Characteristics of Radial Transport in Solid-Liquid Slug Flow

oretically as a.function of Peclet and Reynolds numbers for liquid slugs with ... creasing aspect ratio, but no Reynolds number dependence was detecte...
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Ind. Eng. Chem. Fundam., Vol. 17, NO, I , 1978

39

Characteristics of Radial Transport in Solid-Liquid Slug Flow James S. Vrentas’’ Department of Chemical Engineering, lllinois lnstitute of Technology, Chicago, lllinois 606 16

J. Larry Duda Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802

George D. Lehmkuhl Rockwell International, Golden, Colorado 8040 1

The characteristics of radial heat transfer in solid-liquid slug flow were investigated both experimentally and theoretically as a.function of Peclet and Reynolds numbers for liquid slugs with aspect ratios ranging from 1 to 122. The measured Nusselt numbers are from 1.0 to 3.0 times greater than those for single-phase flow at the same value of the Peclet number. The radial transport enhancement increases with increasing Peclet number and decreasing aspect ratio, but no Reynolds number dependence was detected for a limited Reynolds number range near the creeping flow limit. The heat transfer results for solid-liquid slug flow are compared with recent mass transfer data for gas-liquid slug flow.

Slug flow is defined as the two-phase flow field which results when a tube is filled with alternating moving slugs of two different phases. This flow field has two characteristics which are useful in a number of diverse applications. First of all, the alternating slug configuration reduces the axial dispersion that is typical of laminar flow in tubes. This characteristic has been exploited in automatic clinical analyzers in which slugs of air segment the flowing streams of samples, thereby producing sample integrity by eliminating mixing between samples (Skeggs, 1957). Second, it has been demonstrated both theoretically and experimentally that slug flow significantly enhances radial transfer of heat or mass in tubes. Horvath et al. (1973) have shown that gas-liquid slug flow leads to appreciable increases in radial mass transfer, and the corresponding enhancement of radial heat transfer in gas-liquid slug flow has been demonstrated by Prothero and Burton (1961), Oliver and Wright (1964), and Oliver and Young Hoon (1968). The validity of the radial transport augmentation observed in these experimental studies is supported by a theoretical analysis of unsteady-state heat transfer to a liquid in a cylindrical cavity with circulating flow induced by the relative motion between the fluid and the wall (Duda and Vrentas, 1971b). These results suggest potential applications of slug flow in reactors, dialyzers, heat exchangers, and other tubular flow operations. In addition, slug flow occurs in the pulmonary and peripheral capillaries of the body where slugs of plasma are trapped between red blood cells. Since slug flow provides a useful and convenient method of modifying the axial dispersion and radial transport in tubes, there is interest in obtaining a better understanding of the transport processes in this flow field, particularly the enhancement of radial heat or mass transport. T o date, all studies of slug flow radial transport have been conducted with gas-liquid systems. However, this flow field could also be generated by using solid spheres to segment the liquid in a tube. Ellis (1964a,b) investigated the flow behavior of the simultaneous transfer of spheres and fluid in a cylindrical tube with sphere diameter to pipe diameter ratio of 0.9. P a r t of work carried out a t The Dow Chemical Company, Midland, Mich. 48640.

0019-7874/78/1017-0039$01.00/0

Bauer and DuPuis (1967) and Christopherson and Dowson (1959) studied the transport of spheres in close-fitting tubes filled with fluid. Recently, Kern (1975) described the utilization of sponge-rubber spheres to provide a continuous mechanical cleaning of heat exchanger tubes. Solid-liquid slug flow appears to be a viable alternative to slug flow produced by gas bubbles, and in this paper we study some of the characteristics of a slug flow field induced by solid spheres in a tube. The advantages and disadvantages of solid-liquid slug flow are discussed, and heat transfer data for this flow field are presented. These data complement a previous theoretical study (Duda and Vrentas, 1971b) and allow a more extensive comparison of the enhancement in radial transport with that reported for gas-liquid systems.

