Phase Change in Heat Conduction Problems:

Solidification or Melting Front

 

Atha Khan

Department of Mechanical Engineering, Michigan State University.

ME 812 Conductive Heat Transfer: Midterm Project

 

 

Introduction

 

Heat transfer analysis is more often than not an integral part of any engineering problem. Thermal stresses form a major design liability when present. Heat transfer through conduction, convection and radiation places additional constraints on engineers for designing a device. While these three modes of heat transfer occur simultaneously in most of the natural processes, it is often the case that one mode of heat transfer is significantly dominant over the others. In such situations, the other modes of heat transfer are often ignored. However, the presence of such secondary modes of heat transfer does alter the behavior of a process and therefore leads to uncertainties and anomalies.

 

The present project aims at studying the phase change phenomenon in heat conduction problems. Phase change is an isothermal process in case of pure substances. Ice melting into water at 0 0C after absorbing the latent heat is a common example that can be cited to illustrate phase change. The other examples would be boiling and condensation. In this project, the focus will be on solidification and melting aspect of phase change.

 

Understanding Melting and Solidification

 

When a liquid in a vessel is exposed on the top to a temperature lower than the freezing temperature, the liquid on the top layer will solidify first and slowly the thickness of the solid layer will increase with time. Heat is conducted through the solid region and then convected at the interface. The solid region and the liquid region have completely different thermal properties. If the temperature of the liquid is higher than the freezing temperature it has to be cooled to the freezing temperature by the removal of sensible heat. Any subsequent cooling would initiate the phase change process – freezing or solidification. Similarly if a solid at the freezing temperature is exposed to higher temperature, the solid will begin to melt and if the liquid is not removed then heat has to be conducted through liquid layer and then convected at the interface. [1]

 

Focus of Current Study

 

Phase change problems are encountered in everyday life. These set of problems form some of the most interesting problems in engineering, ecology, medicine etc. The melting of polar ice caps and variation in coast lines has been a subject of interest for environmentalists in the past few decades [2]. Solidification of enzyme and tissue proteins into small capsules has captivated the attention of medical researchers for many years. In engineering there is an endless list of applications where the concept of heat transfer with phase change is observed and studied. These problems include formation of ice on aircraft wings, the casting process in industries, purification of metals [3], study of geophysical phenomena (fusion of glaciers and volcanic eruptions) [4], cooling of electronic equipments [5] and thermal control of space stations and vehicles [6], the freezing of water and other chemicals in pipes etc.

 

This project will mostly deal with problems that come from the manufacturing side of the engineering field, in particular the casting technique. The variety of processes involved in casting and manufacturing in general, provide an excellent opportunity to educate oneself about the phase change process and its importance. Investment casting, continuous casting, injection molding, all involve a phase change which plays an important part in determining the properties of the castings. The rate of cooling is considered to be the most important parameters that influence the strength of the castings. The molten metal when cast into die has to solidify at a fixed rate to obtain the desired qualities. For example, a steel casting cooled too fast, with accelerated phase change or solidification will result in finer grains. (Grain and grain boundaries are terms involved with the study of microstructure of metals). Finer grains offer greater hardness but lesser strength, as the metal becomes brittle. On the other hand, slow rate of cooling ensures that any thermal stresses during the phase change process are relieved. The resulting microstructure consists of coarse grains which have greater strength.

 

When one considers any particular casting system, one can see that the flow is present from the early stages of the process. During the casting, flow generally occurs in the bulk liquid and in the semi-solid regions. Some of the techniques have been successfully used in academic research with the aim of studying the fundamentals of solidification under forced-flow conditions: gravity flow-through systems, mechanical stirring, centrifugal casting, application of a magnetic or electromagnetic field creating the Lorenz force, etc. Computational modeling and simulation are being widely used in the last two decades as cost-saving tools for the prediction and interpretation of the results along with experimental methodologies. Using these two approaches in combination leads to a deeper understanding of the effects of melt flow as a result of natural and forced convection on the solidification phenomenon in metallic alloys, i.e., (1) the morphology of grains and their deflection toward incoming flow, (2) the columnar-to-equiaxed transition and grain morphology, and (3) the change of segregation pattern. [7]

 

Turchin et al. analyzed the solidification of an alloy in a shallow cavity under conditions of forced flow both by fluid-dynamics simulations with solidification included and by experiments. The variation in bulk-flow velocity and initial superheat dramatically changes the macro and microstructure, promoting grain refinement, an equiaxed-to-columnar transition (ECT), the formation of peculiar grain and dendrite morphologies, etc. The solidification parameters during solidification in the shallow cavity under forced-flow conditions were determined by computer simulations and partially compared with the experimental results. The interaction between flow vortices and the progressing solidification front and its effect on structure evolution were also analyzed. They reported an increase in the coarsening constant in the well-known power law, due to the flow velocity under conditions of identical thermal gradient and initial superheat.

