Ind. Eng. Chem. Res. 1989,28, 1899-1903
1899
Mixed Convection Heat Transfer at High Grashof Number in a Vertical Tube Donald D. Joye,* Joseph P. Bushinsky, and Paul E. Saylor Department of Chemical Engineering, Villanova University, Villanova, Pennsylvania 19085 Mixed convection heat transfer in a vertical tube was studied experimentally for Reynolds numbers ranging from 360 to 35 OOO in both upflow and downflow heating a t constant Grashof number under constant wall temperature (CWT) conditions. The results show a turbulent region at Re greater than 15000, an asymptotic region a t Re less than 2000-4000, and a mixed convection region at Re in between. Upflow and downflow heating gave different results in the mixed convection region but gave identical results in the other two. Mixed convection effects existed in the range 0.2-50Gr/Re2 for our experimental conditions (LID = 49.6 and Gr = 1 X 108 based on diameter and film properties). The boundaries for mixed convection behavior are developed based on Re as the independent variable. There is a shift in laminar-turbulent transition for upflow heating, which moves the data past the turbulent bound and accounts for the lower than expected heat-transfer coefficients in this region. Mixed convection flow occurs when both natural (free) and forced convection mechanisms simultaneously and significantly contribute to the heat transfer. The relative contribution of each mechanism depends on the flow regime (laminar or turbulent), the magnitude of the temperature driving force for heat transfer (reflected by the Grashof number), the geometry (internal flow, external flow, channel shape, etc.), and the orientation (vertical, horizontal, angled). The nature of the fluid also has an effect, since the Grashof number (the natural convection parameter) increases for liquids with an increase in temperature but reaches a maximum for gases, because the viscosity of gases increases with temperature and can counteract the density effects. Natural convection currents arise from buoyancy forces and have a different relationship to the forced convection depending on whether the forced flow is vertically up, vertically down, horizontal, or some combination. When the direction of the natural convection currents is the same as the forced flow, the flow is sometimes termed assisting (or aiding); when the currents are opposite, the flow is sometimes termed opposing. This paper discusses the results of experimental investigations of heating in vertical upflow (aiding) and heating in vertical downflow (opposing). The literature for mixed convection flow is extensive because of the many configurational possibilities that exist. A comprehensive review paper by Jackson et al. (1989) summarizesthe experimental and theoretical contributions in vertical, internal, mixed convection flows and the difficulties in this field. Correlations, in particular, exist only for limited cases. The range and usefulness of these correlations are also limited, because of the generally complex behavior of the Nusselt number in these flow situations. The present work is a step toward broadening the scope of experimental investigations, so that more comprehensive correlations can be developed. Mixed convection is important in many situations including process heat transfer, where it is not normally recognized as a potential contributor, nuclear reactor technology (Bishop et al., 1980; Symolou et al., 1987; Ianello et al., 1988),and some aspects of electronic cooling.
Experimental Method An apparatus similar to those used by previous investigators (Colburn and Hougen, 1930; Martinelli et al., 1942;
* Author to whom correspondence
should be addressed.
