Ind. Eng. Chem. Res. 1996, 35, 2399-2403
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Design Criterion for the Heat-Transfer Coefficient in Opposing Flow, Mixed Convection Heat Transfer in a Vertical Tube Donald D. Joye Department of Chemical Engineering, Villanova University, Villanova, Pennsylvania 19085-1681
Mixed convection heat transfer in a vertical tube with opposing flow (downflow heating) was studied experimentally for Reynolds numbers ranging from about 1000 to 30 000 at constant Grashof numbers ranging about 1 1/2 orders of magnitude under constant wall temperature (CWT) conditions. Three correlations developed for opposing mixed convection flows in vertical conduits predicted the data reasonably well, except near and into the asymptote region for which these equations were not designed. A critical Reynolds number is developed here, above which these equations can be used for design purposes regardless of the boundary condition. Below Recrit, the correlations, the asymptote equation should be used for the CWT boundary condition, which is more prevalent in process situations than the uniform heat flux (UHF) boundary condition. Introduction Heat-transfer situations in which natural convection and forced convection heat-transfer mechanisms interact are termed “mixed” or “combined” convection. Vertical flow situations are particularly sensitive to potential interactions between forced and natural convection. In this work, downward internal flow in a heated vertical tube is discussed from a design standpoint (predicting the heat-transfer coefficient). Downflow heating is termed “opposing” flow, because the natural convection currents are in the opposite direction from the forced flow. In contrast, upflow heating is often termed “aiding” flow, because the natural convection currents are in the same direction as the forced flow. Heat-transfer results for upflow heating are quite different than those for downflow heating; hence, these are often treated as two separate cases. Aiding flow shows more complex experimental behavior than opposing flow, and older formulations are being discarded in favor of newer approaches. To date, correlations for the heat-transfer coefficient for this case are being vigorously pursued (Poskas et al., 1994). That improved design equations are needed for opposite flow, vertical, mixed convection situations is acknowledged in the Heat Transfer Design Handbook (Churchill, 1983), where the Churchill method is recommended for the “turbulent” region. The method was found to be satisfactory in predicting the data of Herbert and Sterns (1972). For the laminar region, “a more precise correlation has not yet been established”. In the last 12 years, this field has seen an explosion of work, and much more data and correlations have been published. Correlations for opposing flow, vertical, mixed convection heat transfer have been developed by Jackson and Fewster in Jackson et al. (1989) and Swanson and Catton (1987) for the uniform heat flux (UHF) boundary condition. Both of these equations for opposing flow were tested independently for Grashof number and Reynolds number dependence and found to be quite satisfactory even for the constant wall temperature (CWT) boundary condition, except for the low Reynolds number region (Joye, 1995). In this work, we further test these equations and develop a criterion for their application to design. In process industries, the UHF boundary condition is rare. It occurs mainly with electrical heating or nuclear heating. It can occur, however, in a double-pipe S0888-5885(95)00464-7 CCC: $12.00
heat exchanger (for example) if the temperature-difference driving force between the two fluids is the same throughout the length of the exchanger. The CWT boundary condition occurs with phase change heat transfersboiling or condensation, for example, which have relatively high heat-transfer coefficients and are well represented in the process industries. Also, the constant wall temperature condition is often used in an average sense for most two-fluid heat exchange. Vertical, internal flow heat transfer has importance in vertical shell-and-tube heat exchangers and other chemical process technology where simultaneous heat and mass transfer occur, nuclear power technology (Symolou et al., 1987; Ianello et al., 1988), and some aspects of electronic cooling. There can be significant heat-transfer enhancement in vertical flow situations; hence, they may have advantages for energy savings in process technology. Joye et al. (1989) present data from an experimental study of heat-transfer characteristics for upflow and downflow heating in a vertical tube under a constant wall temperature boundary condition and high Grashof number. Three regions are shown to characterize the heat transfer in this situation. The turbulent flow region occurs at Reynolds numbers greater than about 10 000, where the Nusselt numbers for vertical, mixed convection heat transfer agree with those predicted from well-known forced flow correlations, e.g., the SiederTate or any of the newer relationships now presented in textbooks, e.g., Holman (1990). Mixed convection is not important in this turbulent region, because the hydrodynamic turbulence at these Reynolds numbers is much stronger than the natural convection mechanism and dominates the heat transfer. At low Reynolds numbers, the asymptotic region exists and represents heat transfer and flow conditions where the outlet temperature approaches the wall temperature. This region was anticipated by both Martinelli et al. (1942) and McAdams (1954), where it is discussed in some detail. At Reynolds numbers between the turbulent and the asymptotic regions, the mixed convection region exists, where both natural convection and forced convection mechanisms are of the same order of magnitude and interact in complex ways. The transitions between the three regions are affected primarily by the Grashof number. Since the correlations of Jackson and Fewster, Swanson and Catton, and the Churchill method from the Design Handbook tran© 1996 American Chemical Society
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sition to forced flow turbulence rather well, only the transition to asymptotic behavior needs to be addressed.
