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A semi-empirical correlation for the thermal conductivity of frost
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 5 8 ( 2 0 1 5 ) 2 4 3 e2 5 2
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A semi-empirical correlation for the thermal conductivity of frost* Silvia Negrelli, Christian J.L. Hermes* , 81531990 Curitiba, PR, Brazil Laboratory of Thermodynamics and Thermophysics, Federal University of Parana
article info
abstract
Article history:
The present work is aimed at advancing an improved correlation for the thermal con-
Received 23 February 2015
ductivity of frost. The analysis was conducted based on the weighted geometric mean of
Received in revised form
the thermal resistances of moist air and ice crystals, thus setting the theoretical back-
30 April 2015
ground for a dimensionless model for the thermal conductivity as a function of the
Accepted 26 May 2015
porosity of the frosted medium. Experimental data obtained elsewhere for a wide wall
Available online 22 June 2015
surface temperature span (from 30 C to 4 C) were employed to fit the model, coming up with a semi-empirical correlation for the thermal conductivity of frost valid for porosities
Keywords:
ranging from 0.5 to 0.95. Comparisons of the proposed correlation with the experimental
Frost
data showed that it is able to predict 81% of the data points (153 out of 188) within the 15%
Thermal conductivity
thresholds. Comparisons with other correlations available in the open literature are also
Correlation
reported. © 2015 Elsevier Ltd and IIR. All rights reserved.
lation semi-empirique pour la conductivite Une corre thermique de givre thermique ; Corre lation Mots cles : Givre ; Conductivite
1.
Introduction
Frost often builds-up on evaporators of small-capacity refrigerating appliances, resulting in an increased energy input to accomplish the same refrigerating effect. To mitigate the issues caused by the evaporator frosting, simulationbased designs have been carried out by means of mathematical models for the growth and densification of frosted media
*
in various geometries, such as flat surfaces (Sami and Duong, 1989; Tao et al., 1993; Lee at el., 1997; Na and Webb, 2004; Hermes et al., 2009; Hermes, 2012), parallel plate channels (O'Neal, 1982; Ismail and Salinas, 1999; Luer and Beer, 2000; Cui et al., 2011; Loyola et al., 2014), and tube-fin heat exchangers (Tso et al., 2006; Huang et al., 2008; Silva et al., 2011). Most models, however, rely on empirical correlations for the thermophysical properties of frost, particularly the density and the thermal conductivity, which limit their range of
An abridged version of this manuscript was submitted to be presented at the 24th IIR International Congress of Refrigeration, August 16e22, 2015, Yokohama, Japan. * Corresponding author. Tel.: þ55 41 3361 3239; fax: þ55 41 3361 3123. E-mail address: [email protected] (C.J.L. Hermes). http://dx.doi.org/10.1016/j.ijrefrig.2015.05.021 0140-7007/© 2015 Elsevier Ltd and IIR. All rights reserved.
244
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 5 8 ( 2 0 1 5 ) 2 4 3 e2 5 2
Nomenclature Roman A a, b B C Dv E, F Fo hsv k L M patm pv r Re Rv S T t X
Parameter of Equation (2) Coefficients of Equations (18) and (19) Shape factor of Equation (12) Blending factor of Equation (6) Water vapor diffusivity in air, m2 s1 Parameters of Equation (5) Fourier number, dimensionless Latent heat of sublimation, J kg1 Thermal conductivity, W m1 K1 Binary parameter of Equation (4) Molecular weight, kg mol1 Atmospheric pressure, Pa Vapor pressure of water, Pa Radius, m Reynolds number, dimensionless Gas constant for water vapor, J kg1 K1 Surface area, m2 Temperature, K Time, s Dimensionless frost thickness
Greek a, b, g d ε r u x z
Coefficients of Equation (1) Thickness Frost porosity Density, kg m3 Humidity ratio Blending factor of Equation (10) Dimensionless thermal conductivity
Subscripts 0 Reference state a Moist air air Dry air c Turbulent convection eff Effective f Frost g Geometric mean i Ice m Molecular diffusion max Maximum min Minimum p Parallel association s Serial association t Thermal diffusion tp Triple point v Water vapor w Wall surface
operation. The former was assessed in prior studies, when empirical and semi-empirical correlations were put forward for the frost density (Hayashi et al., 1977; Mao et al., 1992; Yang and Lee, 2004; Kandula, 2011; Hermes et al., 2014; Nascimento et al., 2015). The latter is at the aim of the present paper. Several studies are available in the open literature focused on the thermal conductivity of frost. The most influential ones
are described in Table 1, whereas Table 2 summarizes the key parameters evaluated in the studies reported in Table 1. They are detailed as follows. Yonko and Sepsy (1967), in a pioneering study, advanced an empirical correlation for the thermal conductivity of frost as a quadratic polynomial function of the frost density. The correlation was not able to predict the experimental data satisfactorily, suggesting that the thermal conductivity is dependent on parameters other than the density. Despite of that, their approach influenced the works of Ostin and Andersson (1990) and Sturm et al. (1997), which also employed quadratic polynomials to correlate the thermal conductivity as a function of the frost density, as follows: kf ¼ a þ brf þ gr2f
(1) -1
1
where kf is the thermal conductivity of frost in [W m K ], rf is the frost density in [kg m3], and the coefficients a, b and g are summarized in Table 3. Pitman and Zuckerman (1967) put forward a semiempirical correlation assuming the ice crystals as small spheres connected to each other by ice cylinders. The correlation predictions were compared with experimental data for wall surface temperatures of 88, 27 and 5 C, disregarding the dense frost media typical of the plate-shaped ice crystals observed for wall surface temperatures between 27 and 5 C, as depicted in Fig. 1, which reproduces the frost morphology map due to Kobayashi (1958). Their correlation is as follows: 1 1r 4lnðA þ 1Þ=ðA 1Þ ¼ þ kf ðki ka ÞS þ ka 2Aprðki ka Þ
(2)
where ka and ki are the thermal conductivities of moist air and ice, respectively, r is the radius of the spheres, and S is the surface area of the cylinders. The parameter A is defined as follows: A¼
1þ
4 p½ðki =ka Þ 1r2
1=2 (2.a)
A few years later, Brian et al. (1969) noticed the frost density and the wall surface temperature are key independent parameters that affect the thermal conductivity of the frost media, thus coming up with the following empirical correlation based on their own data, which is valid for frost densities lower than 250 kg m3: kf ¼ 2:401 105 T1:272 þ 3:921 108 rf T1:74 w w
(3)
where Tw is the wall surface temperature in [K], rf is the frost density in [kg m3], and kf is the thermal conductivity of frost in [W m-1 K1]. One year later, Biguria and Wenzel (1970) advanced theoretical models to predict the effective thermal conductivity of frost based on the crystal morphology (e.g., spheres, plates, columns, needles), as depicted in Fig. 1. They found out that none of the theoretical models were able to predict the experimental data satisfactorily using the frost density as the only parameter. In order to improve the model prediction capabilities, they included other key heat and mass transfer parameters into correlation, as the air velocity and humidity, as follows:
245
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Table 1 e Summary of key studies on the thermal conductivity of frost. Author
Year
Origin
Approach
Yonko and Sepsy Pitman and Zuckerman Brian et al. Biguria and Wenzel Dietenberger
1967 1967
USA USA
Empirical Semi-empirical
1969 1970
USA USA
Empirical Empirical
1982
USA
Semi-empirical
Auracher Ostin and Andersson Mao et al. Sturm et al. Mao et al. Sahin Na and Webb
1986 1990
Germany Sweden
Theoretical Empirical
1992 1997 1999 2000 2004
Canada USA Canada Saudi Arabia USA
Empirical Empirical Empirical Semi-empirical Semi-empirical
Yang and Lee Kandula
2004 2010
S. Korea USA
Empirical Theoretical
Background model
Geometry
Independent parameter
Density range
e Spheres and cylinders e e
Flat plate Flat plate
rf rf, Tw
rf < 573 rf < 400
Flat plate Flat plate
rf, Tw rf, Tw
rf < 250 50 < rf < ri
Spheres and cylinders Spheres e
Flat plate
ε, Tw
ra < rf < ri
Flat plate Parallel plate channel Flat plate Flat plate Flat plate Flat plate Flat plate
rf, Tw rf
100 < rf < 600 50 < rf < 680
X, Re, Fo, ua, Ta, Tw rf X, Re, Fo, ua, Ta, Tw ε, Tw rf, Tw
rf < rf < rf < rf < rf <
Flat plate Flat plate
Re, Fo, ua, Ta, Tw ε, Tw
rf < 180 rf < 500
e e e Cylinders Cylinders and plates e Cylinders
200 600 200 250 400
Table 2 e Summary of the parameters considered in the studies reported in Table 1. Author
Year
Yonko and Sepsy Pitman and Zuckerman Brian et al. Biguria and Wenzel Dietenberger Auracher Ostin and Andersson Mao et al. Sturm et al. Mao et al. Sahin Na and Webb Yang and Lee Kandula
Frost density
1967 1967 1969 1970 1982 1986 1990 1992 1997 1999 2000 2004 2004 2010
Wall temperature
X X X X X X X X X X X X X X
þ 3:5719847 104 Ua þ 6:2047771 104 t 8:9475394 105 t2 þ 1:0182528 107 t3 þ 2:6084586 108 t4 4:2023418 102 ua Tw þ 0:11349924tua þ 1:0859212 102 L þ 2:1232614 105 Ua t 2:6856724 105 Tw L (4) where ua is the humidity ratio of the air stream in [kgv kg1 air], Ua is its velocity in [ft s1], t is the time in [minutes], Tw is the
Table 3 e Coefficients of Equation (1).
