Ind. Eng. Chem. Res. 1988,27, 149-153 Miller, R. E.; Stein, S. E. Prepr.-Am. Chem. SOC.,Diu. Fuel Chem. 1979, 24,271. Modell, M. “Reforming of Glucose and Wood at the Critical Conditions of Water”. Technical Report, 1977; ASME Intersociety Conference on Environmental Systems, New York. Moore, J. W.; Pearson, R. G. Kinetics and Mechanism, 3rd ed.; Wiley: New York, 1981. Paulaitis, M. E., Penninger, J. M. L., Gray, R. D., Jr., Davidson, P., Eds. Chemical Engineering at Supercritical Fluid Conditions; Ann Arbor Science Ann Arbor, MI, 1983. Peng, D.-Y.;Robinson, D. B. Ind. Eng. Chem. Fundam. 1976,15,59. Poutsma, M. L.; Dyer, C. W. J . Org. Chem. 1982, 47,4903. Sato, Y.; Yamakawa, T. Znd. Eng. Chem. Fundam. 1985,24(1), 12.
149
Schlosberg, R. H.; Ashe, T. R.; Pancirov, R. J.; Donaldson, M. Fuel 1981a, 60, 155. Schlosberg, R. H.; Davis, W. H., Jr.; Ashe, T. R. Fuel 1981b, 60,201. Simmons, M. B.; Klein, M. T. Ind. Eng. Chem. Fundam. 1985,24(1), 55. Thies, M. C. Ph.D. Thesis, University of Delaware, Newark, 1985. Townsend, S. H.; Klein, M. T. Fuel 1985, 64, 635. Tsekhanskaya, Y. V.; Iomtev, M. B.; Mushinka, E. V. Russ. J. Phys. Chem. 1964, 38, 1173.
Received for review April 20, 1987 Revised manuscript received September 23, 1987 Accepted September 29, 1987
GENERAL RESEARCH Some Studies on Heat Transfer in Diverging-Converging Geometries C. M. Narayanan*+and B. C. Bhattacharyyat Department of Chemical Engineering, Regional Engineering College, Durgapur 713209, India, and Department of Chemical Engineering, Indian Institute of Technology, Kharagpur 721302,India
Transport phenomena in diverging-converging geometries have created a lot of interest recently. The present paper highlights the specific advantages of diverging-converging geometries in the augmentation of heat transfer a t the expense of a negligible increase in the pressure drop penalty. The work is confined to Newtonian, incompressible, and steady-state flow through the annular space between a diverging-converging tube and an outer straight column and also to heat transfer a t constant wall temperature. The work also highlights the dependence of heat-transfer augmentation on the design parameters such as the angle of constriction and segment length. The analysis of transport phenomena (momentum, heat and mass transport) connected with irregular geometries has always generated a lot of interest. Different studies have been made in the past to account for different geometric parameters. However, until very recently, generalized mathematical analyses of velocity, temperature, and concentration distributions in irregular geometries have not been conducted. One of the earliest studies in this connection may be attributed to that of Millsaps and Pohlhausen (19531, who plotted a family of velocity profiles for flow through convergent as well as divergent channels a t different Reynolds numbers on the basis of numerical calculations performed by them on the exact solutions proposed by Jeffery (1915) and Hamel (1916). Among the recent studies, the work of Payatakes et al. (1973) is most noteworthy. They determined the velocity distribution for Newtonian flow through periodically constricted tubes by performing a numerical solution to the flow equations. Flow dynamics for non-Newtonian flow through porous media has been analyzed by Sheffield and Metzner (1976). They have concluded that in such cases the “divergingconverging” character of the flow channels is a very influencing parameter that shall have to be accounted for. Some studies on heat-transfer enhancement in tortuous flow have been reported by Fujita and Hasegawa (1940) and Klepper (1973). Narayanan (1983) and Narayanan and Bhattacharyya (1978) have made elaborate matheRegional Engineering College. t Indian Institute of Technology.
