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A simple methodology to incorporate flashing and variation of thermophysical properties for flow boiling pressure drop in a microchannel
International Journal of Thermal Sciences 132 (2018) 137–145
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
A simple methodology to incorporate flashing and variation of thermophysical properties for flow boiling pressure drop in a microchannel
T
Ashif Iqbal, Manmohan Pandey∗ Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, 781039, Assam, India
A R T I C LE I N FO
A B S T R A C T
Keywords: Flow boiling Pressure drop Flashing Local thermophysical properties Microchannel FC-72
This paper proposes a simple approach to model the complex interplay among the various thermophysical phenomena occurring in flow boiling. In order to assess the need for such a model, flow boiling pressure drop of FC-72 in a single trapezoidal microchannel with a hydraulic diameter of 111 μm is measured by varying heat flux and mass flux. The pressure drop obtained is in the range of 10–45 kPa, which does not compare well with the existing models based on constant properties. Therefore, a new predictive approach is developed for meticulous evaluation of pressure drop across a microchannel with flow boiling. It uses the separated flow model with evaluation of thermophysical properties at local pressure, thus incorporating the effect of flashing on thermodynamic quality, and the effect of heat flux on the two-phase multiplier. Relations based on a modified form of Clausius-Clapeyron equation are employed for evaluation of local thermophysical properties for fluids. This methodology is combined with different empirical correlations from the literature to predict pressure drop. The proposed predictive methodology, comprising close form equations, is physically sound and yet easy to implement, and reproduces large pressure drop experimental data better than the existing methods.
1. Introduction Dissipating high amount of heat flux is an important issue of modern thermal management with the ever increasing demands for high performance and miniaturization. Flow boiling microchannel heat sinks have emerged as one of the most effective solutions for cooling high and ultrahigh heat flux devices such as high performance computer chips, laser diodes and nuclear reactors [1]. Design of these miniature devices requires a proper estimation of two-phase heat transfer, which, in turn, necessitates accurate prediction of pressure drop in flow boiling. The total two-phase pressure drop of a fluid is the sum of the frictional pressure drop, the acceleration pressure drop, and the gravitational pressure drop. The frictional pressure drop can be determined by different two-phase models, and this has been an active research area for the last two decades. In order to calculate two-phase pressure drop, extensive theoretical and experimental studies have been conducted. Since the mechanisms occurring in two-phase flow have not been effectively understood, a number of empirical correlations have been proposed instead [2]. Two-phase pressure drop in microchannels is relatively high as compared to conventional channels, due to their very small sizes and moderate mass fluxes, the latter being so in order to achieve reasonable heat transfer coefficients. Due to the large pressure gradient, the
∗
Corresponding author. E-mail addresses: [email protected] (A. Iqbal), [email protected] (M. Pandey).
https://doi.org/10.1016/j.ijthermalsci.2018.05.046 Received 11 December 2017; Received in revised form 16 April 2018; Accepted 30 May 2018 1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved.
saturation temperature — and hence the thermophysical properties — vary along the length, and the effect of flashing becomes significant. The flashing phenomenon in mini-/microchannels occurs when the pressure at some axial location of the channel drops below the saturation pressure of the fluid, and the liquid at that location becomes superheated temporarily. Dario et al. [3] incorporated the effect of flashing on thermodynamic quality considering a linear pressure profile for heat transfer calculation. Very informative observations were reported by Mirmanto [4,5], by putting pressure sensors along the length of a microchannel. It was observed that the pressure profile along the length was nonlinear, especially at higher heat flux, due to rapid bubble generation and non-uniform distribution of nucleation sites [4]. Pressure drop reported in the experiments [4] was high (up to 75 kPa), therefore, the existing modelling approach, based on system pressure, cannot predict these experimental data accurately. Another group of researchers [6] tried to validate the existing correlations for two-phase pressure drop with their experimental data considering properties at system pressure, but ended up with a significant deviation. Cioncolini and Thome [7] considered the local pressure for evaluating thermophysical properties and calculated two-phase pressure drop, using the homogenous model. The evaporative channel was discretized into equal length subchannels, and the local pressure drops were computed using the local values of thermophysical properties, and the subchannel
International Journal of Thermal Sciences 132 (2018) 137–145
A. Iqbal, M. Pandey
pressure drops were added to get the total pressure drop [7]. In an earlier work [8], pressure drop for flow boiling in a microchannel was predicted using the homogenous model including the effect of local thermophysical properties and flashing. The Clausius-Clapeyron equation, which uses the ideal gas equation for the vapour phase, was employed for evaluation of local thermophysical properties [8]. However, the ideal gas equation is not valid for refrigerants and dielectric fluids, since the compressibility factor is not close to unity. For the electronic cooling applications, dielectric fluids (FC-72 and FC-77) are suitable because they are thermally and chemically stable, compatible with sensitive materials, nonflammable, and practically nontoxic [6,9]. However, there are limited number of studies [6,9–14] on flow boiling pressure drop of dielectric fluids in microchannels, i.e., channels that fulfil condition [10] (Bd 0.5Re = 160 ), under which microscale confinement effects are exhibited. Park et al. [6] investigated the two-phase pressure drop of FC-72 and stated that pressure drop increased with increasing vapor quality and mass flux, and was nearly independent of heat flux for any given exit quality and mass flux. In view of the importance of dielectric fluids, further research on their flow boiling pressure drop characteristics and development of predictive tools for the design of microchannel heat sinks is the need of the hour. In the present work, flow boiling experiments are conducted through a single microchannel using FC-72 and a relatively high pressure drop is observed. For high pressure drop, the effects of flashing and variation of thermophysical properties are significant. Therefore, a new methodology is developed, which is simple but successful (compares well with experimental data) to evaluate flow boiling pressure drop by incorporating the flashing effect and local thermophysical properties. The present approach includes a modified form of Clausius-Clapeyron equation to characterize the saturation line on the P-T plane for fluids whose vapor phase has nearly constant compressibility factor in the range of operation. The equations governing the proposed methodology are derived in an elegant close form, consisting of a system of ODEs, and their simulation is computationally inexpensive.
Table 1 Estimation of error in various parameters measured and derived in the experiments. Parameter
Value/Range
Uncertainty (%)
Top width (Wt ) Bottom width (Wb) Height (H) Length (L) Hydraulic Diameter (DH ) Heated Area (Aht ) Heat Flux (qeff )
245 μm 115 μm 90 μm 20 mm 111 μm 6.74 × 10−7 m2 4–8.5 W/cm2
± 0.43 ± 0.85 ± 1.11 ± 0.50 ± 1.48 ± 0.33 ± 5.01 to ± 2.03
Mass flux (G ) Pressure drop (Δp)
431–690 kg/m2s 10–45 kPa
± 5.32 to ± 2.52 ± 7.02 to ± 3.13
temperature. All devices were connected using silicone tubing (Master Flex). A pre-calibrated syringe pump (Cole Parmer) was used for metering and controlling the mass flow rate of FC 72. A pre-calibrated digital pressure gauge (Deltabar PMD75) was used to measure the pressure drop across the microchannel. A contact probe was used to make electric contact between microheater and DC power supply. K-type thermocouples (Cole-Parmer) were placed inside the inlet and outlet reservoirs. Another K-type thermocouple was placed on the bottom surface (microheater side) of the microchannel to measure the wall temperature. All the thermocouples were connected to a data logger which logs the temperatures. The test sections were fabricated using microfabrication facility available at CEN, IIT Bombay. Doublesided polished (100) p-type, 2″ silicon wafer with resistivity 4–7 Ω-cm was used for fabrication. Microchannel was etched using chemical etching process and a trapezoidal cross-section was obtained as shown in a cross sectional view of Fig. 1(b). It was then bonded with a quartz plate at the top. The fabrication process was the same as that used by Singh et al. [15]. Detailed geometrical specifications of the microchannel employed in the experiment are given in Table 1. The following procedure was adopted while performing the experiments. The flow was started and then an input electric power was provided by supplying a fixed voltage and current using a DC power supply. Continuous monitoring of the inlet and outlet temperatures and pressure drop was undertaken to check for steady state. Once steady state condition was achieved, all relevant parameters such as pressure drop, inlet–outlet temperatures, power supplied and flow rate were recorded. The above procedure was repeated for other values of flow rates and heat fluxes.
