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New models for heat transfer and pressure drop during flow boiling of R407C and R410A in a horizontal microfin tube
International Journal of Thermal Sciences 103 (2016) 57e66
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
New models for heat transfer and pressure drop during flow boiling of R407C and R410A in a horizontal microfin tube P. Rollmann*, K. Spindler Institute of Thermodynamics and Thermal Engineering (ITW), University of Stuttgart, Pfaffenwaldring 6, 70550 Stuttgart, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 February 2015 Received in revised form 22 November 2015 Accepted 23 November 2015 Available online 21 January 2016
The heat transfer of R407C and the pressure drop of R407C and R410A during flow boiling in a horizontal microfin tube were investigated. The measurements were conducted at saturation temperatures between 30 and þ10 C. The heat flux was varied between 1000 and 20,000 W/m2. The mass flux amounted to 25e300 kg/s m2 and the vapor quality was between 0.1 and 1.0. The microfin tube is made of copper with a total fin number of 55 and a helix angle of 15 . The fin height is 0.24 mm and the inner tube diameter at fin root is 8.95 mm. The test tube is 1 m long. It is heated electrically. A total number of 1614 heat transfer measurements and 952 pressure drop measurements was conducted. A new method for deriving correlations is shown. With this new method a new correlation for the Nusselt number is being derived. The mean deviation of the calculated from the measured values is 9.8%. Within the range of ±30% there are 94.2% of the measured values. A new correlation for the total pressure drop is derived. The mean deviation of the calculated from the measured values is 18.1%. Within the range of ±30% there are 83.1% of the measured values. © 2016 Elsevier Masson SAS. All rights reserved.
Keywords: Heat transfer coefficient Pressure drop Flow boiling R407C R410A Microfin tube Nusselt number correlation
1. Introduction There are many correlations for calculating the heat transfer coefficient during flow boiling. The correlation of Chen [1] from 1963 is considered as the first widely accepted correlation for the heat transfer of two-phase flows in smooth tubes. It is limited to vertical tubes. In 1982 the correlation of Shah [2] followed. With this correlation it is possible to determine the heat transfer coefficient in horizontal tubes with a correction factor. More smooth tube correlations followed in 1986 and 1987 by Gungor and Winterton [3, 4] and in 1991 by Liu and Winterton [5]. They were created for refrigerants such as R11, R12, R22 and R113. These correlations also require a correction factor for horizontal tubes [6]. With the advent of internally finned tubes correlations were needed, which may reflect the greater heat transfer coefficient. In 1990 Kandlikar developed a smooth tube correlation [7], which was expanded by amplification factors in 1991 for finned tubes [8].
* Corresponding author. E-mail address: [email protected] (P. Rollmann). http://dx.doi.org/10.1016/j.ijthermalsci.2015.11.010 1290-0729/© 2016 Elsevier Masson SAS. All rights reserved.
In 1999, Cavallini et al. [9] and Thome et al. [10] developed universal correlations, which include the geometric properties of the internally finned tubes. Cavallini et al. additionally differentiate between single component refrigerants and refrigerant mixtures. In 2006, Chamra and Mago [11] adapted Cavallini's fit constants to more refrigerants. Kattan et al. [12] developed a correlation based on flow pattern maps. In 2008, the correlation of Thome et al. was adapted to zeotropic refrigerant mixtures by Zhang et al. [13]. The model of Kattan et al. was expanded to R410A by Wojtan et al. [14] in 2005. Rollmann et al. [15] compared their measured heat transfer coefficients with calculated values by the correlation of Cavallini et al., Shah and Zhang et al. The measured values were also compared with the correlations of Liu and Winterton, Kattan et al. and Cavallini et al. by Rollmann and Spindler [16]. Correlations for the pressure drop during flow boiling have mostly been created for other refrigerants (e. g. R134a) or other pipes (e. g. straight tubes). The first correlations for the calculation of the pressure drop in two-phase flows have been developed around 1950 by Lockhart and Martinelli [17] and Pierre [18]. Often, these correlations were used as the basis for new correlations and they were improved, e. g. by Choi et al. [19].
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Nomenclature
Latin characters Bo boiling number [e] specific heat capacity at constant pressure [J/kg K] cp D diameter [m] Dhyd hydraulic diameter [m] Do outer diameter [m] G mass flux [kg/s m2] h heat transfer coefficient [W/m2 K] L length of the test section [m] m_ mass flow [kg/s] Nu Nusselt number [e] P electrical power [W] Pr Prandtl number [e] q_ heat flux [W/m2] 2 R coefficient of determination [e] Re Reynolds number [e] sk wall thickness [m] t fin height [m] T temperature [ C] U circumference [m] v specific volume [m3/kg] x vapor quality [e] z coordinate in tube length direction [m]
In 1996 Kuo and Wang [20] conducted measurements with R407C in a microfin tube. Therefor, they developed a correlation. Müller-Steinhagen and Heck [21] developed a correlation in 1986. It contains only a few parameters and is therefore very easy to use. Rollmann et al. [15,16] measured the pressure drop of R407C and compared it with the correlations of Pierre, Kuo and Wang, MüllerSteinhagen and Heck as well as Choi et al. Many correlations for the pressure drop, such as the equation proposed by Müller-Steinhagen and Heck [21] need the friction factor for turbulent flows in rough pipes. Since the ColebrookeWhite equation [22] could not be solved analytically, approximations like the well known Blasius equation were used. In 2015, Rollmann and Spindler [23] showed that there is an analytical solution of the ColebrookeWhite equation. Rollmann and Spindler [24] also examined the flow patterns during flow boiling. They found the new flow pattern helix flow and developed a new flow pattern map for horizontal microfin tubes. In recent years, the tube diameter tends to smaller values. In 2013, Wu et al. [25] examined R22 and R410A in five microfin tubes with outside diameter 5 mm. In 2014, Mancin et al. [26] conducted measurements with R134a in a 4 mm tube. Also other refrigerants were examined, such as natural gas in a 12.7 mm tube by Xu et al. [27] and the new refrigerants R1234yf in a 3/800 tube by Mendoza-Miranda et al. [28] as well as R1234ze in a 4 mm tube by Diani et al. [29]. Tables 3 and 4 show that most correlations for the heat transfer coefficient as well as the pressure drop have large deviations and scatter widely. Therefore, new correlations must be developed.
Greek characters b helix angle of fins [rad] DhV specific heat of vaporization [J/kg] Dp pressure drop [N/m2] z friction factor [e] h dynamic viscosity [kg/m s] l thermal conductivity [W/m K] r density [kg/m3] Indices ad calc exp F FR fric H I L m M mom O S V W
adiabatic calculated experimental fluid at fin root frictional heating inlet liquid mean measuring section momentum outlet saturation vapor wall
the pressure drop during flow boiling in horizontal microfin tubes. The measuring section is shown in Fig. 1. There are five measurement planes with four thermocouples at the circumference each. Thus, during one set of measurement parameters (see below) the heat transfer coefficient at 20 positions was calculated. In the following, the heat transfer coefficient is the mean of each plane. The geometrical dimensions of the electrically heated microfin tube are shown in Fig. 2. The apex angle is 25 . The measurements were conducted with R407C at the saturation temperatures 30; 10; 0 and þ10 C. The mass flux was 25; 62.5; 100; 150 and 300 kg/s m2. The heat flux was 1; 3; 5; 10 and 20 kW/m2. The local vapor quality varied between 0.102 and 0.998. The saturation pressure at the inlet of the test section was measured. By a heat balance (subcooled liquid and supplied heat flow) the local specific enthalpy of the refrigerant was calculated. From the saturation pressure and the specific enthalpy the local vapor quality and the saturation temperature were determined with REFPROP [30]. The mass flux was measured with a coriolis mass flow meter ðuncertainty : 0:001$ðm_ in kg=hÞ þ 0:0225Þ. The heat flux was calculated from the electrically supplied heat flow and the heat flow from the environment (uncertainty: 0.0025 $ P þ 0.001 $ measurement range). The pressure drop was measured directly with a piezoelectric pressure transducer (uncertainty: ±0.2775 mbar).
2. Flow boiling test facility A detailed description of the test facility can be found in Ref. [15]. It was designed to measure the heat transfer coefficient and
Fig. 1. Test section.
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Nu ¼
hDFR D q_ ¼ $ FR TW TS lL lL
59
(1)
In the following, the term Measured Nusselt numbers is used for the Nusselt numbers calculated according to equation (1) because they are based on measured heat transfer values. Fig. 3 shows the measured Nusselt number versus the vapor quality for the mass flux 300 kg/s m2, the saturation temperature 10 C and the heat fluxes 1; 3; 5; 10 and 20 kW/m2. All measured values lie on power functions of the form
NuðxÞ ¼ C5 xC6
Fig. 2. Dimensions of the microfin tube.
There are many data published in Ref. [15]. All data are published in Ref. [31]. There, one is able to extract all measured heat transfer coefficients, vapor qualities, mass fluxes, heat fluxes, saturation pressures at the inlet and pressure drops from both, plots and tables.
