THEORETICAL AND EXPERIMENTAL ANALYSIS OF FLOW …liquid subcooling at inflow, flow velocity and pressure) on the incipience boiling in minichannels were also presented. Results were

V Minsk International Seminar “Heat Pipes, Heat Pumps, Refrigerators” Minsk, Belarus , September 8-11, 2003

THEORETICAL AND EXPERIMENTAL ANALYSIS OF FLOW BOILING

HEAT TRANSFER IN MINICHANNEL

Magdalena Piasecka, Mieczyslaw E. Poniewski the Chair of Thermodynamics and Fluid Mechanics

the Kielce University of Technology Al. 1000-lecia P.P. 7, PL – 25-314 Kielce, Poland

fax. (+48 –41) 3424340; E-mail: [email protected]

Abstract Experimental investigations dealt with heat transfer of refrigerant R123 flowing through a vertical minichannel

of 1 mm height with one wall heated uniformly and others approximately adiabatic. An inverse problem was dealt with as, on the basis of temperature measurement on external side of the heating foil (with thermography technique) and the measurement of the electric power supplied to this foil, it was possible to determine local heat transfer coefficients on the contact surface of the heating wall and the fluid flowing along the minichannel. Two models of heat transfer through the heating foil and the isolating glass into the boiling liquid were proposed and discussed: one- and two-dimensional. In order to solve the two-dimensional inverse heat conduction problem the method of thermal polynomials was applied. The boundary value problem in the foil was solved with application of the least square and Trefftz computational techniques. Some exemplary results of numerical calculations made on the basis of experimental data were presented for both models. Particular attention was paid to boiling incipience conditions and nucleate hysteresis phenomena. The hysteresis phenomena observations, including the impact of various factors (the liquid subcooling at inflow, flow velocity and pressure) on the incipience boiling in minichannels were also presented. Results were shown in the form of boiling curves plotted for selected points in the minichannel. The correlation concerning Nusselt number determination for refrigerant R123 boiling incipience in a vertical minichannel was developed. KEYWORDS

flow boiling incipience, nucleation hysteresis, minichannel 1. INTRODUCTION

In recent times, an increasing number of high-tech heat exchange devices are based on heat transfer to liquids flowing in minichannels of various geometries and orientations. These devices include nuclear reactors, sophisticated electronic instruments, internal combustion engines, water-cooled gas turbines, X-ray sources and certain components of particle accelerators. Owing to the change of state, which accompanies boiling, it is possible to simultaneously meet contradictory demands, i.e. to obtain the high heat flux at small temperature difference between the heating surface and the saturated liquid and, at the same time, reduce dimensions of heat transfer systems.

Boiling incipience is not only a general problem in boiling itself, but also a practical one in the evaluation of the safety of equipment ranging from electronics to nuclear reactors. It is known, that under certain conditions, there can occur a considerable rise in wall temperature above the saturation point before boiling begins. This “temperature overshoot” called “superheated excursion”, “nucleation hysteresis”, “zero boiling crisis” is conspicuous when highly wetting dielectric fluids (e. g. refrigerants) are used. There are a lot of surveys providing experimental results and making attempts at finding a theoretical explanation of boiling incipience and “nucleation hysteresis”, which accompanies it [1, 3, 4, 23]. This kind of hysteresis, which occurs in the flow through minichannels, has been investigated recently (c. f. [7, 8, 10, 16, 17, 24]). It was also observed in our experimental investigations and discussed in the paper. The occurrence of “nucleation hysteresis” may result in the destruction of the object being cooled, also in one-phase flow, if the temperature increment required for boiling incipience exceeds the design value. Moreover, in certain cases vapour bubble formation is not admissible, e.g., in cooling

Mieczyslaw E. Poniewski et. al. 228

channels of a nuclear reactor core. Thus, boiling incipience is of fundamental importance for safe operation of devices. 2. EXPERIMENTAL STAND AND PROCEDURE

The main loop of the flowing system for boiling process examination of working fluid includes the following elements (Fig. 1): the test section with a vertical minichannel, rotary pump with an inverter, compensating tank, heat exchangers, flow meters (rotameters), pressure converters and thermocouples located at different points of the installation and the test section.

Fig. 1 The diagram of main loop of the flowing system: #1-test section; #2-rotary pump; #3-inverter, #4-compensating tank; #5, #6, #7-heat exchangers; #8-rotameters; #9-pressure converters

The diagram of the test section is presented in Fig. 2. A minichannel of 1 mm height, 40 mm width,

360 mm length was built in the test section basic part. The heating element for working fluid (R123), which flows along the minichannel from inlet conduit to outlet chamber, is an alloy foil stretched between the front cover and the channel body. The heating element 0.004’’ (approx. 0.1 mm) foil designated as Haynes-230, made of Ni-Cr-W-Mo high-temperature alloy was selected. It is possible to observe changes in the foil surface temperature through the opening covered with glass. One side of the heating foil (between the foil and the glass) was covered with a base coat and liquid crystals paint. Due to thermography [5, 6, 25] and the data acquisition system, Fig. 3, it is possible to measure temperature distribution on the heating wall. The test section rear cover contains channels, to which water is fed, or which are air gaps. Due to these auxiliary channels it is possible to maintain the desirable temperature on the wall, which is treated as quasiadiabatic. There are five thermocouples soldered to the bottom wall of the water channel, Fig. 2. Two other thermocouples with pressure converters are placed at the minichannel inlet and outlet.

Fig. 2 Diagram of the test section: #1-minichannel inlet, #2-glass, #3-liquid crystals, #4-heating foil, #5-minichannel, #6-quasiadiabatic wall, #7- minichannel outlet, #8-water channels, #9-air gaps

Temperature distribution recording on the foil surface is performed by the following system: the test

section with a minichannel, Fig. 2, CCD colour video camera equipped with the device decomposing video signal into the RGB signal and zoom lens with manual control located straight in front of the examined object and illumination system, which consists of two white light sources (fluorescent lamps emitting “cool white light”), located at the same distance and at the same angle with respect to the object examined, Fig. 3. The colour image acquisition system, Fig. 3, allows real time image projection directly on the computer monitor, image preservation in RGB format and its transfer to the computer for further


analysis, simultaneous projection of the real time image on the recorder monitor as well as the image recording on the video tape. The software working with frame grabber and the one processing the measurement data complement the system. The registration of the remaining measurement data is carried out with Keithley 500 A data acquisition station equipped with ViewDac software installed at another computer.

