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Freezing due to direct contact heat transfer including sublimation
International Journal of Refrigeration 25 (2002) 235–242 www.elsevier.com/locate/ijrefrig
Freezing due to direct contact heat transfer including sublimation Kazuo Aoki*, Masayuki Sawada, Masatoshi Akahori Department of Mechanical Engineering, Nagaoka University of Technology, 1603-1, Kamitomioka, Nagaoka, Niigata 940-2188, Japan Received 19 April 2001; received in revised form 3 September 2001; accepted 3 September 2001
Abstract From the energy saving viewpoint, the utilization of LNG cold is very important in the refrigeration industry concerning the low temperature region. In this paper, as a basic study of the freezing due to direct contact, including evaporation, solid–liquid direct contact heat transfer associated with sublimation has been investigated using the dry ice in water experimentally and theoretically. Based on a two-layers-model composed of CO2 vapor and the bulk water around the dry ice, the velocity and temperature fields in two layers was calculated numerically and the calculated results for the freezing condition of the bulk water were compared with the experimental results. # 2002 Published by Elsevier Science Ltd and IIR. Keywords: Heat transfer; Phase change; Solid; Liquid; Sublimation; Modelling; Dry ice; Water
Conge´lation due au transfert de chaleur a` contact direct, y compris par sublimation Re´sume´ Afin de re´aliser des e´conomies d’e´nergie, l’utilisation du gaz naturel lique´fie (GNL) comme source de froid est tre`s importante dans le secteur frigorifique en ge´ne´ral et dans le domaine des tre`s basses tempe´ratures en particulier. Cette communication de´crit une e´tude fondamentale sur la conge´lation re´alise´e par contact direct, par l’e´vaporation, ou par pour cette e´tude la glace carbonique dans l’eau afin d’effectuer des e´tudes the´oriques et expe´rimentales. Ils ont employe´ un mode`le a` deux couches compose´es de vapeur de CO2 et d’eau en masse entourant la glace carbonique ; ils ont calcule´ les vitesses et champs de tempe´rature dans les deux couches de facon nume´rique avant de comparer les re´sultats obtenus par ces calculs, pour la conge´lation de l’eau, avec les re´sultats expe´rimentaux. # 2002 Published by Elsevier Science Ltd and IIR. Mots cle´s : transfert de chaleur ; changement de phase ; solide ; liquide ; sublimation ; mode´lisation ; glace carbonique ; eau
1. Introduction Recently, there has been a jump in the demand of LNG (liquefied natural gas) as an alternative energy * Corresponding author. Tel.: +81-258-47-9729; fax: +81258-47-9770. E-mail address: [email protected] (K. Aoki).
from oil. When LNG is used as gas, it releases a great amount of cold energy due to evaporation (170 C). From the energy saving viewpoint, the utilization of LNG cold is very important for the refrigeration industry concerning low temperature region. As one way of utilizing the LNG cold, we propose an energy storage system with freezing due to direct contact between LNG and the bulk liquid. By pouring LNG drops in liquid,
0140-7007/02/$22.00 # 2002 Published by Elsevier Science Ltd and IIR. PII: S0140-7007(01)00084-6
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Nomenclature d g L q T u,v x y
diameter of the dry ice, m gravity, m/s2 latent heat of sublimation, J/kg heat flux, W/m2 temperature, K velocities in x- and y-direction, respectively, m/s distance along the dry ice surface, m distance perpendicular to the dry ice surface, m
Greek symbols coefficient of volumetric thermal expansion, 1/K v thickness of vapor film layer, m
the cold energy of LNG due to evaporation can be directly transferred to the bulk liquid associated with freezing. In this system, two phase-changes, i.e. the evaporation of LNG and the freezing of the bulk water, occur at the same time. Direct contact heat transfer between two immiscible liquids associated with the evaporation of one of the fluids has an advantage of eliminating a heat transfer surface, and has a higher heat transfer rate and an ability to operate at a small temperature difference. Up to the present, a number of theoretical and experimental studies have been reported on this problem. A critical review of the work on the direct contact heat transfer is given by Sideman [1]. As basic studies for single drop evaporation, Battya et al. [2] and Tadrist et al. [3] studied the vaporization of a moving bubble-droplet. Many researchers investigated the heat transfer coefficient and the evaporation length for direct contact heat transfer with evaporation under the combinations of various liquids [4–7]. The practical applications are found in water desalination, geothermal heat recovery, ocean thermal energy conversion and thermal energy storage systems. However, few have been reported on the freezing due to direct contact including evaporation. In the present paper, as a basic study of the freezing due to direct contact including evaporation, solid–liquid direct contact heat transfer with sublimation has been investigated using the dry ice in water experimentally and theoretically. Main focus was put on the freezing condition of the bulk water. Based on a two-layers-model composed of CO2 vapor and the bulk water around the dry ice, the velocity and temperature fields in both layers was calculated numerically and the calculated results for the
’ l
coordinate transformation variable, angle measured from lowest position of the dry ice, deg thermal conductivity, W/(m.K) viscosity, Pa.s dynamic viscosity, m2/s density, kg/m3 coordinate transformation variable
Subscripts i CO2 vapor–water interface l water v CO2 vapor w dry ice surface 1 bulk
freezing condition of the bulk water were compared with the experimental results.
