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Artificial boiling heat transfer in the free convection to carbonic acid solution
Experimental Thermal and Fluid Science 35 (2011) 645–652
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Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Artificial boiling heat transfer in the free convection to carbonic acid solution S.A. Alavi Fazel a,⇑, A. Arabi Shamsabadi b,1, M.M. Sarafraz c,2, S.M. Peyghambarzadeh c,3 a
Islamic Azad University, Mahshahr Branch, Iran Petroleum University of Technology, Ahvaz, Iran c Chemical Engineering Department, Islamic Azad University, Mahshahr Branch, Mahshahr, Iran b
a r t i c l e
i n f o
Article history: Received 5 July 2010 Received in revised form 27 December 2010 Accepted 27 December 2010 Available online 30 December 2010 Keywords: Heat transfer Convection Negative solubility Gas bubbles
a b s t r a c t Free convection phenomenon has been experimentally investigated around a horizontal rod heater in carbonic acid solution. Because of the tendency of the solution to desorb carbon dioxide gas when temperature is increased, bubbles appear when cylinder surface is heated. The bubbles consists mainly carbon dioxide and also a negligible amount of water vapor. The results present that dissolved carbon dioxide in water significantly enhances the heat transfer coefficient in compare to pure free convection regime. This is mainly due to the microscale mixing on the heat transfer surface, which is induced by bubble formation. In this investigation, experiments are performed at different bulk temperatures between 288 and 333 K and heat fluxes up to 400 kW m2 at atmospheric pressure. Bubble departure diameter, nucleation site density and heat transfer coefficient have been experimentally measured. A model has been proposed to predict the heat transfer coefficient. Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved.
1. Introduction Free convection mechanism plays an important role in many industrial heat transfer processes. There are many developed correlations in the past few decades to predict the free convection heat transfer coefficients for different conditions, however, the effects of the dissolved gases, with negative solubility in liquids have never been considered. Generally, the homogeneous gas/liquid solutions can be classified in two different groups: (1) positive soluble and (2) negative soluble systems, i.e. the slope of solubility saturation curve relative to temperature are positive for the positive soluble and negative for the negative soluble solutions. Carbon dioxide/ water solution is a negative soluble system; accordingly, any heating surface which is exposed to this solution would locally release a fraction of dissolved gas, which is exceeding the saturation level of the gas in liquid. The released gas forms bubbles, analogous to boiling phenomenon, however with different mechanisms. In the boiling phenomenon, the bubbles absorb the latent heat of vaporization, in contrast, in the mentioned system, bubbles absorb the heat of solution. In both cases, the heat transfer coefficient is highly
enhanced in compare to pure convection heat transfer. This enhancement is mainly related to the intense micro-convection; the result of bubble formation and local movements of liquid around the heating surface. Fig. 1 presents the solubility function of carbon dioxide in water versus temperature [1]. Carbonic acid has many industrial applications such as soda pop, as a gas in the medical field, pharmaceutical, cosmetics, oil shale, food processing aid, medical, anesthetic, fuel, industrial, lasers, bottling, contact lens cleaner, engines, hydrolysis of starch, drugs, and welding. The phase-change heat transfer coefficients and pressure loss factors required for the design of boilers and evaporators involve some of the most complex thermo-fluid phenomena. In this investigation, saturated carbon dioxide/water solution has been selected as a negative soluble gas/liquid system and a horizontal rod heater is used as the heating surface. This phenomenon has close interaction to bubble dynamics and convection heat transfer. A brief literature review is accordingly presented. 2. Literature review 2.1. Bubble departure diameter
⇑ Corresponding author. Address: Chemical Engineering department, Islamic Azad University, Mahshahr, Khuzestan Province, Iran. Tel.: +98 9166137305; fax: +98 6113914801. E-mail addresses: [email protected] (S.A. Alavi Fazel), arabishamsabadi@ gmail.com (A.A Shamsabadi), [email protected] (M.M. Sarafraz), [email protected] (S.M. Peyghambarzadeh). 1 Tel.: +98 9163313445. 2 Tel.: +98 9166317313; fax: +98 6522338586. 3 Tel.: +98 9123241450; fax: +98 6522338586.
Bubble departure diameter is known as a key parameter in many phenomena including boiling, bubble column systems, etc. The detailed impact of physical properties such as interfacial tension, heat of evaporation, viscosity and thermal conductivity on the bubble departure diameter are still not well understood. There are many proposed correlations for predicting the bubble departure diameter in boiling phenomenon. The Fritz [2] model is one
0894-1777/$ - see front matter Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2010.12.014
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Nomenclature A Cp db f g k N Nu Pr Ra Pr q Q Re s T
area, m2 heat capacity, J kg1 °C1 bubble departure diameter, m bubble departure frequency, Hz gravity acceleration, m s2 thermal conductivity, W m1 °C1 number of nucleate site Nusselt number, [] reduced pressure, [] Rayleigh number, [] Prandtl number, [] heat flux, W m2 heat, W Reynolds number, [] distance, m temperature, K or °C
Fig. 1. Solubility of carbon dioxide in water [1].