0888-5885/89/2628-1899$01.50/0
Kirschbaum, 1952; Hanratty et al., 1958; Petuckov and Nolde, 1959; Kemeny and Somers, 1962; Mullin and Gerhard, 1977) was employed for this work. Quite similar arrangements have been used to study the mixed convection, vertical flow of air (Brown and Gauvin, 1965), and the flow of liquids in vertical annuli (Beck, 1963) under mixed convection conditions. The test section is shown in Figure 1. Steam condenses in the annulus, and water flows in the central tube. Pressurized steam gives the high Grashof numbers in this work and constant wall temperatures in any run. The wall temperature is somewhat hotter at low flow rates and somewhat cooler (by about 20 "C) at high flow rates, because the steam pressure could not be varied enough to compensate for the greater heat flows at higher Reynolds numbers. Copper-constantan thermocouples were used for the temperature measurement. Wall temperatures at five locations along the central tube were obtained by embedded and silver-soldered constantan wire. Inlet and outlet thermocouples measured the bulk temperature of the water being heated in the central tube. Five elbows each were used at the upper end and at the lower end of the water tube to get a well-mixed exit temperature in both upflow and downflow configurations. A digital readout device was used to obtain all temperature values from the thermocouples. The unit was insulated between and including the inlet/outlet temperature ports to minimize heat losses. Both concentric tubes of the test section were type L, harddrawn copper, 3.2-cm (11/4-in.) and 5-cm (2-in.) nominal size. The LID of the heated section was 49.6. An unheated section of tube preceded and followed the heated section with a length of about 10 diameters from the inlet to the heated section. This is to allow flow disturbances from the i d e t tee to dissipate and for the velocity profile to become reasonably established before heating. After 10 diameters, further changes in the friction factor are small: thus, the major effects of velocity profile development have occurred within this length. That entrance length depends on Reynolds number is recognized, so at the higher and lower Reynolds numbers, this will be more true than at the midrange Reynolds numbers. Flow rates were measured by one of three rotameters. Three had to be used because of the wide variation in Reynolds numbers in these experiments. The steam pressure was pneumatically controlled to be in the range 83-104 kPa (12-15 psig). The water flow was once-through from a supply tank; the water was not recirculated or 0 1989 American Chemical Society
1900 Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 PROCESS WATER IN/OUT
-UNHEATED
STEAM IN
SECTION (UPPER)
-I'
( Leveque Approximation)
( lower bound - Marfinelli,et ai)
t I
CONDENSATE OUT
5
x
w
2
8 103
5
8 lD4
2
5
Re -UNHEATED
THERMOCOUPLE 7
-TUBE
SECTION (LOWER)
CAP
PROCESS OUT/IN WATER
NOTE: PIPES ARE INSULATED BETWEEN THERMOCOUPLES 6 AND 7
Figure 1. Test section detail.
deaerated. Thus, a pump was required not only to move the fluid but also to keep the system under pressure to prevent boiling and to prevent air coming out of solution (degasing). The centrifugal pump provided about 210 kPa (30 psig) at the low flow rates (hottest water, and thus the greatest dange of degasing or boiling) and about half that at the higher flow rates (cooler water). By Henry's law (Foust et al., 1980),this was sufficient to prevent degasing. The vapor pressure of the water (at the wall temperature) was always well below the water pressure. Also, we ran the apparatus with and without the pump (using only the head tank for pressure) and observed marked differences between single-phase and two-phase heat transfer. A set of valves was used to direct water flow to the top of the test section for downflow heating and to the bottom of the test section for upflow heating experiments. Heat-transfer coefficients were calculated from Newton's law of cooling, where the heat rate (W or Btu/h) was calculated from the temperature rise and flow rate of the water. An average temperature difference driving force, defined as the difference between the average bulk temperature and the average wall temperature, was used to calculate the film coefficient. These driving force temperature differences ranged from about 10 to about 110 "C. Dimensionless groups for correlating purposes were calculated based on both the average film temperature and the bulk average temperature. The practice of the literature in mixed convection is not uniform. In this work, Reynolds number, Nusselt number, and Prandtl number were calculated using the bulk average fluid properties, and the Grashof number was calculated using the fluid properties evaluated at the average film temperature, which is the arithmetic average of the bulk fluid temperatwe and
Figure 2. Nusselt plot, upflow and downflow heating in a vertical tube.
the average wall temperature. The Grashof numbers could be translated to other bases where necessary, because all temperatures were recorded, and all kinds of temperature averages were included in the calculated data.
