Ice Melting Driven by Natural Convection in The

Ice melting driven by natural convection in the

vicinity of the density maximum

G.Vieira' , S. L. Braga* & D. Gobiir* 'Mechanic Engineering Department - UCP, Brazil

'Materials and Mechanic Engineering Department - IME, Brazil ^Mechanic Engineering Department - PUC, Brazil 7608 - CNRS - Paris VI, France

Abstract

A numerical investigation is presented concerning the melting problem of a

vertical ice layer in the presence of horizontal temperature gradients. The main heat transfer mechanism is due to natural convection in water near the density maximum. The solution procedure uses a front tracking method which allows us to solve the fluid flow problem and the interface motion separately.

1 Introduction

The phenomena involving solid-liquid phase change are associated with many practical applications of current engineering interest and also in geophysics. Among them, there are, for example, energy storage systems using phase change materials (Stampa and Braga [1]), thermal environmental control, crystal growth processes (Ostrach [2]) and freezing of soil in cold weather regions (Zhang and

Nguyen [3]). Many simulation codes have been designed to solve these problems. The first numerical study regarding natural convection involving phase change of a pure substance and considering the finitevolum e method was done by Sparrow et al. [4]. A purely numerical comparison exercise may be found in Bertrand et al. [5], which compares different numerical procedures applied to a simple

melting problem. The main mathematical difficulty comes from the non- linearity due to the moving of the liquid-solid interface and natural convection effect. Due to extra difficulty introduced by the presence of a density maximum in the temperature range of interest, the problem of ice melting has received much less attention, except in the case of the horizontal layer, where the complexity of the stability problem has justified a number of studies. The literature concerning

heat transfer during the melting process of ice in a confined space appears to be mostly restricted to the cylindrical geometry [6-7]. This kind of problems find many applications such as ice thermal storage techniques for the air-conditioning, production of ice, oceanographic studies dealing with iceberg melting. Fluids like water, tellurium, liquid bismuth, antimony and gallium exhibit a maximum

density near their freezing points. Then, the usual linear approximation of the temperature effect on density is not valid near this region, which becomes the problem more complex. Numerical and experimental studies may be found regarding natural thermal convection with maximum density (Lin and Nansteel [8]; Bennacer et al [9]). When also accounting the influence of the phase

change, the need of more specific testing is clear. Schutz and Beer [10] developed a study, in which they analyzed the fusion of ice in salt water, involving thermosolutal natural convection. Braga and Viskanta [11] and Kowalewsky and Rebow [12] conducted works considering solidification of water in a maximum

density environment inside rectangular cavities. Tsai et al [13] studied the maximum density effect on laminar water pipe flows solidification. This work is motivated by the need to gain a more complete understanding of the heat transfer process during solid-liquid phase change in connection with natural convection and a density maximum. Particularly, it is analyzed the

melting of a vertical ice slab upon a gravitational field.

2 Mathematical modeling

The process being studied occurs inside a rectangular cavity (height //, width L, Fig. 1) containing a pure substance at its fusion temperature, To=T^. Initially, half the material's volume is in a solid state, while the other half is in a liquid state. The solid material is isothermal to the fusion temperature while one of the vertical walls of the cavity is heated. Therefore, the entire process is controlled

by natural convection during the liquid phase. At /*=#, the system receives energy through the vertical wall at the liquid side, heated at T//>7}^. The opposite wall is maintained isothermal at TQ. Horizontal walls are adiabatic. The position of the melting front is defined by a space and time function, c* (z*, t*). This relation is equal to the horizontal distance between the heated wall and the

melting front at height z*. Since the governing equations will be nondimensionalized later, the asterisk superior index (*) has been used to indicate the dimensional variables. LJ

adiabalie surface Liquid Solid Phase Phase

H A//) ^

adiabatic surface Figure 1 - Problem definition, t* =0.

The hypothesis below were assumed: - The flow is laminar and two-dimensional;

- The liquid material is Newtonian and incompressible; - The fluid's physical properties are constant, except for density in the buoyancy force term; - The viscous dissipation is negligible;

- The density change of the material upon melting is neglected; - It is assumed that the velocity of propagation of the melting front is several orders of magnitude smaller than the fluid velocities in the boundary layers on the vertical walls. This suggests that it is possible to divide the process in a number of quasi-static steps, separating, therefore, the melting front motion calculations from the natural convective calculations. Based on the above hypothesis, the governing equations used in the liquid domain can be written in their dimensionless form as following:

V*V=0 (1)

— V^8 (3) Pr

The equations shown above were included in their dimensionless form using as reference length, height //of the cavity and the cinematic viscosity u for velocity and time. Therefore: y = y*/H, z = z*/H, r=7*H/u, t = t*»/H*,

P = P*v/H^ e c(z,t) = c* (z* ,t* )jH , in which V is the dimensionless velocity vector; P is the dimensionless pressure; p^y is the reference density