of the most longstanding models for prediction of the bubble departure diameter in boiling of either pure liquids or liquid mixtures. This correlation is based on force balances on a single bubble departing the solid surface and includes an empirical tuning parameter. Stephan [3] has modified the Fritz [2] model by involving three dimensionless number including the Jacob, Prandtl and Archimedes numbers, which shows some improvements in compare to Fritz [2] model for some systems. Van Stralen and Zijl [4] proposed an empirical model for boiling systems by considering bubble growth mechanisms. This model includes Jacob number and thermal diffusivity of the solution. Cole [5] modified the bubble contact angle and included the effect of pressure through a modified Jacob number. Zeng et al. [6] assumed that the dominate forces leading to bubble detachment would be the unsteady growth and the bouncy forces. They developed their model based on an empirical expression for bubble growth mechanisms. This model can be used only if specific information on the vapor bubble growth parameters is available. In this empirical correlation, it is assumed that the departure time is just one-half the time period, which is same as suggested by McFadden and Grassman [7]. Yang et al. [8] developed a correlation by considering the analogy between nucleate boiling and forced convection heat transfer. The key parameter of this model is given graphically as a function of Jacob number. Recently, Alavi Fazel and Shafaee [9] proposed a new correlation for bubble departure diameter for electrolyte solutions. In this model, the dimensionless Bond number is related to Capillary number. In some of the existing models, the dimensionless Jacob number is involved including Ruckenstein [10], Cole and Rohsenow [11], Cole [5], Van Stralen and Zijl [4] and Stephan [3]
Tb
bulk temperature, °C
Greek letters a heat transfer coefficient, W m2 °C1 h contact angle, degree r surface tension, N m1 q density, kg m3 Subscripts b bubble or bulk c convection l liquid s surface th thermocouples v vapor
correlations. This dimensionless number is a function of surface temperature, which is fundamentally unknown in any given system. However, the surface temperature could be predicted through an iterative procedure by the existing correlations developed for predicting the boiling heat transfer coefficient conjugated with the Newtonian cooling law. A significant error should be expected through this iterative procedure; especially where the surface characteristics in the predictive correlations for boiling heat transfer coefficient is not implicated. These calculations could be more complicated in the sub-cooled pool boiling phenomenon. No specific correlation for prediction of bubble departure diameter for the bubbles which are released by heating a negative soluble gas/liquid system has been found in the past literature. However because of the similarity between the mechanisms of bubble formation in the mentioned systems and the boiling phenomenon, in this article, the correlations for prediction of the bubble departure diameter in boiling phenomenon has been used to predict the bubble diameter. In general, because of the very complex nature of bubble formation phenomenon, development of a theoretical basis model for prediction of the bubble departure diameter it still not possible and requires more extensive researches. In this investigation, the Fritz [2] correlation has been applied to predict the bubble departure diameter by the following mathematical form:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r gðql qm Þ
db ¼ 0:0146h
ð1Þ
which h the contact angle, proposed equal to 45° for water and 35° for mixtures by Fritz [2]. 2.2. Convection heat transfer Convection is the transfer of heat from one point to another within a fluid, gas, or liquid by the mixing of one portion of the fluid with another. In free convection, the motion of the fluid is entirely the result of differences in density resulting from temperature differences; in forced convection, the motion is produced by mechanical means. 2.2.1. Forced convection The Nusselt number, as a key parameter for forced convection heat transfer is known to be a function of the Reynolds number and Prandtl number. There are many proposed correlations in the literature to predict the forced convection heat transfer coefficient
S.A. Alavi Fazel et al. / Experimental Thermal and Fluid Science 35 (2011) 645–652
for different conditions [12]. In this investigation, for forced convection heat transfer around cylinders, a correlation proposed by Churchill and Bernstein [13] has been used by the following mathematical form:
NuD ¼ 0:3 þ
"
1=3 0:62Re1=2 D Pr
½1 þ ð0:4=PrÞ2=3 1=4
ReD 1þ 282; 000
5=8 #4=5 ð2Þ
0:90:3P0:3 r
a q ¼ a0 q0
FðPr Þ
0:68 P2 FðP r Þ ¼ 1:73P0:27 þ 6:1 þ r 1 Pr r 3. Experimental
2.2.2. Free convection heat transfer Free convection occurs when a fluid is in contact with a solid surface of different temperature. Temperatures differences create the density gradients that drive free convection. Here, the Nusselt number in also known as a key parameter for natural convection, which is known to be a function of the Rayleigh number and the Prandtl number. For laminar and turbulent flow on isothermal, horizontal cylinders of diameter D, Churchill and Chu [14] recommends the following equation:
3.1. Experimental setup
(
1=6
0:60 þ
0:387RaD
½1 þ ð0:559=PrÞ9=16 8=27
) 106 6 RaD
ð3Þ
In the above correlation, the fluid properties should be evaluated at the film temperature which is defined by: Tf = (Ts + Tb)/2. Eq. (3) is used to be correlated with the current experimental data. 2.3. Heat transfer through the bubble effected area When a bubble is developing on a heating surface, the heat transfer through the bubble stem is predicted by Mikic and Rohsenow [15], which is the transient conduction around nucleation sites and is calculated by the following equation:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ab ¼ 2 pkl ql C pl f
ð4Þ
In which, f is the bubble departure frequency. The bubble departure frequency can be predicted by some correlations including Zuber [16] correlation:
fdb ¼ 0:59
rgðql qm Þ q2l
0:25 ð5Þ
ð7Þ
Fig. 2 presents a schematic of the experimental equipment used in the present study. This boiling vessel is a vertical hollow cylinder of stainless steel containing 38 L of test liquid connected to a vertical condenser to condense and recycle the evaporated liquid. The whole system is heavily isolated for better control and reduction of the heat loss. The temperature of the liquid inside the tank is constantly monitored and controlled to any set point using a band heater covering the outside of the tank. Before any experiment, the liquid inside the tank is heated to any predetermined temperature. The pressure of the system is also monitored and regulated continuously. The test section is a horizontal rod heater with a diameter of 10.67 mm and a heating length of 99.1 mm which can be observed and photographed through observation glasses. This heater consists of an internally heated stainless steel sheathed rod and four sheathed thermocouples with an exterior diameter of 0.25 mm are entrenched 0.1 mm deep along the circumference of the heater close to the heating surface. The test heater is manufactured by Drew Industrial Chemicals Company according to specifications by Heat Transfer Research Incorporated (HTRI). Some details of the rod heater are given in Fig. 3. A PC-based data acquisition system was used to record all measuring parameters. The input power to the rod heater is precisely equal to the heat flux and could be calculated by the product of electrical voltage, current and cosine of the difference between electrical voltage and current. The average of five readings was used to determine the difference between heating surface and the bulk temperature of each thermocouple. To calculate the real surface temperature by correcting the minor temperature drop due to the small distance between surface and thermocouple location, the Fourier’s conduction equation is used as follow:
2.4. Boiling heat transfer coefficient Despite of fundamental differences between the boiling phenomenon and the gas released phenomenon during heating of carbonic acid, in this investigation, the heat transfer coefficient in two mentioned cases are correlated to highlight the order of differences. There are many predictive correlations for boiling heat transfer coefficient. These correlations are generally empirical or semi-empirical. Vinayak and Balakrishnan [17] has a wide-ranging survey on some correlations including Gorenflo [18], Stephan and Abdelsalam [19] and McNelly [20] for pure boiling liquids. The applicability and constancy of some other correlations such as Boyko-Kruzhilin [21] and Mostinski [22] could be found in some other references [23]. As a general rule, each correlation has some conflicting advantages and disadvantages. Among this complexity, Gorenflo [18] has a major distinction in compare to other existing correlations with two tuning parameters, a0 and q0. These tuning parameters are already found for many different pure boiling systems. These tuning parameters could also be determined empirically. The simplified Gorenflo [18] correlation is given by the following:
ð6Þ
which Pr is the reduced pressure. For water and low boiling point liquids:
where ReD is the Reynolds number based in the cylinder diameter.
NuD ¼
647
Fig. 2. A schematic of experimental equipment.