Comparison of Solid-Liquid and Gas-Liquid Slug Flows The principal disadvantage of using gas bubbles to generate a slug flow field is that the type of two-phase flow field that is produced depends on the properties of the fluids and on other experimental variables. It is therefore not always possible to produce a train of slugs which essentially fill up the tube cross section, leaving a thin liquid film. For example, Taylor (1961) has shown that the liquid film thickness increases as the viscosity of the liquid increases, and it is thus difficult to produce thin films when viscous liquids are used. On the other hand, the film thickness for solid-liquid slug flow can be controlled independently by choosing an appropriate sphere size. Consequently, solid spheres Can be used to enhance heat or mass transfer in viscous liquids and to modify the operation of tubular polymerization reactors, where the residence time distribution problem can be particularly severe in certain instances. The enhancement capabilities of slug flow fields for highly viscous liquids is especially important since operation in the turbulent flow region is often impractical. The major disadvantage of solid-liquid slug flow is the increased pressure drop across the tube for liquid slugs of all sizes. Enhanced radial transport occurs at the cost of increased power requirements, and the liquid slug length and film thickness must be chosen to optimize the overdl performance of the slug flow unit. For gas-liquid slug flow, an increase in

0 1978 American Chemical Society

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Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978

Table I. Summary of Some Possible Boundary Conditions for Slug Flow

Case no.

Momentum boundary condition at ends of liquid slug

1

No drag

2

No slip

3

No drag

4

No slip

Heat or mass transfer boundary condition

Corresponding application

Temperature of gas bubble equals tube wall temperature Temperature of solid sphere equals tube wall temperature No heat or mass flux between adjacent slugs No heat or mass flux between adjacent slugs

pressure drop is observed only for short liquid slugs (Horvath e t al., 1973). Another problem with solid-liquid slug flow is that provisions must be made for recycling the solid spheres. The difficulty in producing uniform liquid slugs seems t o be comparable in the two cases for longer liquid slugs, but it appears to be easier to form short liquid slugs using gas bubbles. The radial transport enhancement that can be expected for slug flow depends on the boundary conditions which exist a t the boundary between the two phases. A summary of some possible momentum and heat or mass transfer boundary conditions for closely fitting solid or gas slugs is presented in Table I. From analytical solutions of the equations of motion in the creeping flow limit for no slip and no drag boundary conditions (Duda and Vrentas, 1971a), it can be shown t h a t radial velocities are greater when there is no drag at the ends of the liquid slugs. Hence, it appears reasonable t o speculate that a no drag momentum boundary condition leads to greater radial transport enhancement than a no slip condition if the heat or mass transfer boundary conditions are identical. If this reasoning is valid, then case 1 in Table I provides the best radial transport enhancement and case 4 the worst. Case 2 has the preferred boundary condition for heat transfer, whereas case 3 has the preferred fluid mechanics boundary condition, and further information is needed before the relative transport rates for these cases can be ascertained. As discussed below, the data of this investigation indicate that case 2 gives better radial transport enhancement. From Table I and the above discussion, we conclude that better radial mass transport enhancement can be realized by using a gas-liquid system if the liquid film is thin. The choice of a n appropriate system for heat transfer depends on the nature of the heat transfer in the gas phase for slug flow. The heat transfer data of Oliver and Young Hoon (1968) provide little insight into this question since these investigators used gas slugs with thick liquid films and a well defined slug flow field was not realized. It is perhaps reasonable to suppose that heat transfer in gas-liquid slug flow corresponds more closely to case 3 than to case 1,and we would thus conclude from the above discusion that better heat transfer enhancement in slug flow is realized using a solid-liquid system.

Theoretical Considerations To investigate radial transport in slug flow, we consider the transfer of heat from a tube wall a t temperature T , to a fluid initially a t temperature To. If 9.4 is the heat transferred to the liquid per unit time, it is evident that where T , is the dimensionless average fluid temperature and where e, the volume fraction of the tube made ineffective by the spheres, can be approximated by the expression

Heat transfer from wall using highly conducting gas phase Heat transfer from wall using highly conducting solid spheres Heat or mass transfer from wall using gas phase impermeable to heat or mass transfer Heat or mass transfer from wall using solid spheres impermeable to heat or mass transfer r)

An average heat transfer coefficient based on a n arithmetic mean temperature difference, (ha)*,can be defined in the usual manner 9A (ha)A =

T R L [ ( T ,- To)

+ ( T ,- T,*)]

(3)

and the following result can easily be derived from eq 1 and 3 (4)

An ideal heat exchanger can be defined as one in which the gas phase or solid phase dividing the liquid slugs is infinitely thin, and the heat transfer coefficient for an ideal exchanger is given by the expression

As pointed out by Horvath e t al. (19731, this average Nusselt number provides a useful indication of the effectiveness of the radial transport process in the tube. The Nusselt number can be readily evaluated if T , is known either from theory or experiment. Finally, we note from eq 5 that the maximum obtainable arithmetic mean Nusselt number for a given Pe and RIL is simply

