A fractal analysis of subcooled flow boiling heat transfer
北海道面包Boqi Xiao
a,b
,Boming Yu
a,*
a
Department of Physics,Huazhong University of Science and Technology,1037Luoyu Road,Wuhan 430074,Hubei,PR China
b
Department of Physics and Electromechanical Engineering,Sanming University,25Jingdong Road,
Sanming 365004,Fujian,PR China
Received 18July 2006;received in revid form 8April 2007
Abstract
A fractal model for the subcooled flow boiling heat transfer is propod in this paper.The analytical expressions for the subcooled flow boiling heat transfer are derived bad on the fractal distribution of nucleation sites on boiling surfaces.The propod fractal model for the subcooled flow boiling heat transfer is found to be a function of wall superheat,liquid subcooling,bulk velocity of fluid (or Reynolds number),fractal dimension,the minimum and maximum active cavity size,the contact angle and physical properties of fluid.No additional/new empirical constant is introduced,and the propod model contains less empirical constants than the conventional models.The propod model takes into account all the pos-sible mechanisms for subcooled flow boiling heat transfer.The model predictions are compared with the existing experi-mental data,and fair agreement between the model predictions and experimental data is found for different bulk flow rates.Ó2007Elvier Ltd.All rights rerved.
Keywords:Subcooled flow boiling;Fractal;Heat transfer
1.Introduction
冲绳民谣The subcooled flow boiling is widely applied in engineering and technology.In the development of mo
dern cooling systems such as internal combustion engines,power engineering,nuclear reactors and microproces-sors,the increasing output of specific power combined with a most compact space and weight saving design leads to high thermal loads on heating surfaces.For such devices,a high coolant power is to be required with limitations on the available surface area and the mass flux of liquid coolant as well as the acceptable wall tem-perature,and a controlled operating mode to the boiling regime is desirable.
In subcooled flow boiling heat transfer it is generally recognized that there are three main mechanisms con-tributing to the wall heat flux (q W ):single-pha heat transfer (q sp ),micro-layer evaporation (q ev ),and the
0301-9322/$-e front matter Ó2007Elvier Ltd.All rights rerved.doi:10.1016/j.ijmultiphaflow.2007.05.001
*
Corresponding author.
E-mail address: (B.
Yu).
International Journal of Multipha Flow 33(2007)
1126–1139
基于沸腾表面得成核集不规则分配而衍生的本文所推荐的
过冷沸腾传热
不规则模型是
由壁面过热丆
过冷流体丆流体体积速率或
雷诺数丆不规则容积最小及幼儿园大班教育笔记
最大主动腔尺寸丆接触角和
流体物理特性而得出的方程。
过冷沸腾广泛的应用到工程技术领域，在现代冷却系统的发展中，如内燃机，发动机，核反应堆与微处理，结合极多的紧凑表面及节省重量的设计的越来越多的输出具体功率导致高热量加载于供热表面。对于这个设备，对于有效表面区域的局限性丆高丠功率是必需的.并且液体流量团也是可接受的壁温度，所以一个受约束的沸腾体系操作模型是值得的.
nsible heat of fluid that occupies the volume evacuated by a departing bubble (q b ).Thus the wall heat flux can be expresd as q W ¼q sp þq ev þq b
ð1Þ
Bowring (1962)obtained the relation between q ev and q b as
e ¼q b =q ev
ð2Þ
The evaporation heat flux (q ev )was given by
q ev ¼q G h L V b fN a
ð3Þ
where V b is the volume of single bubble at departure,f is the bubble departure frequency,N a is the number of active sites per unit area of heated surfaces,h L is latent heat of evaporation of liquid,and q G is the vapor den-sity.The ratio e is found empirically,which was given by the following expression:
e ¼1þ3:2
q c p D T sub
q G h L
1Â105Pa 6p 69:5Â105Pa
ð4a Þe ¼2:39:5Â105Pa 6p 650Â105Pa ð4b Þe ¼2:6p P 50Â105Pa
ð4c Þ
where p is pressure,q is the liquid density,c p is specific heat at constant pressure,D T sub is the subcooling (T S ÀT L )of liquid,and T S is the saturation temperature of liquid,T L is the bulk temperature of liquid.The single-pha heat transfer (q sp )is given by Mikic and Rohnow (1969)as
q sp ¼ð1ÀKN a p D 2b Þh ðT W ÀT L Þ
ð5Þ
where K is the proportional constant for bubble diameter of influence,which is taken to be 1.8by Judd
and Hwang (1976),D b is bubble departure diameter,T W is the wall temperature,and h is the single-pha heat transfer coefficient for forced convection,which can be calculated using the Dittus–Boelter equation (1930)
h ¼0:023Re 0:8Pr 0:4k L
D
ð6Þ
where D is the inner diameter of flow channel,k L is thermal conductivity of liquid,Pr is the Prandtl number of fluid defined by Pr =m /a ,Re is the Reynolds number defined by Re =uD /m ,and m is kinematic viscosity of
fluid,a is thermal diffusivity of fluid,u is the bulk velocity of fluid.
