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A prediction model of surface heat transfer coefficient in insulating packaging with phase change materials
Food Packaging and Shelf Life 24 (2020) 100474
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Food Packaging and Shelf Life journal homepage: www.elsevier.com/locate/fpsl
A prediction model of surface heat transfer coefficient in insulating packaging with phase change materials
T
Liao Pana,b, Xi Chena, Lixin Lua,b,*, Jun Wanga,b, Xiaolin Qiua,b a b
Department of Packaging Engineering, Jiangnan University, Wuxi, 214122, China Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment and Technology, Wuxi, 214122, China
ARTICLE INFO
ABSTRACT
Keywords: Surface heat transfer coefficient Prediction model Insulating packaging Phase change Heat transfer
Surface heat transfer coefficient is one of the crucial parameters for determining the system thermal resistance of insulating packaging. However, the surface heat transfer coefficient which couples with the temperature field in insulating packaging is difficult to experimentally measure and independently calculate. In this study an iterative method based on a semi-infinite phase change model was proposed to solve the coupling parameters of surface heat transfer coefficient and temperature field. Then, the semi-infinite phase change model curves were compared with classic Neumann model and Mehling model. Furthermore, the affecting parameters of surface heat transfer coefficient were discussed through this semi-infinite phase change model, and a concise prediction model of surface heat transfer coefficient was also developed for engineering application. Finally, this concise prediction model was verified by “Ice Melting Method”. The results indicate that the proposed semi-infinite phase change model and iterative method can simultaneously calculate the coupling parameters of surface heat transfer coefficient and the temperature field in insulating packaging, and show a consistent with the classic Neumann model and Mehling model. The key parameters of surface heat transfer coefficient include the thickness and thermal conductivity coefficient of insulating wall, and the excess temperature also significantly influences the surface heat transfer coefficient. Furthermore, the system thermal resistance calculated by the novel concise prediction model has a good agreement with the experimental data of “Ice Melting Method” presented by Burgess.
1. Introduction Insulating packaging with phase change materials (PCMs) can be used for storing temperature-sensitive products (such as fresh foods) without electrical energy during the using process, and show some significant advantages over classic cold chain (Moreno et al., 2014). For insulating packaging design, surface heat transfer coefficient is an essential parameter which significantly influences the system thermal resistance of insulating packaging. The surface heat transfer coefficient is usually identified by boundary-layer theory (Prandtl, 1940), and temperature difference T and critical temperature Tc are two essential parameters during calculating. However, in the situation of insulating packaging, T and Tc are unknown parameters, which are determined by the temperature field of insulating packaging. The insulating packaging is a kind of phase change system. For the determination of the temperature field in phase change system, Neumann proposed the most classic analytical solution of semi-infinite phase change system under the first-class heat exchange boundary
⁎
condition where the heat load directly occurs at the surface of PCM (Zhang, 2009). Then, numerous studies have been conducted on the other phase change systems with the first-class heat exchange boundary condition by using numerical methods, analytical models and experiment (Gowreesunker, Tassou, & Kolokotroni, 2012; Jradi, Gillott, & Riffat, 2013; Prashant, Varuna, & Singal, 2008; Wang, Li, Lin, & Hu, 2018; Zalba, Marin, Cabeza, & Mehling, 2003; Zhou, Zhao, & Tian, 2012). In these research studies, there is no insulating material outside the PCM, and the heat exchange boundary condition of insulating packaging belongs to the natural convection boundary (not the first-class heat exchange boundary condition). Therefore, these models cannot be used to solve the temperature field in insulating packaging. For these reasons, Mehling firstly presented an approximate one-dimensional phase change heat transfer model considering insulating wall for describing the temperature fields in insulating packaging (Mehling & Cabeza, 2008). However, in Mehling’s solution, the surface heat transfer coefficient was assumed to be a known parameter, which is impossible in most cases. The relationship between the surface heat transfer coefficient and the temperature field in insulating packaging is complex. On one hand, the
Corresponding author at: Department of Packaging Engineering, Jiangnan University, No. 1800 Lihu Avenue, Wuxi, 214122, China. E-mail address: [email protected] (L. Lu).
https://doi.org/10.1016/j.fpsl.2020.100474 Received 8 August 2019; Received in revised form 9 January 2020; Accepted 24 January 2020 2214-2894/ © 2020 Elsevier Ltd. All rights reserved.
Food Packaging and Shelf Life 24 (2020) 100474
L. Pan, et al.
Nomenclature T
T
t x b p a k L h s m Gr Pr Nu E g l C A R
Body expansion coefficient/K−1 Kinematic viscosity/m2/s Dimensionless temperature Dimensionless temperature in complex-frequency domain after Laplace transfer Excess temperature/K Density/kg/m3 Maximum permissible errors/K
α υ θ
Temperature/K Temperature difference between environmental temperature and surface temperature of insulating wall/K Time/s Position/m Thickness of insulating wall/m Position of moving boundary/m Thermal diffusivity/m2/s Thermal conductivity/W/(m·K) Latent heat of PCM/×105J/kg Surface heat transfer coefficient/ W/(m2·K) Complex-frequency domain variable of time after Laplace transfer Mass of melted water/kg Grashof number Prandtl number Nusselt number Relative Error/% Acceleration of gravity/m/s2 Characteristic dimension/m Specific heat/J/(kg·K) Superficial area/ m2 System thermal resistance/m2·K/W
ψ ρ ε Subscripts a v b t w s l e m c j i o
surface heat transfer coefficient determines the heat transfer of insulating packaging, then influences the temperature field in insulating packaging. On the other hand, the temperature field in insulating packaging will change the surface temperature of insulating wall and simultaneously change the surface heat transfer coefficient caused by nature convection. Thus, the surface heat transfer coefficient and the temperature field in insulating packaging are a group of coupling parameters. The major objective of this study is to develop a semi-infinite phase change model of insulating packaging, and propose an iterative algorithm to solve the coupling parameters of surface heat transfer coefficient and temperature field in the semi-infinite phase change model. Then, the affecting parameters of surface heat transfer coefficient are discussed through this semi-infinite phase change model, and a group of concise prediction models of surface heat transfer coefficient is also provided for engineering application. Finally, the system thermal resistance which is essential for designing insulating packaging can be calculated through the proposed prediction model.