matical analysis of momentum and heat transport characteristics in irregular geometries with special reference to diverging-converging tubes. Navier-Stokes equation for steady-state, two-dimensional flow (in the annular space between a diverging-converging tube and an outer straight column) in terms of dimensionless stream function (Bird et al., 1960) has been solved numerically by using the line successive overrelaxation method. The values of velocity components so obtained have been substituted in the energy equation so as to derive the temperature profile (also numerically). The concerned equations are
where
E*4 = E*2(E*2) E*2’---a 2
1 a +a2
&*2 r* ar* &*2 ReM = Lvop//.l $* = $ / V J 2 ff* = [Re~Prl-’ V,* = V,/Vo VI* = V I / V o r* = r / L z* = z / L T* = T/To Assuming “no slip” at the solid walls and constant wall temperature, the boundary conditions have been devel-
0888-5885/88/2627-Ol49$01.50/0 0 1988 American Chemical Society
150 Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988
* n
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0
0
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Figure 2. Friction factor (f,) versus Reynolds number (Re,) plot.
I
I I
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'h
Figure 1. Type of geometry
oped. Velocity distributions at the inlet and outlet of the system (AJ and FE in Figure 1) or the inlet and outlet boundary conditions have been developed as part of the solution by using an iterative procedure. This therefore does not neglect the fact that velocity profiles are welldeveloped at the inlet as well as at the outlet. Details of the numerical algorithm have been given elaborately by Narayanan (1983). The theoretical values have been checked against the experimentally measured values of point velocities and point temperatures, and satisfactory agreement has been reported between the two. It has been shown that even a t the low Reynolds number (Re,) range of 200-1300 under study, the velocity profiles are quite flattened (from the parabolic nature). The curves are flat essentially in the center portion, and the velocity falls sharply to zero a t the walls, thus indicating a boundary layer character. The temperature profiles have also been found to indicate a sluggish increase of temperature from outer wall to the bulk and then a sharp increase to the constricted tube wall (heated wall), thus once again indicating a boundary layer character and the absence of any stagnant thermal layer in the neighborhood of the constricted tube wall. The values of pressure drop (over a given length of constricted tube) have also been computed by using the mathematical model (and verified experimentally), and they have been used to prepare the friction factor (f,) versus Reynolds number (Re,) plot. The plot is given in Figure 2.
Estimation of Heat-Transfer Coefficient Since the mathematical model has been well-established as reported in the above given references, the values of the heat-transfer coefficient h, for the diverging-converging
Figure 3. Experimental setup.
geometry are computed mathematically from the temperature distribution reported by Narayanan (1983) by using
where AT2 = T, - Tb, and AT, = T, - Tb,. T b i and Tb,are the values of average bulk temperature a t cross section 1 (corresponding to minimum width of the annular space) and cross section 2 (corresponding to maximum width of annulus), respectively, and Tb is defined as
The above integration has been performed numerically on an IBM 370 computer using Sympson's rule at cross section 1 and trapezoidal rule at cross section 2. To estimate the heat-transfer coefficient experimentally, the setup used has been shown schematically in Figure 3. The type of geometry treated in the work is also shown separately in Figure 1. The constricted tube C is made of I/,,-in.-thick brass. The cross section at the minimum constriction is 3-in. i.d. and that at the maximum constriction is 2-in. i.d. The total length of the constricted column is 36 in., thereby consisting of three segments of
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 151 12 in. each. The angle of constriction is thus approximately 5" (tan a = l/,,) and is kept constant throughout the length of the column. The constricted column is enclosed in a Perspex shell S of i.d. 4 in. and thickness 1 / 4 in. The tank T1is first filled with the test fluid (say, water). The immersion heaters 11,I,, and I3 are switched on to raise the temperature of water (heating fluid) in the well-insulated bath T2to a high level. The pumps P, and Pz are started, the former pumping the test fluid and the latter the heating fluid. The globe valves Gzand G5are opened, and G2is adjusted for a given flow rate of the test fluid (say 100 L/h) as read on the precalibrated rotameter Rz. The test fluid passes up