Simulation of Block Ice Formation With Varying Brine Temperatures

8/8/2019 Simulation of Block Ice Formation With Varying Brine Temperatures

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Simulation of Block Ice Formation with Varying Brine Temperatures

Arsanchai Sukkuea and Kuntinee Maneeratana*

Department of Mechanical Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok 10330 Thailand

Tel: 0-2218-6610, Fax: 0-2252-2889, E-mail: [email protected], [email protected]

Abstract

This paper simulates the formation of block ice with real

brine temperature changes by the finite volume method. The

mathematical model is based upon the explicit heat conduction

equation with the fixed gridand latent heat source approaches.

With hourly brine temperature measurements, 3 different types of

variations linear interpolation, constant average and step are

considered in the 1D and 3D simulations. The main results are

temperature profiles, ice/water fraction and internal energy loss.

The test cases with linear interpolation show that results have

similar overall characteristics to the simulation with constant brine

temperature, but with distinctive temperature variations in the

frozen regions. When the brine temperature variation is changed

to the step approximation, the results differ slightly from the linear

interpolation and may possibly be used for further approximation.

1. Introduction

Ice factories produce block ice for consumption, fishing and

frozen food businesses. Such factories consume huge amount of

electricity in the manufacturing process. A factory in Samut

Sakhon Province can be considered as a typical large ice

manufacturer in Thailand. The block ice section contains two

14.5172-m brine pools, each containing 2,600 ice moulds

(Figure 1). Ice blocks, weighting around 150-160 kg each, are

ready for sale in 40 to 70-hours cycles. As the factory sells ice

blocks to customers sporadically, substantial savings can be

made by increasing equipment efficiencies as well as optimizing

operating conditions to reduce ice oversupply and electricity cost

[1].

In order to effectively control operating conditions, it is

crucial to accurately predict the ice formation within the moulds.

Even though there are many parameters affecting the rates of

block ice formation, such as the brine temperature, brine level,

size of block ice as well as losses, etc. The first obvious choice

for operation control is the brine temperature which only involves

the refrigeration system, does not affect the product sizes and

requires relatively low investment and development.

The factory operates 20 hours every day from 0:00-20:00 hr.

During this period, evaporator piping and fans keep the brine

water temperature within 2 to 12C range (Figure 2). From the

measurements at two opposite ends of a pool, denoted Tb1 and

Tb2, it was found that the brine temperatures were almost uniform,

especially when compared to the average brine temperatures Tb.

It was also observed that the brine temperature increased

significantly outside operating hours.

Figure 1 An ice factory floor (left) and a set of ice moulds (right).

time t, hr

0 24 48 72 96 120

brine

tempera

ture,

C

-12

-10

-8

-6

-4

-2

0

2

Tb

TbT

b1T

bT

b2

Figure 2 Hourly brine temperatures during 1-5 Oct 2004 period

[2]. The Tb is averaged from measurements Tb1 and Tb2 at two

opposite pool ends.

A cost effective and quick method of predicting the ice

formation rate is the numerical computation. The final objective of

this research is to develop a cheap computational program that


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can predict the ice formation with sufficient accuracy, leading to a

better plant control for energy saving, increased efficiency and

profitability [1].

During solidification, the liquid/solid front, which releases

massive latent heat, continuously moves through the domain.

The ice formation, characterised by isothermal phase change

under constant freezing temperature and abrupt property

discontinuity, is highly non-linear and exact solutions of the

mathematical models are extremely difficult to obtain. Thus, the

numerical simulations are popularly employed instead.

The overall literature survey on such numerical simulation

was provided in [3]. The numerical procedures can be

categorized as combinations of two main models, grid and latent

heat representations. The grid consideration may be further

divided into front tracking and fixed grid approaches while the

latent heat is represented by either the temperature-based or the

enthalpy-based methods. By comparing combinations of these

approaches for the finite volume (FV) simulation in a previous

study [3], the fixed grid and the modified fictitious heat schemes

is chosen such that the latent heat increment is calculated from

the fictitious temperature in the freezing region and then the

temperature fields are adjusted.

