1
M N no
P, PR Q
R r
S T T, t u
Z,,, x
= plasma length, meters = molecular weight of gas, kg. per mole =
number of turns in induction coil
= number density of atoms, meters-3 = conduction and convection power, watts = = = = = = = = = = =
radiation power, watts induction power number gas law constant, joules. mole-’. OK-’ radius, meters “skin depth of heating,” meters plasma temperature a t radius r, OK. plasma temperature for simplified model (assumed constant throughout plasma), OK. time, sec. axial velocity of gas out of plasma (assumed to be uniform), meters per second screened nuclear charge number argument of Kelvin function
GREEKLETTERS a y
= =
eo
9 po
u,,
= = =
=
#,I
=
p
=
u w
= =
permittivity of free space, faradays per meter viscosity of gas, kg. per meter per second permeability of free space, henries per meter frequency interval, radians per second turbulence correction factor density of plasma (assuming complete dissociation of gas, p = 6.02 X lOZ3noM), kg. per cu. meter electrical conductivity of plasma, mhos per meter frequency, radians per second
literature Cited
Babat, G. I., J . Znst. Elect. Engrs. 94 (111), 27 (1947). Cannon, H. R., “A Study of an Induction-Coupled Plasma Operating at 400 Kilocycles,” M.S. thesis, Air Force Institute of Technology, Air University, 1962. Petschek, H. E., Rose, P. E., Glick, H. S . , Kane, A., Kantrowitz, A., J . Appl. Phys. 26, 83 (January 1955). Reed, T. B., J . Appl. Phys. 32, 821 (May 1961). Scholz, O., Umschau Wiss. Tech. 23, 716 (1959). Unsold, A., Ann. Phys. 33, 607-16 (1938).
ionization fraction partition function ratio
RECEIVED for review August 22, 1966 ACCEPTED August 25, 1967
ENTRANCE HEAT TRANSFER FROM A PLASMA STREAM IN A CIRCULAR TUBE Engineering Correlations J. R. JOHNSON, N. M. C H O K S I , AND Texas A & M University, College Station, T e x . 77843
P. T . E U B A N K
Experimental values of the Nusselt number were measured and empirically correlated for entrance section heat transfer from a confined plasma jet of three gases-helium, argon, and nitrogen-flowing in a circular tube. The approximate upper limit of Reynolds number was 900 and temperature level was 7000” K. Heat transfer coefficients were higher than in previous investigations because the plasma enters the tube from a nozzle of smaller diameter causing recirculation near the tube wall. The present correlation is compared with empirical equations for warm gases at subplasma temperatures, boundary layer theory predictions, and the plasma correlation of Skrivan and von Jaskowsky. The “hot plasma core” model is discussed in respect to photographs of the plasma jet for each of the three gases.
of the most recent topics in heat transfer involves conplasma streams produced from gases heated to extremely high temperatures. At present, plasma jets are used for propulsion studies (Skifstad, 1961), simulation of space vehicle surrounding during re-entry, and studies of chemical synthesis (Freeman and Skrivan, 1962). A fourth important area of application of plasma heat transfer is power generation from nuclear fusion reactors. Electrical power may be obtained from the plasma produced by the nuclear reaction by heat transfer to the confining walls or direct production of electrical current from concentric magnetic coils surrounding the plasma stream. Good reviews of plasma jet technology have been prepared by Dennis et al. (1953) and Kubanek and Gauvin (1 967) Numerous theoretical and/or experimental plasma heat transfer investigations have been reported, including those of Bro and Steinburg (1962), Emmon (1963), John and Bade (1959), Stokeset al. (1960), and Wethern and Brodkey (1963). T h e objectives of the present study are the experimental measurement of entrance-section heat flux in the radial direcNE
0 fined
34
I&EC PROCESS DESIGN A N D DEVELOPMENT
tion from a 12-kw. gas vortex-stabilized d.c.-plasma jet to the walls of a water-cooled, tubular, segmented calorimeter with three separate gases (helium, argon, and nitrogen), the development of an empirical correlation from the resultant Nusselt number data applicable to the entrance region, and a comparison of the present data and correlation with boundary layer predictions and the results of previous plasma heat transfer investigations (notably, Skrivan and von Jaskowsky, 1965). The objectives of this study differ from those of previous experimental heat transfer work in that the diameter of the heat exchange section is large compared to the nozzle exit. Experimental Apparatus
A detailed description of the present apparatus with operating procedures has been given by Johnson (1966). Plasma Generator. A schematic diagram of the plasma generator and attached heat transfer measurement chamber is shown as Figure 1. The cathode is a 1/2-inch diameter rod of 2% thoriated tungsten. The cathode operates as a micrometer to adjust the arc length or distance between the cathode