3. Derivation of the new heat transfer model The correlations mentioned in Section 1 calculate the heat transfer with large deviations compared to the measured values. As can be seen in Ref. [15], the data points in calculated versus measured values-diagrams scatter widely. This means that the underlying models do not describe the evaporation process correctly. Therefore, a new method for creating correlations is presented here. Although no knowledge of the physics is required, a correlation with low deviation and low scatter can be derived. First step: For all combinations of constant mass flux, heat flux and saturation temperature, the calculated Nusselt number versus the vapor quality is plotted. The Nusselt numbers are calculated with the measured heat transfer coefficient h, the diameter at fin root DFR and the thermal conductivity of the liquid phase lL:
Fig. 3. Nusselt number versus vapor quality for G ¼ 300 kg/s m2, TS ¼ 10 C and the heat fluxes 1; 3; 5; 10 and 20 kW/m2.
(2)
The coefficients of determination R2 are given in Fig. 3. They are nearly 1 for all power functions. This also applies to all mass fluxes, heat fluxes and saturation temperatures at which measurements were conducted and that are not shown. Second step: Since in the first step the coefficients of determination are in all cases nearly 1, the dependence of the Nusselt number on the vapor quality is completely reproduced by the power functions. The constants C5 and C6 are not depending on the vapor quality but they may be functions of the heat flux, the mass flux and the saturation €tze are chosen: temperature. The following ansa
C5 ¼ C5 ðBo; Re; PrÞ and C6 ¼ C6 ðBo; Re; PrÞ
(3)
The dimensionless quantities are calculated as follows:
Bo ¼
hL cp;L GDFR q_ ; Re ¼ ; Pr ¼ GDhV hL lL
(4)
For all combinations of constant mass flux and constant saturation temperature C5 is plotted versus the Boiling number Bo. Fig. 4 shows this for the saturation temperature 10 C and the mass fluxes 25; 62.5; 100; 150 and 300 kg/s m2. Logarithmic trend lines and the coefficients of determination R2 are shown. From the low coefficient of determination R2 ¼ 0.509, it can be seen that the logarithmic function cannot describe C5 versus the Boiling number Bo for the mass flux 25 kg/s m2. For the mass fluxes 62.5e300 kg/s m2, coefficients of determination near 1 are obtained. Therefore, the only restriction for the new Nusselt
Fig. 4. C5 versus the Boiling number for TS ¼ 10 C and G ¼ 25; 62.5; 100; 150 and 300 kg/s m2.
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correlation is made here. It is not suitable for small mass fluxes at which there is a stratified flow. The factor C5 can now be expressed as
C5 ¼ C51 lnðBoÞ þ C52
(5)
The constants C51 and C52 were determined for all logarithmic functions. It was found that the ratio
C3 ¼
C52 C51
(6)
is constant for all measured values. Equations (5) and (6) yield
C5 ðBoÞ ¼ C51 ½lnðBoÞ þ C3
(7)
Fig. 5 shows the exponent C6 versus the Boiling number Bo for the mass flux 300 kg/s m2 and the saturation temperature 10 C. With increasing Boiling number, C6 initially increases. However, the value decreases for the largest Boiling number. This trend cannot be described by power or logarithmic functions. Therefore, the points C6 $ Bo as well as a linear trend line and the coefficient of determination are also shown in Fig. 5. For all combinations of constant saturation temperature and constant mass flux, except for the already excluded mass flux of 25 kg/s m2, this linear trend is obtained. It was also found that all straight lines are lines through origin. From the general linear ansatz
C6 $Bo ¼ C61 $Bo þ C62 |fflfflffl{zfflfflffl} |fflfflfflffl{zfflfflfflffl} |{z} y
m$x
(8)
and with C62 ¼ 0 one obtains
(9)
The exponent is therefore not dependent on the Boiling number. With equations (7) and (9) the Nusselt correlation is
Nuðx; BoÞ ¼ C51 ½lnðBoÞ þ C3 xC61
C51 ðReÞ ¼ C511 ReC512
(10)
(11)
and the coefficients of determination are shown. For the four saturation temperatures, almost identical values for C512 are found. The mean value is 2/3. This results in
C51 ðReÞ ¼ C511 Re2=3
b
C6 ¼ C61
Fig. 6. C51 versus Re for TS ¼ 30; 10; 0 and 10 C.
(12)
Fig. 7 shows the exponent C61 (equation (10)) versus the Reynolds number for the saturation temperatures 30; 10; 0 and 10 C. In this form, no relation between C61 and the Reynolds number can be seen. Fig. 8 shows the product C61 $ Re2 versus the Reynolds number to the power of two Re2 for the four saturation temperatures. In addition, linear trend lines and the coefficients of determination are given. Again a general linear ansatz can be found:
C61 $Re2 ¼ C611 $Re2 þ C612
(13)
Third step: Fig. 6 shows C51 versus the Reynolds number for the saturation temperatures 30; 10; 0 and 10 C. In addition, trend lines of the form
Since the trend lines are straight lines through origin, with C612 ¼ 0 one obtains
Fig. 5. C6 and C6 $ Bo versus Bo for TS ¼ 10 C and G ¼ 300 kg/s m2.
Fig. 7. C61 versus Re for TS ¼ 30; 10; 0 and 10 C.
P. Rollmann, K. Spindler / International Journal of Thermal Sciences 103 (2016) 57e66
C611 ¼
61
C1 þ C2 Pr 2
(18)
That finally yields the new Nusselt correlation
Nuðx; Bo; Re; PrÞ ¼ C4
C1 2=3 þ C ½lnðBoÞ þ C3 x 2 Re Pr 2
C1 þC2 Pr 2
(19) Fifth step: In the previous steps it was not taken into account that the constants C1, C2, C3 and C4 may influence each other in a minor way. They were therefore adjusted to the measured values in a final step:
C1 ¼ 3:7; C2 ¼ 0:71; C3 ¼ 12:17; C4 ¼ 1:2
Fig. 10 shows the calculated Nusselt numbers according to equation (19) versus the measured values for R407C. The mean deviation of the calculated from the measured values is 9.81%. Within the range of ±30% there are 94.18% of the measured values. The values scatter in a minor way.
Fig. 8. C61 $ Re2 versus Re2 for TS ¼ 30; 10; 0 and 10 C.
C61 ¼ C611
(20)
(14)
Thus, the exponent C61 is not dependent on the Reynolds number. With equations (12) and (14) the Nusselt correlation is
Nuðx; Bo; ReÞ ¼ C511 Re2=3 ½lnðBoÞ þ C3 xC611
(15)
Fourth step: The influence of the saturation temperature on the Nusselt number is taken into account by using the Prandtl number. Fig. 9 shows C511 $ Pr2 and C611 $ Pr2 versus Pr2. The quotients C511Pr2/C611Pr2 are nearly equal. With the quotient C4¼ C511/C611 the Nusselt correlation can be transformed:
Nuðx; Bo; ReÞ ¼ C4 C611 Re2=3 ½lnðBoÞ þ C3 xC611
(16)
3.1. Discussion of the new heat transfer model Due to a dryout of the tube wall, there can be a reduction in the heat transfer coefficient with increasing vapor quality. Although measurements up to x ¼ 0.998 were made, this was not observed in any of the own measurements. Because of the temperature glide of R407C, the refrigerant mixture separates during evaporation. If there is a partial dryout of the tube wall, the vapor phase superheats. The vapor quality is calculated from a heat balance. Although there is a calculated vapor quality of 1, the two-phase flow consists of a superheated gas phase and a liquid phase. The helical fins improve the wetting of the tube wall. This leads to large heat transfer coefficients. Mathematically, there are three restrictions for the new Nusselt correlation:
With the linear ansatz
C611 $Pr 2 ¼ C2 $Pr 2 þ C1
(17)
one obtains
Fig. 9. C511 $ Pr2 and C611 $ Pr2 versus Pr2.
Fig. 10. Comparison of the calculated Nusselt numbers according for R407C to equation (19) versus the measured Nusselt numbers.
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The following must apply: [ln(Bo) þ C3] > 0, otherwise Nu 0. Solved for the Boiling number and C3 ¼ 12.17 inserted, yields
q_ Bo ¼ ¼ GDhV
Q_ pDFR L 4m_ Dh V pD2 FR
¼
_ V Dx mDh pDFR L 4m_ Dh V pD2
¼
DFR Dx > 5:1837$106 4L
FR
(21) This Boiling number results in a vapor quality increase over the measuring section (L ¼ 1 m; DFR ¼ 0.0088 m) of Dx ¼ 0.0024. Since almost no liquid is evaporated, there is virtually no flow boiling. However, the new correlation has been created for flow boiling. The following must apply: ððC1 =Pr 2 Þ þ C2 Þ > 0, otherwise Nu 0. Solved for the Prandtl number and C1 ¼ 3.7 as a well as C2 ¼ 0.71 inserted, yields
Pr > 2:2828
(22)
For R407C the run of the curve Pr(T) has a minimum at T ¼ 46.88 C with the value Pr ¼ 2.72. Thus, condition (22) is met for all temperatures. The following must apply:
C1 þ C2 Pr 2
1
(23)
Otherwise Nu(x) is not a power function. Since C1 < 0 and C2 > 0, the expression reaches its theoretical maximum for Pr / ∞, because then C1/Pr2 tends to 0. In this case the term takes the maximum value C2 ¼ 0.71. Thus, condition (23) is always met.
