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Ind. Eng. Chem. Res. 1991, 30, 784-788
Hydrostatic Correction and Pressure Drop Measurement in Mixed Convection Heat Transfer in a Vertical Tube Paul E. Saylor and Donald D. Joye* Department of Chemical Engineering, Villanova University, Villanova, Pennsylvania 19085
In vertical, internal, mixed convection flows at high Grashof numbers and low Reynolds numbers, large radial and axial temperature profiles result in property variations which can lead to significant error in the hydrostatic correction applied to pressure drop measurements. The improvement developed here consists of integrating an equation for the mixed mean temperature as a function of axial distance in order to obtain a better estimate for the volume-average temperature, from which a more accurate density for use in the hydrostatic correction can be determined. The pressure drop calculated in constant wall temperature (CWT) situations using the improved correction shows, relative to purely forced flow, much higher flow resistance for aiding flow and a negative region for opposing flow. The technique is useful for the mixed convection region, unnecessary in the turbulent flow region, and inapplicable in the asymptote region of vertical, internal flows with mixed convection heat transfer.
Introduction In mixed convection heat transfer, the natural convection forces (directed vertically) are the same order of magnitude as the forced flow. Therefore, the situation resulting from the interaction of buoyancy-driven flows with the forced flow can have a profound effect on the velocity profile, the heat-transfer coefficient, and the pressure drop. Vertical, internal mixed convection flows are generally divided into opposing flow, where the buoyancy currents and the forced flow are in opposite directions, and aiding flow, where the buoyancy and forced flows are in the same direction. Heating in upflow and cooling in downflow are therefore aiding flows, while cooling in upflow and heating in downflow are opposing flows. Heating in upflow (aiding) and heating in downflow (opposing) are the situations discussed in this work. In these kinds of flow situations with heat transfer the pressure drop is difficult to determine because of the complex temperature field which depends on both radial and axial distances. This is especially true in mixed convection situations at low Reynolds number, where thermal property changes (viscosity and density in particular) can have a very large effect on the measurements. Pressure measurement, usually by wall taps, is complicated by small values at low flow rates. Computing the pressure drop also involves a small difference between two much larger numbers, regardless of the devices used; hence accurate values are inherently difficult to obtain. Also, when the pressure drop driving the flow in vertical situations is calculated, the measured pressure drop must be corrected for hydrostatic head. The arithmetic average temperature between inlet and outlet has been used to calculate fluid properties from which various corrections were made, including the one for hydrostatic head in the pressure drop calculation. For flow situations that can be truly termed umixednconvection, we show this to be inadequate and develop an improved hydrostatic correction for vertical, internal mixed convection flows. Previous Literature Pressure drop in vertical flows with mixed convection heat transfer has been studied by very few investigators. Anomalous behavior for pressure drop at low Reynolds number and high Grashof number has been shown by
* Author to whom correspondence should be addressed.
Marcucci and Joye (1985). The anomaly appeared as large positive (for downflow heating) and negative (for upflow heating) pressure drop at low Reynolds number when standard hydrostatic corrections were used. Earlier investigators who have reported pressure drop measurements, such as Kemeny and Somers (1962), Bishop et al. (1980),Carr et al. (1973),and Martinelli et al. (1942), did not show this pressure drop anomaly. Negative pressure drops were mentioned by Kemeny and Somers (1962), but these were attributed to experimental error, because of the scatter in the data and the difficulty in measuring small pressure drops. Martinelli et al. (1942) also showed some negative pressure drop points but attributed this to a possibly improper hydrostatic correction. In two recent publications (Morton et al., 1989; Lavine, et al., 1989),negative pressure drops are shown to appear from the governing equations (Morton et al.) and may arise from a cross-stream component of gravity (Lavine et al.) in tilted tubes. Both imply the possibility of an overall negative pressure drop appearing in reverse flow situations (low Reynolds number and high Grashof number). A local negative pressure gradient is thought to drive the reverse flow. The possibility of pressure gradients in opposing directions across a given cross-section of tube is discussed in both papers.