the same temperature, thermal equilibrium is said to exist. For plasma jets a t atmospheric pressure and above, thermal equilibrium is generally assumed, as discussed by Finkelnburg (1965). Recirculation. When a n isothermal j e t passes from a nozzle of diameter D , to a confining tube of greater diameter D , the jet spreads to the tube wall, with a resultant decrease in velocity and increase in pressure as described by Craya and Curtet (1955). If the adverse pressure gradient is of sufficient magnitude, the phenomena of recirculation may occur between D,, the rethe main jet stream and the tube wall. For D circulation rate is approximated by
>>
‘ ~ ! C O O ~ ~ ~ ~
-HEAT
TRANSFER CHAMBER
A
I
Q
P L A S M A GENEqATOR-’
Figure 1. Cross-sectional view of plasma generator and heat-tra nsfer c ha mber
and anode. T h e anode is a nozzle of pure copper which is cooled directly by a high-velocity water stream. A 3/l&ch diameter nozzle was used for argon and helium, and a ‘/r-inch diameter nozzle for nitrogen. T h e gas vortex used to stabilize the jet is created by injecting the gas tangentially between the electrodes, which produces a n arc with a high rate of spin. Heat Transfer Chamber. T h e horizontal heat transfer chamber consists of five heat exchange sections each 1.1875 inches long with a n inside diameter of 1 inch. Each section is a water-cooled calorimeter with inlet and exit water temperatures measured by thermocouples connected to a n automatic temperature recorder. I n addition, four equally spaced thermocouples were installed on the inside wall of each section to provide the average wall temperature. Both cooling water and experimental gas flow rates were measured with rotameters of 0.5% accuracy. T h e maximum water and gas flow rates were 2.00 gallons per minute (per section) and 1.30 standard cubic feet per minute, respectively. With negligible pressure drop through the apparatus, the gas or plasma pressure was atmospheric. T h e instrumentation described above allowed all data for a given steady-state run to be recorded within 5 minutes.
Operating Conditions. T h e operating variables were power input, gas flow rate, and the gas under investigation. Table I provides the operating ranges for these variables together with wall temperatures and heat rates. Diagnostics
Flame Configurations. Figure 2 illustrates typical plasma flame configurations for helium, argon, and nitrogen. T h e flame or “hot plasma core” diameter a t the left side of each photograph equals the nozzle diameter. T h e red helium plasma is conical and 2 to 3 inches long; the blue argon plasma is also conical and 1 to 2 inches long; nitrogen, however, emits a dense white plasma approximately 10 inches long. For all experimental runs, the nitrogen flame length exceeded that of the heat transfer chamber, whereas this was never the case for helium and argon. Equilibrium. Particles present in the plasma stream include molecules, atoms, electrons, and ions of various charge. Each particle possesses energy and hence a temperature. When all the particles a t a given point in the plasma have
Table 1.
where Q, is the volumetric recirculation rate about the eye of the eddy, Q is the over-all forward flow rate through the tube, and Q+ is the relative recirculation rate as given by Becker (1961). With the present apparatus, Equation 1 predicts relative recirculation rates of 1.6 and 1.0 for the smaller and larger nozzles, respectively. Examination of Figure 2 fails to show significant swelling of the hot-plasma-core diameter, which indicates that the predicted recirculation rate may be high. Comparison of Figure 2 and Equation 1 may not, however, be valid, because the jet is not isothermal and the jet diameter equals the hot-plasma-core or flame diameter only a t the tube entrance. Reynolds Number. Different runs were made by varying the gas flow rate and electrical energy supplied to the plasma. T h e Reynolds number
w
4 Re = -
nDuP
is generally used as the criterion for the mode of flow. Neither the mass flow rate, W‘, nor the inside diameter of the heat transfer tube, D, depends on the longitudinal distance, Z , from the heat transfer chamber entrance. T h e temperature, T,and, hence, viscosity of the plasma, p , , decrease with increased Z . Thus, R e is an increasing function of Z . For a given experimental run, all physical properties of the plasma were evaluated for each section a t the bulk mean average plasma temperature of that section. Reynolds numbers from 50 to 900 were thus calculated for 142 runs with the three gases-helium, argon, and nitrogen. Model. In’consideration of the low Reynolds numbers and the presence of recirculation, it is likely that laminar, transitional, and turbulent flow existed simultaneously a t different locations in the fluid. Flow occurs at high velocity because a gas of constant flow rate has been heated to a high temperature and must escape by expansion through the heat transfer chamber. A hot core model was selected which assumes that radial heat transfer occurs from a hot plasma flame (Figure 2) of uniform temperature across a thermal boundary layer to the tube wall. T h e thermal boundary layer should be turbulent
Range of Operating Conditions
Power Input Gas
E, volts
I, amp.