 

The use of ‘chills’ is a common practice in the manufacturing industry when directional solidification is desired. ‘Chills’ are pieces of metals of different shapes and sizes which have lower melting point than the bulk of the metal. Because of their lower melting point, the chills melt and in the process absorb the latent heat from the bulk of the metal. This tends to cool down the molten metal resulting in directional solidification.

 

Mahmoudi and Vynnycky developed a finite-volume model and applied it to compute the fluid flow, temperature and heat flux distribution within the liquid and solid zones in the mould region of a pure copper continuous strip casting machine. A fixed solidification front between solid and liquid was assumed. The model was produced with the commercial program CFX, which allows this non-linear, highly turbulent problem to be simulated using the k-e turbulence model. The effect of casting speed on heat transfer was investigated. The predicted temperature distribution and heat fluxes were compared with experimental measurements, and reasonable agreement was obtained.

 

They observed that an analysis of the heat flux plots for the liquid/ solid and solid/mould interfaces can assist in predicting the location of the true solidification front. The mathematical model was found to be capable of reproducing the temperature-distance curves obtained from experimental work in a continuous strip casting process which is an indication of feasibility for the planned use of Fluent commercial code in this project. 5. The heat transfer coefficient was found to be a parameter of major importance in analyzing the shape and location of the solidification front.  [8]

 

Fig 1: Variation of total heat flux with casting velocity

 
 

 

 

 


Often times when the surface tension of a fluid is strongly dependent on temperature, a gradient in the surface stresses is observed in the fluid. The Marangoni effect (sometimes also called the Gibbs-Marangoni effect) is the mass transfer on, or in, a liquid layer due to surface tension differences. Since a liquid with a high surface tension pulls more strongly on the surrounding liquid than one with a low surface tension, the presence of a gradient in surface tension will naturally cause the liquid to flow away from regions of low surface tension. The surface tension gradient can in turn be caused by concentration gradient .Under earth conditions the effect of gravity inducing density driven convection in a system with a temperature gradient along a fluid/fluid interface is usually much stronger than the Marangoni effect. The effect is also visible (but to a lesser degree) in windshield washer fluid on a windshield, which is also a water/alcohol mix. [9]

 

 

Nota et al. studied the behavior of drops in non-isothermal liquids with solidification front experimentally and numerically. In presence of a temperature gradient, the surface tension driven flows at the drop liquid interface affect the drop/solidification front interaction. Preliminary experiments were carried out to measure the interfacial tension of the liquid–liquid system under investigation at different temperatures and to study the fundamental behavior of a single drop close to the melting front, in presence of a temperature gradient. It was found that heating the system from above and cooling from below, the melting front exhibits a shape deformation as a consequence of the surface tension driven convection around the drop.

 

They also investigated the interaction between free droplets and a melting front. It was observed that when the relative distance between the droplets and the melting front is of the order of the drop radius, the droplets engulfment in the solid is prevented by a layer of liquid between the drops and the solidification front. This experimental evidence supports the assumption that a “pushing force” is induced by Marangoni effect when the droplets approach the solidification front. [10]

 

 

    Fig 2: Computed pressure distribution in presence of
    Marangoni motions: (a) far from the solidification front;
    (b) near the solidification front.

 

 
 

 

 

 

 

 


Melting is the reverse process of solidification wherein the heat is absorbed. The heat transfer at the melting front of a sliding ice bar against a heated wall was studied by Dominic and Marcel. Close contact melting occurs when a solid melts while being in contact with a heat source. The liquid generated at the melting front is squeezed out from under the solid by the pressure maintained in the central section of the film by the weight of the free solid. If the heat source moves relative to the melting solid, the liquid generated at the melting front is also dragged out from under the solid by the action of the moving heat source. The problem of close contact melting has been the subject of a number of investigations related to the fundamentals of heat transfer. Close contact melting is primarily studied because the heat transferred across the melt layer separating the heat surface from the solid phase change material (PCM) is much higher than the heat transferred by convection, which generally occurs in much thicker layers of molten material. As a consequence of the higher heat fluxes, the melting time is considerably reduced.

In most of the previous studies on close contact melting, the process by which the melt is squeezed out of the small gap separating the heat source and the solid was assumed to be quasi-steady and the heat transfer through the liquid film was considered to be conduction dominated. Recent studies suggest however that this last assumption may no longer be valid when a relative motion between the solid PCM and the heat source is imposed. [11]

 

Illustrations

 

Solidification and melting can be analyzed using basic principles of heat transfer taught in the class. Consider the following cases:

 

a) Melting of a solid at freezing temperature when exposed to a higher temperature at the surface

                                                                                                    

h

 

T

 

Fig 3

 
The time taken for a solid to melt can be calculated as follows:

 

To derive these equations the concept of thermal resistance is invoked and solution procedure discussed in the class is used.

 

Heat flow per unit area for a liquid layer x units thick:

 

                                                                                                             (1)

For layer dx to melt in time dt, the energy required per unit area is:

 

q = ρs * hsf * (dx/dt)                                                                                                    (2)

 

where, ρs is density of solid, Tf is the freezing temperature, kl is the thermal conductivity of liquid and hsf is the heat of fusion.