(equal to the average density of the period imposed by wall temperatures) and k is the unitary vector in the vertical direction. The dimensionless temperature is defined by 6 = (r - 7^,]/Ar, where the average temperature given by

Tw=(Tf] + Tpuj)/2 , T is the dimensional temperature and AT" = 7// - Tp^ . The Prandtl number is given by Pr = u / a , being a the thermal diffusivity. At the moving interface, the energy balance equation is:

V6#»=— (4) di

dc where — represents the local velocity of the melting front along /?, the normal dr vector to the interface and i =Ste x Fo with Stefan number given by

Ste - CpATY Lp , being Cp the specific heat and Lp the latent heat. The boundary conditions were specified aty and z. Along z = 0:

0 (5)

Advanced Computational Methods in Heat Transfer VI, C.A. Brebbia & B. Sunden (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-818-X 230 Advanced Computational Methods in Heat Transfer VI w and v being the vertical and horizontal components of the dimensionless velocity vector V. Along the top horizontal wall, where z= 1:

dQ(y,l)/dz = 0; w(y,l) = 0; v(y,l)=0 (6)

Along the left vertical wall, wherey=0:

Q(0,z) = 0,5; w(0,z) = 0; v(0,z) = Q (7)

Along the melting front, where >>=c(z,f):

Q(c(z,t),z) = —0,5; w(c(z,t),z) = 0; v(c(z,t),z) = 0 (8)

Density approximation proposed in the buoyancy term. For a temperature range below 10°C, the approximation proposed by Gebhart and Mollendorf [14] presents good results and is given by the equation:

,/,\

(9)

where y is the phenomenological coefficient; for water: y = 8 x Iff* °C* and q=2; Pref is the maximum fluid density, also denominated p^; T^ is the

temperature of the maximum density, also denominated TM , for water TM=3,98 °C. m this case, the number of Grashof appears modified and is based on the cavity height and on the maximum temperature interval, given by:

g.^.Y.(A^)' - 3- ^ ' i)

The modified Rayleigh number is:

hi order to understand the meaning of the maximum temperature interval

A7^,we should consider that the variation effect of p is approximately symmetric in relation to the maximum density. The formulation, which refers to the maximum density, predicts a bicellular flow. On the other hand, the linear

approximation predicts a unicellular flow. Therefore, as the density variation is approximately symmetric in relation to TM, the greatest cell will generally appear at the side where the TM temperature is of greatest importance. Consequently, the smallest cell will be situated at the wall side where the temperature is nearest to T^. Therefore, the relative density variation, which induces the flow in each of the cells, is directly linked to the intervals between TM and

Advanced Computational Methods in Heat Transfer VI, C.A. Brebbia & B. Sunden (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-818-X Advanced Computational Methods in Heat Transfer VI 23 1 the wall temperatures given by:

TH-TM (12)

TM-TO (is)

As heat transfer at the greatest cell dominates energy transfer at the cavity, the Grashof and Rayleigh number were defined, modified based on A7^ , as:

(14)

The coordinates were transformed with the objective of mapping the irregular space occupied by the liquid in a rectangular computational space. This work uses the same Manseur [15] coordinate transformation method, which was adapted to the phase change problem. The system of coordinates adopted is related to the Cartesian system as shown below:

Z=z e y = ),/C(Z; (15)

where C(Z) = c* (z*)/L, being L the maximum liquid cavity width. Z and Fare the computational dimensionless coordinates.

3 Solution Procedure

The numeric method used in the present work was successfully compared to other researchers' results in the case of materials that do not show maximum density through a benchmark proposed by Gobin and Le Quere (Bertrand et al. [16]). Some results involving water were compared with results from an experimental investigation conducted by Vieira [17] and a good agreement was obtained. The solution procedure is based on the classical quasi-steady and quasi-stationary hypothesis. Within each quasi-static step, the melting front position is fixed for calculation of the steady-state natural convection flow. At the end of this calculation, the shape and motion of the melting front are recalculated before beginning the calculations of natural convection flow in the newly defined melted region for the next quasi-static step. An algebraic coordinate transformation is used, as already mentioned before, to solve the non-linear equations system, eqns. (1,2 and 3) that govern the natural convection in the irregular liquid cavity. The transformed equations are discretized on a rectangular computational domain, using the hybrid scheme (Patankar [18]). The algorithm SIMPLE is used to solve the pressure-velocity coupling. The solution of the discretized equations is obtained with an ADI procedure. The grid defined on the computational domain is irregularly spaced in order to provide better resolution of the temperature and velocity gradients at the solid walls. Among the different grids tested, the 42 X 42 was chosen for the presentation of the results, based on the commitment between result precision and computational time. The distribution of grid nodes chosen is regular according to the vertical direction and geometrical

according to the horizontal direction. Table 1 shows the test cases performed.

