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Semi-empirical correlation of gas cooling heat transfer of supercritical carbon dioxide in microchannels.


When it comes to ozone depletion and global warming, natural refrigerants offer a clear advantage over CFCs and their substitutes. Ohadi and Mo (1997) have provided a detailed analysis of selected properties of natural refrigerants. Recently, [CO.sub.2] has been investigated as an alternative for vapor compression systems. As a nontoxic and noncombustible gas, [CO.sub.2] is ecologically safe, is available in sufficient quantities, and is relatively inexpensive. However, the most important advantage of [CO.sub.2] is its attractive thermal characteristics. Chapter 19 of the 2005 ASHRAE Handbook--Fundamentals provides a comparison of the general physical and chemical properties of [CO.sub.2] to those of other refrigerants. It can be seen in Chapter 19 (ASHRAE 2005) that to match the temperature levels in ordinary heat pump or heating systems, [CO.sub.2] has to work in the subcritical or supercritical region (critical point: 31.1[degrees]C / 73.8bar). In these regions, the thermodynamic and transport properties--in particular, the heat transfer and pressure drop characteristics--of [CO.sub.2] are quite different from those of conventional refrigerants.

Since the thermophysical properties of supercritical fluids show variational values and, even worse, change drastically near the pseudo-critical region, conventional heat transfer correlations do not predict the heat transfer coefficient accurately. In this paper, experimentally measured heat transfer data for gas cooling of [CO.sub.2] are reported and then compared to existing correlations for the supercritical region (or near critical region). This is followed by the introduction of a new semi-empirical correlation developed for this study to better predict the gas cooling heat transfer coefficient of supercritical [CO.sub.2] in microchannels.


The test facility used in this study measured the average gas cooling heat transfer coefficient and pressure drop of refrigerant [CO.sub.2] throughout the test section.

A schematic diagram of the test facility is shown in Figure 1. The microchannel tubes were made of aluminum, with 11 circular ports, and an inner diameter of 0.79 mm.


A detailed introduction of experimental apparatus and the data reduction and experimental procedure, along with the experimental results of heat transfer coefficient and pressure drop, are reported elsewhere (Kuang et al. 2004) and will not be repeated here.


Due to the difficulty of dealing with the wide property variations of supercritical [CO.sub.2], satisfactory analytical methods have not yet been developed. Therefore, empirical generalized correlations based on experimental data are typically used to predict the heat transfer coefficient at supercritical pressure.

Since Bringer and Smith (1957), and Miropolskii and Shitsman (1957) proposed the Dittus-Boelter-type correlation for supercritical water, studies have focused on how to integrate the effects of property variations into the correlation. Considering the thermophysical properties of specific heat and density, Ghajar and Asadi (1986) modified the Dittus-Boelter-type correlation for heat transfer near the critical region. Krasnoshchekov and Protopopov (1966) proposed a correlation that adds the specific heat and density effect into Gnielinski's (1976) correlation. Liao and Zhao (2002) revised the Dittus-Boelter-type correlation by adding the effect of specific heat, density, and the buoyancy force. Pitla et al. (2002) developed a correlation based on mean Nusselt numbers that are calculated using the thermophysical properties at the wall and the bulk temperatures, respectively. Later, Yoon et al. (2003) and Huai et al. (2005) proposed other modified Dittus-Boelter-type correlations based on their own experimental data.

In this paper, Gnielinski's (1976), Krasnoshchekov and Protopopov's (1966), Ghajar and Asadi's (1986), Pitla et al.'s (2002), and Huai et al.'s (2005) correlations were selected for comparison with the present experimental results for pure supercritical [CO.sub.2] gas cooling in micro-channel tubes.

Here, the global Nusselt number is defined as:

Nu [equivalent to] [[bar.h]D/k]

where, [bar.h] is the global heat transfer coefficient of supercritical [CO.sub.2] through the whole test section, D is the hydraulic diameter of the channel, and k is the thermal conductivity of the fluid. The detailed uncertainty analysis showed that the uncertainty of the heat transfer coefficient is within 20% for the presented experimental data (Kuang 2006).

Gnielinski's Correlation

Gnielinski's (1976) correlation is valid for 0.5 < Pr < 2000 and 3000 < Re < 5 x [10.sup.6], given constant thermophysical properties, and is given as

Nu = [[(f/8)(Re - 1000)Pr]/[1.07 + 12.7[square root of ([f/8])]([Pr.sup.[2/3]] - 1)]] (1)

where the friction factor must be calculated from Petukhov et al.'s (1961) formula:

f = [(0.790ln Re - 1.64).sup.-2] (2)

Figure 2 shows a comparison of the results of the experimental data and Gnielinski's correlation at a pressure of 9 MPa. The individual data points are experimental data, and the dashed lines are calculation results based on Gnielinski's correlation. As shown there, the correlation works well only at the lower mass flux regions and the region far from the pseudo-critical temperature. At the high mass flux region, the correlation underpredicts the experimental data up to 54%, especially near the pseudo-critical temperature.


