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Effects of the Stefan number on solidification process with Pr = 1.0, Ra = 10 4 , We = 110-3 , 0 = 1.5, sl = 0.9, gl = 0.005 and gr = 0 0. (a) Velocity field at time = 0.12 for St = 0.01 (left) and 1.0 (right). (b) The shape of the solid phase at the nearly final stage of solidification. (c) Temporal variation of the area ratio. The velocity vector is normalized by U c .
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We present numerical simulations of solidification in an open vertical circular cylinder with the presence of a gas phase and natural convection. The numerical technique used is a two-dimensional front-tracking method combined with an interpolation method. A simple tri-junction condition, in terms of the growth angle ϕgr, is included due to the pre...
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Context 1
... phase change materials yield different growth angles, e.g., in the range of 0 - 28 0 as reported by Satunkin [27]. For instance, the growth angles of water, silicon and germanium are 0 0 , 12 0 and 14 0 [27]. Accordingly, we vary the growth angle in the range of 0 - 15 0 to examine the effect of the triple point on the form of the solid. positive slope, and the liquid-gas front with a negative slope near the triple point (Fig. 8a). As a result, at the center the level of the liquid phase is higher for gr = 15 0 as compared with that for gr = 0 0 . Fig. 8b shows the solid phase shape at the nearly final stage of solidification with gr = 0 0 , 5 0 , 10 0 , and 15 0 . A zero growth angle gr = 0 0 yields a solid phase with a flat surface at the top, i.e., zero slope (the solid line). As the growth angle increases, the magnitude of the slope of the solid top surface near the center increases as shown in Fig. 8b. However, the growth angle has only a minor effect on the solidification rate, and thus the temporal variation of the area ratio of the solid to the cylinder As/Ac is affected slightly by the growth angle, as shown in Fig. 8c. 9a shows the velocity field at time = 0.12 for two Stefan numbers, St = 0.01 (left) and 1.0 (right) with the presence of volume expansion, i.e., sl = 0.9. Circulations exist in the liquid phase because of natural convection. Fig. 9a also indicates that at this time, the solidification interface for St = 1.0 has advanced much farther than for St = 0.01. It can be understood from the equation for the normal velocity of the solidifying interface. Increasing St corresponding to reducing the latent heat released increases the growth rate of the solidification front, and thus reduces the time for completing solidification, as shown in Fig. 9c. Accordingly, the liquid-gas interface in the case of St = 1.0 has gone higher than for St = 0.01 ( Fig. 9a) because of the presence of volume expansion. In addition, decreasing the Stefan number with volume expansion results in steeper solid interfaces at the center as shown in Fig. 9b. Fig. 10a). For a higher Pr, i.e., Pr = 10, the circulation becomes much stronger and slightly deforms liquid-gas interfaces, as shown in the right frame of Fig. 10a and in b. This indicates that increasing the Prandtl number leads to a more deformed surface at the top of the solid phase (Fig. 10b). However, the effect of Pr on the shape of the solid top surface becomes more evident when Pr increases from 1.0 to 10.0 as shown in Fig. 10b. The temporal variation of the ratio of the solid area to the cylinder area, as shown in Fig. 10c, shows that an increase in Pr from 0.01 to 10.0 has a very minor effect on the growth rate of solidification, and the evolution of the solidification interface. Fig. 11a. Natural convection tends to perturb the liquid-gas interface. Accordingly, for Ra = 10 3 the buoyancy force, however, does not dominate over the surface tension force, and thus the liquid-gas is still flat, as shown in Fig. 11a and b. However, increasing Ra to 510 5 the buoyancy force becomes stronger, and thus deforms the liquid-gas interface. As a result, the top surface of the solidified product for 510 5 is not flat anymore, as shown in Fig. 11b. In addition, the strong circulation at the high value of Ra enhances heat transfer in the liquid phase, and thus the solidifying interface propagates faster. However, this effect with these values of the parameters is minor. This effect is clearly shown in Fig. 11c in which increasing Ra in the range of 10 3 to 510 5 has a very minor effect on the area ratio and evolution of the solid phase. is normalized by U c ...