Helium Argon Nitrogen
29.5-41.2 12 .8-25 .8 46.0-68.5
98-272 78-306 63-220
Gas Flow Rate, W , Lb./Hr. 0.644-2.111 2.700-6.744 1.145-6.219
Wall Temp., T,, OF. 75.9-179.9 80.0-131.1 86.1-174.2
VOL 7
NO. 1
Heat Rate,
Q , B.t.u./Hr. 0-5319 0-2023 227-4310
JANUARY 1 9 6 0
35
near the tube entrance owing to recirculation. Further downstream a laminar thermal boundary layer should predict an increasing boundary layer thickness owing to depletion of the flame. Physical Properties. Because of the relatively low plasma temperatures achieved in this investigation, physical properties were taken from references which assumed low per cent ionization. The transport properties-viscosity, w, and frozen or translational thermal conductivity, k,-were obtained from Amdur and Mason (1958) for the temperature range of 1000° to 15,000° K. These authors used intermolecular force constants derived from molecular beam measurements for the calculation of viscosity of nitrogen and the inert gases. T h e translational thermal conductivity was then computed from the equation
Heat Trtlnrfer Analysis H e a t Transfer Coefficient. The heat transfer coefficient, hi, for section i was calculated from the equation (5) where L , is the section length, Qt is the steady state rate of heat absorbed by the cooling water, and Tu,{ is the average wall temperature. The bulk mean average plasma temperature is
The hulk mean plasma temperature Tv,, at the entrance to the ith section is found from the heat balance
where R is the gas constant in calories per gram-OK. For monatomic helium and argon, the specific heat, C p , was assumed constant a t 1.25 and 0.125 cal. per gram-OK., respectively. For nitrogen, C , was determined from the expression
where 8 is the characteristic temperature. Additional thermodynamic and transport coefficient results have been tabulated for argon by Drellishak et al. (1962) and nitrogen by Ahtye and Peng (1962). 36
1 6 E C PROCESS D E S I G N A N D DEVELOPMENT
-kza,] (k) i
(3)
=
[EZ
(7)
and
where HD,t is the bulk mean plasma enthalpy a t the entrance to the ith section, E is the voltage across the electrodes, Z is the current, and H, is the enthalpy of the gas at a suhplasma reference temperature T,. The heat capacity of Equation 8 is understood to include sensible heat, heat of dissociation, and heat of ionization effects. Radiation. Contributions to the heat transfer coefficient of Equation 5 come from forced convection, radiation, and
conduction. T h e last two modes of heat transfer were neglected after the radiation contribution was estimated by the Kramers-Unsold model as used by Tankin and Berry (1964) and preliminary calculations indicated conduction to be unimportant. Calculation of the radiation heat flux to the wall from each cylindrical elemental volume of plasma requires knowledge of the point plasma temperature distribution, T ( r , z ) . An attempt was made to scale the experimental plasma temperature profiles of Grey (1965) to the present system by placing both systems on a dimensionless basis. Radiation contributions so calculated never exceed 3% of the total heat flux with maximum plasma temperatures in the heat transfer chamber of 3990°, 6370°, and 6500' K. for helium, argon, and nitrogen, respectively. Extensive investigations of radiation losses from argon plasmas have been conducted by Barzelay (1966) and by Brown and Ross (1965). Analysis. A total of 40 experimental runs were performed with helium, 38 with argon, and 64 with nitrogen. These data in the form of the Nusselt number as a function of Reynolds number, viscosity ratio, and longitudinal distance from the heat transfer chamber entrance have been deposited with the American Documentation Institute. Dimensional analysis of entrance heat transfer from the apparatus yields for each gas