through the annular space, and hot water is passed through the constricted tube. After sufficient time is given for the system to attain steady state, the inlet and outlet temperatures of the test fluid are recorded by using thermometers Thl and Thz which have f O . l "C accuracy. The average wall temperature of the constricted column is also recorded using precalibrated copper-constantan thermocouples. This measurement is done at six different points on the constricted tube wall so as to ensure that the wall temperature is essentially constant throughout the column. Readings are repeated for other flow rates of 200,300,400,500, and 600 L/h. In each case, the test fluid emerges from the top outlet of the annular space, passes through cooling coils (not shown in the figure), and then enters back into tank T,. As a consequence, the inlet temperature of the test fluid remains essentially constant. The heating fluid leaving the top of the constricted column flows back to bath T,. The bath temperature at the same time is controlled by simmerstats attached to the immersion heaters. The system is thus fully recirculating. Readings are also repeated for other test fluids such as 5%, lo%, and 15% glycerol solutions. The experiment is repeated using a straight tube of diameter D, (thus having the same heat-transfer area per unit length as the constricted tube) in the place of the constricted tube. In this case also, the inlet and outlet temperatures of the test fluid as well as the average outer wall temperature of the inner tube are measured a t different flow rates and also using different test fluids. The values of ha and Re' are then calculated by using the corresponding relations. Since the flow rate of test fluids employed in the experimental study was varied from 200 to 600 L/h, the ranges of Reynolds numbers under study were 380-1900 for water and 300-1700 for 5% glycerol, 200-1300 for 10% glycerol, and 180-1100 for 15% glycerol solutions. Under these conditions, the temperature difference in the fluid between the inlet and outlet of the annulus varied from 13.7 to 8.5 "C for water and 11.5to 7.0 "C for 5% glycerol, 11.0 to 7.0 "C for 10% glycerol, and 10.5 to 6.6 "C for 15% glycerol solutions when the wall temperature of the diverging-converging column was kept constant at 75 "C. As stated earlier, the inlet and outlet temperatures of the test fluids were recorded by using thermometers of *0.1 "C accuracy. The multichannel potentiometric recorder used for recording the output from thermocouples is capable of recording millivolts up to second decimal accuracy. At each flow rate, the experiment was repeated at least thrice to minimize the experimental inaccuracies and also to maintain consistancy in experimental measurements. Observations and Inferences The values of the heat-transfer coefficient (both experimental as well as theoretical) are plotted against Re' as shown in Figures 4. It can be seen that the values of heat-transfer coefficient in constricted geometry are much
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Figure 4. Heat-transfer coefficient (h,) versus Reynolds number (Re') plot.
higher than those for a straight tube of the same heattransfer area per unit length at the same Reynolds number. A t the same time, it can be seen from Figure 2 that the value of the friction factor and therefore the pressure drop in diverging-converging geometry is only slightly above that in the equivalent straight tube at the same Reynolds number. The increase in pressure drop can thus be considered to be negligibly small as compared to the enhancement in the heat-transfer coefficient provided by the constricted geometry. For example, at a flow rate of 100 L/h and when water is used as the test fluid (Re' = 381.34 and Re, = 213.675), the enhancement in ha is 2.1999 times (or approximately 220%), whereas the increase in the value of friction factor f,(which is proportional to the pressure drop at a given flow rate) is only 1.04166 times or approximately 104.2%. This, therefore, indicates that the constricted geometry provides appreciable enhancement in the heat-transfer coefficient as compared with the straight geometry, with negligible increase in the pressure drop penalty. The phenomenon is evidenced both theoretically as well as experimentally. The percentage enhancement in h, (defined as the ratio of the value of h, for constricted geometry to that for the equivalent straight tube geometry multiplied by 100) has also been plotted against Re'as shown in Figure 5. It can be seen that the percentage of enhancement increases with the increase in Reynolds number for both water and the glycerol solutions. However, the enhancement at the same Reynolds number is less for water than for the glycerol
152 Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988
