analysis of freezing of viscous liquid in cylindrical containers

8/14/2019 Analysis of Freezing of Viscous Liquid in Cylindrical Containers

1/25

International Journal of Dynamics of Fluids

ISSN 0973-1784 Vol.2, No.1 (2006), pp. -

Research India Publications

http://www.ripublication.com/ijdf.htm

An Analysis of Freezing of Viscous Liquids in

Cylindrical Containers, Part A: Effects of Natural

Convection

K.K. Rathore1

and S. Srinivasa Murthy*2

Refrigeration and Air - Conditioning Laboratory

Department of Mechanical Engineering

Indian Institute of Technology Madras

Chennai - 600 036 (INDIA)

Abstract

A Numerical analysis of freezing of viscous liquids including the presenceof a mushy region between the solid and liquid phases is presented. The

conservation equations of continuity, momentum, and energy are solved usingthe enthalpy - porosity method. Iso - liquid fraction lines in the mushy region,streamlines and isotherms are obtained. Two cases, namely "conduction -

only", and "conduction - natural convection" are studied. The solid and liquid

interface movements with time are traced. Results show that heat transfer

during the early periods of freezing is dominated by conduction in the solid

formed close to the wall. Natural convection exerts greater influence on the

morphology of the liquid front whereas the solid front is affected slightly.

Key words: Freezing, Natural Convection, Conduction, Mushy Regio.

IntroductionTransient "moving boundary" heat transfer problems are encountered in many

engineering applications such as freezing of foods, solidification of metals, thermal

energy storage, aerodynamic ablation, cryosurgery etc. In the freezing or

solidification of pure substances like water, the phase change takes place at a discrete

temperature, and the solid and liquid phases are separated by a sharp movinginterface. On the other hand, in the case of mixtures, alloys and impure materials,

1Current Address: Sr. Engineer, Advanced Engineering, ASHOK LEYLAND LTD., Chennai-600035

E-mail: [email protected]

*2

Corresponding author: E-mail: [email protected]


2/25

2 K.K. Rathore and S. Srinivasa Murthy

solidification occurs over an extended temperature range, as a result of which, a two-

phase moving region called "mushy" region, separates the solid and liquid phases.

Unlike freezing of pure substances, viscous liquids like, jams, jellies, sugar syrups,

chocolate melts, ice cream mixtures, etc do not exhibit a distinct front separating solid

and liquid phases. Instead, solid is first formed as a permeable, fluid saturatedcrystalline like matrix which co -exists with the liquid phase. The presence of this

mushy region makes a double moving boundary problem, because the freezingprocess is now accompanied by two moving interfaces, i.e. the solidified front and the

liquid front. Also, there are three moving region, i.e. frozen region, mushy region and

liquid region. It is generally recognized that the dynamics of such phase change

process are to a large extent influenced by natural convection.

Many investigators have presented mathematical models for predicting the time-

temperature histories, freezing rates and freezing time of food products. Duda, et al[1] presented a technique for the analysis of unsteady, two-dimensional diffusive heat

or mass transfer problem characterized by moving irregular phase change boundaries.The technique includes an immobilization transformation and a numerical scheme for

the solution. Voller and Cross [2] gave a solution for the moving boundary problemusing the enthalpy method for both two-dimensional and one-dimensional freezing

using explicit and implicit finite difference methods. Gau and Viskanta [3] studied the

role of natural convection on solid liquid phase front during melting and

solidification of Lipowitz metal alloy inside a rectangular cavity. Experiments were

performed for solidification as well as melting. Measured temperature distribution and

fluctuation gave a qualitative indication of natural convection flow regime inside the

cavity. They found that suppression of natural convection in the metal leads to slowermelting rate and higher freezing rate. Voller et.al [4] developed an enthalpyformulation for convection-diffusion phase change problem. As an example they

implemented the method to two test problems i.e. solidification in a cavity under

conduction, and phase change under conduction and natural convection using the

control volume numerical discretization. Voller and Prakash [5] developed a fixed

grid numerical modeling methodology for convection diffusion, mushy region-

phase change problem. The basic feature of this method lies in the representation of

latent heat of evolution, and of flow in the solid-liquid mushy zone. Linear

relationship was used between enthalpy change and temperature. Motion in the mushy

region was simulated using the Darcy law. They also demonstrated an application of

this method through a test problem of freezing in a thermal cavity under naturalconvection. Viskanta [6] made a review of the significance of heat transfer during

melting and solidification of metals and alloys. He stressed that the fundamental

macroscopic transport process in the solid and the melt mainly influence the solid-

liquid interface shape and motion, and not the external heat exchange. The important

role played by buoyancy driven fluid flow was also discussed. Bennon and Incropera

[7] presented a numerical solution to the binary solid-liquid phase change problem

through a continuum model. The solution of the multiconstituent, multiphase problem

was reduced to that of a coupled single-phase problem. Sample calculations were

performed for static solidification in Cartesian co-ordinates to illustrate the

convenience of the solution methodology and to emphasize the significance of bulk