What is sought is the heat transfer coefficient during the incipience of boiling on the surface separating the heating foil and the boiling fluid, Fig. 4. That is an inverse problem, in which with the measurement of temperature at the system internal points (temperature distribution on the heating foil on the side of the glass) and the measurement of the electric power supplied to the heater, one can calculate the requested local heat transfer coefficients on the heating wall surface contacting the boiling fluid [9, 12, 18, 19, 20, 25].

Fig. 3 The diagram of the system of acquisition of measurement data, colour images and their further processing: #1-test section; #2-lighting system; #3-video camera; #4-signal decomposer; #5-Betacam video recorder; #6-monitor; #7-computer with frame grabber and monitor; #8-data acquisition station; #9-computer with monitor

Y 0F Gδδ

Tl,inlet

Z

l,outletTX

L

single phaseforcedconvection

fullydevelopednucleateboiling

incipienceof boiling

liquid crystals

glass

boiling liquid

quasiadiabatic wall

heating foil

T w =T w (hue)

310

320

330

340

0 40 80 120 16hue

T w [K

]

Fig. 4 One- and two-dimensional models diagram

Fig. 5 Calibration curve Tw = Tw (hue)

3. THERMOGRAPHY FOR THE MEASUREMENT OF THE HEATING SURFACE

TEMPERATURE DISTRIBUTION 3.1. Thermography and properties of thermochromic liquid crystals

Liquid crystals can function as temperature indicators. This happens due to the property of light selective reflection from planary oriented thermochromic liquid crystals (the reflection of waves of the same length, i.e. the same hue, dependent on the incidence angle) and the dependence of the hue on the surface temperature. Such manner of stating temperature distribution is called thermography. It provides a


fairly accurate and convenient method used to determine the two-dimensional distribution of the surface temperature [5, 6, 24, 25]. The range of temperatures measured must remain within the active band of liquid crystals. That is, between the lowest temperature when the surface covered with liquid crystals is seen to be red and the highest temperature when purple hue is observed. Hues change in the visible spectrum sequence. Each measurement must be preceded by liquid crystals calibration in order to state the relation between the hue on the surface and the corresponding temperature. The image of temperature fields on the surface covered with liquid crystals changes depending on the angle of observation, it also depends on the incident light spectral composition. Therefore, in the course of the experiments, lighting conditions must be kept constant.

3.2. The liquid crystals surface hue calibration with respect to temperature

In the calibration process, the water of pre-set temperature is fed, in a closed circuit, to a minichannel (#5, Fig. 2) and auxiliary channels (#8, Fig. 2). For a given temperature, a colour image of the heating foil covered with liquid crystals is recorded and the values of RGB components are determined. HSI system is used for colour image processing.The constant lightning conditions lead to constant values of saturation (S) and intensity (I). As a result, three-component RGB signal is converted into single hue (H) equation [5, 6, 24, 25]:

))2/()(3( 1−−−−= BGRBGarctanhue (1)

where: R, G, B - the participation of red, green and blue component. Calibration yields the curve Tw = Tw (hue), which is presented in Fig. 5.

4. EXPERIMENT 4.1. The definition and range of experimental parameters

The variation range of parameters characterising experimental investigations are presented in Table 1. Table 1. The definition and variation range of basic parameters in experimental investigations [25] Quantities set qV [kW/m3] 6.63⋅104 ÷1.17⋅106 ⋅∆⋅= −1)/( FFV AUIq δ (2)

Tw [K] from the hue distribution on the surface QV [m3/s] 3.88⋅10-6 ÷1.90⋅10-5 rotameters readings Quantities

measured pinlet [kPa] 126÷345 pressure converter readings

Tl [K] 1

, )/()(4)( −⋅⋅⋅⋅+= heplwinletll DcGxxqTxT (3) where Dhe = 4Ach/Ohe, Ohe = Hch

qw(x) [kW/m2] 6.7÷118.8 α(x) [kW/mK]

determined with the use of one and two-dimensional model of heat transfer

∆Tsub [K] 23.3÷71.1 inletlsatsub TTT )( −=∆ (4) ∆Tsat [K] )()()( xTxTxT satwsat −=∆ (5)

u [m/s] 0.10÷0.48 chV AQu /= (6) G [kg/m2s] 143÷710 uG l ⋅= ρ (7) Re [-] 618÷2 890 1−⋅⋅= lhDGRe µ (8), where Dh = 2Ach/(Wch+Hch)

Quantities calculated

Nu 1)()( −⋅⋅= lhDxxNu λα (9) 4.2. Uncertainty of the main parameters

Mean square errors of the measured parameters were estimated as follows [25]: foil temperature (liquid crystals thermography) Tw=0.86 K; inlet (outlet) pressure pinlet (poutlet)=0.5 kPa; foil superheating ∆Tsat=1.28 K; inlet liquid subcooling ∆Tsub=0.68 K; capacity of internal heat source qV =4.08 %.


4.3. Experimental procedure The images of colour distribution, Fig. 7, correspond to the temperature distribution on the heating

foil, while increasing and later decreasing the heating power supplied to it. In the first stage of the run, the current supplied to the heating foil increases gradually. Heat transfer between the foil and the working fluid in the channel proceeds by single phase laminar forced convection (Fig. 7 - first six images). It is recognisable as the hue sequence pattern, which indicates a gradual increase in the heating surface temperature (temperatures out of sensitivity range are shown in black). Further increase in heating causes appearance of a “boiling front”, resulting in a sudden drop of the foil temperature, Fig. 7, images from #7 up to #14. It is seen as sharp hue changes of the liquid crystals, inversely to the spectrum sequence. Then black hue returns, which is the evidence of boiling incipience (BI).

Image #8, Fig. 7, was chosen here to give detailed characteristics of boiling front occurrence. Boiling front is seen as a sequence of stages downstream: 1) It starts with the black hue (image bottom). There gradually appear subsequent hues of the visible spectrum (red, yellow, green, blue and navy blue), That points to a gradual increase in the heating surface temperature. Heat transfer between the heating surface and the liquid in the channel takes place as single phase forced convection. 2) At the image top sharp hue changes on the heating foil take place inversely to the spectrum sequence. The black hue returns. It indicates a sudden drop in the heating surface temperature at constant heat flux. It is when „nucleation hysteresis” phenomena, accompanying nucleate boiling incipience take place.

The boiling incipience (BI) is identified with the maximum value of the heating surface temperature. A sharp temperature drop follows further out. It is impossible to detect BI with the naked eye. You have to use the calibration curve (an exemplary one is presented in Fig. 5). In turn, in order to establish the exact location of BI, it is necessary to determine the maximum of foil temperature along the channel, Fig. 6.