2. Sublimation patterns and freezing When the dry ice contacts with the water having a higher temperature than the sublimation point of the dry ice, CO2 vapor, due to the sublimation, generates around the dry ice. By observation, the sublimation is divided into two patterns, the film-state sublimation and the nucleate-state sublimation. Fig. 1 shows the photographs of two sublimation patterns around the dry ice column horizontally set up in the water and the ethanol, respectively. In the film-state sublimation shown in Fig. 1(a), CO2 vapor due to the sublimation of the dry ice flows in a film state around the dry ice column. Since the vapor flow causes a liquid boundary flow in the bulk water, a two-layer flow composed of vapor and liquid phases exists around the dry ice column. In the nucleatestate sublimation shown in Fig. 1(b), the CO2 gas phase, due to the sublimation of the dry ice, becomes the large number of tiny bubbles which frequently form and separate on the dry ice surface like the nucleate boiling phenomenon. The difference in the two sublimation patterns strongly depends on viscosity and surface tension of the liquid and the sublimation heat flux on the surface. In this paper, we consider only the film-state sublimation because it generally occurs for the dry ice– water direct contact system. Fig. 2 is a schematic diagram showing the freezing of the bulk water for the filmstate sublimation using the dry ice. Since the vapor phase generated by the sublimation acts as an insulator,
K. Aoki et al. / International Journal of Refrigeration 25 (2002) 235–242
237
Fig. 1. Sublimation patterns of dry ice in liquid. Fig. 1. Sublimation de la glace carbonique dans l’eau.
the freezing does not occur under a higher bulk temperature condition. When the bulk temperature decreases and the temperature at the interface between vapor and the bulk water attains at 0 C, the freezing of the bulk water occurs as a thin ice plate at the lower stagnation point of ’=0 . As the bulk temperature drops further, the interface temperature becomes below 0 C and a rigid ice layer due to the freezing begins to form at the interface and rapidly grows along the angle because of the change of the interface flow.
3. Analysis 3.1. Physical model Fig. 3 shows the schematic diagram showing the sublimation of the dry ice column set up in the water. When the dry ice contacts with the water having a higher temperature than sublimation point of dry ice, CO2 vapor film layer is formed around the dry ice. Since the vapor flow causes a liquid boundary flow in the bulk water, the model is considered as a two-layers flow around the dry ice which is composed of CO2 vapor and the bulk water. In this paper, we calculate only the freezing condition of the bulk water corresponding to Fig. 2(a) and (b) and the growth of a rigid ice layer
Fig. 2. Schematic diagram showing the freezing of the bulk water for the film-state sublimation. Fig. 2. Sche´ma de la conge´lation de l’eau en masse entourant la glace carbonique lors de formation du film de CO2 par sublimation.
corresponding to Fig. 2(c) and (d) is out of consideration because of the change of the two-layers-flow. In analysis, we introduce the following assumptions: 1. The thickness of vapor film produced by the sublimation of the dry ice is very small, compared to the dry ice diameter. 2. The temperature of the dry ice is constant at the sublimation temperature (78.5 C). 3. The water density depends on temperature. 4. There is no slip condition at the vapor-water interface. 5. The vapor film flow and the liquid boundary layer flow are laminar. 6. The system is quasi-steady state and the volumetric change of the dry ice due to the sublimation is not considered because of its small change. 3.2. Basic equation 3.2.1. Vapor film layer Assuming the boundary layer approximation, the conservation equations of mass, momentum and energy in the vapor film are expressed by
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@uv @vv þ ¼0 @x @y
ð1Þ
uv
@uv @uv v @2 uv l v 2x þ g sin þ vv ¼ @x @y v @y2 v d
ð2Þ
uv
@Tv @Tv lv @2 Tv þ vv ¼ @x @y v cpv @y2
ð3Þ
where u and v respectively represent the velocities of xaxis and y-axis directions, the viscosity, the density, d the diameter of the dry ice, cp the specific heat at constant pressure, l the thermal conductivity, T the temperature, and the subscripts v and l denote the CO2 vapor and the water, respectively. Fig. 3. Schematic diagram showing the sublimation of the dry ice column in the water.
3.2.2. Water boundary layer Similarly, in the water boundary layer @ul @vl þ ¼0 @x @y
ð4Þ
@ul @ul l @ ul 2x ul þ vl ¼ þ g ð T T Þsin l l1 d @x @y l @y2
ð5Þ
3.3.3. Outer boundary of water flow layer At the outer boundary of the water flow layer, the boundary conditions are given as
ð6Þ
y ! 1:
2
ul
Fig. 3. Sche´ma montrant la sublimation de la glace carbonique en colonne dans l’eau.