Air Vertical Bottom Top Insulating wall Solid phase Liquid phase Environment Melting point Critical temperature The surface number of insulating packaging Inside Outside
(3) The PCM and insulating material are isotropic; (4) Volume changing, convection, and thermal radiation during the melting process are ignored (Aduda, 1996); (5) The PCM and insulating material properties are assumed to be independent of temperature. 2.1. Mathematical analysis 2.1.1. Heat transfer function The heat transfer process of the semi-infinite phase change model can be presented mathematically by applying the Fourier law of heat conduction as
Ts = Tm, p < x < 2T T al 2l = l , 0 < x < p ,t>0 x t 2T Tw w aw = , b
2. Model development
Where aj =
kj j Cj
(1)
, j = w, l, s .
The boundary conditions can be shown applying the energy balance
The insulating packaging is constituted by insulating wall, PCM and products. This is a multilayer finite 3-dimensional phase change heat transfer system. According to Mehling’s research (Mehling & Cabeza, 2008), the heat transfer only occurs in the insulating wall and the liquid field of PCM before the complete melting of PCM. Besides, the heat exchange area of insulating wall is much larger at the face than at the edge and corner. Thus, Mehling simplified the insulating packaging into a one-dimensional semi-infinite phase change model with natural convection boundary. As shown in Fig. 1, the main component of the one-dimensional phase change model is a semi-infinite PCM that is covered by insulating material; the insulating material is heated by natural convection. Initially, the temperature of the solid phase and insulating material are isothermal (Ts = Tw =Tm). The following assumptions are made in the analysis: (1) Heat transfer occurs only in the x direction; (2) The PCM area is simplified into a semi-infinite region due to the constant temperature of solid phase before melting;
Fig. 1. The semi-infinite phase change model with natural convection boundary. 2
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L. Pan, et al.
at each interface as
l (p ,
Tl (p , t ) = Tm Tl x
kl
= x= p
kl
lL
kw
= h [Te
,t>0
Tw ( b, t )]
(2)
x= b
w
The initial conditions can be easily obtained as
s
Tw = Tm, b < x < 0 Ts = Tm, 0 < x < ,t=0 p=0
2.1.2. Surface heat transfer coefficient According to boundary-layer theory, the surface heat transfer coefficient of natural convection above the insulating wall can be calculated through a Nusselt number (Yang & Tao, 2008) as shown in Eq. (4). where Nu = , Gr = g a a T , and Pr = a / a . C and n are two parameters that can be determined by experiments (According to Refs. Shimoyama, Horibe, and Haruki (2018) and Guangsong and Zhiyong (2009)). l is equal to the height of heat exchange surface when the surface is vertical, and l is equal to the ratio of area and perimeter of heat exchange surface when the surface is horizontal (Yang & Tao, 2008). In addition, the air parameters ( a , a , and ka ) under critical temperature Tc can be obtained by referencing to the previous research studies (Yang & Tao, 2008). The critical temperature Tc is approximately equal to the average temperature of the environment temperature and the surface temperature of insulating packaging (Yang & Tao, 2008). In Eq. (4), T and Tc are unknown parameters coupling with the temperature field solution of the heat transfer function in Section 2.1.1.
l3
2
w (x , s ) =
=
Tj
Tm
Te
Tm
kl kw
2
al
= h [1
w
x2 2
h s
al aw
l
2
w
x2
(8)
s ) = 0,
s
l (x ,
s ) = 0, 0 < x < p
l (p ,
s) = 0
l
lL
= x=p
Te
l (0,
s) =
w (0,
x
= kw x=0
=h
x
b
x
w
kw
x= b
Tm
p
s)
w
x
1 s
x=0 w(
b, s )
aw / al sinh (b s / aw )
cosh (p s / al ) sinh (x s / al ) k
=
l
,0
w
,
t t
b
(10)
aw / al cosh (p s / al ) sinh (x s / aw )
sinh (p s / al )[kw s / aw sinh (b s / aw ) + h cosh (b s / aw )] + cosh (p s / al ) kl s / al cosh (b s / aw ) + h l aw / al sinh (b s / aw ) kw
=
(9)
Then, the solution of differential equation systems can be obtained
as
(11)
where p can be determined by
kl s/ al h lL = p s sinh (p s /al )[k w s / a w sinh (b s /a w ) + h cosh (b s / a w )] Te Tm
= 0, p < x <
x2
(6)
(7)
p s
l
kl
tioned governing equations, the boundary conditions and the initial conditions can be written as Eqs. (1)–(3), respectively. s 2
b, t )]
w (x ,
l
kw
l (x , s ) =
w(
s
x2
k sinh (p s / al )[kw s / aw sinh (b s / aw ) + h cosh (b s / aw )] + cosh (p s / al ) kl s / al cosh (b s / aw ) + h l
sinh (p s / al ) cosh (x s / al )
x=0
x= b
kl
, and j = w, l, s , e . Furthermore, the aforemen-
sinh (p s / al ) cosh (x s / aw )
h s
x
x=0
x
,t>0
= 0, b < x < 0 ,t=0 = 0, 0 < x < p=0
aw
In order to simplify the derivation process, a dimensionless temperature j is invited. j
x
t)
w
The Laplace transform of Eqs. (5)–(7) leads to the following ordinary differential equation systems in terms of parameter p after substituting for the initial conditions:
2.2. Model solution
Where
w (0,
= kw
L [p (t )] = P (s ) =
(4)
C (Gr Pr) n
dp Tm dt
It should be noted that there is a moving boundary p in Eqs. (5)–(7) that depends on time t. Zhu (2006) presented a pseudo-steady-state approximation for this type of moving boundary problem that can be used to find the solution for the one-dimensional problem. Based on Zhu’s pseudo-steady-state approximation, the optimal interface was assumed to be moving slowly in comparison with the diffusion of heat; thus, p can still be held as a constant during the Laplace transform in Eq. (8).
(3)
h = Nuka/l
t) =
l
w
kw
lL
Te
x=p
l (0,
kl
t) = 0 =
x
dp dt
Tl (0, t ) = Tw (0, t ) T T kl l = kw w x x=0 x x=0 Tw x
l
+ cosh (p s/ al ) kl s / al cosh (b s / a w )
,t>0
k
+ hk l
(5)
w
a w / al sinh (b s / a w ) (12)
3
Food Packaging and Shelf Life 24 (2020) 100474
L. Pan, et al.
The inverse Laplace transformation of Eqs. (11) and (12) is extremely difficult to perform. Thus, a numerical-integration formula presented by Crump (1976) is applied here to solve the problem of inverse Laplace transformation. This numerical-integration formula has been shown to be sufficiently accurate in many analogous situations (Narayanan, 1982; Peng, Chen, Liu, & Fu, 2008). In this numericalintegration formula, the relationship between a time domain function f (t ) and its complex-frequency domain function F (s ) is applied as follows:
f (t ) =
e pt 2t
1 F (s ) + 2
N
( 1)n Re F s + n=1
N
( 1)n Im F s +
+
(2n
n= 1
1) 2t
n i t
i
(13)
The error of the formula can be written as
E
e
4st f
(5t )
Fig. 2. The model solution process.