Experimental Method A typical jacketed, vertical-tube apparatus was used in this work to generate experimental data. Water flowed in the central tube, and steam condensed in the annular jacket. The test section is shown schematically in Figure 1. A water-over-carbontetrachloride manometer was used for the pressure drop measurements. All temperature measurements were made with copper-constantan thermocouples. Five equally spaced thermocouples consisting of constantan wire were silver-soldered into grooves cut into the copper tube wall. Wall temperatures were used for film coefficient and Nusselt number calculations. Well-mixed outlet temperatures were obtained by the use of elbows in the line. The unit was insulated between inlet and outlet and included inlet and outlet unheated sections corresponding to about 10 diameters. The length-to-diameter ratio (L/D) of the heated section was 49.6, with pressure taps spaced slightly larger (2 cm on each side). Flow rates were measured by rotameters. The water flow was once through from a supply tank and was kept under pressure (210 kPa or 30 psig at the inlet at low flow rates).
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It was about half that value at the high flow rates. In all cases this was sufficient to prevent degassing or boiling from occurring. Constant steam pressure gave constant wall temperature conditions and an essentially constant Grashof number with properties based on the film temperature. Another thermocouple was used to measure ambient temperature in the manometer lines and the horizontal manometer line cooling sections, which were metal tubes about 30 cm long placed horizontally between the respective tap (upper or lower) on the test section and the fitting connection to the flexible, plastic manometer lines. These tubes, shown in Figure 1, minimized heat migration effects into the vertical, ambient-temperature legs of the manometer and helped to avoid melting the plastic tube lines. Thermocouple measurements inside the manometer lines showed ambient temperature up to about 2 cm from the fitting, where the temperature increased slightly. However, both manometer lines ran down, initially, from the junction, and whatever temperature effects there might have been canceled. Thus no significant temperature migration into the vertical manometer lines occurred. Pressure Drop Calculation The equation for calculating the measured pressure drop, including the hydrostatic correction, is given by Martinelli et al. (1942)and Kemeny and Somers (1962). For the upflow heating situation in Figure 1, it takes the form -AP = PI - P? = - b M - PW,AMB)gh + (PW,AMB - PW,INT)gX (1) where P is pressure, p is density, X is the distance between pressure taps, h is the manometer reading (left leg - right leg in Figure 11, subscript M refers to manometer fluid, subscript W is water, INT refers to water flowing in the tube, AMB refers to ambient conditions outside the tube, and p is the average density of water flowing inside the tube. The first term is called the manometer hp; the second term is called the leg correction or hydrostatic correction, which depends on knowing the correct average density of fluid flowing in the tube. The quantity PI - Pz is the pressure drop causing flow the pressure drop corrected for hydrostatic head in this case), which should always be a positive number for the upflow case shown.
Figure 2. Pressure drop data for vertical upflow and downflow heating including anomaly at low flow rates.
Results Using Standard Hydrostatic Correction The pressure drop data calculated by using the standard hydrostatic correction are shown in Figure 2, where the anomaly appears. The standard hydrostatic correction consists of using the density at the arithmetic average of inlet and outlet temperatures. The data points are shown in comparison to the pressure drop expected on the basis of isothermal fluid at the bulk average temperature (the solid, curved line). A viscosity correction based on wall temperature accounts for property variation. The expected pressure drop is the same regardless of the flow direction. The friction factor was calculated from Churchill's equation, which covers all flow regimes with a single equation, as recommended in Perry's Chemical Fngineers' Handbook (Sakiadis, 1984);the original reference is Churchill (1977). Reynolds numbers corresponding to the flow rates are shown on the plot. The slight difference in Reynolds numbers for upflow and downflow at the same volume throughput is due to the difference in inlet conditions and bulk average viscosity. At Reynolds numbers below about 5000, a deviation from the expected curve occurs. This happens at somewhat higher flow rates for downflow than for upflow. At zero flow the pressure drop should be zero, but the data show two intercepts whose absolute value appears to be the same. This value corresponds to the difference in pressure between the water in the manometer line at ambient temperature and the water in the tube at essentially the wall temperature. A hydrostatic correction based on the arithmetic average temperature is clearly inadequate in this region, and an improved correction is needed. Improved Hydrostatic Correction The true hydrostatic correction depends on the volume-average density. Since density depends on temperature, temperature profiles could be used to determine this. In the appendix material of Martinelli et al. (1942) an equation for calculating the mixed mean temperature a t any axial location in the tube is presented for upflow heating in constant wall temperature (CWT) cases. The equation is an approximate relationship that described their data well and is reproduced as follows: Tb,z,L = Tw- (T,- To) ex~[-(rr(Nu)/F~(Gz))(z/L)~'~l (2)