@Tl @Tl ll @2 Tl þ vl ¼ @x @y l cpl @y2
ul ¼ 0;
@ul ¼ 0; @y
Tl ¼ Tl1 ;
@Tl ¼0 @y ð9Þ
where is the coefficient of volumetric thermal expansion, and Tl1 is the temperature of the bulk water 3.3. Boundary and interfacial conditions 3.3.1. Surface of the dry ice The boundary conditions on the surface of the dry ice are given as, using the relationship between the sublimation rate and the heat flux on the surface of the dry ice y¼0:
uv ¼ 0;
lv @Tv j ; vv ¼ v L @y w
3.3.4. The lower stagnation point of j=0 By considering the symmetrical condition for the temperature and velocity fields at the lower stagnation point of ’=0 , ’¼0:
Tv ¼ Tw
ð10Þ
3.4. Coordinate transformation In calculation, we introduce the coordinate transformation from the physical space to a calculation space. Non-dimensional coordinate variables and are determined by the following equations. ¼
Tv ¼ Tl ;
@uv @ul ¼ l ; @y @y @Tv @Tl ¼ ll lv @y @y
@v @l ¼ ¼0 @x @x
ð7Þ
3.3.2. CO2 vapor–water interface Assuming no slip and continuity boundary conditions at the CO2 vapor-water interface
uv ¼ ul ;
@Tv @Tl ¼ ¼ 0; @x @x
where v and l are the thickness of the vapor film layer and the water boundary layer, respectively.
where L is the latent heat of sublimation, and the subscript w denotes the surface of the dry ice.
y ¼ v :
uv ¼ ul ¼ 0;
x ; d
¼
y v ðxÞ
ð11Þ
v
ð8Þ
By using these variables, the physical space (x, y) is transformed to the calculation space ((x, y), (x, y)) as shown in Fig. 4.
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¼1: ll
ul ¼ uv ;
l
@ul @uv ¼ v ; @ @
Tl ¼ Tv ;
@Tl @Tv ¼ lv @ @ ul ¼ 0;
¼1:
@ul ¼ 0; @
Tl ¼ Tl1 ;
@Tl ¼0 @ ’¼0:
ð19Þ
@uv @ul ¼ ¼ 0; @ @ @v @l ¼ ¼0 @ @
ð20Þ
uv ¼ ul ¼ 0;
@Tv @Tl ¼ ¼ 0; @ @
ð21Þ
Fig. 4. Schematic diagram in Z–x plane. Fig. 4. Sche´ma du phe´nome`ne dans un syste`me de coordonne´s sans dimension –.
Substituting Eq. (11) into Eqs. (1)–(10), the governing equations and the boundary conditions are transformed as follows. 3.4.1. Vapor film layer 1 @uv @v @uv 1 @vv þ ¼0 d @ dv @ @ v @
ð12Þ
uv @uv @v 1 @uv þ vv uv d @ d @ v @ v 1 @2 uv l v þ gsinð2Þ ¼ v 2v @2 v
ð13Þ
uv @Tv @v 1 @Tv lv 1 @2 Tv þ vv uv ¼ d @ v @ d @ v cpv 2v @2
ð14Þ
1 @ul @v @ul 1 @vl þ ¼0 d @ dv @ @ v @
ð15Þ
ul @ul @v 1 @ul þ vl ul d @ v @ d @ l 1 @2 ul þ gðTl Tl1 Þsinð2Þ l 2v @2
ð16Þ
ul @Tl @v 1 @Tl ll 1 @2 Tl þ vl ul ¼ d @ l cpl 2v @2 d @ v @
ð17Þ
3.4.3. Boundary conditions ¼0:
uv ¼ 0;
vv ¼
lv @Tv j ; v Lv @ ¼0
Using a finite difference method, the governing equations are discretized and the discrete algebraic equations obtained are numerically solved by using SOR method. In calculation, the thickness of the vapor film layer v is first assumed, and the velocity and the temperature fields in all the calculation area are calculated by using the assumed thickness. Next, the velocity on the surface of the dry ice is compared with that obtained by temperature distribution. If two velocities are not equal, the thickness of the vapor film layer v is then modified by the following equation. vw ¼
lv Tv;1 Tv;0 v Lv
ð22Þ
The calculation is repeatedly executed until the values of vw, u, v, T, and v converge.
4. Experimental apparatus and procedure
3.4.2. Water boundary layer
¼
3.5. Calculation procedure
Tv ¼ Tw
ð18Þ
Fig. 5 shows the experimental apparatus for the experiments of solid (dry ice)–liquid (water) direct contact heat transfer. The experimental apparatus is composed of the test part, the cooling system to keep the bulk water temperature constant and the temperature measuring system. The dry ice column having dimensions of 45 mm in diameter 90 mm in length is used as a test sample. The test sample is inserted into bulk water controlled at the constant temperature and the sublimation state of the dry ice and the freezing of the bulk water are observed by using a video camera. The temperature distributions in CO2 vapor film and the bulk water layer in a lower stagnation point of ’=0 are measured using T-type thermocouples of 0.1 mm in diameter and the data are recorded by a data logger.
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Fig. 5. Experimental apparatus. Fig. 5. Appareil expe´rimental.
The location of the CO2 vapor-water interface and its temperature are obtained from these temperature distributions.