3.2. Validation of the new heat transfer model with values from literature In the literature, few data points for the heat transfer coefficient are published. Often, only diagrams of the form “calculated versus measured value” are shown. From these, no certain test parameters can be assigned to the measured values. Therefore, the dimensionless quantities Nu, Bo, Re and Pr cannot be calculated. Measured values of Lallemand et al. [32] and Passos et al. [33] were tested with the new correlation. Table 1 compares the tube geometry and the parameter ranges of the test series. Both authors used a microfin tube with a diameter at fin root of 11.98 mm. The new correlation was created on the basis of measured values in a tube with diameter at fin root of 8.95 mm. Only the publication of Kuo and Wang [20] was found in which the experiments were also made with R407C in a tube with the same diameter at fin root. However, there may be a systematic error. On the one hand, very low heat transfer coefficients were measured. On the other hand, the measured values are almost independent of
Fig. 11. Comparison of the calculated heat transfer coefficients for R407C according to the new correlation versus the measured values of Passos et al.
the vapor quality or decrease from the vapor quality x ¼ 0.35 on with increasing vapor quality. Figs. 11 and 12 show the calculated heat transfer coefficients according to the new correlation versus the measured values of Passos et al. and Lallemand et al. Furthermore, points are shown which have been calculated with the new correlation and an additional pre-factor of 1.5. In both cases the values are on straight lines. This means that the correlation describes the heat transfer process during flow boiling well. The pre-factor can be explained by the different tube geometries. The larger helix and apex angle as well as the larger numbers of fins increase the heat transfer area. The heat transfer is improved. However, there are too few measurements to model the factor C4 with different tube geometries. With the pre-factor 1.5, the mean deviations are 3.81% (Passos et al.) and 10.26% (Lallemand et al.). Within the range of ±30% there are 98.1% and 95.3% of the measured values. 3.3. Applicability of the new heat transfer model using other tube geometries and refrigerants The new Nusselt correlation was derived for the refrigerant R407C in one microfin tube. It is expected that the runs of the
Table 1 Comparison of the tube geometry and the parameter ranges of Rollmann and Spindler, Lallemand et al. and Passos et al. Rollmann and Spindler
Lallemand et al.
Passos et al.
Diameter at fin root [mm] Fin height [mm] Helix angle [ ] Apex angle [ ] Number of fins
8.95 0.24 15 25 55
11.98 0.22 and 0.25a 20 40 70
11.98 0.25 30 65 65
Vapor quality Heat flux [kW/m2] Mass flux [kg/s m2] Saturation temperature [ C]
0.10e1.00 1e20 62.5e300 30 to 10
0.08e0.98 10 100e250 10.2
0.09e0.96 10 200 9.72
a
Alternating.
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63
Since the densities of the gas phase and the liquid phase hardly vary, despite of the temperature increase by the temperature glide, they can be considered as constant. The vapor quality increases approximately linearly. The function
xðzÞ ¼
xO xI z þ xI L
(29)
can be applied. The integration yields
Dp ¼
Fig. 12. Comparison of the calculated heat transfer coefficients for R407C according to the new correlation versus the measured values of Lallemand et al.
curves shown in Figs. 3e9 are qualitatively the same for other refrigerants and tube geometries. The constants C1eC4 must be adjusted experimentally. 4. Derivation of the new pressure drop model The total pressure drop of a diabatic two-phase flow in a horizontal microfin tube is the sum of the momentum pressure drop Dpmom and the frictional pressure drop Dpfric:
Dp ¼ Dpmom þ Dpfric
1 2L zG ½ðxI þ xO ÞðvV vL Þ þ 2vL 4 D
(30)
In the measurements, the friction factor was independent of the Reynolds number and was therefore kept constant. The new value z ¼ 0.05 was found. Fig. 13 shows the calculated over the measured pressure drop for the refrigerants R407C and R410A. The mean deviation of the calculated from the measured values is 18.1%. Within the range of ±30% there are 83.1% of the measured values. In 5.6% of the measurements, the measured value was smaller than the accuracy of the pressure transducer. These measurements were not taken into account. For comparison, the changes in the densities of the gas and the liquid phase due to the temperature glide were considered. For the densities and instead of the linear function for the vapor quality (equation (29)), polynomials of degree six were chosen. In addition, the momentum pressure drop was taken into account. The mean deviation is then 17.1%. However, the equation becomes unwieldy and for each calculation an integration is necessary. The new correlation calculates the total pressure drop and is applicable for R407C as well as for R410A. The parameter range G ¼ 25 to 300 kg/m2 s, q_ ¼ 1000 to 20,000 W/m2, TS ¼ 30 to þ10 C and x ¼ 0.1e1.0 is covered.
(24)
The local frictional pressure drop can be calculated with the DarcyeWeisbach equation:
dpfric 1 * G2 vm ¼ z 2 dz D
(25)
The momentum pressure drop only slightly contributes to the total pressure drop. It can be taken into account by adjusting the friction factor. The resulting error is small. That results in
dp 1 G2 vm ¼ z dz 2 D
(26)
The total pressure drop is obtained by integration over the length L of the test section. Here, z can be considered as constant.
Dp ¼
1 G2 z 2 D
ZL vm ðzÞdz
(27)
z¼0
The mean specific volume is calculated according to the homogeneous model:
vm ðzÞ ¼
xðzÞ 1 xðzÞ þ rV ðzÞ rL ðzÞ
(28) Fig. 13. Calculated over measured pressure drop for the refrigerants R407C and R410A.
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4.1. Discussion of the new pressure drop model In the new correlation for the pressure drop, the outlet vapor quality can be replaced:
Dx ¼ xO xI ¼
Q_ _ mDh V
(31)
_ FR L as well as m_ ¼ GðpD2FR =4Þ, the With equation (31) and Q_ ¼ qpD new correlation for the pressure drop becomes:
Dp ¼
1 2 L zG 2 DFR
Dp ¼
1 2 L zG 2 DFR
2L q_ þ xI ðvV vL Þ þ vL DFR GDhV
(32)
2L Bo þ xI ðvV vL Þ þ vL DFR
(33)
It can be seen that the pressure drop is not proportional to the square of the mass flux. The mass flux is also in the denominator of the Boiling number and weakens the influence on the pressure drop. In correlations for the pressure drop of single-phase flows, this attenuation is included in the friction factor which is then not constant. The new correlation can be simplified for adiabatic applications. With Bo ¼ 0 one obtains:
Dpad ¼
1 2 L zG ½x ðv vL Þ þ vL 2 DFR I V
(34)
For the boundary values xI ¼ 0 and xI ¼ 1 equation (34) becomes the equations for the adiabatic single-phase pressure drop:
Bo ¼ 0; xI ¼ 0 0 Dpad;L ¼
1 21 L 1 L zG ¼ zr u2 2 rL DFR 2 L L DFR
(35)
Bo ¼ 0; xI ¼ 1 0 Dpad;V ¼
1 21 L 1 L zG ¼ zr u2 2 rV DFR 2 V V DFR
(36)
4.2. Validation of the new pressure drop model with values from literature Measurement values from the publications of Kuo and Wang [20] as well as Passos et al. [33] were tested with the new correlation. Table 2 compares the tube geometries and parameter ranges of the test series. Both, Kuo and Wang and Passos et al. used a microfin tube and conducted experiments with heat supply. Kuo and Wang also conducted experiments under adiabatic conditions. Fig. 14 shows the comparison of the values calculated with the new correlation and the measurement values of Kuo and Wang.
Fig. 14. Comparison of the pressure drop values calculated with the new correlation and the measurement values of Kuo and Wang for R407C.
The mean deviation of the calculated from the measured values is 20.2%. Within the range of ±30% there are 72.2% of the measured values. Both, the measurement values with heat supply as well as the measurements under adiabatic conditions can be calculated. This confirms the assumption that the momentum pressure drop only slightly contributes to the total pressure drop. Fig. 15 shows the comparison of the values calculated with the new correlation and the measurement values of Passos et al. The mean deviation of the calculated from the measured values is 18.7%. Within the range of ±30% there are 77.8% of the measured values. The largest deviations arise with the test tube with the small diameter at fin root 6.48 mm. With decreasing diameter, the influence of the diameter on the pressure drop increases. Kandlikar defines the transition to mini channels at Dhyd ¼ 3 mm [34]. Not all geometric parameters for the calculation of the hydraulic diameter are available. Compared to the smooth tube, the area where the fluid flows through decreases, while the wetted circumference increases. With Dhyd ¼ 4A=U the hydraulic diameter of microfin tubes is substantially smaller than the diameter at fin root. The measuring tube with the diameter at fin root of 6.48 mm may have a hydraulic diameter in the order of the transition to mini channels. Correlations for internally structured mini channels are not known.
Table 2 Comparison of the tube geometries and the parameter ranges of Rollmann and Spindler, Kuo and Wang as well as Passos et al. Rollmann and Spindler
Kuo and Wang
Passos et al.
Diameter at fin root [mm] Fin height [mm] Helix angle [ ] Apex angle [ ] Number of fins
8.95 0.24 15 25 55
8.92 0.2 18 Not specified 60
11.98 0.25 30 65 65
Vapor quality Heat flux [kW/m2] Mass flux [kg/m2 s] Saturation temperature [ C]
0.10e1.00 1e20 62.5e300 30 to 10
0.09e0.79 0 and 10 100e300 1.7
0.09e0.96 10 200 9.72
6.48 0.15 18 50 50
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65
based on the often confirmed DarcyeWeisbach equation, it can be assumed that it is valid for different tube geometries and refrigerants and is applicable by adjusting the friction factor z. 5. Conclusions and outlook
Fig. 15. Comparison of the pressure drop values calculated with the new correlation and the measurement values of Passos et al. for R407C.