NuN ) 0.15(GrPr)1/3/(1 + (0.492/Pr)9/16)16/27
Theoretical Background The correlation of Jackson and Fewster in Jackson et al. (1989) is
Nu/Nu(forced) ) (1 + 4500GrbarD/Re2.625Pr0.5)b0.31 (1) where the subscript “b” refers to properties evaluated at the bulk average temperature and GrbarD is defined by the integrated density term
GrbarD ) F(F - Fbar)gD3/µ2
(2)
and
∫TT F dT)/(Tw - Tb)
Fbar ) (
b
w
(3)
and the forced flow model is the Petukhov-Kirillov model given by the following:
Nu(forced) )
one developed by Churchill and Chu:
RePrCf/2 12.7(xCf/2)(Pr2/3 - 1) + 1.07
(4)
(10)
The asymptote relationship for CWT conditions at low Reynolds numbers was developed by Martinelli et al. (1942) and McAdams (1954). The asymptote is a result of losing ∆T driving force at the exit and, consequently, would not be expected to hold for a UHF boundary condition, which maintains ∆T driving force. The asymptote equation takes various forms, but fundamentally, it is
Nuasymptote ) (2/π)Gz
(11)
where Gz is the well-known Graetz number. When Re is used as an independent parameter as we prefer, the asymptote is also Prandtl number dependent,
Nuasymptote ) 0.5RePrD/L
(12)
which translates to
Nuasymptote/Pr1/3φv0.14 ) 0.5RePr2/3D/L
(13)
in the equivalent Sieder-Tate style formulation.
where
Cf ) 1/(3.64 log Re - 3.28)2
(5)
The correlation of Swanson and Catton (1987) for flow in vertical channels of rectangular cross section is shown below. This has an additional term and slightly different Prandtl number dependence. The Reynolds number formulation originally contained the well-known hydraulic diameter; thus, this equation is also applicable to flow in tubes.
Nu ) 0.0115Re0.8Pr0.5{1 + [1 - (696/Re0.8) + 8300Gr/Re2.6(Pr0.5 + 1)]0.39} (6) These correlations use, as a base case, the turbulent forced flow only correlation from which increases in heat transfer due to natural convection effects can be followed. The Sieder-Tate equation with viscosity correction is often preferred as the base case,
Nu/Pr1/3φv0.14 ) 0.023Re0.8
(7)
For heating situations, the viscosity correction factor, φv, is often folded into a slightly different power for Pr. The Swanson and Catton base case shows this preference; Jackson and Fewster choose the PetukhovKirillov relationship, which differs only slightly from the Sieder-Tate one. Thus, the base cases are all comparable. In the Churchill (1983) method, the Nusselt number is determined by taking the cube root of the sum of the cubes of the natural convection Nusselt number and the forced convection Nusselt number.