5. Results and discussion Direct contact heat transfer associated with evaporation has many advantages due to higher effective heat transfer. However, the vapor phase due to the evaporation often acts as insulation for freezing of the bulk water. Hence, it is important to clarify the effect of the vapor phase on the freezing of the bulk water. Fig. 6 shows velocity profiles in the vapor and water layers against the distance from the dry ice surface, taking the angle, ’ , as a parameter. The thickness of the vapor layer shown by the dotted line in this figure increases with the angle because of an increase in the total sublimation quantity along the flow direction. The velocity of vapor flow increases with the angle, it takes
the maximum value at an angle of about ’=120 , and then it decreases with the angle. Corresponding to this, the velocity of water is the maximum at the angle of ’ =120 , but the change in the velocity is very small, compared with that in the vapor velocity. Fig. 7 shows the heat flux, qi, at the interface between the vapor and water layers as well as the heat flux, qw, on the dry ice surface against the angle, ’, taking the bulk water temperature as a parameter. The former corresponds to the heat flux supplied from the bulk water, and the latter the sublimation heat flux. The difference between the two heat fluxes means the heat carried away by the vapor flow. Both heat fluxes decrease with increasing the angle, ’ , and the difference between two heat fluxes become small because the vapor layer acting as an insulation becomes thicker along the flow direction. Fig. 8 shows the interface temperature against the angle, ’ , taking the bulk water temperature, Tl1, as a parameter. The interface temperature decreases with decreasing Tl1 and it becomes below 0 C at the condition of Tl1=2 C. Also, the interface temperature decreases with increasing ’ and it reaches the minimum at the lower stagnation point, ’=0 . This means that the freezing starts at the lower stagnation point. This result corresponds to the experimentally observed result of the appearance of freezing. In the appearance of the water freezing, the temperature distribution at the lower stagnation point becomes
Fig. 6. Velocity profiles in the vapor and water layers as a parameter of the angle ’. Fig. 6. Vitesses dans les couches de vapeur et d’eau en fonction de l’angle j.
Fig. 7. Heat fluxes against the angle ’. Fig. 7. Flux thermiques selon l’angle j.
K. Aoki et al. / International Journal of Refrigeration 25 (2002) 235–242
Fig. 8. Vapor-water interface temperature against the angle ’. Fig. 8. Tempe´rature a` l’interface vapeur-eau selon l’angle j.
very important. Fig. 9 shows the comparison between the calculated and the experimental results for the temperature distributions in vapor and water layers at the angle of ’=0 . The temperature rapidly changes in the vapor layer and the experimental results agree well with the calculated results. Fig. 10 shows the comparison between the calculated and the experimental results for the interface temperature. The interface temperature decreases with decreasing the bulk temperature. Under the non-freezing conditions, the calculated results of the interface temperature are in good agreement with the experimental results. Once the freezing occurs at the interface, it is difficult to measure the interface temperature correctly because the freezing layer causes the change of the flow state in the vapor and water layers and leads to the change of the temperature distributions. Accordingly, under the freezing conditions, two interface temperatures measured before and after the formation of the freezing layer are plotted on Fig. 10. The model presented here is out of consideration for the formation of the freezing layer, so the calculated results are compared with only the interface temperatures measured before the formation of the freezing layer. The predicted bulk temperature where the freezing occurs is about 2.5 C and it corresponds to the experimental result showing about 2.8 C for the appearance of the water freezing. Also, these results well correspond to the observed results for the appearance of the water freezing.
241
Fig. 9. Comparison between the calculated and the experimental results for temperature profile. Fig. 9. Comparaison entre les re´sultats calcule´s et mesure´s pour le profil de tempe´rature.
Fig. 10. Comparison between the calculated and the experimental results for interface temperature. Fig. 10. Comparaison entre les re´sultats calcule´s et mesure´s pour les tempe´ratures a` l’interface.
6. Conclusions As the basic study of the utilization of LNG cold, the freezing due to direct contact heat transfer including sublimation has been investigated experimentally and
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theoretically. The results obtained here are summarized as follows; (1) The sublimation was divided into two patterns, film-state sublimation and nucleate-state sublimation. (2) In the film-state sublimation pattern, a two-layersflow model composed of CO2 vapor and the bulk water around the dry ice was presented, and the velocity and temperature fields in the two layers were calculated numerically. (3) The interface temperature between the vapor and water layers decreased with increasing ’ and it reached the minimum at the lower stagnation point, ’=0 . The water freezing started from the lower stagnation point. The calculated results of the interface temperature at the lower stagnation point were in agreement with the experimental results and the appearance of the water freezing was connected with the temperature of the bulk liquid.
Acknowledgements This research was partially supported by a grant from Japan Society for the Promotion of Science (JSPSRFTF97P01003)
References [1] Sideman S. Direct contact heat transfer between immiscible liquids. Adv Chem Eng 1966;6:207–86. [2] Battya P, Vijay R, Raghavan Seetharamu K N. Parametric studies on direct contact evaporation of a drop in immiscible liquid. Int J Heat Mass Transfer 1984;27(2): 263–72. [3] Tadrist L, Shehu Diso I, Santini R, Pantaloni J. Vaporization of a liquid by direct contact in another immiscible liquid. Int J Heat Mass Transfer 1987;30(9):1773–85. [4] Sideman S, Taitel Y. Direct-contact heat transfer with change of phase: Evaporation of drops in an immiscible liquid medium. Int J Heat Mass Transfer 1964;7:1273– 89. [5] Prakash CB, Pinder KL. Direct contact heat transfer between two immiscible liquids during vaporization. Part I: measurement of heat transfer coefficient; Part II: total evaporation time. Can J Chem Eng 1967;45:210– 20. [6] Adams AES, Pinder KL. Average heat transfer coefficient during the direct evaporation of a liquid drop. Can J Chem Eng 1972;50:707–13. [7] Selecki A, Gradon L. Equation of motion of an expanding vapour drops in an immiscible liquid medium. Int J Heat Mass Transfer 1976;19:925–9.