Yonko and Sepsy Ostin and Andersson Sturm et al. (156 < rf < 600) Sturm et al. (rf < 156)
Year
a
b
1967 0.02422 7.214 10 1.1797 106 1990 0.00871 4.39 102 1.05 106 1997 0.138 1.01 103 3.233 106 0.023
Humidity
X
X
X
X X X
X
X
X X X X X X X
X
X
X X X X X
X X X X X
wall surface temperature in [R], and L is a binary parameter which may be 1 or 1 depending on the flow regime. According to Biguria and Wenzel (1970), Equation (4) was able to predict their own experimental data within ±25% error bounds. Dietenberger (1982) devised a semi-empirical model to predict the thermal conductivity considering two different frost structures, a low-density array of spherical and cylindrical ice crystals, which provides the minimum threshold for the thermal conductivity, and a porous matrix comprised of air bubbles imprisoned in high-density ice plates, corresponding to the maximum threshold. The model blends the two structures based on the frost porosity, ε, as follows:
g 4
1997
Velocity
X X X X X X X X X X X X X X
kf ¼ 0:23376438 þ 1:0342876 104 T1:3 w þ 18:007637ua
Author
Air Temperature
2.34 102
e
1 kf ¼ ðð3E 1Þ kmin þ ð3F 1Þ kmax 4 1=2 2 þ ðð3E 1Þ kmin þ ð3F 1Þ kmax Þ þ 8kmin kmax where F ¼ 1-E, and
(5)
246
Sheath
0.2
Needle
0.3
Sectorated plate
0.1 0
where hsv is the latent heat of sublimation in [J kg1], ri is the density of ice in [kg m3], patm is the atmospheric pressure in [bar], Tw is wall surface temperature in [K], and the index “0” refers to the reference state (1 bar, 273.16 K). Mao et al. (1992) put forward the following dimensionless correlation, which was later revisited by Mao et al. (1999) and Yang and Lee (2004) (Iragorry et al., 2004):
Dendrite
0.4
Plate
Supersaturation degree [g kg-1]
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 5 8 ( 2 0 1 5 ) 2 4 3 e2 5 2
Sheath
Ttp Tw 0:216 1:093 0:699 kf ¼ 0:011$X0:37 u2:044 Re Fo a ki Ta Tw
Hollow column Solid column 0
-10
-5
-15
-20
-25
where X is the dimensionless frost thickness, Re and Fo are the Reynolds number and the Fourier number, respectively, and Ttp is the temperature of the triple point of water. Their correlation is limited to densities lower than 200 kg m3. Equation (7) was later improved by Mao et al. (1999), coming up with the following expression (Iragorry et al., 2004):
-30
Temperature [°C] Fig. 1 e Illustration of Kobayashi's (1958) map of frost morphology.
Ttp Tw 0:737 0:375 0:18 kf ¼ 6:534 104 $X0:048 ua0:004 Re Fo ki Ta Tw
"
2 ε E1 ε E1 E ¼ 13:6ðE2 E1 Þðε E1 Þ 1 þ 3 E3 E1 E2 E1 # 2 ðε E1 Þ þ 2ðE3 E1 ÞðE2 E1 Þ
(7)
(8)
2
kmax
a =ki 1 2ε 1k 2þka =ki
þ ε 1 εÞki þ εka ¼ ð1 εÞki 1ka =ki 1 þ ε 2þka =ki "
kmin ¼ ð1 εÞ
(5.a)
(5.b)
(5.c)
where E1 ¼ 0.1726(Tf/273), E2 ¼ 0.751 and E3 ¼ 1.051. Despite of its theoretical background, the model tends to overpredict the experimental data for high densities (Sahin, 2000). Auracher (1986) assumed the effective thermal conductivity of frost has contributions of both thermal (kt) and molecular (km) diffusion, as follows: kf ¼ kt þ km
(6)
The former is set as a blend of serial and parallel thermal resistance association, 1 C 1C ¼ þ kt ks kp
(6.a)
where ks and kp are the thermal conductivities obtained from serial and parallel thermal association, respectively, and the blending factor C is calculated from the following empirical expression: C ¼ 0:42ð0:1 þ 0:995rf Þ
(6.b)
The contribution due to the molecular diffusion, on the other hand, is calculated as follows: km ¼ 1:958 109 hsv 1:28 p Tw 0 patm T0
! ri rf 6145 exp 24:02 Tw ri 0:58$rf
1 1 ¼ kf df
Zdf 0
1 dy kf ðTÞ
(9)
where df is the frost layer thickness (m), and kf(T) is the local thermal conductivity, calculated from:
#
ka ki ½3 þ 2εðka =ki 1Þ þ εki 1 εÞka þ εki i 1 3 ε kak=k a =ki
Sahin (2000) developed a model for the early crystal growth stages based on a local energy balance, and vapor mass diffusion and desublimation considering column-shaped ice crystals. The thermal conductivity is expressed as follows:
(6.c)
hsv patm pv0 hsv 1 1 exp Rv T0 T T01:94 R2v T1:06
3 0:963 þ 1:202 10 εri þ þð1 εÞ 1:0465 þ 0:017Tf 105
kf ðTÞ ¼ 0:131 106 ð1 εÞ
(9.a) where ε is the frost porosity, T is the local frost temperature in [K], pv0 is the vapor pressure of water at the reference temperature T0 (273.16 K), and Rv is the gas constant for water vapor. The model is able to predict the temperature variation along the ice crystals and the frost density over time for the early stages of frost formation, when the density of the frosted medium is low. In addition, the model is limited to columnshaped crystal morphologies, being not suitable for applications with wall surface temperatures typical of dendritic and plate-shaped ice crystals, i.e., from 19 C to 10 C according to the morphology map illustrated in Fig. 1. Na and Webb (2004) proposed an empirical correlation blending the minimum and maximum thermal resistances calculated according to the models presented in Sanders (1974). The blending factor was correlated based on their own experimental data for different wall surface temperature ranges, spanning three different frost morphologies: columns (10 < Tw < 4 C), plates (21 < Tw < 10 C), and columns (Tw < 21 C), according to the map for ice crystal morphology. The correlation is as follows:
kf ¼ kp xf þ ks 1 xf
(10)
where kp and ks are the thermal conductivity obtained from parallel and series thermal association, respectively. The
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 5 8 ( 2 0 1 5 ) 2 4 3 e2 5 2
blending factor xf is calculated from the following empirical expressions: 10 < Tw < 4o C (10.a) xf ¼ 0:283 þ exp 0:020$rf xf ¼ 0:140 þ 0:919 exp 0:0142$rf
21 < Tw < 10+ C (10.b)
Tw < 21+ C xf ¼ 0:0107 þ 0:419 exp 0:00424$rf
(10.c)
where rf is the frost density in [kg m3]. In the same year, Yang and Lee (2004) proposed a dimensionless empirical correlation for the thermal conductivity of frost, among others derived for key heat and mass transfer parameters related to frost growth and densification. Their correlation is as follows: 0:512 Ta Ttp kf 0:619 ¼ 1:184 102 ðln ReÞ Fo0:084 u0:086 exp a ki Ta Tw (11) where Re and Fo are the Reynolds number and the Fourier number, respectively, and Ttp is the temperature of the triple point of water. They reported an agreement between equation (11) and their own data within ±7% error bounds, although their correlation is valid for a limited range of application, with air temperatures spanning from 5 to 15 C, and wall surface temperatures ranging from 35 to 15 C. Recently, Kandula (2010) included the mass diffusion and turbulent convection into the model of Zehner and Schlunder (1970), coming up with the following fully theoretical model for predicting the thermal conductivity of frost: pffiffiffiffiffiffiffiffiffiffiffi " pffiffiffiffiffiffiffiffiffiffiffi 2 1 ε kf ð1 zÞB 1 Bþ1 þ ¼1 1εþ ln 1 zB zB 2 keff ð1 zBÞ2 # B1 (12) 1 zB where B = 2.5(1/e-1)10/9 is a shape factor related to porosity, z = keff/ki is a dimensionless thermal conductivity, keff = ka þ kc þ km is the combination of the air thermal conductivity, ka, the turbulent convection, kc = 0.00568$Ua, and the conductivity due to mass diffusion, km, which is calculated from: km ¼ hsv Dv
" # Ma =Mv hsv 1 1 p exp v0 Rv T0 Tf R2v T3f
(12.a)
crystal growth stages, being not suitable for dense frosted media. Finally, the literature also shows that the frost density is not the only parameter which affects the thermal conductivity, since the ice crystal morphology is also influenced by the wall surface temperature, as illustrated in Fig. 1. The present work is therefore aimed at putting forward, by means of a theoretical-experimental approach, an improved correlation for the thermal conductivity of frost formed on flat surfaces. The theoretical analysis was conducted to set the scales of the problem, which served as the basis for a semiempirical correlation for the effective thermal conductivity of frosted media as a function of the frost density and the wall surface temperature.