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Fig. 3. The rod heater.
s T s T b ¼ ðT th T b Þ ðq=AÞ k
ð8Þ
where s is the distance between the thermocouple location and heat transfer surface in meter, k is the thermal conductivity of the heater material in W m1 K1, q/A is the heat flux in W m2, T is the temperature in K and the subscripts ‘‘s’’, ‘‘b’’, ‘‘th’’ presents surface, bulk and thermocouples respectively. The average temperature difference was the arithmetic average of the four thermocouple locations. The boiling heat transfer coefficient, a, is calculated by:
a¼
q=A ðT s T b Þave:
ð9Þ
where q/A is the heat flux in W m2, a in W m2 K1 and the average temperature difference in K. Note that the vapor pressure of water is negligible in compare to carbon dioxide at same temperatures. This means that the generated gas bubbles only includes carbon dioxide and the water content inside the bubbles is insignificant. Fig. 4 compares the vapor pressures [24]. The horizontal dash line shows that water vapor pressure. For each experiment, picture of heating surface was taken using a high speed camera. A high speed video recorder was also used to record the nucleate site density, bubble departure frequency and also bubble departing diameter. 3.2. Experimental procedure
from the test solution. Following this, carbon dioxide gas has been continuously fed into bottom of the tank and the upper valve has been opened to pass the gas out. This procedure has been continued for 1 h to ensure the saturation condition, meanwhile the tank band heater was switched on and the temperature of the system allowed rising to the predetermined temperature. This procedure presents a homogeneous condition right through. Then the electric power was slowly supplied to the rod heater and increased gradually to any constant predetermined heat flux. Data acquisition system, video equipment including a digital camera were simultaneously switched on to record the required parameters including the rod heater temperature, bulk temperature, heat flux and also all visual information. Some runs were repeated twice and even thrice to ensure the reproducibility of the experiments. Each experiment has been performed at a short period of time; consequently the deviation from saturation condition could be ignored. The carbon dioxide supplement has been reconnected to the test vessel before each next run to compensate the stripped gas. Note that the flow rate of released gas was small due to the small heating area. 4. Results and discussion 4.1. Visual records The experimental apparatus prepared facilities to take photograph from heating surface during bubble formation. These photographs can be useful in understanding of bubble behavior near the heated surface and also measure the gas bubble departure diameters and nucleation site density. Fig. 5 typically presents the heat transfer surface during heating operation. Spherical bubbles are obvious on the heating surface, while a number of growing bubbles are attached on the heating surface. To determine an accurate bubble diameter for each photo, all bubbles inside any photo are analyzed by photo-analysis software and the arithmetic average bubble diameter has been considered as the average bubble diameter for any specific bulk temperature and heat flux. 4.2. Bubble departure diameter Fig. 6 presents the measured bubble departure diameter as a function of heat flux at different bulk temperatures. The bubbles
Initially, the entire system including the rod heater and the inside of the tank were cleaned and the distilled water was introduced. The vacuum pump is then turned on and the pressure of the system is kept low approximately to 10 kPa for 5 h to allow all the dissolved gases mainly the dissolved air has been stripped
Fig. 4. The vapor pressure of water and carbon dioxide as a function of temperature [24].
Fig. 5. Typical heating surface at Tb = 50 °C and q/A = 46.64 kW m2.
S.A. Alavi Fazel et al. / Experimental Thermal and Fluid Science 35 (2011) 645–652
Fig. 6. Measured bubble diameter as a function of heat flux at different bulk temperatures.
are almost uniform in size between 2.1 and 2.2 mm in diameter. However some very small bubbles are also observable on the heat transfer area. The very small bubbles are the results of bubble breakage and are not considered in the measurements. The standard deviations of bubble distribution are approximately equal to 0.05 mm and the bubbles are assumed to be uniform in size in this investigation. Fig. 6 indicates that the bubble departure diameter has inverse proportionality to heat flux at any given bulk temperature. This is due to higher bubble frequency, which limits the growing time of each bubble, which means smaller bubble size. In addition, the bulk temperatures has also inverse proportionality to bubble departure diameter at any given heat flux. This phenomenon can be related to smaller surface tension at higher temperatures. Fritz [2] model has found to have good agreement with experimental data. Note that the Fritz [2] model is apparently independent of heat flux; but surface temperature is a function of heat flux according to the Newton’s cooling law. The contact angle of h = 39° in the Fritz [2] model offers the best agreement with experimental data. Fig. 7 typically compares the predicted bubble diameters by Fritz [2] model and the experimental data at the bulk temperature of 15 °C. 4.3. Heat transfer coefficient Fig. 8 presents the measured heat transfer coefficient as a function of bulk temperature and heat flux for carbonic acid heating phenomenon. This figure which is enforced with 3D wire extrapolation states that the heat transfer coefficient increases with increasing heat flux at any constant bulk temperature; while, it is a weak function of bulk temperature.
649
Fig. 8. Experimental values of heat transfer coefficient as a function of heat flux and bulk temperature for carbonic acid heating phenomenon.
Fig. 9. Typical comparison between calculated forced, free, and boiling heat transfer coefficients and the experimental values.
Fig. 9 typically correlates the experimental data with predicting correlations for (1) boiling phenomenon, by Gorenflo [18], (2) forced convection, by Churchill and Bernstein [13] and (3) free convection, by Churchill and Chu [14]. Correlations express that the actual heat transfer coefficient in carbonic acid heating is bounded between the boiling heat transfer coefficient and the forced convection. The free convection correlation states sever under-predictions, which mean that free convection mechanism has practically no contribution in the total amount of heat transfer. 5. Modeling From the Newton‘s cooling law, it is well known that the convective heat transfer is proportional to the area and the thermal driving force, i.e.:
q A ¼ a A DT
ð10Þ
In the presence of bubbles on the heating surface, the heating area can be divided by two different zones: (1) Ab, the area affected by bubbles and (2) Ac, the convective heat transfer area, as indicated in Fig. 10. Each zone has the individual extent of heat transfer:
Fig. 7. Typical comparison between the measured and predicted bubble diameter at T = 15 °C.
qc Ac ¼ ac Ac DT
ð11Þ
qb Ab ¼ ab Ab DT
ð12Þ
which the subscripts ‘‘c’’ and ‘‘b’’ stands for ‘‘convection’’ and ‘‘bubble affected’’ areas respectively. Additionally, the following equation is clearly established:
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Fig. 10. Heat transfer surface and the division of surface.
A ¼ Ac þ Ab
ð13Þ
Summing up Eqs. (11) and (12) yields:
Q ¼ q A ¼ qc Ac þ qb Ab
ð14Þ
By combining Eqs. (11), (12), and (14) and ignoring the heat of solution, which is clearly insignificant, the total amount of heat transfer can be calculated:
Q ¼ q A ¼ aa Ab DT þ ac Ac DT
ð15Þ
Assuming the affected areas by spherical bubbles are equal to the projected area of the bubbles, Ab can be calculated simply by:
Ab ¼ N
p 4
2
db
ð16Þ
where N is the number of bubble nucleation. Consequently, the nucleation site density could be calculated by dividing both sides of Eq. (16) by the total area:
N 4 Ab ¼ A pd2b A
ð17Þ
where db, NA and A stands for bubble diameter in meter, nucleation site density in site m2 and area respectively. Combining Eqs. (15) and (17) yields:
a ¼ ac þ ðab ac Þ
2 N pdb A 4
ð18Þ
where a stands for total heat transfer coefficient, ac convective heat transfer coefficient and ab the heat transfer coefficient in the bubble affected area all in W m2 K1 units. In this article, the equation proposed by Churchill and Bernstein [13] which is applicable to forced convection around the horizontal cylinders presented by Eq. (2) has been implemented. To calculate the liquid convective velocity, it is assumed that any departed spherical bubble moves the liquid on the heating surface by the equal volume divided to the projected area. Considering the bubble departure frequency f, the velocity can be calculated by the following combination:
Fig. 11. Calculated bubble velocity as a function of heat flux at various bulk temperatures.