Freezing due to direct contact heat transfer including sublimation Kazuo Aoki*, Masayuki Sawada, Masatoshi Akahori Department of Mechanical Engineering, Nagaoka University of Technology, 1603-1, Kamitomioka, Nagaoka, Niigata 940-2188, Japan Received 19 April 2001; received in revised form 3 September 2001; accepted 3 September 2001
Abstract From the energy saving viewpoint, the utilization of LNG cold is very important in the refrigeration industry concerning the low temperature region. In this paper, as a basic study of the freezing due to direct contact, including evaporation, solid–liquid direct contact heat transfer associated with sublimation has been investigated using the dry ice in water experimentally and theoretically. Based on a two-layers-model composed of CO2 vapor and the bulk water around the dry ice, the velocity and temperature fields in two layers was calculated numerically and the calculated results for the freezing condition of the bulk water were compared with the experimental results. # 2002 Published by Elsevier Science Ltd and IIR. Keywords: Heat transfer; Phase change; Solid; Liquid; Sublimation; Modelling; Dry ice; Water
Conge´lation due au transfert de chaleur a` contact direct, y compris par sublimation Re´sume´ Afin de re´aliser des e´conomies d’e´nergie, l’utilisation du gaz naturel lique´fie (GNL) comme source de froid est tre`s importante dans le secteur frigorifique en ge´ne´ral et dans le domaine des tre`s basses tempe´ratures en particulier. Cette communication de´crit une e´tude fondamentale sur la conge´lation re´alise´e par contact direct, par l’e´vaporation, ou par pour cette e´tude la glace carbonique dans l’eau afin d’effectuer des e´tudes the´oriques et expe´rimentales. Ils ont employe´ un mode`le a` deux couches compose´es de vapeur de CO2 et d’eau en masse entourant la glace carbonique ; ils ont calcule´ les vitesses et champs de tempe´rature dans les deux couches de facon nume´rique avant de comparer les re´sultats obtenus par ces calculs, pour la conge´lation de l’eau, avec les re´sultats expe´rimentaux. # 2002 Published by Elsevier Science Ltd and IIR. Mots cle´s : transfert de chaleur ; changement de phase ; solide ; liquide ; sublimation ; mode´lisation ; glace carbonique ; eau
1. Introduction Recently, there has been a jump in the demand of LNG (liquefied natural gas) as an alternative energy * Corresponding author. Tel.: +81-258-47-9729; fax: +81258-47-9770. E-mail address: [email protected] (K. Aoki).
from oil. When LNG is used as gas, it releases a great amount of cold energy due to evaporation (170 C). From the energy saving viewpoint, the utilization of LNG cold is very important for the refrigeration industry concerning low temperature region. As one way of utilizing the LNG cold, we propose an energy storage system with freezing due to direct contact between LNG and the bulk liquid. By pouring LNG drops in liquid,
0140-7007/02/$22.00 # 2002 Published by Elsevier Science Ltd and IIR. PII: S0140-7007(01)00084-6
236
K. Aoki et al. / International Journal of Refrigeration 25 (2002) 235–242
Nomenclature d g L q T u,v x y
diameter of the dry ice, m gravity, m/s2 latent heat of sublimation, J/kg heat flux, W/m2 temperature, K velocities in x- and y-direction, respectively, m/s distance along the dry ice surface, m distance perpendicular to the dry ice surface, m
Greek symbols coefficient of volumetric thermal expansion, 1/K v thickness of vapor film layer, m
the cold energy of LNG due to evaporation can be directly transferred to the bulk liquid associated with freezing. In this system, two phase-changes, i.e. the evaporation of LNG and the freezing of the bulk water, occur at the same time. Direct contact heat transfer between two immiscible liquids associated with the evaporation of one of the fluids has an advantage of eliminating a heat transfer surface, and has a higher heat transfer rate and an ability to operate at a small temperature difference. Up to the present, a number of theoretical and experimental studies have been reported on this problem. A critical review of the work on the direct contact heat transfer is given by Sideman [1]. As basic studies for single drop evaporation, Battya et al. [2] and Tadrist et al. [3] studied the vaporization of a moving bubble-droplet. Many researchers investigated the heat transfer coefficient and the evaporation length for direct contact heat transfer with evaporation under the combinations of various liquids [4–7]. The practical applications are found in water desalination, geothermal heat recovery, ocean thermal energy conversion and thermal energy storage systems. However, few have been reported on the freezing due to direct contact including evaporation. In the present paper, as a basic study of the freezing due to direct contact including evaporation, solid–liquid direct contact heat transfer with sublimation has been investigated using the dry ice in water experimentally and theoretically. Main focus was put on the freezing condition of the bulk water. Based on a two-layers-model composed of CO2 vapor and the bulk water around the dry ice, the velocity and temperature fields in both layers was calculated numerically and the calculated results for the
’ l
coordinate transformation variable, angle measured from lowest position of the dry ice, deg thermal conductivity, W/(m.K) viscosity, Pa.s dynamic viscosity, m2/s density, kg/m3 coordinate transformation variable
Subscripts i CO2 vapor–water interface l water v CO2 vapor w dry ice surface 1 bulk
freezing condition of the bulk water were compared with the experimental results.