2. Experimental setup and procedure A schematic of the experimental setup employed in the measurements is shown in Fig. 1(a). The setup consists of a syringe pump, a differential pressure gauge, the test section, a DC power supply, and a data logger to record inlet/outlet temperatures of FC 72 and surface
Fig. 1. (a) Schematic of the experimental setup (b) Cross section of the test section. 138
International Journal of Thermal Sciences 132 (2018) 137–145
A. Iqbal, M. Pandey
Fig. 2. (a) Heat loss calibration at no flow condition (b) comparison of friction factor correlation [17] with experimental data in no heat condition. Fluid: FC-72; DH = 111 μm.
quality is chosen as the total pressure drop is the pressure difference between the inlet and outlet headers. As seen from Fig. 3(a)–(b), total pressure drop increases with an increase in the heat flux or with the exit vapor quality for all the mass flux values. Here, the exit vapor quality was calculated considering the flashing effect. Markal et al. [18] observed a similar trend and explained that, due to increase in the heat flux, evaporation momentum force increases, which results in the fact that extended bubble toward the inlet of the channel begins to apply more force, and consequently, flow resistance increases. The force opposed to evaporation force is due to the inertia of the liquid and it increases with increasing mass flux. For very low flow rates (between 2.0 and 3.63 ml/min) and for very small channel dimensions (150 × 150 μm2), the interaction of these forces becomes significant [18]. In the present study, the channel dimension ( DH = 111 μm) and flow rates (0.2–0.4 ml/min) are quite low, thus resulting in observations similar to those reported previously [18]. Park et al. [6] stated that heat flux negligibly affected the pressure drop under identical mass flux and vapor quality conditions, indicating that most of the pressure drop was caused by friction. However, their calculation of thermodynamic vapor quality was based on the effect of heat flux only, without considering the effect of flashing. This assumption is valid in case of low pressure drops only. The pressure drops in their experiments were large, causing significant effect of flashing on the vapor quality. In the later part of the present study, an approach to evaluate thermodynamic quality with the effect of flashing is discussed. In this approach, the pressure profile is obtained using the separated flow model and the thermophysical properties are evaluated at local pressures.
Measurement uncertainties of the parameters are listed in Table 1. 3. Data reduction To estimate heat loss, a constant power was applied to the microheater, resulting in increase of the temperature of the test section. After steady state was reached, the wall temperatures on the test section together with that of the ambient were recorded and correlated with the power input to the microheater. This procedure was repeated for several input powers, as shown in Fig. 2(a). A linear relation qloss = 0.0385ΔTwa − 0.2372 was fitted to the data points. Here, ΔTwa is the average wall-to-ambient temperature difference (ΔTwa = Tw − Tamb). This method is the same as that used by Lee and Garimella [16]. Downstream of the syringe pump, the fluid gets heated in the inlet header before entering the microchannel. The heat utilized to heat the fluid inside the header is given by equation (1).
˙ p (Tin − Tsyringe ) qheader = mc
(1)
The effective heat flux was assumed to be uniform along the microchannel length and is given by equation (2). (Here qinput is the electric power supplied to the microheater.)
qeff = (qinput − qheader − qloss )/ Aht
(2)
The channel shape is trapezoidal and, therefore heat transfer area for the 3-side heated walls is given by equation (3) and the friction factor used for this calculation is given by equation (4) [17]. Equation (4) was compared with the present experimental data for adiabatic flow and showed a good prediction (Fig. 2(b)).
Aht = [2 ((wt − wb)/2)2 + H 2 + Wb] L
(3)
fRe = 11.43 + 0.80 exp (2.67Wb/Wt )
(4)
5. Existing approach for evaluation of two-phase pressure drop for miniature channel
Fig. 3(a)–(b) show the variation of total pressure drop with the wall heat flux and the exit vapor quality, respectively. Here, the exit vapor
The two-phase pressure drop in steady state can be expressed as the sum of frictional, gravitational and acceleration components. In twophase flow through microchannels, commonly observed flow patterns in the literature were slug flow and annular flow [19]. The homogeneous equilibrium model of two-phase flow assumes no slip between
4. Experimental result and discussion
Fig. 3. Experimental total pressure drop with the (a) wall heat flux and (b) the exit vapor quality. Working Fluid: FC-72; DH = 111 μm; G = 431–690 kg/m2s; qeff = 4–8.5 W/cm2. 139
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A. Iqbal, M. Pandey
the phases and hence cannot accurately model these flow patterns [20]. Therefore, most of the literature used the separated flow model to obtain good agreement with experimental data because of its applicability with observed flow patterns (slug and annular flow) [21–28]. Frictional pressure drop component evaluated from separated flow model was expressed by equation (4) and acceleration pressure drop was given by equation (5). Thermodynamic equilibrium quality (x) was determined from the energy balance without the effect of flashing and given by equation (6). The exit quality (xo) was found by putting z = L .
small. It is observed that almost all the correlations are able to predict the data with reasonably good accuracy. In Fig. 4(b), the predictions of the same correlations (listed in Table 2) with the existing modelling approach are compared with experimental data from the literature [4] where the pressure drop was relatively high. It is observed that the existing approach, with any of the correlations, exhibits poor prediction of the experimental data for these cases, the percentage deviation being larger for higher pressure drop. The existing approach neglects the effect of local properties and the effect of flashing — assumptions valid for low pressure drop cases only. MAE (%) for each model is listed in Table 2 and given by equation (10).
xo
(Δptp ) = LTP / x o f
∫ 2G2vf (1 − xo)2φf2/Dh dx
(4a)
0 2 ⎡ xo ⎛ vg ⎞
)2
(1 − x o (Δptp ) = G 2vf ⎢ ⎜ ⎟ + a α v 1 − αo ⎣ o⎝ f⎠
x=
qeff Ph GAhfg
MAE (%) = ⎤ − 1⎥ ⎦
(5)
(z − LSP )
(6)
0.5
(v v ) f
g
0.5
(7)
⎛ (1 − x ) x ⎞ ⎝ ⎠
0.5
(8)
Evaluation of accelerational pressure drop requires a relation for the void fraction. In most of the literature related to miniature channels [21–24,28], Zivi's void fraction correlation, given by equation (9), has been used. 2
⎛ 1 − x o ⎞ ⎛ vg ⎞ α 0 = 1/ ⎜1 + ⎛ ⎜ ⎟ ⎜ ⎝ x 0 ⎠ ⎝ vf ⎠ ⎝ ⎜
⎟
Δpexp
× 100 (10)
6. A new approach to predict two-phase pressure drop As it has been observed, the predictions of the existing approach are good for low pressure drop cases but poor for relatively higher pressure drop cases. Therefore, the study proposes a new approach for accurate prediction of pressure drop for flow boiling of water, dielectric fluids and refrigerants, especially in case of high pressure drop. In most of the experiments, it was assumed that coolant enters the heat sink in subcooled condition and maintains a single-phase liquid state until it reaches zero thermodynamic equilibrium quality ( x = 0). Microchannels can be divided into two regions, single-phase and twophase regions, as shown in Fig. 5.
3⎞
⎟ ⎟ ⎠
Δppred − Δpexp
⁃ For relatively high pressure drop cases, assumption of system pressure instead of local pressure to evaluate thermophysical properties contributes a part of the total error. ⁃ Effect of flashing, i.e., the effect of pressure drop in the calculation of thermodynamic quality was not considered in the existing approach. Hence, the vapor quality is underpredicted, which results in the underestimation of the pressure drop. ⁃ Estimation of two-phase frictional pressure drop for boiling channels requires a different approach as opposed to non-boiling channels. This effect was incorporated in C by Kim and Mudawar (2013), thus resulting in the lowest MAE among all models listed in Table 2.
and the In frictional pressure drop, the two-phase multiplier Martinelli parameter (Χvv ) [20] for laminar vapor and laminar liquid were expressed by equation (7) and equation (8), respectively. All experimental data used [4,23] falls into the laminar vapor and laminar liquid region.
Χvv = ⎛ μf μ ⎞ g ⎝ ⎠
∑
The causes for this high margin of errors in the calculation of twophase pressure drop in a microchannel, in case of high pressure drop, are the following.
φf2
φf2 = 1 + C / Χvv + 1/ Χvv2
1 N
(9)
In the existing approach to pressure drop modeling, the Chisholm parameter (C) was modified to predict the pressure drop. All the modified expressions for the Chisholm parameter (C) are listed in Table 2. In Fig. 4(a), pressure drops predicted with the existing approach using different correlations [21–27] are compared with experimental data from the literature [23] where the pressure drop was Table 2 List of modified Chisholm parameter (C ) from literature. Authors [Reference]
Equations for Chisholm parameter (Liquid laminar and gas laminar)
MAE
MAE
High pressure drop (%)
Low pressure drop (%)
Mishima and Hibiki [21]
C = 21(1 − e319Dh)
43.73
17.35
Lee and Lee [22]
0.726 C = 6.185 × 10−2Refo
42.24
16.13
Qu and Mudawar [23]
C = 21(1 − e−319Dh)(0.00418G + 0.0613)
41.67
12.15
Lee and Mudawar [24]
0.047 We 0.60 C = 2.16Refo fo
40.12
11.95
Li and Wu [25]
Bd < 0.1, C = 5.6Bd 0.28
43.35
15.25
37.79
10.73
40.05
11.36
Kim and Mudawar [26]
Cboiling = 3.5 ×
0.44 Su 0.5 (ρ / ρ ) 0.48 10−5Refo go f g
(
0.52 Bo PH × ⎡1 + 530Wefo PF ⎢ ⎣
1.09
)
⎤ ⎥ ⎦
Sugo = ρg σDh / μg2 Wefo = G 2Dh / ρf σ
Bo = qeff / Ghfg Li and Hibiki [27]
0.42 0.21 C = 41.7Nμ0.66 TP ReTP x
Retp = GDh /μTP 0.5
Nμtp = μtp/ ⎛ρTP σ σ ⎞ gΔρ ⎠ ⎝ 1/μTP = (1 − x )/μf + x /μ g
ρTP = (1 − x ) ρf + xρg
140
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A. Iqbal, M. Pandey
Fig. 4. Comparison of the existing correlations for (a) Low pressure drop case with experimental data from Qu and Mudawar [23]; Water, DH = 349 μm and G = 255 kg/m2s (b) High pressure drop case with experimental data from Mirmanto [4], water, DH = 438 μm and G = 805 kg/m2s.
thermophysical properties can be evaluated at the system pressure. However, if the pressure drop is high, then these properties play a significant role in the evaluation of the pressure drop. To evaluate these thermophysical properties, the local saturation temperature is evaluated from the local pressure by using the Clausius-Clapeyron relation which considers the ideal gas approximation for the vapor phase. Iqbal and Pandey [8] used it for water for which the compressibility factor is close to unity. For a non-ideal gas condition, a modified form of Clausius-Clapeyron relation is derived, considering the compressibility factor (Z) by equation (15), where pref and Tref are the reference pressure and temperature, respectively. Thermophysical properties hf and vg vary significantly with local pressure. For evaluation of vg , modified ideal gas equation (pvg = ZRT ) is considered. Here Z is considered as constant since the variation along the length is not significant (2–3%). Change of vg and hf with respect to the pressure is given by equations
Fig. 5. Schematic diagram of the model.