Table 1. Numerical Tests

TEST 7X°C) AT i (°C) J7L«(°C) Ste Pr Grm,(ay.7Q)

u (&y.72) (&7.J4 Gr(Eq.l6) / 8 4 4 0,1 10,71 GW 5,69x10* II 8 0,1 10,71 Gr: 2,84x10*

III 16 12 12 %2 10,00 Gr^t: 5,88xltf

For the test II it was used the usual linear approximation of the temperature effect on density. Therefore, in this case it appears the conventional number of Grashof, given by:

&„ = *'** f'^ (16)

The values of the employed physical properties are defined in Table 2. These values were obtained through the work of Gebhart et al. [19] and also based on average temperature of each temperature interval considered. The cavity considered in the testing has H=L=0,1 m. For the acceleration of gravity g, it was used 7 0,0m//.

Table 2. Thermophysical Properties

Tff=8°C 7),=760C

PROPERTIES 7W°c 3^=0*C U =^7^ WC =>7^=^"C Thermal conductivity: k (W/m°K) 0,6 0,6

^ecz/fc /%%#. C,, (J/Kg°K) 420J 47P7

Kinematic viscosity: v (nf/s) l,5xlO~* 7,4*70-* Density: p (Kg/nf) P%P2 PPP,#0

Latent heat: Z^(J/Kg) j,j%70^ J,Jjc70^ Thermal diffusivity: a(nf/s) 7,4*70^ A4%70-7 AT(°C) 8 16

4 Results and Discussion

According to Lin and Nansteel [8], the fact that the density of water attains a maximum value is considered significant, specially in the following range of temperatures: 0 < TC < 7// < 20 °C. Therefore, in this study the temperature was

maintained within this range. The results show (Tests I and III) how the ice fusion process with water in presence of natural convection is affected by the increase of the heated wall temperature, while the cold wall is maintained at 0 °C. It is interesting to observe how the size and orientation of the convective

cells is directly related to the relation between the wall temperature and the maximum density temperature, 7^ Normally, when flow is bicellular, the largest cell appears at the side where the temperature interval in relation to TM is mostly important The correspondent temperature to the maximum density is

located at the isothermal that separates both cells. The interface format is directly related to the direction of the dominating cell. The velocity of the melting front is greater at the lower part of the cavity, when the isothermal values of the dominating cell are smaller than T^ The opposite happens when the isothermal values of the dominating cells are greater than TM- In this case, the melting front has its velocity increased in the upper part of the cavity. In the case of wall heated at 8°C, the influence of the maximum density is more evident. As expected, the counterclockwise cell is located near the cold wall and the clockwise cell is

located near the heated wall. So as time goes by, the ice layer becomes thinner at the lower region of the cavity. In the case of wall heated at 16°C, the main cell is located near the heated wall and follows a clockwise pattern. A smaller and counterclockwise cell is found at the lower part of the cavity, next to the solid- liquid interface. This was predicted as observed in Table 1, for the case:

ATwv=ATj =12°C. Therefore, AT2 =4°C. This means that the smallest cell appears next to the heated wall where the temperature related to TM is mostly important. Regarding the interface, an opposite behavior to the previous case cited above may be observed. In this case, the fluid layer near the interface achieves its greatest temperature at the upper part of the cavity. This heated fluid, normal to the interface, is responsible for the accelerated fusion of ice at this point.

(a) T=0( t*= 0 mm) (b) r = 0,01000 (t*= 120 min) (c) T = 0,01500 (t*= 180 min)

Figure 2 - Progression of flow structure and temperature distribution for AT=8°C, considering the phenomenon of density maximum (Test I)

234 Advanced Computational Methods in Heat Transfer VI

BIT

(a; r = 0 (/*= 0 min) (b) T =0,01000 (/*= 120 min) (c) r - 0,01500 (f*s 180 min)

Figure 3 -Progression of flow structure and temperature distribution for AT=8°C, without considering the phenomenon of density maximum (Test II)

(a) r = 0 (/*= 0 min) (b) T = 0,00400 (t*= 24 min) (c) T = 0,00600 (t*= 36 min)

Figure 4 — Progression of flow structure and temperature distribution for AT=16°C, considering the phenomenon of density maximum (Test III) The Tests I and II, presented in Figs. 2 and 3, respectively, confirm the idea that

in the neighbourhood of the maximum density, a linear variation of density with respect to temperature is no acceptable (Fig. 3).

5 Conclusion

The results show that the non-linear variation of density in the buoyancy term affects the heat transfer, the movement of the interface and the flow structure. Such phenomenon creates a complex structure of two contra-rotating cells and

shall be considered when predicting the solid-liquid interface movement. For this reason, the usual linear approximation of the temperature effect on density, used in conventional analysis, must be replaced by another density equation of state.

Acknowledgement

The first author wishes to acknowledge support for this study by the Research Aid Foundation of The State of Rio de Janeiro, Brazil (Funda$ao de Amparo a Pesquisa do Estado do Rio de Janeiro -FAPERJ) under research grant E-26/151.276/98.

References

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