T w =T w (x)

315

320

325

330

335

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

Distance x [m]

T w [K

]

#8

single phase convection

BI

Fig. 6 Foil temperature dependence on the distance along the channel vertical extension, image #8, Fig. 7

When the heat flux supplied to the foil increases, the “boiling front” moves in the direction opposite to the direction of the liquid flow in the channel. It can be observed clearly while analysing images from #7 up to #14, Fig. 7. When we continue increasing the heat flux, and the heating wall temperature exceeds the sensitive range of liquid crystals, another hue change is observed on the heating surface in the visible spectrum sequence (Fig. 7, image #13). It is accompanied by pressure increase in the channel, flow fluctuations, sharp increase in the liquid temperature in the flow core (observed at the channel outlet) and flow resistance fluctuations. All these phenomena point to the increase in the amount of vapour phase in the boiling vapour-liquid mixture and the occurrence of developed nucleate boiling (Fig. 7, images from #13 up to #33). Then the current supplied to the foil is gradually reduced following the occurrence of saturated navy blue hue on the foil surface (Fig. 7, images from #34 up to #51). Mild hue changes, in the direction opposite to the spectrum sequence are observed to accompany the decrease in the current supplied to the foil. As a result, heat transfer returns to single phase forced convection. The heat flux decrease makes the boiling “fading away” start at the channel inlet. At the end of the run, black hue of the heating foil returns, which means the temperature drops below that of the liquid crystal active band.


1# #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 #15 #16 #17 #18

#19 #20 #21 #22 #23 #24 #25 #26 #27 #28 #29 #30 #31 #32 #33 #34 #35 #36

#37 #38 #39 #40 #41 #42 #43 #44 #45 #46 #47 #48 #49 #50 #51

Fig. 7 Images of temperature distribution on the heating wall, QV =8.0⋅10-6 m3/s; G =298 kg/m2s; u =0.2 m/s; Re=1240; pinlet =142 kPa; ∆Tsub=27.5 K; qV=6.63⋅104÷7.46⋅105 kW/m3; qw=6.7÷75.8·kW/m2 [25]

4. ONE-DIMENSIONAL MODEL

One-dimensional model put forward in [16, 17, 24] and modified in [9, 12, 18, 20, 25] is based on the equations of energy balance for individual control volumes of the test section. The analysis was carried out in respect of isolated volumes, separately for the foil and the glass.

Assumptions for one-dimensional model: 1) The liquid crystal layer on the heating foil is very thin, therefore it is disregarded in considerations; 2) In the fluid - wall system under investigation, the state is steady;


3) Changes in the temperature of the foil and the liquid as well as the fluid velocity along the axis Z (along the channel width perpendicular to the liquid flow) are disregarded, Fig. 8;

4) Temperature of the working fluid Tl(x) is determined on the basis of enthalpy balance, equation (3), Table 1;

5) The investigations take into account the dimension along the flow direction – X, but the dimension perpendicular to it – Y, is related only to the foil or glass thickness, Fig. 8.

Fig. 8 presents heat fluxes transferred to control volume of foil or glass, per minichannel width, as the product of heat fluxes density and either the thickness or the elementary length of each layer.

Fig. 8 The detailed energy balance in one-dimensional model

The heat balance equation for control volumes can be presented as follows: 1) for foil:

(10) dxqdxqqdxqqFGG yFyFFdxxFFVFxF ⋅+⋅+⋅=⋅⋅+⋅ ++++ δδδδδδ ,,,,

2) for glass: (11) Gdxx,Gy,GGx,G qdxqq

Gδδ δ ⋅=⋅+⋅ ++

3) for the foil-glass interface:

(12) dxqdxqGG yGyF ⋅=⋅ ++ δδ ,,

We assume:

2

2

2

2

dxTd

dxTd GF = (13)

which constitutes the extension of the condition, which results from equation (12). It can be applied owing to the small thickness of the partition, in which conduction process is accounted for. At the same time, it is assumed that the derivative of the temperature gradient on the heating foil is equal to the temperature gradient on the foil surface, read from liquid crystals hue distribution. As a result of transformations and after the introduction of the dependences resulting from Fourier’s law, the local heat transfer coefficient is calculated from the equation:

[ ](x)T(x)Tδqdx

Td)δλδλ(x)α lFFV2F

2

GGFF1 −⎥⎦

⎤⎢⎣

⎡⋅+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⋅⋅+⋅= /( 14)


In equation (14) it is necessary to determine the second derivative of the heating wall temperature along the minichannel length. The method of the least squares was applied to the function TF = TF (x) polynomial approximation.

Local heat flux density could be obtained from the equation:

⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⋅+⋅+⋅= 2

F2

GGFFFVw dxTd)(q)x(q δλδλδ (15)

5. TWO-DIMENSIONAL MODEL 5.1. Problem presentation and calculation algorithm

The assumptions made about the two-dimensional model are the same as about the one-dimensional model (from #1 to #4). In the two-dimensional model the dimension perpendicular to the flow direction is also taken into account [9, 12, 18, 19, 20, 25]. All the parameters describing physical properties – given, measured or assumed, are the same as in the one-dimensional model. The process of heat transfer in a two-layer wall, shown in Fig. 4, is described with the following equations: 1) in foil:

F

V2F

2

2F

2 qy T

x T

λ−=

∂∂

+∂∂ (16)

2) in glass:

(17)

0 2

2

2

2

=∂∂

+∂∂

yT

xT GG

and the boundary conditions concerning TG and TF are shown in Fig. 9.

Fig. 9 Boundary conditions for a two-layer wall


We shall use harmonic polynomials in the solution of the problem under consideration. Harmonic polynomials un(x,y) which satisfy equation (17) are defined as the real and imaginary part of the holomorphic function , respectively, i.e. ( )niyx +

( )( )( )( )⎪⎩

⎪⎨⎧

++= +

niyxniyxyxu n

n

n evenfor Im oddfor Re),(

21

2 (18)

Here are some of the first polynomials un: . Since they satisfy the heat equation (eq. 17), they are also called the heat functions. The unknown T

,...yxu,xu,yu,u 223210 1 −====

G and TF will be approximated with a linear combination of the harmonic polynomials un(x,y):

(19) ∑=

=≈N

iiiGG yxucyxyxT

0

),(),(),( θ

(20) ∑=

=≈−M

jjjFF yxudyxyxuyxT

0

),(),(),(~),( θ

where 2

F

V yλ2

q)y,x(u~ −= is a particular solution of eq. (16).