Method I
For each section (constant 2 ) plots were prepared of In N u us. In R e with the viscosity ratio as a parameter as discussed in
detail by Choksi (1967). T h e slope B of these graphs varied little with viscosity ratio but showed a definite decrease with increased values of 2. For simplicity, a n average value of B (Table 11) was chosen for each gas. Values of C were found in an analogous manner from graphs of In Nu us. In ( , u p / p w ) . Again, an average value of C was selected for each gas; all values were negative and a definite decrease was noted with increased values of 2. Next, a correlation between sections was initiated with computation of A
(g)
from Equation 11 and the average
values of B and C. Arithmetic average values of A for each section were then plotted as In A us. In Z (Figure 3). T h e resultant parabolic curve led to the equation
+
A = A1 exp ( A z Z * ~ AZ*)
(13)
where Z* = In (Z/D).T h e constants, A I , A * , and A B , appearing in Table 11, were calculated by least square techniques for each gas. 100
80
(9)
60
where Nu = hD/kp is the Nusselt number, (kp/jiTv) is the ratio of bulk mean plasma viscosity to the viscosity of the gas a t the
40
CPPP
wall temperature, and P r = - is the Prandtl number. kP
T h e range of Nusselt number values for this investigation was 0 to 170. The Reynolds number and viscosity ratio have essentially replaced the experimental variables of flow rate W and temperature or energy level, respectively. As the Prandtl number and ( D n / D ) were constant for each of the three gases. Equation 9 reduces to
(
0 ARGON
20 A HELIUM
+NITROGEN
IC
a
::,3
c
NU = NU R e , - ? -
(E)
4
C
Nu = A(Re)B or In Nu = In A
+ B In R e f C In , B,
T h e coefficients A = A
p)
\4
(12) 0.4
and C were found for each
0.8 1.0
Figure 3.
Table 11.
No. of Data Points
Ai
4
2
6
VD
gas by two methods.
Gas
0.6
Variation of A with (Z/D)
Summary of Methods
-A2
A3
B
-c
0.759 0.851 0.689
0.600 0.434 0.255
1.547 1.338 1.611
34.4 30.3 26.2
1.234 1.003 0.814
1.283 0.991 0.215
0.965 1.246 1.595
50.6 59.6 42.1
(AAPD)
METHOD I
Helium Nitrogen
200 190 320
3.73 11.5 81.7
0.797 0.931 0.725
Helium Argon Nitrogen
200 190 320
7.41 15.73 17.52
0,849 0.845 0.668
Argon
METHOD I1
VOL. 7
NO. 1
JANUARY 1 9 6 0
37
Combination of Equations 11 and 13 yields the final form of the correlation by Method I :
Table 111. Gas
Helium Argon Nitrogen Nusselt numbers calculated from this equation were compared with corresponding experimental values for each run. Each comparison provided a value of the absolute per cent deviation (APD), APD =
1 calculated Nu - experimental Nu , . .. - ~ X 100% I experimental Nu I
(15)
which was averaged arithmetically to yield the average absolute per cent deviation (AAPD) values of Table 11. Method II
A second method as followed by Skrivan and von Jaskowsky (1965) was also used to correlate the data. This method differs from the first in two important aspects: evaluation of physical properties and order of evaluation of the coefficients, A
(3 -
, B , and
C for Equation 11.