(14)
However, the solution of Eqs. (11)–(13) includes an unknown surface heat transfer coefficient. Thus, combining Eqs. (11)–(13) and the surface heat transfer coefficient of natural convection of Eq. (4), an alternative surface heat transfer coefficient identification method which involves a hybrid utilization of a heat transfer model and an iterative algorithm is developed. The temperature fields in insulating packaging can also be simultaneously calculated through this iterative algorithm. The iterative process is shown in Fig. 2. The iterative results of both temperature fields in PCM and surface heat transfer coefficient of the model results are shown in Fig. 3. The parameters for calculation are show in Table 1.
semi-infinite phase change model will be used to quantificationally analyze the influencing parameters of surface heat transfer coefficient in insulating packaging. 2.3.2. Influencing parameters of surface heat transfer coefficient By using the proposed semi-infinite phase change model, the influencing parameters of surface heat transfer coefficient are discussed in this section. The model parameters used in this section are listed in Table 2. Fig. 6 shows that the surface heat transfer coefficient increases with the decrease of insulating wall thickness (b) and the increase of thermal conductivity in insulating wall (kw). The surface heat transfer coefficient also significantly increases while the excess temperature ( = Te Tm ) increases. However, the main thermal parameters of PCM (kl and L) show an inapparent effect on the surface heat transfer coefficient.
2.3. Model discussion 2.3.1. Compare with the classic models Neumann model and Mehling model are two classic models which have similar condition discussed in this research. Thus, these two classic models are used to verify the proposed model.
2.4. Prediction model of surface heat transfer coefficient Although the proposed semi-infinite phase change model provides an effective method to calculate the surface heat transfer coefficient in insulating packaging, this analysis model is unsuitable for engineering application due to the complex iterative process. The parameters discussion indicates that the key parameters of surface heat transfer coefficient are the excess temperature (ψ), thermal conductivity coefficient (kw) and thickness (b) of insulating wall. Furthermore, the ratio of b and kw is called the material thermal resistance which has been identified in literatures (Yang & Tao, 2008). Therefore, the surface heat transfer coefficient of insulating packaging can be approximatively calculated by the excess temperature (ψ) and the material thermal resistance (b/kw). Besides, the surface heat transfer coefficients on vertical, bottom and top surfaces of packaging are quite different due to the nature convection. A concise prediction model of surface heat transfer coefficient on different packaging surfaces is established as Eq. (15) based on the calculated results of semi-infinite phase change model. Both the correlation coefficients in Eq. (15) and the fitting surfaces in Fig. 7 indicate that these regression formulas can effectively predict the surface heat transfer coefficients of insulating packaging. Based on these prediction models, the system thermal resistance of insulating packaging can be easily predicted according to the definition of system thermal resistance in Eq. (16) (Pan, Lu, & Wang, 2014).
(1) Compare with Neumann model Neumann model presents a semi-infinite phase change system under the first heat exchange boundary condition (means h ), and there is no insulating wall outside PCM (means b = 0) (Zhang, 2009). Thus, the proposed iterative solution will simplify into the Neumann model solution when b = 0 and h . Fig. 4 shows the comparison between simplified solution of proposed model and Neumann analytical model (the values of calculation parameters are all referenced to Table 1). Both the moving process of solid-liquid interface and the temperature field of PCM show a significant consistency. (2) Compare with Mehling model Based on the linear hypothesis, Mehling’s model can calculated an approximate one-dimensional phase change process with insulating wall. However, the surface heat transfer coefficient in Mehling model must be a known parameter. Therefore, a presupposed surface heat transfer coefficient which is obtained through the proposed semi-infinite phase change model is substituted into Mehling’s model. Fig. 5 shows the semi-infinite phase change model also shows a good agreement with Mehling model in both moving process of solidliquid interface and the temperature field of PCM (the values of calculation parameters are all referenced to Table 1). According to the aforementioned comparison, the proposed semiinfinite phase change model demonstrates a good veracity at predicting the phase change process of PCM in insulating packaging. Thus, this 4
h v = (e
0.7014b / kw
+ 2.334)(0.1015
0.0037 ), (R2 = 0.9957)
h b = (e
0.7808b / kw
+ 2.116)(0.1405
0.0052 ), (R2 = 0.9957)
h t = (e
0.5898b / kw
+ 2.813)(0.0574
0.0021 ), (R2 = 0.9958)
(15)
Food Packaging and Shelf Life 24 (2020) 100474
L. Pan, et al.
Fig. 3. Iterative results. Table 1 Model parameters for calculation examples. kw
kl
0.03
0.5
10
b
L
w
0.10
3
20
l
1000
Cw
Cl
Tm
8000
4000
273
Fig. 5. Proposed semi-infinite phase change model compared to the Mehling model.
Table 2 Model parameters for model discussion.
Fig. 4. Simplified solution of proposed model compared to the Neumann solution. 5
Level
kw
kl
I II III
0.05 0.50 5.00
0.5 5.0 50.0
10 30 50
b
L
0.10 0.06 0.02
3 2 1
w
20
l
1000
Cw
Cl
Tm
8000
4000
273
Food Packaging and Shelf Life 24 (2020) 100474
L. Pan, et al.
Fig. 6. Influencing parameters of surface heat transfer coefficient.
R=
1 6
+
Ao, j h v, b, t
j=1
where, A =
b ¯ w Ak
water (HACH-272-56, Sinopharm Chemical Reagent Co., Ltd). The thermophysical properties of deionized water were determined with a thermal conductivity measurer (KD2 Pro, Decagon, USA) and a DSC (Q2000, TA, USA); the results are shown in Table 4.
(16)
Ai Ao is the equivalent area presented by Qian (2009).
3. Experiments
3.2. Equipment
Burgess’s “Ice Melting Method” (Burges, 1999) is the most classic method to calculate the system thermal resistance of insulating packaging. Thus, a group of “Ice Melting Method” experiments were used to verify the the proposed concise prediction models.
A programmable temperature chamber (THS-D7C-100AS, KSON, Taiwan) which can control temperature in the range of 203 K∼423 K with an accuracy of ± 0.2 K is used to control the environment temperature during the “Ice Melting Method” experiment. The mass of melted ice was measured by an electronic balance (PL6001E, Mettler Toledo, Switzerland) with an accuracy of ± 0.1 g in the range of 0∼6200 g. Besides, A distributed optical fiber temperature measuring system with ten optical fiber (EVO-SD-5 & FPI-HR-2-F2-SCAI-V & FOTL-SD, FISO, Canada) was used to record temperature during the “Ice Melting Method” experiment, and the accuracy is ± 0.1 K in the range of 263 K–393 K.