The unknown coefficients ci (i = 0,1,..., N) and dj (j = 0,1, ...,M) are obtained with the least square and Trefftz techniques. In the computational procedure the approximation of TG will be obtained first and TF next. Provided TF(x,y) on the boundary y = δG+δF is known, we are able to compute the heat transfer coefficient α2 (or α2’) from the following boundary condition:

( ))(),(

),( 2 xTxT

yxT

lFGFFGF

F −+=∂

+∂− δδα

δδλ (21)

5.2. Application of the least square computational technique The function TG(x,y) satisfies the eq. (17) and the boundary conditions (Fig. 9):

00 ==∂

∂y

yTG at (22) 00 ==

∂∂

xx

TG at (23)

Lxx

TG ==∂

∂at0 (24) (25) kGkG TxT =δ ),(

The unknown coefficients ci (i = 0 ,1,..., N) are taken to minimize the error functional whose form is dependent on the conditions (22)-(25):

(∫∫∫ −+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂+⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

====

L

kGGLx

G

x

GL

y

GG dxTxdy

xxdx

yJ

G

0

2

0

22

00

2

0

),( δθθθθ δ

) (26)

Substituting (19) for in the eq. (26) we get: )y,x(Gθ

∫ ∑∫ ∑∫ ∑∫ ∑ ⎟⎠

⎞⎜⎝

⎛ −+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=== == == =

L

k

N

iGii

N

i Lx

ii

N

i x

ii

L N

i y

iiG dxTxucdy

xucdy

xucdx

yucJ

GG

0

2

00

2

00

2

0 00

2

0 0

),( δδδ

(27)


The minimum of the functional JG leads to the solution of the following set of equations:

N,...,1,0ifor0cJ

i

G ==∂∂ (28)

which can be written as:

∫

∫ ∫∫ ∫∑

=

=⎥⎥⎦

⎤+

∂∂

∂∂

+⎢⎢⎣

⎡

∂∂

∂∂

+∂∂

∂∂

=======

L

Gjk

L

GjGiLx

j

Lx

iL

x

j

x

i

y

j

y

iN

ii

dxxuT

dxxuxudyxu

xudy

xu

xudx

yu

yuc

GG

0

0 00 0 00000

),(

),(),(

δ

δδδδ

(29)

Similarly, the values of the coefficients dj (j=0,1,...M) appearing in (20), are determined on the basis of boundary conditions subject to TF (see Fig. 9). In this case, the error functional has the form:

[ ] ∫∫∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂∂

+∂

∂−++−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂=

==

+

==

L

y

GG

y

FF

L

GGGFLx

F

x

FF dx

yydxxxdy

xxJ

GG

FG

G 0

22

0

22

0

),(),(δδ

δδ

δ

θλθλδθδθθθ

(30) Minimisation of JF leads to the set of equations:

M,...,1,0jfor0dJ

j

F ==∂∂ (31)

where substituting gives: ∑=

+=M

0jjjF )y,x(ud)y,x(u~)y,x(θ

( )∫∫∫

∫∫∫∑

−+∂

∂∂∂

+∂∂

∂

∂−=

=∂∂

∂

∂++

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂

∂+

∂∂

∂

∂

====

==

+

=====

L

GGGGi

L

y

G

yGF

L

yy

jF

L

y

i

y

jFGi

L

GjLx

i

Lx

j

x

i

x

jM

jj

dxxuxxudxyy

udxyu

yu

dxyu

yu

dxxuxudyxu

xu

xu

xu

d

GGGG

GG

FG

G

000

2

0

2

0000

),(~),(),(~~

),(),(

δδθδθ

λλλ

λδδ

δδδδ

δδ

δδ

δ

(32)

5.3. Application of Treffz computational technique We now use Green’s formula:

( ) ∫∫∫ ⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

−=∆−∆ΓΩ

dsnn

dxdy ϕψψϕϕψψϕ (33)

where 2

2

2

2

yx ∂∂

+∂∂

=∆ is a Laplace operator and n∂∂ means differentiation along the inward normal.

Consider the function w(x,y), (x,y)∈Ω, which satisfies the eq. (16) and the boundary condition:

321 AnwAwA =

∂∂

+ on Γ (34)

where A1, A2, A3 are the given functions on Γ.


The function w(x,y) can be approximated as follows:

∑=

+≈N

0iii )y,x(ua)y,x(u~)y,x(w (35)

Since and uw ~−=ϕ ju=ψ satisfy the eq. (17), then substituting them in (33) we have:

Nj dsn

uwunu

uw jj ..., 2, 1, 0)~()~(

Γ

==⎟⎟⎠

⎞⎜⎜⎝

⎛∂

−∂−

∂

∂−∫ (36)

If we take A1 ≠ 0 and A2 = 0 in (23) and nua

n)u~w( i

N

ii ∂∂

=∂−∂ ∑

=0

in (36), we obtain:

Nj dsuAA

nu

dsnuua j

N

i

iji ..., 2, 1, ~

Γ 1

3

1 Γ

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂

∂=

∂∂

∫∑ ∫=

(37)

On the other hand, if we take A1 = 0 and A2 ≠ 0 in (34) and in (36), we come to: ∑=

=−N

iiiuauw

0

~

..., 2, 1, ~

Γ 2

3

1 Γ

Nj dsnu

AAuds

nu

ua j

N

i

jii =⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

−=∂

∂∫∑ ∫

=

(38)

Equations (37) and (38) are called Trefftz equations for boundary condition of the first and second kind, respectively. It is possible to derive the Trefftz equations for the third kind boundary condition [13]. The right-hand side of (37) contains constant a0, the computation of which will be shown in the included numerical example. If Γ consists of two components Γ1 and Γ2 , so that on Γ1 we have the first kind boundary condition and on Γ2 the second kind boundary condition, we apply the transformations on Γ1 and Γ2 leading to (37) and (38). In both cases we obtain the system of equations with the unknown ai, i = 1, 2,..., N. The unknown coefficients c1, ..., cN are obtained from the set of equations:

∫∫∑ ∫==== ==

∂

∂=

⎪⎭

⎪⎬⎫

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂−

∂

∂

⎪⎩

⎪⎨⎧

+⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂+

∂

∂ L

y

jk

x

ji

Lx

ji

N

i

L

y

jGi

y

jii dx

yu

Tdyxu

yuxu

yLudxyu

xuyu

xucG

G

G 00 01 0 0

),0(),(),()0,(δ

δ

δ

δ

(39) The constant c0 is chosen so that the function θG would satisfy the condition , i.e.: kGkG TxT =),( δ

),(θ0 GkGk

k xTc δ−= (40)

The unknown coefficients d1, ..., dM are obtained from the set of equations:

( ) ∫∫

∫∑ ∫

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂

∂+

∂∂

−=

=⎪⎭

⎪⎬⎫

⎥⎦

⎤⎢⎣

⎡∂∂

−∂∂

+⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

+∂

∂

===

+

=== ==

L

yy

G

F

GGi

L

y

iGk

x

ij

Lx

ij

M

j

L

y

iGj

y

jGij

dxyu

yxudx

yuxuT

dyxuyu

xuyLudx

yuxu

yu

xud

GGG

FG

GGG

00

01 0

~θ),(,(~

),0(),(),(),(

δδδ

δδ

δδδ

λλδδ

δδ

(41)


The constant d0 satisfies the equation:

),(θ0 GkFkk xTd δ−= (42)

Integration in (29), (32), (39), (41) is easy because the integrands are polynomials.