Context 2
... phase change materials yield different growth angles, e.g., in the range of 0 - 28 0 as reported by Satunkin [27]. For instance, the growth angles of water, silicon and germanium are 0 0 , 12 0 and 14 0 [27]. Accordingly, we vary the growth angle in the range of 0 - 15 0 to examine the effect of the triple point on the form of the solid. positive slope, and the liquid-gas front with a negative slope near the triple point (Fig. 8a). As a result, at the center the level of the liquid phase is higher for gr = 15 0 as compared with that for gr = 0 0 . Fig. 8b shows the solid phase shape at the nearly final stage of solidification with gr = 0 0 , 5 0 , 10 0 , and 15 0 . A zero growth angle gr = 0 0 yields a solid phase with a flat surface at the top, i.e., zero slope (the solid line). As the growth angle increases, the magnitude of the slope of the solid top surface near the center increases as shown in Fig. 8b. However, the growth angle has only a minor effect on the solidification rate, and thus the temporal variation of the area ratio of the solid to the cylinder As/Ac is affected slightly by the growth angle, as shown in Fig. 8c. 9a shows the velocity field at time = 0.12 for two Stefan numbers, St = 0.01 (left) and 1.0 (right) with the presence of volume expansion, i.e., sl = 0.9. Circulations exist in the liquid phase because of natural convection. Fig. 9a also indicates that at this time, the solidification interface for St = 1.0 has advanced much farther than for St = 0.01. It can be understood from the equation for the normal velocity of the solidifying interface. Increasing St corresponding to reducing the latent heat released increases the growth rate of the solidification front, and thus reduces the time for completing solidification, as shown in Fig. 9c. Accordingly, the liquid-gas interface in the case of St = 1.0 has gone higher than for St = 0.01 ( Fig. 9a) because of the presence of volume expansion. In addition, decreasing the Stefan number with volume expansion results in steeper solid interfaces at the center as shown in Fig. 9b. Fig. 10a). For a higher Pr, i.e., Pr = 10, the circulation becomes much stronger and slightly deforms liquid-gas interfaces, as shown in the right frame of Fig. 10a and in b. This indicates that increasing the Prandtl number leads to a more deformed surface at the top of the solid phase (Fig. 10b). However, the effect of Pr on the shape of the solid top surface becomes more evident when Pr increases from 1.0 to 10.0 as shown in Fig. 10b. The temporal variation of the ratio of the solid area to the cylinder area, as shown in Fig. 10c, shows that an increase in Pr from 0.01 to 10.0 has a very minor effect on the growth rate of solidification, and the evolution of the solidification interface. Fig. 11a. Natural convection tends to perturb the liquid-gas interface. Accordingly, for Ra = 10 3 the buoyancy force, however, does not dominate over the surface tension force, and thus the liquid-gas is still flat, as shown in Fig. 11a and b. However, increasing Ra to 510 5 the buoyancy force becomes stronger, and thus deforms the liquid-gas interface. As a result, the top surface of the solidified product for 510 5 is not flat anymore, as shown in Fig. 11b. In addition, the strong circulation at the high value of Ra enhances heat transfer in the liquid phase, and thus the solidifying interface propagates faster. However, this effect with these values of the parameters is minor. This effect is clearly shown in Fig. 11c in which increasing Ra in the range of 10 3 to 510 5 has a very minor effect on the area ratio and evolution of the solid phase. is normalized by U c ...