2. Sublimation patterns and freezing When the dry ice contacts with the water having a higher temperature than the sublimation point of the dry ice, CO2 vapor, due to the sublimation, generates around the dry ice. By observation, the sublimation is divided into two patterns, the film-state sublimation and the nucleate-state sublimation. Fig. 1 shows the photographs of two sublimation patterns around the dry ice column horizontally set up in the water and the ethanol, respectively. In the film-state sublimation shown in Fig. 1(a), CO2 vapor due to the sublimation of the dry ice flows in a film state around the dry ice column. Since the vapor flow causes a liquid boundary flow in the bulk water, a two-layer flow composed of vapor and liquid phases exists around the dry ice column. In the nucleatestate sublimation shown in Fig. 1(b), the CO2 gas phase, due to the sublimation of the dry ice, becomes the large number of tiny bubbles which frequently form and separate on the dry ice surface like the nucleate boiling phenomenon. The difference in the two sublimation patterns strongly depends on viscosity and surface tension of the liquid and the sublimation heat flux on the surface. In this paper, we consider only the film-state sublimation because it generally occurs for the dry ice– water direct contact system. Fig. 2 is a schematic diagram showing the freezing of the bulk water for the filmstate sublimation using the dry ice. Since the vapor phase generated by the sublimation acts as an insulator,
K. Aoki et al. / International Journal of Refrigeration 25 (2002) 235–242
237
Fig. 1. Sublimation patterns of dry ice in liquid. Fig. 1. Sublimation de la glace carbonique dans l’eau.
the freezing does not occur under a higher bulk temperature condition. When the bulk temperature decreases and the temperature at the interface between vapor and the bulk water attains at 0 C, the freezing of the bulk water occurs as a thin ice plate at the lower stagnation point of ’=0 . As the bulk temperature drops further, the interface temperature becomes below 0 C and a rigid ice layer due to the freezing begins to form at the interface and rapidly grows along the angle because of the change of the interface flow.
3. Analysis 3.1. Physical model Fig. 3 shows the schematic diagram showing the sublimation of the dry ice column set up in the water. When the dry ice contacts with the water having a higher temperature than sublimation point of dry ice, CO2 vapor film layer is formed around the dry ice. Since the vapor flow causes a liquid boundary flow in the bulk water, the model is considered as a two-layers flow around the dry ice which is composed of CO2 vapor and the bulk water. In this paper, we calculate only the freezing condition of the bulk water corresponding to Fig. 2(a) and (b) and the growth of a rigid ice layer
Fig. 2. Schematic diagram showing the freezing of the bulk water for the film-state sublimation. Fig. 2. Sche´ma de la conge´lation de l’eau en masse entourant la glace carbonique lors de formation du film de CO2 par sublimation.
corresponding to Fig. 2(c) and (d) is out of consideration because of the change of the two-layers-flow. In analysis, we introduce the following assumptions: 1. The thickness of vapor film produced by the sublimation of the dry ice is very small, compared to the dry ice diameter. 2. The temperature of the dry ice is constant at the sublimation temperature (78.5 C). 3. The water density depends on temperature. 4. There is no slip condition at the vapor-water interface. 5. The vapor film flow and the liquid boundary layer flow are laminar. 6. The system is quasi-steady state and the volumetric change of the dry ice due to the sublimation is not considered because of its small change. 3.2. Basic equation 3.2.1. Vapor film layer Assuming the boundary layer approximation, the conservation equations of mass, momentum and energy in the vapor film are expressed by
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@uv @vv þ ¼0 @x @y
ð1Þ
uv
@uv @uv v @2 uv l v 2x þ g sin þ vv ¼ @x @y v @y2 v d
ð2Þ
uv
@Tv @Tv lv @2 Tv þ vv ¼ @x @y v cpv @y2
ð3Þ
where u and v respectively represent the velocities of xaxis and y-axis directions, the viscosity, the density, d the diameter of the dry ice, cp the specific heat at constant pressure, l the thermal conductivity, T the temperature, and the subscripts v and l denote the CO2 vapor and the water, respectively. Fig. 3. Schematic diagram showing the sublimation of the dry ice column in the water.
3.2.2. Water boundary layer Similarly, in the water boundary layer @ul @vl þ ¼0 @x @y
ð4Þ
@ul @ul l @ ul 2x ul þ vl ¼ þ g ð T T Þsin l l1 d @x @y l @y2
ð5Þ
3.3.3. Outer boundary of water flow layer At the outer boundary of the water flow layer, the boundary conditions are given as
ð6Þ
y ! 1:
2
ul
Fig. 3. Sche´ma montrant la sublimation de la glace carbonique en colonne dans l’eau.