6. EXPERIMENTAL RESULTS 6.1 Boiling curves

Boiling curves are obtained on the basis of data, while increasing and later decreasing the heat flux supplied to the heating foil. They are constructed for the selected points in the minichannel where heat flux density depends on the heating surface superheating ∆Tsat. The minimum and the maximum values of the heat flux supplied to the foil are limited by the sensitivity range of liquid crystals.

Fig. 10 shows a typical shape of a boiling curve. While increasing the heat flux density (from point A to point BI - nucleate boiling incipience), the heat transfer between the heating foil and subcooled liquid flowing upward the minichannel proceeds by means of single phase forced convection. In the foil adjacent area, the liquid becomes superheated, point BI, whereas in the flow core it remains subcooled. The increase in the heat flux density results in vapour nuclei activation on the channel heating surface. Boiling incipience comes spontaneously for the superheating ∆Tsat = 25 ÷ 30 K [25], point BI. Spontaneous nucleation causes the heating surface temperature drop, for almost constant heat flux density. It is visible as drop from point BI to point C. Further increase in the heat flux density leads to developed nucleate boiling, section C-D. Decreasing the heat flux, starting at point D, proceeds along the same line in boiling curves, in the opposite direction, section D-C. Leap heating surface temperature decrease results from vapour bubbles spontaneous formation in the wall adjacent layer. The bubbles function as internal heat sinks, absorbing a significant amount of energy transferred to the liquid [2, 9, 15, 20, 21, 22, 25].

Fig. 10 Typical boiling curve, parameters as for Fig. 7 The shape of experimental boiling curve shown in Fig. 11 is different from a typical boiling curve

(Fig. 10). The differences are clearly seen in the region of developed nucleate boiling. Such boiling curves demonstrate two-stage, “stepped” course of nucleation hysteresis in the region of developed nucleate boiling. So far similar hysteresis course, corresponding to II kind hysteresis, has manifested itself in pool


boiling investigations, carried out for capillary-porous, metal and fibrous micro-surfaces, manufactured with various techniques [23]. The above-mentioned boiling curves are characterized by smaller temperature drops, ∆Tsat = 15 ÷ 20 K [25]. For such boiling curves, in nucleate boiling, the increase in the heating surface superheating while the heat flux grows, is carried out at far lower densities of the heat flux, sections B-C, than when the heat flux diminishes, sections C-D. For decreasing densities of the heat flux, much higher values of the heat transfer coefficients are obtained. It means that nucleation centres activated while the heat flux density was growing, still remain active when the flux decreases. The boiling curve, Fig. 11, was plotted in the system qw = qw (Tw-Tl), not in qw = qw (Tw-Tsat). Such forms of boiling curves can be often found in the literature on the subject [2]. Observations [25] lead to the statement that at elevated pressures, pinlet > 200 kPa, and high subcooling at the inlet, ∆Τsub > 40K, the local value of the saturation temperature might be higher than the heating surface temperature. At the same time, the “boiling front” is observed to be correctly moving, at heat flux increase, in the direction opposite to the flow direction. That means that the temperature of the superheated wall adjacent layer is much higher than the liquid temperature in the core, and the mean temperature of the boiling liquid is lower than the local saturation temperature. Thus, relying on the temperature difference ∆Tsat = Tw - Tsat would lead to a false conclusion that the surface is not superheated when compared with the liquid mean temperature. The instance given here refers to a vapor bubble, generated on the heating surface, which at the same time, condenses intensively in the flow core because of the low liquid temperature. Therefore, it becomes a very active heat sink, which results in the heating surface temperature drop below the local liquid saturation temperature.

Fig. 11 Exemplary boiling curves with II kind hysteresis: QV =1.34⋅10-5 m3/s; G =501 kg/m2s; u =0.34 m/s; Re=2 015; pinlet =247 kPa; ∆Tsub=41.3 K; qV=1.17⋅105÷8.01⋅105·kW/m3; qw=11.9÷81.4 kW/m2

6.2. Heat transfer coefficient calculation

The results of heat transfer coefficient calculations obtained with one- and two-dimensional models are compared. Exemplary calculation results are presented in the form of foil temperature dependence on the distance along the channel length (Fig. 12 – a, b, c), heat transfer coefficient dependence on the distance along the channel length, calculated according to one-dimensional model (Fig. 12 – d, e, f) and to two-dimensional model with the application of the least square (Fig. 13 – a, b, c) and Trefftz techniques (Fig. 13 – d, e, f).


a) T w =T w (x )

315

320

325

330

335

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

Distance x [m]

T w [K

]1.43E+0.51.50E+0.51.58E+0.51.66E+0.51.74E+0.51.83E+0.51.91E+0.52.00E+0.52.09E+0.52.19E+0.52.28E+0.52.47E+0.5

q V [kW/m3]

d) α1=α1(x )

300

400

500

600

700

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

Distance x [m]

1 [W

/m2 K]

1.43E+0.51.50E+0.51.58E+0.51.66E+0.51.74E+0.51.83E+0.51.91E+0.52.00E+0.52.09E+0.52.19E+0.52.28E+0.52.47E+0.5

q V [kW/m3]

b) T w=T w (x )

315

320

325

330

335

340

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

Distance x [m]

T w [K

]

1.38E+0.51.47E+0.51.54E+0.51.62E+0.51.70E+0.51.78E+0.51.86E+0.51.96E+0.52.04E+0.5

q V [kW/m3]

e) α1=α1(x )

200

300

400

500

600

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

Distance x [m]

1 [W

/m2 K]

1.38E+0.51.47E+0.51.54E+0.51.62E+0.51.70E+0.51.78E+0.51.86E+0.51.96E+0.52.04E+0.5

q V [kW/m3]

c) T w=T w (x )