A reasonable but idealized theoretical analysis of heat transfer in solid-liquid slug flow can be carried out by introducing the following assumptions. (1)The fluid is Newtonian and the flow is laminar. (2) The properties of the fluid are independent of temperature. (3) The temperature and velocity fields are axially symmetric. (4) The fluid motion in the tube is assumed to be established before heating or cooling of the fluid begins. ( 5 ) Viscous dissipation is considered negligible. (6) The spheres are approximated by tightly fitting cylindrical plugs with a height equal to the diameter of the spheres. The curvature and rolling motion of the spheres are hence ignored, and no account is taken of the thin liquid film which separates the sphere from the tube wall. (7) End effects a t the entrance and exit of the heated tube are neglected. Such effects result because all of the liquid segment confined between two spheres does not make or break contact with the heated wall at the same instant of time. These end effects could possibly be important if the length of the liquid slug is a signficant fraction of the length of the heated section of the tube. (8) Highly conducting spheres are used so that the sphere temperature is equal to the wall temperature. With these assumptions, analysis of solid-liquid slug flow

Ind. Eng. Chem. Fundam., Vol. 17, No. 1. 1978

41

Table 11. Summary of Experiments Fluid

Pe x 10-3

Re

Br

X

10"

100-cSt silicone fluid 1000-cSt silicone fluid

Single Phase Flow in Tube 4.64-17.4 4.21-17.8 0.70-1.68 6.35-15.3

0.06-0.92 1.14--6.32

100-cSt silicone fluid 1000-cSt silicone fluid

Slug Flow in Tube 9.55-15.2 10.1-15.8 6.44-15.8 0.7 8-1.95

0.31-0.79 1.37-8.65

Slug Flow Experiments of Horvath e t al. (1973) 34-510 20-300

is equivalent to the calculation of the velocity and temperature fields in a cylindrical cavity with a uniformly translating wall (Duda and Vrentas, 1971a,b). The dimensionless average temperature of the fluid in the cavity has a functional dependence of the form T, = T,(p,t,Pe,ReI

(7)

so, as noted previously by Horvath, et al. (1973), the average Nusselt number for the tube based on an arithmetic mean temperature difference must have the following functional form

G

Nu = Nu $, - , Pe,Re

)

An analytical solution has been derived (Duda and Vrentas, 1971a,b) for the cylindrical cavity problem for Re = 0, and results have been presented for p = 1 and Peclet numbers ranging from 0 to 400. These calculations can easily be extended to higher Peclet numbers, but computations for larger values of $ are difficult because the eigenvalues of the heat transfer problem lie closer together as p is increased. Furthermore, the analytical solution cannot of course be modified to include higher Reynolds numbers because of the nonlinearity introduced in the inertial terms of the equations of motion. These difficulties could presumably be overcome by utilizing finite-difference or weighted residual methods to analyze the heat and momentum transfer problem in the cavity. Gross and Aroesty (1972) review numerical solutions for slug flow. However, it would be difficult to include all of the possibly important effects (for example, rotation of the spheres and end effects) even when numerical solutions are used. Thus, it was felt that it would be useful to collect experimental data on heat transfer in solid-liquid slug flow. These data complement the theoretical results, and the combination of theory and experiment provides heat transfer results for values of /3 ranging from 1 to 122. In addition, the data presented here complement the experimental data of Horvath et al. (1973) since average Nusselt numbers are measured for higher values of p and since the variation of Nu with Re a t constant Pe was studied. Also, the present data and those of Horvath et al. (1973) can be used to compare the radial transport enhancements for cases 2 and 3. Finally, slug lengths which are nearly one-half of the tube length are used, and some idea of the importance of end effects can be ascertained. Experimental Section Heat transfer data for solid-liquid slug flow were collected using a modification of the type of heat exchanger employed by Bonilla e t al. (1953) and Metzner e t al. (1957). The horizontal heat exchanger was fed by a gear pump which withdrew fluid from a constant head tank in which the fluid was cooled to the inlet temperature. Stainless steel spheres were introduced after the pump a t regular intervals, and the fluid and