In addition, appropriate transient and interface

approximations must be selected. From available numerical

techniques, 4 temporal schemes explicit, fully implicit, Crank-

Nicolson and pseudo-implicit [4] and 3 interface conductivity

approximations arithmetic, harmonic means and solid

conductivity are considered and used for 1D and 2D test cases

[3]. The papers conclude that the best practical choices are the

explicit temporal scheme for least computational expense and the

solid interface approximation, which slightly overestimates the

conductivity as the freezing front progresses across the saturated

cells.

This paper expands the previous studies [3-4] using real-life

brine temperatures for 1D and 3D test cases. As the steel mould

is made from 1.5 to 5.5-mm-thick steel plate of which the heat

conductivity is much higher than ice, the brine temperature is

used as the prescribed temperature of the domain.

In addition, it was found in a previous study [4] that the

highly non-linear, transient 3D simulations uses up a lot of

memory and CPU as well as takes a very long time such that

they usually took roughly 2 days to run the program on a

personal computer. Thus, real-time, on-site simulations for each

block ice are not economically appropriate. Besides, there is no

need to predict the formation rate at an extremely high accuracy

as some extra supply availability are built into the plant control in

case of unexpected customer demands. Thus, the second

objective of this paper is to study the simulation results in order to

identifying a simple brine temperature variation that can be used

in the rough estimation of ice formation rates by tabulated data

for on-site control.

2. Mathematical Model and Simulation

The energy conservation and Fouriers law of heat

conduction are employed as the mathematical model.

( )i i

H Tkt x x

=

, (1)

where H, t, T, kand xiare enthalpy, time, temperature, thermal

conductivity and coordinates. The enthalpy His calculated from:

if ,ref

T

S FT

H c dT T T =


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where ri is the position vector and the superscript denotes the

location of the property. The gradient vector ( )Pix at cell Pis

calculated by ensuring a least square fit of through P and

neighbouring nodes Qfas:

3 31 1

( )( )

( ) ( )

ff f P Q f nb nbi jP

f ff fi

d d d

d x d

= =

=

, (5)

where = ff Q P

i i id r r is the distance vector between Pand Q

f. For

the conductivity k at cell faces, the solid value Sk are used as

recommended in [3-4]. The diffusion flux through the face f into

an adjacent node Qf

is approximated using the orthogonal

correction method as:

( ) ( )f fQ P

f f f f f i ii f f f

i i

S dT T T T k dS k S

x d x S d

+

. (6)

With the explicit scheme, the equation (3) becomes:

,0

0 ,0

1 1

( )( ) ( ) ( )( )

fP n n

P Q f

i if f i

cV S T S T T k T k S d

t d x d

= =

. (7)

By assembling equation (7) for all cells with initial and boundary

conditions, a system of simultaneous equations =[ ] [ ] [ ] A T b is

formed with nodal temperature [ ]T as unknowns.

Before a new time step, the phase status of a cell is

checked. If the node is liquid and the nodal temperature PT

drops lower than the freezing temperature FT , it becomes

saturated. The node is tagged and the temperature is then

reassigned to FT . The latent heat incrementPQ is calculated

from the fictit ious sensible heat such that ( )P P PFQ c T T V = .

TheP

Q is added to the accumulated latent heatP

Q for

subsequent time steps until the PQ equals the total latent heat

PLV of the control volume. At this stage, the control volume

becomes solid, the tag on the cell is removed and the latent heat

increment is no longer calculated (Figure 4).

Input data

Data initilisation

Start time increment loop

Start solution loop

Check phase status of each nodeand specify appropriate properties

Formulate & solve the energy equation

For saturated nodes, calculated latent heatfrom fictitious sensible heat

Prescribed number of time step is reached

stop

NO

Figure 4 Explicit solution algorithm.

3. 1D Case Study

A 0.135-m-long domain of water (shaded area in Figure 5)

with unit square cross section is initially at temperature

= 40 Ci

T . Then, the boundary temperature at = 0.135 mx is

suddenly lowered to the brine temperature Tb (Figure 2) while the

other boundary condition at = 0x is symmetry plane. The

freezing occurs at 0 CFT = with 338 kJ/kgL = while other

properties are shown in Table 1. The domain is discretised into

50 uniform cells and = 1 st which satisfy both the explicit time

step restriction of < 2( ) /2t c x k and the 1-cell-deep freezing

front requirement [3].

Table 1 Material properties of water and ice [3].

