivation of eq 9, the diffusivity can be evaluated by using the asymptotic solution of eq 14
Dividing eq 9 by eq 15 gives
By solving eq 13 in terms of a series for k l , we obtain
+ ( 1 - “)( YTV D) ~ + . . .] It can easily be shown that eq 8 is accurate to 99% provided that t I 0.2333L2/D. Therefore, the diffusivity can be evaluated by rearranging eq 8 to the following form D = - - 4L2 d In C1,ouJdt T2
(10)
where y is an empirical constant. By applying the material balance as expressed in eq 6 , eq 10 can be rewritten as
=L c1=
> 0, x = L , C1 = yCl,out
at t
Section A-A
(at t 2 0.2333L2/D)
(9)
By applying eq 9 to experimental data, it was found, however, that D is inversely proportional to the flow rate of the purge gas (Rhodes, 1962; Schefflan, 1971; Hsieh, 1974). This behavior may be explained by the fact that the boundary condition at the top of the cell as expressed by eq 4 is a relatively poor approximation. Physically, it is known that the concentration of the sample gas a t x = L approaches zero only at very high gas flow rates. For the sake of convenience, let us call the
12
(17)
FL
and consequently
D‘ YVD _ D - 1-2
yVD (z3) ( z) + ... - 2 ( 2 - e) 12 r+)3 FL 2
+
(18)
Equation 18 shows that as F approaches infinity, D’ is equal to D . Also, by truncating the right-hand side of eq 18 after the first two terms, it is seen that [(D’lD) - 11 is inversely proportional to F , and y is obtained as L 2 dD’
Y=-2VD2d(llF)
212
Ind. Eng. Chem. Fundam., Vol. 17, No. 3, 1978
Table I. Diffusivity of the N2-CH4 System curve 1 2
Purge G ~ flow S Rate. cmVmln 330
4
161 111 85
5
72
3
temp, K 298.2 323.2 348.2 373.2 384.2
I Figure 3. Output from gas density balance as a function of time.
temp, K
, , , , , , , , , , , , , ,
298.2 313.2 333.2 353.2 373.2
~
0 1 2 3 4 5 6 7 8 9 101112131415 Reciprocal Purge Gas Flow Rate, lOOO/F, min/cm'
Figure 4. Apparent diffusivity vs. purge gas flow rates. It should be noted that y is always less than 1.A value of 0.91 is chosen best representing the data in Figure 4. For F greater than 2.0 cm3/s the variable yVD/FL2 is less than 0.120. Hence eq 1 8 converges rapidly. By applying the mathematical model to experimental results, the diffusivity D can be determined as follows. A series of runs employing different flow rates of the purge gas are made under isothermal, isobaric, and constant initial concentration conditions. D' is then evaluated by using eq 9. The binary diffusivity D is then obtained by extrapolating D' to infinite flow rate of the purge gas. Results Experimental results on the diffusivities of N2-CH4 and N2-CH3C1 systems are summarized in Tables I and 11. The diffusivity was obtained in the following manner. Take NzCH4 at 50 "C as an example. D' was measured at five flow rates varying between 71.5 and 330.0 cm3/min. Since the output voltage of the gas density balance, E , is directly proportional to Cl,out,the logarithm of E was plotted as a function o f t as shown in Figure 3. D' was then determined by the leastsquares method from the slope of these lines. Finally, D' was plotted as a function of the reciprocal of the purge gas flow rate as shown in Figure 4. D was then determined by an extrapolation also using the least-squares method to infinite flow rate. It was found to be 0.250 cm2/s for the present case. An error analysis (Hsieh, 1974) showed that the maximum experimental error was f 2 % . Discussion Mueller and Cahill (1964) reported diffusivities of the NTCH4 system at 25,79.4, and 109.3 "C to be 0.216,0.287, and 0.332 cm*/s, respectively. At 25 "C, the diffusivity from the present investigation was found to be 0.214 cm2/s, whereas the diffusivities at 79.4 and 109.3 "C were interpolated from the present results to be 0.290 and 0.336 cm%, respectively. The agreement of these two studies is therefore within 1.2%. For the N r C H 4 system, the diffusivity was also calculated by evaluating the collision integral (Hirschfelder et al., 1964)
