An experimental and theoretical study of solidification in a free-convection flow inside a vertical annular enclosure

International Journal of Heat and Mass Transfer 55 (2012) 655–664

Contents lists available at SciVerse ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

An experimental and theoretical study of solidification in a free-convection flow inside a vertical annular enclosure Z. Lipnicki a, B. Weigand b,⇑ a b

Institute of Environmental Engineering, University of Zielona Góra, 65-516 Zielona Góra, Poland Institut für Thermodynamik der Luft- und Raumfahrt, Universität Stuttgart, Pfaffenwaldring 31, 70569 Stuttgart, Germany

a r t i c l e

i n f o

Article history: Received 28 June 2011 Received in revised form 18 October 2011 Accepted 21 October 2011 Available online 22 November 2011 Keywords: Solidification PCM Free convection Vertical annular enclosure Experiments

a b s t r a c t The natural convection and solidification in an annular enclosure has been studied experimentally and theoretically. Here the inner cylinder of the annular enclosure was cooled below the solidification temperature of the water, while the outer cylinder was kept on a uniform temperature well above 0 °C. The problem of the unsteady growth of the ice-layer on the inner cold cylindrical surface is studied theoretically and an approximate solution has been obtained for the quasi-steady development of the ice-layer thickness. In addition, it has been found that the influence of the contact layer between the frozen layer and the cold surface is of significant importance for the solidification process. Results are presented and compared between the experimental and the analytical investigation. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The storage of heat energy based on phase transition (solidification or melting) is a very important process, which can be utilized in many practical applications. The main advantage of this method is the relatively small size of the heat storage accumulator and a stable temperature of the phase transition. A phase change material (PCM), which is used in heat accumulators, is a substance with a high energy of transformation (latent heat) associated with storing and releasing large amounts of heat. The problem is that PCMs are normally poor heat conductors and therefore affect the rate of heat flow i.e. the heat flow from the surface of the PCM to the surrounding. High thermal resistance increases the duration of the heat exchange process and it is therefore appropriate to evaluate the influence of the external geometry of the PCM on the phenomenon of solidification or melting of these materials. Thus, the freezing of liquids inside a vertical annular channel with a free convection flow is of technical importance for these type of applications. Laminar and turbulent free convection flows in a vertical channel without solidification were investigated in a lot of research papers e.g. in [1] for laminar flow and in [2,3] for turbulent flow. A good literature review on the free convection in vertical spaces can be found in [4]. Benard convection between two horizontal ori⇑ Corresponding author. Tel.: +49 711 685 63590. E-mail address: [email protected] (B. Weigand). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.10.044

ented flat plates, heated from below and cooled from above, with the dynamics of solidification was investigated in [5]. On the other side, many theoretical and experimental studies have been performed for free convection flow and heat transfer with internal solidification. A good overview of the existing literature in this field is provided in [6–9]. However, no experimental study is known in literature which investigates the time-dependent growth of an ice-layer subjected to free convection flow inside a vertical annular enclosure for which one wall is cooled and the other wall is kept on a constant temperature, higher than the fusion temperature of the liquid. The study of the problem of solidification in a vertical annular enclosure should also take into account the thermal resistance of the contact layer between the cooled inner wall and the solidified layer, because the thin contact layer will influence dramatically the freezing process. The analysis of the role of the contact layer in the solidification process was recently studied experimentally and theoretically by Loulou et al. [10–12], Lipnicki [13], and Lipnicki et al. [14]. This research shows that the role of the contact layer on the solidification process is of great importance. Therefore, the influence of the contact layer between the cooled wall and the frozen crust on the solidification process is one of the focus in this paper.

2. Formulation and solution of the problem Fig. 1 shows a schematic sketch of the problem under investigation. The object of the present investigation is to study the