LL-.A-I-__L---l___-i--~ 200 600 800 ZOO 2,000
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-
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f" test fluid water 5% glycerol 10% glycerol 15% glycerol
tan a = l I c 0.11 0.13 0.176 0.188
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0.078 0.103 0.141 0.159
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-
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21
solutions. For glycerol solutions, it further increases with the increase in concentration from 5% to 15%. As reported by Narayanan (1983) and Narayanan and Bhattacharyya (1978), the velocity as well as the temperature distribution in diverging-converging geometry exhibits a boundary layer character. Therefore, formation of a stagnant fluid film at the walls is absent, or if formed its thickness is too low. This reduces the resistance to heat transport, and thereby the heat-transfer coefficient gets enhanced. Since the resistance to heat transport has been reduced in this way, the overall pressure drop in the system does not increase appreciably and remains reasonably low. Moreover, the converging-diverging character of the geometry may cause a type of pressure recovery (analagous to that in a venturi tube) which may also be responsible for keeping the pressure drop penalty within reasonable limits, though the transfer coefficient has been appreciably enhanced. For low viscous fluids like water, the thickness of such a stagnant film at the wall would be low in the straight tube itself, but with more viscous fluids like the glycerol solutions, the film would be much thicker in a straight tube. Hence, in the constricted geometry when such a film is made absent or its thickness is reduced considerably, the percentage of enhancement achieved in the transfer coefficient would be much more appreciable for the viscous fluids like the glycerol solutions than that for water. To study the effect of angle of constriction on heattransfer coefficient, the values of h, (theoretically computed) have been plotted against tan a , at the same Reynolds number, Re', as shown in Figure 6. It can be seen that the value of h, decreases as tan a decreases until it becomes equal to that for the straight tube when a = 0. However, it can also be seen from Table I that when tan a becomes much higher, as in the case of tan a = 1/6, the pressure drop in constricted geometry also becomes substantially larger than that in the equivalent straight tube. This leads to the conclusion that though h, increases with an increase in CY, an optimum value of a could be chosen at which the heat-transfer coefficient is appreciably large, but at the same time the pressure drop penalty is kept within reasonable limits. To study the effect of segment length L on h, at a given values of h, (theoretangle of constriction (tan a =
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ically computed) are plotted against L, keeping the effective diameter of annular space De constant, as shown in Figure 7 . It can be seen that for a given angle of constriction and for a given effective diameter of the annulus, the decrease in the segment length or increase in the periodicity of constriction increases the heat-transfer coefficient at any given value of Re'. This could be due to the fact that as the periodicity of constriction increases, the path of the fluid through the system becomes much more tortuous, thus providing more intimate contact among fluid elements. Consequently, the transfer coefficient gets enhanced. Also, as indicated earlier, the converging-diverging nature of the geometry causes periodical changes in the direction of flow of the fluid as well as changes in average velocity of fluid from section to section.
Ind. Eng. Chem. Res. 1988,27, 153-156
This effect becomes much more pronounced when the periodicity of constriction is increased, and as a result the thermal boundary layer resistance is further reduced and the heat-transfer coefficient gets much more enhanced. In conclusion, it may be stated here that divergingconverging geometries exhibit highly attractive features toward the augmentation of heat-transfer efficiency. They can be recommended for the improved design of heattransfer equipment such as heat exchangers, condensers, evaporators, etc. The increased cost of fabrication of such geometries should necessarily be compensated by the excellent increase in the transfer efficiency attainable at the same operating cost (since the corresponding increase in pressure drop is negligibly small). Such variable-area heat exchangers can also be specially attractive to systems incorporating low temperature differences such as the OTEC (Ocean Thermal Energy Conversion) power plants (Narayanan and Bhattacharyya, 1985). This field undoubtedly invites more elaborate work.