6.1. Single-phase pressure drop In diabatic flow, the single phase pressure drop is significantly influenced by the variation of kinematic viscosity with temperature, can be evaluated by equation (11), which is a linear fit to property data. All dimensions are in SI unit. ( ν (T ) at m2s and T at K)
ν (T ) = 1.135 × 10−6 − 2.5872 × 10−9T
(16) and (17). Detailed derivations of the expressions for
qeff PH GAcp
(11)
dvg
dhf
(12)
dp
0
=
Z 2RT ⎡ RT − 1⎤ ⎥ p2 ⎢ h fg ⎦ ⎣
= Cv, f
(16)
ZRT 2 + vf phfg
(17)
{
} ( ){ ( ) ( ){
x 2vg α2
} ⎞⎠ ⎤⎥⎦ ⎟
}
(18)
GACp (Tsat − Tin ) PH qeff
and
(15)
2(1 − x ) vf (1 − x )2vf ∂α ⎡ 2flo G2vf 2 2 dx 2xvg ⎢ Dh φfo + G dz α − (1 − α ) + ∂x p (1 − α )2 − = ⎣ (1 − x )2vf x 2vg ⎡1 + G 2 ⎡ x 2 dvg + ∂α − 2 ⎤⎤ ∂p x α dp α (1 − α )2 ⎢ ⎥ ⎣ ⎦⎦ ⎣
(13)
where LSP is the length from the inlet to the saturation point and is given by equation (14) and fSP can be calculated from equation (4) for a trapezoidal microchannel.
LSP =
dp
dp −⎛ ⎞ ⎝ dz ⎠
LSP
∫ (fSP G2vf /DH ) dz
dvg
−1
The pressure drop in single phase region is evaluated by equation (13).
Δpsp =
,
⎡ 1 ZR ⎛ p ⎞ ⎤ T=⎢ − ln Tref hfg ⎜ pref ⎟ ⎥ ⎝ ⎠⎦ ⎣
dp
z + Tin
dp
modification of Clausius-Clapeyron equation are given in Appendix A.
From energy balance, the axial temperature profile can be expressed by equation (12)
T=
dhf
dhf dp ⎞ dx 1 ⎛ qeff PH = − ⎜ ⎟ dz hfg ⎝ GA dp dz ⎠
(14)
6.2. Two-phase pressure drop
(19)
In the two-phase region, boiling causes vapor generation, resulting in an increase in pressure gradient as compared to purely liquid flow. Consider the channel shown in Fig. 5. Considering the properties (vg and h f ) within the saturated region as functions of the local pressure, the pressure gradient [20] and the thermodynamic vapor quality x can be expressed by equations (18) and (19), respectively. The second term on the right hand side of equation (19) accounts for the axial change in quality due to flashing that results from enthalpy changes with
Two-phase pressure drop calculation in a microchannel has been a challenging topic for research community due to the complicated nature of boiling phenomenon. In the two-phase region, the pressure and hence the saturation temperature decrease along the length, thus causing change in thermophysical properties along the axial direction. If the pressure drop is low then, to a reasonable approximation, 141
International Journal of Thermal Sciences 132 (2018) 137–145
A. Iqbal, M. Pandey
Fig. 6. Comparison of the existing correlations with present experimental data (a) Considering properties at system pressure (b) Considering the effect of flashing and local thermophysical properties. Working Fluid: FC-72; DH = 111 μm; G = 431–690 kg/m2s; qeff = 4–8.5 W/cm2.
quality, the frictional pressure drop in the presence of a heat flux is higher than that in an unheated channel [20]. Therefore, a further modification was required on the other models [21–25,27] in order to improve their predictions. An empirical equation (20) relating the value of φfo2 in heated and unheated tubes for the steam-water system was reported in the literature [30]. This correlation was combined with the aforementioned models [21–25,27]. Then the results of these modified models were compared with the experimental data. From Fig. 7 and Table 3, it is observed that the effect of heat flux on φfo2 improves the predictions of the models. Equation (20) was not incorporated in the model of Kim and Mudawar [26] whose Chisholm parameter already contained the heat flux effect.
pressure. Zivi's void fraction correlation (equation (9)) is used to determine the void fraction. Equation (18) and equation (19) are coupled and were solved using 4th order Runge-Kutta method (ode45 function of MATLAB). 6.3. Comparison of the new approach with experimental results Experiments were conducted on a single microchannel with DH = 111 μm with mass flux range 431–690 kg/m2s and heat flux range 4–8.5 W/cm2. In the experiments, outlet pressure was considered to be atmospheric. In Fig. 6(a), experimental data were compared with the predictions of the existing modelling approach using different correlations (listed in Table 2), where properties were evaluated at system pressure and effect of flashing was not considered. It was observed that almost all the correlations predict the data with high MAE (36–50%). The reason for this prediction with such error in the two-phase pressure drop is an accumulation of uncertainties from the different parameters associated with the pressure drop calculation. Experimental two-phase pressure drop was in the range of 10–45 kPa, which, for FC-72 as the working fluid, causes a change in the saturation temperature in the range of about 4–9 °C along the length. Therefore, if the system pressure was considered instead of local pressure, the uncertainties in vg and hf will be in the range of 12–25% and 9–15%, respectively. Existing modelling approach did not incorporate the effect of flashing, and consequently, the thermodynamic quality ( x ) calculated with these models will be less than the actual value. Therefore, uncertainties associated with vg , hf and x , caused considerable error in the prediction of two-phase pressure drop. The present approach was applied to the existing correlations (listed in Table 3) and compared with the experimental data in Fig. 6(b), which showed significant improvement in their prediction. It was found that the new approach reduces the range of MAE for the all the correlations to 8.4–36%. As discussed earlier, the present approach is based on the evaluation of thermophysical properties (vg and hf ) at local pressure, and the effect of flashing on the thermodynamic quality. It is observed that incorporation of the effect of local properties and flashing improved the prediction to a great extent. Table 3 shows a comparison of MAE (%) of pressure drop predictions using different Ccorrelations. It reveals that the correlation of Kim and Mudawar [26], which incorporated the effect of heat flux on C , achieves the lowest MAE (8.4%), when combined with local thermophysical properties and flashing. This C-correlation is based on an earlier analysis [29] of the fundamental differences in flow structure between flow boiling and adiabatic flows. The difference is mainly due to the existence of entrained droplets in the vapor core for boiling flows and their absence from both condensing and adiabatic flows. Thus, the effect of heat flux on the pressure drop appears to be significant. At lower thermodynamic
0.7
q ⎡φ2 ⎤ ⎡1 + 4.4 × 10−3 ⎛ eff ⎞ ⎤ = ⎡φfo2 ⎤ ⎢ fo⎥ ⎢ ⎥ ⎢ G ⎠ ⎥ ⎝ ⎦ ⎣ ⎦heated tube ⎣ ⎦unheated tube ⎣
(20)
7. Conclusion This paper gives a new predictive approach for flow boiling pressure drop in microchannels. Experiments were conducted to assess the need for such a method. In case of large pressure drop, the effect of local thermophysical properties and flashing on the pressure gradient and the effect of heat flux on the two-phase multiplier are significant. The proposed methodology successfully incorporates the interplay between hydrodynamics, thermodynamics, heat transfer and phase change through simple close form equations. The contributions of the present study are as follow.