656

Z. Lipnicki, B. Weigand / International Journal of Heat and Mass Transfer 55 (2012) 655–664

Nomenclature a Bi B cp Fo Grw k H h L Nu Pr q_ r, z R RO ~r

heat diffusivity (m2/s) Biot number, hCON R=kS (–) cooling parameter (–) specific heat at constant pressure (J/(kg K)) Fourier number, at=R2 (–) Grasshof number based on w, gbðT O  T F Þw3 =m2 (–) heat conductivity (W/(mK)) height of the annular channel (m) 2 _ heat-transfer coefficient, q=ðT O  T F Þ (W/(m K)) latent heat (J/kg) Nusselt number based on R, hR=kL (–) Prandtl number, m=a (–) heat flux (W/m2) coordinates (m) radius of the inner cylinder (m) radius of the outer cylinder (m) dimensionless radial coordinate r/R (–)

time-dependent growth of the frozen crust inside a vertical annular enclosure with free convection flow inside. The study focuses mainly on the development of the frozen layer inside the annular channel. It is assumed that the liquid (see Fig. 1) with density q, viscosity m and specific heat cp has the average temperature Tb. The frozen layer with thickness D is formed on the inner cooled cylinder, which has the height H and the radius R. The coordinate system with the coordinate z, located on the central axis, and the radial coordinate r having its origin on the axis of symmetry of the cylinder. The constant temperature of the surface of the inner cylinder is equal to TW which is less than the fusion temperature TF of the liquid. The outer cylinder is kept at the const. temperature TO. There is a constant layer between the crust and the inner copper cylinder, which is called the contact layer. Across the contact layer, there is the temperature difference ðT  T W Þ; where T is the temperature of the inner surface of the frozen layer (see Fig. 1). As a result of the solidification process, the solidification interface grows from the cold surface with the velocity od/ot. In the following, we want to derive an approximation solution for

RO

g

δ Δ

R

r TO

TF TW

q m D  D d

s ~d

Rayleigh number based on w, Grw Pr (–) Stefan number, cp ðT O  T F Þ=L (–) time (s) temperature (K) liquid fusion temperature (K) temperature of the outer cylinder (K) wall temperature (K) annular gap width, RO  d (m) thermal expansion coefficient (1/K) density (kg/m3) kinematic viscosity (m2/s) frozen layer thickness (m) averaged thickness of the frozen layer (m) position of solidification front (m) dimensionless time, SteFo (–) dimensionless position of the solidification front, ðR þ DÞ=R (–)

the ice-layer thickness d by considering the energy equation at the interface

kL

  oT L  oT s  od þ ks ¼ qs L ;  ot or r¼d or r¼d

kL

 oT L  ¼ hðT F  T O Þ; or r¼d

liquid solid

 0:238  0:442 H RO w d

ð3Þ

where the Rayleigh number is based on the gap width w and on the driving temperature potential between outer cylinder and ice-layer interface

w

cold wall

ð2Þ

where h is the heat-transfer coefficient between the ice–liquid interface and the flowing water. TF is the interface temperature and TO is the temperature of the outer cylinder. Natural convection in vertical spaces has been studied both experimentally and numerically by a number of researchers in the past. A good literature review can be found in [4]. A lot of the investigations focused on rectangular geometries and also on heat flux boundary conditions at the walls. Some selected correlations for these cases can be found in [15]. For annular enclosures it had been found that the radius ratio between the outer and the inner cylinder has a strong effect on flow development in the enclosure and thus on the heat transfer coefficient. Furthermore, the thermal boundary condition affects the values of the Nusselt number. Kumar and Kalam [16] give a good review of the literature on this aspect and provide also correlations for the heat transfer coefficient. For the present investigation, the flow in the enclosure stays laminar and the radius ratio is smaller than 1.5. Therefore, we use for predicting the heat transfer coefficient the correlation by de Vahl Davis and Thomas [17].

Nuw ¼ 0:286Ra0:258 Pr 0:006 w

T.

ð1Þ

where L is the latent heat. In the annular enclosure free convection flow takes place. The free convection flow can be characterized by expressing the heat flux at the solid–liquid interface by a heat transfer coefficient

H q

contact layer

Raw Ste t T TF TO TW w b

solidification front

Fig. 1. Sketch of the geometry of the annular channel.

Raw ¼ Grw Pr ¼

gbðT O  T F Þw3

m2

Pr

ð4Þ

In the present case the annular channel width w = (RO  d) is changing with time and so are the values of the Rayleigh and the Nusselt

657

Z. Lipnicki, B. Weigand / International Journal of Heat and Mass Transfer 55 (2012) 655–664

number. For t = 0, w = (RO  R) and d ¼ 0; RO =R ¼ 1:5. From Eq. (3) one obtains h as

