9
Freezing Time Software
Freezing Time Software
Prediction of the total time in stagnant ambient air for water in bare metal pipes
From: A. McDonald, B. Bschaden, E. Sullivan, R. Marsden, “Mathematical Simulation of the Freezing Time of Water in Small Diameter Pipes”, Appl. Thermal Eng., under review (Manuscript #: ATE-2014-5523)
Prediction of the total time in stagnant ambient air for water in insulated metal pipes
From: A. McDonald, B. Bschaden, E. Sullivan, R. Marsden, “Mathematical Simulation of the Freezing Time of Water in Small Diameter Pipes”, Appl. Thermal Eng., under review (Manuscript #: ATE-2014-5523)
1 Copyright © 2014 by CSME
Proceedings of The Canadian Society for Mechanical Engineering International Congress 2014
CSME International Congress 2014
June 1-4, 2014, Toronto, Ontario, Canada
FREEZING TIME OF WATER IN SMALL DIAMETER TUBES AND PIPES IN RESIDENTIAL AND COMMERCIAL HVAC APPLICATIONS
Erik Sullivan, E.I.T. Department of Mechanical Engineering
University of Alberta Edmonton, Canada [email protected]
Dr. André McDonald, P. Eng. Department of Mechanical Engineering
University of Alberta Edmonton, Canada
Abstract— The cooling and freezing of stagnant water in cylindrical pipes were investigated under conditions of varying ambient air temperatures. A one-dimensional transient heat conduction mathematical model was developed to estimate the freezing time of the liquid in the pipe. The model was based on the separation of variables method for a finite length-scale solidification problem, and was verified with the experimental transient temperature data. The model was applied to predict the cooling and freezing behavior of water in copper and polyvinyl chloride (PVC) pipes of various inner diameters. The results of the model, and their agreement with experimental data, suggest that a separation of variables method for a finite length-scale heat conduction problem is an acceptable method to predict the total cooling and freezing time for water in pipes of small diameters.
Keywords - Insulation; Moving boundary problem; Pipes; Solidification; Water
I. INTRODUCTION
Piping systems subjected to low-temperature environments are vulnerable to freezing and bursting, which resulted in insurance claims over five billion dollars over the past decade [1]. The cooling and freezing of liquids in pipes are of particular importance in applications that involve heating, ventilating, and air-conditioning (HVAC) in residential and commercial buildings located in cold regions. An unexpected event, such as failure of the heating system, exposes uninsulated piping to low ambient temperatures. When the fluids become quiescent in these pipes, stagnant fluids will cool and freeze more rapidly than if they were flowing through the pipe. To mitigate these risks, the standard solution is to insulate the pipes [2-3]. Although this is typical in commercial applications, it is sometimes not the case in residential homes due to the high cost of insulation materials. Some of the most common pipe materials found in commercial and residential HVAC systems are copper, steel and polyvinyl chloride (PVC) [4]. There are several other types of plastics used in HVAC applications, such as chlorinated polyvinyl chloride (CPVC), cross-linked polyethylene (PEX), and acrylonitrile butadiene
styrene (ABS), all of which have similar thermal conductivities to that of PVC [5-7]. A research focus on the prediction of the total time required to freeze water in copper and PVC or other plastic pipes for various ambient temperatures will aid in the development of procedures and/or monitoring systems to prevent a catastrophic failure such as bursting from occurring in HVAC applications.
The phenomenon of solidification of stagnant water in cylindrical pipes has received attention from numerous investigators [8-13]. When pipes are exposed to low-temperature environments, the water will experience cooling, supercooling of the liquid phase below the fusion temperature, solidification at the fusion temperature, and further cooling of the solid ice as shown in Fig. 1 [8, 13]. Gilpin [9] has shown that supercooled water in pipes is characterized by the growth of thin plate-like solid crystals known as dendritic ice, which form in advance of the annular solid ice. The formation of either dendritic ice or solid annular ice in the pipe may result in flow blockage [9] and bursting of the pipe due to the generation of large internal pressures.