Nomenclature A = heat-transfer area, cm2 C, = specific heat, cal/(g.K) D = inside diameter of outer straight column = 2R, cm D1 = minimum inside diameter of constricted tube = 2R1, cm D2 = maximum inside diameter of constricted tube = 2Rz, cm De = effective diameter of the annulus = (D2- D:)/Ds, cm D, = surface diameter of constricted tube = (1 + cot2 a)1/2 (02 - D12)/2L, cm D, = volumetric diameter of constricted tube = [(D; - D?)/3L tan aI1I2,cm f,= friction factor for constricted geometry = (-AI’)@ Dv)/ ~ P Vm2 L h, = heat-transfer coefficient, call (cm2.s.K) K = thermal conductivity, cal/(cm.s.K) L = length of each segment of constricted tube, cm (characteristic length) Pr = Prandtl number = C,p/K Q = volumetric flow rate, cm3/s Re’ = surface Reynolds number = 4Qp/rDep ReM = modified Reynolds number = LV,,p/p Re, = volumetric Reynolds number = 4 Q p / r ( D + Dv)p
153
rw(z) = distance from the axis to the wall of the constricted tube at any z, cm r* = dimensionless radial coordinate = r/L
T * = dimensionless fluid temperature at any point = TIT, Tb= average bulk temperature at any cross section, K T,= wall temperature of constricted tube, K To = average temperature at the entrance constriction, K (characteristic temperature) V , = average velocity based on D, = 4Q/7(D2- D:), cm/s V , = average velocity at the entrance constriction, cm/s (characteristic velocity) V,* = dimensionless radial component of fluid velocity = vr/ v o
V,* = dimensionless axial component of fluid velocity = V,/ V, z* = dimensionless axial coordinate = z/L Greek Symbols a = angle of constriction a’ = thermal diffusivity = K / p C , a* = a’/VJ = l/(ReMPr) p = fluid density, g/cm3
+* = dimensionless stream function = +/ V J 2
p
= viscosity, P
Literature Cited Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. Fujita, Y.; Hasegawa, S. Bull. JSME 1970, 13, 697. Hamel, G. Jahresber. L. Dt. Mathematiker-Vereinigung. 1916,34, 25 Jeffery, G. B. Phil. Mag. 1915, 29, 455. Klepper, 0. H. AIChE Symp. Ser. 1973, 69, 131. Millsaps, K.; Pohlhausen, K. J. Aerosol Sci. 1953,20, 187. Narayanan, C. M. ”Momentum and Heat Transfer Studies in Irregular Geometry”. PbD. Thesis, Indian Institute of Technology, Kharagpur, 1983. Narayanan, C. M.; Bhattacharyya, B. C. Proc. Annu. Sess. IIChE, 31st 1978, 24.
Narayanan, C. M.; Bhattacharyya, B. C. Reg. J . Energy Heat Mass Transfer, 1985, 7, 39. Payatakes, A. C.; Tien, C.; Turian, R. M. AIChE J . 1973, 19, 67. Sheffield, R. E.; Metzner, A. B. AIChE J . 1976, 22, 736. Received for review October 28, 1985 Accepted September 21, 1987
Dissolving Pulps from Wheat Straw by Soda-Anthraquinone Pulping Mohamed A. Abou-State,*+Ahmed M. El-Masry,$and Naglaa Y. S. Mostafat Department of Chemistry, Faculty of Science, Cairo University, Giza, A.R. Egypt, a n d Department of Chemistry, Faculty of Science, Zagazig University, A.R. Egypt
Dissolving pulps are obtained by subjecting prehydrolyzed Egyptian wheat straw to soda-anthraquinone pulping. The pulps obtained are satisfactorily bleached by the CEH sequence, while in the absence of anthraquinone an additional chlorite step should be applied to raise the degree of whiteness. A very important effect of anthraquinone takes place in the fine structure of the pulp whereby it increases the affinities toward water and alkali, lowers the crystallinity, and results in considerable improvement in the reactivity toward xanthation. Soda-anthraquinone pulps with suitable a-cellulose content can be obtained by using the appropriate concentration of acid in prehydrolysis to hydrolyze a greater amount of the lower molecular weight carbohydrates before the soda-anthraquinone pulping. Extensive studies have been carried out on the effect of anthraquinone on pulp and paper from pulpwood. The presence of anthraquinone during alkaline pulping of wood chips results in highly beneficial effects on the properties of pulp and paper obtained. It leads to better deliginifi+Cairo University. t Zagazig University.
cation (Hassan et al., 1981; Germer et al., 1983; Fossum et al., 1980; Sjoeholm and Wikbald, 1980; Ivanova et al., 1980), increased carbohydrate stabilization (Germer et al., 1983), higher yield (Hassan et al., 1981; Fossum et al., 1980; Ivanova et al., 1980; Virkola, 1981; Goel et al., 1980; Kleppe, 19811, better strength properties (Hassan et al., 1981; Germer and Caluzin, 1982; Cameron et al., 1982; Bogomolov et al., 1981; Li et al., 1984), lower alkali consumption, and decreased cooking chemical charge (Germer and Ca0 1988 American Chemical Society