Equation 2 can be integrated along the normalized axial
786 Ind. Eng. Chem. Res., Vol. 30,No. 4, 1991
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Measurement error can play a significant role in any experimental investigation. The manometer scale in our equipment could be read to f0.05 in. (the scale read in tenths of an inch); therefore a maximum error in the manometer measurement would translate to f0.1 in. or f1.5 mm of water (pressure, not manometer reading) at maximum density. This error has to do with reading the manometer and is independent of Reynolds number. The error in the average temperature used in the leg correction calculation is difficult to judge, but say fl OC is reasonable; this translates to fl mm of water at maximum density. This leg correction error depends on Reynolds number and goes to zero (essentially) for Re greater than about 1OOOO. This is where the improved procedure gives the same average temperature as the arithmetic mean of inlet and outlet. Thus the worst case errors are on the order of f2.5 mm of water at maximum density, and they occur at the lowest Reynolds numbers. Errors at high Reynolds numbers would be f1.5 mm of water. Corresponding error bars are shown in Figure 3. Other measurement errors are insignificant compared to these. Negative Pressure Drop in Downflow Heating. In downflow heating the velocity profile distortion theory shows that the velocity gradient can become zero at the wall, or even negative, and thus predicts backflow in the boundary zone near the wall. This would eventually result in a negative pressure drop if substantial back motion existed. The same situation is also possible under CWT conditions and is shown by the data in Figure 3. Recent literature also shows evidence of both flow reversal and negative pressure drop under certain conditions. Lavine et al. (1989)report flow visualization studies of reversal situations in angled tubes but do not give measuremetts of pressure drop. Swanson and Catton (1987) also show flow visualization in a vertical duct. Morton et al. (1989)show reversals and negative pressure drop from numerical studies and velocity and temperature profiles from experiments, but no pressure drop measurements, Yao (1987)in a series of papers gives arguments for flow reversal possibilities and impossibilities from linear stability analyses, but does not discuss negative pressure drops. What does the negative pressure drop mean? The negative pressure drop appears to be due to a local phenomenon at and about the entrance region. We believe it relates to the backflow velocity and negative velocity gradient at the wall. Thus pressure drop becomes negative. However, the backflow situation is also unstable, so turbulent eddies are likely. These turbulent eddies may coexist with the backflow rather than eliminating it. At lower Grashof number the buoyancy forces may not be strong enough to generate backflow against the forced flow direction. Increased Pressure Drop in Upflow Heating. Before the improved hydrostatic correction for the volume average density, the upflow heating pressure drop showed negative values which are known to be incorrect. After the improved hydrostatic correction was made, the curves changes relative position, and upflow heating pressure drops were then higher and positive. This fits qualitatively with the velocity profile distortion theory in UHF cases. For CWT conditions the theory is not well developed, but Figure 3 shows pressure drop can be several orders of magnitude higher than what might be expected.
Range of Applicability for the Improved Procedure The range of applicability for the improved hydrostatic correction for the pressure drop in vertical, internal mixed
Ind. Eng. Chem. Res., Vol. 30,No. 4, 1991 787 ,023R e o ' 8 d
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density for the hydrostatic correction term in the pressure drop equation. This procedure should be used for all genuinely "mixed" convection situations, but is not necessary at the higher Reynolds number end of the mixed convection region defined by heat-transfer results. It does not apply in the asymptote region, because the axial temperature profiles become too distorted for the equation of Martinelli et al. In these cases the wall temperature could be used as the average fluid temperature. 2. The hydrostatic correction may be calculated using the correct average temperature of fluid in the tube. The density profile need not be determined first. 3. The pressure drop in the lower Reynolds number region of vertical, aiding flow, mixed convection heat transfer under.constant wall temperature conditions can be orders of magnitude higher than those expected on the basis of forced flow considerations alone. 4. There is a negative pressure drop in our data for downflow heating (opposing flow) in the mixed convection zone at low Reynolds numbers and high Grashof numbers. This is an evidence for and a consequence of reverse flow in the tube.
h
Figure 4. Range of applicability of the improved procedure relative
Nomenclature
to heat-transfer results.