Figure 1. Transient temperature trace of a typical freezing event in a tube or
pipe [13]
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Sugawara et al. [10] and Oiwake et al. [11] conducted both numerical analyses and experimental studies to quantify the changes in internal pipe pressure during the freezing process, while Gordon [8] presented experimental results of pressures required to cause bursting of a variety of pipes used in building applications. The numerical analyses of Sugawara et al. [10] and Oiwake et al. [11] and the experimental work of Gordon [8] included the case where the pressure increase in the freezing fluid depressed the fusion point. They admitted that the continuous freezing point depression rendered analysis of the problem difficult, forcing the use of incremental time steps and numerical code.
The modeling of solidification phase change problems has been the subject of extensive research investigations [14-21]. More recently, model studies have focused on the determination of the time required for freezing of materials. Pham [18] has extended Planck’s equation [19] to include sensible heat effects during the freezing of food-based materials with simple shapes. Other investigators such as Cleland et al. [20] and Hung [21] used numerical methods, coupled with experimental data, to predict the freezing time of irregularly shaped bodies of food. While these studies on the prediction of freezing time through the use of either simple models or numerical techniques are useful for solid food material, they do not enable the determination of the transient temperature distribution in the material or estimation of the transient solidification front during freezing in pipes in HVAC applications.
The present study examines the freezing of water in pipes common to the HVAC industry. A transient, one-dimensional heat conduction model was developed to estimate the freezing time of water in a pipe and experimental data from Gordon [8] was used to validate the analytical model.
II. MATHEMATICAL MODEL
A. Cooling Time
Cooling of the liquid with an ambient temperature, T∞, from an initial temperature (Ti) to its freezing point, Tf, will occur before initiation of freezing. The time required to cool the liquid from its initial temperature to the freezing point can be determined by considering a one-dimensional transient heat conduction model in cylindrical co-ordinates for the cooling of the stagnant fluid in the pipe. Assuming constant properties, the governing equation is
inn
L2
0 ,11
Rrt
T
r
Tr
rr
,
(1)
where Rinn is the inside diameter of the pipe or tube and αL is the thermal diffusivity of the liquid. The boundary and initial conditions are
0
,0
r
tT, (2)
TtRTU
r
tRTk ,
,inn
innL , (3)
TTrT i0, . (4)
Özişik [23] has solved Eq. (1) with the boundary and initial conditions of Eqs. (2) - (4) to give
t
RJu
rJ
R
TTuTtrT n
n nn
n 2L
1 inn0220
inn
i exp2
,
, (5)
where L/ kUu , kL is the thermal conductivity of the liquid,
and the eigenvalues, βn, are given by
inn0inn1 RuJRJ nnn . (6)
Based on the final temperature, T(r,t), the cooling time is estimated from Eq. (5). The overall heat transfer coefficient (U) during this single-phase heat transfer problem can be estimated by utilizing experimental data available from the work of others. The pipe material and outside convection and radiation heat transfer coefficient (Uconvection/radiation) constituted the overall heat transfer coefficient at the outer surface of the pipe. In general, and for free convection/radiation, which is typical in HVAC applications, with regards to energy loss from pipes,
/radiationconvection
11
UR
U . (7)
The thermal resistance of the pipe material and any given insulation is
insul o,
insul
insul inn,
insul o,
pipe o,pipe
pipe inn,
pipe o,
insulpipe 2
ln
2
ln
Dk
D
D
Dk
D
D
RRR
. (8)
B. Freezing Time
A one-dimensional transient heat conduction model, in cylindrical co-ordinates, was used to estimate the freezing time of the liquid in the pipes. Figure 2 shows a schematic of the model used in the analysis. The pipe was of finite length and was filled with quiescent liquid at the melting point. In order to induce solidification, the surrounding ambient air temperature was less than the melting point of the liquid. It was assumed that the growth of the solid layer was axisymmetric and that all properties were constant.