Nuw h¼ kL w

ð5Þ

hR Nu ¼ kL

0:012  0:442  0:442  0:238  RO R Ro d d :  H R R R R ð6Þ

From this equation one sees explicitly the dependence of the Nusselt number on the geometry of the annular enclosure. One also notices that the term ðRO =R  d=RÞ0:012 is extremely small for the here investigated radius ratios of the annular enclosure of 1.5 and might be neglected. In the following, only a model for predicting the circumferential and axial averaged ice-layer thickness will be developed. Such a model is very valuable for designers of latent heat storages, because the time dependent change in the averaged ice-layer thickness and thus, the time dependent release of heat can be predicted very easily. In the solid layer, the temperature can be calculated from the quasi-steady 1D energy equation and the boundary conditions given by

  1 o oT s r ¼ 0; r or or  r ¼ d : T ¼ TF : r ¼ R : T ¼ T;

~d ¼ d ; R

 oh o~r 

~r ¼d~

¼

1 ; ~d ln ~d

where : ~r ¼

Ste ¼

B¼

ks T F  T W ; kL T O  T F

Nu ¼

hR ; kL

Bi ¼

hCON R ; ks

s ¼ Ste Fo;

~d ¼ d ; R

h¼

T T : TF  T ð8Þ

In the following, we are only interested in the calculation of a circumferential and z-averaged value of the ice-layer thickness as a function of time. Special focus will be given to the influence of the thermal contact layer formed between the inner cooled copper cylinder and the ice-layer (see Fig. 1). The thermal contact layer occurs at the interface between the ice and the surface of the inner copper cylinder. The mechanisms of the development of the contact layer between a solidifying liquid are not known in detail. The influence of all the parameters like roughness, thickness, purity of the wall surface, surface topology,. . . on the thermal contact resistance are considered together. The work on the thermal resistance of the contact layer written by Loulou et al. [10–12] analyzes a cumulative effect of all parameters. Of course, a qualitative analysis is carried out. The contact layer is considered to be thin, because its thickness is much thinner compared to the thickness of the solidification layer. The heat flux through the contact layer can be defined by q_ ¼ hCON ðT  T W Þ, where hCON is the heat transfer coefficient for the contact layer. The existence of a contact layer introduces an additional high thermal contact resistance into the heat transfer process. The values of the heat transfer coefficient for the contact layer can only be determined experimentally. This will be discussed later in the paper. From Eqs. (1) and (8) one obtains

dd T F  T 1 þ hðT O  T F Þ ¼ ks ¼ hCON ðT  T W Þ: dt d ln Rd

ð9Þ

Fo ¼

at R2

a¼

;

ks ~ k;

qs cp

~ ¼ kL ; k ks

    dd~ 1 1 1 Nu þ 1 þ Nu ¼ B : ~d ln ~d ds Bi ~d ln ~d Bi

ð12Þ

In this equation, the Bi number represents the effect of the contact layer on the heat transfer, whereas Nu represents the effect of the free convection inside the liquid flow. Satisfying the initial condition

~d ¼ 1 for

s ¼ 0;

ð13Þ

results in the following integral for the ice-layer thickness

s¼

Z

~d

~d ln ~d þ 1=Bi d~d B  Nuð~d ln ~d þ 1=BiÞ

~d

1 ~d ln ~d þ 1=Bi ~ dd: B Nu ðNu  Bi1 Þ  ~d ln ~d

1

Z 1

ð7Þ

r ; R

cp ðT O  T F Þ ; L

into Eq. (9) results in

¼

The assumption of a quasi-steady behavior is justified here, because of the low values of the Stefan number Ste ¼ cp ðT O  T F Þ=L . Solving Eq. (7), the temperature distribution inside the crust and the temperature gradient at the interface can be obtained

qs L

ð10Þ

ð11Þ

¼ 0:286Gr0:258 Pr0:264 R

ln ~r ; ln ~d

q L dd h þ ðT O  T F Þ: T  T W ¼ s hCON dt hCON Introducing the following dimensionless quantities

We rearrange this equation in order to obtain the Nusselt number based on the radius of the inner cylinder. This results finally in

h¼

From Eq. (9) one obtains

ð14Þ

From the above equation it follows, that a necessary condition for the existence of the solidification process is to satisfy the algebraic inequality: B=Nu > 1=Bi . Because of the strong nonlinear dependence on ~ d, the above given integral in Eq. (14) cannot be solved analytically. However, it is straightforward to solve this integral numerically. Figs. 2 and 3 show the temperature distribution in the frozen crust for several non-dimensional times s as well as the development of the thickness of the frozen crust for selected parameters. Fig. 2 shows the shape of the temperature profile in the solid crust as well as the temperature change in the outer flow. It can be seen from Fig. 3 that the ice-layer continuously increases in thickness with time. For a higher value of the Nusselt number, a thinner ice-layer will be obtained, because of the increased heat transfer from the liquid to the solid–liquid interface. The development of the ice-layer thickness depends on the wall temperature, the mean fluid temperature (given by the parameter B) and the thermal contact resistance (given by the parameter 1/ Bi). This can be seen from Eq. (14). These effects are illustrated in