• The existing approach of pressure drop modeling — using properties • • • • • 142
evaluated at the system pressure — is found to give fairly accurate predictions for small pressure drop cases, For large pressure drop cases, the existing modelling approach does not predict pressure drop accurately. For such cases, MAE was 36–50% for the present experiments and 37–44% of the data from the literature [4]. The effect of local thermophysical properties has been incorporated by using a modified form of Clausius-Clapeyron equation derived for fluids whose vapor phase has compressibility factor nearly constant over the range of operation. The present pressure drop modeling approach — incorporating the effect of local properties and flashing — significantly improves the pressure drop predictions in case of relatively large pressure drop. Further modification of the flow boiling pressure drop models — using a modified two-phase multiplier for boiling — further improves their prediction. Comparison with the present experimental observations reveals that
International Journal of Thermal Sciences 132 (2018) 137–145
A. Iqbal, M. Pandey
Table 3 List of MAE in pressure drop calculation for different C with the different approaches. Authors [Reference]
Equations for Chisholm parameter (Liquid laminar and gas laminar)
MAE (S.P) (%)
MAE (L.P) (%)
MAE (L.P &H.F.E) (%)
Mishima and Hibiki [21]
C = 21(1 − e−0.319Dh)
50.89
35.96
18.67
Lee and Lee [22]
0.726 C = 6.185 × 10−2Refo
46.85
16.62
9.45
44.86
21.85
11.24 12.76
Qu and Mudawar [23]
C = 21(1 −
Lee and Mudawar [24]
0.047 We 0.60 C = 2.16Refo fo
46.85
26.79
Li and Wu [25]
For Bd < 0.1, C = 5.6Bd 0.28
46.72
27.28
14.65
36.71
8.41
8.41
43.08
16.33
9.26
Kim and Mudawar [26]
e−0.319Dh)(0.00418G
Cboiling = 3.5 × Sugo =
+ 0.0613)
0.44 Su 0.5 (ρ / ρ ) 0.48 10−5Refo go f g
ρg σDh / μg2
Wefo =
G 2Dh / ρf
(
0.52 Bo PH × ⎡1 + 530Wefo PF ⎢ ⎣
1.09
)
⎤ ⎥ ⎦
σ
Bo = q" / Ghfg Li and Hibiki [27]
0.42 0.21 C = 41.7Nμ0.66 TP ReTP x
Retp = GDh /μTP Nμtp = μtp/ ⎛ρTP σ ⎝ 1/μTP = (1 − x )/μf + x /μ g
σ gΔρ
0.5
⎞ ⎠
ρTP = (1 − x ) ρf + xρg
S.P. = System Pressure; L.P. = Local Pressure; H.F.E. = Heat Flux Effect.
in Table 2) to 8.4–18.7%. The aim of the present work was to develop a new approach for prediction of flow boiling pressure drop in microchannels by relaxing some of the assumptions made in the existing predictive methods. As demonstrated, in case of high pressure drop, the accuracy of prediction of any of the existing correlations can be improved by combining them with this approach. Comparison of various correlations was not within the scope of the present work. The new pressure-drop modelling approach can be used to improve the accuracy of heat transfer correlations for flow boiling in microchannels since the accurate prediction of local heat transfer coefficient requires a good estimation of axial pressure profile for evaluation of thermophysical properties and thermodynamic quality.
Acknowledgement Micro-fabrication facility for the fabrication of microchannels was provided by IIT Bombay, at its CEN (Center for Excellence in Nanotechnology) and Microfluidics Lab, Department of Mechanical Engineering. Fig. 7. Comparison of the existing correlations with present experimental data considering effect of flashing as well as effect of heating in the two-phase multiplier. Working Fluid: FC-72; DH = 111 μm; G = 431–690 kg/m2s; qeff = 4–8.5 W/cm2.
the new approach reduces the MAE for the all the correlations (listed Nomenclature A Bd
Area of Cross Section, m2
Bo Cp DH f G H h hfg L MAE
Boiling number Specific heat, kJ/kg K Hydraulic diameter, m Friction factor Mass flux, kg/m2 s Height Enthalpy, J/kg latent heat of vaporization Characteristic length, m Mean Absolute Error
⎛ρ − ρ ⎞ gL2 ⎜ f g⎟ ⎝ ⎠ σ
(Bond number)
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Nμ Viscosity number p Pressure, N/m2 pref Reference Pressure, N/m2 PF Wetted perimeter, m PH Heated perimeter, m Δp Pressure drop, N/m2 q Heat input, W qeff Effective Heat Flux W/m2 R Universal gas constant, J/Kkg Refo Liquid only Reynolds number Rego Gas only Reynolds number Su Suratman number T Temperature, K Tf Mean fluid temperature, K Tref Reference Temperature, K v Specific volume, m3/kg W Width, m vG 2L We (Weber number) σ x Thermodynamic equilibrium quality z Stream-wise coordinate, m Z Compressibility Factor Greek symbols α μ ν ρ σ φ Χ Subscripts
Void Fraction Co efficient of Dynamic viscosity, Ns/m2 Co efficient of Kinematic viscosity, m2/s Density, kg/m3 Surface Tension, N/m Two-phase multiplier Martinelli parameter
a b exp in f fo g Header pred o sat SP TP t w vv
Accelerational bottom experimental (measured) Inlet Liquid Liquid only Saturated vapor Inlet header predicted Outlet Saturation Single-phase Two-phase Top Wall Laminar liquid-laminar vapor
Appendix A. Modified Clausius–Clapeyron Relation Let p and T be the saturation vapor pressure and saturation temperature respectively. The Clausius–Clapeyron relation for the equilibrium between liquid and vapor is given by equation (A1)
hfg dp = dT T (vg − vf )
(A1)
Since vg ≫ vf
hfg dp = dT Tvg
(A2)
Now vg can be evaluated with equation (A3), which account the compressibility factor in the ideal gas equation.
vg =
ZRT p
(A3)
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phfg dp = dT ZRT 2
(A4)
Equation (A4) is the modified form of Clausius-Clapeyron equation. dhf dvg Derivation of dp and dp Considering vapor phase as an ideal gas, vg can be expressed by equation (A5). After differentiating equation (A5) we get equation (A6)
vg =
dvg dp
ZRT p
=
(A5)
ZR dT ZRT − p dp p2
From equation (A4) & (A6) we will get
dvg dp
=
(A6) dvg dp
given by equation (A8)
Z 2RT
⎡ RT − 1⎤ ⎥ p2 ⎢ ⎦ ⎣ hfg
(A8)
Again liquid saturated enthalpy can be expressed by equation (A9) and
hf − href = Cv, f (T − Tref ) + vf (p − pref ) dhf dp dhf dp
= Cv, f
= Cv, f
phfg
dp
will be given by equation (A11). (A9)
dT + vf dp ZRT 2
dhf
(A10)
+ vf
(A11)
Component Packaging Technology. VOL, 32 2009 (4). [16] P.S. Lee, S.V. Garimella, Saturated flow boiling heat transfer and pressure drop in silicon microchannel arrays, Int. J. Heat Mass Tran. 51 (2008) 789–806. [17] H.Y. Wu, P. Cheng, Friction factors in smooth trapezoidal silicon microchannels with different aspect ratios, Int. J. Heat Mass Tran. 46 (2003) 2519–2525. [18] B. Markal, O. Aydin, M. Avci, An experimental investigation of saturated flow boiling heat transfer and pressure drop in square microchannels, Int. J. Refrigeration 65 (2016) 1–11. [19] W. Qu, I. Mudawar, Transport phenomena in two-phase micro-channel heat sinks, J. Electron, Packag. Trans. ASME 126 (2004) 213–224. [20] J.G. Collier, J.R. Thome, Convective Boiling and Condensation, third ed., Oxford University Press, Oxford, 1994, pp. 69–70. [21] K. Mishima, T. Hibiki, Some characteristics of air–water two-phase flow in small diameter vertical tubes, Int. J. Multiphas. Flow 22 (1996) 703–712. [22] H.J. Lee, S.Y. Lee, Pressure drop correlations for two-phase flow within horizontal rectangular channels with small heights, Int. J. Multiphas. Flow 27 (2001) 783–796. [23] W. Qu, I. Mudawar, Measurements and prediction of pressure drop in two-phase micro-channel heat sinks, Int. J. Heat Mass Tran. 46 (2003) 2737–2753. [24] J. Lee, I. Mudawar, Two-phase flow in high-heat-flux micro-channel heat sink for refrigeration cooling applications: Part I––pressure drop characteristics, Int. J. Heat Mass Tran. 48 (2005) 928–940. [25] W. Li, Z. Wu, Generalized adiabatic pressure drop correlation in evaporative micro/ mini-channels, Exp. Therm. Fluid Sci. 35 (2011) 866–872. [26] S.M. Kim, I. Mudawar, Universal approach to predicting two-phase frictional pressure drop for mini/micro-channel saturated flow boiling, Int. J. Heat Mass Tran. 58 (2013) 718–734. [27] X. Li, T. Hibiki, Frictional pressure drop correlation for two-phase flows in mini and micro single-channels, Int. J. Multiphas. Flow 90 (2017) 29–45. [28] A. Iqbal, M. Pandey, Modeling of pressure drop in microchannel flow boiling, Proc. 22nd Nat. and 11th Int. ISHMT-ASME Heat Mass Transfer Conf. Kharagpur, India, 2013 Paper No: 366. [29] S.M. Kim, I. Mudawar, Flow condensation in parallel micro-channels – Part 1: experimental results and assessment of pressure drop correlations, Int. J. Heat Mass Tran. 55 (2012) 971–983. [30] N.V. Tarasova, A.I. Leont’ev, V.L. Hlopuskin, V.M. Orlov, Pressure drop of boiling subcooled water and steam-water mixture flowing in heated channels, Paper 133 Presented at 3rd Int. Heat Trans. Conf., Chicago, 4 1966, pp. 178–183.