A, = cross-sectional area of tube (m2) C, = fluid heat capacity (kJ/(kg.K)) D = tube diameter (m) f = Fanning friction factor, dimensionless F1 = Martinelli function, tabulated g = gravitational acceleration (m/s2) GrfD= Grashof number based on film temperature and tube diameter, p2D3gflAT/p2 Gz = Graetz number, mC,/kL h = film heat-transfer coefficient (W/(m2-K)) k = fluid thermal conductivity (W/(mk)) L = heated length of tube (mm) m = mass flow rate, (kg/s) N u = Nusselt number, h D / k P = pressure (kPa, or mm of water) AP = pressure drop (mm of water) Q = volumetric flow rate (m3/s) Pr = Prandtl number, pC / k Re = Reynolds number, L f u p / p sp gr = specific gravity, dimensionless T = temperature (K) AT = temperature difference (K) T b = bulk average temperature (radial mixed mean) or mixing
convection flows can be developed from the corresponding heat-transfer results, shown in Figure 4. These results are discussed more fully elsewhere (Joye et al., 1989);we show them here to point out the asymptote region and the turbulent flow regions; the mixed convection region is that area in between. The Grashof number of these experiments is 1 X 108 based on film temperature properties and 5 X lo7 based on bulk temperature properties. The improved pressure drop procedure should be used in the mixed convection region defined by the bounds shown for the particular set of Grashof number conditions in both aiding and opposing flow situations. For the region to the right of the right-side markings up to the turbulent flow bound, it gives essentially the same results as the standard correction using the arithmetic average temperature. The standard correction is easier to use and therefore would be preferred in turbulent flow situations. The improved hydrostatic correction does not apply in the asymptote region, Le., the region to the left of the left-side markings up to the asymptote bound. In this region the shape of the axial temperature profile approaches a step function and the curvature is too strong for the two-thirds power in Martinelli's equation. The first point at Re = 1220 for downflow heating in Figure 3 fits that description and therefore may not be accurately represented by the improved procedure. In these cases (close to or on the asymptote) the average wall temperature may give a better estimate for the volumeaverage fluid temperature in the tube than the improved procedure. For example, the wall temperature estimate gave a pressure drop of -0.5 mm of water, (shown in Figure 3) instead of about +1.0 mm of water by the improved procedure. Conclusions 1. An improved hydrostatic correction for calculating pressure drop in vertical, internal mixed convection flow situations at low Reynolds numbers has been developed. It consists of obtaining a better estimate of bulk average temperature of fluid in the tube by integrating an equation for mixed mean temperature in vertical tube situations developed by Martinelli et al. (1942). This average temperature rather than the arithmetic average temperature between inlet and outlet is then used to find the average
8
cup temperature at axial location (K) = volume-average fluid temperature in tube (K) Tb,L= mixing cup temperature at tube exit (K) X = axial between pressure taps on tube (mm) Tb,av
Greek Symbols
p = volume expansion coefficient of the fluid (1/K) $v = viscosity ratio, p / p w p = fluid viscosity (cP) p, = fluid viscosity at the wall temperature (cP) p = fluid density (kg/m3) p = fluid density at the bulk average temperature (kg/m3) Subscripts
b = bulk values for fluid properties and temperatures D = (Grashof number) evaluated with the tube diameter as the characteristic length f = fluid properties evaluated at the film temperature, (Twd Tbulk)/2
w = values at the wall of the tube
Literature Cited Bishop, A. A.; Willis, J. M.; Markley, R. A. Effects of Buoyancy on Laminar, Vertical Upward Flow Friction Factors in Cylindrical
Ind. Eng. Chem. Res. 1991, 30, 788-792
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Tubes. Nucl. Eng. Des. 1980,62, 365-369. Carr, A. D.; Connor, M. A.; Buhr, H. 0. Velocity, Temperature and Turbulence Measurements in Air for Pipe Flow with Combined Free and Forced Convection. ASME J. Heat Transfer 1973,95, 445-452.