3.1. Materials The insulating containers are made from expanded polystyrene (EPS, provided by Suzhou Antek Packaging Co., Ltd.). The properties of these containers provided by Antek are shown in Table 3. The PCM used in the “Ice Melting Method” experiment is deionized 6
Food Packaging and Shelf Life 24 (2020) 100474
L. Pan, et al.
Fig. 7. Fitting surfaces of surface heat transfer coefficient.
(2) The experimental cells were placed in the temperature chamber under 253 K until the water completely froze (the temperatures in the center of experimental cells were below 273 K); (3) The environmental temperature in chamber was gradually increased to 273 K and kept for 24 h to reach equilibrium (still in solid phase); (4) The environmental temperature was sharply increased to 313 K and kept for 24 h; (5) The mass of melted water was measured by the electronic balance; (6) The “Ice Melting Method” experiments were repeated three times.
Table 3 The properties of insulating containers. Sample
Outside Size/m
b
kw
ρw
S1 S2 S3
0.250 × 0.215 × 0.150 0.340 × 0.220 × 0.185 0.280 × 0.230 × 0.310
0.030 0.017 0.055
0.039 0.038 0.037
23.84 19.46 28.12
Table 4 The thermophysical properties of pure water under melting temperature. kl
ρl
Tm
L
0.57
1.00 × 103
273.07
3.31 × 105
4. Results and discussion According to Burgess’s “Ice Melting Method” (Burges, 1999), the system thermal resistance can be calculated by Eq. (16). On the other hand, the system thermal resistance of insulating packaging can be expressed as the summation of the material thermal resistance and the surface thermal resistance as Eq. (17).
3.3. Test system Three optical fibers (diameter 1 mm) were respectively fixed in the center of all of the three insulating containers. These experimental cells were placed in the temperature chamber. A computer was used to record the temperature change during the test. The constitution of the test system is shown in Fig. 8.
R=
t mL
(17)
Table 5 shows the comparison between predicted system thermal resistances and test system thermal resistances. The predicted system thermal resistances are calculated through the proposed surface heat transfer coefficients prediction model in Eq. (15) and definition of system thermal resistance in Eq. (16). And the test system thermal resistances are obtained by the Burgess’s “Ice Melting Method”. The results indicate that the predicted system thermal resistances have a good
3.4. Methods (1) The insulating containers were filled to 90 % volume with deionized water; 7
Food Packaging and Shelf Life 24 (2020) 100474
L. Pan, et al.
Fig. 8. Schematic diagram of the test system.
accurately predict the surface heat transfer coefficients in insulating packaging and provide a significant convenience for insulating packaging design.
Table 5 The comparation between predicted system thermal resistances and test system thermal resistance. Sample
m (standard deviation)
R - Prediction
R - Ice Melting Method
E
S1 S2 S3
1.404 ( ± 0.0295) 2.948 ( ± 0.0411) 1.324 ( ± 0.0235)
7.3487 3.3576 7.3951
7.4367 3.5418 7.8860
1.2 5.2 6.2
Acknowledgements The authors would like to express their appreciation for the financial support via the Fundamental Research Funds for Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment and Technology (FMZ201805) and the Fundamental Research Funds for the Central Universities (JUSRP11827).
agreement with the test system thermal resistances. Table 3 also shows that the predicted system thermal resistances are all lower than test results. This deviation is mainly due to the simplification of the prediction models where the predicted surface heat transfer coefficient is the maximal surface heat transfer coefficient on the vertical insulating wall. Whereas the surface convection on the vertical insulating wall is gradually varied with the vertical height. Nevertheless, the higher predicted surface heat transfer coefficient caused by simplification brings to a more dependable design of insulating packaging. Therefore, the proposed concise models have an accurate prediction of surface heat transfer coefficients in insulating packaging and provide a significant convenience for insulating packaging design.
References Aduda, B. O. (1996). Effective thermal conductivity of loose particulate systems. Journal of Materials Science, 31, 644l–6448. Burges, G. (1999). Practical thermal resistance and ice requirement calculations for insulating packages. Packaging Technology and Science, 12(2), 75–80. Crump, K. (1976). Numerical inversion of Laplace transforms using Fourier series approximation. ACM Transactions on Mathematical Software, 23, 89–96. Gowreesunker, B., Tassou, S., & Kolokotroni, M. (2012). Improved simulation of phase change processes in applications where conduction is the dominant heat transfer mode. Energy and Buildings, 47, 353–359. Guangsong, H., & Zhiyong, Y. (2009). Parameters determination in numerical inversion of Laplace transforms. Journal of Lanzhou University (Natural Sciences), 45, 116–119. Jradi, M., Gillott, M., & Riffat, S. (2013). Simulation of the transient behavior of encapsulated organic and inorganic phase change materials for low-temperature energy storage. Applied Thermal Engineering, 59, 211–222. Mehling, H., & Cabeza, L. (2008). Heat and cold storage with PCM. Berlin Heidelberg: Springer-Verlag. Moreno, P., Miró, L., Solé, A., Barreneche, C., Solé, C., Martorell, I., et al. (2014). Corrosion of metal and metal alloy containers in contact with phase change materials (PCM) for potential heating and cooling applications. Applied Energy, 125, 238–245. Narayanan, G. V. (1982). Numerical operational methods for time dependent linear problems. International Journal for Numerical Methods in Engineering, 18(12), 1829–1854. Pan, L., Lu, L. X., & Wang, J. (2014). Prediction model for the shelf life of insulating package. Packaging Engineering, 35(5), 27–30 136. Peng, F., Chen, Y., Liu, Y., & Fu, Y. (2008). Numerical inversion of Laplace transforms in viscoelastic problems by fourier series expansion. Chinese Journal of Theoretical and Applied Mechanics, 40(2), 215–221. Prandtl, G. L. (1940). On the calculation of the boundary layer. Aeronautical Journal, 44(359), 795–801. Prashant, V., Varuna, & Singal, S. (2008). Review of mathematical modeling on latent heat thermal energy storage systems using phase-change material. Renewable and Sustainable Energy Reviews, 12, 999–1031. Qian, J. (2009). Mathematical models for insulating packages and insulating packaging solutions. USA: University of Memphis. Shimoyama, R., Horibe, A., & Haruki, N. (2018). An experimental study of flow and heat transfer characteristics of natural convection heat transfer from a horizontal heated surface with a heated cylinder. Heat and Mass Transfer, 54(1), 185–192. Wang, Z. W., Li, B., Lin, Q. B., & Hu, C. Y. (2018). Two-phase molecular dynamics model to simulate the migration of additives from polypropylene material to food.