Figure 2. Schematic of solidification in a circular tube or pipe
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The governing equations of the temperature distribution in the solid and liquid phases are
trrTtrT infL 0 ,, , (9)
innin
s
s2s ,
11Rrtr
t
T
r
Tr
rr
, (10)
where αs is the thermal diffusivity of the solidified water and rin(t) is the transient location of the solid-liquid interface (Fig. 2). The boundary and initial conditions are
fins , TtrT , (11)
TtRTU
r
tRTk ,
,inns
innss , (12)
fs 0, TrT , (13)
innin 0 Rr . (14)
Conservation of energy at the solid-liquid interface will yield another boundary condition, the interface energy equation, which establishes the problem as a free boundary problem. The solid-liquid interface energy equation is
dt
drL
r
trTk in
sins
s,
, (15)
where ks is the conductivity of solid water, ρs is the density of solid water, and L is the latent heat of fusion.
The initial condition of Eq. (13) suggests that the axisymmetric solid forms instantaneously at t = 0. This assumption serves to simplify the mathematical development of the model. However, Gilpin [14] has shown that ice will form initially as plate-like dendritic crystals that are distributed throughout the water and will be eventually engulfed by solid as the freezing event proceeds.
Using the separation of variables method and superposition, the temperature distribution in the solid phase, Ts(r,t) is
1
2s0
in0
in00
in
inn
s
inn
in
ffs
exp
ln
ln
,
kkk
k
kkk trY
rY
rJrJa
r
r
R
k
R
rU
TTUTtrT
, (16)
where λk is the separation constant.
Knowledge of the transient location of the solid-liquid interface, rin(t), during the freezing event will enable estimation of the liquid freezing time within a pipe of inner
radius, Rinn. Differentiation of Eq. (16) at r = rin and substitution into the solid-liquid interface energy equation of Eq. (15) produced
dt
drL
e
rJ
rYrY
rJ
a
R
k
R
rUr
TTU
k
k
t
k
kk
k
kkk
ins
1in1
in1in0
in0
inn
s
inn
inin
f
s
2s
ln
. (17)
The ice thickness as a function of time is
trRt ininn . (18)
A MATLAB (MathWorks, Inc., Natick, MA, USA) code
was used to solve the expressions for the first 6 eigenvalues of λk. The values of time that corresponded to each rin value were calculated using Eq. (17). The differential equation in Eq. (17) was estimated by a first-order finite difference approximation. The infinite series summation in Eq. (17) was found to be insignificant, with a dominant first term on the order of approximately 10-70. This simplified the expression to
dt
dr
R
k
R
rULr
TTUk in
inn
s
inn
inins
fs
ln
. (19)
Equation (19) gives a solution for the time corresponding to the equally-spaced discretized ice thickness, χ. A water-ice system was investigated in this study. General properties of solid ice that were used in the model are shown in Table 1 [12].