References [1] S.T. Kadam, R. Kumar, Twenty first century cooling solution: microchannel heat sinks, Int. J. Thermal Sciences 85 (2014) 73–92. [2] Y. Xu, X. Fang, A new correlation of two-phase frictional pressure drop for evaporating flow in pipes, Int. J. Refrigeration 35 (2012) 2039–2050. [3] E.R. Dário, J.C. Passos, M.L. Sánchez Simón, L. Tadrist, Pressure drop during flow boiling inside parallel microchannels, Int. J. Refrigeration 72 (2016) 111–123. [4] M. Mirmanto, Heat transfer coefficient calculated using a linear pressure gradient assumption and measurement for flow boiling in microchannels, Int. J. Heat Mass Tran. 79 (2014) 269–278. [5] M. Mirmanto, Local pressure measurements and heat transfer coefficients of flow boiling in a rectangular microchannel, Heat Mass Tran. 52 (2016) 73–83. [6] C.Y. Park, C.Y. Jang, B. Kim, Y. Kim, Flow boiling heat transfer coefficients and pressure drop of FC-72 in microchannels, Int. J. Multiphas. Flow 39 (2012) 45–54. [7] A. Cioncolini, J.R. Thome, Pressure drop prediction in annular two-phase flow in macroscale tubes and channels, Int. J. Multiphas. Flow 89 (2017) 321–330. [8] A. Iqbal, M. Pandey, Two-phase pressure drop modeling in microchannels with effect of slip, surface tension and variable physical properties, Proc. 42nd Nat. Conf. Fluid Mechanics Fluid Power, Surathkal, 2015 India, Paper No: 276. [9] Y. Jang, C. Park, Y. Lee, Y. Kim, Flow boiling heat transfer coefficients and pressure drops of FC-72 in small channel heat sinks, Int. J. Refrig. 31 (2008) 1033–1041. [10] T. Harirchian, S.V. Garimella, A comprehensive flow regime map for microchannel flow boiling with quantitative transition criteria, Int. J. Heat Mass Tran. 53 (2010) 2694–2702. [11] T. Harirchian, S.V. Garimella, Microchannel size effects on local flow boiling heat transfer to a dielectric fluid, Int. J. Heat Mass Tran. 51 (2008) 3724–3735. [12] T. Harirchian, S.V. Garimella, Effects of channel dimension, heat flux, and mass flux on flow boiling regimes in microchannels, Int. J. Multiphas. Flow 35 (2009) 349–362. [13] T. Chen, S.V. Garimella, Local heat transfer distribution and effect on instabilities during flow boiling in a silicon microchannel heat sink, Int. J. Heat Mass Tran. 54 (2011) 3179–3190. [14] A. Megahed, I. Hassan, Two-phase pressure drop and flow visualization of FC-72 in a silicon microchannel heat sink, Int. J. Heat Fluid Flow 30 (2009) 1171–1182. [15] S.G. Singh, R. Bhide, B. Puranik, S.P. Duttagupta, A. Agrawal, Two phase flow pressure drop characteristics in trapezoidal silicon microchannels, IEEE Trans.
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
A simple methodology to incorporate flashing and variation of thermophysical properties for flow boiling pressure drop in a microchannel
T
Ashif Iqbal, Manmohan Pandey∗ Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, 781039, Assam, India
A R T I C LE I N FO
A B S T R A C T
Keywords: Flow boiling Pressure drop Flashing Local thermophysical properties Microchannel FC-72
This paper proposes a simple approach to model the complex interplay among the various thermophysical phenomena occurring in flow boiling. In order to assess the need for such a model, flow boiling pressure drop of FC-72 in a single trapezoidal microchannel with a hydraulic diameter of 111 μm is measured by varying heat flux and mass flux. The pressure drop obtained is in the range of 10–45 kPa, which does not compare well with the existing models based on constant properties. Therefore, a new predictive approach is developed for meticulous evaluation of pressure drop across a microchannel with flow boiling. It uses the separated flow model with evaluation of thermophysical properties at local pressure, thus incorporating the effect of flashing on thermodynamic quality, and the effect of heat flux on the two-phase multiplier. Relations based on a modified form of Clausius-Clapeyron equation are employed for evaluation of local thermophysical properties for fluids. This methodology is combined with different empirical correlations from the literature to predict pressure drop. The proposed predictive methodology, comprising close form equations, is physically sound and yet easy to implement, and reproduces large pressure drop experimental data better than the existing methods.
1. Introduction Dissipating high amount of heat flux is an important issue of modern thermal management with the ever increasing demands for high performance and miniaturization. Flow boiling microchannel heat sinks have emerged as one of the most effective solutions for cooling high and ultrahigh heat flux devices such as high performance computer chips, laser diodes and nuclear reactors [1]. Design of these miniature devices requires a proper estimation of two-phase heat transfer, which, in turn, necessitates accurate prediction of pressure drop in flow boiling. The total two-phase pressure drop of a fluid is the sum of the frictional pressure drop, the acceleration pressure drop, and the gravitational pressure drop. The frictional pressure drop can be determined by different two-phase models, and this has been an active research area for the last two decades. In order to calculate two-phase pressure drop, extensive theoretical and experimental studies have been conducted. Since the mechanisms occurring in two-phase flow have not been effectively understood, a number of empirical correlations have been proposed instead [2]. Two-phase pressure drop in microchannels is relatively high as compared to conventional channels, due to their very small sizes and moderate mass fluxes, the latter being so in order to achieve reasonable heat transfer coefficients. Due to the large pressure gradient, the
∗
Corresponding author. E-mail addresses: [email protected] (A. Iqbal), [email protected] (M. Pandey).
https://doi.org/10.1016/j.ijthermalsci.2018.05.046 Received 11 December 2017; Received in revised form 16 April 2018; Accepted 30 May 2018 1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved.
saturation temperature — and hence the thermophysical properties — vary along the length, and the effect of flashing becomes significant. The flashing phenomenon in mini-/microchannels occurs when the pressure at some axial location of the channel drops below the saturation pressure of the fluid, and the liquid at that location becomes superheated temporarily. Dario et al. [3] incorporated the effect of flashing on thermodynamic quality considering a linear pressure profile for heat transfer calculation. Very informative observations were reported by Mirmanto [4,5], by putting pressure sensors along the length of a microchannel. It was observed that the pressure profile along the length was nonlinear, especially at higher heat flux, due to rapid bubble generation and non-uniform distribution of nucleation sites [4]. Pressure drop reported in the experiments [4] was high (up to 75 kPa), therefore, the existing modelling approach, based on system pressure, cannot predict these experimental data accurately. Another group of researchers [6] tried to validate the existing correlations for two-phase pressure drop with their experimental data considering properties at system pressure, but ended up with a significant deviation. Cioncolini and Thome [7] considered the local pressure for evaluating thermophysical properties and calculated two-phase pressure drop, using the homogenous model. The evaporative channel was discretized into equal length subchannels, and the local pressure drops were computed using the local values of thermophysical properties, and the subchannel
International Journal of Thermal Sciences 132 (2018) 137–145
A. Iqbal, M. Pandey
pressure drops were added to get the total pressure drop [7]. In an earlier work [8], pressure drop for flow boiling in a microchannel was predicted using the homogenous model including the effect of local thermophysical properties and flashing. The Clausius-Clapeyron equation, which uses the ideal gas equation for the vapour phase, was employed for evaluation of local thermophysical properties [8]. However, the ideal gas equation is not valid for refrigerants and dielectric fluids, since the compressibility factor is not close to unity. For the electronic cooling applications, dielectric fluids (FC-72 and FC-77) are suitable because they are thermally and chemically stable, compatible with sensitive materials, nonflammable, and practically nontoxic [6,9]. However, there are limited number of studies [6,9–14] on flow boiling pressure drop of dielectric fluids in microchannels, i.e., channels that fulfil condition [10] (Bd 0.5Re = 160 ), under which microscale confinement effects are exhibited. Park et al. [6] investigated the two-phase pressure drop of FC-72 and stated that pressure drop increased with increasing vapor quality and mass flux, and was nearly independent of heat flux for any given exit quality and mass flux. In view of the importance of dielectric fluids, further research on their flow boiling pressure drop characteristics and development of predictive tools for the design of microchannel heat sinks is the need of the hour. In the present work, flow boiling experiments are conducted through a single microchannel using FC-72 and a relatively high pressure drop is observed. For high pressure drop, the effects of flashing and variation of thermophysical properties are significant. Therefore, a new methodology is developed, which is simple but successful (compares well with experimental data) to evaluate flow boiling pressure drop by incorporating the flashing effect and local thermophysical properties. The present approach includes a modified form of Clausius-Clapeyron equation to characterize the saturation line on the P-T plane for fluids whose vapor phase has nearly constant compressibility factor in the range of operation. The equations governing the proposed methodology are derived in an elegant close form, consisting of a system of ODEs, and their simulation is computationally inexpensive.