0.214 0.250 0.283 0.323 0.339
eviation
IDTbmeasdX'100
0.216 0.250 0.285 0.321 0.338
0.93 0.00 0.71 0.62 0.29
Table 11. Diffusivity of the NZ-CH&l System
nmc, Mlnuter
:::I,
diffusivity, cm% calcd from Kihara measd potential
diffusivity, cm2/s calcd from the effective measd potential model 0.129 0.140 0.157 0.179 0.193
0.126 0.138 0.155 0.173 0.191
%
eviation,
1 Dj /LJrd
X
2.33 1.42 1.27 3.36 1.04
based on the Kihara potential model (Kihara, 1953). For the NrCH3C1 system, the diffusivity was calculated by using the effective potential model (Hirschfelder et al., 1964). Force constants needed in these calculations were determined by the least-squares fitting of viscosity data. Details of these calculations have been described by Hsieh (1974). Results are included in Tables I and 11. It is observed that the maximum deviation between the experimental and calculated values was less than 3.4%. Conclusions Binary diffusivities of N r C H 4 and NrCH3Cl systems were measured by using a modified Stefan cell. Experiments were carried out a t temperatures between 25 and 111"C. Results for the NZ-CH4 system agree well with those reported by Mueller and Cahill. Also, results for both systems compare satisfactorily with predictions from two intermolecular potential models. Nomenclature C1 = concentration of the sample gas in the diffusion cell Cl,o = initial concentration of the sample gas in the diffusion cell Cl,out = concentration of the sample gas in the effluent stream D = diffusivity D' = apparent diffusivity F = volumetric flow rate of the purge gas k, = therootofeq 13 L = length of the diffusion cell t = time V = free volume of the diffusion cell x = axial coordinate measured from the bottom of the diffusion cell y = a dimensionless constant defined by eq 10 Literature Cited Carslaw, H. S., Jaeger, J. C., "Conduction of Heat in Solids", 2nd ed, Oxford University Press, Oxford, 1959. Frost, A. C., Amick, E. H., Ind. Eng. Chem. Fundam., 12, 129 (1973). Guillemin, C. L.,Auricourt, F., Blaise. P., J. Gas Chromafogr..4, 338 (1966). Guillemin, C. L., Auricourt, F., Blaise, P., 2.Anal. Chem., 227, 260 (1967). Hirschfelder, J. O.,Curtiss, C. F., Bird, R. B., "Molecular Theory of Liquids and Gasses", 2nd ed, Wiley, New York, N.Y., 1964. Hsieh, M.-W., Doctoral Dissertation. Columbia University, New York, N.Y., 1974. Kihara, T., Rev. Mod. Phys., 25, 831 (1953). Loschmidt, J., Sitzber. Akad. Wiss. Wien, 61, 367 (1870). Martin, A. J. P., James, A. T., Biochem. J., 63, 138 (1956).
Ind. Eng. Chem. Fundam., Vol. 17, No. 3, 1978 Mueller, C. R., Cahill, R. W., J. Chem. Phys., 40, 651 (1964). Nerheim, A. G., Anal. Chem., 35, 1640 (1963). Rhodes. R. P., Doctoral Dissertation, Columbia University, New York. N.Y., 1962. Schefflan, R. A., Doctoral Dissertation, Columbia University, New York, N.Y., 1971.
213
Stefan, J., Sitzber. Akad. Wiss. Wien, 88, 385 (1874). Walsh, J. T.,J. Gas Chromatogr., 5, 232 (1967).
Received for review November 29,1977 Accepted May 4,1978
The Augmentation of the Heat Transfer Coefficient in Turbulent Flow in Annular Passages by Transverse Flow Ali M. El-Nashar Department of Mechanical Engineering, University of Mansoura, Mansoura, Egypt
Flow and heat transfer measurements were made in an annulus having a diameter ratio of 0.5 with fluid injection at the inner (porous) tube and heat transfer at the outer tube. The Reynolds number of the main flow upstream of the test section before injection started ranged from 0.9 X IO5 to 2.15 X IO5. The injection ratio VwlUo,, ranged from 0 to 0.022. An increase in the heat transfer coefficient of approximately 30% over that for no injection was achieved for a Reynolds number of 2.0 X IO5 and for an injection ratio of 0.022.