5. Conclusions A semi-infinite phase change model solved by an iterative method was proposed to calculate the coupling parameters of surface heat transfer coefficient and temperature field. Then the key parameters of surface heat transfer coefficient in insulating packaging was analyzed through this semi-infinite phase change model. Based on these key affecting parameters, a group of concise models for predicting the surface heat transfer coefficient were presented by multiple nonlinear regression. Finally, system thermal resistances predicted by the concise formulas were compared with the test system thermal resistance which were obtained by the classic “Ice Melting Method”. The results indicate that the proposed semi-infinite phase change model and iterative method can simultaneously calculate the coupling parameters of surface heat transfer coefficient and the temperature field in insulating packaging and show a consistent with the classic Neumann model and Mehling model. The key affecting parameters of surface heat transfer coefficient are the excess temperature (ψ), thermal conductivity coefficient (kw) and thickness (b) of insulating wall. Furthermore, the predicted system thermal resistances have a good agreement with the test system thermal resistances. Therefore, the proposed concise models can
8
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L. Pan, et al. International Journal of Heat and Mass Transfer, 122, 694–706. Yang, S. M., & Tao, W. S. (2008). Heat transfer. Beijing: Higher Education Press. Zalba, B., Marin, J., Cabeza, L., & Mehling, H. (2003). Review on thermal energy storage with phase change: Materials, heat transfer analysis and applications. Applied Thermal Engineering, 23, 251–283. Zhang, R. Y. (2009). The technology of phase change materials and energy storage system.
Beijing: Science Press. Zhou, D., Zhao, C., & Tian, Y. (2012). Review on thermal energy storage with phase change materials (PCMs) in building applications. Applied Energy, 92, 593–605. Zhu, S. (2006). A new analytical-approximation formula for the optimal exercise boundary of american put options. International Journal of Theoretical and Applied Finance, 9(7), 1141–1177.
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Food Packaging and Shelf Life journal homepage: www.elsevier.com/locate/fpsl
A prediction model of surface heat transfer coefficient in insulating packaging with phase change materials
T
Liao Pana,b, Xi Chena, Lixin Lua,b,*, Jun Wanga,b, Xiaolin Qiua,b a b
Department of Packaging Engineering, Jiangnan University, Wuxi, 214122, China Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment and Technology, Wuxi, 214122, China
ARTICLE INFO
ABSTRACT
Keywords: Surface heat transfer coefficient Prediction model Insulating packaging Phase change Heat transfer
Surface heat transfer coefficient is one of the crucial parameters for determining the system thermal resistance of insulating packaging. However, the surface heat transfer coefficient which couples with the temperature field in insulating packaging is difficult to experimentally measure and independently calculate. In this study an iterative method based on a semi-infinite phase change model was proposed to solve the coupling parameters of surface heat transfer coefficient and temperature field. Then, the semi-infinite phase change model curves were compared with classic Neumann model and Mehling model. Furthermore, the affecting parameters of surface heat transfer coefficient were discussed through this semi-infinite phase change model, and a concise prediction model of surface heat transfer coefficient was also developed for engineering application. Finally, this concise prediction model was verified by “Ice Melting Method”. The results indicate that the proposed semi-infinite phase change model and iterative method can simultaneously calculate the coupling parameters of surface heat transfer coefficient and the temperature field in insulating packaging, and show a consistent with the classic Neumann model and Mehling model. The key parameters of surface heat transfer coefficient include the thickness and thermal conductivity coefficient of insulating wall, and the excess temperature also significantly influences the surface heat transfer coefficient. Furthermore, the system thermal resistance calculated by the novel concise prediction model has a good agreement with the experimental data of “Ice Melting Method” presented by Burgess.
1. Introduction Insulating packaging with phase change materials (PCMs) can be used for storing temperature-sensitive products (such as fresh foods) without electrical energy during the using process, and show some significant advantages over classic cold chain (Moreno et al., 2014). For insulating packaging design, surface heat transfer coefficient is an essential parameter which significantly influences the system thermal resistance of insulating packaging. The surface heat transfer coefficient is usually identified by boundary-layer theory (Prandtl, 1940), and temperature difference T and critical temperature Tc are two essential parameters during calculating. However, in the situation of insulating packaging, T and Tc are unknown parameters, which are determined by the temperature field of insulating packaging. The insulating packaging is a kind of phase change system. For the determination of the temperature field in phase change system, Neumann proposed the most classic analytical solution of semi-infinite phase change system under the first-class heat exchange boundary
⁎
condition where the heat load directly occurs at the surface of PCM (Zhang, 2009). Then, numerous studies have been conducted on the other phase change systems with the first-class heat exchange boundary condition by using numerical methods, analytical models and experiment (Gowreesunker, Tassou, & Kolokotroni, 2012; Jradi, Gillott, & Riffat, 2013; Prashant, Varuna, & Singal, 2008; Wang, Li, Lin, & Hu, 2018; Zalba, Marin, Cabeza, & Mehling, 2003; Zhou, Zhao, & Tian, 2012). In these research studies, there is no insulating material outside the PCM, and the heat exchange boundary condition of insulating packaging belongs to the natural convection boundary (not the first-class heat exchange boundary condition). Therefore, these models cannot be used to solve the temperature field in insulating packaging. For these reasons, Mehling firstly presented an approximate one-dimensional phase change heat transfer model considering insulating wall for describing the temperature fields in insulating packaging (Mehling & Cabeza, 2008). However, in Mehling’s solution, the surface heat transfer coefficient was assumed to be a known parameter, which is impossible in most cases. The relationship between the surface heat transfer coefficient and the temperature field in insulating packaging is complex. On one hand, the
Corresponding author at: Department of Packaging Engineering, Jiangnan University, No. 1800 Lihu Avenue, Wuxi, 214122, China. E-mail address: [email protected] (L. Lu).
https://doi.org/10.1016/j.fpsl.2020.100474 Received 8 August 2019; Received in revised form 9 January 2020; Accepted 24 January 2020 2214-2894/ © 2020 Elsevier Ltd. All rights reserved.