4.3. Applicability of the new pressure drop model using other tube geometries and refrigerants The new correlation for the pressure drop was derived for the refrigerants R407C and R410A in one microfin tube. Because it is
A total number of 1614 heat transfer measurements and 952 pressure drop measurements was conducted. It was shown in previous publications [15,16] that correlations from literature for the heat transfer and the pressure drop during flow boiling often calculate the heat transfer coefficient with large deviations from the measured values. In calculated versus measured values-diagrams the data points scatter widely. A new correlation for the Nusselt number during flow boiling of R407C in a horizontal microfin tube was derived. The data points in the calculated versus measured values-diagram (10) scatter in a minor way. The mean deviation is 9.81%. Within the range of ±30% there are 94.18% of the measured values. Table 3 summarizes the results of the correlations from literature and the new correlation. Also, a new correlation for the pressure drop of R407C and R410A during flow boiling in a horizontal microfin tube was derived. The data points in the calculated versus measured values-diagram (13) scatter in a minor way. The mean deviation is 18.1%. Within the range of ±30% there are 83.1% of the measured values. Table 4 summarizes the results of the correlations from literature and the new correlation. Rollmann et al. [31] tested the Kandlikar correlation [8]. It is planned to verify this correlation and obtain the enhancement parameters listed therein. It is also planned to verify the new correlation with more data sets and obtain the constants C1eC4 for more refrigerants. This will be done as soon as sufficient measurement data of low GWP refrigerants are available and these refrigerants come into the market.
Table 3 Summary of the correlations for the heat transfer coefficient during flow boiling. Correlation
Within the range ±30%
Percentage of measurements
49.9% 38.7% 86.5% 24.8% 79.5% 30.6%
17.8% 47.6% 26.9% 77.3% 27.5% 63.6%
100% 100% 100% 100% 100% 79.1%
171.9% 148.6% 53.0% 36.9%
2.2% 2.5% 52.2% 55.8%
100% 87.4% 100% 87.4%
9.8%
94.2%
85.1%
Mean deviation
Shah Shah with pre-factor Liu and Winterton Liu and Winterton with pre-factors Cavallini et al. Cavallini et al. with pre-factors and without q_ ¼ 1 kW=m2 Kattan et al. Kattan et al. with x < 0.85 Zhang et al. Zhang et al. with x < 0.85 Rollmann and Spindler
Table 4 Summary of the correlations for the pressure drop during flow boiling. Correlation
Mean deviation
Within the range ±30%
Percentage of measurements
Pierrea Choi et al.a Kuo and Wanga VDI Heat Atlasa Müller-Steinhagen and Hecka Müller-Steinhagen and Heck with pre-factor 1.35a
96.7% 22.2% 27.9% 29.0% 38.4% 31.3%
22.7% 76.1% 77.1% 84.3% 45.4% 73.2%
100% 51.3% 100% 32.0% 100% 100%
Rollmann and Spindlerb
18.1%
83.1%
94.4%
a b
R407C. R407C and R410A.
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Acknowledgments This work was supported by the German Bundesministerium für Wirtschaft (BMWi) through the Arbeitsgemeinschaft industrieller Forschungsvereinigungen (AiF). The project was accompanied by a €ltetechnik e.V. (FKT), Frankworking party of the Forschungsrat Ka furt a. M., Germany. References [1] J.C. Chen, Correlation for boiling heat transfer to saturated fluids in convective flow, Ind. Eng. Chem. Process Des. Develop. 5 (3) (1966) 322e329, http:// dx.doi.org/10.1021/i260019a023. [2] M.M. Shah, Chart correlation for saturated boiling heat transfer equations and further study, ASHRAE Trans. 88 (1) (1982) 185e196. [3] K.E. Gungor, R.H.S. Winterton, A general correlation for flow boiling in tubes and annuli, Int. J. Heat Mass Transfer 29 (3) (1986) 351e358, http://dx.doi.org/ 10.1016/0017-9310(86)90205-X. [4] K.E. Gungor, R.H.S. Winterton, Simplified general correlation for saturated flow boiling and comparisons of correlations with data, Chem. Eng. Res. Des. 65 (1987) 148e156. [5] Z. Liu, R.H.S. Winterton, A general correlation for saturated and subcooled flow boiling in tubes and annuli, based on a nucleate pool boiling equation, Int. J. Heat Mass Transfer 34 (11) (1991) 2759e2766, http://dx.doi.org/ 10.1016/0017-9310(91)90234-6. [6] A. Bejan, A.D. Kraus, Heat Transfer Handbook, J. Wiley, New York, 2003. [7] S.G. Kandlikar, A general correlation for saturated two-phase flow boiling heat transfer inside horizontal and vertical tubes, J. Heat Transfer 112 (1) (1990) 219e228, http://dx.doi.org/10.1115/1.2910348. [8] S.G. Kandlikar, A model for correlating flow boiling heat transfer in augmented tubes and compact evaporators, J. Heat Transfer 113 (4) (1991) 966e972, http://dx.doi.org/10.1115/1.2911229. [9] A. Cavallini, D. Del Col, L. Doretti, G.A. Longo, L. Rossetto, Refrigerant vaporisation inside enhanced tubes: a heat transfer model, Heat Technol. 17 (2) (1999) 29e36. [10] J.R. Thome, N. Kattan, D. Favrat, Evaporation in microfin tubes: a generalized prediction model, Proc. Int. Eng. Found. 3 (1999) 239e245. [11] L.M. Chamra, P.J. Mago, Modelling of evaporation heat transfer of pure refrigerants and refrigerant mixtures in microfin tubes, Proc. Instn. Mech. Engr., C: J. Mech. Eng. Sci. 221 (4) (2007) 443e447, http://dx.doi.org/10.1243/ 0954406JMES131. [12] N. Kattan, J.R. Thome, D. Favrat, Flow boiling in horizontal tubes: part 3 e development of a new heat transfer model based on flow pattern, J. Heat Transfer 120 (1) (1998) 156e165, http://dx.doi.org/10.1115/1.2830039. [13] X. Zhang, C. Ji, X. Yuan, Prediction method for evaporation heat transfer of non-azeotropic refrigerant mixtures flowing inside internally grooved tubes, Appl. Therm. Eng. 28 (14e15) (2008) 1974e1983, http://dx.doi.org/10.1016/ j.applthermaleng.2007.12.009. [14] L. Wojtan, T. Ursenbacher, J.R. Thome, Investigation of flow boiling in horizontal tubes: part IIddevelopment of a new heat transfer model for stratifiedwavy, dryout and mist flow regimes, Int. J. Heat Mass Transfer 48 (14) (2005) 2970e2985, http://dx.doi.org/10.1016/j.ijheatmasstransfer.2004.12.013. [15] P. Rollmann, K. Spindler, H. Müller-Steinhagen, Heat transfer, pressure drop and flow patterns during flow boiling of R407C in a horizontal microfin tube, Heat Mass Transfer 47 (8) (2011) 951e961, http://dx.doi.org/10.1007/s00231011-0857-x. €rmeübergang und Druckverlust beim [16] P. Rollmann, K. Spindler, Wa €mungssieden von R407C e Vergleich verschiedener Korrelationen, in: Stro
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€lte- und Klimatechnischer Verein (Ed.), Deutsche Ka €lte-KlimaDeutscher Ka Tagung 2010 Magdeburg, vol. II.1, 2010. R.W. Lockhart, R.C. Martinelli, Proposed correlation of data for isothermal two-phase, two component flow in pipes, Chem. Eng. Progr. 45 (1) (1949) 39e48. B. Pierre, Flow resistance with boiling refrigerants e part 1, ASHRAE J. 6 (9) (1964) 58e65. J.Y. Choi, M.A. Kedzierski, P.A. Domanski (Eds.), Generalized Pressure Drop Correlation for Evaporation and Condensation in Smooth and Micro-fin Tubes, 2001. C.-S. Kuo, C.-C. Wang, Horizontal flow boiling of R22 and R407C in a 9.52 mm micro-fin tube, Appl. Therm. Eng. 16 (8e9) (1996) 719e731, http://dx.doi.org/ 10.1016/1359-4311(95)00076-3. H. Müller-Steinhagen, K. Heck, A simple friction pressure drop correlation for two-phase flow in pipes, Chem. Eng. Process. Process Intensif. 20 (6) (1986) 297e308, http://dx.doi.org/10.1016/0255-2701(86)80008-3. C.F. Colebrook, C.M. White, Experiments with fluid friction in roughened pipes, Proc. Roy. Soc. A: Math. Phys. Eng. Sci. 161 (906) (1937) 367e381, http://dx.doi.org/10.1098/rspa.1937.0150. P. Rollmann, K. Spindler, Explicit representation of the implicit colebrook white equation, Case Stud. Therm. Eng. 5 (2015) 41e47, http://dx.doi.org/ 10.1016/j.csite.2014.12.001. P. Rollmann, K. Spindler, A new flow pattern map for flow boiling in microfin tubes, Int. J. Multiph. Flow 72 (2015) 181e187, http://dx.doi.org/10.1016/ j.ijmultiphaseflow.2015.01.003. Z. Wu, Y. Wu, B. Sunden, W. Li, Convective vaporization in micro-fin tubes of different geometries, Exp. Therm. Fluid Sci. 44 (2013) 398e408, http:// dx.doi.org/10.1016/j.expthermflusci.2012.07.012. S. Mancin, A. Diani, L. Rossetto, R134a flow boiling heat transfer and pressure drop inside a 3.4mm ID microfin tube, Energy Proc. 45 (2014) 608e615, http://dx.doi.org/10.1016/j.egypro.2014.01.065. B. Xu, Y. Shi, D. Chen, Investigation on the heat transfer characteristics during flow boiling of liquefied natural gas in a vertical micro-fin tube, Cryogenics 60 (2014) 68e75, http://dx.doi.org/10.1016/j.cryogenics.2014.01.009. J. Mendoza-Miranda, J. Ramirez-Minguela, V. Munoz-Carpio, J. Navarro-Esbri, Development and validation of a micro-fin tubes evaporator model using R134a and R1234yf as working fluids, Int. J. Refrigeration 50 (2015) 32e43, http://dx.doi.org/10.1016/j.ijrefrig.2014.10.021. A. Diani, S. Mancin, L. Rossetto, R1234ze(E) flow boiling inside a 3.4mm ID microfin tube, Int. J. Refrigeration 47 (2014) 105e119, http://dx.doi.org/ 10.1016/j.ijrefrig.2014.07.018. E.W. Lemmon, M.L. Huber, M.O. McLinden, NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties e REFPROP, 2013. €rmeübergang und DruckP. Rollmann, K. Spindler, H. Müller-Steinhagen, Wa €mungssieden von Ka €ltemittelgemischen in einem horizonverlust beim Stro talen innenstrukturierten Rohr (Final report of AiF project no. 15084), Forschungsrat K€ altetechnik e.V., Frankfurt/M, 2010. change locaux au M. Lallemand, C. Branescu, P. Haberschill, Coefficients d'e bullition du R22 et du R407C dans des tubes horizontaux, lisse ou cours de l'e , Int. J. Refrigeration 24 (1) (2001) 57e72, http://dx.doi.org/ micro-ailete 10.1016/S0140-7007(00)00064-5. J.C. Passos, V.F. Kuser, P. Haberschill, M. Lallemand, Convective boiling of R407c inside horizontal microfin and plain tubes, Exp. Therm. Fluid Sci. 27 (6) (2003) 705e713, http://dx.doi.org/10.1016/S0894-1777(02)00308-4. S.G. Kandlikar, Fundamental issues related to flow boiling in minichannels and microchannels, Exp. Therm. Fluid Sci. 26 (2e4) (2002) 389e407, http:// dx.doi.org/10.1016/S0894-1777(02)00150-4.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