a

77-122 21-122 0.36-44

spheres entered a 180 cm calming section where the flow field becomes fully developed before the fluid and the spheres enter the heated test section. The sphere spacing was measured using a photoelectric timer located a t a glass viewing section. The spheres and fluid flowed in a 0.95-cm i.d. tube, and the steel spheres were sized so that there was less than a 0.0025-cm clearance between the spheres and the wall of the tube. After leaving the test section, the fluid passed through a mixing chamber and then flowed back to the tank or through a liquid bypass used for flow rate measurements. All data were obtained with a 122-cm heat exchange section and LID equal to 128. The heat exchanger loop was constructed so that the exit of the heat exchanger was located next to the entrance. The spheres were transferred from the exit line to the tube inlet by a reciprocating piston which was connected perpendicularly to the inlet and outlet lines. At the exit of the heat exchanger, the fluid pressure forced a sphere into a cavity in the piston, and the piston transferred the sphere to the inlet pipe where it joined the inlet fluid. The spacing of the spheres was controlled by the frequency of the reciprocating piston, and the apparatus was designed so that mixing of the inlet and outlet fluids was minimized. The 0.95-cm i.d. heat transfer tube was surrounded by 5.1-cm i.d. and 6.4-cm i.d. tubes, and Freon TF (CC12F-CClFp) vapor was admitted to the two larger tubes. The inner annulus served as a heating jacket and the outer annulus acted as tin isothermal insulator. The wall temperature maintained by the condensing Freon T F vapor was monitored by four thermistors which were imbedded into the wall of the 0.95-cm tube. Thermistors were also used to determine the inlet temperature of the fluid. However, it was not possible to obtain reliable outlet temperatures from thermistors placed in the exit mixing section when data were collected with the steel spheres. Consequently, all heat transfer coefficients reported in this study are based on heat flows deduced from the measured condensate rate. All of the experiments were conducted using Dow Corning 210 silicone fluid, a stable liquid available in a wide range of viscosity. A modest variation of the Reynolds number was achieved by using silicone fluids with kinematic viscosities of 100 and 1000 cSt. The silicone fluid could be considered Newtonian for the conditions of this study, and, in addition, the viscosity of this fluid is a relatively weak function of temperature, a desirable property for heat transfer studies. Heat transfer data were collected both with and without spheres, the latter set of data serving as a control for the slug flow data. The experiments of this study are summarized in Table 11,which also includes a summary of the mass transfer experiments conducted by Horvath et al. (1973). Results a n d Discussion The laminar, single-phase flow experiments provide a basis against which the slug flow experiments can be compared and

42

Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978

2:

2:

2

19

Ii

Nu Pep

t

00105 (Gr

I!

F i g u r e 2. Comparison of single-phase flow data with correlation of Oliver (1962) which includes the effect of natural convection. IZ

natural convection on laminar flow heat transfer in horizontal tubes can be estimated from the following empirical correlation of Oliver (1962)

I

+ 0.0105(GrPr)0.75

!

I

6

1

8

10 pe

I

12 10-3

I

14

16

18

F i g u r e 1. Experimental heat transfer results for single-phase flow and slug flow in a tube. The data represented by the open circles were obtained with 100-cSt fluid and the solid circles represent 1000-cSt fluid data. LIR = 256.

also serve as a check on the experimental apparatus and procedures. A theoretical analysis of heat transfer to a Newtonian fluid in laminar flow inside a pipe with constant wall temperature, constant physical properties, fully developed velocity field, and negligible viscous dissipation was first presented by Graetz (1885).For values of PeDIL greater than 10, the theoretical calculations are represented within 5% by the equation (Norris and Streid, 1940) Nu = 1.62 (Pe

F)’3

with a Nusselt number again based on an arithmetic mean temperature difference. If the experiments are designed to minimize the secondary effects due to viscous dissipation, natural convection, and temperature dependence of the viscosity, then the reliability of the experiments can be ascertained by comparing the data against the predictions of eq 9. For the range of Brinkman numbers used here (see Table 11),it follows that the temperature rise due to viscous dissipation can be disregarded. The effect of the distortion of the velocity distribution, caused by the radial variation in the fluid viscosity, on the heat transfer can be approximated by using the Sieder-Tate (1936) correction factor. In this study, the maximum effect predicted by the Sieder-Tate correction is about 7%. If we suppose that the Sieder-Tate correction gives a reliable estimate of the effect of variable viscosity on the heat transfer process, then no systematic deviation of the data from eq 9 is anticipated since the predicted correction is of the same magnitude as the estimated experimental error. The effect of