Results and Discussion Experimental conditions gave a Prandtl number (an uncontrolled parameter) varying from 2.6 at the low velocities to 6.0 at high velocities, since water was the test fluid. The corresponding viscosity ratio (4" = viscosity at the bulk average temperature divided by that at the wall temperature) varied from about 1.7 to 2.8. The Grashof number based on diameter and fluid properties evaluated at the film temperature averaged about 1 X lo8. The high value at low flow rate was about 1.2 X lo8, and the low value at high flow rate was about 4.1 X 10'. Values for the Grashof number were constant to &20% except at Re greater than about 15000, but natural convection effects are negligible here anyway. Thus, the data presented are for essentially constant Grashof number, which is based on diameter rather than length, because this has been found to be more convenient by most other investigators in this field. Nusselt Plot. Figure 2 shows the results plotted in the Nusselt format, where mixed convection heat transfer may be characterized by three regions. The first region is a turbulent flow region above about 15000 Reynolds number. Mixed convection effects are not significant, and results are independent of the flow direction. Kirschbaum (1952) and Petuckov and Nolde (1959) show data having similar trends. The second region, in which mixed convection effects are strongly apparent, spans the Reynolds number range from about 2000 to 15000, where heattransfer coefficients (cf. Figure 2) in downflow are about double those in upflow. The downflow Nusselt numbers are always higher, because this situation is inherently unstable, and turbulence is always present in the shear layer between the boundary region near the wall and the core fluid (Scheele et al., 1960; Scheele and Hanratty, 1963). The lower limit of this region is not the same for upflow as it is for downflow. In upflow, the region begins at about Re = 2000; in downflow, it begins at about Re = 4000. The third region is the asymptotic region, which
Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 1901 exists at Reynolds numbers below the previous limits, GrfD“ IxIOB b Downflow ( I s t ) where downflow and upflow give the same results once L/D = 49.6 n Downflow (2nd) 5 . again. The data shown are for two different investigations 0 Upflow ( I SI) using two separate pieces of equipment. The data show two transition points between the regions, one from asymptotic to mixed convection (different in downflow heating than in upflow heating) and one from mixed convection to the turbulent region. That bounds exist for mixed convection flows was an idea first brought together by Metais and Eckert (1964) and has received wide distribution in the heat-transfer textbooks but was never intended to be a final analysis. Our data fall outside the range of this chart, because of the high Grashof numbers in this work. More importantly, the Reynolds number ,001 ’ ’ range for mixed convection established in the present work 5 IO4 2 IO2 2 5 lo3 2 5 Reynolds Number, Re is much broader than the limits indicated by this chart. These bounds need to be redefined by much more comFigure 3. Colburn j-factor plot, upflow and downflow heating in a prehensive, uniform, and systematic data. This has not vertical tube. been forthcoming to date, because of numerous experimental difficulties. One of the most important is the Most attempts to correlate mixed convection data are difference between CWT and UHF (uniform heat flux) based on the forced flow. For example, the correlation of wall conditions in the laminar region. CWT conditions give Jackson and Fewster, shown below, for downflow heating the asymptote; UHF conditions give a limiting Nusselt is typical (see Jackson et al., 1989): number instead. Therefore, the bounds will be different Nu/Nu(forced) = (1 4500Gr/Re2.625Pr0.5)$31(1) for the two cases, except for the turbulent flow region. The bound equations are given in terms of Reynolds where subscript “b” refers to fluid properties evaluated at number in Figure 2. The turbulent bound is represented the bulk average temperature. This equation fits our data by the usual Sieder-Tate relationship plus the approxifor downflow heating fairly well when the equivalent bulk mation (dotted curve) for Re less than 8000. The asaverage temperature Grashof number (Grb = 5 X 10’) is ymptote bound, 2Gz/r, was developed in McAdams (1954) used. A similar equation developed by Swanson and from earlier