New models for heat transfer and pressure drop during flow boiling of R407C and R410A in a horizontal microfin tube P. Rollmann*, K. Spindler Institute of Thermodynamics and Thermal Engineering (ITW), University of Stuttgart, Pfaffenwaldring 6, 70550 Stuttgart, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 February 2015 Received in revised form 22 November 2015 Accepted 23 November 2015 Available online 21 January 2016
The heat transfer of R407C and the pressure drop of R407C and R410A during flow boiling in a horizontal microfin tube were investigated. The measurements were conducted at saturation temperatures between 30 and þ10 C. The heat flux was varied between 1000 and 20,000 W/m2. The mass flux amounted to 25e300 kg/s m2 and the vapor quality was between 0.1 and 1.0. The microfin tube is made of copper with a total fin number of 55 and a helix angle of 15 . The fin height is 0.24 mm and the inner tube diameter at fin root is 8.95 mm. The test tube is 1 m long. It is heated electrically. A total number of 1614 heat transfer measurements and 952 pressure drop measurements was conducted. A new method for deriving correlations is shown. With this new method a new correlation for the Nusselt number is being derived. The mean deviation of the calculated from the measured values is 9.8%. Within the range of ±30% there are 94.2% of the measured values. A new correlation for the total pressure drop is derived. The mean deviation of the calculated from the measured values is 18.1%. Within the range of ±30% there are 83.1% of the measured values. © 2016 Elsevier Masson SAS. All rights reserved.
Keywords: Heat transfer coefficient Pressure drop Flow boiling R407C R410A Microfin tube Nusselt number correlation
1. Introduction There are many correlations for calculating the heat transfer coefficient during flow boiling. The correlation of Chen [1] from 1963 is considered as the first widely accepted correlation for the heat transfer of two-phase flows in smooth tubes. It is limited to vertical tubes. In 1982 the correlation of Shah [2] followed. With this correlation it is possible to determine the heat transfer coefficient in horizontal tubes with a correction factor. More smooth tube correlations followed in 1986 and 1987 by Gungor and Winterton [3, 4] and in 1991 by Liu and Winterton [5]. They were created for refrigerants such as R11, R12, R22 and R113. These correlations also require a correction factor for horizontal tubes [6]. With the advent of internally finned tubes correlations were needed, which may reflect the greater heat transfer coefficient. In 1990 Kandlikar developed a smooth tube correlation [7], which was expanded by amplification factors in 1991 for finned tubes [8].
* Corresponding author. E-mail address: [email protected] (P. Rollmann). http://dx.doi.org/10.1016/j.ijthermalsci.2015.11.010 1290-0729/© 2016 Elsevier Masson SAS. All rights reserved.
In 1999, Cavallini et al. [9] and Thome et al. [10] developed universal correlations, which include the geometric properties of the internally finned tubes. Cavallini et al. additionally differentiate between single component refrigerants and refrigerant mixtures. In 2006, Chamra and Mago [11] adapted Cavallini's fit constants to more refrigerants. Kattan et al. [12] developed a correlation based on flow pattern maps. In 2008, the correlation of Thome et al. was adapted to zeotropic refrigerant mixtures by Zhang et al. [13]. The model of Kattan et al. was expanded to R410A by Wojtan et al. [14] in 2005. Rollmann et al. [15] compared their measured heat transfer coefficients with calculated values by the correlation of Cavallini et al., Shah and Zhang et al. The measured values were also compared with the correlations of Liu and Winterton, Kattan et al. and Cavallini et al. by Rollmann and Spindler [16]. Correlations for the pressure drop during flow boiling have mostly been created for other refrigerants (e. g. R134a) or other pipes (e. g. straight tubes). The first correlations for the calculation of the pressure drop in two-phase flows have been developed around 1950 by Lockhart and Martinelli [17] and Pierre [18]. Often, these correlations were used as the basis for new correlations and they were improved, e. g. by Choi et al. [19].
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Nomenclature
Latin characters Bo boiling number [e] specific heat capacity at constant pressure [J/kg K] cp D diameter [m] Dhyd hydraulic diameter [m] Do outer diameter [m] G mass flux [kg/s m2] h heat transfer coefficient [W/m2 K] L length of the test section [m] m_ mass flow [kg/s] Nu Nusselt number [e] P electrical power [W] Pr Prandtl number [e] q_ heat flux [W/m2] 2 R coefficient of determination [e] Re Reynolds number [e] sk wall thickness [m] t fin height [m] T temperature [ C] U circumference [m] v specific volume [m3/kg] x vapor quality [e] z coordinate in tube length direction [m]
In 1996 Kuo and Wang [20] conducted measurements with R407C in a microfin tube. Therefor, they developed a correlation. Müller-Steinhagen and Heck [21] developed a correlation in 1986. It contains only a few parameters and is therefore very easy to use. Rollmann et al. [15,16] measured the pressure drop of R407C and compared it with the correlations of Pierre, Kuo and Wang, MüllerSteinhagen and Heck as well as Choi et al. Many correlations for the pressure drop, such as the equation proposed by Müller-Steinhagen and Heck [21] need the friction factor for turbulent flows in rough pipes. Since the ColebrookeWhite equation [22] could not be solved analytically, approximations like the well known Blasius equation were used. In 2015, Rollmann and Spindler [23] showed that there is an analytical solution of the ColebrookeWhite equation. Rollmann and Spindler [24] also examined the flow patterns during flow boiling. They found the new flow pattern helix flow and developed a new flow pattern map for horizontal microfin tubes. In recent years, the tube diameter tends to smaller values. In 2013, Wu et al. [25] examined R22 and R410A in five microfin tubes with outside diameter 5 mm. In 2014, Mancin et al. [26] conducted measurements with R134a in a 4 mm tube. Also other refrigerants were examined, such as natural gas in a 12.7 mm tube by Xu et al. [27] and the new refrigerants R1234yf in a 3/800 tube by Mendoza-Miranda et al. [28] as well as R1234ze in a 4 mm tube by Diani et al. [29]. Tables 3 and 4 show that most correlations for the heat transfer coefficient as well as the pressure drop have large deviations and scatter widely. Therefore, new correlations must be developed.
Greek characters b helix angle of fins [rad] DhV specific heat of vaporization [J/kg] Dp pressure drop [N/m2] z friction factor [e] h dynamic viscosity [kg/m s] l thermal conductivity [W/m K] r density [kg/m3] Indices ad calc exp F FR fric H I L m M mom O S V W
adiabatic calculated experimental fluid at fin root frictional heating inlet liquid mean measuring section momentum outlet saturation vapor wall
the pressure drop during flow boiling in horizontal microfin tubes. The measuring section is shown in Fig. 1. There are five measurement planes with four thermocouples at the circumference each. Thus, during one set of measurement parameters (see below) the heat transfer coefficient at 20 positions was calculated. In the following, the heat transfer coefficient is the mean of each plane. The geometrical dimensions of the electrically heated microfin tube are shown in Fig. 2. The apex angle is 25 . The measurements were conducted with R407C at the saturation temperatures 30; 10; 0 and þ10 C. The mass flux was 25; 62.5; 100; 150 and 300 kg/s m2. The heat flux was 1; 3; 5; 10 and 20 kW/m2. The local vapor quality varied between 0.102 and 0.998. The saturation pressure at the inlet of the test section was measured. By a heat balance (subcooled liquid and supplied heat flow) the local specific enthalpy of the refrigerant was calculated. From the saturation pressure and the specific enthalpy the local vapor quality and the saturation temperature were determined with REFPROP [30]. The mass flux was measured with a coriolis mass flow meter ðuncertainty : 0:001$ðm_ in kg=hÞ þ 0:0225Þ. The heat flux was calculated from the electrically supplied heat flow and the heat flow from the environment (uncertainty: 0.0025 $ P þ 0.001 $ measurement range). The pressure drop was measured directly with a piezoelectric pressure transducer (uncertainty: ±0.2775 mbar).