This equation represents available heat transfer data in horizontal tubes reasonably well for values of LID greater than 70 and values of the Graetz number larger than xNu. Calculations based on eq 10 show that deviations from eq 9 as high as 15%can be expected for the experimental conditions utilized in this study. Since this is about twice the estimated experimental error, discernible deviations of the experimental data from the predictions of eq 9 can be expected, particularly at the lower values of the Peclet number and the higher values of the Grashof number. The experimental results for heat transfer to fully developed laminar flow in a tube in the absence of spheres are compared with the predictions of eq 9 in the lower part of Figure 1. The data obtained with the 1000-cSt fluid are in excellent agreement with the theory, whereas the Nusselt numbers obtained with the 100-cSt fluid fall above the theoretical curve, particularly a t low Peclet numbers. In Figure 2, the data are compared with the empirical correlation presented by Oliver (1962) which includes the effect of natural convection. The good agreement with this correlation supports the conclusion that the increased heat transfer with the lower viscosity fluid a t the lower flow rates is due to natural convection. I t is thus reasonable to conclude from this agreement of the data with theory that the apparatus and experimental techniques can be used to generate reliable heat transfer data. Slug flow heat transfer data for steel spheres and 100 and 1000-cSt silicone oils are compared with the single phase flow data in Figure 1. Data were collected for six values of p (21,41, 52, 77,96, and 122), but only data for three p values are presented in this figure. Measured condensate rates were conby introducing approximate corrections for the verted to ( h , ) ~ heat picked up by the steel spheres and for the volume fraction of the tube made ineffective by the spheres. Since these corrections were typically of the order of 10%each and of opposite sign, it was felt that negligible errors were introduced in the calculated Nusselt numbers. Furthermore, calculations show that the spheres are effectively heated to the wall temperature in a small fraction of the total length of the heat exchanger, and the desired temperature boundary condition for the solid spheres is in effect for the major portion of the heat transfer process. Finally, from approximate calculations it was concluded that secondary effects due to viscous dissipation,

Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978

natural convection, and temperature dependence of viscosity were small in the slug flow experiments. From Figure 1,it is evident that solid-liquid slug flow leads to a significant enhancement in the heat transfer rate. This radial transport enhancement increases with increasing Peclet number and decreasing (3, and, for the conditions of this study, the measured Nusselt numbers are 1.9-3.0 times greater than those for single phase flow a t the same value of Pe. Furthermore, the slope of the line representing the Nu vs. Pe data increases as $ decreases. Horvath e t al. (1973) observed qualitatively equivalent results for mass transfer in gas-liquid slug flow. The present results show that quite significant heat transfer enhancement (approximately two times) persists for liquid slugs which have a length 60 times the tube diameter. The largest @ utilized in this study is approximately three times the largest (3 used in the gas-liquid experiments. From eq 8, we see that Nu depends on Re as well as on Pe and p for fixed LIR, and hence both (3 and Re should be used as parameters on the slug flow curves in Figure l. However, a tenfold variation of the Reynolds number a t constant Pe and B showed no discernible change in Nu, and hence no Reynolds number dependence was detected for the limited Reynolds number range (0.78-15.8) studied here. This observation is discussed further below. For a given geometry (fixed LIR and UR),the nature of the heat transfer process is governed by Re, which compares the viscous and inertial terms in the equations of motion, and by Pe, which compares the conductive and convective terms in the energy equation. Results from a theoretical study of heat transfer in a cylindrical cavity (Duda and Vrentas, 1971b) with Re = 0 and = 1 show that Nu increases with increasing Pe in a manner qualitatively similar to the behavior exhibited by the data of this investigation. Furthermore, the theory shows that the transition from conduction dominated heat transfer to convection dominated heat transfer is effectively complete a t a Peclet number of 400. In other words, for Pe > 400, the instantaneous or local Nusselt number and the shape of the asymptotic temperature profile are independent of the Peclet number. If we suppose that this Peclet number independence holds over appropriate ranges of Re and @, then the basic mechanism of heat or mass transfer is the same for the present data and for the data of Horvath et al. (1973),even though the present experiments cover a much lower Peclet number range (6400-15 800) than the gas-liquid experiments (34 000510 000). Although the transition of the heat transfer process from the conductive mode of transfer a t low Pe to the convective mode a t high Pe is of theoretical interest, only the high Peclet number region is of practical importance in slug flow since the maximum heat transfer enhancement is desired. It is useful if slug flow heat transfer experiments for a fixed geometry are designed so that Re and Pe can be varied independently since this procedure will permit the separation of thermal and hydrodynamic effects. For example, it is of interest to deduce the dependence of Nu on Pe for fixed fluid mechanics or to examine the effect of a changing velocity field on the heat transfer process for fixed Pe. Experiments of this type can of course be carried out by varying the Prandtl number by using fluids with different properties. An effort was made in this study to ascertain the effect of a variation of the fluid motion on the heat transfer process by using fluids with different viscosities, and, as noted above, no Reynolds number dependence of Nu was observed for a modest Reynolds number range near the creeping flow limit. The experiments of Horvath et al. (1973) were conducted over a much wider Reynolds number range, but a single fluid was used and separation of the thermal and hydrodynamic effects is not possible. Over the limited Reynolds number range covered here, we can conclude that there is an insignificant change in the heat transfer process and presumably a small deviation of the