Fig. 12. Measured nucleation site density as a function of thermal driving force at different bulk temperatures.
u¼
p d3 6
b
p d2 4
b
f ¼
2 db f 3
ð19Þ
Fig. 11 presents the calculated bubble velocity as a function of heat flux at various bulk temperatures. When bubbles are released, there are two parallel mechanisms transferring the heat from the heating surface through the bubbles: (a) conduction heat transfer mechanism between bubbles and the heat transfer area and (b) the heat of solution which is absorbed during bubble detachment. The first mechanism is already quantified by Mikic and Rohsenow [15] presented by Eq. (4), while the second mechanism can be ignored because of low order of magnitude. The nucleation site density has been measured by analyzing the photos. Fig. 12 presents the nucleation site density as a function of thermal driving force, which is the thermal degree of supersaturation. The figure states that the nucleation site density increases with increasing the temperature difference between surface and bulk, Ts Tb, at various constant bulk temperatures. However the impact of bulk temperature at any constant thermal driving forces seems to be irregular. It is well known that an increase in boiling heat flux is always associated with an increase in both numbers of active nucleation sites and frequency of bubble emission. A change in the waiting period is mainly responsible for an increase of frequency by increasing temperature difference. In other words, when heat flux is increased, the time needed to heat up the immediate environment of the nucleation site is significantly shorter. The change in the growth time is not as significant, although the following tendency was confirmed: short growth times are related to small diameters and large temperature differences [25]. Fig. 13 presents the measured nucleation site density as a function of heat flux at different bulk temperatures. Clearly, nucleation site density is increased linearly with heat flux and all data tend to pass on a single line transient through the origin of coordination with slope of about 0.804. It is very difficult to theoretically predict the bubble nucleation site density because of the complicated bubble dynamics, especially at high heat fluxes. Many investigations on the two-phase flow formulations have been performed based on the drift flux and the two-fluid models. In the recent model, the field equations are expressed by the mass, momentum and energy equations for each phase [26]. Note that the phase interaction term appears in each of the averaged balance equations. However reliable and accurate relation for prediction of the bubble nucleate site density is still not fully developed. However, there are many empirical correlations for estimation of the nucleation site density for the boiling phenomenon. Bubble formation caused by the heating of carbonic acid has different mechanisms; consequently it is not
S.A. Alavi Fazel et al. / Experimental Thermal and Fluid Science 35 (2011) 645–652
Fig. 13. Experimental values of nucleation site density as a function of heat flux for different measured bulk temperatures between 288 and 333 K.
651
Fig. 17. Comparison between experimental data, new model and different heat transfer mechanisms at Tb = 333 K.
logical to use mentioned correlation for the current systems. In this investigation, the experimentally measured nucleation sites are correlated to different parameters including heat flux, surface and bulk temperatures, gas solubility and also the derivation of gas solubility curve relative to temperature. It is experimentally found that heat flux has the best proportionality to nucleation site density by the following proportional constant:
N ¼ 0:8036ðq=AÞ A
ð20Þ
where NA is the nucleation site density in site m2 and q/A is the heat flux in W m2 unit. 6. Model validation Fig. 14. Comparison between experimental data, new model and different heat transfer mechanisms at Tb = 288 K.
Experimental heat transfer coefficients of carbonic acid solution measured in this study are used to examine the developed model. Figs. 14–17 typically compare the experimental data and the predicted values. A relatively good agreement is clear. The arithmetic average, absolute arithmetic average and RMS error on all the 106 data points are measured to 13%, 17% and 27% respectively. 7. Conclusion
Fig. 15. Comparison between experimental data, new model and different heat transfer mechanisms at Tb = 293 K.
Heat transfer coefficient around a horizontal rod heater in carbonic acid solution has been measured in various heat fluxes and bulk temperatures. Because of the tendency of the solution to dissolve carbon dioxide gas when temperature is increased, bubbles appear when cylinder surface is heated. The departing bubbles significantly enhance the heat transfer coefficient in compare to pure free convection regime due to microscale mixing on the heat transfer surface. The Fritz [2] model presents good agreement with experimental data. The active nucleation site density in the mentioned system has linear proportionality to heat flux at any bulk temperature. A model has been developed which presents good agreement with the experimental data for mentioned systems. Acknowledgements The authors gratefully acknowledge the financial support provided by Islamic Azad University - Mahshahr Branch to perform research project entitled: ‘‘Pool boiling heat transfer in glycerol aqueous solutions at atmospheric pressure’’. References
Fig. 16. Comparison between experimental data, new model and different heat transfer mechanisms at Tb = 323 K.
[1] R.H. Perry, D.W. Green, Perry’s Chemical Engineers’ Handbook, 8th ed., McGraw-Hill, City of Publication, 2008. Table 2-122. [2] W. Fritz, Berechunung des maximal volumens von Dampfblasen, Phys. Z. 36 (1935) 379–384.
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[3] K. Stephan, cited by: U. Wenzel, Saturated Pool Boiling and Subcooled Flow Boiling of Mixtures, Ph.D. thesis, University of Auckland, New Zealand, 1992. [4] S.J.D. Van Stralen, W. Zijl, Fundamental developments in bubble dynamics, 6th Int. Heat Transfer Conference, vol. 6, Toronto, 1978, 429–450. [5] R. Cole, Bubble frequencies and departure volumes at subatmospheric pressures, AIChE J. 13 (1967) 779–783. [6] L.Z. Zeng, J.F. Klausner, R. Mei, A unified model for the prediction of bubble detachment diameters in boiling systems, part 1: pool boiling, Int. J. Heat Mass Transfer 36 (1993) 2261–2270. [7] P.W. McFadden, P. Grassmann, The relation between bubble frequency and diameter during nucleate pool boiling, Int. J. Heat Mass Transfer 5 (1961) 169– 173. [8] C. Yang, Y. Wu, X. Yuan, C. Ma, Study on bubble dynamics for pool nucleate boiling, Int. J. Heat Mass Transfer 43 (2000) 203–208. [9] S.A. Alavi Fazel, S.B. Shafaee, Bubble dynamics for nucleate pool boiling of electrolyte solutions, J. Heat Transfer ASME 132 (8) (2010) 081502-1–081502-7. [10] R. Ruckenstein, Recent trends in boiling heat transfer research, Appl. Mech. Rev. 17 (1964) 663–672. cited by: N. Zuber. [11] R. Cole, W.M. Rohsenow, Correlation of bubble departure diameters for boiling of saturated liquids, Chem. Eng. Prog. Symp. Ser. 65 (92) (1966) 211–213. [12] John H. Lienhard IV, John H. Lienhard V, A Heat Transfer Textbook, third ed., Phlogiston Press, Cambridge, Massachusetts, 2003. [13] S.W. Churchill, M. Bernstein, A correlating equation for forced convection from gases and liquids to a circular cylinder in crossflow, J. Heat Transfer, Trans. ASME, Ser. C 99 (1977) 300–306. [14] S.W. Churchill, H.H.S. Chu, Correlating equations for laminar and turbulent free convection from a horizontal cylinder, Int. J. Heat Mass Transfer 18 (1975) 1049–1053.