δ 0 . 05

θ

δ 0.02

1.2 1.0

θb

0.8

τ = 0.02

0.6

τ = 0.05

0.4 0.2 0

r 0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Fig. 2. Temperature distribution in the frozen crust for B = 8.18; Nu = 4.29; 1/ Bi = 0.22.

658

Z. Lipnicki, B. Weigand / International Journal of Heat and Mass Transfer 55 (2012) 655–664

a

b

δ 3

2

δ

δ 1.8

25

1.6

0.04

0.02

10

0.08

0.06

Nu=10 1.2

5 0

1.4

15

dδ dτ

0

Nu=8

Nu=4.29

20

asymptotic solution

1

dδ dτ

1.0

τ

τ 0.02

0

0.1

0.04

0.06

0.08

0.1

Fig. 3. Development of the frozen crust over time for (a) B = 8.18; 1/Bi = 0.22; Nu = 4.29 and (b) B = 8.18; 1/Bi = 0.22; Nu = 4.29/10.

a

b

δ

δ

1.3 1.2

1/Bi = 0.2

1/Bi = 0.3

1.1 1/Bi = 0.4

1.0 0.9 0.8 0

0.002

0.004

0.006

0.008

0.01

Fig. 4. Development of the frozen crust over time for (a) B = 7; Nu = 10 and (b) 1/Bi = 0.2; Nu = 10.

a

b δ

δ

1.3

1.025

1.25

1.02

1.2

1.015

Bi=const

Bi=const

1.15

1.01

1.1

Bi(τ) 1.005

1.05 Bi(τ)

1 0

1 0.005

0.01

0.015

0.02

0

0.0004

0.0008

Fig. 5. Frozen layer thickness obtained by various calculation methods for Bi0 = 7.2, Bi = 3.5, B = 7.8, Nu = 6.8, sste = 0.001; (a) – full time; (b) – in the beginning.

Fig. 4. Fig. 4(a) shows that for given values of the cooling parameter B and the Nusselt number Nu, the increase of the thermal contact resistance 1/Bi results in a decreasing ice-layer thickness. Fig. 4(b) shows that for fixed values of the thermal contact resistance 1/Bi and given values of the Nusselt number Nu, the thickness of solid layer grows with increasing values of B. 2.1. The start of the solidification process (asymptotic solution) At the beginning of the solidification process, one can assume that the frozen layer is very thin (D << 1; ~ d ! 1Þ: Thus, by neglecting the small logarithmic term in Eq. (14) one obtains

ds ¼

1 d~d: B Bi  Nu

ð15Þ

Assuming a constant Bi number, one can easily solve this equation analytically. This results, by satisfying the initial condition according to Eq. (13), in

~d ¼ 1 þ ðB Bi  NuÞs:

ð16Þ

This equation can be used to calculate the thickness of the frozen crust only for very small times. As shown in Fig. 3(a) the differences between the exact and the asymptotic solutions are strongly increasing with time. Fig. 3(a) and (b) shows an increase in the thickness of the frozen layer with time. The rate of increase of the frozen layer decreases with time and will reach eventually zero for larger times. The reason for this is that the thermal resistance of the solidified layer increases with time. One can note that the Bi number, which determines the properties of the contact layer, varies with time, especially at the

659

Z. Lipnicki, B. Weigand / International Journal of Heat and Mass Transfer 55 (2012) 655–664

measuring cylinder observation

Δh

overflow channel water

g test section (central unit)

Venturi nozzle

piezometer

ethylene glycol

water pump water

pump illumination water inlet

Venturi nozzle ethylene glycol

water

refrigeration unit

refrigeration unit

Fig. 6. Schematic drawing of the experimental apparatus.

coolant flow (ethylene glycol) observation

measuring cylinder

bolt with flow holes thermocouples

plexiglass

.

water

water flow

test section ice layer

ΔV

g

.

ice-layer

subsection 1 inner copper cylinder

g water

water flow

subsection 2

H illumination

subsection 3 coolant flow (ethylene glycol)

Fig. 7. Schematic drawing of the test section.