43

P e x 10-3

Figure 3. Theoreticaland experimental results for slug flow. LIR = 256.

velocity field from the creeping flow streamline pattern. Indeed, there is some support for such a conclusion in the literature. As the Reynolds number is increased from 0 to m, the flow field in a cavity changes from a viscous eddy to an inviscid core with a thin boundary layer. Burggraf (1966), Bozeman and Dalton (1973), and Nallasamy and Prasad (1977) obtained finite-difference solutions for flow in a square cavity, and they observed a slow change of the streamline pattern as the Reynolds number was increased from the creeping flow limit. Only a modest change in the creeping flow streamline pattern was observed a t Re = 100, and only a t very high Reynolds numbers (Re > 30 000) is the transition from viscous dominated flow to inertia dominated flow effectively complete. Furthermore, Weiss and Florsheim (1965) showed that low Reynolds number solutions of flow in rectangular cavities are in reasonable agreement with experiments conducted a t Re = 150. A theoretical analysis of the entrance flow problem for a tube (Vrentas and Duda, 1973) shows that the velocity field a t Re = 100 is significantly different than the creeping flow velocity distribution. Apparently, the transition from a flow field dominated by viscous effects to one for which inertial effects are dominant occurs over a much wider Reynolds number range for recirculating flows than for single pass flows. Consequently, it is reasonable t o expect that the velocity field for closed streamline flows, such as slug flow, is weakly dependent on Re in the Reynolds number range 0-100, and only a slight dependence of Nu on Re can be expected under such conditions. Hence, as a first approximation, the lower part of Figure 3 of Horvath et al. (1973) can be regarded as showing the variation of Nu with Pe with fluid motion dominated by viscous effects, and this figure thus corresponds to Figure 1 of this study. Clearly, data a t much higher Reynolds numbers

44

Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978

3€

32

28 Nu

24

2c

I€ IO I 0’

12

I 34

L I I I 001

, 002

1

1 l 1 l I I 005 01

1 02

I

,

I l l l l 05 I

I lP

Figure 4. Effect of p on the Nusselt number for slug flow at Pe = 10 000. The point at 0 = 1 is calculated from theory and the remaining points are from the experimental data. L / R = 256.

are needed to determine the effect of the velocity field on the heat transfer process as the inertia dominated flow asymptote is approached. Owing to a limitation on the speed of the sphere changer, reliable data could not be obtained a t values of /3 less than 20. However, theoretical calculations for the case fl = 1 are available, and results of this calculation, which are shown in Figure 3, combined with the experimental results, provide a satisfactory representation of solid-liquid slug flow heat transfer for /3 ranging from 1 to 122. The theoretical results agree with the previous experimental observation that Nu becomes a stronger function of Pe as /3 decreases. The upper curve in Figure 3 is based on eq 6 and represents the upper limit for the Nusselt number which would be realized if the fluid leaving the heat exchanger were at the same temperature as the wall. Figure 4 illustrates the variation of Nu with /3 a t Pe = 10 000. The change of Nu with /3 appears to be more pronounced at lower values of p; similar results were reported by Horvath et al. (1973). Above, it was pointed out that the large sphere spacings used here may lead to significant end effects. It is difficult to assess the magnitude of such end effects, but, since the data of this study appear to show a smooth transition from the larger to the smaller slug lengths, it is probable that these effects play a relatively minor role in the heat transfer process. The experiments of this study are an example of case 2 in Table I, whereas the experiments of Horvath et al. (1973) are an example of case 3. Comparison of these two cases was carried out by using Figure 4 of Horvath et al. (1973) to adjust the data to the LIR used in this study and by converting the present data to a Nusselt number based on the logarithmic mean temperature difference. In Figure 5 , and in this figure only, Nu is a logarithmic mean Nusselt number. It is evident from Figure 5 that case 2 provides superior radial transport enhancement a t comparable /3 and Pe, a t least in the flow region dominated by viscous effects.