TABLE I. GENERAL PROPERTIES OF SOLID ICE [12]
Property Value Melting point, Tf 0oC
Thermal diffusivity, αs 1.194 x 10-6 m2/s Specific heat capacity, cp,s 2010 J/kg-K Thermal conductivity, ks 2.2 W/m-K
Density, ρs 917 kg/m3 Latent heat of fusion, L 334,000 J/kg
III. RESULTS & DISCUSSION
Free convection and radiation will occur at the outer surface of the pipe during cooling in stagnant air. This generates difficulties in estimating the overall heat transfer coefficient in stagnant air due to the various air parameters required to calculate a corresponding heat transfer coefficient. Mcquiston et al. [24] has also suggested that the determination of overall heat transfer coefficients over bare pipes exposed to free convection is complicated due to the lack of knowledge of the transient surface temperature of the pipe upon which the heat transfer coefficients will depend. Gordon [8] conducted
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several tests in which temperature traces (temperature curves as functions of time) of cooling and freezing water in several types of tubes and pipes were generated. Consequently, the results from Gordon [8] were used to calculate the combined free convection and radiation heat transfer coefficients, Uconvection/radiation, which was experienced throughout the entirety of his tests. The assumption was made that the experimental methods by Gordon [8] were identical for all tests that were conducted, resulting in a constant combined free convection and radiation heat transfer coefficient. The results of one test by Gordon [8] were used to estimate the combined free convection and radiation heat transfer coefficient. This was accomplished by substituting the known ambient temperature, the time required to freeze the water in the pipe, and the pipe diameter into Eq. (16), to find Uconvection/radiation. Given that pipe resistance will affect the overall heat transfer coefficient, U (see Eq. (7)), the results from a test in Type M copper tube was used since the thermal conductivity of copper is high (401 W/m-K [25]), resulting in low thermal resistance across the metal tube. This would ensure that the value of U obtained from Eq. (16) would be equal to Uconvection/radiation. The results from a test in ½” Type M copper tube, where the freezing time was calculated to be 25 mins at an ambient temperature of -17.78°C [8], was used. The resulting Uconvection/radiation value that was calculated was 46.5 W/m2-K. This value is higher than those reported by Mcquiston et al. [24], which were on the order of 15 W/m2-K. The orientation of the pipes in Gordon’s study [8] was vertical while Mcquiston et al. [24] studied a horizontal arrangement of tubes and pipes. This orientation difference may have generated a U value that is approximately three times higher than that observed by Mcquiston et al. [24]. Orientation of the tubes and pipes do have an effect on the overall heat transfer coefficient because it introduces turbulence in vertical orientations as shown by Popeil [26], which as a result, increases the convective heat transfer coefficient.
To validate the mathematical model that was developed, the transient temperature data presented by Gordon [8] was used. Figure 3 shows experimentally generated transient temperature traces [8], in which the cooling time and freezing time of water in ½” and ¾” PVC Schedule 40 pipes were being compared. The cooling time of the water in the pipe was taken to be the time required for the water temperature to decrease from the initial temperature to the supercooled temperature (-2°C). The supercooled temperature of -2°C was chosen based on the results observed by McDonald et al. [13] where the freezing of water in steel pipes was examined. In Gordon’s study [8], the ambient temperature in the freezer was not constant during the cooling phase of the water in the pipes (see Fig. 3). However, the liquid cooling model uses a boundary condition of the third kind, Eq. (3), that assumes a constant ambient temperature. In order to obtain an estimate of the liquid cooling time, the start of the cooling phase was assumed to be where the ambient temperature was within 30% of the final ambient temperature as shown in Fig. 3. Transient ambient temperatures could be incorporated into the mathematical model. However, this would generate a more
Figure 3. Analysis of transient temperature data from for ½” and ¾” PVC
Schedule 40 pipes [8]
complex mathematical solution and would likely require the use of Green’s functions. This will be investigated in a future study. The end time for freezing, as shown in Fig. 3, is the point where the slope of the temperature starts to decrease rapidly. This is an indication of completed ice growth.
In order to use the model to calculate the liquid cooling times and freezing times, the inner diameter of the pipes and, for PVC, which has higher thermal resistance than copper, the corresponding thermal conductivity was needed. The inner and outer diameters of the PVC Schedule 40 pipe was obtained from ASTM Standard D 1725 [27]. From the work of Osswald et al. [7], it was determined that the thermal conductivity of PVC varies with temperature, but remains fairly constant from -25°C to 25°C at a value of 0.15 W/m-K, which is also within the range presented by Gordon [8]. The R-value of the PVC pipes was calculated by using Eq. (8). The overall U value was then calculated by using Eq. (9) and the combined free convection and radiation heat transfer coefficient that was determined from Gordon’s results with ½” copper tube. The total overall heat transfer coefficient for 12.7 mm (½”) and 19.1 mm (¾”) PVC pipe was calculated to be 23.32 and 23.23 W/m2-K, respectively. The liquid cooling time was calculated by using Eq. (5) and the freezing time, by using Eqs. (16) - (19), which are shown tabulated in Table 2. Results for cooling and freezing in a ¾” copper tube is also presented in Table 2.