@Tl @Tl ll @2 Tl þ vl ¼ @x @y l cpl @y2
ul ¼ 0;
@ul ¼ 0; @y
Tl ¼ Tl1 ;
@Tl ¼0 @y ð9Þ
where is the coefficient of volumetric thermal expansion, and Tl1 is the temperature of the bulk water 3.3. Boundary and interfacial conditions 3.3.1. Surface of the dry ice The boundary conditions on the surface of the dry ice are given as, using the relationship between the sublimation rate and the heat flux on the surface of the dry ice y¼0:
uv ¼ 0;
lv @Tv j ; vv ¼ v L @y w
3.3.4. The lower stagnation point of j=0 By considering the symmetrical condition for the temperature and velocity fields at the lower stagnation point of ’=0 , ’¼0:
Tv ¼ Tw
ð10Þ
3.4. Coordinate transformation In calculation, we introduce the coordinate transformation from the physical space to a calculation space. Non-dimensional coordinate variables and are determined by the following equations. ¼
Tv ¼ Tl ;
@uv @ul ¼ l ; @y @y @Tv @Tl ¼ ll lv @y @y
@v @l ¼ ¼0 @x @x
ð7Þ
3.3.2. CO2 vapor–water interface Assuming no slip and continuity boundary conditions at the CO2 vapor-water interface
uv ¼ ul ;
@Tv @Tl ¼ ¼ 0; @x @x
where v and l are the thickness of the vapor film layer and the water boundary layer, respectively.
where L is the latent heat of sublimation, and the subscript w denotes the surface of the dry ice.
y ¼ v :
uv ¼ ul ¼ 0;
x ; d
¼
y v ðxÞ
ð11Þ
v
ð8Þ
By using these variables, the physical space (x, y) is transformed to the calculation space ((x, y), (x, y)) as shown in Fig. 4.
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¼1: ll
ul ¼ uv ;
l
@ul @uv ¼ v ; @ @
Tl ¼ Tv ;
@Tl @Tv ¼ lv @ @ ul ¼ 0;
¼1:
@ul ¼ 0; @
Tl ¼ Tl1 ;
@Tl ¼0 @ ’¼0:
ð19Þ
@uv @ul ¼ ¼ 0; @ @ @v @l ¼ ¼0 @ @
ð20Þ
uv ¼ ul ¼ 0;
@Tv @Tl ¼ ¼ 0; @ @
ð21Þ
Fig. 4. Schematic diagram in Z–x plane. Fig. 4. Sche´ma du phe´nome`ne dans un syste`me de coordonne´s sans dimension –.
Substituting Eq. (11) into Eqs. (1)–(10), the governing equations and the boundary conditions are transformed as follows. 3.4.1. Vapor film layer 1 @uv @v @uv 1 @vv þ ¼0 d @ dv @ @ v @
ð12Þ
uv @uv @v 1 @uv þ vv uv d @ d @ v @ v 1 @2 uv l v þ gsinð2Þ ¼ v 2v @2 v
ð13Þ
uv @Tv @v 1 @Tv lv 1 @2 Tv þ vv uv ¼ d @ v @ d @ v cpv 2v @2
ð14Þ
1 @ul @v @ul 1 @vl þ ¼0 d @ dv @ @ v @
ð15Þ
ul @ul @v 1 @ul þ vl ul d @ v @ d @ l 1 @2 ul þ gðTl Tl1 Þsinð2Þ l 2v @2
ð16Þ
ul @Tl @v 1 @Tl ll 1 @2 Tl þ vl ul ¼ d @ l cpl 2v @2 d @ v @
ð17Þ
3.4.3. Boundary conditions ¼0:
uv ¼ 0;
vv ¼
lv @Tv j ; v Lv @ ¼0
Using a finite difference method, the governing equations are discretized and the discrete algebraic equations obtained are numerically solved by using SOR method. In calculation, the thickness of the vapor film layer v is first assumed, and the velocity and the temperature fields in all the calculation area are calculated by using the assumed thickness. Next, the velocity on the surface of the dry ice is compared with that obtained by temperature distribution. If two velocities are not equal, the thickness of the vapor film layer v is then modified by the following equation. vw ¼
lv Tv;1 Tv;0 v Lv
ð22Þ
The calculation is repeatedly executed until the values of vw, u, v, T, and v converge.
4. Experimental apparatus and procedure
3.4.2. Water boundary layer
¼
3.5. Calculation procedure
Tv ¼ Tw
ð18Þ
Fig. 5 shows the experimental apparatus for the experiments of solid (dry ice)–liquid (water) direct contact heat transfer. The experimental apparatus is composed of the test part, the cooling system to keep the bulk water temperature constant and the temperature measuring system. The dry ice column having dimensions of 45 mm in diameter 90 mm in length is used as a test sample. The test sample is inserted into bulk water controlled at the constant temperature and the sublimation state of the dry ice and the freezing of the bulk water are observed by using a video camera. The temperature distributions in CO2 vapor film and the bulk water layer in a lower stagnation point of ’=0 are measured using T-type thermocouples of 0.1 mm in diameter and the data are recorded by a data logger.