Table 1 Estimation of error in various parameters measured and derived in the experiments. Parameter
Value/Range
Uncertainty (%)
Top width (Wt ) Bottom width (Wb) Height (H) Length (L) Hydraulic Diameter (DH ) Heated Area (Aht ) Heat Flux (qeff )
245 μm 115 μm 90 μm 20 mm 111 μm 6.74 × 10−7 m2 4–8.5 W/cm2
± 0.43 ± 0.85 ± 1.11 ± 0.50 ± 1.48 ± 0.33 ± 5.01 to ± 2.03
Mass flux (G ) Pressure drop (Δp)
431–690 kg/m2s 10–45 kPa
± 5.32 to ± 2.52 ± 7.02 to ± 3.13
temperature. All devices were connected using silicone tubing (Master Flex). A pre-calibrated syringe pump (Cole Parmer) was used for metering and controlling the mass flow rate of FC 72. A pre-calibrated digital pressure gauge (Deltabar PMD75) was used to measure the pressure drop across the microchannel. A contact probe was used to make electric contact between microheater and DC power supply. K-type thermocouples (Cole-Parmer) were placed inside the inlet and outlet reservoirs. Another K-type thermocouple was placed on the bottom surface (microheater side) of the microchannel to measure the wall temperature. All the thermocouples were connected to a data logger which logs the temperatures. The test sections were fabricated using microfabrication facility available at CEN, IIT Bombay. Doublesided polished (100) p-type, 2″ silicon wafer with resistivity 4–7 Ω-cm was used for fabrication. Microchannel was etched using chemical etching process and a trapezoidal cross-section was obtained as shown in a cross sectional view of Fig. 1(b). It was then bonded with a quartz plate at the top. The fabrication process was the same as that used by Singh et al. [15]. Detailed geometrical specifications of the microchannel employed in the experiment are given in Table 1. The following procedure was adopted while performing the experiments. The flow was started and then an input electric power was provided by supplying a fixed voltage and current using a DC power supply. Continuous monitoring of the inlet and outlet temperatures and pressure drop was undertaken to check for steady state. Once steady state condition was achieved, all relevant parameters such as pressure drop, inlet–outlet temperatures, power supplied and flow rate were recorded. The above procedure was repeated for other values of flow rates and heat fluxes.
2. Experimental setup and procedure A schematic of the experimental setup employed in the measurements is shown in Fig. 1(a). The setup consists of a syringe pump, a differential pressure gauge, the test section, a DC power supply, and a data logger to record inlet/outlet temperatures of FC 72 and surface
Fig. 1. (a) Schematic of the experimental setup (b) Cross section of the test section. 138
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Fig. 2. (a) Heat loss calibration at no flow condition (b) comparison of friction factor correlation [17] with experimental data in no heat condition. Fluid: FC-72; DH = 111 μm.
quality is chosen as the total pressure drop is the pressure difference between the inlet and outlet headers. As seen from Fig. 3(a)–(b), total pressure drop increases with an increase in the heat flux or with the exit vapor quality for all the mass flux values. Here, the exit vapor quality was calculated considering the flashing effect. Markal et al. [18] observed a similar trend and explained that, due to increase in the heat flux, evaporation momentum force increases, which results in the fact that extended bubble toward the inlet of the channel begins to apply more force, and consequently, flow resistance increases. The force opposed to evaporation force is due to the inertia of the liquid and it increases with increasing mass flux. For very low flow rates (between 2.0 and 3.63 ml/min) and for very small channel dimensions (150 × 150 μm2), the interaction of these forces becomes significant [18]. In the present study, the channel dimension ( DH = 111 μm) and flow rates (0.2–0.4 ml/min) are quite low, thus resulting in observations similar to those reported previously [18]. Park et al. [6] stated that heat flux negligibly affected the pressure drop under identical mass flux and vapor quality conditions, indicating that most of the pressure drop was caused by friction. However, their calculation of thermodynamic vapor quality was based on the effect of heat flux only, without considering the effect of flashing. This assumption is valid in case of low pressure drops only. The pressure drops in their experiments were large, causing significant effect of flashing on the vapor quality. In the later part of the present study, an approach to evaluate thermodynamic quality with the effect of flashing is discussed. In this approach, the pressure profile is obtained using the separated flow model and the thermophysical properties are evaluated at local pressures.
Measurement uncertainties of the parameters are listed in Table 1. 3. Data reduction To estimate heat loss, a constant power was applied to the microheater, resulting in increase of the temperature of the test section. After steady state was reached, the wall temperatures on the test section together with that of the ambient were recorded and correlated with the power input to the microheater. This procedure was repeated for several input powers, as shown in Fig. 2(a). A linear relation qloss = 0.0385ΔTwa − 0.2372 was fitted to the data points. Here, ΔTwa is the average wall-to-ambient temperature difference (ΔTwa = Tw − Tamb). This method is the same as that used by Lee and Garimella [16]. Downstream of the syringe pump, the fluid gets heated in the inlet header before entering the microchannel. The heat utilized to heat the fluid inside the header is given by equation (1).
˙ p (Tin − Tsyringe ) qheader = mc
(1)
The effective heat flux was assumed to be uniform along the microchannel length and is given by equation (2). (Here qinput is the electric power supplied to the microheater.)
qeff = (qinput − qheader − qloss )/ Aht
(2)
The channel shape is trapezoidal and, therefore heat transfer area for the 3-side heated walls is given by equation (3) and the friction factor used for this calculation is given by equation (4) [17]. Equation (4) was compared with the present experimental data for adiabatic flow and showed a good prediction (Fig. 2(b)).
Aht = [2 ((wt − wb)/2)2 + H 2 + Wb] L
(3)
fRe = 11.43 + 0.80 exp (2.67Wb/Wt )
(4)
5. Existing approach for evaluation of two-phase pressure drop for miniature channel
Fig. 3(a)–(b) show the variation of total pressure drop with the wall heat flux and the exit vapor quality, respectively. Here, the exit vapor
The two-phase pressure drop in steady state can be expressed as the sum of frictional, gravitational and acceleration components. In twophase flow through microchannels, commonly observed flow patterns in the literature were slug flow and annular flow [19]. The homogeneous equilibrium model of two-phase flow assumes no slip between
4. Experimental result and discussion
Fig. 3. Experimental total pressure drop with the (a) wall heat flux and (b) the exit vapor quality. Working Fluid: FC-72; DH = 111 μm; G = 431–690 kg/m2s; qeff = 4–8.5 W/cm2. 139
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the phases and hence cannot accurately model these flow patterns [20]. Therefore, most of the literature used the separated flow model to obtain good agreement with experimental data because of its applicability with observed flow patterns (slug and annular flow) [21–28]. Frictional pressure drop component evaluated from separated flow model was expressed by equation (4) and acceleration pressure drop was given by equation (5). Thermodynamic equilibrium quality (x) was determined from the energy balance without the effect of flashing and given by equation (6). The exit quality (xo) was found by putting z = L .
small. It is observed that almost all the correlations are able to predict the data with reasonably good accuracy. In Fig. 4(b), the predictions of the same correlations (listed in Table 2) with the existing modelling approach are compared with experimental data from the literature [4] where the pressure drop was relatively high. It is observed that the existing approach, with any of the correlations, exhibits poor prediction of the experimental data for these cases, the percentage deviation being larger for higher pressure drop. The existing approach neglects the effect of local properties and the effect of flashing — assumptions valid for low pressure drop cases only. MAE (%) for each model is listed in Table 2 and given by equation (10).
xo
(Δptp ) = LTP / x o f
∫ 2G2vf (1 − xo)2φf2/Dh dx
(4a)
0 2 ⎡ xo ⎛ vg ⎞
)2
(1 − x o (Δptp ) = G 2vf ⎢ ⎜ ⎟ + a α v 1 − αo ⎣ o⎝ f⎠
x=
qeff Ph GAhfg
MAE (%) = ⎤ − 1⎥ ⎦
(5)
(z − LSP )
(6)
0.5
(v v ) f
g
0.5
(7)
⎛ (1 − x ) x ⎞ ⎝ ⎠
0.5
(8)
Evaluation of accelerational pressure drop requires a relation for the void fraction. In most of the literature related to miniature channels [21–24,28], Zivi's void fraction correlation, given by equation (9), has been used. 2
⎛ 1 − x o ⎞ ⎛ vg ⎞ α 0 = 1/ ⎜1 + ⎛ ⎜ ⎟ ⎜ ⎝ x 0 ⎠ ⎝ vf ⎠ ⎝ ⎜
⎟
Δpexp
× 100 (10)
6. A new approach to predict two-phase pressure drop As it has been observed, the predictions of the existing approach are good for low pressure drop cases but poor for relatively higher pressure drop cases. Therefore, the study proposes a new approach for accurate prediction of pressure drop for flow boiling of water, dielectric fluids and refrigerants, especially in case of high pressure drop. In most of the experiments, it was assumed that coolant enters the heat sink in subcooled condition and maintains a single-phase liquid state until it reaches zero thermodynamic equilibrium quality ( x = 0). Microchannels can be divided into two regions, single-phase and twophase regions, as shown in Fig. 5.