2.
Theoretical model
The effective thermal conductivity of porous media depends on the thermal conductivities of each phase (solid and fluid), on the volume fraction between the fluid phase and the porous medium (i.e., porosity), and on the array formed by particles and pores. In the case of frost, the porosity ε can be calculated from: ε¼
ri rf ri ra
(13)
where ra (z1.35 kg m3) and ri (z920 kg m3) are the densities of moist air and ice, respectively. For high porosity frost, the thermal conductivity approaches that of moist air (ka z 0.0225 W m-1 K1), whereas for low porosity frost, the thermal conductivity tends to that of ice (ki z 2.0 W m-1 K1). In addition, the array plays an important role upon the effective thermal conductivity. The limiting cases are the parallel (kp) and serial (ks) association of the thermal resistances of the ice and air phases, illustrated in Fig. 2, and calculated respectively as follows (Sanders, 1974): kp ¼ ka ε þ ki ð1 εÞ
(14)
1 ε 1ε ¼ þ ks ka ki
(15)
Alternatively, Nield and Bejan (2006) proposed the weighted geometric mean of ka and ki, as follows: ε
where Dv is the mass diffusion coefficient of water vapor, Ma and Mv are the molecular weights of air and water vapor, respectively, Tf is the frost surface temperature in [K], and Ua is the air velocity in [m s1]. Kandula (2010) has also observed that the value of kc can be approximated as kc z ka with satisfactory agreement. The model predictions were compared with experimental data obtained elsewhere, showing satisfactory results for low frost densities. The literature survey reveals that most of the correlations available in the open literature for the thermal conductivity of frost holds a strong empirical character, being restricted to the range of the experimental data used in the fitting process. In addition, the theoretical models have been developed for early
247
1ε
kg ¼ ka ki
(16)
Fig. 3 illustrates Equations (14)e(16), where one can note that kf / ki where ε / 0 (left corner) and that kf/ka where ε/1 (right corner) independently of the model (parallel, serial, geometric mean). It is worth noting that kp is the maximum allowed thermal conductivity, whereas ks is the minimum one, suggesting that the parallel array is related to the column-shaped crystal morphology, whereas the serial one is regarded with the plate-shaped ice morphologies, as depicted in Fig. 2. The geometric mean provides intermediate values between ks and kp. Experimental data obtained from the open literature is also plot in Fig. 3. All data points (but four) lie in between the thresholds defined by parallel and geometric mean model.
248
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 5 8 ( 2 0 1 5 ) 2 4 3 e2 5 2
Heat flux
Ice columns Air
Heat flux Ice plates Air A Cold surface
Cold surface
Fig. 2 e Schematic of parallel (left) and serial (right) thermal resistance association.
1 Auracher -48°C to -23°C
parallel
Auracher -23°C to -13°C Auracher 11°C Brian et al. -18ºC
kf / ki
Brian et al. -6ºC Hayashi et al.
geometric
0.1
Na and Webb -6°C Na and Webb -13°C
Na and Webb 24°C
serial
Ostin and Andersson Pitman and Zuckermann -27°C Pitman and Zuckermann -6°C Yonko and Sepsy
0.01 0
0.2
0.4
0.6
0.8
1
ε Fig. 3 e Illustration of Equations (14)e(16) with experimental data from the open literature.
Only four data points lie in the region below the geometric mean, which one might attribute to measurement uncertainties.
mathematically. Thus making kf ¼ kg and dividing Equation (16) by ki, and taking the logarithm in both sides, it follows that: log
3.
Semi-empirical correlation
The dependence of the thermal conductivity with the porosity is quite obvious in Fig. 3. Since the thresholds are related to the morphology, which in turn is strongly dependent on the wall surface temperature, and mildly dependent of the supersaturation degree, as can be seen in Fig. 1, the correlation for the thermal conductivity of frost proposed in this work shall include both. The former was taken into account by fitting the model to different temperature ranges, as proposed by Na and Webb (2004), whereas the latter was accounted for by means of the frost porosity, which carries information on the supersaturation degree, as it is a function of the modified Jakob number and the square root of time (Hermes et al., 2014). An inspection of Fig. 3 reveals that most of the experimental data follow the trend described by the geometric mean, in spite of an offset and a small slope that can be corrected empirically. Since there is no room in the geometric mean for empirical fitting, Equation (16) had to be modified
kf ka ¼ εlog ki ki
(17)
Adding two linear (slope and intercept) coefficients into Equation (17), it follows that: log
kf ka ¼ loga þ b$εlog ki ki
(18)
Re-arranging Equation (18) in a more elegant form, it follows that: bε kf ka ¼a ki ki
(19)
where the “a” (intercept) and “b” (slope) coefficients of Equation (19) are fitted to the experimental data according to the frost morphology, as depicted in Table 4, which in turn is embedded in the range of the wall surface temperature, namely needles and sheaths (10 < Tw < 4 C), plates and dendrites (19 < Tw < 10 C), and sheaths (30 < Tw < 19 C) (see Fig. 1). A similar approach was adopted by Na and Webb (2004), who proposed empirical expressions for the thermal
249
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Table 4 e Coefficients of Equations (18) and (19). Dataset# (i) (ii) (iii)
Temperature range
Morphology
Intercept (a)
Slope (b)
log a
10 < Tw < 4 C 19 < Tw < 10 C 30 < Tw < 19 C
Needles and sheaths Plates and dendrites Sheaths
1.576 1.594 1.035
0.797 0.761 0.797
0.1976 0.2025 0.0149
1 parallel
k f / ki
out. The error embedded in the frost correlation, Eε, propagates to the thermal conductivity, Ekf, as follows:
-10ºC to -4ºC -19ºC to -10ºC -30°C to -19°C experimental fit
Ekf ¼
vkf Eε vε
(20)
where the derivative can be obtained analytically from Equation (19), yielding
0.1
vkf ka ¼ bkf log vε ki
geometric
Equation (19) shows that the highest errors were observed for the lowest temperatures and porosities (i.e., 0.5) in the interval. Noting that Hermes et al. (2014) correlated the frost porosity by means of a semi-empirical model, obtaining errors within the ±2% bounds, one can show, based on Equation (21), that errors up to 2.5% can be observed for Tw ¼ 30 C, up to 4.0% for Tw ¼ 19 C, and up to 3.5% for 10 C.
serial
0.01 0.5
0.6
0.7
0.8
0.9
(21)
1
ε Fig. 4 e Best fitting of Equation (19) spanning three different temperature ranges: ¡10 < Tw < ¡4 C (needles and sheaths), ¡19 < Tw < ¡10 C (plates and dendrites), and ¡30
4.
Discussion
A close inspection of Table 4 reveals that datasets (i) and (iii) presented the same slope, which is probably due to the similar frost morphologies (column-shaped ice crystals) observed for both datasets. Since the intercept and slope coefficients diverge from the unity, one may not expect Equation (19) to be valid in the limits where ε / 0 and ε / 1. Fig. 4 plots the best fits for the three temperature ranges, where one can observe that Equation (19) is able to predict the experimental trends quite satisfactorily. It is worth noting that two competing effects resulting from the wall surface temperature take place, namely the molecular diffusion and the frost morphology. Fig. 3 suggests that the former overrules the latter, thus explaining why the high temperature data (from 10 to 4 C) are close to the parallel array, the low temperature data (from 30 to 19 C) are near the geometric
conductivity aimed at different frost morphologies (see Equations 10.ae10.c). It is worth noting that some experimental data points plotted in Fig. 3 do not provide information on the wall surface temperature and, therefore, had to be discarded. In total, 188 data points from Pitman and Zuckerman (1967), Brian et al. (1969), Auracher (1986), and Na and Webb (2004) have been considered in the fitting process. The resulting fitting coefficients are shown in Table 4. Since the thermal conductivity of frost depends on the frost porosity, which in turn is calculated from empirical correlations, an error propagation analysis must be carried
Table 5 e Overall comparison between the correlations and experimental data. Author
Number of experimental data points
Yonko and Sepsy (1967) Pitman and Zuckerman (1967) Brian et al. (1969) Dietenberger (1982) Auracher (1986) Ostin and Andersson (1990) Sturm et al. (1997) Na and Webb (2004) Kandula (2010) Present work a
Correlation predictions
10 < Tw < 4 C
19 < Tw < 10 C
30 < Tw < 19 C
RMS
e 6 12 e 0 e e 12 e e
e e 17 e 74 e e 24 e e
e 10 0 e 25 e e 8 e e
0.519
Correlation not evaluated due to its dependence on parameters other than porosity and temperature.