theory and represents essentially fully deCatton (1987) has an additional term and slightly different veloped flow for constant wall temperature (CWT) heat Pr dependence and fits downflow heating data somewhat transfer. This condition occurs when the outlet temperbetter. These kinds of equations do not fit upflow data, ature is approximately equal to the wall temperature, even with a minus sign substituted for the plus sign in eq giving rise to a zero driving force at the exit. Lowering the 1, though it seems they should. The problems are disReynolds numbers by decreasing flow rate does not change cussed by Jackson et al. (1989). The cause of the difficulty the exit temperature significantly and just results in less can be traced to the transition shift and the stabilization heat transfer and lower Nusselt numbers according to the of turbulence in the layer near the wall in aiding flow cases. asymptotic line. On a Reynolds number plot, this bound That aiding flow stabilizes turbulence is shown more is P r dependent. definitively by Carey and Gebhart (1983) and KrishnaAs an approximation to the lower bound equation, the murthy and Gebhart (1989). Leveque relationship can be used. The true lower bound The range of mixed convection taken from the transition equation is more complex and requires trial-and-error for points in Figure 2 can be expressed in terms of GrlRe2. its predictions, but it differs from Leveque only in a small This range is from 0.2 to about 50 for our data, which region near the asymptote (see Figure 2). The use of the straddle the theoretical value of 1.0 (Incropera and DeWitt, Leveque approximation for the heat-transfer coefficient 1985). Marcucci and Joye (1985) show this range to be in all mixed convection cases would result in a large error LID dependent. at high Grashof numbers, e.g., when temperature driving Colburn j-Factor Plot. Relative to the turbulent forces are large (30-100 “C). For a given Reynolds number bound in Figure 2, one sees that the downflow data follow in the mixed convection region defined above, exactly what the bound closely, but the upflow data go beyond it. The Grashof number results in significant mixed convection reason for this is more apparent from a Colburn j-factor effects has yet to be systematically explored. By extrapplot than from a Nusselt plot. This plot is shown in Figure olating the data shown in Martinelli et al. (1942), it appears 3. Clearly, there is a major shift in the laminar-turbulent to be where GrPrDIL is about 100. This is a relatively transition point, evidenced by the “dip” region on this plot. small number, and it may be that natural convection efThe lowest point is shifted from a Re of 2100 to about fects in mixed convection vertical tube flow cannot be 10000. Downflow heating, by contrast, shows no “dip” ignored under any circumstances. region at all but rather only a deviation from the turbulent line below Reynolds numbers of about 15000. This tranThese results also imply that natural convection corresition shift explains one of the problems of upflow heating lations cannot be used alone in mixed convection flow, no heat transfer, in that mixed convection Nusselt numbers matter how small the Reynolds numbers. This may result are higher than expected (from forced flow) at low Reyfrom the high Grashof number of our data. Whether this nolds numbers (laminar region) but lower than expected is general for all situations in vertical flow, mixed convection awaits further work. The value of N ~ / P r ~ / ~ 4 , 0 . at ~ ~higher Reynolds numbers (7000-15000 here). Marcucci and Joye (1985) showed this shift point for upflow heating for purely natural convection was calculated for water on to be a function of Grashof number and LID in similar a vertical surface according to the usual recommendations experiments in four vertical tubes of L I D ranging from 60 (Incropera and DeWitt, 1985) and is about 36. This lies between the upflow and downflow data in the laminar, to 114. mixed convection region. The corresponding asymptotic region is also shown; its ’
1
’
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1902 Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989
n Downflow, 2 n d Gr Pr D/Llw = 8 4 x IO‘
IO
,/’
/y
%I
Gr Pr D/LIw = 5.5 x IO’(Mullin
3
Gr Pr D/Ll, = I a 1O4(MarlmellI ,el I
O5
2
5
’
10
2
5
io2
I
01)
1
,
2
a Gerhordl
5
2
5
103
Graetz Number, G z
Figure 4. Graetz plot, upflow and downflow heating in a vertical tube compared with theory.