Δ

R

Fig. 9. Measurement method of the mean ice-layer thickness.

Fig. 8. Details of the annular channel (a) – the two copper cylinders; (b) – top view.

660

Z. Lipnicki, B. Weigand / International Journal of Heat and Mass Transfer 55 (2012) 655–664

Table 1 Summary of the experimental parameters. Run

Sub-section

I

1 2 3

II

TW (°C)

1/Bi

93.77 93.77 93.26

194,000 194,000 191,000

5.43 5.43 5.40

0.11 0.16 0.40

3.13 4.69 13.44

90.21 90.21 89.69

170,000 170,000 165,000

5.24 5.24 5.21

0.05 0.09 0.32

0.0135 0.0135 0.0135

16.63 17.34 17.87

93.34 93.34 93.34

191,000 191,000 191,000

5.40 5.40 5.40

0.35 0.40 0.42

0.0123

14.61

89.49

492,000

7.06

0.14

B

h

3.0 4.2 8.7

4.0 4.0 3.0

0.0123 0.0123 0.0092

5.84 8.18 22.64

1 2 3

1.0 1.5 4.3

2.5 2.5 2.5

0.0077 0.0077 0.0077

III

1 2 3

9.4 9.8 10.1

4.4 4.4 4.4

IV

–

7.5

4.0

beginning of the solidification process. This fact was stated in [14]. At the beginning of the solidification process we can approximate Bi according to the experimental results of Wang and Matthyes [18] by

sst

s  sst ;

ð17Þ

where sst is the time where a constant contact resistance is reached. Inserting Eq. (17) into Eq. (15) results in

    ps  Nu ds ¼ d~d; B Bi0 sin 0:85

ð18Þ

sst

which has to satisfy the initial condition (13). Thus for a variable Bi number one obtains the following asymptotic solution for small times

   ~d ¼ 1  Nus þ BBio sst 1  cos 0:85p s for sst 0:85p

s  sst



Nuw(t=0)

Ste

  ps for BiðsÞ ¼ Bi0 sin 0:85



Raw(t=0)

TO (°C)

ð19Þ

In the beginning, when the thermal resistance (1=Bi) changes with time, the growth of the frozen crust is slow (see Fig. 5). Than a strong increase of the frozen layer with time can be observed. The growth rate decreases for larger times as depicted in Fig. 3.

The temperature values expected are reported in Table 1. After the water and the ethylene glycol had reached their desired temperatures, the experiment was started. The water was coloured by Methylene blue in order to clearly observe the water-ice interface. Additionally, a neon light mounted below the channel was used to obtain a good contrast between the ice-layer and the water flow. During the run several photographs were taken at different times and evaluated afterwards. The temperature of the ethylene glycol, the water inlet and exit temperature as well as the wall temperature of the copper cylinders at several positions were monitored during the run. The inlet temperature of the water varied normally by not more than 0.1 °C whereas the inlet temperature of the ethylene glycol varied normally by not more than 0.2 °C during the experiment. Because the photographs were enlarged afterwards, the local ice-layer thickness could be measured with good accuracy of the order of less than 0.1 mm. The accuracy of the calibrated thermocouples was of the order of 0.1 °C. For measuring averaged values of the ice-layer thickness, the volume expansion DV of the water during phase change has been determined by a small tube mounted on top of the test section (see Fig. 9). The volume change in the small tube (measuring cylinder) was then used to recalculate the average ice-layer thickness (see Eq. (20)).

3. Experimental investigation

 ¼ R þ D 3.1. Experimental apparatus A schematic outline of the experimental apparatus is given in Fig. 6. The apparatus mainly consists of a test section, two refrigeration units and two circulation systems for water and for coolant. Fig. 6 shows a schematic drawing of the test section, while Fig. 8 shows two photographs of details of the test section. The test section (see also Figs. 6–8) was installed vertically. Between the two concentric vertical copper cylinders the enclosure creates the annular channel. The dimensions of the annular channel are as follows: outer diameter 2RO ¼ 200 mm, inner diameter 2R = 134 mm, height H ¼ 170 mm. For the observation and measurement of the icelayer thickness, the top and the bottom of the test section were covered by transparent perspex plates, allowing for photographic registration of the ice-layers. On both copper cylinders (at the bottom, in the middle and at the top) thermocouples have been placed for measuring the temperature. 3.2. Operation procedure In the annular channel the water is cooled from the inner cylinder by cold ethylene glycol and is heated from the outside by a water flow. Before the experiment was started the water as well as the ethylene glycol were cooled down to some desired values.