3⎞
⎟ ⎟ ⎠
Δppred − Δpexp
⁃ For relatively high pressure drop cases, assumption of system pressure instead of local pressure to evaluate thermophysical properties contributes a part of the total error. ⁃ Effect of flashing, i.e., the effect of pressure drop in the calculation of thermodynamic quality was not considered in the existing approach. Hence, the vapor quality is underpredicted, which results in the underestimation of the pressure drop. ⁃ Estimation of two-phase frictional pressure drop for boiling channels requires a different approach as opposed to non-boiling channels. This effect was incorporated in C by Kim and Mudawar (2013), thus resulting in the lowest MAE among all models listed in Table 2.
and the In frictional pressure drop, the two-phase multiplier Martinelli parameter (Χvv ) [20] for laminar vapor and laminar liquid were expressed by equation (7) and equation (8), respectively. All experimental data used [4,23] falls into the laminar vapor and laminar liquid region.
Χvv = ⎛ μf μ ⎞ g ⎝ ⎠
∑
The causes for this high margin of errors in the calculation of twophase pressure drop in a microchannel, in case of high pressure drop, are the following.
φf2
φf2 = 1 + C / Χvv + 1/ Χvv2
1 N
(9)
In the existing approach to pressure drop modeling, the Chisholm parameter (C) was modified to predict the pressure drop. All the modified expressions for the Chisholm parameter (C) are listed in Table 2. In Fig. 4(a), pressure drops predicted with the existing approach using different correlations [21–27] are compared with experimental data from the literature [23] where the pressure drop was Table 2 List of modified Chisholm parameter (C ) from literature. Authors [Reference]
Equations for Chisholm parameter (Liquid laminar and gas laminar)
MAE
MAE
High pressure drop (%)
Low pressure drop (%)
Mishima and Hibiki [21]
C = 21(1 − e319Dh)
43.73
17.35
Lee and Lee [22]
0.726 C = 6.185 × 10−2Refo
42.24
16.13
Qu and Mudawar [23]
C = 21(1 − e−319Dh)(0.00418G + 0.0613)
41.67
12.15
Lee and Mudawar [24]
0.047 We 0.60 C = 2.16Refo fo
40.12
11.95
Li and Wu [25]
Bd < 0.1, C = 5.6Bd 0.28
43.35
15.25
37.79
10.73
40.05
11.36
Kim and Mudawar [26]
Cboiling = 3.5 ×
0.44 Su 0.5 (ρ / ρ ) 0.48 10−5Refo go f g
(
0.52 Bo PH × ⎡1 + 530Wefo PF ⎢ ⎣
1.09
)
⎤ ⎥ ⎦
Sugo = ρg σDh / μg2 Wefo = G 2Dh / ρf σ
Bo = qeff / Ghfg Li and Hibiki [27]
0.42 0.21 C = 41.7Nμ0.66 TP ReTP x
Retp = GDh /μTP 0.5
Nμtp = μtp/ ⎛ρTP σ σ ⎞ gΔρ ⎠ ⎝ 1/μTP = (1 − x )/μf + x /μ g
ρTP = (1 − x ) ρf + xρg
140
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A. Iqbal, M. Pandey
Fig. 4. Comparison of the existing correlations for (a) Low pressure drop case with experimental data from Qu and Mudawar [23]; Water, DH = 349 μm and G = 255 kg/m2s (b) High pressure drop case with experimental data from Mirmanto [4], water, DH = 438 μm and G = 805 kg/m2s.
thermophysical properties can be evaluated at the system pressure. However, if the pressure drop is high, then these properties play a significant role in the evaluation of the pressure drop. To evaluate these thermophysical properties, the local saturation temperature is evaluated from the local pressure by using the Clausius-Clapeyron relation which considers the ideal gas approximation for the vapor phase. Iqbal and Pandey [8] used it for water for which the compressibility factor is close to unity. For a non-ideal gas condition, a modified form of Clausius-Clapeyron relation is derived, considering the compressibility factor (Z) by equation (15), where pref and Tref are the reference pressure and temperature, respectively. Thermophysical properties hf and vg vary significantly with local pressure. For evaluation of vg , modified ideal gas equation (pvg = ZRT ) is considered. Here Z is considered as constant since the variation along the length is not significant (2–3%). Change of vg and hf with respect to the pressure is given by equations
Fig. 5. Schematic diagram of the model.
6.1. Single-phase pressure drop In diabatic flow, the single phase pressure drop is significantly influenced by the variation of kinematic viscosity with temperature, can be evaluated by equation (11), which is a linear fit to property data. All dimensions are in SI unit. ( ν (T ) at m2s and T at K)
ν (T ) = 1.135 × 10−6 − 2.5872 × 10−9T
(16) and (17). Detailed derivations of the expressions for
qeff PH GAcp
(11)
dvg
dhf
(12)
dp
0
=
Z 2RT ⎡ RT − 1⎤ ⎥ p2 ⎢ h fg ⎦ ⎣
= Cv, f
(16)
ZRT 2 + vf phfg
(17)
{
} ( ){ ( ) ( ){
x 2vg α2
} ⎞⎠ ⎤⎥⎦ ⎟
}
(18)
GACp (Tsat − Tin ) PH qeff
and
(15)
2(1 − x ) vf (1 − x )2vf ∂α ⎡ 2flo G2vf 2 2 dx 2xvg ⎢ Dh φfo + G dz α − (1 − α ) + ∂x p (1 − α )2 − = ⎣ (1 − x )2vf x 2vg ⎡1 + G 2 ⎡ x 2 dvg + ∂α − 2 ⎤⎤ ∂p x α dp α (1 − α )2 ⎢ ⎥ ⎣ ⎦⎦ ⎣
(13)
where LSP is the length from the inlet to the saturation point and is given by equation (14) and fSP can be calculated from equation (4) for a trapezoidal microchannel.
LSP =
dp
dp −⎛ ⎞ ⎝ dz ⎠
LSP
∫ (fSP G2vf /DH ) dz
dvg
−1
The pressure drop in single phase region is evaluated by equation (13).
Δpsp =
,
⎡ 1 ZR ⎛ p ⎞ ⎤ T=⎢ − ln Tref hfg ⎜ pref ⎟ ⎥ ⎝ ⎠⎦ ⎣
dp
z + Tin
dp
modification of Clausius-Clapeyron equation are given in Appendix A.
From energy balance, the axial temperature profile can be expressed by equation (12)
T=
dhf
dhf dp ⎞ dx 1 ⎛ qeff PH = − ⎜ ⎟ dz hfg ⎝ GA dp dz ⎠
(14)
6.2. Two-phase pressure drop
(19)
In the two-phase region, boiling causes vapor generation, resulting in an increase in pressure gradient as compared to purely liquid flow. Consider the channel shown in Fig. 5. Considering the properties (vg and h f ) within the saturated region as functions of the local pressure, the pressure gradient [20] and the thermodynamic vapor quality x can be expressed by equations (18) and (19), respectively. The second term on the right hand side of equation (19) accounts for the axial change in quality due to flashing that results from enthalpy changes with
Two-phase pressure drop calculation in a microchannel has been a challenging topic for research community due to the complicated nature of boiling phenomenon. In the two-phase region, the pressure and hence the saturation temperature decrease along the length, thus causing change in thermophysical properties along the axial direction. If the pressure drop is low then, to a reasonable approximation, 141
International Journal of Thermal Sciences 132 (2018) 137–145
A. Iqbal, M. Pandey
Fig. 6. Comparison of the existing correlations with present experimental data (a) Considering properties at system pressure (b) Considering the effect of flashing and local thermophysical properties. Working Fluid: FC-72; DH = 111 μm; G = 431–690 kg/m2s; qeff = 4–8.5 W/cm2.
quality, the frictional pressure drop in the presence of a heat flux is higher than that in an unheated channel [20]. Therefore, a further modification was required on the other models [21–25,27] in order to improve their predictions. An empirical equation (20) relating the value of φfo2 in heated and unheated tubes for the steam-water system was reported in the literature [30]. This correlation was combined with the aforementioned models [21–25,27]. Then the results of these modified models were compared with the experimental data. From Fig. 7 and Table 3, it is observed that the effect of heat flux on φfo2 improves the predictions of the models. Equation (20) was not incorporated in the model of Kim and Mudawar [26] whose Chisholm parameter already contained the heat flux effect.