The current single-phase heat transfer coefficient correlations, such as Gnielinski's correlation, assume single-phase and constant thermodynamic properties. However, in the neighborhood region of the pseudo-critical temperature, the thermodynamic properties of [CO.sub.2] do not qualify as a "constant" value, as shown in Figure 3 (data taken from Engineering Equation Solver [EES] software). The specific heat, as well as the density, have varying values in the region. This phenomenon explains why there is a larger deviation between the experimental data and the correlation results in the region of the pseudo-critical temperature.


Krasnoshchekov and Protopopov's Correlation

Krasnoshchekov and Protopopov's formula (1966) is given as:

[Nu.sub.b] = [Nu.sub.b]'[([[rho].sub.w]/[[rho].sub.b]).sup.0.3][([bar.Cp]/[Cp.sub.b]).sup.0.4] (3)

[bar.Cp] = ([H.sub.b] - [H.sub.w])/([T.sub.b] - [T.sub.w]) (4)

where subscript b and w represent the value evaluated at the bulk flow temperature and the wall temperature, respectively. H is enthalpy, [bar.Cp] is average integral specific heat, and [Nu.sub.b]' is defined from Gnielinski's (1976) formula, Equation 1.

Figure 4 shows the comparison between experimental data and Krasnoshchekov and Protopopov's correlation. Data points with different shapes represent different pressures. As shown in the figure, the experimental data match the correlation well in the region with low mass flux/heat transfer coefficients. But otherwise the correlation generally underpredicts the experimental data. The maximum deviation is up to +86%.


Ghajar and Asadi's Correlation

Ghajar and Asadi (1986) proposed the following correlation for heat transfer in the near-critical region:

[Nu.sub.b] = a[Re.sup.b][Pr.sup.c][([[rho].sub.w]/[[rho].sub.b]).sup.d][([bar.Cp]/[Cp.sub.b]).sup.n] (5)

where the average integral specific heat [bar.Cp] is defined as in Equation 4; a, b, c, and d are curve-fitted constants; and n is suggested as a value of 0.4.

The results of the comparison between the experimental data and Ghajar and Asadi's formula are shown in Figure 5. In the figure, the deviation between the curve-fitting correlation and the experimental data is within 35% for the majority of the data. The performance of this correlation is better than that for Krasnoshchekov and Protopopov's (1966) correlation, but it is still not accurate enough to predict the heat transfer coefficient.


Pitla et al.'s Correlation

Pitla et al. (2002) developed a new correlation to predict the heat transfer coefficient of super-critical carbon dioxide during in-tube cooling, as shown in Equation 6:

Nu = ([N[u.sub.wall] + N[u.sub.bulk]]/2)[[k.sub.wall]/[k.sub.bulk]] (6)

where Gnielinski's (1976) correlation, Equation 1, is used to calculate both the Nusselt numbers N[u.sub.wall] and N[u.sub.bulk]. The subscripts wall and bulk indicate that the thermophysical properties are evaluated at the wall temperature and bulk flow temperature, respectively.

Figure 6 shows the results of the comparison between experimental data and Pitla et al.'s correlation. Data points with different shapes designate different pressure levels. Unlike Krasnoshchekov and Protopopov's (1966) correlation (Figure 4), the experimental data scatter both above and below Pitla et al.'s correlation results. The deviation is up to +85%.


Huai et al.'s Correlation

Based on their experimental data on gas cooling of supercritical carbon dioxide in multiport minichannels, Huai et al. (2005) proposed a correlation for heat transfer coefficient, as given by Equation 7:

Nu = 0.022186R[e.sup.0.8]P[r.sup.0.3][([[rho].sub.w]/[rho]).sup.1.4652][([bar.Cp]/C[p.sub.w]).sup.0.0832] (7)

Again, the average integral specific heat [bar.Cp] is defined as in Equation 4, and the subscript w indicates the value measured at wall temperature.

Figure 7 shows results of the comparison between experimental data and Huai et al.'s correlation. As shown there, Huai et al.'s correlation predicted 90% of experimental data within an error of 25%.



When compared against the existing correlations, Ghajar and Asadi's (1986) curve-fitting method and Huai et al.'s (2005) correlations were found to most closely predict the present experimental heat transfer data among the various correlations examined. Huai et al.'s formula is also based on a curve-fitting criterion, with the difference between the two being the definition of the ratio of specific heat. Ghajar and Asadi defined this ratio as the average integral specific heat over the specific heat at bulk flow temperature, while Huai et al. defined the ratio as the average integral specific heat over the specific heat at wall temperature.