character is essentially a constant j factor (about 0.019), which provides a convenient limiting case for the asymptote. Because of the j-factor definition, this value is L I D and Pr dependent. Graetz Plot. Figure 4 is a Graetz plot which shows our data and the corresponding theory of Martinelli et al. (1942),which was developed for upflow heating only. The Graetz plot is preferred by many for discussing laminar flow situations. The asymptote bound (curve 2) was first defined in this manner for CWT conditions (McAdams, 1954). Curve 1is the lower bound equation for zero natural convection (zero Grashof number). Curve 3 is Martinelli’s prediction for our Grashof number conditions. The downflow heating data are above curve 3, but the upflow data are well described by curve 3. The theory of Martinelli et al. applied to our conditions includes an adjustment to the GrPrDIL parameter, which is based on the wall temperature and not the film temperature. Curve 4 is a line representing the data of Mullin and Gerhard (1977) from similar experiments under lower Grashof number conditions. Curve 5 is a representation of Martinelli’s data at still lower Gr. The theory of Martinelli et al. describes the upflow data fairly well, but replacing the positive sign in the cube root term with a negative one for downflow heating will clearly give erroneous predictions for these kinds of situations. This has been noted by Mullin and Gerhard and is recognized by others. Jackson et al. (1989) discuss other works under different conditions that do not show this effect. Conclusions 1. Heat-transfer data in vertical, mixed convection flow in tubes under CWT conditions show three distinct regions with respect to Reynolds number at constant Grashof number. The three regions are turbulent flow, where natural convection effects are insignificant; a mixed convection region, where natural convection effects are important; and an asymptotic region, where the flow is fully developed in a heat-transfer sense. 2. The mixed convection region spans a range of Gr/Re2 from 0.2 to 50 for the experimental conditions presented; the theoretical threshold of 1.0 is close to the lower limit of this range, but an upper limit exists also. 3. Heat-transfer coefficients in the mixed convection region reported in this work for downflow heating are roughly double those for upflow heating. In both cases,
the Nusselt number is approximately independent of the forced flow parameter (Reynolds number). The coefficients are the same in the other two regions. 4. The laminar-turbulent transition point at a Reynolds number of 2100 for isothermal flow in tubes is profoundly affected by natural convection in upflow heating as evidenced by the shift in the “dip” region in a Colburn j-factor plot. In downflow heating, by contrast, no dip is evident. 5. Asymptote, lower laminar and turbulent boundaries exist in CWT mixed convection heat transfer. In the case of upflow heating (aiding flow), the turbulent bound is traversed, owing the shift in laminar-turbulent transition point. Nomenclature C, = fluid heat capacity, kJ/(kg.K) D = tube diameter, m F,, F2 = Martinelli functions, tabulated g = gravitational acceleration, m/s2 Grm = Grashof number based on film temperature and tube diameter, p2D3gpAT/p2 Gz = Graetz number, mC,/kL h = film heat-transfer coefficient, W/ (m2.K) k = fluid thermal conductivity, W/(m.K) L = heated length of tube, m m = mass flow rate, kg/s Nu = Nusselt number, h D / k Pr = Prandtl number, p C / k Re = Reynolds number, L f u p / p T = temperature, K Tb, T , = average temperature, bulk and wall, respectively, K AT = temperature difference (K),average wall to average bulk fluid for Grashof number, unless otherwise noted u = average fluid velocity, m/s Greek Symbols