a
Points ±15% 28 a
0.170 0.178 0.172 0.386 0.464 0.179 0.202 0.111
108 95 129 91 18 92 87 153
250
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 5 8 ( 2 0 1 5 ) 2 4 3 e2 5 2
0.3
Predicted kf [W m-1 K-1]
+15%
Yonko and Sepsy (1967) Brian et al. (1969) -15%
0.2
Dietenberger (1982) Auracher (1986)
Ostin and Andersson (1990) Sturm et al. (1997) Na and Webb (2004)
0.1
Kandula (2010) Present work
-10
0 0
0.1
0.2
Measured kf [W
0.3
m-1 K-1]
0.5
Predicted kf [W m-1 K-1]
Yonko and Sepsy (1967)
0.4
Brian et al. (1969)
+15%
Dietenberger (1982) Auracher (1986)
0.3
Ostin and Andersson (1990)
-15%
Sturm et al. (1997)
0.2
Na and Webb (2004) Kandula (2010)
0.1 -19
Present work
0
0
0.1
0.2
0.3
0.4
0.5
Measured kf [W m-1 K-1] 0.3 +15%
Yonko and Sepsy (1967)
Predicted kf [W m-1 K-1]
Brian et al. (1969) -15% 0.2
Dietenberger (1982) Auracher (1986)
Ostin and Andersson (1990) Sturm et al. (1997) 0.1
Na and Webb (2004) Kandula (2010)
Present work
-30
0.1
0.2
0.3
Measured kf [W m-1 K-1] Fig. 5 e Comparison between the experimental data and some key correlations available in the literature.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 5 8 ( 2 0 1 5 ) 2 4 3 e2 5 2
mean, and the data for the plates and dendrites (range 19 to 10 C) are in between. Table 5 compares the predictions of the correlations due to Yonko and Sepsy (1967), Brian et al. (1969), Dietenberger (1982), Auracher (1986), Ostin and Andersson (1990), Sturm et al. (1997), Na and Webb (2004) and Kandula (2010) against all 188 experimental data points. One should note that the numbers in the middle of the table represent the number of data points from each dataset in each temperature (morphology) range. The numbers in the right-side represent the RMS (root mean square) errors and the number of data points predicted by the proposed model (Equation (19)) within the ±15% bounds. It can be seen that the models due to Pitman and Zuckerman (1967), Mao et al. (1992, 1999) and Yang and Lee (2004) have not been taken into account as they depend upon parameters other than the density and the wall surface temperature, not available in most of the dataset. In the same fashion, the model due to Sahin (2000) has not been included into the analysis, as it is applicable for the early stages of the frost formation only. One can see that the proposed correlation presented an RMS error ~11%, a figure significantly smaller than that observed for the other models (>17%). In total, 153 out of 188 data points (~81%) fall within the ±15% error bounds, as depicted in Fig. 5, where the predictions of the correlations with all experimental data for different temperature ranges are compared. One can also see in Fig. 5 that, for 10 < Tw < 4 C, the correlation of Auracher (1986) tends to overpredict the experimental data, whereas the correlations of Brian et al. (1969), Dietenberger (1982), Ostin and Andersson (1990) and Sturm et al. (1997) underpredict the data points for high porosity (low density). The empirical correlation of Yonko and Sepsy (1967) follows a different pattern, underpredicting the thermal conductivity at high porosities, and doing the contrary at low porosities. In this temperature range, not only equation (19) but also the models of Na and Webb (2004) and Kandula (2010) have shown satisfactory predictions for most of the data points. For 19 < Tw < 10 C, the correlations of Auracher (1986), Dietenberger (1982) and Equation (19) have all shown good predictions for the whole dataset. The correlations of Brian et al. (1969), Sturm et al. (1997) and Na and Webb (2004), however, tend to underpredict the experimental data points, whereas the correlations of Yonko and Sepsy (1967) and Kandula (2010) follow the opposite behavior. Finally, for 30 < Tw < 19 C, most of the correlations evaluated in this work presented satisfactory results, but those of Yonko and Sepsy (1967), Ostin and Andersson (1990) and Sturm et al. (1997). At all, equation (19) is the only one whose predictions are within the ±15% thresholds independently of the temperature range.
5.
Conclusions
A semi-empirical correlation was advanced based on 188 experimental data points obtained from the open literature with wall surface temperatures ranging from 30 to 4 C, and frost porosities spanning from 0.5 to ~0.95. The
251
correlation predictions were compared with the experimental data, when an 11% RMS error was observed, with 153 out of 188 (81%) data points lying within the ±15% error band. The correlations due to Yonko and Sepsy (1967), Brian et al. (1969), Dietenberger (1982), Auracher (1986), Ostin and Andersson (1990), Sturm et al. (1997), Na and Webb (2004) and Kandula (2010) were also compared with the experimental data, showing poorer prediction capabilities than that proposed in this study, with RMS errors higher than 17%. Among all correlations evaluated in this study, Equation (19) has shown to be the only one whose predictions are within the ±15% thresholds independently of the temperature range.
Acknowledgments This work was carried out under the auspices of the Brazilian Government funding agency CNPq (Grant No. 441603/2014-9). Ms. S. Negrelli duly acknowledges the CAPES agency, Government of Brazil, for supporting her MSc education at the . Federal University of Parana
references
Auracher, H., 1986. Effective thermal conductivity of frost. In: Proceedings of the International Symposium of Heat and Mass Transfer in Refrigeration Cryogenics, Dubrovnik, pp. 285e302. Biguria, G., Wenzel, L.A., 1970. Measurement and correlation of water frost thermal conductivity and density. I&EC Fundam. 9, 129e137. Brian, P.L.T., Reid, R.C., Brazinsky, I., 1969. Cryogenic Frost Properties. Cryogenic Technology, pp. 205e212, 10/11. Cui, J., Li, W.Z., Liu, Y., Jiang, Z.Y., 2011. A new time and spacedependent model for predicting frost formation. App. Therm. Eng. 31, 447e457. Dietenberger, M.A., 1982. Generalized correlation of the water frost thermal conductivity. Int. J. Heat Mass Transf. 26, 607e619. Hayashi, Y., Aoki, A., Adashi, S., Hori, K., 1977. Study of frost properties correlating with frost formation types. ASME J. Heat Transf. 99, 239e245. Hermes, C.J.L., 2012. An analytical solution to the problem of frost growth and densification on flat surfaces. Int. J. Heat Mass Transf. 55, 7346e7351. Hermes, C.J.L., Loyola, F.R., Nascimento, V.S., 2014. A semiempirical correlation for the frost density. Int. J. Refrigeration 46, 100e104. Hermes, C.J.L., Piucco, R.O., Melo, C., Barbosa Jr., J.R., 2009. A study of frost growth and densification on flat surfaces. Exp. Therm. Fl. Sci. 33, 371e379. Huang, J.M., Hsieh, W.C., Ke, X.J., Wang, C.C., 2008. The effects of frost thickness on the heat transfer of finned tube heat exchanger subject to the combined influence of fan types. App. Therm. Eng. 28, 728e737. Iragorry, J., Tao, Y.-X., Jia, S., 2004. A critical review of properties and model for frost formation analysis. J. HVAC&R Res. 10, 393e420. Ismail, K.A.R., Salinas, C.S., 1999. Modeling of frost formation over parallel cold plates. Int. J. Refrigeration 22, 425e441. Kandula, M., 2011. Correlation of water frost porosity in laminar flow over flat surfaces. Spec. Top. Rev. Porous Media 3, 79e87.