Nu3 ) NuN3 + NuF3
(8)
The forced flow correlation is one developed by Churchill,
NuF ) 0.0357RePr1/3/(1 + Pr-4/5)5/6 ln(Re/7) (9) but it gives results only marginally different from eq 7 and eqs 4 and 5. The natural convection correlation is
Experimental Method The apparatus employed for the present study was identical to that of previous investigations (Joye, 1995; Joye et al., 1989), except that inlet water was preheated in an attempt to lower the temperature driving force and, hence, Grashof number in this investigation. This was only partly successful because the increased temperature altered the properties of the fluid which tended to increase the Grashof number, so the competing effects largely canceled one another. However, we were able to get two runs below (and one run above) previous data as well as corroborating the previous work. Heating was provided by steam condensing in the annulus of a copper-copper, double-pipe heat exchanger, shown in Figure 1. Steam pressure was controlled by a Nash vacuum pump and control valve system. A vacuum gauge measured the (vacuum) pressure in the jacket, and a pressure gauge measured positive steam pressure when that was used. Five (copper-constantan) thermocouples were embedded in the pipe wall to measure the wall temperature. The pipe wall was grooved, and constantan wire was silverbrazed in the groove. Excess metal was then filed smooth to minimize interference with the condensate film. The vacuum pressures in the jacket varied from 99.6- to 30.2-kPa absolute pressure (0.5-21-mm Hg vacuum, respectively). A run at 345-kPa absolute (35 psig) steam pressure was included to extend the range of the present data. Any particular value of steam pressure could be held constant by the control system. The steam side is not important in and of itself, since the wall temperature were measured directly. Constant wall temperature (an average) could be maintained and checked easily. At low pressures, the steam can often condense at the inlet only, and when this happened, the steam valves were adjusted to provide sufficient steam flow. The lowest pressures were at the limit of the apparatus capability. Wall temperatures and, hence, Grashof numbers were changed by adjusting the steam pressure within the limits of the equipment.
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Figure 2. Downflow heating data compared with the correlations of Jackson et al. and Swanson and Catton. Shell-side steam pressure ) 345 kPa absolute (GrbD ) 6.0 × 107).
Figure 1. Experimental apparatus.
The inlet/outlet temperatures were measured by thermowell-type thermocouple probes. The unit was insulated to minimize heat loss, particularly important at the outlet. Mixing elbows were used to get a wellmixed outlet temperature in both upflow and downflow. Two rotameters were used to measure the flow rates at high throughput and at low throughput. The L/D of the heated section was 49.6; the inside diameter of the central tube was 0.032 m (1.265 in.). Water in the tube was prevented from boiling or degassing by pump pressures of 308-515 kPa absolute (30-60 psig), which is much higher than needed. Water temperature rise varied from 5 to about 40 °C, and the temperature difference driving force from wall to bulk average fluid varied by about the same order. Heat-transfer data including all dimensionless groups, coefficients, etc., were calculated on spreadsheet software. Heat-transfer coefficients (film coefficients) were calculated from Newton’s Law of Cooling, where the heat rate (W or Btu/h) was calculated from the temperature rise and the flow rate of water, and an arithmetic average temperature driving force was used. (The logmean temperature driving force was also used for comparison.) All dimensionless groups (Nusselt number, Prandtl number, Reynolds number, and Grashof number) were based on the arithmetic average (bulk fluid) temperature. Other bases were also used for the Grashof number, since the Jackson and Fewster correlation requires it by using its own definition. Normally the Grashof number based on properties evaluated at the film temperature, defined as the arithmetic mean of average wall temperature and bulk average fluid temperature, is preferred by the author. As shown in Table 1 of Joye (1995), these different Grashof number bases differ essentially by a constant, so the
Figure 3. Downflow heating data compared with the correlations of Jackson et al. and Swanson and Catton. Shell-side steam pressure ) 30.2 kPa absolute (GrbD ) 2.8 × 106).
choice is arbitrary. However, many investigators prefer the same temperature basis for all dimensionless groups, and thus, the bulk average temperature basis is so used here. Results and Discussion Figures 2 and 3 show how the data of the present investigation compare with the correlations. This comparison is typical of that in a previous work (Joye, 1995), where the constancy of Gr is about the same for each individual run (it varied from run to run from about 8% standard error to about 22%, typically). In the highest Gr run, the standard error was about 36%, because we had a difficult time controlling the large rate of heat transfer. (Standard error is defined here as one standard deviation divided by the mean.) From these data and those of others, forced convection correlations alone will be in significant error for vertical, internal flows with heat transfer if Re is less than about 12 000. For Reynolds numbers above 12 000, the “0.023” equation predicts a bit low for the data, but this is typical, and others have used slightly different coefficients, e.g., 0.027, to get a better fit.
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is described by the equation below:
Recrit ) C*(GrbD[L/D]3)1/3/Pr2/3 ) C*GrbL1/3/Pr2/3 (14)
Figure 4. Reynolds number criterion for the CWT boundary condition used to select the correlation or asymptote equation for the Nusselt number calculation.