[15] B.B. Mikic, W.M. Rohsenow, A new correlation of pool boiling data including the effect of heat surface characteristics, ASME J. Heat Transfer 91 (1969) 245– 250. [16] N. Zuber, The dynamics of vapor bubbles in nonuniform temperature fields, Int. J. Heat Mass Transfer 2 (1961) 83–98. [17] G. Vinayak Rao, A.R. Balakrishnan, Heat transfer in nucleate pool boiling of multicomponent mixtures, Exp. Therm. Fluid Sci. 29 (2004) 87–103. [18] D. Gorenflo, Pool boiling, in: VDI Heat Atlas (Chapter Ha), 1993. [19] K. Stephan, K. Abdelsalam, Heat transfer correlation for natural convection boiling, Int. J. Heat Mass Transfer 23 (1980) 73–87. [20] M.J. McNelly, A correlation of rates of heat transfer to nucleate boiling of liquids, J. Imp. Coll. Chem. Eng. Soc. 7 (1953) 18–34. [21] Boyko-Kruzhilin, Heat transfer and hydraulic resistance during condensation of steam in a horizontal tube and in a bundle of tubes, Int. J. Heat Mass Transfer 10 (1967) 361. [22] I.L. Mostinski, Application of the rule of corresponding states for calculation of heat transfer and critical heat flux, Teploenergetika 4 (1963) 66. [23] F. Taboas, M. Valle‘s, M. Bourouis, A. Coronas, Pool boiling of ammonia/ water and its pure components: Comparison of experimental data in the literature with the predictions of standard correlations, Int. J. Refrig. 30 (5) (2007) 1–11. [24] R.H. Perry, D.W. Green, Perry’s Chemical Engineers’ Handbook, 8th ed., McGraw-Hill, City of Publication, 2008. Table 2-10. [25] E. Hahne, U. Grigull, Heat Transfer in Boiling, Hemisphere Publishing Corporation, 1977. [26] T. Hibiki, M. Ishii, Active nucleation site density in boiling systems, Int. J. Heat Mass Transfer 46 (2003) 2587–2601.
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Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Artificial boiling heat transfer in the free convection to carbonic acid solution S.A. Alavi Fazel a,⇑, A. Arabi Shamsabadi b,1, M.M. Sarafraz c,2, S.M. Peyghambarzadeh c,3 a
Islamic Azad University, Mahshahr Branch, Iran Petroleum University of Technology, Ahvaz, Iran c Chemical Engineering Department, Islamic Azad University, Mahshahr Branch, Mahshahr, Iran b
a r t i c l e
i n f o
Article history: Received 5 July 2010 Received in revised form 27 December 2010 Accepted 27 December 2010 Available online 30 December 2010 Keywords: Heat transfer Convection Negative solubility Gas bubbles
a b s t r a c t Free convection phenomenon has been experimentally investigated around a horizontal rod heater in carbonic acid solution. Because of the tendency of the solution to desorb carbon dioxide gas when temperature is increased, bubbles appear when cylinder surface is heated. The bubbles consists mainly carbon dioxide and also a negligible amount of water vapor. The results present that dissolved carbon dioxide in water significantly enhances the heat transfer coefficient in compare to pure free convection regime. This is mainly due to the microscale mixing on the heat transfer surface, which is induced by bubble formation. In this investigation, experiments are performed at different bulk temperatures between 288 and 333 K and heat fluxes up to 400 kW m2 at atmospheric pressure. Bubble departure diameter, nucleation site density and heat transfer coefficient have been experimentally measured. A model has been proposed to predict the heat transfer coefficient. Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved.
1. Introduction Free convection mechanism plays an important role in many industrial heat transfer processes. There are many developed correlations in the past few decades to predict the free convection heat transfer coefficients for different conditions, however, the effects of the dissolved gases, with negative solubility in liquids have never been considered. Generally, the homogeneous gas/liquid solutions can be classified in two different groups: (1) positive soluble and (2) negative soluble systems, i.e. the slope of solubility saturation curve relative to temperature are positive for the positive soluble and negative for the negative soluble solutions. Carbon dioxide/ water solution is a negative soluble system; accordingly, any heating surface which is exposed to this solution would locally release a fraction of dissolved gas, which is exceeding the saturation level of the gas in liquid. The released gas forms bubbles, analogous to boiling phenomenon, however with different mechanisms. In the boiling phenomenon, the bubbles absorb the latent heat of vaporization, in contrast, in the mentioned system, bubbles absorb the heat of solution. In both cases, the heat transfer coefficient is highly
enhanced in compare to pure convection heat transfer. This enhancement is mainly related to the intense micro-convection; the result of bubble formation and local movements of liquid around the heating surface. Fig. 1 presents the solubility function of carbon dioxide in water versus temperature [1]. Carbonic acid has many industrial applications such as soda pop, as a gas in the medical field, pharmaceutical, cosmetics, oil shale, food processing aid, medical, anesthetic, fuel, industrial, lasers, bottling, contact lens cleaner, engines, hydrolysis of starch, drugs, and welding. The phase-change heat transfer coefficients and pressure loss factors required for the design of boilers and evaporators involve some of the most complex thermo-fluid phenomena. In this investigation, saturated carbon dioxide/water solution has been selected as a negative soluble gas/liquid system and a horizontal rod heater is used as the heating surface. This phenomenon has close interaction to bubble dynamics and convection heat transfer. A brief literature review is accordingly presented. 2. Literature review 2.1. Bubble departure diameter
⇑ Corresponding author. Address: Chemical Engineering department, Islamic Azad University, Mahshahr, Khuzestan Province, Iran. Tel.: +98 9166137305; fax: +98 6113914801. E-mail addresses: [email protected] (S.A. Alavi Fazel), arabishamsabadi@ gmail.com (A.A Shamsabadi), [email protected] (M.M. Sarafraz), [email protected] (S.M. Peyghambarzadeh). 1 Tel.: +98 9163313445. 2 Tel.: +98 9166317313; fax: +98 6522338586. 3 Tel.: +98 9123241450; fax: +98 6522338586.