TABLE II. COMPARISON OF LIQUID COOLING AND FREEZE TIMES
Gordon Results [8] Model Prediction Liquid
Cooling Time (min)
Freeze Time (min)
TOTAL (min)
Liquid Cooling
Time (min)
Freeze Time (min)
TOTAL (min)
Type M Cu Tube
Ambient Temperature, T∞ = -6.67oC
¾” 15 105 120 13 91 104 PVC
Sch. 40 Pipe
Ambient Temperature, T∞ = -12.22oC
½” 10 75 85 10 74 84 ¾” 20 110 130 17 99 116
PVC Sch. 40
Pipe Ambient Temperature, T∞ = -17.78oC
½” 10 50 60 9 51 60
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The liquid cooling and freezing times presented in Table 2 for the mathematical model are within 15% of the experimental results presented by Gordon [8]. To apply the mathematical models presented in this study further to HVAC applications, various pipe diameters, ambient temperatures, and pipe materials were investigated. The diameters that were investigated were 6.4 to 102 mm (¼” - 4”) for copper and 12.7 to 102 mm (½” - 4”) for PVC, as recommended by ASHRAE [28] for HVAC applications. Figures 4 and 5 show curves of the model prediction of the total time required for the cooling and freezing of water in completely filled metal and PVC pipes of various inner diameters from an initial temperature of 20°C, for U-values obtained from Mcquiston et al. [24] for horizontal pipes, and for various ambient air temperatures. The results of Fig. 4 for bare, uninsulated metal pipes show that for pipes with inner diameters less than 40 mm, the total times required for cooling and freezing of the water were close when the stagnant ambient air temperature is -15°C or less. The curves nearly coalesce for smaller pipe inner diameters at low stagnant ambient air temperatures. Given the high thermal conductivity of metal pipes, their resistance to heat transfer is negligible. This will enable the use of the curves of Fig. 4 with a variety of metal pipes, not just copper only.
For PVC pipes, the thermal resistance could not be neglected, and by comparing Figs. 4 and 5, the total times were 1.5 times longer than those of metal pipes. This is a result of the low thermal conductivity and increased wall thickness of PVC plastic pipes. The model results presented in Figs. 4 and 5 were generated under the assumption that the internal pressure in the pipe is relieved as the water froze. It is possible that in practical applications, as the water freezes, the pipe may be blocked, causing an increase in the internal pressure in the pipe through the freezing event. The Clapeyron-Clausius equation will show that as the internal pressure on the water increases, the fusion temperature of the water will decrease. Therefore, no distinct solidification plateau, similar to that shown in Fig. 1, will be present. The effects of internal pressurization will form the basis of future work beyond this study. Lastly, it is known that if a heating failure were to occur in a residential or commercial building, the ambient temperature would not be constant and would require time to cool to an ambient temperature shown in Figs. 4 and 5. However, these figures serve as conservative scenarios, assuming that the ambient temperature of the building would fall instantaneously to the steady-state ambient temperature.
To increase the total time required for the cooling and freezing of the water in the pipe, a common practice in commercial applications is to insulate the pipe. There are a variety of insulation materials that are available on the market for use in industrial HVAC applications. However, ASHRAE Standard 90.1-2013 [29] for energy efficiency in buildings specifies that for pipes with less than 100 mm nominal diameters that contain hot water between 41°C and 60°C or cold water between 4°C and 16°C, the insulation thickness should be between 15 and 40 mm and have a thermal conductivity of 0.030 to 0.040 W/m-K. For this study, it was
Figure 4. Model prediction of the total (cooling + freezing) time in stagnant
ambient air for water in uninsulated, horizontal metal pipes as a function of pipe diameter [13]
Figure 5. Model prediction of the total (cooling + freezing) time in stagnant
ambient air for water in horizontal polyvinyl chloride (PVC) pipes as a function of pipe diameter
assumed that a 25 mm thick insulation is used to protect the pipes thermally, and the thermal conductivity of the insulation is 0.035 W/m-K, identical to the work of McDonald et al. [13]. Equation (8) shows the expression to calculate the resistance due to insulation for cylindrical pipes and tubes. The values obtained from Eq. (8) for the resistance due to insulation was used in Eq. (7) to estimate the overall heat transfer coefficient. The heat transfer coefficient due to the combination of free convection and radiation was obtained from Mcquiston et al. [24]. The total times required to cool and freeze water in insulated pipes are shown in Fig. 6. The figure shows clearly that the use of insulation increases the total time required to cool and freeze the water by nearly an order of magnitude when compared to bare, uninsulated pipes, identical to the conclusions of McDonald et al. [13]. While this observation is intuitive, it has also been observed experimentally by other investigators [8].