Property Water Ice

(W/m K)k 0.556 2.220

(kJ/kg K)c 4.226 1.762

3(kg/m ) 1000 1000*

*Approximate to the water value to ensure mass conservation

bT

= 40 Ci

T

0.27 m

x

symmetry plane

bT

a monitoring node

Figure 5 1D problem descriptions.

As the brine temperatures are measured on the hour and the

hourly data is used in the plant control [1], the appropriate

estimation of brine temperatures during the hour must be first

considered.

3.1 Linear Interpolation of Brine Temperature

With linear interpolation of hourly brine temperatures in

Figure 2, the boundary condition can be considered a fairly close

approximation of the real brine temperature and can be used as

the base case. It takes 77.3 hours before all water are frozen.

Figure 6 shows the temperature profiles at various time instants

while Figure 7 displays the temperature changes with time at

monitoring nodes distributed throughout the domain (Figure 5),

the estimated ice thickness from the calculated latent energy [4]

and associated rate as well as the internal energy and cooling

load. The energy loss at an instant time t is obtained from the

difference between the current and initial internal energy values.

The results exhibit many characteristics of the simulation

with constant brine temperature [4]. The rate of heat transfer is

predominantly controlled by the position of the freezing front.

Liquid cells cool down slowly; once a control volume is frozen, its

temperature drops rapidly such that the temperature gradient in

the ice is almost linear and the freezing of the next cell starts

shortly afterwards (Figure 6). The freezing front advances at a

slowing rate and exhibits similar characteristics to the decreasing

cooling load. The rate of ice formation or thickness changes is

still exhibit the cyclic numerical errors from the fixed grid scheme

because the front has to wait for a short interval before a new


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node starts to freeze as the front progresses from one control

volume to the next [4].

Due to the varying values of the brine temperature, nodal

temperatures fluctuate with the brine temperature. From Figure 7,

this fluctuation can be seen to diffuse deeply into the domain.

The temperature variation is particularly pronounced in the frozen

region but is reduced with increasing distances from the

boundary. This variation also triggers some small amount of heat

transfer into the domain through the boundary when the brine

temperature increases. In short, the brine temperature cycles can

be observed in both the ice thickness and energy plots.

positionx, m

0.000 0.025 0.050 0.075 0.100 0.125

tempe

ratureT

,C

-10

0

10

20

30

401 hr

10 hr

20 hr

40 hr

80 hr

Figure 6 1D case study 1: Temperature distributions.

tempera

tureT

,C

-10

0

10

20

30

40

increasingx

ice

thicknesss,

m

0.000

0.025

0.050

0.075

0.100

0.125

ch

angera

teds/dt,mm

/hr

-5

0

5

10

15

20

25

time t, hr

0 20 40 60 80

energy

lossU

,MJ/m

2

0

20

40

60

80

coo

ling

loa

ddU/dt,kW/m

2

0

2

4

6

8

thickness

thickness

change rate

energy loss

cooling load

Figure 7 1D case study 1: Temperatures at monitored nodes

(Figure 5), ice thickness and energy plots.

3.2 Constant Average Brine Temperature

As it takes 77.3 hours for fully freezing all water when the

brine temperature variation is linearly interpolated, the averaged

brine temperature over 78 hours of -5.996C is used for the next

comparison (Figure 8). When the average brine temperature is

used, only the result characteristics of problems with constant

boundary temperature [4] are obtained with many differences due

to the brine temperature variations. The time taken for the water

to be fully frozen is slightly shorter at 76.8 hours. The difference

in ice thickness is quite large when compared with results from

the interpolated brine temperature simulation. The difference in

energy loss is also significant.

These differences make the use of average brine

temperature an inappropriate approximation even though its use

would greatly simplify the plant controls. Thus the step brine

temperature is considered next.

tempera

tureT

,C

-10

0

10

20

30

40

increasingx

thickness

difference,

%

-20

0

20

40

60

80

100

ice

thickness

differe

nce,

mm

-6

-4

-2

0

2

4

6

time t, hr

0 20 40 60 80

los

sdifference,

%

-5

0

5

10

15

energyloss

difference,

MJ/m

2

-2.0

-1.0

0.0

1.0

2.0

savg

sinterp

100(savg

sinterp

)/sinterp

100(Uavg

Uinterp

)/Uinterp

Uavg

Uinterp

Figure 8 1D case study 2: Temperatures at monitored nodes as

well as ice thickness and energy plots as compared against the

linear interpolation case study 1.