(AAPD) 32.2 32.3 24.0
three gases, a large increase in the (AAPD) was observed. An average value of B for only helium and argon, however, did not change the accuracy of the correlation (Table 111). Nitrogen dissociation prior to ionization provides a stable hot core over a wide range of energy levels and flow rates Examination of Figure 2 and other flame photographs indicates that the nitrogen flame configuration is less sensitive than that of the inert gases to changes in the flow rate or Reynolds number. The significantly higher value of A1 for nitrogen is also attributed to flame configuration, as the thermal boundary layer thickness is less than that of the inert gases, because a much higher percentage of nitrogen molecules are dissociated and/or ionized which broadens the hot core zone. Warm Gas Extrapolation
With Method 11, all physical properties for all heat transfer sections of a given run are determined from the bulk mean temperature T p , lat the inlet to the first section. Thus, the Reynolds number of Equation 2 and the viscosity ratio do not vary with Z for a steady state run and calculational procedures are simplified. The correlation procedure of Method I1 is initiated by plotting In Nu us. Z* for each run. Curves similar in shape to those of Figure 3 were obtained with the present data. Equation 14 was rewritten as Nu = A’1[exp(A*Z**
Final Coefficient Values Ai B 5 .OO 0.517 7.70 0.517 64.2 0.255
+ AaZ*)]
(1 6)
Common correlation equations for heat transfer from a warm gas at subplasma temperature to the walls of the confining tube are Laminar flow. fiu = 1.86(Re Pr) 1/3
(g)1’3(
Turbulent flow. Nu = 0.026 Reo.* Pr1i3
:y’O4
(18)
(:>””
Here R u is the average Nusselt number over a length of tube
L, p b is the bulk mean average viscosity of the gas, and flow is
run, values of A‘1, A B , and A3 may be determined from a least square fit of the curve. The constants, A2 and A3 (Table 11), are arithmetic average values for all runs. T h e exponents, B and C, were then evaluated as in Method I. Finally, constant A1 was adjusted for each gas to provide the lowest (AAPD) between calculated and experimental Nusselt numbers (Table 11). Comparisons of Methods I and I1 illustrate that they provide coefficients of the same sign and approximate magnitude. Less change in A1 between the three gases is noted with Method 11, whereas the exponents, B and C, are somewhat more consistent by Method I. Because of the better correlation with experimental data points (AAPD), the results of Method I were selected.
assumed to be fully developed. If the present heat transfer chamber had been of considerable length, heat transfer between the gas now cooled to subplasma temperatures would be observed downstream. The Nusselt number would be a continuous function of Z for Z[(O, L ) . The present plasma correlation for entrance flow should thus extrapolate to Equation 18 or 19 for high values of (Z/D). Likewise, the coefficients A(Z/D), B, and C should extrapolate to the analogous coefficients of Equation 18 or 19. Figure 3 illustrates the continuity of A(Z/D). As expected, higher values of Nu were obtained for entrance plasma heat transfer than predicted by either Equation 18 or 19. Present values of B are closer to the laminar correlation (Equation 18), although turbulent flow would exist a t the exit of a greatly elongated tube with all the experimental runs because of the effect of cooling on gas viscosity. Exponents B and C are discussed below from a boundary layer theory viewpoint.
Correlation between Gases
Boundary Layer Theory Results
where A ’ I = A1(Re)B
(k)“.
Since
A’1
is constant for each
An attempt was next made to find a single value for each of the coefficients--81, A*, As, B, and C-applicable to all three gases without significant increases of the (AAPD) values of Table 11. The result was the final correlation equation Nu = Al[e~p(-O.818Z*~f 0.764Z*)]ReB
(E)
(17 )
with values of A1 and B given in Table 111. Because little variation existed in AB, AS, and C for the three gases, these quantities were averaged to yield the numerical values of Equation 1’7. With an average value of B for all 38
l&EC PROCESS DESIGN A N D DEVELOPMENT
The boundary layer was taken as the hot gas region between the hot plasma core of Figure 2 and the tube wall. If the thermal boundary layer thickness, 8T is assumed to equal the momentum boundary layer thickness, 6, or Pr = 1, the equation h =
3 (k/6) 2
-
results when the Pohlhausen temperature and velocity profiles are assumed across the thickness of the boundary layer. Equation 20 provides only a rough approximation of h from simplified boundary layer theory.
Table IV.
Comparison of Coefficients with Those of the Boundary layer Equation Equation A ( Z / D) B C 17 A I exp (-0.818 2*2 0,517 -1.50 0.764 Z * ) (argon, helium)
+
0.255 25
(nitrogen)
0.323 ( Z / D ) - 1 ' 2
0.5
theory for parallel flow over a thin, flat plate. Neglecting tube curvature, this equation is next compared with the empirical Equation 17. When the two equations are rewritten in the form of Equation 11, the coefficients A(Z/D), B, and Cof Table IV are obtained. The present value of B is in excellent agreement with boundary layer theory for the inert gases but not nitrogen.