2. Flow boiling test facility A detailed description of the test facility can be found in Ref. [15]. It was designed to measure the heat transfer coefficient and
Fig. 1. Test section.
P. Rollmann, K. Spindler / International Journal of Thermal Sciences 103 (2016) 57e66
Nu ¼
hDFR D q_ ¼ $ FR TW TS lL lL
59
(1)
In the following, the term Measured Nusselt numbers is used for the Nusselt numbers calculated according to equation (1) because they are based on measured heat transfer values. Fig. 3 shows the measured Nusselt number versus the vapor quality for the mass flux 300 kg/s m2, the saturation temperature 10 C and the heat fluxes 1; 3; 5; 10 and 20 kW/m2. All measured values lie on power functions of the form
NuðxÞ ¼ C5 xC6
Fig. 2. Dimensions of the microfin tube.
There are many data published in Ref. [15]. All data are published in Ref. [31]. There, one is able to extract all measured heat transfer coefficients, vapor qualities, mass fluxes, heat fluxes, saturation pressures at the inlet and pressure drops from both, plots and tables.
3. Derivation of the new heat transfer model The correlations mentioned in Section 1 calculate the heat transfer with large deviations compared to the measured values. As can be seen in Ref. [15], the data points in calculated versus measured values-diagrams scatter widely. This means that the underlying models do not describe the evaporation process correctly. Therefore, a new method for creating correlations is presented here. Although no knowledge of the physics is required, a correlation with low deviation and low scatter can be derived. First step: For all combinations of constant mass flux, heat flux and saturation temperature, the calculated Nusselt number versus the vapor quality is plotted. The Nusselt numbers are calculated with the measured heat transfer coefficient h, the diameter at fin root DFR and the thermal conductivity of the liquid phase lL:
Fig. 3. Nusselt number versus vapor quality for G ¼ 300 kg/s m2, TS ¼ 10 C and the heat fluxes 1; 3; 5; 10 and 20 kW/m2.
(2)
The coefficients of determination R2 are given in Fig. 3. They are nearly 1 for all power functions. This also applies to all mass fluxes, heat fluxes and saturation temperatures at which measurements were conducted and that are not shown. Second step: Since in the first step the coefficients of determination are in all cases nearly 1, the dependence of the Nusselt number on the vapor quality is completely reproduced by the power functions. The constants C5 and C6 are not depending on the vapor quality but they may be functions of the heat flux, the mass flux and the saturation €tze are chosen: temperature. The following ansa
C5 ¼ C5 ðBo; Re; PrÞ and C6 ¼ C6 ðBo; Re; PrÞ
(3)
The dimensionless quantities are calculated as follows:
Bo ¼
hL cp;L GDFR q_ ; Re ¼ ; Pr ¼ GDhV hL lL
(4)
For all combinations of constant mass flux and constant saturation temperature C5 is plotted versus the Boiling number Bo. Fig. 4 shows this for the saturation temperature 10 C and the mass fluxes 25; 62.5; 100; 150 and 300 kg/s m2. Logarithmic trend lines and the coefficients of determination R2 are shown. From the low coefficient of determination R2 ¼ 0.509, it can be seen that the logarithmic function cannot describe C5 versus the Boiling number Bo for the mass flux 25 kg/s m2. For the mass fluxes 62.5e300 kg/s m2, coefficients of determination near 1 are obtained. Therefore, the only restriction for the new Nusselt
Fig. 4. C5 versus the Boiling number for TS ¼ 10 C and G ¼ 25; 62.5; 100; 150 and 300 kg/s m2.
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correlation is made here. It is not suitable for small mass fluxes at which there is a stratified flow. The factor C5 can now be expressed as
C5 ¼ C51 lnðBoÞ þ C52
(5)
The constants C51 and C52 were determined for all logarithmic functions. It was found that the ratio
C3 ¼
C52 C51
(6)
is constant for all measured values. Equations (5) and (6) yield
C5 ðBoÞ ¼ C51 ½lnðBoÞ þ C3
(7)
Fig. 5 shows the exponent C6 versus the Boiling number Bo for the mass flux 300 kg/s m2 and the saturation temperature 10 C. With increasing Boiling number, C6 initially increases. However, the value decreases for the largest Boiling number. This trend cannot be described by power or logarithmic functions. Therefore, the points C6 $ Bo as well as a linear trend line and the coefficient of determination are also shown in Fig. 5. For all combinations of constant saturation temperature and constant mass flux, except for the already excluded mass flux of 25 kg/s m2, this linear trend is obtained. It was also found that all straight lines are lines through origin. From the general linear ansatz
C6 $Bo ¼ C61 $Bo þ C62 |fflfflffl{zfflfflffl} |fflfflfflffl{zfflfflfflffl} |{z} y
m$x
(8)
and with C62 ¼ 0 one obtains
(9)
The exponent is therefore not dependent on the Boiling number. With equations (7) and (9) the Nusselt correlation is
Nuðx; BoÞ ¼ C51 ½lnðBoÞ þ C3 xC61
C51 ðReÞ ¼ C511 ReC512
(10)
(11)
and the coefficients of determination are shown. For the four saturation temperatures, almost identical values for C512 are found. The mean value is 2/3. This results in
C51 ðReÞ ¼ C511 Re2=3
b
C6 ¼ C61
Fig. 6. C51 versus Re for TS ¼ 30; 10; 0 and 10 C.
(12)
Fig. 7 shows the exponent C61 (equation (10)) versus the Reynolds number for the saturation temperatures 30; 10; 0 and 10 C. In this form, no relation between C61 and the Reynolds number can be seen. Fig. 8 shows the product C61 $ Re2 versus the Reynolds number to the power of two Re2 for the four saturation temperatures. In addition, linear trend lines and the coefficients of determination are given. Again a general linear ansatz can be found:
C61 $Re2 ¼ C611 $Re2 þ C612
(13)
Third step: Fig. 6 shows C51 versus the Reynolds number for the saturation temperatures 30; 10; 0 and 10 C. In addition, trend lines of the form
Since the trend lines are straight lines through origin, with C612 ¼ 0 one obtains
Fig. 5. C6 and C6 $ Bo versus Bo for TS ¼ 10 C and G ¼ 300 kg/s m2.
Fig. 7. C61 versus Re for TS ¼ 30; 10; 0 and 10 C.
P. Rollmann, K. Spindler / International Journal of Thermal Sciences 103 (2016) 57e66
C611 ¼
61
C1 þ C2 Pr 2
(18)
That finally yields the new Nusselt correlation
Nuðx; Bo; Re; PrÞ ¼ C4
C1 2=3 þ C ½lnðBoÞ þ C3 x 2 Re Pr 2
C1 þC2 Pr 2
(19) Fifth step: In the previous steps it was not taken into account that the constants C1, C2, C3 and C4 may influence each other in a minor way. They were therefore adjusted to the measured values in a final step:
C1 ¼ 3:7; C2 ¼ 0:71; C3 ¼ 12:17; C4 ¼ 1:2
Fig. 10 shows the calculated Nusselt numbers according to equation (19) versus the measured values for R407C. The mean deviation of the calculated from the measured values is 9.81%. Within the range of ±30% there are 94.18% of the measured values. The values scatter in a minor way.
Fig. 8. C61 $ Re2 versus Re2 for TS ¼ 30; 10; 0 and 10 C.
C61 ¼ C611
(20)
(14)
Thus, the exponent C61 is not dependent on the Reynolds number. With equations (12) and (14) the Nusselt correlation is
Nuðx; Bo; ReÞ ¼ C511 Re2=3 ½lnðBoÞ þ C3 xC611
(15)
Fourth step: The influence of the saturation temperature on the Nusselt number is taken into account by using the Prandtl number. Fig. 9 shows C511 $ Pr2 and C611 $ Pr2 versus Pr2. The quotients C511Pr2/C611Pr2 are nearly equal. With the quotient C4¼ C511/C611 the Nusselt correlation can be transformed:
Nuðx; Bo; ReÞ ¼ C4 C611 Re2=3 ½lnðBoÞ þ C3 xC611
(16)
3.1. Discussion of the new heat transfer model Due to a dryout of the tube wall, there can be a reduction in the heat transfer coefficient with increasing vapor quality. Although measurements up to x ¼ 0.998 were made, this was not observed in any of the own measurements. Because of the temperature glide of R407C, the refrigerant mixture separates during evaporation. If there is a partial dryout of the tube wall, the vapor phase superheats. The vapor quality is calculated from a heat balance. Although there is a calculated vapor quality of 1, the two-phase flow consists of a superheated gas phase and a liquid phase. The helical fins improve the wetting of the tube wall. This leads to large heat transfer coefficients. Mathematically, there are three restrictions for the new Nusselt correlation:
With the linear ansatz
C611 $Pr 2 ¼ C2 $Pr 2 þ C1
(17)
one obtains
Fig. 9. C511 $ Pr2 and C611 $ Pr2 versus Pr2.