3.3 Step Brine Temperature

If the step values, in which the brine temperature is kept

constant during the hour and jumps to the next measured value,

are used instead of the linear interpolation of hourly brine

temperatures, the results such as temperature distributions and

ice thickness, etc. are still similar but with more stepping


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shapes (Figure 9). It takes 77.5 hours to fully freeze all water.

The difference in ice thickness is small when compared with the

results from interpolated brine temperature and fall within the -

0.592 0.900 mm range with the average and SD values of -

0.010 mm and 0.310 mm. If the percentage difference in ice

thickness is considered, the difference is within the range of -6.87

1.614% but with the average and SD values of -0.115% and

0.761% during the 80-hours period. The differences in internal

energy losses are also acceptable when compared with the

results from interpolated brine temperature. The differences fall

within the -0.230 0.309 MJ/m2

range with the average and SD

values of -0.014 MJ/m2

and 0.117 MJ/m2. If the percentage

difference in loss internal energy is considered, the difference is

within the range of -2.862 0.757% but with the average and SD

values of -0.080% and 0.383% only.

tempera

tureT

,C

-10

0

10

20

30

40

increasingx

thickness

diffe

rence,

%

-6

-4

-2

0

2

4

ice

thickness

difference,

mm

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

time t, hr

0 20 40 60 80

loss

difference,

%

-3

-2

-1

0

1

energy

loss

difference,

MJ/m

2

-0.50

-0.25

0.00

0.25

0.50

sstep

sinterp

100(sstep

sinterp

)/sinterp

100(Ustep

Uinterp

)/Uinterp

Ustep

Uinterp

Figure 9 1D case study 3: Temperatures at monitored nodes as

well as ice thickness and energy plots as compared against the

linear interpolation case study 1.

4. 3D Case Study

The developed program [4] is employed to study the freezing

process in industrial ice block manufacturing with the actual size

of ice block shown in Figure 10a. Initially, temperature of water is

= 40 CiT throughout. The boundary conditions on the top end

is assumed to be = 0 CbT . Due to symmetry, only one-fourth of

the total area is modelled. The discretised domain consists of 26

13 120 cells while 10 st = which satisfy the combined cell

size/time step restrictions [4].

4.1 Linear Interpolation of Brine Temperature

With linear interpolation of hourly brine temperatures in

Figure 2, the boundary condition can be considered a close

approximation of the real brine temperature and used as the

reference case. It takes just under 89.5 hours before all water are

frozen.

0.52 m

1.2 m

0.27 m0.56 m

a

b

c

d

e

a) block geometry b) monitored cells

Figure 10 Geometry and domain of ice block.

Selected locations in the ice block are monitored (Figure

10b) and the temperatures and ice fractions at these cells are

shown in Figure 11. It is noted that the values at cells b and care

similar with only small delays for c, indicating that the side

freezing fronts dominates the solidification process while the

freezing front from the bottom exerts much less influences.

Although the uppermost-centre node d experiences a sharp

temperature drop early on, it still remains unfrozen until the last

moments due to the fact that the ambient temperature is not

sufficiently low to properly induce an effective freezing process.

Coupled with the fact that it takes more than 10 hours longer for

the domain to be fully frozen when compared to the 1D test case,

the importance of the ambient temperature on the determination

of overall freezing duration is clearly demonstrated.

For overall results, the loss of internal energy and fraction of

ice in the block are displayed. The characteristic fluctuations of

results with the brine temperature variations are clearly observed.

4.2 Average Brine Temperature

As previous, the averaged brine temperature over 90 hours

of -6.46C is used for the next comparison. It takes just under

88.5 hours before all water are frozen in keeping with the trend in

1D simulations. As in the 1D case, the differences in water

fraction and energy loss are quite large when compared with the

results from interpolated brine temperature changes as shown in

Figure 12.