From Figures 2 and 3, one can see that both the Jackson et al. and the Swanson and Catton correlations fit the data fairly well, except at low Reynolds numbers (below about 4000). The Churchill method from the Heat Transfer Design Handbook, section 2.5.10(c) (1983), fits the data well down to Reynolds numbers of about 7000sthe nominally “turbulent” region for which this equation was designed. Its predictions are just slightly below the other two. The line is not shown in the figures, because the overlap with the lower correlation is quite close. The first two correlations were developed with uniform heat flux boundary condition in mind; in the Swanson and Catton theory, this is specific. The data in Figures 2 and 3 show that the boundary condition, or wall condition, does not seem to matter for most of the Reynolds number range. Where it clearly does matter is in the low Reynolds number regime, below about 4000. The data for the constant wall temperature boundary condition follow the asymptote in this region, as they should. The correlations take no notice of the asymptote shown in the figures. The Churchill method was developed for CWT conditions but does not show the asymptote, because it was designed as a turbulent flow model and is not valid for low Re conditions. The other two correlations were developed for UHF conditions, which would not show this asymptote. Under such conditions, the driving force temperature difference takes a value consistent with the heat flux, which is constant. In order to transfer heat to the fluid whose temperature is rising, the wall temperature must also rise; therefore, the outlet temperature can never approach the wall temperature under the UHF boundary condition. The heat-transfer coefficient is determined by flow conditions, which then fixes the Twall for UHF conditions. It is possible that the Nusselt number for natural convection, e.g., eq 10, may be an appropriate asymptote for this situation, as implied by Churchill (1983), but this has yet to be demonstrated conclusively. Figure 4 shows the critical Reynolds number, Recrit, as a function of mixed convection parameters. This was developed by finding the Reynolds number at the intersection of the correlation line and the asymptote (the correlation lines give close to identical intersections; an average value was used here). The line in Figure 4
which can be derived from eqs 1 and 12, using the Sieder-Tate formulation without viscosity correction and with Pr0.5 for the forced flow term in eq 1, ignoring all but the Grashof number term in eq 1 and algebraically solving for the Reynolds number of intersection. From the intersections of asymptote and correlation lines at GrbD ) 1.6 × 106 to 6.0 × 107, the constant C* ) 0.36 ( 4% standard error (data from Joye (1995) and Wojnovich (1995)). If the film property Grashof number is used (either GrfD or GrfL), the constant becomes Cf* ) 0.24. This derived from the average differences between the two Grashof number temperature bases from Joye (1995). One can see from Figure 4 that the intersections of the asymptote with experimental data fall uniformly below the line of intersection derived above. This is because the data approach the asymptote in a curved fashion, gradually, and generally lie somewhat below the correlations near the asymptote region as in Figures 2 and 3. Using eq 14 will be conservative. Recrit gives the value of the Reynolds number where the highest heat-transfer enhancement, defined by Nu/ Nuforced flow, occurs. This may be the optimum operating point from an economics standpoint, because it gives the best enhancement at a low pressure drop. Further studies are needed to verify this, however. The criterion of eq 14 can be used for design in the following way. Once the inlet and outlet temperatures of the fluid are known, the bulk average temperature can be calculated. The wall temperature is needed for the Grashof number calculation. This could be known, or estimated, using the steam temperature with or without an estimate for the temperature drop of the condensate film. The dimensions of the tube are used with the Prandtl number to calculate Recrit according to eq 14. If the design Re is above this, then any of the three correlations can be used, as is, regardless of the boundary condition. If the Reynolds number is below Recrit, then the asymptote (eqs 12 or 13) should be used if the boundary condition is CWT. If the boundary condition is UHF (electrical, nuclear, or constant ∆T along the tube), either the Jackson and Fewster correlation or the Swanson and Catton correlation can be used. In a two-fluid exchange with vertical flow, the fluids can be introduced in two ways, countercurrent or cocurrent. If the operation is countercurrent (which is often favored), the