Context 3
... phase change materials yield different growth angles, e.g., in the range of 0 - 28 0 as reported by Satunkin [27]. For instance, the growth angles of water, silicon and germanium are 0 0 , 12 0 and 14 0 [27]. Accordingly, we vary the growth angle in the range of 0 - 15 0 to examine the effect of the triple point on the form of the solid. positive slope, and the liquid-gas front with a negative slope near the triple point (Fig. 8a). As a result, at the center the level of the liquid phase is higher for gr = 15 0 as compared with that for gr = 0 0 . Fig. 8b shows the solid phase shape at the nearly final stage of solidification with gr = 0 0 , 5 0 , 10 0 , and 15 0 . A zero growth angle gr = 0 0 yields a solid phase with a flat surface at the top, i.e., zero slope (the solid line). As the growth angle increases, the magnitude of the slope of the solid top surface near the center increases as shown in Fig. 8b. However, the growth angle has only a minor effect on the solidification rate, and thus the temporal variation of the area ratio of the solid to the cylinder As/Ac is affected slightly by the growth angle, as shown in Fig. 8c. 9a shows the velocity field at time = 0.12 for two Stefan numbers, St = 0.01 (left) and 1.0 (right) with the presence of volume expansion, i.e., sl = 0.9. Circulations exist in the liquid phase because of natural convection. Fig. 9a also indicates that at this time, the solidification interface for St = 1.0 has advanced much farther than for St = 0.01. It can be understood from the equation for the normal velocity of the solidifying interface. Increasing St corresponding to reducing the latent heat released increases the growth rate of the solidification front, and thus reduces the time for completing solidification, as shown in Fig. 9c. Accordingly, the liquid-gas interface in the case of St = 1.0 has gone higher than for St = 0.01 ( Fig. 9a) because of the presence of volume expansion. In addition, decreasing the Stefan number with volume expansion results in steeper solid interfaces at the center as shown in Fig. 9b. Fig. 10a). For a higher Pr, i.e., Pr = 10, the circulation becomes much stronger and slightly deforms liquid-gas interfaces, as shown in the right frame of Fig. 10a and in b. This indicates that increasing the Prandtl number leads to a more deformed surface at the top of the solid phase (Fig. 10b). However, the effect of Pr on the shape of the solid top surface becomes more evident when Pr increases from 1.0 to 10.0 as shown in Fig. 10b. The temporal variation of the ratio of the solid area to the cylinder area, as shown in Fig. 10c, shows that an increase in Pr from 0.01 to 10.0 has a very minor effect on the growth rate of solidification, and the evolution of the solidification interface. Fig. 11a. Natural convection tends to perturb the liquid-gas interface. Accordingly, for Ra = 10 3 the buoyancy force, however, does not dominate over the surface tension force, and thus the liquid-gas is still flat, as shown in Fig. 11a and b. However, increasing Ra to 510 5 the buoyancy force becomes stronger, and thus deforms the liquid-gas interface. As a result, the top surface of the solidified product for 510 5 is not flat anymore, as shown in Fig. 11b. In addition, the strong circulation at the high value of Ra enhances heat transfer in the liquid phase, and thus the solidifying interface propagates faster. However, this effect with these values of the parameters is minor. This effect is clearly shown in Fig. 11c in which increasing Ra in the range of 10 3 to 510 5 has a very minor effect on the area ratio and evolution of the solid phase. is normalized by U c ...
Context 4
... phase change materials yield different growth angles, e.g., in the range of 0 - 28 0 as reported by Satunkin [27]. For instance, the growth angles of water, silicon and germanium are 0 0 , 12 0 and 14 0 [27]. Accordingly, we vary the growth angle in the range of 0 - 15 0 to examine the effect of the triple point on the form of the solid. positive slope, and the liquid-gas front with a negative slope near the triple point (Fig. 8a). As a result, at the center the level of the liquid phase is higher for gr = 15 0 as compared with that for gr = 0 0 . Fig. 8b shows the solid phase shape at the nearly final stage of solidification with gr = 0 0 , 5 0 , 10 0 , and 15 0 . A zero growth angle gr = 0 0 yields a solid phase with a flat surface at the top, i.e., zero slope (the solid line). As the growth angle increases, the magnitude of the slope of the solid top surface near the center increases as shown in Fig. 8b. However, the growth angle has only a minor effect on the solidification rate, and thus the temporal variation of the area ratio of the solid to the cylinder As/Ac is affected slightly by the growth angle, as shown in Fig. 8c. 9a shows the velocity field at time = 0.12 for two Stefan numbers, St = 0.01 (left) and 1.0 (right) with the presence of volume expansion, i.e., sl = 0.9. Circulations exist in the liquid phase because of natural convection. Fig. 9a also indicates that at this time, the solidification interface for St = 1.0 has advanced much farther than for St = 0.01. It can be understood from the equation for the normal velocity of the solidifying interface. Increasing St corresponding to reducing the latent heat released increases the growth rate of the solidification front, and thus reduces the time for completing solidification, as shown in Fig. 9c. Accordingly, the liquid-gas