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Kandula, M., 2010. Effective thermal conductivity of frost considering mass diffusion and eddy convection. Spec. Top. Rev. Porous Media 1, 321e336. Kobayashi, T., 1958. On the habit of snow crystals artificially produced at low pressures. J. Met. Soc. Jpn. 36, 193e208. Lee, K.S., Kim, W.S., Lee, T.H., 1997. A one-dimensional model for frost formation on a cold flat surface. Int. J. Heat Mass Transf. 40, 4359e4365. Loyola, F.R., Nascimento, V.S., Hermes, C.J.L., 2014. Modeling of frost build-up on parallel-plate channels under supersaturated air-frost interface conditions. Int. J. Heat Mass Transf. 79, 790e795. Luer, A., Hans, B., 2000. Frost deposition in a parallel plate channel under laminar flow conditions. Int. J. Therm. Sci. 39, 85e95. Mao, Y., Besant, R.W., Rezkallah, K.S., 1992. Measurement and correlations of frost properties with airflow over a flat plate. ASHRAE Trans. 98, 65e77. Mao, Y., Besan, R.W., Chen, H., 1999. Frost characteristics and heat transfer on a flat plate under freezer operating conditions. Part I ASHRAE Trans. 105, 231e251. Na, B., Webb, R.L., 2004. New model for frost growth rate. Int. J. Heat Mass Transf. 47, 925e936. Nascimento, V.S., Loyola, F.R., Hermes, C.J.L., 2015. A study of frost build-up on parallel plate channels. Exp. Therm. Fl. Sci. 60, 328e336. Nield, D.A., Bejan, A., 2006. Convection in Porous Media. Springer. O'Neal, D.L., 1982. The Effects of Frost Formation on the Performance of a Parallel Plate Heat Exchanger. Department of Mechanical Engineering, Purdue University, West LafayetteIN, USA. PhD thesis. Ostin, R., Andersson, S., 1990. Frost growth parameters in a forced air stream. Int. J. Heat Mass Transf. 34, 1009e1017.
Pitman, D., Zuckerman, B., 1967. Effective thermal conductivity of snow at -88 , -27 , and -5 C. J. Appl. Phys. 38, 2698e2699. Sahin, A.G., 2000. Effective thermal conductivity of frost during the crystal growth period. Int. J. Heat Mass Transf. 43, 539e553. Sami, S.M., Duong, T., 1989. Mass and heat transfer during frost growth. ASHRAE Trans. 158e165. Sanders, C.T., 1974. Frost Formation: the Influence of Frost Formation and Defrosting on the Performance of Air Coolers. Dept. of Mechanical, Maritime and Materials Engineering, TU Delft, The Netherland. PhD thesis. Silva, D.L., Hermes, C.J.L., Melo, C., 2011. First-principles simulation of frost accumulation on fan-supplied tube-fin evaporators. App. Therm. Eng. 31, 2616e2621. Sturm, M., Holmgen, J., Konig, M., Morris, K., 1997. The thermal conductivity of seasonal snow. Glaciology 43, 26e41. Tao, Y.X., Besant, R.W., Mao, Y., 1993. Characteristics of frost growth on a flat plate during the early growth period. In: ASHRAE Trans.: Symposia, CH-93-2-2, pp. 746e753. Tso, C.P., Cheng, Y.C., Lai, A.C.K., 2006. Dynamic behavior of a direct expansion evaporator under frosting condition. Part I. Distributed model. Int. J. Refrigeration 29, 611e623. Yang, D.-K., Lee, K.-S., 2004. Dimensionless correlations of frost properties on a cold plate. Int. J. Refrigeration 27, 89e96. Yonko, J.D., Sepsy, C.F., 1967. An investigation of the thermal conductivity of frost while forming on a flat horizontal plate. ASHRAE Trans. 2043, I1.1eI1.9. Zehner, P., Schlunder, E.U., 1970. Thermal conductivity of granular materials at moderate temperatures. Chem. Ingr. Tech. 42, 933e941.
Available online at www.sciencedirect.com
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journal homepage: www.elsevier.com/locate/ijrefrig
A semi-empirical correlation for the thermal conductivity of frost* Silvia Negrelli, Christian J.L. Hermes* , 81531990 Curitiba, PR, Brazil Laboratory of Thermodynamics and Thermophysics, Federal University of Parana
article info
abstract
Article history:
The present work is aimed at advancing an improved correlation for the thermal con-
Received 23 February 2015
ductivity of frost. The analysis was conducted based on the weighted geometric mean of
Received in revised form
the thermal resistances of moist air and ice crystals, thus setting the theoretical back-
30 April 2015
ground for a dimensionless model for the thermal conductivity as a function of the
Accepted 26 May 2015
porosity of the frosted medium. Experimental data obtained elsewhere for a wide wall
Available online 22 June 2015
surface temperature span (from 30 C to 4 C) were employed to fit the model, coming up with a semi-empirical correlation for the thermal conductivity of frost valid for porosities
Keywords:
ranging from 0.5 to 0.95. Comparisons of the proposed correlation with the experimental
Frost
data showed that it is able to predict 81% of the data points (153 out of 188) within the 15%
Thermal conductivity
thresholds. Comparisons with other correlations available in the open literature are also
Correlation
reported. © 2015 Elsevier Ltd and IIR. All rights reserved.
lation semi-empirique pour la conductivite Une corre thermique de givre thermique ; Corre lation Mots cles : Givre ; Conductivite
1.
Introduction
Frost often builds-up on evaporators of small-capacity refrigerating appliances, resulting in an increased energy input to accomplish the same refrigerating effect. To mitigate the issues caused by the evaporator frosting, simulationbased designs have been carried out by means of mathematical models for the growth and densification of frosted media
*
in various geometries, such as flat surfaces (Sami and Duong, 1989; Tao et al., 1993; Lee at el., 1997; Na and Webb, 2004; Hermes et al., 2009; Hermes, 2012), parallel plate channels (O'Neal, 1982; Ismail and Salinas, 1999; Luer and Beer, 2000; Cui et al., 2011; Loyola et al., 2014), and tube-fin heat exchangers (Tso et al., 2006; Huang et al., 2008; Silva et al., 2011). Most models, however, rely on empirical correlations for the thermophysical properties of frost, particularly the density and the thermal conductivity, which limit their range of
An abridged version of this manuscript was submitted to be presented at the 24th IIR International Congress of Refrigeration, August 16e22, 2015, Yokohama, Japan. * Corresponding author. Tel.: þ55 41 3361 3239; fax: þ55 41 3361 3123. E-mail address: [email protected] (C.J.L. Hermes). http://dx.doi.org/10.1016/j.ijrefrig.2015.05.021 0140-7007/© 2015 Elsevier Ltd and IIR. All rights reserved.
244
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 5 8 ( 2 0 1 5 ) 2 4 3 e2 5 2
Nomenclature Roman A a, b B C Dv E, F Fo hsv k L M patm pv r Re Rv S T t X
Parameter of Equation (2) Coefficients of Equations (18) and (19) Shape factor of Equation (12) Blending factor of Equation (6) Water vapor diffusivity in air, m2 s1 Parameters of Equation (5) Fourier number, dimensionless Latent heat of sublimation, J kg1 Thermal conductivity, W m1 K1 Binary parameter of Equation (4) Molecular weight, kg mol1 Atmospheric pressure, Pa Vapor pressure of water, Pa Radius, m Reynolds number, dimensionless Gas constant for water vapor, J kg1 K1 Surface area, m2 Temperature, K Time, s Dimensionless frost thickness
Greek a, b, g d ε r u x z
Coefficients of Equation (1) Thickness Frost porosity Density, kg m3 Humidity ratio Blending factor of Equation (10) Dimensionless thermal conductivity
Subscripts 0 Reference state a Moist air air Dry air c Turbulent convection eff Effective f Frost g Geometric mean i Ice m Molecular diffusion max Maximum min Minimum p Parallel association s Serial association t Thermal diffusion tp Triple point v Water vapor w Wall surface