wall situation is closer to a UHF boundary condition; it is exactly a UHF boundary condition only if the temperature driving forces are the same at the entrance and exit. If the flow is cocurrent (which sometimes it must be due to other constraints), the boundary condition is very close to a CWT boundary condition. In crossflow and shell-and-tube exchangers, where the temperature profiles are mixed countercurrent and cocurrent, or crosscurrent, the constant wall temperature condition would be used in an average sense. Occasionally, it may happen that the boundary condition is clearly neithersthen for Reynolds numbers below Recrit, one must estimate between the asymptote line and the correlation of choice. Conclusions 1. The correlations of Jackson and Fewster and Swanson and Catton show a good fit of the data with
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respect to Reynolds numbers from the asymptote region right into the forced convection turbulent region but do not fit the asymptote region at all. The Churchill method also fits the data well at Reynolds numbers greater than about 7000. 2. All three correlations fit the data well for GrbD variations ranging 1 1/2 orders of magnitude from about 2 × 106 to 6 × 107. 3. Use of the Recrit criterion allows the designer to appropriately use the two correlations for opposing flow, vertical, mixed convection heat transfer, depending upon the Reynolds number, the Grashof number, and the boundary condition at the wall. Acknowledgment I thank the following students who assisted in the data collection and reduction: Matt Wojnovich, Chris Lynch, and Jim Harten. Nomenclature Cp ) fluid heat capacity, kJ/(kg K) D ) tube diameter, m g ) gravitational acceleration, m/s2 GrbD ) Grashof number based on properties evaluated at the average bulk fluid temperature and tube diameter, F2gD3β∆T/µ2 Gz ) Graetz number, wCp/kL h ) film heat-transfer coefficient, W/(m2 K) k ) fluid thermal conductivity, W/(m K) L ) heated length of tube, m Nu ) Nusselt number, hD/k Pr ) Prandtl number, µCp/k Re ) Reynolds number, DvF/µ T ) temperature, K ∆T ) temperature difference (K) of average wall to average bulk fluid for the Grashof number, unless otherwise noted v ) average fluid velocity, m/s w ) mass flow rate, kg/s Greek Symbols β ) volume expansivity, 1/K µ ) viscosity, Pa s F ) density, kg/m3 Subscripts b ) fluid properties evaluated at the bulk average temperature D ) (Grashof number) evaluated with tube diameter as the characteristic length f ) fluid properties evaluated at the average film temperature ((average wall temperature + bulk average fluid temperature)/2)
L ) (Grashof number) evaluated with tube length as the characteristic length
Literature Cited Churchill, S. W. Combined Free and Forced Convection in Channels. In Heat Transfer Design Handbook; Schlu¨nder, E., Ed.; Hemisphere: New York, 1983; Section 2.5.10. Herbert, L. S.; Sterns, U. J. Heat Transfer in Vertical Tubess Interaction of Forced and Free Convection. Chem. Eng. J. 1972, 4, 46-52. Holman, J. P. Heat Transfer, 7th ed.; McGraw-Hill: New York, 1990. Ianello, V.; Suh, K. Y.; Todreas, N. E. Mixed Convection Friction Factors and Nusselt Numbers in Vertical Annular and Subchannel Geometries. Int. J. Heat Mass Transfer 1988, 31, 2175-2189. Jackson, J. D.; Cotton, M. A.; Axcell, B. P. Studies of Mixed Convection in Vertical Tubes: A Review. Int. J. Heat Fluid Flow 1989, 10 (1), 2-15. Joye, D. D. Comparison of Correlations and Experiment in Opposing Flow, Mixed Convection Heat Transfer in a Vertical Tube with Grashof Number Variation. Int. J. Heat Mass Transfer 1995, in press. Joye, D. D.; Bushinsky, J. P.; Saylor, P. E. Mixed Convection Heat Transfer at High Grashof Number in a Vertical Tube. Ind. Eng. Chem. Res. 1989, 28 (12), 1899-1903. 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. Inst. Chem. Eng. 1942, 38, 493-530. McAdams, W. H. Heat Transmission, 3rd ed.; McGraw-Hill: New York, 1954; pp 229-235. Poskas, P.; Adomaitis, J. E.; Vilemas, J.; Bartkus, G. Development of Turbulent Heat Transfer over the Length of Vertical Flat Channel under a strong Influence of Buoyancy. In Proceedings 10th International Heat Transfer Conference, Brighton, UK; Hewitt, G. F., Ed.; IChemE: Rugby, UK, 1994; Vol. 5, Paper 12-NM-26, pp 555-560. Swanson, L. W.; Catton, I. Surface Renewal Theory for Turbulent Mixed Convection in Vertical Ducts. Int. 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. Wojnovich, M. J. Mixed Convection Heat Transfer in a Vertical Tube with Grashof Number Variation and Heated Inlet Fluid. M.S. Thesis, Department of Chemical Engineering, Villanova University, Villanova, PA, 1995.
Received for review July 25, 1995 Accepted February 14, 1996X IE950464J
X Abstract published in Advance ACS Abstracts, June 1, 1996.