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Nomenclature T
T
t x b p a k L h s m Gr Pr Nu E g l C A R
Body expansion coefficient/K−1 Kinematic viscosity/m2/s Dimensionless temperature Dimensionless temperature in complex-frequency domain after Laplace transfer Excess temperature/K Density/kg/m3 Maximum permissible errors/K
α υ θ
Temperature/K Temperature difference between environmental temperature and surface temperature of insulating wall/K Time/s Position/m Thickness of insulating wall/m Position of moving boundary/m Thermal diffusivity/m2/s Thermal conductivity/W/(m·K) Latent heat of PCM/×105J/kg Surface heat transfer coefficient/ W/(m2·K) Complex-frequency domain variable of time after Laplace transfer Mass of melted water/kg Grashof number Prandtl number Nusselt number Relative Error/% Acceleration of gravity/m/s2 Characteristic dimension/m Specific heat/J/(kg·K) Superficial area/ m2 System thermal resistance/m2·K/W
ψ ρ ε Subscripts a v b t w s l e m c j i o
surface heat transfer coefficient determines the heat transfer of insulating packaging, then influences the temperature field in insulating packaging. On the other hand, the temperature field in insulating packaging will change the surface temperature of insulating wall and simultaneously change the surface heat transfer coefficient caused by nature convection. Thus, the surface heat transfer coefficient and the temperature field in insulating packaging are a group of coupling parameters. The major objective of this study is to develop a semi-infinite phase change model of insulating packaging, and propose an iterative algorithm to solve the coupling parameters of surface heat transfer coefficient and temperature field in the semi-infinite phase change model. Then, the affecting parameters of surface heat transfer coefficient are discussed through this semi-infinite phase change model, and a group of concise prediction models of surface heat transfer coefficient is also provided for engineering application. Finally, the system thermal resistance which is essential for designing insulating packaging can be calculated through the proposed prediction model.
Air Vertical Bottom Top Insulating wall Solid phase Liquid phase Environment Melting point Critical temperature The surface number of insulating packaging Inside Outside
(3) The PCM and insulating material are isotropic; (4) Volume changing, convection, and thermal radiation during the melting process are ignored (Aduda, 1996); (5) The PCM and insulating material properties are assumed to be independent of temperature. 2.1. Mathematical analysis 2.1.1. Heat transfer function The heat transfer process of the semi-infinite phase change model can be presented mathematically by applying the Fourier law of heat conduction as
Ts = Tm, p < x < 2T T al 2l = l , 0 < x < p ,t>0 x t 2T Tw w aw = , b
2. Model development
Where aj =
kj j Cj
(1)
, j = w, l, s .
The boundary conditions can be shown applying the energy balance
The insulating packaging is constituted by insulating wall, PCM and products. This is a multilayer finite 3-dimensional phase change heat transfer system. According to Mehling’s research (Mehling & Cabeza, 2008), the heat transfer only occurs in the insulating wall and the liquid field of PCM before the complete melting of PCM. Besides, the heat exchange area of insulating wall is much larger at the face than at the edge and corner. Thus, Mehling simplified the insulating packaging into a one-dimensional semi-infinite phase change model with natural convection boundary. As shown in Fig. 1, the main component of the one-dimensional phase change model is a semi-infinite PCM that is covered by insulating material; the insulating material is heated by natural convection. Initially, the temperature of the solid phase and insulating material are isothermal (Ts = Tw =Tm). The following assumptions are made in the analysis: (1) Heat transfer occurs only in the x direction; (2) The PCM area is simplified into a semi-infinite region due to the constant temperature of solid phase before melting;
Fig. 1. The semi-infinite phase change model with natural convection boundary. 2
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at each interface as
l (p ,
Tl (p , t ) = Tm Tl x
kl
= x= p
kl
lL
kw
= h [Te
,t>0
Tw ( b, t )]
(2)
x= b
w
The initial conditions can be easily obtained as
s
Tw = Tm, b < x < 0 Ts = Tm, 0 < x < ,t=0 p=0
2.1.2. Surface heat transfer coefficient According to boundary-layer theory, the surface heat transfer coefficient of natural convection above the insulating wall can be calculated through a Nusselt number (Yang & Tao, 2008) as shown in Eq. (4). where Nu = , Gr = g a a T , and Pr = a / a . C and n are two parameters that can be determined by experiments (According to Refs. Shimoyama, Horibe, and Haruki (2018) and Guangsong and Zhiyong (2009)). l is equal to the height of heat exchange surface when the surface is vertical, and l is equal to the ratio of area and perimeter of heat exchange surface when the surface is horizontal (Yang & Tao, 2008). In addition, the air parameters ( a , a , and ka ) under critical temperature Tc can be obtained by referencing to the previous research studies (Yang & Tao, 2008). The critical temperature Tc is approximately equal to the average temperature of the environment temperature and the surface temperature of insulating packaging (Yang & Tao, 2008). In Eq. (4), T and Tc are unknown parameters coupling with the temperature field solution of the heat transfer function in Section 2.1.1.
l3
2
w (x , s ) =
=
Tj
Tm
Te
Tm
kl kw
2
al
= h [1
w
x2 2
h s
al aw
l
2
w
x2
(8)
s ) = 0,
s
l (x ,
s ) = 0, 0 < x < p
l (p ,
s) = 0
l
lL
= x=p
Te
l (0,
s) =
w (0,
x
= kw x=0
=h
x
b
x
w
kw
x= b
Tm
p
s)
w
x
1 s
x=0 w(
b, s )
aw / al sinh (b s / aw )
cosh (p s / al ) sinh (x s / al ) k
=
l
,0
w
,
t t
b
(10)
aw / al cosh (p s / al ) sinh (x s / aw )
sinh (p s / al )[kw s / aw sinh (b s / aw ) + h cosh (b s / aw )] + cosh (p s / al ) kl s / al cosh (b s / aw ) + h l aw / al sinh (b s / aw ) kw
=
(9)
Then, the solution of differential equation systems can be obtained
as
(11)
where p can be determined by
kl s/ al h lL = p s sinh (p s /al )[k w s / a w sinh (b s /a w ) + h cosh (b s / a w )] Te Tm
= 0, p < x <
x2
(6)
(7)
p s
l
kl
tioned governing equations, the boundary conditions and the initial conditions can be written as Eqs. (1)–(3), respectively. s 2
b, t )]
w (x ,
l
kw
l (x , s ) =
w(
s
x2
k sinh (p s / al )[kw s / aw sinh (b s / aw ) + h cosh (b s / aw )] + cosh (p s / al ) kl s / al cosh (b s / aw ) + h l
sinh (p s / al ) cosh (x s / al )
x=0
x= b
kl
, and j = w, l, s , e . Furthermore, the aforemen-
sinh (p s / al ) cosh (x s / aw )
h s
x
x=0
x
,t>0
= 0, b < x < 0 ,t=0 = 0, 0 < x < p=0
aw
In order to simplify the derivation process, a dimensionless temperature j is invited. j
x
t)
w
The Laplace transform of Eqs. (5)–(7) leads to the following ordinary differential equation systems in terms of parameter p after substituting for the initial conditions:
2.2. Model solution
Where
w (0,
= kw
L [p (t )] = P (s ) =
(4)
C (Gr Pr) n
dp Tm dt
It should be noted that there is a moving boundary p in Eqs. (5)–(7) that depends on time t. Zhu (2006) presented a pseudo-steady-state approximation for this type of moving boundary problem that can be used to find the solution for the one-dimensional problem. Based on Zhu’s pseudo-steady-state approximation, the optimal interface was assumed to be moving slowly in comparison with the diffusion of heat; thus, p can still be held as a constant during the Laplace transform in Eq. (8).