In Ghajar and Asadi's (1986) correlation (Equation 5), two terms are added into the original Dittus-Boelter-type heat transfer correlations. One is the density ratio and the other is the specific heat ratio. Generally, the density ratio accounts for the effect of density gradient and buoyancy. The specific heat ratio here may account for the effect of variational specific heat along the cross section of the tube.

However, it may be more important to consider the effect of the variational specific heat along the whole length of the tube. Based on such an assumption, the mean specific heat along the whole test section is defined as follows:

[bar.Cp] = [([] - [H.sub.out])/([] - [T.sub.out]) (8)

With the modification of mean specific heat in Equation 8, a new correlation can be developed to predict the heat transfer coefficient at the supercritical region. Fitting the data via curve-fitting, yields the following correlation:

Nu = 0.001546R[e.sup.1.054]P[r.sup.0.653][([[rho].sub.w]/[rho]).sup.0.367][( [bar.Cp]/Cp).sup.0.4] (9)

Figure 8 shows a comparison of the experimental data with the proposed correlation. In the figure, the proposed correlation predicts the experimental data much better than the previously reviewed correlations. Most (91%) of the data are within an error of 15%. The proposed correlation should be safely valid for hydraulic diameters between 0.5 mm and 2 mm. For smaller diameters the tube roughness, among other factors, may affect the results.



The present study compared five existing empirical models with the experimental results of the present study at the supercritical region for [CO.sub.2] gas cooling. It was found that the conventional heat transfer correlation (Gnielinski's [1976] correlation) for forced convection in-tube flow failed to predict the experimental data near the pseudo-critical region where the thermophysical properties have variational values.

Both Krasnoshchekov and Protopopov's (1966) correlation and Pitla et al.'s (2002) correlation showed large deviations from the present experimental data, which may suggest that they are not valid for predicting the heat transfer of supercritical cooling in microchannel tubes. Ghajar and Asadi's (1986) correlation and Huai et al.'s (2005) correlation showed fair comparison with the experimental data, with deviations of 35% and 25%, respectively. Both empirical models employed the curve-fitting method and the modified Dittus-Boelter-type heat transfer correlations.

A new empirical correlation was proposed in the present study to predict the heat transfer of the supercritical gas cooling process in microchannel tubes. This new model is based on Ghajar and Asadi's (1986) correlation, but it introduces a modified average specific heat to account for the effect of fluctuation along the whole test section. The new predicting model agrees very well with the experimental data, within an error of 15% for most (91%) of the experimental data. Comparison with more extensive databases (as they become available) will further verify the applicability of the proposed correlation for gas cooling of supercritical [CO.sub.2].


a,b,c,d = curve fitting constant

Cp = specific heat

D = hydraulic diameter of the microchannels

f = friction factor

h = heat transfer coefficient

G = mass flux

H = enthalpy

k = thermal conductivity

n = constant index

Nu = Nusselt number

P = pressure

Pr = Prandtl number

Re = Reynolds number

T = temperature

[rho] = density


b, bulk = value evaluated at temperature of bulk flow

w, wall = value evaluated at wall temperature

cal = value calculated with correlations

exp = value measured in experiments

p-c = pseudo-critical

in = inlet

out = outlet


ASHRAE. 2005, ASHRAE Handbook--Fundamentals, SI ed. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

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Pitla, S., E. Groll, and S. Ramadhyani. 2002. New correlation to predict the heat transfer coefficient during in-tube cooling of turbulent supercritical [CO.sub.2]. International Journal of Refrigeration 25(7):887-95.

Yoon, S.H., J.H. Kim, Y.W. Huang, M.S. Kim, M. Kyoungdoug, and K. Yongchan. 2003. Heat transfer and pressure drop characteristics during the in-tube cooling process of carbon dioxide in the supercritical region. International Journal of Refrigeration 26(8):857-64.

Guohua Kuang, PhD

Michael Ohadi, PhD


Serguie Dessiatoun

Received June 29, 2007; accepted July 11, 2008

Guohua Kuang is a research assistant and Serguie Dessiatoun is a research scientist in the Department of Mechanical Engineering at the University of Maryland, College Park, MD. Michael Ohadi is a professor of mechanical engineering at the University of Maryland and provost at the Petroleium Institute, Abu Dhabi, UAE.
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Author:Kuang, Guohua; Ohadi, Michael; Dessiatoun, Serguie
Publication:HVAC & R Research
Article Type:Report
Geographic Code:1USA
Date:Nov 1, 2008
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