 

Equating and rearranging,

 

[h (TTf) / ρs  hsf]* dt = [1+ (x h / kl)] dx                                                             (3)

 

Introduce,

 

x` = (x h / kl) and t` = [h2 (TTf) / ρs  hsf kl] * t

 

Substituting in eq (3) we get,

 

t` = (1+x`) dx`                                                                                                             (4)

 

The RHS of eq (3) changes to

 

x` = (x h / kl)

dx` = dx (h / kl)

[1+ (x h / kl)] dx  = (kl/ h)[1+x`]dx`

ð      

The LHS of eq (3) changes to:

 

(kl/ h) t`

 

Equation (4) can be integrated to get,

 

 t` = x` + 0.5 x`2                                                                                                                                               (5)

 

Equation (5) gives the time required for a given depth to melt.

 

 

b) Solidification of liquid at freezing temperature when it is exposed to colder temperature at the surface

 

This case can be formulated in the same manner as the previous case. The kl is replaced by ks which is the thermal conductivity of the solid.

 

c) Solidification of liquid when it is above the freezing temperature

 

In the previous cases it was assumed that the substance is at freezing temperature. This was done to perform a simple analysis neglecting the effect of sensible heat and only considering the latent heat of fusion.

 

However, a more practical approach would be to analyze the solidification of a liquid above the freezing temperature. In this case the liquid is first cooled to freezing temperature by removing sensible heat and solidified by removing the latent heat.

 

 

 

Tl

 

Tf

 
In this case convection is encountered at the liquid – solid interface also.

 

Heat removed for cooling the liquid: hc (TlTf)                                                       (1)

 

Heat removed for freezing: q = ρs * hsf * (dx/dt)                                                        (2)

 

The total heat should pass through the solid layer and to the outside:

 

{(Tf - T)/[(x/ks) + (1/ h)]} = [(TlTf)/[(1/ hl)] + ρs * hsf * (dx/dt)                              (3)

Introducing,

T` = (TlTf)/ (Tf - T), x` = (x h / ks), t` = [h2 (Tf  - T) / ρs  hsf ks] * t, H`=( hl/ h)

Equation (3) reduces to

1/(x`+1) = H`T` + dx`/dt`

Rearranging and integrating from 0 to t we get

t` = (1/H`T`)2 ln [(1-H`T`)/(1-H`T`(1+x`))] –( x`/H`T`)                                               (4)

Equation (4) gives the time required for the liquid to get frozen upto a certain depth.

 

These cases can be used as a basis for analysis of solidification of metals in casting process. A continuous casting process is intended to be modeled in Fluent CFD Package.

 

 

 

References

 

1) C.P. Kothandaraman, Heat and Mass Transfer, Text book, 1992.

 

2) C. Gordon, C. Cooper, C. A. Senior, H. Banks,J. M. Gregory, T. C. Johns , J. F. B. Mitchell, R. A. Wood, The simulation of SST, sea ice extents and ocean heat transports in a version of the Hadley Centre coupled model without flux adjustments, Climate Dynamics (2000) 16:147±168.

 

3) J.M. Khodadadi, Y. Zhang, Effects of buoyancy-driven convection on melting within spherical containers, Int. J. Heat Mass Transfer 44 (2001) 1605–1618.

 

4) M.K. Moallemi, B.W. Webb, R. Viskanta, An experimental and analytical study of close contact melting, J. Heat Transfer 108 (1986) 894–899.

 

5) I. Sezai, A.A. Mohamad, Natural convection in a rectangular cavity heated from below and cooled from the top as well as the sides, Phys. Fluids ,12 (2) (2000) 432–443.

 

6) M. Ibrahim, P. Sokolov, T. Kerslake, C. Tolbert, Experimental and computational investigations of phase change Thermal energy storage canisters, J. Sol. Energy Eng,122 (2000) 176–182.

 

7) A.N. Turchin, D.G. Eskin, and l. Katgerman, Solidification under Forced-Flow Conditions in a Shallow Cavity, The Minerals, Metals & Materials Society and ASM International 2007

 

8) J. Mahmoudi and M. Vynnycky, Modelling of fluid flow, heat transfer and solidification in

the strip casting of copper base alloy(II). Heat transfer, Scandinavian Journal of Metallurgy 2001; 30: 30–40

 

9) Takaji Kuroda, The Marangoni Effect and Its Artistic Application, Society for Promotion of Space Science, 203–204, 2000

 

10) F. Nota, R. Savino, S. Fico, The interaction between drops and solidification front in presence of Marangoni effect, Acta Astronautica 59 (2006) 20 – 31

 

11) Dominic Groulx, Marcel Lacroix, Study of close contact melting of ice from a sliding heated flat plate, International Journal of Heat and Mass Transfer 49 (2006) 4407–4416