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Figure 6. Model prediction of the total (cooling + freezing) time in stagnant
ambient air for water in insulated metal pipes as a function of pipe diameter [13]
IV. CONCLUSIONS
This study focused on the estimation of the cooling and freezing time of water in cylindrical pipes in residential and commercial HVAC applications. Models that were based on the separation of variables method were developed and used to estimate the cooling and freezing times of water in both bare and insulated pipes. The models were validated with experimental transient temperature data of water during the cooling and freezing processes in bare copper tubes and PVC Schedule 40 pipes from the work of Gordon [8].
The estimates of the cooling and freezing times of the water that were obtained from the models were in good agreement with the experimental measurements by Gordon [8]. The models were then used to predict the total time to cool and freeze water for various pipe diameters of common pipe materials used in the HVAC industry. It was shown that PVC pipes have longer total freeze times than metal pipes, but most uninsulated pipes will freeze completely in approximately a day or less when subjected to cold ambient temperatures. Consequently, during a failure of heating equipment, such as furnaces or water heaters, immediate repair is required if outdoor temperatures are below 0°C for an extended period of time, or the pipes will need to be insulated to increase the total cooling and freezing time.
Future work and extensions of this study and model are possible. In particular, the effects of internal pressurization on the freezing time could be modeled analytically. This extension of the model could be coupled with pressure vessel analysis to estimate the burst pressures to predict failure in piping where freezing may occur. Other liquid material systems could also be explored. Water is a unique material in that its density decreases as it experiences a phase change from liquid to solid.
V. ACKNOWLEDGEMENTS
The authors gratefully acknowledge the assistance of B. Bschaden with MATLAB programming. Funding for this project was provided by the Natural Sciences and Engineering Research Council of Canada and Cenovus Energy Incorporated.
REFERENCES [1] Karnowski, S., “Deep Freeze May Have Cost Economy About $5 billion”,
Time magazine (Jan. 10, 2014), sec. Business & Money [2] Li, Y. F., Chow, W. K., Optimum insulation-thickness for thermal and
freezing protection, Appl. Energ., 80 ( 2005) 23-33. [3] Kecebas, A., Determination of insulation thickness by means of exergy
analysis in pipe insulation. Energ. Convers. Manage., 58 (2012) 76-83. [4] Stanford, H. W., HVAC water chillers and cooling towers, Marcel Dekker,
Inc., New York, 2003, pp. 48. [5] McDonald, A. G., Magande, H. L., Introduction to thermo-fluids systems,
John Wiley and Sons, Inc., West Sussex, 2012, pp. 76. [6] Wypych, G., Handbook of Polymers, ChemTec Publishing, Toronto, 2012, pp.
68. [7] Osswald, T. A., Baur, E., Brinkmann, S., Oberbach, K., Schmachtenberg, E.,
International Plastics Handbook - The Resource for Plastics Engineers, 4th ed., Hanser Publishers, 2006, pp. 69, 733.
[8] Gordon, J., An investigation into freezing and bursting water pipes in residential construction, Research Report No. 96-1, Building Research Council, School of Architecture, University of Illinois-Urbana-Champaign, 2006, pp. 1-51.