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te

mpera

tureT

,C

-10

0

10

20

30

40

pt a

ce

llwa

ter

frac

tion

0.00

0.25

0.50

0.75

1.00

time t, hr

0 20 40 60 80

totalenergy

lossU

,M

J

0

5

10

15

20

25

totalwa

ter

frac

tionf

0.0

0.2

0.4

0.6

0.8

1.0

energy loss

water fraction

pt e

pt b

pt d

pt e

pt a pt b

pt d

pt c

pt c

Figure 11 3D case study 1: Temperature & ice fraction at

monitoringcells as well as total energy loss and water fraction.

te

mpera

tureT

,C

-10

0

10

20

30

40

pt a

ce

llwa

ter

frac

tion

difference,

%

0

20

40

60

80

100

time t, hr

0 20 40 60 80

energy

loss

difference,

MJ

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

totalfrac

tion

difference

,%

-6

-4

-2

0

2

4

6

8

pt e

pt b

pt d

pt e

pt a

pt b pt d

pt c

pt c

100(favg

finterp

)

Uavg

Uinterp

100(favg

finterp

)



4.3 Step Brine Temperature

If the step values are used instead of the linear interpolation

of hourly brine temperatures, the overall results are quite similar

even though some differences in some individual cells are

comparatively larger than others (Figure 13). The difference in

total water fraction is small when compared with the results from

interpolated brine temperature such that the differences fall within

the -0.531 0.678% range with the average and SD values of -

0.001% and 0.221% or in the order of 1/10th compared to the

constant brine temperature simulation (Figure 12). For the totalinternal energy loss, the differences fall within the -0.106 0.093

MJ range with the average and SD of -0.002 MJ and 0.038 MJ,

respectively.

5. Conclusion

The ice formation simulation with real hourly brine

temperatures by the finite volume method with the fixed grid and

the latent heat by fictitious sensible heat schemes is successfully

performed for 1D and 3D test cases. Three different temperature

estimations between the hours are considered the linear

interpolation, constant average and step values. The linear

interpolation best emulates the real changes but requires more

detailed analyses for on-site plant control. The average brine

temperature, while is able to predict the overall trend, can not

capture the variations within the domain during the freezing

period. Hence, it is not suitable for further uses in the real

optimisation of plant operating conditions. The results from the

step approximation differ slightly from those of linear interpolation

and can probably be used for the existing hourly plant control [1]

instead.

The future works includes the analyses of simulated data for

better ice formation approximation which is in immediate demandfor the plant control as the current approximation method by

tabulate energy loss data [1] was found the overestimate the

number of ready-for-sell ice blocks by some 200 to 400 blocks

out of the total number of 2600 [5]. Later, the programs should be

used to study effects of various parameters, such as the brine

and ambient temperatures, brine level, mould thickness and sizes

on the ice formation rate so that a more efficient plant control and

design can be formulated and obtained.


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te

mpera

tureT

,C

-10

0

10

20

30

40

pt a

ce

llwa

ter

frac

tion

difference,

%

-10.0

-7.5

-5.0

-2.5

0.0

2.5

5.0

time t, hr

0 20 40 60 80

energy

loss

difference,

MJ

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

totalfrac

tion

difference

,%

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

pt e

pt b

pt d

pt e

pt apt b

pt d

pt c

pt c

100(fstep

finterp

)

UstepUinterp

100(fstep

finterp

)



6. Acknowledgements

Special thanks are due to Dr. Naebboon Hoonchareon and

Miss Thitima Lertpiya from the Department of Electrical

Engineering, Chulalongkorn University as well as the Siam

Scholars Co., Ltd., Bangkok.

References

1. Lertpiya, T. Energy Management System for Block-Ice

Factory using TOD and TOU Tariff, M. Eng. Thesis,

Department of Electrical Engineering, Faculty of

Engineering, Chulalongkorn University, 2005.

2. Hoonchareon, N. and Lertpiya, T. Private Communication,

2004.

3. Prapainop, R. and Maneeratana, K. Simulation of ice

formation by the finite volume method, Songklanakarin

Journal of Science and Technology, vol. 26, no. 1, pp. 55-

70, 2004.

4. Meneeratana, K. Simulation of ice formation by the

unstructured finite volume method, Proceedings of the 1st

E-NETT Conference. Ambassador City Jomtien, Chonburi,

code ECB02, pp. 211-216, 11-13 May 2005.

5. Hoonchareon, N. Private Communication, 2007.

Documents

Simulation of Block Ice Formation With Varying Brine Temperatures