(3)
h = Nuka/l
t) =
l
w
kw
lL
Te
x=p
l (0,
kl
t) = 0 =
x
dp dt
Tl (0, t ) = Tw (0, t ) T T kl l = kw w x x=0 x x=0 Tw x
l
+ cosh (p s/ al ) kl s / al cosh (b s / a w )
,t>0
k
+ hk l
(5)
w
a w / al sinh (b s / a w ) (12)
3
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L. Pan, et al.
The inverse Laplace transformation of Eqs. (11) and (12) is extremely difficult to perform. Thus, a numerical-integration formula presented by Crump (1976) is applied here to solve the problem of inverse Laplace transformation. This numerical-integration formula has been shown to be sufficiently accurate in many analogous situations (Narayanan, 1982; Peng, Chen, Liu, & Fu, 2008). In this numericalintegration formula, the relationship between a time domain function f (t ) and its complex-frequency domain function F (s ) is applied as follows:
f (t ) =
e pt 2t
1 F (s ) + 2
N
( 1)n Re F s + n=1
N
( 1)n Im F s +
+
(2n
n= 1
1) 2t
n i t
i
(13)
The error of the formula can be written as
E
e
4st f
(5t )
Fig. 2. The model solution process.
(14)
However, the solution of Eqs. (11)–(13) includes an unknown surface heat transfer coefficient. Thus, combining Eqs. (11)–(13) and the surface heat transfer coefficient of natural convection of Eq. (4), an alternative surface heat transfer coefficient identification method which involves a hybrid utilization of a heat transfer model and an iterative algorithm is developed. The temperature fields in insulating packaging can also be simultaneously calculated through this iterative algorithm. The iterative process is shown in Fig. 2. The iterative results of both temperature fields in PCM and surface heat transfer coefficient of the model results are shown in Fig. 3. The parameters for calculation are show in Table 1.
semi-infinite phase change model will be used to quantificationally analyze the influencing parameters of surface heat transfer coefficient in insulating packaging. 2.3.2. Influencing parameters of surface heat transfer coefficient By using the proposed semi-infinite phase change model, the influencing parameters of surface heat transfer coefficient are discussed in this section. The model parameters used in this section are listed in Table 2. Fig. 6 shows that the surface heat transfer coefficient increases with the decrease of insulating wall thickness (b) and the increase of thermal conductivity in insulating wall (kw). The surface heat transfer coefficient also significantly increases while the excess temperature ( = Te Tm ) increases. However, the main thermal parameters of PCM (kl and L) show an inapparent effect on the surface heat transfer coefficient.
2.3. Model discussion 2.3.1. Compare with the classic models Neumann model and Mehling model are two classic models which have similar condition discussed in this research. Thus, these two classic models are used to verify the proposed model.
2.4. Prediction model of surface heat transfer coefficient Although the proposed semi-infinite phase change model provides an effective method to calculate the surface heat transfer coefficient in insulating packaging, this analysis model is unsuitable for engineering application due to the complex iterative process. The parameters discussion indicates that the key parameters of surface heat transfer coefficient are the excess temperature (ψ), thermal conductivity coefficient (kw) and thickness (b) of insulating wall. Furthermore, the ratio of b and kw is called the material thermal resistance which has been identified in literatures (Yang & Tao, 2008). Therefore, the surface heat transfer coefficient of insulating packaging can be approximatively calculated by the excess temperature (ψ) and the material thermal resistance (b/kw). Besides, the surface heat transfer coefficients on vertical, bottom and top surfaces of packaging are quite different due to the nature convection. A concise prediction model of surface heat transfer coefficient on different packaging surfaces is established as Eq. (15) based on the calculated results of semi-infinite phase change model. Both the correlation coefficients in Eq. (15) and the fitting surfaces in Fig. 7 indicate that these regression formulas can effectively predict the surface heat transfer coefficients of insulating packaging. Based on these prediction models, the system thermal resistance of insulating packaging can be easily predicted according to the definition of system thermal resistance in Eq. (16) (Pan, Lu, & Wang, 2014).
(1) Compare with Neumann model Neumann model presents a semi-infinite phase change system under the first heat exchange boundary condition (means h ), and there is no insulating wall outside PCM (means b = 0) (Zhang, 2009). Thus, the proposed iterative solution will simplify into the Neumann model solution when b = 0 and h . Fig. 4 shows the comparison between simplified solution of proposed model and Neumann analytical model (the values of calculation parameters are all referenced to Table 1). Both the moving process of solid-liquid interface and the temperature field of PCM show a significant consistency. (2) Compare with Mehling model Based on the linear hypothesis, Mehling’s model can calculated an approximate one-dimensional phase change process with insulating wall. However, the surface heat transfer coefficient in Mehling model must be a known parameter. Therefore, a presupposed surface heat transfer coefficient which is obtained through the proposed semi-infinite phase change model is substituted into Mehling’s model. Fig. 5 shows the semi-infinite phase change model also shows a good agreement with Mehling model in both moving process of solidliquid interface and the temperature field of PCM (the values of calculation parameters are all referenced to Table 1). According to the aforementioned comparison, the proposed semiinfinite phase change model demonstrates a good veracity at predicting the phase change process of PCM in insulating packaging. Thus, this 4
h v = (e
0.7014b / kw
+ 2.334)(0.1015
0.0037 ), (R2 = 0.9957)
h b = (e
0.7808b / kw
+ 2.116)(0.1405
0.0052 ), (R2 = 0.9957)
h t = (e
0.5898b / kw
+ 2.813)(0.0574
0.0021 ), (R2 = 0.9958)
(15)
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Fig. 3. Iterative results. Table 1 Model parameters for calculation examples. kw
kl
0.03
0.5
10
b
L
w
0.10
3
20
l
1000
Cw
Cl
Tm
8000
4000
273
Fig. 5. Proposed semi-infinite phase change model compared to the Mehling model.