W m2 K

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qL DV : R2 þ qL  qS pH

ð20Þ

Inside the inner cylinder the cold ethylene glycol was held at a desired temperature. Using a Venturi nozzle the flow rate of the glycol was measured. Similarly, the water flow rate from the second refrigeration unit was measured using a Venturi nozzle. 4. Results and discussion Table 1 shows a summary of the test conditions for the individual runs. The test section was subdivided into three subsections (bottom, middle, top) as it can be seen in Fig. 9, each containing roughly 1/3 of the total height of the annular cylinder. In each of the subsections, the wall and fluid temperatures have been measured and are reported in Table 1. From the observation of the frozen layer it can be seen that the ice-layer thickness (the white layer in Fig. 10 on the inner cylinder) is almost uniform over the height of the test section. However, there might be small changes in the thickness of the ice-layer in the circumferential direction (The dark blue shape near the outer cylinder are only reflections. They have no meaning.). As it is well known, the development of the ice-layer thickness, depends on both the wall temperature, the mean fluid temperature (given by the parameter B) and the thermal resistance of the contact layer (given by the parameter 1/Bi). By carrying out the measurements

Z. Lipnicki, B. Weigand / International Journal of Heat and Mass Transfer 55 (2012) 655–664

Frozen time 200 s

Frozen time 990 s

Frozen time 1570 s

Frozen time 1880 s

Frozen time 2550 s

Frozen time 2740 s

Frozen time 2950 s

Frozen time 3000 s

661

Fig. 10. Photographs of the frozen layer for different times – run I.

it can be seen that the wall temperature varies along the cylinder height (see Figs. 11 and 12). Therefore, the vertical annular channel is divided into three subsections: top-1, middle-2, and bottom-3, in which the wall temperature is nearly constant. Changes in the wall temperature as a function of height might be related to the changes in the thin contact layer as a function of height. This introduces different thermal resistances over height. However, the driving temperature potential for heat transfer from the ice interface to the free convection flow (TF  TO) is not affected by these changes, because at the ice-water interface TF is equal to 0 °C. The above-men-

tioned parameter 1/Bi can be established from Eq. (14) by matching the experimental measured ice-layer thickness to the predicted values. Table 1 shows data for some runs which were carried out within the present investigation. The dimensionless quantities used in this table are defined in the nomenclature. Heat transfer coefficients, h, the parameter, B, the Nusselt numbers, Nuw(t=0), and the Rayleigh numbers, Raw(t=0) are calculated on basis of Eq. (3). The heat flow resistance 1/Bi was determined by comparing the theoretical curves with the experiments (see Figs. 13–15). Furthermore, it was noted that the solidification fronts are similar

662

Z. Lipnicki, B. Weigand / International Journal of Heat and Mass Transfer 55 (2012) 655–664

a

b Δ mm

7 6 5 4 3 2 1 00

T, °C 6

top

2

down

outside cylinder

middle

-2

top internal cylinder

-6 1000

2000

t, s 4000

3000

-10

middle down

t, s 0

1000

3000

2000

4000

Fig. 11. Frozen layer thickness – (a) and temperature – (b) as a function of time (run I).

b

a Δ mm 6

4

4

0

T, °C top, middle, down (outside cylinder) top middle

-4

2

down

-8

t, s

0 0

1500

3000

4500

internal cylinder

t, s

6000

1500

0

3000

4500

6000

Fig. 12. Frozen layer thickness – (a) and temperature – (b) as a function of time (run II).

a

b δ

δ 1.06

1.06

experiment

1.04

1/Bi = 0.11

1.04

1/Bi = 0.12 1/Bi = 0.14

1.02

0.001

0.0015

experiment

1

τ 0.0005

1/Bi = 0.19

1.02

1 0

1/Bi = 0.16

0.002

τ 0

0.0005

0.001

0.0015

0.002

Fig. 13. Comparison between theoretical and experimental data for: run I Section 1 – (a); run I Section 2 – (b).

a

b

δ

δ 1.1

1.08

1.08

1.06

1/Bi=0.05

1.06 1.04

1/Bi=0.06

1.04

1/Bi = 0.40

1.02

experiment

1.02 experiment

1 0

0.001

τ 0.002

1

τ 0

0.001

0.002

0.003

Fig. 14. Comparison between theoretical and experimental data for: run I Section 3 – (a); run II Section 1 – (b).