315

320

325

330

335

340

345

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

Distance x [m]

T w [K

]

1.38E+0.51.50E+0.51.57E+0.51.66E+0.51.74E+0.51.82E+0.51.91E+0.52.00E+0.52.09E+0.5

q V [kW/m3]

f) α1=α1(x )

200

300

400

500

600

700

800

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

Distance x [m]

1 [W

/m2 K]

1.38E+0.51.50E+0.51.57E+0.51.66E+0.51.74E+0.51.82E+0.51.91E+0.52.00E+0.52.09E+0.5

q V [kW/m3]

Fig. 12 a), b), c) Foil temperature dependence on the distance along the channel length;

d), e), f) Heat transfer coefficient dependence on the distance along the channel length, calculated according to one-dimensional model


a) α2=α2(x )

300

400

500

600

700

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

Distance x [m]

2 [W

/m2 K]

1.43E+0.51.50E+0.51.58E+0.51.66E+0.51.74E+0.51.83E+0.51.91E+0.52.00E+0.52.09E+0.52.19E+0.52.28E+0.52.47E+0.5

q V [kW/m3]

d) α2

'=α2'(x )

300

400

500

600

700

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

Distance x [m]

2' [W/m

2 K]

1.43E+0.51.50E+0.51.58E+0.51.66E+0.51.74E+0.51.83E+0.51.91E+0.52.00E+0.52.09E+0.52.19E+0.52.28E+0.52.47E+0.5

q V [kW/m3]

b) α2=α2(x )

200

300

400

500

600

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

Distance x [m]

2 [W

/m2 K]

1.38E+0.51.47E+0.51.54E+0.51.62E+0.51.70E+0.51.78E+0.51.86E+0.51.96E+0.52.04E+0.5

q V [kW/m3]

e) α2

'=α2'(x )

200

300

400

500

600

700

800

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

Distance x [m]

2' [W/m

2 K]

1.38E+0.51.50E+0.51.57E+0.51.66E+0.51.74E+0.51.82E+0.51.91E+0.52.00E+0.52.09E+0.5

q V [kW/m3]

c) α2=α2(x )

200

300

400

500

600

700

800

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

Distance x [m]

2 [W

/m2 K]

1.38E+0.51.50E+0.51.57E+0.51.66E+0.51.74E+0.51.82E+0.51.91E+0.52.00E+0.52.09E+0.5

q V [kW/m3]

f) α2

'=α2'(x )

200

300

400

500

600

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

Distance x [m]

2' [W/m

2 K]

1.38E+0.51.47E+0.51.54E+0.51.62E+0.51.70E+0.51.78E+0.51.86E+0.51.96E+0.52.04E+0.5

q V [kW/m3]

Fig. 13 Heat transfer coefficient dependence on the distance along the channel length, calculated

according to two-dimensional model: a), b), c) with the application of the least square technique; d), e), f) with the application of Trefftz technique


Experimental parameters for the results of exemplary calculations are (the pre-set value qV is given in the figure): Figs. 12a, 12d, 13a and 13d: QV =1.4⋅10-5 m3/s; G =524 kg/m2s; u = 0.35 m/s; Re=2 100;

pinlet =180 kPa; ∆Tsub=33.0 K; qV=8.28⋅104÷9.06⋅105 kW/m3; qw=8.4÷92.0 kW/m2; Figs. 12b, 12e, 13b and 13e: QV =1.1⋅10-5 m3/s; G =412 kg/m2s; u =0.28 m/s; Re=1 646;

pinlet =190 kPa; ∆Tsub=36.0 K; qV=8.07⋅104÷8.73⋅105 kW/m3; qw=8.2÷88.6 kW/m2; Figs. 12c, 12f, 13c and 13f: QV = 1.0⋅10-5 m3/s; G = 373 kg/m2s; u = 0.25 m/s; Re = 1 540;

pinlet = 170 kPa; ∆Tsub = 36.0 K; qV=7.57⋅104÷8.00⋅105 kW/m3; qw=7.7÷81.3 kW/m2.

6. 3. The impact of pressure, liquid subcooling at the inlet and flow velocity on boiling incipience The analysis of the impact of pressure on boiling incipience was carried out on the basis of eleven examples grouped in the series from 1 to 3, Table 2. Each series covered experiments performed at constant liquid flow velocity and variable inlet pressures. The impact of the liquid flow velocity was investigated in the same fashion, series from 4 to 5, Table 2. Each of these series comprised experiments carried out at constant inlet pressure and variable liquid flow velocities. Table 2. Experimental parameters of the analysed runs [25]

Series

No. u [m/s] QV [m3/s] pinlet [kPa] ∆Tsub [K] Series

No. pinlet

[kPa] u [m/s] QV [m3/s] Re [-]

1 0.24 9.40·106330 297 237

64.5 38.9 52.7

4 320-330

0.24 0.18 0.15 0.10

9.40·106

7.00·106

5.92·106

3.88·106

1 474 1 099

935 621

2 0.15 5.92·106

345 298 271 213

71.1 53.0 45.5 35.0

5 269

0.24 0.18 0.15 0.10

9.40·106

7.00·106

5.92·106

3.88·106

1 476 1 102

934 618

3 0.10 3.88·106

321 298 269 239

61.0 53.1 45.4 39.3

6 239 (237)

0.24 0.15 0.10

9.40·106

5.92·106

3.88·106

1 472 931 628

The analysis of the experimental images of the heating foil temperature distributions, for the runs presented in the Table 2, leads to the following conclusions [11, 21, 22, 25]: Inlet pressure increasing is always connected with the increase of the inlet liquid subcooling. It results

in enlargement of heating surface temperature drop in „nucleation hysteresis”, accompanying boiling incipience.

In spite of increasing pressure in the system, the nucleate boiling incipience starts at almost a constant distance from the channel inlet. The distance of the boiling incipience occurrence from the inlet grows when the liquid flow velocity increases.

In most cases observed, the “boiling front” is manifested over a longer channel section, the higher liquid flow rate is.

The liquid flow velocity value does not significantly affect the temperature drop accompanying the “nucleation hysteresis”.

For low liquid velocities and low pressures, the “boiling front” is very unstable. Under such conditions, boiling incipience is characterised by small heating surface temperature drops. They occur over a very short channel section. The higher the flow rate and pressure are, the larger is the temperature drop, and at the same time, it appears over a longer channel section.