Table 2 Model parameters for model discussion.
Fig. 4. Simplified solution of proposed model compared to the Neumann solution. 5
Level
kw
kl
I II III
0.05 0.50 5.00
0.5 5.0 50.0
10 30 50
b
L
0.10 0.06 0.02
3 2 1
w
20
l
1000
Cw
Cl
Tm
8000
4000
273
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L. Pan, et al.
Fig. 6. Influencing parameters of surface heat transfer coefficient.
R=
1 6
+
Ao, j h v, b, t
j=1
where, A =
b ¯ w Ak
water (HACH-272-56, Sinopharm Chemical Reagent Co., Ltd). The thermophysical properties of deionized water were determined with a thermal conductivity measurer (KD2 Pro, Decagon, USA) and a DSC (Q2000, TA, USA); the results are shown in Table 4.
(16)
Ai Ao is the equivalent area presented by Qian (2009).
3. Experiments
3.2. Equipment
Burgess’s “Ice Melting Method” (Burges, 1999) is the most classic method to calculate the system thermal resistance of insulating packaging. Thus, a group of “Ice Melting Method” experiments were used to verify the the proposed concise prediction models.
A programmable temperature chamber (THS-D7C-100AS, KSON, Taiwan) which can control temperature in the range of 203 K∼423 K with an accuracy of ± 0.2 K is used to control the environment temperature during the “Ice Melting Method” experiment. The mass of melted ice was measured by an electronic balance (PL6001E, Mettler Toledo, Switzerland) with an accuracy of ± 0.1 g in the range of 0∼6200 g. Besides, A distributed optical fiber temperature measuring system with ten optical fiber (EVO-SD-5 & FPI-HR-2-F2-SCAI-V & FOTL-SD, FISO, Canada) was used to record temperature during the “Ice Melting Method” experiment, and the accuracy is ± 0.1 K in the range of 263 K–393 K.
3.1. Materials The insulating containers are made from expanded polystyrene (EPS, provided by Suzhou Antek Packaging Co., Ltd.). The properties of these containers provided by Antek are shown in Table 3. The PCM used in the “Ice Melting Method” experiment is deionized 6
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Fig. 7. Fitting surfaces of surface heat transfer coefficient.
(2) The experimental cells were placed in the temperature chamber under 253 K until the water completely froze (the temperatures in the center of experimental cells were below 273 K); (3) The environmental temperature in chamber was gradually increased to 273 K and kept for 24 h to reach equilibrium (still in solid phase); (4) The environmental temperature was sharply increased to 313 K and kept for 24 h; (5) The mass of melted water was measured by the electronic balance; (6) The “Ice Melting Method” experiments were repeated three times.
Table 3 The properties of insulating containers. Sample
Outside Size/m
b
kw
ρw
S1 S2 S3
0.250 × 0.215 × 0.150 0.340 × 0.220 × 0.185 0.280 × 0.230 × 0.310
0.030 0.017 0.055
0.039 0.038 0.037
23.84 19.46 28.12
Table 4 The thermophysical properties of pure water under melting temperature. kl
ρl
Tm
L
0.57
1.00 × 103
273.07
3.31 × 105
4. Results and discussion According to Burgess’s “Ice Melting Method” (Burges, 1999), the system thermal resistance can be calculated by Eq. (16). On the other hand, the system thermal resistance of insulating packaging can be expressed as the summation of the material thermal resistance and the surface thermal resistance as Eq. (17).
3.3. Test system Three optical fibers (diameter 1 mm) were respectively fixed in the center of all of the three insulating containers. These experimental cells were placed in the temperature chamber. A computer was used to record the temperature change during the test. The constitution of the test system is shown in Fig. 8.
R=
t mL
(17)
Table 5 shows the comparison between predicted system thermal resistances and test system thermal resistances. The predicted system thermal resistances are calculated through the proposed surface heat transfer coefficients prediction model in Eq. (15) and definition of system thermal resistance in Eq. (16). And the test system thermal resistances are obtained by the Burgess’s “Ice Melting Method”. The results indicate that the predicted system thermal resistances have a good
3.4. Methods (1) The insulating containers were filled to 90 % volume with deionized water; 7
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Fig. 8. Schematic diagram of the test system.
accurately predict the surface heat transfer coefficients in insulating packaging and provide a significant convenience for insulating packaging design.
Table 5 The comparation between predicted system thermal resistances and test system thermal resistance. Sample
m (standard deviation)
R - Prediction
R - Ice Melting Method
E
S1 S2 S3
1.404 ( ± 0.0295) 2.948 ( ± 0.0411) 1.324 ( ± 0.0235)
7.3487 3.3576 7.3951
7.4367 3.5418 7.8860
1.2 5.2 6.2
Acknowledgements The authors would like to express their appreciation for the financial support via the Fundamental Research Funds for Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment and Technology (FMZ201805) and the Fundamental Research Funds for the Central Universities (JUSRP11827).
agreement with the test system thermal resistances. Table 3 also shows that the predicted system thermal resistances are all lower than test results. This deviation is mainly due to the simplification of the prediction models where the predicted surface heat transfer coefficient is the maximal surface heat transfer coefficient on the vertical insulating wall. Whereas the surface convection on the vertical insulating wall is gradually varied with the vertical height. Nevertheless, the higher predicted surface heat transfer coefficient caused by simplification brings to a more dependable design of insulating packaging. Therefore, the proposed concise models have an accurate prediction of surface heat transfer coefficients in insulating packaging and provide a significant convenience for insulating packaging design.
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5. Conclusions A semi-infinite phase change model solved by an iterative method was proposed to calculate the coupling parameters of surface heat transfer coefficient and temperature field. Then the key parameters of surface heat transfer coefficient in insulating packaging was analyzed through this semi-infinite phase change model. Based on these key affecting parameters, a group of concise models for predicting the surface heat transfer coefficient were presented by multiple nonlinear regression. Finally, system thermal resistances predicted by the concise formulas were compared with the test system thermal resistance which were obtained by the classic “Ice Melting Method”. The results indicate that the proposed semi-infinite phase change model and iterative method can simultaneously calculate the coupling parameters of surface heat transfer coefficient and the temperature field in insulating packaging and show a consistent with the classic Neumann model and Mehling model. The key affecting parameters of surface heat transfer coefficient are the excess temperature (ψ), thermal conductivity coefficient (kw) and thickness (b) of insulating wall. Furthermore, the predicted system thermal resistances have a good agreement with the test system thermal resistances. Therefore, the proposed concise models can
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