663

Z. Lipnicki, B. Weigand / International Journal of Heat and Mass Transfer 55 (2012) 655–664

a

b δ

δ

1.1

1.1

1.08

1.08

1.06

1.06

1/Bi=0.09

1/Bi = 0.32

1.04

1.04

1.02

1.02 experiment

experiment

1

τ 0

0.001

0.002

0.003

τ

1 0.001

0

0.002

0.003

Fig. 15. Comparison between theoretical and experimental data for: run II Section 2 – (a); run II Section 3 – (b).

for all zones in each run. All these parameters are included in Table 1. The Stefan number given in Table 1 is normally lower than 0.014. This shows the validity of the assumption of a quasi-steady ice-layer growth. Figs. 11–15 show the experimental results and compare them with theoretical predictions. Figs. 11 and 12 show first the time development of the mean  as well as the temperature records in the ice-layer thickness D three subsections as a function of time for the runs I and II. In Figs. 11 and 12(b) the temperature development on the inner cold cylinder is shown and also the temperature development of the warmer outer cylinder. It might be noted in Fig. 11(b) that the temperature of the inner cold cylinder (middle/top) shows a clear transient behavior for the first 2000 s. After this, the temperature is approximately constant over time. The Figs. 13–15 show for the runs I, II comparisons between the theoretical and experimental data for the dimensionless quantity ~ d ¼ ðR þ DÞ=R as a function of time. Fig. 13 shows the effect of a varying contact resistance (varying 1/Bi) on the shape of the frozen crust. It can be seen that the experimental results can be matched by the approximate solution by setting 1/Bi  0.16. In Fig. 14 one can see that a different value of 1/Bi has to be taken for Section 3, where the wall temperature of the cooled cylinder is quite different. This shows nicely that during the solidification process the contact resistance in the present case strongly influences the freezing process. This is related to the fact that the developing ice-layer is formed around a cylinder where due to the density change between solid and liquid an expansion occurs, which can’t be compensated by a movement of the frozen crust (as for example in the case of freezing on a flat plate). Fig. 15 shows a comparison between experimental and theoretical results for run II for two subsections. One can see again that the theoretical model predicts the ice-layer growth very accurately if 1/Bi is adapted. For other liquids (not water) where the density changes between solid and liquid might be much smaller, the effect of the contact resistance might be much smaller. However, this should always be tested by experiments. As stated before, the thermal resistance of the contact layer is very important because it determines the solidification process of the liquid. A theoretical determination of this parameter is very difficult, because of the large number of influencing parameters. The values of the thermal contact layer resistance for the individual runs are summarized in Table 1. These values have been obtained from the experiments. On the basis of the measurements one can also recalculate the dependence of the thermal resistance on the wall temperature of the cooled inner copper cylinder. In Fig. 16 the thermal resistance between the inner copper cylinder and the ice-layer is shown as a function of the wall temperature. When the wall temperature decreases, the thermal resistance of the contact layer grows. This can be explained by the stronger undercooling of the water

1/Bi 0.6

0.4

0.2

TW ,°C -12

-8

-4

0

Fig. 16. Dependency of the heat contact resistance on the wall temperature.

and the resulting stronger initial freezing speed of the fluid, resulting in a thicker contact layer (besides other effects) between the inner cooled wall and the ice-layer. As one can see from the figure the measurement data are scattered. This can be explained by the fact that the thermal resistance is a very complex parameter, also depending on the purity of the wall surface, the grade of wetting of the wall etc., whose impact is difficult to determine. For fixed temperatures of the internal surface of the cylinder (after the initial solidification) the thermal resistance of the contact layer is approximately constant. As can be seen in Fig. 16 the thermal resistance of the contact layer depends on the surface temperature. Based on the experiments, it is shown that the parameter (1/Bi) is rather complex and also difficult to determine. 5. Conclusions The development of the thickness of the frozen layer in a free convection flow inside a cooled annular enclosure depends on the way of heating/cooling (B, Nu) and also on the thermal contact resistance between the flowing liquid and the cold wall (1/Bi). At the start of the solidification process, when the thermal contact resistance (1/Bi) is time dependent, the increase of the thickness of the frozen layer is slow. After this a strong increase in the thickness of the frozen layer with time is observed. For larger times the thermal resistance of the contact layer can be considered constant. In this paper, the thermal resistance of the contact layer was determined experimentally. Experimental results are shown for the average ice-layer thickness in the annular channel. These values are compared to theoretical predictions and the agreement is general quite well. The present paper shows for the first time experimental results on the behavior of a freezing front in an annular enclosure with an inner cooled cylinder. The presented experimental and theoretical study on solidification in a free-convection flow inside a vertical annular enclosure is