3/25

Effects of Natural Convection 3

and mushy region fluid motion for macroscopic phase change behavior. Numerical

calculation was performed for an aqueous ammonium chloride solution in a

rectangular cavity. A comparative study was made between the diffusion-dominated

binary solidification and binary solidification with advection. Based on the results

they concluded that advection had a significant influence on phase change behavior inbinary systems. Even though growth of the solid layer was comparable for both the

cases, the morphology of the liquidus, temperature and liquid compositions differeddramatically. This confirmed the inability of diffusion-dominated models to describe

binary phase change problems. Cao and Poulikakos [8] performed an experimental

study of solidification of binary mixtures (NH4Cl H20) in a rectangular cavity

cooled from the top wall and observed the complex flow, heat and mass transfer

phenomena. These phenomena drastically affect the solidification process and,

therefore the growth rate and structure of the resulting solid. Samarskii et.al [9] madea review of numerical techniques for the solution of heat and mass transfer problems

with solid/liquid phase change. Tan and Leong [10] studied the effect of walltemperature and aspect ratio on the solid liquid interface during solidification inside

a rectangular enclosure. Three different types of n paraffins namely, n octadecane,n heptadecane & n hexadecane were used. They found that using an enclosure

with a higher aspect ratio produces a flatter phase front compared to that of lower

aspect ratio. A qualitative analysis revealed that the natural convection in the liquid of

lower aspect ratio had a stronger influence on the phase front. Also, they found that a

lower wall temperature resulted in higher solidification rate and the phase front had a

more prominent curve at the top of the test cell. Giangi et.al [11] presented numerical

and experimental studies on unsteady natural convection during freezing of water in adifferentially heated cube shaped cavity. Numerical methods with boundary fitted gridas well as the enthalpy porosity fixed grid were used. Both numerical models

showed good agreement with the experimental data for pure convection during initial

periods while the discrepancies between numerical predictions and the experiments

became significant at higher times.

The literature reveals that conduction heat transfer has been widely considered

during freezing/solidification studies, and the effect of natural convection has been

usually neglected. The presence of a partially frozen mushy region is generally

ignored. The variation of thermo physical properties (thermal conductivity &

specific heat) across the mushy region due to the variation in liquid fraction is

important but often neglected. This paper presents a rigorous analysis of the freezingprocess taking into account the above aspects. The thermal characteristics of freezing

in conduction convection mode is compared with conduction only. A liquid

food model, Tylose gel (23% methyl cellulose & 77% water) is taken as an example.

Formulation of ProblemThe physical model is shown in Fig 1. Initially the viscous liquid in the cylindrical

container is at temperature Ti, which is above its freezing point. The container is

instantaneously exposed to a temperature lower than the freezing temperature. As

freezing proceeds, three distinct regions are formed; a solid region, a mushy region


4/25

4 K.K. Rathore and S. Srinivasa Murthy

consisting of solid crystals dispersed in the liquid, and a liquid region. Natural

convection currents are set both in the liquid region and mushy region. The frozen

region grows continuously till all the liquid is solidified. The mushy region grows

initially and later diminishes as the frozen region encroaches into it. The liquid region

diminishes continuously till it vanishes completely.

Figure 1: Schematic of model showing physical feature of the problem with applied

thermal boundary condition.

The following assumptions are made to facilitate the problem formulation and

solution:

The fluid flow in the liquid region developed due to natural convection islaminar since the viscosity of the liquid is high and the temperature gradientsare low.

The liquid is Newtonian and incompressible. Thermal conductivity and specific heat are functions of temperature in the

mushy region, and are constant for solid and liquid regions. Across the mushy

region the values of thermo-physical properties at the solid and liquid

interfaces are equal to those of solid and liquid respectively. In the absence of

exact variation, and for simplicity, linear variations are assumed.

7 K H G H Q V L W \ Y D U L D W L R Q G X H W R W K H S K D V H F K D Q J H L V Q H J O L J L E O H L H l = s, as there

is negligible expansion due to freezing.


5/25

Effects of Natural Convection 5

Flow in the mushy region follows the Darcy law. Since, it has been found inliterature [4] that Darcy source technique for velocity correction has greater

physical significance and also realistic.Darcy law states that flow in a porous medium is proportional to the

pressure gradient i.e.

( ) ( )= a Ku grad P (1): K H U H W K H S H U P H D E L O L W \ . L V D I X Q F W L R Q R I S R U R V L W \

Z K L F K L V H T X D O W R

the elemental liquid fraction. As the porosity decreases the permeability

and the superficial velocity (ua) also decrease down to a limiting value of

zero when the mush becomes completely solid

The flow is axi-symmetric i.e. occurs only along the r and z directions.

The Boussinesq approximation for natural convection flow is applicablesince the variation in density with respect to the reference density is small.

The cylindrical container is fully filled with no vapor phase present.

The enthalpy porosity technique [5] is used for formulation of the problem. In

this technique, the liquid interface is not tracked explicitly, instead the fraction of the

cell volume that is in the liquid form, is associated with each cell in the domain. The

liquid fraction is computed at each iteration based on an enthalpy balance.

The enthalpy of the material is computed as the sum of the sensible enthalpy h and

latent heat + L H

H = h + +

Where,

ref

T

ref P

T

h h C dT = + (3)

and,

refh is the reference enthalpy

refT is the reference temperature

PC is the specific heat at constant pressure

In order to establish a mushy phase change, the latent heat contribution is specified as

a function of temperature,( )

H f T

=. The nature of the latent heat evolution in the

mushy region is dependent on the local liquid fraction( ) temperature relationship.In the current work a simple linear form is chosen. The liquid fraction, is definedas,

0

1

S

L

S

S L

L S

for T T

for T T

T Tfor T T T

T T

analysis of freezing of viscous liquid in cylindrical containers

Documents