Churchill, S. W. Friction Factor Equation Spans All Fluid Flow Regimes. Chem. Eng. 1977, 84 (24), 91-92. Hanratty, T. J.; Rosen, E. M.; Kabel, R. L. Effect of Heat Transfer on Flow Field at Low Reynolds Numbers in Vertical Tubes. Ind. Eng. Chem. 1958,50 (5), 815-820. 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. Kemeny, G. A.; Somers, E. V. Combined Free and Forced Convective Flow in Vertical Circular Tubes-Experiments with Water and Oil. ASME J. Heat Transfer 1962,84,339-346. Lavine, A. S.; Kim, M. Y.; Shores, C. N. Flow Reversal in Opposing Mixed Convection Flow in Inclined Pipes. ASME J. Heat Transfer 1989, 111, 114-120. 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. Inst. Chem. Eng. 1942,38,493-530. Morton, B. R.; Ingham, D. B.; Keen, D. J.; Hegg, P. J. Recirculating Combined Convection in Laminar Pipe Flow. ASME J. Heat Transfer 1989, 111, 106-113. Sakiadis, B. Fluid and Particle Mechanics. In Perry's Chemical Engineers' Handbook, 6th ed.; Perry, R. H., Green, D. W., Male ney, J. O., Eds.; McGraw-Hill: New York, 1984; Chapter 5, p 24. Scheele, G. F.; Greene, H. L. Laminar-Turbulent Transition for Nonisothermal Pipe Flow. AIChE J. 1966,12 (4), 737-740. 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. Enhanced Heat Transfer Due to Secondary Flows in Mixed Turbulent Convection. ASME J. Heat Transfer 1987, 109, 943-946. Yao, L. S. Linear Stability Analysis for Opposing Mixed Convection in a Vertical Pipe. Int. J. Heat Mass Transfer 1987,30,810-811. Zeldin, B.; Schmidt, F. W. Developing Flow with Combined Forced-Free Convection in an Isothermal Vertical Tube. ASME J. Heat Transfer 1972,94, 211-233. Received for review May 2, 1990 Accepted October 23, 1990
Reaction between Chitosan and Cellulose on Biodegradable Composite Film Formation Jun Hosokawa,*JMasashi Nishiyama,?Kazutoshi Yoshihara,?Takamasa Kubo,?and Akira Terabe* Government Industrial Research Institute Shikoku, 2-3-3 Hananomiya-cho, Takamatsu 761,Japan, and Aicello Chemical Company Limited, Ishimakihon-machi, Toyohashi 441 -11, J a p a n
It was found that the film formation of composite films is accompanied by cross-linking between cellulose and chitosan, the composite films obtained thus becoming insoluble in water. In this paper the cross-linking reaction is discussed mainly from the point of view of biodegradability, swelling, and strength of the composite film. An increase in the carbonyl groups of cellulose enhances cross-linkings in the composite film formation, while an increase in carboxyl groups does not. Though only trace amounts of carbonyl groups exist in cellulose, they evidently play an important role in cross-linking to chitosan. Further, the free amine form of the primary amino group of chitosan reacts more easily with cellulose than does the amine salt form. These facts suggest that one of the cross-linking structures between cellulose and chitosan in the stage of film formation originates from Schiff base formation between carbonyl groups of cellulose and primary amino groups of chitosan. Plastics made from petroleum have come into wide use throughout the world. With increased applications, the disposal of waste plastics has become a serious problem due to the lack of landfill dump sites and harmful smoke problems when they are burned in incinerators. Therefore, the development of new plastics that can be degraded by microorganisms in soil and seawater has recently been undertaken (Otey et al., 1980; Stepto and Tomka, 1987; Kunioka et al., 1989). We reported a novel and biodegradable composite film derived from chitosan and homogenized cellulose that has a high oxygen-gas barrier capacity and is hydrophilic but insoluble in water (Hosokawa et al., 1990). The period of its biodegradation can be controlled by adjusting the conditions of film formation. We and collaborating companies are now planning to apply this biodegradable material as a binder in nonwoven fabrics, films, sponge type moldings, and so on (Hosokawa, 1990). We wrote in the previous report that the biodegradability of the composite film can be controlled by adjusting the degree of oxidation of the cellulosic material and other Government Industrial Research Institute Shikoku.
* Aicello Chemical Company Limited.
conditions of film formation. We proposed that the biodegradability of the composite film is related to the cross-linking structure between chitosan and homogenized cellulose. However, little is known about the mechanism of reaction between chitosan and cellulose in the step of the composite film formation. We shall, therefore, discuss the mechanism of the cross-linking in this report.
Experimental Section A. Materials. Commercial-grade chitosan (chitosan 10B, Katokichi Co.; origin, prawn; degree of deacetylation, 99.8%; viscosity, 200 CPat 0.5% concentration in 0.5% acetic acid at 25 "C) was used to prepare the composite films used in this study. Fine cellulosic fiber (homogenized cellulose) was obtained by Daicel Chemical Industries, Ltd. The homogenized cellulose named Micro-fibril-cellulose from Daicel was made from bleached pulp and was offered as a 4 % suspension in water. Micro-fibril-cellulosehas a diameter of