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Fig. 5. Experimental apparatus. Fig. 5. Appareil expe´rimental.
The location of the CO2 vapor-water interface and its temperature are obtained from these temperature distributions.
5. Results and discussion Direct contact heat transfer associated with evaporation has many advantages due to higher effective heat transfer. However, the vapor phase due to the evaporation often acts as insulation for freezing of the bulk water. Hence, it is important to clarify the effect of the vapor phase on the freezing of the bulk water. Fig. 6 shows velocity profiles in the vapor and water layers against the distance from the dry ice surface, taking the angle, ’ , as a parameter. The thickness of the vapor layer shown by the dotted line in this figure increases with the angle because of an increase in the total sublimation quantity along the flow direction. The velocity of vapor flow increases with the angle, it takes
the maximum value at an angle of about ’=120 , and then it decreases with the angle. Corresponding to this, the velocity of water is the maximum at the angle of ’ =120 , but the change in the velocity is very small, compared with that in the vapor velocity. Fig. 7 shows the heat flux, qi, at the interface between the vapor and water layers as well as the heat flux, qw, on the dry ice surface against the angle, ’, taking the bulk water temperature as a parameter. The former corresponds to the heat flux supplied from the bulk water, and the latter the sublimation heat flux. The difference between the two heat fluxes means the heat carried away by the vapor flow. Both heat fluxes decrease with increasing the angle, ’ , and the difference between two heat fluxes become small because the vapor layer acting as an insulation becomes thicker along the flow direction. Fig. 8 shows the interface temperature against the angle, ’ , taking the bulk water temperature, Tl1, as a parameter. The interface temperature decreases with decreasing Tl1 and it becomes below 0 C at the condition of Tl1=2 C. Also, the interface temperature decreases with increasing ’ and it reaches the minimum at the lower stagnation point, ’=0 . This means that the freezing starts at the lower stagnation point. This result corresponds to the experimentally observed result of the appearance of freezing. In the appearance of the water freezing, the temperature distribution at the lower stagnation point becomes
Fig. 6. Velocity profiles in the vapor and water layers as a parameter of the angle ’. Fig. 6. Vitesses dans les couches de vapeur et d’eau en fonction de l’angle j.
Fig. 7. Heat fluxes against the angle ’. Fig. 7. Flux thermiques selon l’angle j.
K. Aoki et al. / International Journal of Refrigeration 25 (2002) 235–242
Fig. 8. Vapor-water interface temperature against the angle ’. Fig. 8. Tempe´rature a` l’interface vapeur-eau selon l’angle j.
very important. Fig. 9 shows the comparison between the calculated and the experimental results for the temperature distributions in vapor and water layers at the angle of ’=0 . The temperature rapidly changes in the vapor layer and the experimental results agree well with the calculated results. Fig. 10 shows the comparison between the calculated and the experimental results for the interface temperature. The interface temperature decreases with decreasing the bulk temperature. Under the non-freezing conditions, the calculated results of the interface temperature are in good agreement with the experimental results. Once the freezing occurs at the interface, it is difficult to measure the interface temperature correctly because the freezing layer causes the change of the flow state in the vapor and water layers and leads to the change of the temperature distributions. Accordingly, under the freezing conditions, two interface temperatures measured before and after the formation of the freezing layer are plotted on Fig. 10. The model presented here is out of consideration for the formation of the freezing layer, so the calculated results are compared with only the interface temperatures measured before the formation of the freezing layer. The predicted bulk temperature where the freezing occurs is about 2.5 C and it corresponds to the experimental result showing about 2.8 C for the appearance of the water freezing. Also, these results well correspond to the observed results for the appearance of the water freezing.
241
Fig. 9. Comparison between the calculated and the experimental results for temperature profile. Fig. 9. Comparaison entre les re´sultats calcule´s et mesure´s pour le profil de tempe´rature.
Fig. 10. Comparison between the calculated and the experimental results for interface temperature. Fig. 10. Comparaison entre les re´sultats calcule´s et mesure´s pour les tempe´ratures a` l’interface.
6. Conclusions As the basic study of the utilization of LNG cold, the freezing due to direct contact heat transfer including sublimation has been investigated experimentally and
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theoretically. The results obtained here are summarized as follows; (1) The sublimation was divided into two patterns, film-state sublimation and nucleate-state sublimation. (2) In the film-state sublimation pattern, a two-layersflow model composed of CO2 vapor and the bulk water around the dry ice was presented, and the velocity and temperature fields in the two layers were calculated numerically. (3) The interface temperature between the vapor and water layers decreased with increasing ’ and it reached the minimum at the lower stagnation point, ’=0 . The water freezing started from the lower stagnation point. The calculated results of the interface temperature at the lower stagnation point were in agreement with the experimental results and the appearance of the water freezing was connected with the temperature of the bulk liquid.
Acknowledgements This research was partially supported by a grant from Japan Society for the Promotion of Science (JSPSRFTF97P01003)
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