interface in the case of St = 1.0 has gone higher than for St = 0.01 ( Fig. 9a) because of the presence of volume expansion. In addition, decreasing the Stefan number with volume expansion results in steeper solid interfaces at the center as shown in Fig. 9b. Fig. 10a). For a higher Pr, i.e., Pr = 10, the circulation becomes much stronger and slightly deforms liquid-gas interfaces, as shown in the right frame of Fig. 10a and in b. This indicates that increasing the Prandtl number leads to a more deformed surface at the top of the solid phase (Fig. 10b). However, the effect of Pr on the shape of the solid top surface becomes more evident when Pr increases from 1.0 to 10.0 as shown in Fig. 10b. The temporal variation of the ratio of the solid area to the cylinder area, as shown in Fig. 10c, shows that an increase in Pr from 0.01 to 10.0 has a very minor effect on the growth rate of solidification, and the evolution of the solidification interface. Fig. 11a. Natural convection tends to perturb the liquid-gas interface. Accordingly, for Ra = 10 3 the buoyancy force, however, does not dominate over the surface tension force, and thus the liquid-gas is still flat, as shown in Fig. 11a and b. However, increasing Ra to 510 5 the buoyancy force becomes stronger, and thus deforms the liquid-gas interface. As a result, the top surface of the solidified product for 510 5 is not flat anymore, as shown in Fig. 11b. In addition, the strong circulation at the high value of Ra enhances heat transfer in the liquid phase, and thus the solidifying interface propagates faster. However, this effect with these values of the parameters is minor. This effect is clearly shown in Fig. 11c in which increasing Ra in the range of 10 3 to 510 5 has a very minor effect on the area ratio and evolution of the solid phase. is normalized by U c ...
Context 5
... phase change materials yield different growth angles, e.g., in the range of 0 - 28 0 as reported by Satunkin [27]. For instance, the growth angles of water, silicon and germanium are 0 0 , 12 0 and 14 0 [27]. Accordingly, we vary the growth angle in the range of 0 - 15 0 to examine the effect of the triple point on the form of the solid. positive slope, and the liquid-gas front with a negative slope near the triple point (Fig. 8a). As a result, at the center the level of the liquid phase is higher for gr = 15 0 as compared with that for gr = 0 0 . Fig. 8b shows the solid phase shape at the nearly final stage of solidification with gr = 0 0 , 5 0 , 10 0 , and 15 0 . A zero growth angle gr = 0 0 yields a solid phase with a flat surface at the top, i.e., zero slope (the solid line). As the growth angle increases, the magnitude of the slope of the solid top surface near the center increases as shown in Fig. 8b. However, the growth angle has only a minor effect on the solidification rate, and thus the temporal variation of the area ratio of the solid to the cylinder As/Ac is affected slightly by the growth angle, as shown in Fig. 8c. 9a shows the velocity field at time = 0.12 for two Stefan numbers, St = 0.01 (left) and 1.0 (right) with the presence of volume expansion, i.e., sl = 0.9. Circulations exist in the liquid phase because of natural convection. Fig. 9a also indicates that at this time, the solidification interface for St = 1.0 has advanced much farther than for St = 0.01. It can be understood from the equation for the normal velocity of the solidifying interface. Increasing St corresponding to reducing the latent heat released increases the growth rate of the solidification front, and thus reduces the time for completing solidification, as shown in Fig. 9c. Accordingly, the liquid-gas interface in the case of St = 1.0 has gone higher than for St = 0.01 ( Fig. 9a) because of the presence of volume expansion. In addition, decreasing the Stefan number with volume expansion results in steeper solid interfaces at the center as shown in Fig. 9b. Fig. 10a). For a higher Pr, i.e., Pr = 10, the circulation becomes much stronger and slightly deforms liquid-gas interfaces, as shown in the right frame of Fig. 10a and in b. This indicates that increasing the Prandtl number leads to a more deformed surface at the top of the solid phase (Fig. 10b). However, the effect of Pr on the shape of the solid top surface becomes more evident when Pr increases from 1.0 to 10.0 as shown in Fig. 10b. The temporal variation of the ratio of the solid area to the cylinder area, as shown in Fig. 10c, shows that an increase in Pr from 0.01 to 10.0 has a very minor effect on the growth rate of solidification, and the evolution of the solidification interface. Fig. 11a. Natural convection tends to perturb the liquid-gas interface. Accordingly, for Ra = 10 3 the buoyancy force, however, does not dominate over the surface tension force, and thus the liquid-gas is still flat, as shown in Fig. 11a and b. However, increasing Ra to 510 5 the buoyancy force becomes stronger, and thus deforms the liquid-gas interface. As a result, the top surface of the solidified product for 510 5 is not flat anymore, as shown in Fig. 11b. In addition, the strong circulation at the high value of Ra enhances heat transfer in the liquid phase, and thus the solidifying interface propagates faster. However, this effect with these values of the parameters is minor. This effect is clearly shown in Fig. 11c in which increasing Ra in the range of 10 3 to 510 5 has a very minor effect on the area ratio and evolution of the solid phase. is normalized by U c ...