0
Effect of longitudinal Position
At the plasma entrance, 2 = 0, the diameter of the hot plasma core equals that of the n0zzle-~/16 inch for argon and helium and l / 4 inch for nitrogen. Here, Equation 20 may be written as
h, = 3kp/(D
- D,)
Equation 25 assumes the boundary layer thickness is zero a t 2 = 0. Because the boundary layer thickness of the present D,)/2 at 2 = 0, the coefficient A(Z/D) of Table work is (D IV is of a different form and magnitude. Figures 4 and 5
-
(21) 4c
or ie=300
Re.600
Equation 22 predicts values for Nuo of 48/13 and 4 for the inert gases and nitrogen, respectively. Extrapolation of the experimental data of Nu us. 2 yields intercepts of 1 to 4 in approximate agreement with Equation 22. At any location 2,
IP/P"= 6 . 5
)rp/)r.=3.5
PC
la $ 8
\
l-6 Y In
solution of the integral momentum equation with the Pohlhausen velocity profile results in the equation
z 4
(24)
2
I\
where ReZ =
(g)
Re, for parallel flow over a thin flat plate
as given by Schlichting (1960). Combination of Equations 23 and 24 provides Nu = 0.323
[g]"'
z/Re
I
4
- (-)
PPPP
4
6
z/D
/'D
'/D
Figure 4. Comparison of Skrivan's correlation with present correlation for argon 0 Present investigation X
---
where tube curvature is neglected. Moore (1952) has discussed the development of the densityviscosity parameter (pppp/pwpw) from the laminar boundary layer equations for compressible flow over a flat plate, including temperature variations. This ratio is an inverse function of the bulk mean plasma enthalpy and should be added to the independent variables-(D/Z) and Re-of Equation 25 to improve the boundary layer model. Both Equation 17 of the present work and the correlation equation of Skrivan use the viscosity or thermal conductivity ratio as the energy parameter. If Nu
0
-z
Skrivan's correlation Equation 25
Re = 3 0 0 Jp/Pw
*6 5
a
PWPW
use of the ideal gas equation
P P = -
RT
and the temperature dependence of viscosity from the kinetic energy theory of dilute gases I
4
6
results in Nu
- (E)-"
Equation 25 was derived from laminary boundary layer
6
4
z/D
'/D
2
4
6
'/D
Figure 5. Comparison of Skrivan's correlation with present correlation for nitrogen 0 Present investigation X
---
Skrivan's correlation Equation 25
VOL. 7
NO. 1
JANUARY 1968
39
compare Nusselt numbers calculated from the boundary layer equation and the present correlation, Equation 17, for typical values of R e and ( f i P / p R . ) . These diagrams for argon and nitrogen correspond to the third, fourth, and fifth heat transfer sections of the experimental apparatus. Recirculation and other turbulent effects are probably responsible for the higher Nu values of the present work. Agreement in the magnitude of Nu depends on ( Z I D ) ,Re, ( p P / p l v ) , and the gas. Of equal importance is the slope of In Nu us. In ( Z / D ) , which was nearly constant and equal to -1.67 for all the experimental curves. Linear approximation of these curves is comparable with drawing a straight line through the last three points of the argon and nitrogen curves of Figure 3. The slope of -1.67 is considerably higher than the boundary layer value of -0.50 but is in agreement with that of Skrivan and von Jaskowsky also based on experimental results. The latter curves were calculated from the correlation equation
of Skrivan. A direct comparison of Nusselt numbers with the present investigation cannot be made, as different experimental systems are employed; Skrivan used a diverging transition nozzle of length S to match the heat transfer chamber to the plasma generator. As a result, the inlet to the first heat transfer section of Skrivan, ( S / D ) = 2.50, closely corresponds to the inlet to the third heat transfer section, ( Z I D ) = 2.38, of the present work for a 1-inch tube. As in the comparison of present data to boundary layer equations, a complete analogy cannot be drawn because of physical differences in the first two sections which affect the data of the last three sections. Equation 30 was rewritten as
Nomenclature
dimensionless coefficient dependent on ( Z / D ) constants 4 ( R e )%P/PW) constant Reynolds number exponent viscosity ratio exponent heat capacity, cal./g.-OK. (B.t.u./lb.-OR.) inside tube diameter, inches voltage drop across electrodes enthalpy, B.t.u./lb. heat transfer coefficient, B.t.u./hr.-sq. ft.-OR. current, amperes thermal conductivity, B.t.u./hr.-ft.-OR. or cal./sec.-cm.-OK. length of heat transfer chamber, inches Nusselt number Prandtl number volumetric flow rate, cu. ft./hr. ; heat rate, B.t.u./hr. gas constant, cal./g.-OK. Reynolds number radial distance from the tube center nozzle length of Skrivan absolute temperature, OK. or OR. mass flow rate, Ib./hr. axial distance from entrance to heat transfer chamber GREEKLETTERS 6
e P
5 7r
P
=
boundary layer thickness
= characteristic temperature, OK.