7. BOILING INCIPIENCE CORRELATION

While working out the results of experimental investigations, the following similarity numbers were used [9, 11, 20, 25]:

• Nusselt number:

l

hBI1,BI λ

DαNu

⋅= (43)

• Reynolds number – eq. 9, Table 1, • boiling number:

lvl

BI w,

hρuq

Bo⋅⋅

= (44)

• Prandtl number:

l

pll

λcµ

Pr⋅

= (45)

Out of two models discussed earlier, the one-dimensional model was chosen to determine the local heat transfer coefficient in eq. 14. The basic physical properties of the liquid are assumed on the basis of the literature [14] or by introducing approximation polynomials worked out by the authors.

In accordance with the requirements of dimensional analysis, applying multiple variable regression analysis, the dependence for Nusselt number was written as follows [20, 25]:

(49) 261640

BI PrBo)(Re232Nu ... −⋅⋅⋅= In order to estimate regression parameters values, 464 measurement results for refrigerant R123 nucleate boiling incipience in the minichannel were accounted for. The data were collected for BI points, where the constant heat flux was supplied to the heating surface. It shifted together with the “boiling front” in the direction opposite to the liquid flow direction in the channel as the heat flux supplied to the foil increased. NuBI number calculated from equation (49) – Fig. 14, was compared with Nusselt number, determined experimentally, eq. (43). Standard error of calculated Nusselt number, amount to approx. 0.4, whereas determination coefficient R2, being the measure of regression line matching accuracy (in the logarithmic system) – approx. 0.7. The equation analysed by the authors demonstrates congruence with over 99.6 % of observable “boiling fronts”, the tolerance being ±25 %. The dependence shown in equation (49) refers to the following range of similarity numbers: 6.06 ≤ NuBI 17.09; 580 Re 2 960; 1.34⋅10

≤ ≤ ≤-4 Bo 7.00⋅10≤ ≤ -4; 5.49 ≤ Pr 6.22. ≤

Fig. 14 Correlation (eq. 49) for predicting refrigerant 123 boiling incipience in the minichannel


8. ANALYSIS OF EXPERIMENTAL DATA AND NUMERICAL CALCULATIONS. CONCLUSIONS

In the minichannel flow boiling, considerable heat transfer enhancement takes place at boiling incipience. It is observed as a sharp increase in the heat transfer coefficient.

• In general, the following regularity should be noted: heat transfer coefficient values obtained for one-dimensional model are higher than those in two-dimensional model. The differences are thought to result from simplifications assumed in one-dimensional model, which does not cover the temperature gradient in the direction perpendicular to the fluid flow. Thus, we may assume that one-dimensional model can be successfully applied to work out the results of experimental investigations concerning heat transfer in the liquid flow through a vertical minichannel. It happens so because the local values of heat transfer coefficient are not greatly affected, provided that a very thin heater is applied.

• Detailed analysis with computational techniques applied to solve the two-dimensional model shows that in the majority of examples, the values of the local heat transfer coefficient obtained for one-dimensional model (Figs. 12d-f) are higher than those for the two-dimensional one, solved with the use of the least square technique (Figs. 13a-c). On the other hand, the values of heat transfer coefficient computed with Treffz equations are generally close to, or locally higher than those resulting from one-dimensional model. The greatest divergences are observed for the highest settings qV (Figs 13d, 13f).

• The local heat transfer coefficient dependence on the distance from the inlet, determined from two-dimensional model with the use of the least square technique, demonstrates polynomial character, with smoothed extremes, typical of those functions (Figs. 13a-c). “Polynomial smoothing” leads to averaged or rounded results. Yet, they seem to be far less sensitive to entrance data errors, especially when compared with two-dimensional model results and Treffz equations computing, but also with one-dimensional method.

• The sensitivity of Treffz technique results from the fact that experimental data are directly applied in order to determine constants in the approximations of glass and foil temperature functions. The one-dimensional model relies on the least square technique to determine the polynomial dependence of the heating surface temperature on the distance. In the two-dimensional model, where the least square is applied, the coefficients of heat polynomials are determined due to the minimisation of the functional. It specifies to what extent an approximation, which is sought, adjusts to the conditions assumed, i.e. – experimental data. In both above-mentioned methods, contrary to Treffz equations, experimental data are not used directly.

• The phenomenon of “nucleation hysteresis”, also called “zero boiling crisis”, was observed. It is characteristic of the boiling of a fluid of a small wetting angle. It is accompanied by a considerable heating surface temperature drop (up to 50 K). The appearance of boiling incipience is associated with the activation of some nuclei on the heated wall. It manifests itself as the sudden decrease in the wall temperature, as bubbles behave to be acting like heat sinks. That results in the diminishing of the local heating surface temperature.

• Apart from familiar shapes of boiling curves with one-step nucleation hysteresis available in the literature, a two-stepped course of the phenomenon was observed. Such a nucleation hysteresis was earlier reported in a few experimental investigations concerning surfaces with capillary porous coverings. Two-step nucleation hysteresis in the minichannel is accompanied by a phenomenon similar to II kind hysteresis, characteristic of nucleate pool boiling on developed micro-surfaces.

• The numbers chosen by the authors to develop correlations, namely Re, Pr and Bo, yielded the best results. They are generally believed to describe boiling process in the channel. A relatively high value of determination coefficient (≈0,7) and low values of standard errors (≈0,04) confirm a good choice of similarity numbers for the sake of heat transfer calculations in minichannel boiling. The result seems very satisfying, as for the majority of the equations, almost all measurements (99÷100%) show congruence with calculations with small tolerance ±25%.


NOMENCLATURE A – cross-section, m2

A1,A2,A3 – functions a, c – coefficients cpl – specific heat of liquid, J/kg K D – diameter, m d – coefficient G – liquid mass velocity, kg/m2 s H – height, m hlv – latent heat of vaporization, J/kg I – current supplied to the heating foil, A i – natural number, square root of –1 J – error functional j, k – natural numbers L – length, m M, N – total number of heat polynomials in an approximation temperature n – natural number O – perimeter, m p – pressure, N/m2

QV – volumetric flow rate, m3/s q – heat flux density,W/m2

qV – capacity of internal heat source, W/m3

T – temperature, K Tk – temperature at measurement points k, K u – liquid velocity, m/sun – harmonic polynomials u~ – particular solution of Poisson’s equation W – width, m w – solution of Poisson’s equation X, Y, Z – axes of Cartesian coordinate system x – distance from the channel inlet, m x, y – locations, m

Greek α(x) − local heat transfer coefficient, W/m2 K α1(x) − local heat transfer coefficient, obtained from one-dimensional model, W/m2 K α2/α2’(x) − local heat transfer coefficient, obtained from two-dimensional model with application of the least square technique/

the Treffz technique, W/m2 K Γ − boundary of the region Ω δ − width, m ∆ − Laplace’s operator ∆Tsat − surface superheating, Tw − Tsat, K ∆Tsub − inlet liquid subcooling (Tsat − Tl)inlet, K ∆U − the voltage drop across the foil, V θ − approximate temperature λ − thermal conductivity, W/m K µ − dynamic viscosity, kg/m s ρ − density, kg/m3

ψϕ , − real functions Ω − region Indexes BI − boiling incipience ch − inichannel F − foil G − glass h − hydraulic he − heated l − liquid / mixture of liquid and vapour sat − saturation w − wall

References

1. Bar-Cohen, A., Hysteresis phenomena at the onset of nucleate boiling, Proc. Engineering Foundation Conference on Pool and External Flow Boiling, Santa Barbara, CA, USA,1992, pp. 1-14.