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Citations
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... Accordingly, filling this gap is the main purpose of the present study because of its importance in academic and engineered applications [42][43][44][45]. We here use a two-dimensional front-tracking method, for tracking interfaces, combined with an interpolation technique, for dealing with the non-slip boundary on the solid surface, [25,[46][47][48] to investigate the deformation with the solidification of a liquid drop attached to a vertical cold wall. Various parameters are investigated to reveal their effects on the solidification process. ...
... The numerical method used in the present study has been extensively used in our previous works. The validations of the method were also carried out [25,[46][47][48]51]. However, to ease reference, we here present the solidification result of a drop sessile on the plate and compare 10 with the experiments of Zhang et al. [16]. ...
... Other validation can be found elsewhere, e.g. [25,[46][47][48]. ...
Solidification of a liquid drop on a vertical wall is a typical phase change heat transfer problem that exists widely in natural and engineering situations. In this study, we present the fully resolved two-dimensional simulations of such a problem by a front-tracking/finite difference method. Because of gravity, the liquid drop assumed stick to the cold wall shifts to the bottom during solidification. Numerical results show that the conical tip at the solidified drop top is still available with the presence of volume expansion, but the tip location is shifted downward, resulting in an asymmetric drop after complete solidification. The tip shift, height and shape of the solidified drop are investigated under the influences of various parameters such as the Prandtl number Pr, the Stefan number St, the Bond number, the Ohnesorge number Oh, and the density ratio of the solid to liquid phases ρsl. We also consider the effects of the growth angle ϕgr (at the triple point) and the initial drop shape (in terms of the contact angle ϕ0) on the solidification process. The most influent parameter is Bo whose increase in the range of 0.1–3.16 makes the drop more deformed with the tip shift linearly increasing with Bo. The tip shift also increases with an increase in ϕ0. However, increasing Oh (from 0.001 to 0.316), St (from 0.01 to 1.0) or ρsl (from 0.8 to 1.2) leads to a decrease in the tip shift. Concerning time for completing solidification, an increase in Bo, ϕgr or ϕ0, or a decrease in St, or ρsl results in an increase in the solidification time. The effects of these parameters on the drop height are also investigated
... The growth angle at the trijunction (i.e. triple point) at which three phases meet [12,13] is defined as ...
... In the present study, we present a direct numerical simulation of a water drop that is sessile on or pendant from a cold plate, and solidifies. The methods used are the front tracking technique to represent the interface and an interpolation technique to deal with the no-slip boundary conditions [16][17][18][19]. We focus on the effects of the Bond number with various initial water drop shapes, i.e., various contact angles. ...
... This confirms and supports the accuracy of the method used in the present study. Other validations can be found elsewhere (e.g., [16][17][18][19]21]). at = 0.4 (Fig. 3b), the gravity force has already balanced with the surface tension force acting on the liquid-gas front, and thus no downward flow appears within the liquid phase. ...
This paper presents a direct numerical work on the freezing process of a water drop that is either sessile on or pendant from a cold plate.
The numerical technique used is an axisymmetric front-tracking method to represent interfaces that separate different phases. The sessile
drop corresponds to positive Bond numbers Bo (i.e., Bo > 0), and the pendant drop represents the other values of Bo. Numerical results
show that pendant drops break up into liquid drops when gravity dominates the force induced by surface tension at Bo < 0. That is, a
decrease in Bo enhances the breakup of the freezing drop. The breakup also depends significantly on the initial shape of the drop in terms
of the contact angle at the plate f0, that is, increasing f0 induces breakup. In addition, the drop rapidly completes freezing due to breakup.
In the case of non-breakup, the increase in Bo reduces the solidified drop height and decreases the time to complete solidification. The
freezing process also consumes minimal time with small f0. The solidified drop after solidification has a cone near the axis of symmetry
due to volume expansion of water upon solidification. This shape of the solidified drop is in accordance with the experimental observation.