operation. The former was assessed in prior studies, when empirical and semi-empirical correlations were put forward for the frost density (Hayashi et al., 1977; Mao et al., 1992; Yang and Lee, 2004; Kandula, 2011; Hermes et al., 2014; Nascimento et al., 2015). The latter is at the aim of the present paper. Several studies are available in the open literature focused on the thermal conductivity of frost. The most influential ones
are described in Table 1, whereas Table 2 summarizes the key parameters evaluated in the studies reported in Table 1. They are detailed as follows. Yonko and Sepsy (1967), in a pioneering study, advanced an empirical correlation for the thermal conductivity of frost as a quadratic polynomial function of the frost density. The correlation was not able to predict the experimental data satisfactorily, suggesting that the thermal conductivity is dependent on parameters other than the density. Despite of that, their approach influenced the works of Ostin and Andersson (1990) and Sturm et al. (1997), which also employed quadratic polynomials to correlate the thermal conductivity as a function of the frost density, as follows: kf ¼ a þ brf þ gr2f
(1) -1
1
where kf is the thermal conductivity of frost in [W m K ], rf is the frost density in [kg m3], and the coefficients a, b and g are summarized in Table 3. Pitman and Zuckerman (1967) put forward a semiempirical correlation assuming the ice crystals as small spheres connected to each other by ice cylinders. The correlation predictions were compared with experimental data for wall surface temperatures of 88, 27 and 5 C, disregarding the dense frost media typical of the plate-shaped ice crystals observed for wall surface temperatures between 27 and 5 C, as depicted in Fig. 1, which reproduces the frost morphology map due to Kobayashi (1958). Their correlation is as follows: 1 1r 4lnðA þ 1Þ=ðA 1Þ ¼ þ kf ðki ka ÞS þ ka 2Aprðki ka Þ
(2)
where ka and ki are the thermal conductivities of moist air and ice, respectively, r is the radius of the spheres, and S is the surface area of the cylinders. The parameter A is defined as follows: A¼
1þ
4 p½ðki =ka Þ 1r2
1=2 (2.a)
A few years later, Brian et al. (1969) noticed the frost density and the wall surface temperature are key independent parameters that affect the thermal conductivity of the frost media, thus coming up with the following empirical correlation based on their own data, which is valid for frost densities lower than 250 kg m3: kf ¼ 2:401 105 T1:272 þ 3:921 108 rf T1:74 w w
(3)
where Tw is the wall surface temperature in [K], rf is the frost density in [kg m3], and kf is the thermal conductivity of frost in [W m-1 K1]. One year later, Biguria and Wenzel (1970) advanced theoretical models to predict the effective thermal conductivity of frost based on the crystal morphology (e.g., spheres, plates, columns, needles), as depicted in Fig. 1. They found out that none of the theoretical models were able to predict the experimental data satisfactorily using the frost density as the only parameter. In order to improve the model prediction capabilities, they included other key heat and mass transfer parameters into correlation, as the air velocity and humidity, as follows:
245
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 5 8 ( 2 0 1 5 ) 2 4 3 e2 5 2
Table 1 e Summary of key studies on the thermal conductivity of frost. Author
Year
Origin
Approach
Yonko and Sepsy Pitman and Zuckerman Brian et al. Biguria and Wenzel Dietenberger
1967 1967
USA USA
Empirical Semi-empirical
1969 1970
USA USA
Empirical Empirical
1982
USA
Semi-empirical
Auracher Ostin and Andersson Mao et al. Sturm et al. Mao et al. Sahin Na and Webb
1986 1990
Germany Sweden
Theoretical Empirical
1992 1997 1999 2000 2004
Canada USA Canada Saudi Arabia USA
Empirical Empirical Empirical Semi-empirical Semi-empirical
Yang and Lee Kandula
2004 2010
S. Korea USA
Empirical Theoretical
Background model
Geometry
Independent parameter
Density range
e Spheres and cylinders e e
Flat plate Flat plate
rf rf, Tw
rf < 573 rf < 400
Flat plate Flat plate
rf, Tw rf, Tw
rf < 250 50 < rf < ri
Spheres and cylinders Spheres e
Flat plate
ε, Tw
ra < rf < ri
Flat plate Parallel plate channel Flat plate Flat plate Flat plate Flat plate Flat plate
rf, Tw rf
100 < rf < 600 50 < rf < 680
X, Re, Fo, ua, Ta, Tw rf X, Re, Fo, ua, Ta, Tw ε, Tw rf, Tw
rf < rf < rf < rf < rf <
Flat plate Flat plate
Re, Fo, ua, Ta, Tw ε, Tw
rf < 180 rf < 500
e e e Cylinders Cylinders and plates e Cylinders
200 600 200 250 400
Table 2 e Summary of the parameters considered in the studies reported in Table 1. Author
Year
Yonko and Sepsy Pitman and Zuckerman Brian et al. Biguria and Wenzel Dietenberger Auracher Ostin and Andersson Mao et al. Sturm et al. Mao et al. Sahin Na and Webb Yang and Lee Kandula
Frost density
1967 1967 1969 1970 1982 1986 1990 1992 1997 1999 2000 2004 2004 2010
Wall temperature
X X X X X X X X X X X X X X
þ 3:5719847 104 Ua þ 6:2047771 104 t 8:9475394 105 t2 þ 1:0182528 107 t3 þ 2:6084586 108 t4 4:2023418 102 ua Tw þ 0:11349924tua þ 1:0859212 102 L þ 2:1232614 105 Ua t 2:6856724 105 Tw L (4) where ua is the humidity ratio of the air stream in [kgv kg1 air], Ua is its velocity in [ft s1], t is the time in [minutes], Tw is the
Table 3 e Coefficients of Equation (1).
Yonko and Sepsy Ostin and Andersson Sturm et al. (156 < rf < 600) Sturm et al. (rf < 156)
Year
a
b
1967 0.02422 7.214 10 1.1797 106 1990 0.00871 4.39 102 1.05 106 1997 0.138 1.01 103 3.233 106 0.023
Humidity
X
X
X
X X X
X
X
X X X X X X X
X
X
X X X X X
X X X X X
wall surface temperature in [R], and L is a binary parameter which may be 1 or 1 depending on the flow regime. According to Biguria and Wenzel (1970), Equation (4) was able to predict their own experimental data within ±25% error bounds. Dietenberger (1982) devised a semi-empirical model to predict the thermal conductivity considering two different frost structures, a low-density array of spherical and cylindrical ice crystals, which provides the minimum threshold for the thermal conductivity, and a porous matrix comprised of air bubbles imprisoned in high-density ice plates, corresponding to the maximum threshold. The model blends the two structures based on the frost porosity, ε, as follows:
g 4
1997
Velocity
X X X X X X X X X X X X X X
kf ¼ 0:23376438 þ 1:0342876 104 T1:3 w þ 18:007637ua
Author
Air Temperature
2.34 102
e
1 kf ¼ ðð3E 1Þ kmin þ ð3F 1Þ kmax 4 1=2 2 þ ðð3E 1Þ kmin þ ð3F 1Þ kmax Þ þ 8kmin kmax where F ¼ 1-E, and
(5)
246
Sheath
0.2
Needle
0.3
Sectorated plate
0.1 0
where hsv is the latent heat of sublimation in [J kg1], ri is the density of ice in [kg m3], patm is the atmospheric pressure in [bar], Tw is wall surface temperature in [K], and the index “0” refers to the reference state (1 bar, 273.16 K). Mao et al. (1992) put forward the following dimensionless correlation, which was later revisited by Mao et al. (1999) and Yang and Lee (2004) (Iragorry et al., 2004):
Dendrite
0.4
Plate
Supersaturation degree [g kg-1]
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 5 8 ( 2 0 1 5 ) 2 4 3 e2 5 2
Sheath
Ttp Tw 0:216 1:093 0:699 kf ¼ 0:011$X0:37 u2:044 Re Fo a ki Ta Tw
Hollow column Solid column 0
-10
-5
-15
-20
-25
where X is the dimensionless frost thickness, Re and Fo are the Reynolds number and the Fourier number, respectively, and Ttp is the temperature of the triple point of water. Their correlation is limited to densities lower than 200 kg m3. Equation (7) was later improved by Mao et al. (1999), coming up with the following expression (Iragorry et al., 2004):
-30
Temperature [°C] Fig. 1 e Illustration of Kobayashi's (1958) map of frost morphology.