Bubble departure diameter is known as a key parameter in many phenomena including boiling, bubble column systems, etc. The detailed impact of physical properties such as interfacial tension, heat of evaporation, viscosity and thermal conductivity on the bubble departure diameter are still not well understood. There are many proposed correlations for predicting the bubble departure diameter in boiling phenomenon. The Fritz [2] model is one
0894-1777/$ - see front matter Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2010.12.014
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Nomenclature A Cp db f g k N Nu Pr Ra Pr q Q Re s T
area, m2 heat capacity, J kg1 °C1 bubble departure diameter, m bubble departure frequency, Hz gravity acceleration, m s2 thermal conductivity, W m1 °C1 number of nucleate site Nusselt number, [] reduced pressure, [] Rayleigh number, [] Prandtl number, [] heat flux, W m2 heat, W Reynolds number, [] distance, m temperature, K or °C
Fig. 1. Solubility of carbon dioxide in water [1].
of the most longstanding models for prediction of the bubble departure diameter in boiling of either pure liquids or liquid mixtures. This correlation is based on force balances on a single bubble departing the solid surface and includes an empirical tuning parameter. Stephan [3] has modified the Fritz [2] model by involving three dimensionless number including the Jacob, Prandtl and Archimedes numbers, which shows some improvements in compare to Fritz [2] model for some systems. Van Stralen and Zijl [4] proposed an empirical model for boiling systems by considering bubble growth mechanisms. This model includes Jacob number and thermal diffusivity of the solution. Cole [5] modified the bubble contact angle and included the effect of pressure through a modified Jacob number. Zeng et al. [6] assumed that the dominate forces leading to bubble detachment would be the unsteady growth and the bouncy forces. They developed their model based on an empirical expression for bubble growth mechanisms. This model can be used only if specific information on the vapor bubble growth parameters is available. In this empirical correlation, it is assumed that the departure time is just one-half the time period, which is same as suggested by McFadden and Grassman [7]. Yang et al. [8] developed a correlation by considering the analogy between nucleate boiling and forced convection heat transfer. The key parameter of this model is given graphically as a function of Jacob number. Recently, Alavi Fazel and Shafaee [9] proposed a new correlation for bubble departure diameter for electrolyte solutions. In this model, the dimensionless Bond number is related to Capillary number. In some of the existing models, the dimensionless Jacob number is involved including Ruckenstein [10], Cole and Rohsenow [11], Cole [5], Van Stralen and Zijl [4] and Stephan [3]
Tb
bulk temperature, °C
Greek letters a heat transfer coefficient, W m2 °C1 h contact angle, degree r surface tension, N m1 q density, kg m3 Subscripts b bubble or bulk c convection l liquid s surface th thermocouples v vapor
correlations. This dimensionless number is a function of surface temperature, which is fundamentally unknown in any given system. However, the surface temperature could be predicted through an iterative procedure by the existing correlations developed for predicting the boiling heat transfer coefficient conjugated with the Newtonian cooling law. A significant error should be expected through this iterative procedure; especially where the surface characteristics in the predictive correlations for boiling heat transfer coefficient is not implicated. These calculations could be more complicated in the sub-cooled pool boiling phenomenon. No specific correlation for prediction of bubble departure diameter for the bubbles which are released by heating a negative soluble gas/liquid system has been found in the past literature. However because of the similarity between the mechanisms of bubble formation in the mentioned systems and the boiling phenomenon, in this article, the correlations for prediction of the bubble departure diameter in boiling phenomenon has been used to predict the bubble diameter. In general, because of the very complex nature of bubble formation phenomenon, development of a theoretical basis model for prediction of the bubble departure diameter it still not possible and requires more extensive researches. In this investigation, the Fritz [2] correlation has been applied to predict the bubble departure diameter by the following mathematical form:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r gðql qm Þ
db ¼ 0:0146h
ð1Þ
which h the contact angle, proposed equal to 45° for water and 35° for mixtures by Fritz [2]. 2.2. Convection heat transfer Convection is the transfer of heat from one point to another within a fluid, gas, or liquid by the mixing of one portion of the fluid with another. In free convection, the motion of the fluid is entirely the result of differences in density resulting from temperature differences; in forced convection, the motion is produced by mechanical means. 2.2.1. Forced convection The Nusselt number, as a key parameter for forced convection heat transfer is known to be a function of the Reynolds number and Prandtl number. There are many proposed correlations in the literature to predict the forced convection heat transfer coefficient
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for different conditions [12]. In this investigation, for forced convection heat transfer around cylinders, a correlation proposed by Churchill and Bernstein [13] has been used by the following mathematical form:
NuD ¼ 0:3 þ
"
1=3 0:62Re1=2 D Pr
½1 þ ð0:4=PrÞ2=3 1=4
ReD 1þ 282; 000
5=8 #4=5 ð2Þ
0:90:3P0:3 r
a q ¼ a0 q0
FðPr Þ
0:68 P2 FðP r Þ ¼ 1:73P0:27 þ 6:1 þ r 1 Pr r 3. Experimental
2.2.2. Free convection heat transfer Free convection occurs when a fluid is in contact with a solid surface of different temperature. Temperatures differences create the density gradients that drive free convection. Here, the Nusselt number in also known as a key parameter for natural convection, which is known to be a function of the Rayleigh number and the Prandtl number. For laminar and turbulent flow on isothermal, horizontal cylinders of diameter D, Churchill and Chu [14] recommends the following equation:
3.1. Experimental setup
(
1=6
0:60 þ
0:387RaD
½1 þ ð0:559=PrÞ9=16 8=27
) 106 6 RaD
ð3Þ
In the above correlation, the fluid properties should be evaluated at the film temperature which is defined by: Tf = (Ts + Tb)/2. Eq. (3) is used to be correlated with the current experimental data. 2.3. Heat transfer through the bubble effected area When a bubble is developing on a heating surface, the heat transfer through the bubble stem is predicted by Mikic and Rohsenow [15], which is the transient conduction around nucleation sites and is calculated by the following equation:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ab ¼ 2 pkl ql C pl f
ð4Þ
In which, f is the bubble departure frequency. The bubble departure frequency can be predicted by some correlations including Zuber [16] correlation:
fdb ¼ 0:59
rgðql qm Þ q2l
0:25 ð5Þ
ð7Þ
Fig. 2 presents a schematic of the experimental equipment used in the present study. This boiling vessel is a vertical hollow cylinder of stainless steel containing 38 L of test liquid connected to a vertical condenser to condense and recycle the evaporated liquid. The whole system is heavily isolated for better control and reduction of the heat loss. The temperature of the liquid inside the tank is constantly monitored and controlled to any set point using a band heater covering the outside of the tank. Before any experiment, the liquid inside the tank is heated to any predetermined temperature. The pressure of the system is also monitored and regulated continuously. The test section is a horizontal rod heater with a diameter of 10.67 mm and a heating length of 99.1 mm which can be observed and photographed through observation glasses. This heater consists of an internally heated stainless steel sheathed rod and four sheathed thermocouples with an exterior diameter of 0.25 mm are entrenched 0.1 mm deep along the circumference of the heater close to the heating surface. The test heater is manufactured by Drew Industrial Chemicals Company according to specifications by Heat Transfer Research Incorporated (HTRI). Some details of the rod heater are given in Fig. 3. A PC-based data acquisition system was used to record all measuring parameters. The input power to the rod heater is precisely equal to the heat flux and could be calculated by the product of electrical voltage, current and cosine of the difference between electrical voltage and current. The average of five readings was used to determine the difference between heating surface and the bulk temperature of each thermocouple. To calculate the real surface temperature by correcting the minor temperature drop due to the small distance between surface and thermocouple location, the Fourier’s conduction equation is used as follow:
2.4. Boiling heat transfer coefficient Despite of fundamental differences between the boiling phenomenon and the gas released phenomenon during heating of carbonic acid, in this investigation, the heat transfer coefficient in two mentioned cases are correlated to highlight the order of differences. There are many predictive correlations for boiling heat transfer coefficient. These correlations are generally empirical or semi-empirical. Vinayak and Balakrishnan [17] has a wide-ranging survey on some correlations including Gorenflo [18], Stephan and Abdelsalam [19] and McNelly [20] for pure boiling liquids. The applicability and constancy of some other correlations such as Boyko-Kruzhilin [21] and Mostinski [22] could be found in some other references [23]. As a general rule, each correlation has some conflicting advantages and disadvantages. Among this complexity, Gorenflo [18] has a major distinction in compare to other existing correlations with two tuning parameters, a0 and q0. These tuning parameters are already found for many different pure boiling systems. These tuning parameters could also be determined empirically. The simplified Gorenflo [18] correlation is given by the following:
ð6Þ
which Pr is the reduced pressure. For water and low boiling point liquids:
where ReD is the Reynolds number based in the cylinder diameter.