Fig. 10. Comparison of the calculated Nusselt numbers according for R407C to equation (19) versus the measured Nusselt numbers.
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The following must apply: [ln(Bo) þ C3] > 0, otherwise Nu 0. Solved for the Boiling number and C3 ¼ 12.17 inserted, yields
q_ Bo ¼ ¼ GDhV
Q_ pDFR L 4m_ Dh V pD2 FR
¼
_ V Dx mDh pDFR L 4m_ Dh V pD2
¼
DFR Dx > 5:1837$106 4L
FR
(21) This Boiling number results in a vapor quality increase over the measuring section (L ¼ 1 m; DFR ¼ 0.0088 m) of Dx ¼ 0.0024. Since almost no liquid is evaporated, there is virtually no flow boiling. However, the new correlation has been created for flow boiling. The following must apply: ððC1 =Pr 2 Þ þ C2 Þ > 0, otherwise Nu 0. Solved for the Prandtl number and C1 ¼ 3.7 as a well as C2 ¼ 0.71 inserted, yields
Pr > 2:2828
(22)
For R407C the run of the curve Pr(T) has a minimum at T ¼ 46.88 C with the value Pr ¼ 2.72. Thus, condition (22) is met for all temperatures. The following must apply:
C1 þ C2 Pr 2
1
(23)
Otherwise Nu(x) is not a power function. Since C1 < 0 and C2 > 0, the expression reaches its theoretical maximum for Pr / ∞, because then C1/Pr2 tends to 0. In this case the term takes the maximum value C2 ¼ 0.71. Thus, condition (23) is always met.
3.2. Validation of the new heat transfer model with values from literature In the literature, few data points for the heat transfer coefficient are published. Often, only diagrams of the form “calculated versus measured value” are shown. From these, no certain test parameters can be assigned to the measured values. Therefore, the dimensionless quantities Nu, Bo, Re and Pr cannot be calculated. Measured values of Lallemand et al. [32] and Passos et al. [33] were tested with the new correlation. Table 1 compares the tube geometry and the parameter ranges of the test series. Both authors used a microfin tube with a diameter at fin root of 11.98 mm. The new correlation was created on the basis of measured values in a tube with diameter at fin root of 8.95 mm. Only the publication of Kuo and Wang [20] was found in which the experiments were also made with R407C in a tube with the same diameter at fin root. However, there may be a systematic error. On the one hand, very low heat transfer coefficients were measured. On the other hand, the measured values are almost independent of
Fig. 11. Comparison of the calculated heat transfer coefficients for R407C according to the new correlation versus the measured values of Passos et al.
the vapor quality or decrease from the vapor quality x ¼ 0.35 on with increasing vapor quality. Figs. 11 and 12 show the calculated heat transfer coefficients according to the new correlation versus the measured values of Passos et al. and Lallemand et al. Furthermore, points are shown which have been calculated with the new correlation and an additional pre-factor of 1.5. In both cases the values are on straight lines. This means that the correlation describes the heat transfer process during flow boiling well. The pre-factor can be explained by the different tube geometries. The larger helix and apex angle as well as the larger numbers of fins increase the heat transfer area. The heat transfer is improved. However, there are too few measurements to model the factor C4 with different tube geometries. With the pre-factor 1.5, the mean deviations are 3.81% (Passos et al.) and 10.26% (Lallemand et al.). Within the range of ±30% there are 98.1% and 95.3% of the measured values. 3.3. Applicability of the new heat transfer model using other tube geometries and refrigerants The new Nusselt correlation was derived for the refrigerant R407C in one microfin tube. It is expected that the runs of the
Table 1 Comparison of the tube geometry and the parameter ranges of Rollmann and Spindler, Lallemand et al. and Passos et al. Rollmann and Spindler
Lallemand et al.
Passos et al.
Diameter at fin root [mm] Fin height [mm] Helix angle [ ] Apex angle [ ] Number of fins
8.95 0.24 15 25 55
11.98 0.22 and 0.25a 20 40 70
11.98 0.25 30 65 65
Vapor quality Heat flux [kW/m2] Mass flux [kg/s m2] Saturation temperature [ C]
0.10e1.00 1e20 62.5e300 30 to 10
0.08e0.98 10 100e250 10.2
0.09e0.96 10 200 9.72
a
Alternating.
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63
Since the densities of the gas phase and the liquid phase hardly vary, despite of the temperature increase by the temperature glide, they can be considered as constant. The vapor quality increases approximately linearly. The function
xðzÞ ¼
xO xI z þ xI L
(29)
can be applied. The integration yields
Dp ¼
Fig. 12. Comparison of the calculated heat transfer coefficients for R407C according to the new correlation versus the measured values of Lallemand et al.
curves shown in Figs. 3e9 are qualitatively the same for other refrigerants and tube geometries. The constants C1eC4 must be adjusted experimentally. 4. Derivation of the new pressure drop model The total pressure drop of a diabatic two-phase flow in a horizontal microfin tube is the sum of the momentum pressure drop Dpmom and the frictional pressure drop Dpfric:
Dp ¼ Dpmom þ Dpfric
1 2L zG ½ðxI þ xO ÞðvV vL Þ þ 2vL 4 D
(30)
In the measurements, the friction factor was independent of the Reynolds number and was therefore kept constant. The new value z ¼ 0.05 was found. Fig. 13 shows the calculated over the measured pressure drop for the refrigerants R407C and R410A. The mean deviation of the calculated from the measured values is 18.1%. Within the range of ±30% there are 83.1% of the measured values. In 5.6% of the measurements, the measured value was smaller than the accuracy of the pressure transducer. These measurements were not taken into account. For comparison, the changes in the densities of the gas and the liquid phase due to the temperature glide were considered. For the densities and instead of the linear function for the vapor quality (equation (29)), polynomials of degree six were chosen. In addition, the momentum pressure drop was taken into account. The mean deviation is then 17.1%. However, the equation becomes unwieldy and for each calculation an integration is necessary. The new correlation calculates the total pressure drop and is applicable for R407C as well as for R410A. The parameter range G ¼ 25 to 300 kg/m2 s, q_ ¼ 1000 to 20,000 W/m2, TS ¼ 30 to þ10 C and x ¼ 0.1e1.0 is covered.
(24)
The local frictional pressure drop can be calculated with the DarcyeWeisbach equation:
dpfric 1 * G2 vm ¼ z 2 dz D
(25)
The momentum pressure drop only slightly contributes to the total pressure drop. It can be taken into account by adjusting the friction factor. The resulting error is small. That results in
dp 1 G2 vm ¼ z dz 2 D
(26)
The total pressure drop is obtained by integration over the length L of the test section. Here, z can be considered as constant.
Dp ¼
1 G2 z 2 D
ZL vm ðzÞdz
(27)
z¼0
The mean specific volume is calculated according to the homogeneous model:
vm ðzÞ ¼
xðzÞ 1 xðzÞ þ rV ðzÞ rL ðzÞ
(28) Fig. 13. Calculated over measured pressure drop for the refrigerants R407C and R410A.
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P. Rollmann, K. Spindler / International Journal of Thermal Sciences 103 (2016) 57e66
4.1. Discussion of the new pressure drop model In the new correlation for the pressure drop, the outlet vapor quality can be replaced:
Dx ¼ xO xI ¼
Q_ _ mDh V
(31)
_ FR L as well as m_ ¼ GðpD2FR =4Þ, the With equation (31) and Q_ ¼ qpD new correlation for the pressure drop becomes:
Dp ¼
1 2 L zG 2 DFR
Dp ¼
1 2 L zG 2 DFR
2L q_ þ xI ðvV vL Þ þ vL DFR GDhV
(32)
2L Bo þ xI ðvV vL Þ þ vL DFR
(33)
It can be seen that the pressure drop is not proportional to the square of the mass flux. The mass flux is also in the denominator of the Boiling number and weakens the influence on the pressure drop. In correlations for the pressure drop of single-phase flows, this attenuation is included in the friction factor which is then not constant. The new correlation can be simplified for adiabatic applications. With Bo ¼ 0 one obtains:
Dpad ¼
1 2 L zG ½x ðv vL Þ þ vL 2 DFR I V
(34)
For the boundary values xI ¼ 0 and xI ¼ 1 equation (34) becomes the equations for the adiabatic single-phase pressure drop:
Bo ¼ 0; xI ¼ 0 0 Dpad;L ¼
1 21 L 1 L zG ¼ zr u2 2 rL DFR 2 L L DFR
(35)
Bo ¼ 0; xI ¼ 1 0 Dpad;V ¼
1 21 L 1 L zG ¼ zr u2 2 rV DFR 2 V V DFR
(36)
4.2. Validation of the new pressure drop model with values from literature Measurement values from the publications of Kuo and Wang [20] as well as Passos et al. [33] were tested with the new correlation. Table 2 compares the tube geometries and parameter ranges of the test series. Both, Kuo and Wang and Passos et al. used a microfin tube and conducted experiments with heat supply. Kuo and Wang also conducted experiments under adiabatic conditions. Fig. 14 shows the comparison of the values calculated with the new correlation and the measurement values of Kuo and Wang.