Ttp Tw 0:737 0:375 0:18 kf ¼ 6:534 104 $X0:048 ua0:004 Re Fo ki Ta Tw
"
2 ε E1 ε E1 E ¼ 13:6ðE2 E1 Þðε E1 Þ 1 þ 3 E3 E1 E2 E1 # 2 ðε E1 Þ þ 2ðE3 E1 ÞðE2 E1 Þ
(7)
(8)
2
kmax
a =ki 1 2ε 1k 2þka =ki
þ ε 1 εÞki þ εka ¼ ð1 εÞki 1ka =ki 1 þ ε 2þka =ki "
kmin ¼ ð1 εÞ
(5.a)
(5.b)
(5.c)
where E1 ¼ 0.1726(Tf/273), E2 ¼ 0.751 and E3 ¼ 1.051. Despite of its theoretical background, the model tends to overpredict the experimental data for high densities (Sahin, 2000). Auracher (1986) assumed the effective thermal conductivity of frost has contributions of both thermal (kt) and molecular (km) diffusion, as follows: kf ¼ kt þ km
(6)
The former is set as a blend of serial and parallel thermal resistance association, 1 C 1C ¼ þ kt ks kp
(6.a)
where ks and kp are the thermal conductivities obtained from serial and parallel thermal association, respectively, and the blending factor C is calculated from the following empirical expression: C ¼ 0:42ð0:1 þ 0:995rf Þ
(6.b)
The contribution due to the molecular diffusion, on the other hand, is calculated as follows: km ¼ 1:958 109 hsv 1:28 p Tw 0 patm T0
! ri rf 6145 exp 24:02 Tw ri 0:58$rf
1 1 ¼ kf df
Zdf 0
1 dy kf ðTÞ
(9)
where df is the frost layer thickness (m), and kf(T) is the local thermal conductivity, calculated from:
#
ka ki ½3 þ 2εðka =ki 1Þ þ εki 1 εÞka þ εki i 1 3 ε kak=k a =ki
Sahin (2000) developed a model for the early crystal growth stages based on a local energy balance, and vapor mass diffusion and desublimation considering column-shaped ice crystals. The thermal conductivity is expressed as follows:
(6.c)
hsv patm pv0 hsv 1 1 exp Rv T0 T T01:94 R2v T1:06
3 0:963 þ 1:202 10 εri þ þð1 εÞ 1:0465 þ 0:017Tf 105
kf ðTÞ ¼ 0:131 106 ð1 εÞ
(9.a) where ε is the frost porosity, T is the local frost temperature in [K], pv0 is the vapor pressure of water at the reference temperature T0 (273.16 K), and Rv is the gas constant for water vapor. The model is able to predict the temperature variation along the ice crystals and the frost density over time for the early stages of frost formation, when the density of the frosted medium is low. In addition, the model is limited to columnshaped crystal morphologies, being not suitable for applications with wall surface temperatures typical of dendritic and plate-shaped ice crystals, i.e., from 19 C to 10 C according to the morphology map illustrated in Fig. 1. Na and Webb (2004) proposed an empirical correlation blending the minimum and maximum thermal resistances calculated according to the models presented in Sanders (1974). The blending factor was correlated based on their own experimental data for different wall surface temperature ranges, spanning three different frost morphologies: columns (10 < Tw < 4 C), plates (21 < Tw < 10 C), and columns (Tw < 21 C), according to the map for ice crystal morphology. The correlation is as follows:
kf ¼ kp xf þ ks 1 xf
(10)
where kp and ks are the thermal conductivity obtained from parallel and series thermal association, respectively. The
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 5 8 ( 2 0 1 5 ) 2 4 3 e2 5 2
blending factor xf is calculated from the following empirical expressions: 10 < Tw < 4o C (10.a) xf ¼ 0:283 þ exp 0:020$rf xf ¼ 0:140 þ 0:919 exp 0:0142$rf
21 < Tw < 10+ C (10.b)
Tw < 21+ C xf ¼ 0:0107 þ 0:419 exp 0:00424$rf
(10.c)
where rf is the frost density in [kg m3]. In the same year, Yang and Lee (2004) proposed a dimensionless empirical correlation for the thermal conductivity of frost, among others derived for key heat and mass transfer parameters related to frost growth and densification. Their correlation is as follows: 0:512 Ta Ttp kf 0:619 ¼ 1:184 102 ðln ReÞ Fo0:084 u0:086 exp a ki Ta Tw (11) where Re and Fo are the Reynolds number and the Fourier number, respectively, and Ttp is the temperature of the triple point of water. They reported an agreement between equation (11) and their own data within ±7% error bounds, although their correlation is valid for a limited range of application, with air temperatures spanning from 5 to 15 C, and wall surface temperatures ranging from 35 to 15 C. Recently, Kandula (2010) included the mass diffusion and turbulent convection into the model of Zehner and Schlunder (1970), coming up with the following fully theoretical model for predicting the thermal conductivity of frost: pffiffiffiffiffiffiffiffiffiffiffi " pffiffiffiffiffiffiffiffiffiffiffi 2 1 ε kf ð1 zÞB 1 Bþ1 þ ¼1 1εþ ln 1 zB zB 2 keff ð1 zBÞ2 # B1 (12) 1 zB where B = 2.5(1/e-1)10/9 is a shape factor related to porosity, z = keff/ki is a dimensionless thermal conductivity, keff = ka þ kc þ km is the combination of the air thermal conductivity, ka, the turbulent convection, kc = 0.00568$Ua, and the conductivity due to mass diffusion, km, which is calculated from: km ¼ hsv Dv
" # Ma =Mv hsv 1 1 p exp v0 Rv T0 Tf R2v T3f
(12.a)
crystal growth stages, being not suitable for dense frosted media. Finally, the literature also shows that the frost density is not the only parameter which affects the thermal conductivity, since the ice crystal morphology is also influenced by the wall surface temperature, as illustrated in Fig. 1. The present work is therefore aimed at putting forward, by means of a theoretical-experimental approach, an improved correlation for the thermal conductivity of frost formed on flat surfaces. The theoretical analysis was conducted to set the scales of the problem, which served as the basis for a semiempirical correlation for the effective thermal conductivity of frosted media as a function of the frost density and the wall surface temperature.
2.
Theoretical model
The effective thermal conductivity of porous media depends on the thermal conductivities of each phase (solid and fluid), on the volume fraction between the fluid phase and the porous medium (i.e., porosity), and on the array formed by particles and pores. In the case of frost, the porosity ε can be calculated from: ε¼
ri rf ri ra
(13)
where ra (z1.35 kg m3) and ri (z920 kg m3) are the densities of moist air and ice, respectively. For high porosity frost, the thermal conductivity approaches that of moist air (ka z 0.0225 W m-1 K1), whereas for low porosity frost, the thermal conductivity tends to that of ice (ki z 2.0 W m-1 K1). In addition, the array plays an important role upon the effective thermal conductivity. The limiting cases are the parallel (kp) and serial (ks) association of the thermal resistances of the ice and air phases, illustrated in Fig. 2, and calculated respectively as follows (Sanders, 1974): kp ¼ ka ε þ ki ð1 εÞ
(14)
1 ε 1ε ¼ þ ks ka ki
(15)
Alternatively, Nield and Bejan (2006) proposed the weighted geometric mean of ka and ki, as follows: ε
where Dv is the mass diffusion coefficient of water vapor, Ma and Mv are the molecular weights of air and water vapor, respectively, Tf is the frost surface temperature in [K], and Ua is the air velocity in [m s1]. Kandula (2010) has also observed that the value of kc can be approximated as kc z ka with satisfactory agreement. The model predictions were compared with experimental data obtained elsewhere, showing satisfactory results for low frost densities. The literature survey reveals that most of the correlations available in the open literature for the thermal conductivity of frost holds a strong empirical character, being restricted to the range of the experimental data used in the fitting process. In addition, the theoretical models have been developed for early
247
1ε
kg ¼ ka ki
(16)
Fig. 3 illustrates Equations (14)e(16), where one can note that kf / ki where ε / 0 (left corner) and that kf/ka where ε/1 (right corner) independently of the model (parallel, serial, geometric mean). It is worth noting that kp is the maximum allowed thermal conductivity, whereas ks is the minimum one, suggesting that the parallel array is related to the column-shaped crystal morphology, whereas the serial one is regarded with the plate-shaped ice morphologies, as depicted in Fig. 2. The geometric mean provides intermediate values between ks and kp. Experimental data obtained from the open literature is also plot in Fig. 3. All data points (but four) lie in between the thresholds defined by parallel and geometric mean model.
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Heat flux
Ice columns Air
Heat flux Ice plates Air A Cold surface
Cold surface
Fig. 2 e Schematic of parallel (left) and serial (right) thermal resistance association.
1 Auracher -48°C to -23°C
parallel
Auracher -23°C to -13°C Auracher 11°C Brian et al. -18ºC
kf / ki
Brian et al. -6ºC Hayashi et al.
geometric
0.1
Na and Webb -6°C Na and Webb -13°C
Na and Webb 24°C
serial
Ostin and Andersson Pitman and Zuckermann -27°C Pitman and Zuckermann -6°C Yonko and Sepsy
0.01 0
0.2
0.4
0.6
0.8
1
ε Fig. 3 e Illustration of Equations (14)e(16) with experimental data from the open literature.
Only four data points lie in the region below the geometric mean, which one might attribute to measurement uncertainties.
mathematically. Thus making kf ¼ kg and dividing Equation (16) by ki, and taking the logarithm in both sides, it follows that: log
3.
Semi-empirical correlation
The dependence of the thermal conductivity with the porosity is quite obvious in Fig. 3. Since the thresholds are related to the morphology, which in turn is strongly dependent on the wall surface temperature, and mildly dependent of the supersaturation degree, as can be seen in Fig. 1, the correlation for the thermal conductivity of frost proposed in this work shall include both. The former was taken into account by fitting the model to different temperature ranges, as proposed by Na and Webb (2004), whereas the latter was accounted for by means of the frost porosity, which carries information on the supersaturation degree, as it is a function of the modified Jakob number and the square root of time (Hermes et al., 2014). An inspection of Fig. 3 reveals that most of the experimental data follow the trend described by the geometric mean, in spite of an offset and a small slope that can be corrected empirically. Since there is no room in the geometric mean for empirical fitting, Equation (16) had to be modified
kf ka ¼ εlog ki ki
(17)
Adding two linear (slope and intercept) coefficients into Equation (17), it follows that: log
kf ka ¼ loga þ b$εlog ki ki
(18)
Re-arranging Equation (18) in a more elegant form, it follows that: bε kf ka ¼a ki ki
(19)
where the “a” (intercept) and “b” (slope) coefficients of Equation (19) are fitted to the experimental data according to the frost morphology, as depicted in Table 4, which in turn is embedded in the range of the wall surface temperature, namely needles and sheaths (10 < Tw < 4 C), plates and dendrites (19 < Tw < 10 C), and sheaths (30 < Tw < 19 C) (see Fig. 1). A similar approach was adopted by Na and Webb (2004), who proposed empirical expressions for the thermal
249
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Table 4 e Coefficients of Equations (18) and (19). Dataset# (i) (ii) (iii)
Temperature range
Morphology
Intercept (a)
Slope (b)
log a
10 < Tw < 4 C 19 < Tw < 10 C 30 < Tw < 19 C
Needles and sheaths Plates and dendrites Sheaths
1.576 1.594 1.035
0.797 0.761 0.797
0.1976 0.2025 0.0149
1 parallel
k f / ki
out. The error embedded in the frost correlation, Eε, propagates to the thermal conductivity, Ekf, as follows:
-10ºC to -4ºC -19ºC to -10ºC -30°C to -19°C experimental fit
Ekf ¼
vkf Eε vε
(20)
where the derivative can be obtained analytically from Equation (19), yielding
0.1
vkf ka ¼ bkf log vε ki
geometric
Equation (19) shows that the highest errors were observed for the lowest temperatures and porosities (i.e., 0.5) in the interval. Noting that Hermes et al. (2014) correlated the frost porosity by means of a semi-empirical model, obtaining errors within the ±2% bounds, one can show, based on Equation (21), that errors up to 2.5% can be observed for Tw ¼ 30 C, up to 4.0% for Tw ¼ 19 C, and up to 3.5% for 10 C.