p = volume expansion coefficient of the fluid, 1/K & = viscosity ratio, p = fluid viscosity, Pa-s p, = fluid viscosity at the average wall temperature, Paas p = fluid density, kg/m3 Literature Cited Beck, F. Warmeubergang und Druckverlust in senkrechten konzentrischen und exzentrischen Ringspalten bei erzwungener Stromung und freier Konvektion. Chem.-Ing.-Tech. 1963, 35(12), 837-844. Bishop, A. A.; Willis, J. M.; Markley, R. A. Effects of Bouyancy on Laminar, Vertical Upward Flow Friction Factors in Cylindrical Tubes. Nuclear Eng. Des. 1980, 62, 365-369. Brown, C. K.; Gauvin, W. H. Combined Free and Forced Convection-I. Aiding Flow and 11. Opposing Flow. Can. J . Chem. Eng. 1965,43, 306-318. Carey, V. R.; Gebhart, B. The Stability and Disturbance Amplification Characteristics of Vertical Mixed Convection Flow. J . Fluid Mech. 1983, 127, 185-201. Colburn, A. F.; Hougen, 0. A. Studies in Heat Transmission. Ind. Eng. Chem. 1930, 22(5),522-539. Foust, A. S.; Wenzel, L. A.; Clump, C. W.; Maus, L.; Andersen, L. B. Principles of Unit Operations, 2nd ed.; Wiley: New York, 1980. Hanratty, T. J.; Rosen, E. M.; Kabel, R. L. Effect of Heat Transfer on Flow Field a t Low Reynolds Number in Vertical Tubes. Ind. Eng. Chem. 1958,50(5), 815-820. Ianello, V.; Suh, K. Y.; Todreas, N. E. Mixed Convection Friction Factors and Nusselt Numbers in Vertical Annular and Subchanne1 Geometries. Int. J. Heat Mass Transfer 1988,31,2175-2189. Incropera, F. P.; DeWitt, D. P. Fundamentals of Heat and Mass Transfer, 2nd ed.; Wiley: New York, 1985. Jackson, J. D.; Cotton, M. A,; Axcell, B. P. Studies of Mixed Convection in Vertical Tubes: A Review. Int. J.Heat Mass Transfer, in press. Kemeny, G. A.; Somers, E. V. Combined Free and Forced-Convective Flow in Vertical Circular Tubes-Experiments with Water and
Ind. Eng. Chem. Res. 1989,28, 1903-1907 Oil. ASME J. Heat Transfer 1962,84, 339-346. Kirschbaum, E. Neues zum Warmeubergang mit und ohne Anderung des Aggregatzustandes. Chem.-1ng.-Tech. 1952,24(7), 393-400. Krishnamurthy, R.; Gebhart, B. An Experimental Study of Transition to Turbulence in Vertical Mixed Convection Flows. ASME J. Heat Transfer 1989, 111,121-130. Marcucci, B. J.; Joye, D. D. Experimental Study of Transitions in Mixed-Convection, Vertical Upflow Heating of Water in Tubes. In Fundamentals of Forced and Mixed Convection; Kulacki, F. A., Boyd, R. D., Eds.; Heat Transfer/Denver, ASME Publications: New York, 1985; HTD Vol. 42, pp 131-139. Martinelli, R. C.; Southwell, C. J.; Alves, G.; Craig, H. L.; Weinberg, E. B.; Lansing, N. F.; Boelter, L. M. K. Heat Transfer and Pressure Drop for a Fluid Flowing in the Viscous Region Through a Vertical Pipe. Trans. Am. Znst. Chem. Eng. 1942, 38, 493-530. McAdams, W. H. Heat Transmission,3rd. ed.; McGraw-Hill: New York, 1954; pp 229-235. Metais, B.; Eckert, E. R. G. Forced, Mixed, and Free Convection Regimes. ASME J. Heat Transfer 1964,86, 295-296. Mullin, T. E.; Gerhard, E. R. Heat Transfer to Water in Downward Flow in a Uniform Wall Temperature Vertical Tube a t Low
1903
Graetz Numbers. ASME J. Heat Transfer 1977, 99, 586-589. Petuckov, B. S.; Nolde, L. D. Heat Transfer in the Visco-gravitational Flow of Liquid in Pipes. Teploenergetika 1959,6, 72-80 (in Russian). Scheele, G. F.; Hanratty, T. J. Effect of Natural Convection Instabilities on Rates of Heat Transfer a t Low Reynolds Numbers. AZChE J. 1963, 9(2), 183-185. Scheele, G. F.; Rosen, E. M.; Hanratty, T. J. Effect of Natural Convection on Transition to Turbulence in Vertical Pipes. Can. J . Chem. Eng. 1960,38, 67-73. Swanson, L. W.; Catton, I. Surface Renewal Theory for Turbulent Mixed Convection in Vertical Ducts. Znt. J . Heat Mass Transfer 1987, 30(11), 2271-2279. Symolou, P. D.; Todreas, N. E.; Rohsenow, W. M. Criteria for the Onset of Flow Recirculation and Onset of Mixed Convection in Vertical Rod Bundles. ASME J. Heat Transfer 1987, 109(1), 138-145.