viscosity, g./cm.-sec. included within the range = constant (3.1416) = density, lb./cu. ft. or g./cc. = =
SUPERSCRIPTS
+, *
=
denotes dimensionless quantities
SUBSCRIPTS where the thermal conductivity ratio was replaced by the corresponding viscosity ratio in accordance with Equation 3. Equation 31 may be considered as a shift of the Z-axis in order to superimpose the system of Skrivan on the present. As presently noted, such a superposition assumes that the diverging nozzle of Skrivan and the first two sections of this study have the same effect on the fluid mechanics and heat transfer in sections further donmstream. Figures 4 and 5 show the slope of the two experimental curves to be in excellent agreement. The higher Nu values of the present work may again be attributed to recirculation caused by sudden expansion. Agreement is improved at higher Re or flow rate and at higher ( f i p / p w ) or plasma temperature.
Conclusions
The empirical equation Nu = Al[exp(-O.B18Z**
+ 0.764Z*)]ReE
(!?)-1.m
for entrance heat transfer from confined argon, helium, and nitrogen plasma streams to a circular tube wall has been presented for estimation of Nusselt numbers for R e 5 0-900, Z / D 5 0-6, ( p P / p F v ) 5 1-10, and (D,/D) l/4. The correlation is in agreement with that of Skrivan in regard to the variation of Nu with ( Z I D ) . Values of the Reynolds number exponent B of 0.52 for argon and helium are in agreement with boundary layer results. 40
l&EC PROCESS DESIGN A N D DEVELOPMENT
b
P>P
= = = = =
T S
=
i, k n 0
T W
= = =
gas property evaluated a t bulk mean temperature section number nozzle pertains to entrance conditions a t Z = 0 plasma property evaluated a t bulk mean temperature recirculation outer edge of boundary layer thermal wall
literature Cited
Ahtye, W. F., Peng, T., Natl. Aeron. Space Admin., NASA Tech. Note D-1303 (July 1962). Amdur, I., Mason, E. A., Phvs. Fluids 1, 370 (1958). Barzelay, M. E., Am. Inst. Aeron. Astronaut. J . 4, 815 (1966). Becker, H. A., D. Sc. thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1961. Bro, P., Steinburg, S., A m . Rocket SOC., J . 32, 528 (1962). Brown, E. A., Ross, P. A , , Am. Inst. Aeron. Astronaut. J . 3, 660 (1965). Choksi, N. M., M. S. thesis, Texas A&M University, College Station, Tex., 1967. Craya, A,, Curtet, R., Compt. Rend. 241, 621 (1955). Dennis, P. R., Smith C. R., Gates, D. W., Bond, J. B., Natl. Aeron. Suace Admin., NASA SP-5033 (October 1953). Drellishak,‘K. S., Knopp, C. F., Cambel, A. B., Gas Dynamics Lab., Northwestern University, Evanston, Ill. Rept. A-3-62 (1962). Emmon, H. W., in “Modern Developments in Heat Transfer,” W. E. Ibele, Ed., pp. 401-78, Academic Press, New York, 196:; Finkelnburg, IV., in “High Temperature, a Tool for the Future, pp. 39-44, 200, Stanford Research Institute, Menlo Park, Calif., 1956.
Freeman, M. P., Skrivan, J. F., A.Z.Ch.E. J . 8, 450 (1962). Grey, J., ZSA Trans. 4, 102 (1965). John, R. R., Bade, W. L., A m . Rocket Soc. J . 29, 523 (1959). Johnson, J. R., M.S. thesis, Texas A&M University, College Station, Tex., 1966. Kubanek, G. R., Gauvin, W. H., 6lst National Meeting of A.1.Ch.E ., Houston, Tex., February 1967. Moore, L. L., J . Aeron. Sci. 19, 505 (1952). Schlichting, H., “Boundary Layer Theory,” 4th ed., McGraw-Hill, New York, 1960. Skifstad, J . G., Jet Propulsion Center, Purdue University, Lafayette, Ind., Interim Report Contract Nor 1100 (17) (August 1961). Skrivan, J. F., von Jaskowsky, W., IND. ENG.CHEM.PROCESS DESIGN DEVELOP. 4, 371 (1965). Stokes, C. W., Knipe, W. W., Streng, L. A., J . Electrochem. Soc. 107, 35 (1960).