664

Z. Lipnicki, B. Weigand / International Journal of Heat and Mass Transfer 55 (2012) 655–664

an example of an indirect method which can also been used to determine the thermal resistance of the contact layer for other solidification processes. Acknowledgements Both authors would like to thank Mr. Kaczmarek and Mr. Gomaa for their various help and support in this investigation. References [1] W. Edler, Laminar free convection in a vertical slot, J. Fluid Mechanics 23 (1965) 77–98. [2] W. Edler, Turbulent free convection in a vertical slot, J. Fluid Mechanics 23 (1965) 99–111. [3] M.A. Azouni, Evolution of nonlinear flow structure with aspect ratio, Int. Com. Heat Mass Transfer 14 (1987) 447–456. [4] I. Catton, Natural Convection in Enclosures, in: Proc. of the 6th Heat Transfer Conference, Toronto, Canada 6, 1978, pp. 13–43. [5] S.H. Davis, U. Mueller, C. Dietsche, Pattern selection in single-component systems coupling Benard convection and solidification, J. Fluid Mechanics 144 (1984) 133–151. [6] R. Viskanta, Heat transfer during melting and solidification of metals, J. Heat Transfer 110 (1988) 1205–1219. [7] R. Viskanta, Phase-change heat transfer in solar heat storage, in: G.A. Lane (Ed.), Latent Heat Transfer, CRS Press, Int. Bocaratod, Florida, 1983, pp. 153– 221.

[8] S.O. Neumann, H. Beer, Morphology of ice layers in turbulent straight walled diffusers flows, in: 7th Int. Symposium on Therm. Eng. Sci. for Cold Regions, Seoul Korea, 2001. [9] B. Weigand, J. Braun, S.O. Neumann, K.J. Rinck, Freezing in forced convection flows inside ducts: a review, Heat Mass Transfer 32 (1997) 341–352. [10] T. Loulou, E.A. Artyukhin, J.P. Bardon, Solidification of molten tin drop on a nickel substrate, in: 10th Int. Heat Transfer Conference, Brighton, UK, vol. 4, 1998, pp. 73–78. [11] T. Loulou, E.A. Artyukhin, J.P. Bardon, Estimation of thermal contact resistance during the first stages of metal solidification process: I – experiment principle and modelisation, Int. J. Heat Mass Transfer 42 (1999) 2119–2127. [12] T. Loulou, E.A. Artyukhin, J.P. Bardon, Estimation of thermal contact resistance during the first stages of metal solidification process: II – experimental setup and results, Int. J. Heat Mass Transfer 42 (1999) 2129–2142. [13] Z. Lipnicki, Role of the contact layer between liquid and solid on solidification process, Int. J. Heat Mass Transfer 46 (2003) 2149–2154. [14] Z. Lipnicki, B. Weigand, A. Bydałek, On the effect of a variable thermal contact resistance on the solidification process, Arch. Metall. Mater. 50 (2005) 1055– 1064. [15] W. Kays, M. Crawford, B. Weigand, Convective Heat and Mass Transfer, Mc Graw Hill, Higher Education, 2005. [16] R. Kumar, M.A. Kalam, Laminar thermal convection between vertical coaxial isotermal cylinders, Int. J. Heat Mass Transfer 34 (1991) 513–524. [17] R. W. Thomas, G. de Vahl Davis, Natural convection annular and rectangular cavities – a numerical study, in: Proc. 4th Int. Heat Transfer Conference, Paris 4, Paper N.C. 2.4, 1970. [18] G.X. Wang, E.F. Matthyes, Interfacial thermal contact during rapid solidification on substrate, in: Proc. 10th Int. Heat Transfer Conference, Brighton, UK 4, 1994, pp. 169–174.