serial
0.01 0.5
0.6
0.7
0.8
0.9
(21)
1
ε Fig. 4 e Best fitting of Equation (19) spanning three different temperature ranges: ¡10 < Tw < ¡4 C (needles and sheaths), ¡19 < Tw < ¡10 C (plates and dendrites), and ¡30
4.
Discussion
A close inspection of Table 4 reveals that datasets (i) and (iii) presented the same slope, which is probably due to the similar frost morphologies (column-shaped ice crystals) observed for both datasets. Since the intercept and slope coefficients diverge from the unity, one may not expect Equation (19) to be valid in the limits where ε / 0 and ε / 1. Fig. 4 plots the best fits for the three temperature ranges, where one can observe that Equation (19) is able to predict the experimental trends quite satisfactorily. It is worth noting that two competing effects resulting from the wall surface temperature take place, namely the molecular diffusion and the frost morphology. Fig. 3 suggests that the former overrules the latter, thus explaining why the high temperature data (from 10 to 4 C) are close to the parallel array, the low temperature data (from 30 to 19 C) are near the geometric
conductivity aimed at different frost morphologies (see Equations 10.ae10.c). It is worth noting that some experimental data points plotted in Fig. 3 do not provide information on the wall surface temperature and, therefore, had to be discarded. In total, 188 data points from Pitman and Zuckerman (1967), Brian et al. (1969), Auracher (1986), and Na and Webb (2004) have been considered in the fitting process. The resulting fitting coefficients are shown in Table 4. Since the thermal conductivity of frost depends on the frost porosity, which in turn is calculated from empirical correlations, an error propagation analysis must be carried
Table 5 e Overall comparison between the correlations and experimental data. Author
Number of experimental data points
Yonko and Sepsy (1967) Pitman and Zuckerman (1967) Brian et al. (1969) Dietenberger (1982) Auracher (1986) Ostin and Andersson (1990) Sturm et al. (1997) Na and Webb (2004) Kandula (2010) Present work a
Correlation predictions
10 < Tw < 4 C
19 < Tw < 10 C
30 < Tw < 19 C
RMS
e 6 12 e 0 e e 12 e e
e e 17 e 74 e e 24 e e
e 10 0 e 25 e e 8 e e
0.519
Correlation not evaluated due to its dependence on parameters other than porosity and temperature.
a
Points ±15% 28 a
0.170 0.178 0.172 0.386 0.464 0.179 0.202 0.111
108 95 129 91 18 92 87 153
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i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 5 8 ( 2 0 1 5 ) 2 4 3 e2 5 2
0.3
Predicted kf [W m-1 K-1]
+15%
Yonko and Sepsy (1967) Brian et al. (1969) -15%
0.2
Dietenberger (1982) Auracher (1986)
Ostin and Andersson (1990) Sturm et al. (1997) Na and Webb (2004)
0.1
Kandula (2010) Present work
-10
0 0
0.1
0.2
Measured kf [W
0.3
m-1 K-1]
0.5
Predicted kf [W m-1 K-1]
Yonko and Sepsy (1967)
0.4
Brian et al. (1969)
+15%
Dietenberger (1982) Auracher (1986)
0.3
Ostin and Andersson (1990)
-15%
Sturm et al. (1997)
0.2
Na and Webb (2004) Kandula (2010)
0.1 -19
Present work
0
0
0.1
0.2
0.3
0.4
0.5
Measured kf [W m-1 K-1] 0.3 +15%
Yonko and Sepsy (1967)
Predicted kf [W m-1 K-1]
Brian et al. (1969) -15% 0.2
Dietenberger (1982) Auracher (1986)
Ostin and Andersson (1990) Sturm et al. (1997) 0.1
Na and Webb (2004) Kandula (2010)
Present work
-30
0.1
0.2
0.3
Measured kf [W m-1 K-1] Fig. 5 e Comparison between the experimental data and some key correlations available in the literature.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 5 8 ( 2 0 1 5 ) 2 4 3 e2 5 2
mean, and the data for the plates and dendrites (range 19 to 10 C) are in between. Table 5 compares the predictions of the correlations due to Yonko and Sepsy (1967), Brian et al. (1969), Dietenberger (1982), Auracher (1986), Ostin and Andersson (1990), Sturm et al. (1997), Na and Webb (2004) and Kandula (2010) against all 188 experimental data points. One should note that the numbers in the middle of the table represent the number of data points from each dataset in each temperature (morphology) range. The numbers in the right-side represent the RMS (root mean square) errors and the number of data points predicted by the proposed model (Equation (19)) within the ±15% bounds. It can be seen that the models due to Pitman and Zuckerman (1967), Mao et al. (1992, 1999) and Yang and Lee (2004) have not been taken into account as they depend upon parameters other than the density and the wall surface temperature, not available in most of the dataset. In the same fashion, the model due to Sahin (2000) has not been included into the analysis, as it is applicable for the early stages of the frost formation only. One can see that the proposed correlation presented an RMS error ~11%, a figure significantly smaller than that observed for the other models (>17%). In total, 153 out of 188 data points (~81%) fall within the ±15% error bounds, as depicted in Fig. 5, where the predictions of the correlations with all experimental data for different temperature ranges are compared. One can also see in Fig. 5 that, for 10 < Tw < 4 C, the correlation of Auracher (1986) tends to overpredict the experimental data, whereas the correlations of Brian et al. (1969), Dietenberger (1982), Ostin and Andersson (1990) and Sturm et al. (1997) underpredict the data points for high porosity (low density). The empirical correlation of Yonko and Sepsy (1967) follows a different pattern, underpredicting the thermal conductivity at high porosities, and doing the contrary at low porosities. In this temperature range, not only equation (19) but also the models of Na and Webb (2004) and Kandula (2010) have shown satisfactory predictions for most of the data points. For 19 < Tw < 10 C, the correlations of Auracher (1986), Dietenberger (1982) and Equation (19) have all shown good predictions for the whole dataset. The correlations of Brian et al. (1969), Sturm et al. (1997) and Na and Webb (2004), however, tend to underpredict the experimental data points, whereas the correlations of Yonko and Sepsy (1967) and Kandula (2010) follow the opposite behavior. Finally, for 30 < Tw < 19 C, most of the correlations evaluated in this work presented satisfactory results, but those of Yonko and Sepsy (1967), Ostin and Andersson (1990) and Sturm et al. (1997). At all, equation (19) is the only one whose predictions are within the ±15% thresholds independently of the temperature range.
5.
Conclusions
A semi-empirical correlation was advanced based on 188 experimental data points obtained from the open literature with wall surface temperatures ranging from 30 to 4 C, and frost porosities spanning from 0.5 to ~0.95. The
251
correlation predictions were compared with the experimental data, when an 11% RMS error was observed, with 153 out of 188 (81%) data points lying within the ±15% error band. The correlations due to Yonko and Sepsy (1967), Brian et al. (1969), Dietenberger (1982), Auracher (1986), Ostin and Andersson (1990), Sturm et al. (1997), Na and Webb (2004) and Kandula (2010) were also compared with the experimental data, showing poorer prediction capabilities than that proposed in this study, with RMS errors higher than 17%. Among all correlations evaluated in this study, Equation (19) has shown to be the only one whose predictions are within the ±15% thresholds independently of the temperature range.
Acknowledgments This work was carried out under the auspices of the Brazilian Government funding agency CNPq (Grant No. 441603/2014-9). Ms. S. Negrelli duly acknowledges the CAPES agency, Government of Brazil, for supporting her MSc education at the . Federal University of Parana
references
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