NuD ¼
647
Fig. 2. A schematic of experimental equipment.
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Fig. 3. The rod heater.
s T s T b ¼ ðT th T b Þ ðq=AÞ k
ð8Þ
where s is the distance between the thermocouple location and heat transfer surface in meter, k is the thermal conductivity of the heater material in W m1 K1, q/A is the heat flux in W m2, T is the temperature in K and the subscripts ‘‘s’’, ‘‘b’’, ‘‘th’’ presents surface, bulk and thermocouples respectively. The average temperature difference was the arithmetic average of the four thermocouple locations. The boiling heat transfer coefficient, a, is calculated by:
a¼
q=A ðT s T b Þave:
ð9Þ
where q/A is the heat flux in W m2, a in W m2 K1 and the average temperature difference in K. Note that the vapor pressure of water is negligible in compare to carbon dioxide at same temperatures. This means that the generated gas bubbles only includes carbon dioxide and the water content inside the bubbles is insignificant. Fig. 4 compares the vapor pressures [24]. The horizontal dash line shows that water vapor pressure. For each experiment, picture of heating surface was taken using a high speed camera. A high speed video recorder was also used to record the nucleate site density, bubble departure frequency and also bubble departing diameter. 3.2. Experimental procedure
from the test solution. Following this, carbon dioxide gas has been continuously fed into bottom of the tank and the upper valve has been opened to pass the gas out. This procedure has been continued for 1 h to ensure the saturation condition, meanwhile the tank band heater was switched on and the temperature of the system allowed rising to the predetermined temperature. This procedure presents a homogeneous condition right through. Then the electric power was slowly supplied to the rod heater and increased gradually to any constant predetermined heat flux. Data acquisition system, video equipment including a digital camera were simultaneously switched on to record the required parameters including the rod heater temperature, bulk temperature, heat flux and also all visual information. Some runs were repeated twice and even thrice to ensure the reproducibility of the experiments. Each experiment has been performed at a short period of time; consequently the deviation from saturation condition could be ignored. The carbon dioxide supplement has been reconnected to the test vessel before each next run to compensate the stripped gas. Note that the flow rate of released gas was small due to the small heating area. 4. Results and discussion 4.1. Visual records The experimental apparatus prepared facilities to take photograph from heating surface during bubble formation. These photographs can be useful in understanding of bubble behavior near the heated surface and also measure the gas bubble departure diameters and nucleation site density. Fig. 5 typically presents the heat transfer surface during heating operation. Spherical bubbles are obvious on the heating surface, while a number of growing bubbles are attached on the heating surface. To determine an accurate bubble diameter for each photo, all bubbles inside any photo are analyzed by photo-analysis software and the arithmetic average bubble diameter has been considered as the average bubble diameter for any specific bulk temperature and heat flux. 4.2. Bubble departure diameter Fig. 6 presents the measured bubble departure diameter as a function of heat flux at different bulk temperatures. The bubbles
Initially, the entire system including the rod heater and the inside of the tank were cleaned and the distilled water was introduced. The vacuum pump is then turned on and the pressure of the system is kept low approximately to 10 kPa for 5 h to allow all the dissolved gases mainly the dissolved air has been stripped
Fig. 4. The vapor pressure of water and carbon dioxide as a function of temperature [24].
Fig. 5. Typical heating surface at Tb = 50 °C and q/A = 46.64 kW m2.
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Fig. 6. Measured bubble diameter as a function of heat flux at different bulk temperatures.
are almost uniform in size between 2.1 and 2.2 mm in diameter. However some very small bubbles are also observable on the heat transfer area. The very small bubbles are the results of bubble breakage and are not considered in the measurements. The standard deviations of bubble distribution are approximately equal to 0.05 mm and the bubbles are assumed to be uniform in size in this investigation. Fig. 6 indicates that the bubble departure diameter has inverse proportionality to heat flux at any given bulk temperature. This is due to higher bubble frequency, which limits the growing time of each bubble, which means smaller bubble size. In addition, the bulk temperatures has also inverse proportionality to bubble departure diameter at any given heat flux. This phenomenon can be related to smaller surface tension at higher temperatures. Fritz [2] model has found to have good agreement with experimental data. Note that the Fritz [2] model is apparently independent of heat flux; but surface temperature is a function of heat flux according to the Newton’s cooling law. The contact angle of h = 39° in the Fritz [2] model offers the best agreement with experimental data. Fig. 7 typically compares the predicted bubble diameters by Fritz [2] model and the experimental data at the bulk temperature of 15 °C. 4.3. Heat transfer coefficient Fig. 8 presents the measured heat transfer coefficient as a function of bulk temperature and heat flux for carbonic acid heating phenomenon. This figure which is enforced with 3D wire extrapolation states that the heat transfer coefficient increases with increasing heat flux at any constant bulk temperature; while, it is a weak function of bulk temperature.
649
Fig. 8. Experimental values of heat transfer coefficient as a function of heat flux and bulk temperature for carbonic acid heating phenomenon.
Fig. 9. Typical comparison between calculated forced, free, and boiling heat transfer coefficients and the experimental values.