Received f o r review April 5, 1989 Revised manuscript received August 24, 1989 Accepted September 11, 1989
Critical Locus and Partial Molar Volume Studies of the Benzaldehyde-Carbon Dioxide Binary System Neil R. Foster, Stuart J. Macnaughton, Rodney P. Chaplin,* and P. Tony Wells School of Chemical Engineering & Industrial Chemistry, University of New South Wales, P.O.Box 1, Kensington, N S W , Australia 2033
A fundamental experimental study of the behavior of the benzaldehyde-carbon dioxide binary system has been undertaken. The critical locus for the system has been accurately determined for concentrations up to 1.5 mol % benzaldehyde. Partial molar volume data were also obtained, which exhibited characteristically large negative values near the critical region. The Peng-Robinson equation of state in combination with van der Waals mixing rules proved to be only semiquantitative in the prediction of partial molar volumes. Binary systems comprising supercritical carbon dioxide and both liquid and solid solutes have been extensively studied over the last decade, and a comprehensive phase classification scheme has been compiled (Williams, 1981; Johnston, 1983; McHugh, 1983). In this study, the behavior of the system benzaldehyde-carbon dioxide has been investigated at supercritical conditions. The benzaldehyde-carbon dioxide system is of considerable interest due to the possible commercial extraction of benzaldehyde from Australian flora using supercritical carbon dioxide. The study comprised two interlinked parts. In the first section, the supercritical behavior of the binary benzaldehyde-carbon dioxide system was investigated with the purpose of enabling the system to be classified according to its phase behavior. The investigation was limited to low concentrations of benzaldehyde in carbon dioxide. The objective of the second section was to measure the partial molar volume of benzaldehyde in carbon dioxide at infinite dilution. The justification for this investigation extends beyond its physical significance. The partial molar volume can be used to test the ability of an equation of state to predict derivative-based thermodynamicproperties (Eckert et al., 1983). In particular the Peng-Robinson equation of state may be able to predict the shape of the partial molar property correctly provided that the system can be described by adequate mixing rules. Eckert et al. (1983) have shown that this equation of state is the most useful in this region, although lack of adequate mixing rules is still a major limitation. 0888-5885/89/2628-1903$01.50/0
Theoretical Background To enable a pressure-explicit cubic equation of state to predict partial molar volumes, a derivative form must be used (Edmister, 1974). The ability of an equation of state to maintain accuracy under differentiation is a better measure of its usefulness than the ability to match P, V, and T data. Most of the important thermodynamic properties of mixtures such as entropy and enthalpy can only be determined by differentiating cubic equations of state. While the prediction of such bulk properties relies essentially on the form of the equation of state, the prediction of partial molar properties is also heavily influenced by the choice of mixing rules. The partial molar volume at infinite dilution of a solute, VZm,can be determined from the standard thermodynamic relation
The symbol y1 refers to the mole fraction of component 1 in the supercritical phase. In this series of experiments, the magnitude of Vzmwas determined both experimentally and also calculated by using the Peng-Robinson equation of state. The method adopted for the calculation of partial molar volumes requires that the experimental procedure be designed such that the values for the derivative (dV/dy!)T,p at infinite dilution may be obtained. A t this low dilution, the relationship between molar volume and composition is sufficiently linear such that graphical determination of 8 1989 American Chemical Society