Introduction There has been much interest for the past two decades in the subject of increasing the heat transfer coefficient from heated surfaces. Most of the heated surfaces are cooled by fluids that are made to flow past the heated surface in a direction tangential to the surface. An alternative method, which is the subject of the present investigation, is to inject part of the coolant normal to the heated surface. In an annular arrangement with one heated surface, injection could be achieved by making the unheated tube of a porous material through which injected fluid passes. If one considers such an annular arrangement, the injected flow-through the inner tube of the annulus-may be expected to distort the main axial velocity profile and to move the region of maximum velocity closer to the heated surface, which forms the outer tube, and thus increases the outer wall shear stress and heat transfer coefficient. This would result in an improvement in the cooling capability of the system additional to that caused by increasing the coolant flow rate. The purpose of the study reported herein is to experimentally verify and quantify this effect. Experimental Apparatus and Test Procedure 1. Test Section Description. An annular arrangement with a diameter ratio of 0.5 was selected for this study. In order to reduce the experimental difficulties, the apparatus was designed so the outer wall of the annulus was heated; flow was injected through the inner (porous) wall. A schematic of the annular test section is shown in Figure 1. It consists of a 2-in. 0.d. stainless steel pipe 23 f t long, the middle 19 in. of which are cut and replaced by two sintered bronze porous pipes joined together. Each of these porous pipes has an outside diameter of 2 in. and a length of 9.5 in, The inner pipe is plugged at one end and connected to an injection air blower at the other end. The outer tube is a Pyrex tube. 4 in. i.d., 4.5 in. o.d., and 10 f t in length. The inside of the Pyrex tube was given a 0.001-in. thick nickel-chromium coating. Joulean heating was generated in this coating under a dc electric potential difference imposed at the ends of the glass tube. A 2 in. thick fiber glass insulation was placed on the glass tube. The maximum heat generated was 6 kW, which 0019-7874/78/1017-0213$01.00/0
corresponded to a maximum heat flux of 2000 Btu/(h ft2). The main air in the annulus was provided by the main air blower having a capacity of 850 cfm at a pressure head of 4.6 cmHg which corresponds to a maximum average axial velocity U,, in the annulus of 220 ft/s (with no injection) and a Reynolds number of Re = 2.2 X lo5. The injection air was provided by a blower having a capacity of 216 cfm at a pressure head of 28.3 cmHg which corresponds to a maximum injection velocity V, = 4.4 ftls. The experimental apparatus was designed so that fully developed flow and heat transfer would be established upstream of the position where injection starts. Hence any change in the wall shear stress and heat transfer coefficient in t$e test section (where injection takes place) would be due to mass injection. 2. Measurements. Both velocity and temperature distribution in the annulus were measured using 10-pm diameter tungsten hot-wire probes. Each probe was attached to a traversing mechanism using a micrometer to adjust the position of the probe inside the annular space. A Disa Model 55A01 constant-temperature anemometer was used in the tests. Each probe was calibrated at a range of operating temperatures by placing it side by side with a pitot static tube in a calibration unit with controllable air temperature. It was possible to find the air velocity at any location in the annular space by knowing the probe dc voltage and the air temperature a t that location and then referring to the calibration curves for that particular probe. For surveying the temperature of the air, the hot-wire probes were calibrated against copper-constantan thermocouples in the same calibration unit. The wire resistance was measured at different air temperatures and a calibration curve was drawn for each probe. The temperature of the outer surface of the Pyrex tube was measured at several axial and radial locations using copper-constantan thermocouples attached to the surface using an epoxy resin. The pressure drop in the main flow direction between two sections each 6 in. away from the middle section of the porous tube was measured using an F. Fuess Model 134B micromanometer having a sensitivity of 0.05 mmHzO. The heat flux was estimated from the dc current and voltage after the heat losses were subtracted. The voltage at several
0 1978 American Chemical Society