Acknowledgment The authors are indebted to W. L. Sigelko for assistance in the experimental work. Nomenclature Br = Brinkman number, p U a 2 / k A T , C ; = specific heat capacity a t constant specific volume D = diameter of tube

I

I

IO4

lo5 Pe

Figure 5. Comparison of cases 2 and 3. The solid lines on the left of t h e graph represent data of this study and the solid lines on the right are adjusted data of Horvath et al. (1973). The dotted lines represent extrapolations of the latter data. A logarithmic mean Nusselt number is used in this figure. All data adjusted to L / R = 256.

Gr = Grashof number, D3p2gaAT,/p2 g = gravitational acceleration (ha)* = actual average heat transfer coefficient based on arithmetic mean temperature difference (h,)I = idealized average heat transfer coefficient based on arithmetic mean temperature difference k = thermal conductivity L = length of heat exchanger 1 = spacing between spheres ~ Nu = arithmetic mean Nusselt number, ( h , ) Dlk Pe = Peclet number, RePr Pr = Prandtl number, C; p l k q~ = heat transferred to liquid per unit time R = radius of tube Re = Reynolds number, DU,p/p T , = dimensionless average exit liquid temperature, T,* TolT, - To T,* = average exit liquid temperature T o = inlet liquid temperature T , = wall temperature AT, = arithmetic mean temperature difference t = dimensionless time, t * U,/R t* =time U , = average velocity of liquid in tube

Greek Letters N = thermal coefficient of volumetric expansion /3 = aspect ratio, l/R t = volume fraction of tube made ineffective by spheres p = liquid viscosity p = liquid density Literature Cited Bauer, A. B., DuPuis, R. A,, J. Appl. Mech., 34, 538 (1967). Bonilla, C. F., Cewi, A., Colven, T. J., Wang, S. J., Chem. Eng. Prog. Symp. Ser., 49 (9,127 (1953). Bozeman, J. D., Dalton, C., J. Comput. Phys., 12, 348 (1973). Burggraf, 0. R., J. Fluid Mech., 24, 113 (1966). London, Ser. A, 251, 550 Christopherson. D. G., Dowson, D., Proc. Roy. SOC. (1 959). Duda, J. L., Vrentas, J. S., J. FluidMech., 45, 247 (1971a). Duda, J. L., Vrentas, J. S., J. NuidMech., 45, 261 (1971b). Ellis, H. S., Can. J. Chem. Eng., 42, 1 (1964a). Ellis, H. S., Can. J. Chem. Eng., 42, 155 (1964b). Graetz, L., Ann. Phys., 25, 337 (1885). Gross, J. F., Aroesty, J., Biorheology, 9, 225 (1972). Howath, C., Solomon, B. A,. Engasser, J.-M., lnd. Eng. Chem. Fundam., 12, 431 (1973). Kern, W. I., Chem. Eng., 82, 139 (Oct. 13, 1975). Metzner, A. B.. Vaughn, R. D., Houghton, G. L., AICMJ., 3, 92 (1957). Nallasamy, M., Prasad, K. K.. J. FluidMech., 79, 391 (1977). Norris, R. H., Streid, D. D., Trans. A.S.M.E.,36, 525 (1940).

Ind. Eng. Chern. Fundam. Vol. 17, No. 1, 1978 Oliver, D. R., Chem. Eng. Sci., 17, 335 (1962). Oliver, D. R., Wright, S.J., Brit. Chem. Eng.. 9, 590 (1964). Oliver, D. R., Young Hoon. A.. Trans. Inst. Chem. Eng., 46, T116 ( 1968). Prothero. J., Burton, A. C., Biophys. J., 1, 565 (1961). Sieder, E. N., Tate, G. E., Ind. Eng. Chem.. 28, 1429 (1936). Skeggs, L. J., Am. J. Clin. Pathol., 28, 311 (1957).