[9] Gilpin, R., The effects of dendritic ice formation in water pipes, Int. J. Heat Mass Transfer 20 (1977) 693-699.
[10] Sugawara, M., Seki, N., Kimoto, K., Freezing limit of water in a closed circular tube, Wärme und Stoffübertragung 17 (1983) 187-192.
[11] Oiwake, S., Saito, H., Inaba, H., Tokura, I., Study on dimensionless criterion of fracture of closed pipe due to freezing of water, Wärme und Stoffübertragung 20 (1986) 323-238.
[12] Akyurt, M., Zaki, G., Habeebullah, B., Freezing phenomena in ice-water systems, Energy Conversion Management 43 (2002) 1773-1789.
[13] McDonald, A., Bschaden, B., Sullivan, E., Marsden, R., Mathematical Simulation of the Freezing Time of Water in Small Diameter Pipes, Appl. Thermal Eng., under review (Manuscript #: ATE-S-14-00077)
[14] Gilpin, R., The influence of natural convection on dendritic ice growth, J. Crystal Growth 36 (1976) 101-108.
[15] Stefan, J., Über die theorie der eisbildung, insbesondere über die eisbildung in polarmaere, Annalen der Physik und Chemie 42 (1891), 269-286 (in German).
[16] Poots, G., On the application of integral-methods to the solution of problems involving the solidification of liquids initially at the fusion temperature, Int. J. Heat Mass Transfer 5 (1962) 525-531.
[17] Muehlbauer, J., Sunderland, J., Heat conduction with freezing or melting, Appl. Mech. Rev. 18 (1965) 951-959.
[18] Pham, Q., Extension to Planck’s equation for predicting freezing times of foodstuffs of simple shapes, Int. J. Refrigeration 7 (1984) 377-383.
[19] Planck, R., Beitrage zur Berechnung und Bewertung der Gefriergeschwindigkeit von Lebensmitteln, Beihefte Z ges Kalte-Ind 3 (1941) 1-16 (in German).
[20] Cleland, D., Cleland, A., Earle, R., Byrne, S., Prediction rates of freezing, thawing, and cooling in solids of arbitrary shape using a finite element method, Int. J. Refrigeration 7 (1984) 6-13.
[21] Hung, Y., Prediction of cooling and freezing times, Food Technol. 44 (1990) 137-153.
[22] Delgado, A., Sun, D., Heat and mass transfer models for predicting freezing processes – A review, J. Food Eng. 47 (2001) 157-174.
[23] Özisik, M., Heat Conduction, Second ed., John Wiley and Sons, Inc., New York, 1993, pp. 14, 116-118, 284-309.
[24] McQuiston, F., Parker, J., Spitler, J., Heating, Ventilating, and Air Conditioning – Analysis and Design, Fifth ed., John Wiley and Sons, Inc., New York, 2000, pp. 155.
[25] Cengel, Y. A., Ghajar, A. J., Heat and mass transfer : fundamentals & applications, 4th ed, McGraw-Hill, New York, 2011, pp. 868.
[26] Popiel, C. O., Free Convection Heat Transfer from Vertical Slender Cylinders: A Review. Heat Transfer Engineering, (2008) 29(6), 521-536.
[27] American Society for Testing and Materials (ASTM), “Standard Specification for Poly(Vinyl Chloride) (PVC) Plastic Pipe, Schedules 40, 80, and 120”, ASTM D 1785, pp. 68-69.
[28] American Society of Heating, Refrigeration, and Air-conditioning Engineers, ASHRAE Handbook - Fundamentals, American Society of Heating, Refrigeration, and Air-conditioning Engineers, Atlanta, 2009, pp. 22.1-22.11.
[29] American Society of Heating, Refrigeration, and Air-conditioning Engineers, ASHRAE Standard 90.1 – Energy Standard for Buildings except Low-rise Residential Buildings, American Society of Heating, Refrigeration, and Air-conditioning Engineers, Atlanta, 2013, pp. 81.