Fig. 9 typically correlates the experimental data with predicting correlations for (1) boiling phenomenon, by Gorenflo [18], (2) forced convection, by Churchill and Bernstein [13] and (3) free convection, by Churchill and Chu [14]. Correlations express that the actual heat transfer coefficient in carbonic acid heating is bounded between the boiling heat transfer coefficient and the forced convection. The free convection correlation states sever under-predictions, which mean that free convection mechanism has practically no contribution in the total amount of heat transfer. 5. Modeling From the Newton‘s cooling law, it is well known that the convective heat transfer is proportional to the area and the thermal driving force, i.e.:
q A ¼ a A DT
ð10Þ
In the presence of bubbles on the heating surface, the heating area can be divided by two different zones: (1) Ab, the area affected by bubbles and (2) Ac, the convective heat transfer area, as indicated in Fig. 10. Each zone has the individual extent of heat transfer:
Fig. 7. Typical comparison between the measured and predicted bubble diameter at T = 15 °C.
qc Ac ¼ ac Ac DT
ð11Þ
qb Ab ¼ ab Ab DT
ð12Þ
which the subscripts ‘‘c’’ and ‘‘b’’ stands for ‘‘convection’’ and ‘‘bubble affected’’ areas respectively. Additionally, the following equation is clearly established:
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Fig. 10. Heat transfer surface and the division of surface.
A ¼ Ac þ Ab
ð13Þ
Summing up Eqs. (11) and (12) yields:
Q ¼ q A ¼ qc Ac þ qb Ab
ð14Þ
By combining Eqs. (11), (12), and (14) and ignoring the heat of solution, which is clearly insignificant, the total amount of heat transfer can be calculated:
Q ¼ q A ¼ aa Ab DT þ ac Ac DT
ð15Þ
Assuming the affected areas by spherical bubbles are equal to the projected area of the bubbles, Ab can be calculated simply by:
Ab ¼ N
p 4
2
db
ð16Þ
where N is the number of bubble nucleation. Consequently, the nucleation site density could be calculated by dividing both sides of Eq. (16) by the total area:
N 4 Ab ¼ A pd2b A
ð17Þ
where db, NA and A stands for bubble diameter in meter, nucleation site density in site m2 and area respectively. Combining Eqs. (15) and (17) yields:
a ¼ ac þ ðab ac Þ
2 N pdb A 4
ð18Þ
where a stands for total heat transfer coefficient, ac convective heat transfer coefficient and ab the heat transfer coefficient in the bubble affected area all in W m2 K1 units. In this article, the equation proposed by Churchill and Bernstein [13] which is applicable to forced convection around the horizontal cylinders presented by Eq. (2) has been implemented. To calculate the liquid convective velocity, it is assumed that any departed spherical bubble moves the liquid on the heating surface by the equal volume divided to the projected area. Considering the bubble departure frequency f, the velocity can be calculated by the following combination:
Fig. 11. Calculated bubble velocity as a function of heat flux at various bulk temperatures.
Fig. 12. Measured nucleation site density as a function of thermal driving force at different bulk temperatures.
u¼
p d3 6
b
p d2 4
b
f ¼
2 db f 3
ð19Þ
Fig. 11 presents the calculated bubble velocity as a function of heat flux at various bulk temperatures. When bubbles are released, there are two parallel mechanisms transferring the heat from the heating surface through the bubbles: (a) conduction heat transfer mechanism between bubbles and the heat transfer area and (b) the heat of solution which is absorbed during bubble detachment. The first mechanism is already quantified by Mikic and Rohsenow [15] presented by Eq. (4), while the second mechanism can be ignored because of low order of magnitude. The nucleation site density has been measured by analyzing the photos. Fig. 12 presents the nucleation site density as a function of thermal driving force, which is the thermal degree of supersaturation. The figure states that the nucleation site density increases with increasing the temperature difference between surface and bulk, Ts Tb, at various constant bulk temperatures. However the impact of bulk temperature at any constant thermal driving forces seems to be irregular. It is well known that an increase in boiling heat flux is always associated with an increase in both numbers of active nucleation sites and frequency of bubble emission. A change in the waiting period is mainly responsible for an increase of frequency by increasing temperature difference. In other words, when heat flux is increased, the time needed to heat up the immediate environment of the nucleation site is significantly shorter. The change in the growth time is not as significant, although the following tendency was confirmed: short growth times are related to small diameters and large temperature differences [25]. Fig. 13 presents the measured nucleation site density as a function of heat flux at different bulk temperatures. Clearly, nucleation site density is increased linearly with heat flux and all data tend to pass on a single line transient through the origin of coordination with slope of about 0.804. It is very difficult to theoretically predict the bubble nucleation site density because of the complicated bubble dynamics, especially at high heat fluxes. Many investigations on the two-phase flow formulations have been performed based on the drift flux and the two-fluid models. In the recent model, the field equations are expressed by the mass, momentum and energy equations for each phase [26]. Note that the phase interaction term appears in each of the averaged balance equations. However reliable and accurate relation for prediction of the bubble nucleate site density is still not fully developed. However, there are many empirical correlations for estimation of the nucleation site density for the boiling phenomenon. Bubble formation caused by the heating of carbonic acid has different mechanisms; consequently it is not
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Fig. 13. Experimental values of nucleation site density as a function of heat flux for different measured bulk temperatures between 288 and 333 K.
651
Fig. 17. Comparison between experimental data, new model and different heat transfer mechanisms at Tb = 333 K.
logical to use mentioned correlation for the current systems. In this investigation, the experimentally measured nucleation sites are correlated to different parameters including heat flux, surface and bulk temperatures, gas solubility and also the derivation of gas solubility curve relative to temperature. It is experimentally found that heat flux has the best proportionality to nucleation site density by the following proportional constant:
N ¼ 0:8036ðq=AÞ A
ð20Þ
where NA is the nucleation site density in site m2 and q/A is the heat flux in W m2 unit. 6. Model validation Fig. 14. Comparison between experimental data, new model and different heat transfer mechanisms at Tb = 288 K.
Experimental heat transfer coefficients of carbonic acid solution measured in this study are used to examine the developed model. Figs. 14–17 typically compare the experimental data and the predicted values. A relatively good agreement is clear. The arithmetic average, absolute arithmetic average and RMS error on all the 106 data points are measured to 13%, 17% and 27% respectively. 7. Conclusion
Fig. 15. Comparison between experimental data, new model and different heat transfer mechanisms at Tb = 293 K.
Heat transfer coefficient around a horizontal rod heater in carbonic acid solution has been measured in various heat fluxes and bulk temperatures. Because of the tendency of the solution to dissolve carbon dioxide gas when temperature is increased, bubbles appear when cylinder surface is heated. The departing bubbles significantly enhance the heat transfer coefficient in compare to pure free convection regime due to microscale mixing on the heat transfer surface. The Fritz [2] model presents good agreement with experimental data. The active nucleation site density in the mentioned system has linear proportionality to heat flux at any bulk temperature. A model has been developed which presents good agreement with the experimental data for mentioned systems. Acknowledgements The authors gratefully acknowledge the financial support provided by Islamic Azad University - Mahshahr Branch to perform research project entitled: ‘‘Pool boiling heat transfer in glycerol aqueous solutions at atmospheric pressure’’. References
Fig. 16. Comparison between experimental data, new model and different heat transfer mechanisms at Tb = 323 K.
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