pressure. Zivi's void fraction correlation (equation (9)) is used to determine the void fraction. Equation (18) and equation (19) are coupled and were solved using 4th order Runge-Kutta method (ode45 function of MATLAB). 6.3. Comparison of the new approach with experimental results Experiments were conducted on a single microchannel with DH = 111 μm with mass flux range 431–690 kg/m2s and heat flux range 4–8.5 W/cm2. In the experiments, outlet pressure was considered to be atmospheric. In Fig. 6(a), experimental data were compared with the predictions of the existing modelling approach using different correlations (listed in Table 2), where properties were evaluated at system pressure and effect of flashing was not considered. It was observed that almost all the correlations predict the data with high MAE (36–50%). The reason for this prediction with such error in the two-phase pressure drop is an accumulation of uncertainties from the different parameters associated with the pressure drop calculation. Experimental two-phase pressure drop was in the range of 10–45 kPa, which, for FC-72 as the working fluid, causes a change in the saturation temperature in the range of about 4–9 °C along the length. Therefore, if the system pressure was considered instead of local pressure, the uncertainties in vg and hf will be in the range of 12–25% and 9–15%, respectively. Existing modelling approach did not incorporate the effect of flashing, and consequently, the thermodynamic quality ( x ) calculated with these models will be less than the actual value. Therefore, uncertainties associated with vg , hf and x , caused considerable error in the prediction of two-phase pressure drop. The present approach was applied to the existing correlations (listed in Table 3) and compared with the experimental data in Fig. 6(b), which showed significant improvement in their prediction. It was found that the new approach reduces the range of MAE for the all the correlations to 8.4–36%. As discussed earlier, the present approach is based on the evaluation of thermophysical properties (vg and hf ) at local pressure, and the effect of flashing on the thermodynamic quality. It is observed that incorporation of the effect of local properties and flashing improved the prediction to a great extent. Table 3 shows a comparison of MAE (%) of pressure drop predictions using different Ccorrelations. It reveals that the correlation of Kim and Mudawar [26], which incorporated the effect of heat flux on C , achieves the lowest MAE (8.4%), when combined with local thermophysical properties and flashing. This C-correlation is based on an earlier analysis [29] of the fundamental differences in flow structure between flow boiling and adiabatic flows. The difference is mainly due to the existence of entrained droplets in the vapor core for boiling flows and their absence from both condensing and adiabatic flows. Thus, the effect of heat flux on the pressure drop appears to be significant. At lower thermodynamic
0.7
q ⎡φ2 ⎤ ⎡1 + 4.4 × 10−3 ⎛ eff ⎞ ⎤ = ⎡φfo2 ⎤ ⎢ fo⎥ ⎢ ⎥ ⎢ G ⎠ ⎥ ⎝ ⎦ ⎣ ⎦heated tube ⎣ ⎦unheated tube ⎣
(20)
7. Conclusion This paper gives a new predictive approach for flow boiling pressure drop in microchannels. Experiments were conducted to assess the need for such a method. In case of large pressure drop, the effect of local thermophysical properties and flashing on the pressure gradient and the effect of heat flux on the two-phase multiplier are significant. The proposed methodology successfully incorporates the interplay between hydrodynamics, thermodynamics, heat transfer and phase change through simple close form equations. The contributions of the present study are as follow.
• The existing approach of pressure drop modeling — using properties • • • • • 142
evaluated at the system pressure — is found to give fairly accurate predictions for small pressure drop cases, For large pressure drop cases, the existing modelling approach does not predict pressure drop accurately. For such cases, MAE was 36–50% for the present experiments and 37–44% of the data from the literature [4]. The effect of local thermophysical properties has been incorporated by using a modified form of Clausius-Clapeyron equation derived for fluids whose vapor phase has compressibility factor nearly constant over the range of operation. The present pressure drop modeling approach — incorporating the effect of local properties and flashing — significantly improves the pressure drop predictions in case of relatively large pressure drop. Further modification of the flow boiling pressure drop models — using a modified two-phase multiplier for boiling — further improves their prediction. Comparison with the present experimental observations reveals that
International Journal of Thermal Sciences 132 (2018) 137–145
A. Iqbal, M. Pandey
Table 3 List of MAE in pressure drop calculation for different C with the different approaches. Authors [Reference]
Equations for Chisholm parameter (Liquid laminar and gas laminar)
MAE (S.P) (%)
MAE (L.P) (%)
MAE (L.P &H.F.E) (%)
Mishima and Hibiki [21]
C = 21(1 − e−0.319Dh)
50.89
35.96
18.67
Lee and Lee [22]
0.726 C = 6.185 × 10−2Refo
46.85
16.62
9.45
44.86
21.85
11.24 12.76
Qu and Mudawar [23]
C = 21(1 −
Lee and Mudawar [24]
0.047 We 0.60 C = 2.16Refo fo
46.85
26.79
Li and Wu [25]
For Bd < 0.1, C = 5.6Bd 0.28
46.72
27.28
14.65
36.71
8.41
8.41
43.08
16.33
9.26
Kim and Mudawar [26]
e−0.319Dh)(0.00418G
Cboiling = 3.5 × Sugo =
+ 0.0613)
0.44 Su 0.5 (ρ / ρ ) 0.48 10−5Refo go f g
ρg σDh / μg2
Wefo =
G 2Dh / ρf
(
0.52 Bo PH × ⎡1 + 530Wefo PF ⎢ ⎣
1.09
)
⎤ ⎥ ⎦
σ
Bo = q" / Ghfg Li and Hibiki [27]
0.42 0.21 C = 41.7Nμ0.66 TP ReTP x
Retp = GDh /μTP Nμtp = μtp/ ⎛ρTP σ ⎝ 1/μTP = (1 − x )/μf + x /μ g
σ gΔρ
0.5
⎞ ⎠
ρTP = (1 − x ) ρf + xρg
S.P. = System Pressure; L.P. = Local Pressure; H.F.E. = Heat Flux Effect.
in Table 2) to 8.4–18.7%. The aim of the present work was to develop a new approach for prediction of flow boiling pressure drop in microchannels by relaxing some of the assumptions made in the existing predictive methods. As demonstrated, in case of high pressure drop, the accuracy of prediction of any of the existing correlations can be improved by combining them with this approach. Comparison of various correlations was not within the scope of the present work. The new pressure-drop modelling approach can be used to improve the accuracy of heat transfer correlations for flow boiling in microchannels since the accurate prediction of local heat transfer coefficient requires a good estimation of axial pressure profile for evaluation of thermophysical properties and thermodynamic quality.
Acknowledgement Micro-fabrication facility for the fabrication of microchannels was provided by IIT Bombay, at its CEN (Center for Excellence in Nanotechnology) and Microfluidics Lab, Department of Mechanical Engineering. Fig. 7. Comparison of the existing correlations with present experimental data considering effect of flashing as well as effect of heating in the two-phase multiplier. Working Fluid: FC-72; DH = 111 μm; G = 431–690 kg/m2s; qeff = 4–8.5 W/cm2.
the new approach reduces the MAE for the all the correlations (listed Nomenclature A Bd
Area of Cross Section, m2
Bo Cp DH f G H h hfg L MAE
Boiling number Specific heat, kJ/kg K Hydraulic diameter, m Friction factor Mass flux, kg/m2 s Height Enthalpy, J/kg latent heat of vaporization Characteristic length, m Mean Absolute Error
⎛ρ − ρ ⎞ gL2 ⎜ f g⎟ ⎝ ⎠ σ
(Bond number)
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International Journal of Thermal Sciences 132 (2018) 137–145
A. Iqbal, M. Pandey
Nμ Viscosity number p Pressure, N/m2 pref Reference Pressure, N/m2 PF Wetted perimeter, m PH Heated perimeter, m Δp Pressure drop, N/m2 q Heat input, W qeff Effective Heat Flux W/m2 R Universal gas constant, J/Kkg Refo Liquid only Reynolds number Rego Gas only Reynolds number Su Suratman number T Temperature, K Tf Mean fluid temperature, K Tref Reference Temperature, K v Specific volume, m3/kg W Width, m vG 2L We (Weber number) σ x Thermodynamic equilibrium quality z Stream-wise coordinate, m Z Compressibility Factor Greek symbols α μ ν ρ σ φ Χ Subscripts
Void Fraction Co efficient of Dynamic viscosity, Ns/m2 Co efficient of Kinematic viscosity, m2/s Density, kg/m3 Surface Tension, N/m Two-phase multiplier Martinelli parameter
a b exp in f fo g Header pred o sat SP TP t w vv
Accelerational bottom experimental (measured) Inlet Liquid Liquid only Saturated vapor Inlet header predicted Outlet Saturation Single-phase Two-phase Top Wall Laminar liquid-laminar vapor
Appendix A. Modified Clausius–Clapeyron Relation Let p and T be the saturation vapor pressure and saturation temperature respectively. The Clausius–Clapeyron relation for the equilibrium between liquid and vapor is given by equation (A1)
hfg dp = dT T (vg − vf )
(A1)
Since vg ≫ vf
hfg dp = dT Tvg
(A2)
Now vg can be evaluated with equation (A3), which account the compressibility factor in the ideal gas equation.
vg =
ZRT p
(A3)
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International Journal of Thermal Sciences 132 (2018) 137–145
A. Iqbal, M. Pandey
phfg dp = dT ZRT 2
(A4)
Equation (A4) is the modified form of Clausius-Clapeyron equation. dhf dvg Derivation of dp and dp Considering vapor phase as an ideal gas, vg can be expressed by equation (A5). After differentiating equation (A5) we get equation (A6)
vg =
dvg dp
ZRT p
=
(A5)
ZR dT ZRT − p dp p2
From equation (A4) & (A6) we will get
dvg dp
=
(A6) dvg dp
given by equation (A8)
Z 2RT
⎡ RT − 1⎤ ⎥ p2 ⎢ ⎦ ⎣ hfg
(A8)
Again liquid saturated enthalpy can be expressed by equation (A9) and
hf − href = Cv, f (T − Tref ) + vf (p − pref ) dhf dp dhf dp
= Cv, f
= Cv, f
phfg
dp
will be given by equation (A11). (A9)
dT + vf dp ZRT 2
dhf
(A10)
+ vf
(A11)
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