Fig. 14. Comparison of the pressure drop values calculated with the new correlation and the measurement values of Kuo and Wang for R407C.
The mean deviation of the calculated from the measured values is 20.2%. Within the range of ±30% there are 72.2% of the measured values. Both, the measurement values with heat supply as well as the measurements under adiabatic conditions can be calculated. This confirms the assumption that the momentum pressure drop only slightly contributes to the total pressure drop. Fig. 15 shows the comparison of the values calculated with the new correlation and the measurement values of Passos et al. The mean deviation of the calculated from the measured values is 18.7%. Within the range of ±30% there are 77.8% of the measured values. The largest deviations arise with the test tube with the small diameter at fin root 6.48 mm. With decreasing diameter, the influence of the diameter on the pressure drop increases. Kandlikar defines the transition to mini channels at Dhyd ¼ 3 mm [34]. Not all geometric parameters for the calculation of the hydraulic diameter are available. Compared to the smooth tube, the area where the fluid flows through decreases, while the wetted circumference increases. With Dhyd ¼ 4A=U the hydraulic diameter of microfin tubes is substantially smaller than the diameter at fin root. The measuring tube with the diameter at fin root of 6.48 mm may have a hydraulic diameter in the order of the transition to mini channels. Correlations for internally structured mini channels are not known.
Table 2 Comparison of the tube geometries and the parameter ranges of Rollmann and Spindler, Kuo and Wang as well as Passos et al. Rollmann and Spindler
Kuo and Wang
Passos et al.
Diameter at fin root [mm] Fin height [mm] Helix angle [ ] Apex angle [ ] Number of fins
8.95 0.24 15 25 55
8.92 0.2 18 Not specified 60
11.98 0.25 30 65 65
Vapor quality Heat flux [kW/m2] Mass flux [kg/m2 s] Saturation temperature [ C]
0.10e1.00 1e20 62.5e300 30 to 10
0.09e0.79 0 and 10 100e300 1.7
0.09e0.96 10 200 9.72
6.48 0.15 18 50 50
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65
based on the often confirmed DarcyeWeisbach equation, it can be assumed that it is valid for different tube geometries and refrigerants and is applicable by adjusting the friction factor z. 5. Conclusions and outlook
Fig. 15. Comparison of the pressure drop values calculated with the new correlation and the measurement values of Passos et al. for R407C.
4.3. Applicability of the new pressure drop model using other tube geometries and refrigerants The new correlation for the pressure drop was derived for the refrigerants R407C and R410A in one microfin tube. Because it is
A total number of 1614 heat transfer measurements and 952 pressure drop measurements was conducted. It was shown in previous publications [15,16] that correlations from literature for the heat transfer and the pressure drop during flow boiling often calculate the heat transfer coefficient with large deviations from the measured values. In calculated versus measured values-diagrams the data points scatter widely. A new correlation for the Nusselt number during flow boiling of R407C in a horizontal microfin tube was derived. The data points in the calculated versus measured values-diagram (10) scatter in a minor way. The mean deviation is 9.81%. Within the range of ±30% there are 94.18% of the measured values. Table 3 summarizes the results of the correlations from literature and the new correlation. Also, a new correlation for the pressure drop of R407C and R410A during flow boiling in a horizontal microfin tube was derived. The data points in the calculated versus measured values-diagram (13) scatter in a minor way. The mean deviation is 18.1%. Within the range of ±30% there are 83.1% of the measured values. Table 4 summarizes the results of the correlations from literature and the new correlation. Rollmann et al. [31] tested the Kandlikar correlation [8]. It is planned to verify this correlation and obtain the enhancement parameters listed therein. It is also planned to verify the new correlation with more data sets and obtain the constants C1eC4 for more refrigerants. This will be done as soon as sufficient measurement data of low GWP refrigerants are available and these refrigerants come into the market.
Table 3 Summary of the correlations for the heat transfer coefficient during flow boiling. Correlation
Within the range ±30%
Percentage of measurements
49.9% 38.7% 86.5% 24.8% 79.5% 30.6%
17.8% 47.6% 26.9% 77.3% 27.5% 63.6%
100% 100% 100% 100% 100% 79.1%
171.9% 148.6% 53.0% 36.9%
2.2% 2.5% 52.2% 55.8%
100% 87.4% 100% 87.4%
9.8%
94.2%
85.1%
Mean deviation
Shah Shah with pre-factor Liu and Winterton Liu and Winterton with pre-factors Cavallini et al. Cavallini et al. with pre-factors and without q_ ¼ 1 kW=m2 Kattan et al. Kattan et al. with x < 0.85 Zhang et al. Zhang et al. with x < 0.85 Rollmann and Spindler
Table 4 Summary of the correlations for the pressure drop during flow boiling. Correlation
Mean deviation
Within the range ±30%
Percentage of measurements
Pierrea Choi et al.a Kuo and Wanga VDI Heat Atlasa Müller-Steinhagen and Hecka Müller-Steinhagen and Heck with pre-factor 1.35a
96.7% 22.2% 27.9% 29.0% 38.4% 31.3%
22.7% 76.1% 77.1% 84.3% 45.4% 73.2%
100% 51.3% 100% 32.0% 100% 100%
Rollmann and Spindlerb
18.1%
83.1%
94.4%
a b
R407C. R407C and R410A.
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Acknowledgments This work was supported by the German Bundesministerium für Wirtschaft (BMWi) through the Arbeitsgemeinschaft industrieller Forschungsvereinigungen (AiF). The project was accompanied by a €ltetechnik e.V. (FKT), Frankworking party of the Forschungsrat Ka furt a. M., Germany. References [1] J.C. Chen, Correlation for boiling heat transfer to saturated fluids in convective flow, Ind. Eng. Chem. Process Des. Develop. 5 (3) (1966) 322e329, http:// dx.doi.org/10.1021/i260019a023. [2] M.M. Shah, Chart correlation for saturated boiling heat transfer equations and further study, ASHRAE Trans. 88 (1) (1982) 185e196. [3] K.E. Gungor, R.H.S. Winterton, A general correlation for flow boiling in tubes and annuli, Int. J. Heat Mass Transfer 29 (3) (1986) 351e358, http://dx.doi.org/ 10.1016/0017-9310(86)90205-X. [4] K.E. Gungor, R.H.S. Winterton, Simplified general correlation for saturated flow boiling and comparisons of correlations with data, Chem. Eng. Res. Des. 65 (1987) 148e156. [5] Z. Liu, R.H.S. Winterton, A general correlation for saturated and subcooled flow boiling in tubes and annuli, based on a nucleate pool boiling equation, Int. J. Heat Mass Transfer 34 (11) (1991) 2759e2766, http://dx.doi.org/ 10.1016/0017-9310(91)90234-6. [6] A. Bejan, A.D. Kraus, Heat Transfer Handbook, J. Wiley, New York, 2003. [7] S.G. Kandlikar, A general correlation for saturated two-phase flow boiling heat transfer inside horizontal and vertical tubes, J. Heat Transfer 112 (1) (1990) 219e228, http://dx.doi.org/10.1115/1.2910348. [8] S.G. Kandlikar, A model for correlating flow boiling heat transfer in augmented tubes and compact evaporators, J. Heat Transfer 113 (4) (1991) 966e972, http://dx.doi.org/10.1115/1.2911229. [9] A. Cavallini, D. Del Col, L. Doretti, G.A. Longo, L. Rossetto, Refrigerant vaporisation inside enhanced tubes: a heat transfer model, Heat Technol. 17 (2) (1999) 29e36. [10] J.R. Thome, N. Kattan, D. Favrat, Evaporation in microfin tubes: a generalized prediction model, Proc. Int. Eng. Found. 3 (1999) 239e245. [11] L.M. Chamra, P.J. Mago, Modelling of evaporation heat transfer of pure refrigerants and refrigerant mixtures in microfin tubes, Proc. Instn. Mech. Engr., C: J. Mech. Eng. Sci. 221 (4) (2007) 443e447, http://dx.doi.org/10.1243/ 0954406JMES131. [12] N. Kattan, J.R. Thome, D. Favrat, Flow boiling in horizontal tubes: part 3 e development of a new heat transfer model based on flow pattern, J. Heat Transfer 120 (1) (1998) 156e165, http://dx.doi.org/10.1115/1.2830039. [13] X. Zhang, C. Ji, X. Yuan, Prediction method for evaporation heat transfer of non-azeotropic refrigerant mixtures flowing inside internally grooved tubes, Appl. Therm. Eng. 28 (14e15) (2008) 1974e1983, http://dx.doi.org/10.1016/ j.applthermaleng.2007.12.009. [14] L. Wojtan, T. Ursenbacher, J.R. Thome, Investigation of flow boiling in horizontal tubes: part IIddevelopment of a new heat transfer model for stratifiedwavy, dryout and mist flow regimes, Int. J. Heat Mass Transfer 48 (14) (2005) 2970e2985, http://dx.doi.org/10.1016/j.ijheatmasstransfer.2004.12.013. [15] P. Rollmann, K. Spindler, H. Müller-Steinhagen, Heat transfer, pressure drop and flow patterns during flow boiling of R407C in a horizontal microfin tube, Heat Mass Transfer 47 (8) (2011) 951e961, http://dx.doi.org/10.1007/s00231011-0857-x. €rmeübergang und Druckverlust beim [16] P. Rollmann, K. Spindler, Wa €mungssieden von R407C e Vergleich verschiedener Korrelationen, in: Stro
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