Tankin, R. S., Berry, J. M., Phys. Fluids 7, 1620 (1964). Wethern, R. J., Brodkey, R. S., A.Z.Ch.E. J . 9, 49 (1963). RECEIVED for review April 19, 1967 ACCEPTED September 21, 1967 IYork supported by the National Aeronautics and Space Xdministration through NASA Grant 32550 for Space Technology Project No. 3. Material supplementary to this article has been deposited as Document No. 9655 with ,4DI Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington, D. C. A copy may be secured by citing the document number and by remitting $2.50 for photoprints or $1.75 for 35-mm. microfilm. Advance payment is required. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress.
CONCENTRATION POLARIZATION EFFECTS IN R E V E R S E OSMOSIS U S I N G POROUS C E L L U L O S E ACETATE M E M B R A N E S SHOJl
K I M U R A A N D S. S O U R I R A J A N
Diuision of Applied Chemistry, .Vational Research Council of Canada, Ottawa, Canada
Analytical expressions have been developed to predict concentration polarization effects in reverse osmosis from the data on membrane specifications given in terms of the pure water permeability constant, A, and the solute transport parameter, DAM/K6. The results are illustrated for a set of Loeb-Sourirajan type porous cellulose acetate membranes using 0.5M aqeuous sodium chloride solution as the feed at an operating pressure of 102 atm. The case considered here is for turbulent and laminar flow in rectangular channels between flat parallel membranes. Some factors relating to the economic analysis of the process are also indicated. HE practical importance of the problem of concentration Tpolarization in reverse osmosis has been recognized, and several analytical studies have been reported with particular reference to saline water conversion (Brian, 1965a, 1965b, 1965c; Gill et al., 1965, 1966a, 1966b; Johnson et al., 1966; Sherwood et al.,1965; Srinivasan et al.,1967). These studies assume that the membrane exhibits either complete salt rejection or incomplete salt rejection a t a constant level. Such a n assumption is, in general, invalid, since salt rejection can vary widely depending on feed concentration and feed flow rate, even a t the same operating pressure. A more desirable approach to the subject is given by Sherwood et al. (1967), who have coupled the equations of solute and solvent transport through the membrane to the theory of concentration polarization. A similar approach is offered by the results of the Kimura-Sourirajan analysis (Kimura and Sourirajan, 1967) of the experimental reverse osmosis data obtained with the Loeb-Sourirajan type porous cellulose acetate membranes. The latter analysis, however, is different from that of Sherwood et al. (1967). The Kimura-Sourirajan analysis is based on a generalized pore diffusion model applicable for the entire possible range of solute separation. I t gives rise to a set of basic equations relating the pure water permeability constant, A , the transport of solvent water, iVB, the transport of solute, ArA, the solute transport parameter, DA.M/Kb, and the mass transfer coefficient, k, all of which can be determined from a single set of
experimental pure water permeability, product rate, and solute separation data obtained from laboratory cells. T h e parameter A is obtained from the pure water permeability data, and hence is independent of any solute under consideration. T h e parameter D A w / K 6 depends, of course, on the nature of the solute. Both A and D,,*/’KG are dependent on the porous structure of the membrane surface, and hence they are different for different membranes; and both are functions of operating pressure. Further, a t a given operating pressure, DA,/K6 is independent of feed concentration and feed flow rate. Also, the values of k are well correlated by a generalized log-log. plot of .VRe us. .VSh/-1rgo0.33 for the type of apparatus used. T h e distinguishing feature of the above analysis is the fact that the interconnected parameters A and D A,JKG specify a particular membrane-solution system a t the given operating pressure with reference to this separation process. Using the above parameters, the membrane performance-i.e., the variations of solute separation and membrane throughput rate as a function of feed concentration and feed flow rate-can be predicted, provided the applicable mass transfer correlations are available. This has been illustrated for the systems [NaC1-H20], [NaNOa-H20], [Na2S04-H20], [MgC12-H20], [MgS04--H20],and glycerolwater (Kimura and Sourirajan, 1967; Sourirajan and Kimura, 1967). Hence the above analysis and correlations are valid at least for the above system. The object of this paper is to illustrate how the concenVOL. 7
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