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Taylor,G. I., J. NuidMech., 10, 161 (1961). Vrentas, J. S . , Duda, J. L., Appl. Sci. Res., 28, 241 (1973). Weiss, R. F.. Florsheim, B. H.,fhys. fluids, 8, 1631 (1965)

Received for reuieb June 15, 1977 Accepted November 10,1977

Generalized Vapor Pressure Equation for Nonpolar Substances Mateo Gomez-Nieto and George Thodos' Northwestern University, Evanston, lllinois 6020 1

A reduced vapor pressure equation of the form In PR = N -I-P / T R mf y TR" has been applied to 113 nonpolar substances which include 95 hydrocarbons of all types and 18 inorganic substances. In this study the exponent n was taken to be 7.0, as established in an earlier study associated with the normal paraffins. The involvement of the critical point eliminates parameter cy, requiring knowledge of the three parameters 13, 7 ,and rn. Relationships for these vapor pressure parameters have been developed and require for their establishment knowledge of the normal point, critical temperature, and critical pressure of the substance. Vapor pressures calculated with predicted parameters were compared with corresponding values reported in the literature to produce an average deviation of 0.97% (6290 points) for these 113 organic and inorganic substances. Whenever available, vapor pressure values between the critical point and the triple point were considered in these comparisons.

The vapor pressure behavior of normal alkanes ranging from methane through n -eicosane has been recently reported ( 5 3 ) using the expression in reduced form

where the exponent n was found to be essentially constant at a value of 7.0 for these normal alkanes. When eq 1 is applied to the critical point, it yields the following boundary condition (2)

a+P+y=O

If cy is substituted into eq 1,this reduced vapor pressure relationship takes the form In P R = p

[ T R r n -11 + Y [ T R "

- 11

(3)

Equation 3 conforms more directly with the corresponding states principle, when compared with eq 1,since the boundary condition imposed through eq 2 forces the form of eq 3 to satisfy the critical point condition. In addition, the involvement of only the three parameters, P, ?, and m in this relationship, makes this expression more amenable to a nonlinear regression analysis, when compared with eq 1. In the treatment of normal alkanes, the characterization parameter, s , was used to correlate the vapor pressure coefficients P and y and exponent m. This parameter represents the negative of the slope existing between the critical point and the normal boiling point on a In PRvs. 1 / T R coordinate system and can be expressed as s =

Tb ~

In P,

(4)

Tc - T b The present investigation undertakes a similar study that extends beyond normal alkanes to study the behavior of light hydrocarbons of all types, including saturated isomeric paraffins, olefins, diolefins, acetylenes, naphthenes, and aromatics. Furthermore, in order to extend the scope of this 0019-7874/78/1017-0045$01.00/0

study, several inorganic substances were also included. Altogether 113 substances, including the 20 light normal alkanes through n-eicosane, were considered in this investigation. For all these substances, vapor pressure data were obtained from an exhaustive literature search. The screening process of these data that followed considered a careful examination of these literature reported values and the rejection of those that were not internally consistent with the most credited and recent references. From a total of 6443 vapor pressure values, 153 points associated with nine substances obtained from 11 different references were rejected. These 11 references are not reported. In this screening process, the values from a reference were either included in their entirety or were completely rejected. The sources of data used are reported in Tables I, 11, and 111, along with the basic physical properties associated with each substance. The International Practical Temperature Scale of 1968 ( 1 ) has been applied throughout this study. This scale utilizes the relationship, T(K) = t("C) 273.15, and the following reference points: neon ( T b = 27.102 K), oxygen ( T b = 90.188 K), water (Tt = 273.16 K, t i = 0.01 "C), and others. Whenever necessary, the proper corrections were incorporated to the literature vapor pressure values.

+

Establishment of Coefficients ,f3 a n d y a n d Exponent m The vapor pressure values corresponding to each substance were normalized using the critical constants presented in Tables 1-111. Applying a nonlinear regression analysis on eq 3 and involving the vapor pressure values of each substance, the parameters of this equation were optimized to produce the values of @, y,and m reported in Tables 1-111. Table I includes 18 typical nonpolar inorganic substances ranging in complexity from the monatomic gases, including helium (4He), to carbon tetrachloride and sulfur trioxide. This listing is extended in Table I1 to include 51 saturated alkanes of which 20 are straight chain hydrocarbons. Table I11 presents five other types of hydrocarbons including 11 monoolefins, 8 diolefins, 6 acetylenes, 10 naphthenes, and 9 aromatics. For each

0 1978 American Chemical Society