2. Bohdal, T., Development of Bubbly Boiling in Channel Flow, Experimental Heat Transfer, 2001, vol. 14, No. 3, pp. 199-215.

3. Brauer, H., Mayinger, F., Onset of nucleate boiling and hysteresis effects under forced convection and pool boiling, Proc. Engineering Foundation Conference on Pool and External Flow Boiling, Santa Barbara, CA, USA,1992, pp. 15-36.

4. Celata, G.P., Cumo, M., Setaro, T., Hysteresis phenomena in subcooled flow boiling of well-wetting fluids, Experimental Heat Transfer, 1992, vol. 5, No. 4, pp. 253-275.

5. Hay, J.L., Hollingsworth, D.K., A comparison of trichromic systems for use in the calibration of polymer-dispersed thermochromic liquid crystals, Experimental Thermal and Fluid Science, 1996, vol. 12, pp. 1-12.

6. Hay, J.L., Hollingsworth, D.K., Calibration of micro-encapsulated liquid crystals using hue angle and a dimensionless temperature, Experimental Thermal and Fluid Science, 1998, vol. 18, pp. 251-257.


7. Lazarek, G.M., Black, S.H., Evaporative heat transfer, pressure drop and critical heat flux in a small vertical tube, International Journal of Heat Mass Transfer, 1992, vol. 25, pp. 945-960.

8. Orozco, J., Hanson, C., A study of mixed convection boiling heat transfer in narrow gaps, 1992, ASME, HTD-vol. 206-2, pp. 81-85.

9. Piasecka, M., Hozejowska, S., Poniewski, M.E., Experimental evaluation of flow boiling incipience of subcooled fluid in a narrow channel, International Journal of Heat and Fluid Flow, 2003, in print.

10. Simon, T. W., Forced convection boiling on small regions, Thermal Eng. Proc., ASME/JSME, 1991, vol. 2, pp. 259-268.

11. Piasecka, M., Poniewski, M.E., The onset of refrigerant R-123 flow boiling in a narrow vertical channel, Archives of Thermodynamics, 2003, vol. 24, No. 2, in print.

12. Piasecka, M., Hozejowska, S., Poniewski, M.E., Determination of local flow boiling heat transfer coefficient in narrow channel, Archives of Thermodynamics, 2003, vol. 24, No. 2, in print.

13. Collatz, L., Numerical methods of solving differential equations, PWN, Warsaw, Poland, 1960. 14. McLinden, M.O., Tables and diagrams for the refrigeration industry. Thermodynamic and Physical

Properties, Int. Institute of Refrigeration, France, 1995. 15. Bilicki, Z., The relation between the experiment and theory for nucleate forced boiling, Proc. 4th

World Conf. on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, Brussels, Belgium, 1997, vol. 2, pp. 571-578.

16. Chin, Y., Hollingsworth, D. K., Witte L. C., A study of convection in an asymmetrically heated duct using liquid crystal thermography, Proc. AIAA/ASME Thermophysics and Heat Conf., Albuquerque, USA, 1998, ASME, HTD 357-2, pp. 63-70.

17. Chin, Y., Hollingsworth, D.K., Witte L.C., Investigation of flow boiling incipience in a narrow rectangular channel using liquid crystal thermography, Proc. AIAA/ASME Thermophysics and Heat Conf., Albuquerque, USA, 1998, ASME, HTD 357-3, pp. 71-78.

18. Hozejowska, S., Piasecka, M., Poniewski, M. E., One- and two-dimensional models for local flow boiling heat transfer coefficient determination in narrow channel, Proc. 3rd Int. Conf. on Transport Phenomena in Multiphase Systems, Baranow Sandomierski, Poland, 2002, pp. 307-314.

19. Hozejowski, L., Hozejowska, S., Piasecka, M., Application of harmonic polynomials as complete solutions of Laplace equation in an inverse heat conduction problem, Annual Sci. Conf. GAMM’2002, Stuttgart, Germany, 2002, Book of Abstracts, in print.

20. Piasecka, M., Hozejowska, S., Poniewski, M.E., Experimental evaluation of flow boiling incipience of subcooled fluid in a narrow channel, Proc. 5th Int. Conf. on Boiling Heat Transfer, Montego Bay, Jamaica, 2003, pp. 47-64.

21. Piasecka, M., Poniewski, M.E, Flow boiling incipience of subcooled Freon 123 in narrow rectangular channel, Proc. 3rd Int. Conf. on Transport Phenomena in Multiphase Systems, Baranow Sandomierski, Poland, 2002, pp. 419-424.

22. Piasecka, M., Poniewski, M.E., Hysteresis phenomena at the onset of subcooled nucleate flow boiling in microchannels, Proc. 1st Int. Conf. on Microchannels and Minichannels, Rochester, USA, 2003, ASME, pp. 581-588.

23. Poniewski, M.E., Wisniewski, M.L., Wojcik, T.M., Boiling heat transfer on metal fibrous porous surfaces - experiment, model and verification, Proc. 5th Int. Conf. on Boiling Heat Transfer, Anchorage, USA, 2000, vol. 2, pp. 772-788.

24. Chin, Y., An experimental study on flow boiling in narrow channel: from convection to nucleate boiling, PhD thesis, University of Houston, Houston, USA, 1997.

25. Piasecka, M., Theoretical and experimental investigations into flow boiling heat transfer in a narrow channel, PhD thesis (in Polish), Kielce University of Technology, Kielce, Poland, 2002.

The work was carried out as a part of the grant KBN 8T10B 00519


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THEORETICAL AND EXPERIMENTAL ANALYSIS OF FLOW …liquid subcooling at inflow, flow velocity and pressure) on the incipience boiling in minichannels were also presented. Results were