As discusd by Basu et al.(2002,2005a,b),a quantitative prediction of subcooled flow boiling heat flux from a superheated wall bad on Eqs.(1)–(6)requires the knowledge of veral additional empirical constants becau each of the quantities D b (or V b ),f and N a contains veral empirical constants,which usually have no
physical meanings.On the other hand,the calculation of subcooled flow boiling heat transfer so far lacks the connsus as to which t of empirical constants is to be ud since different authors ud different correlations.
Until now no united mechanistic model is available becau boiling is a very complex and elusive process.
From the earlier literature review of the available models for prediction of wall heat flux in flow boiling,it
is evident that in most cas not all the mechanisms have been taken into account.Some studies ignored
the contribution of heat transfer due to liquid circulation caud by bubbles disrupting the boundary layer and only considered q sp and q ev as the sum of the wall heat flux.The models by Larn and Tong (1969),Ahmad (1970),Hancox and Nicol (1971),Maroti (1977),Lahey (1978),Chatoorgoon et al.(1992),Zeitoun
(1994)fall into the above category.Most of them do not calculate q ev directly,but indirectly by knowin
g q W and calculating q sp .In most studies some models were developed as a part of the modeling for void frac-tion,the independent validation of q W partitioning has never been carried out though the overall model val-idation for void fraction prediction has been justified.Since most of the correlations were developed at high pressures and high velocity conditions,at low pressures the comparison on the above models with experimen-tal data shows great discrepancies.
From the above brief review it is en that a mechanistic model has not yet been developed,in which every component of wall heat flux should be determined independently.The modeling should be such that the
B.Xiao,B.Yu /International Journal of Multipha Flow 33(2007)1126–11391127
壁热流量可以表示成
蒸发热流量
单个脱离气泡的体积
加热表面的单元面积的主动集百分数
液体蒸发潜在蒸发热
蒸气密度经验数
恒压下比热
运动粘度
热扩散率热传导率
物理意义
计算
远缺乏共识
相关
作者和谐的
难以预测文献预测
显然
忽略了由于液体流通而使气泡扰乱边界层的传热贡献
扰乱
以上此类直接计算
空隙率
独立验证分区
整体模型验证
相关
独立决定
empirical relations ud reprent the physical process of subcooledflow boiling phenomena.In this paper,we attempt to develop a mechanistic model for subcooledflow boiling bad on the fractal characteristics of sizes of active cavities on heated surfaces,and on the available relations that the volume of single bubble at depar-ture and the bubble relea frequency are related to active sizes.
2.Fractal characters of nucleation sites on boiling surfaces for subcooledflow boiling
This work is devoted to deriving a subcooledflow boiling model bad on the fractal distribution of nucle-ation sites N a on heated surfaces.We consider the active cavities formed on the heated surface are analogous to pores in porous media.Yu and Cheng(2002a)found that the cumulative number(N)of pores in porous media with the diameter greater than and equal to a particular value,D s,obeys the following fractal scaling law:
NðD L P D sÞ¼ðD s;max=D sÞd f with D s;min6D s6D s;maxð7aÞwhere D s,max is the maximum diameter of pores in porous media,D s,min is the minimum diameter of pores in porous media,D s is the diameter of a pore,and d f is the fractal dimension.If active cavities formed on the heated surface are analogous to pores in pores media,the cumulative number of active cavities with diameters greater than and equal to D c can also be described by Eq.(7a)with N and D s replaced by N a and D c ,
N aðD L P D cÞ¼ðD c;max=D cÞd f with D c;min6D c6D c;maxð7bÞwhere D c is the diameter of a active cavity,D c,max and D c,min are respectively the maximum and minimum diameters of active cavity.The total number of nucleation sites(N a,tot)per unit area from the minimum active cavity to the maximum active cavity can be obtained from Eq.(7b)as
N a;tot¼
D c;max
D c;min
d f
ð8Þ
0.00.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1
2
3
4
5
6
7
8
9
10
d
f
=-slope=1.81
L
n
(
N
a
)
Ln (D c)
1128 B.Xiao,B.Yu/International Journal of Multipha Flow33(2007)1126–1139
描绘
不规则特性
释放速率
致力于源于
类似气孔多孔介质累积
过冷沸腾的成核集沸腾表面的不规则特性
大于或等于
服从于
不规则尺寸
类似于
各自的
获得
有效腔直径
单位面积成核集总数
盒子数理论下的成核集不规则尺寸的判断
The minimum active cavity radius R min and the maximum active cavity radius R max are given by Hsu(1962) for nucleation site distribution:
R min¼
d
C1
1À
h S
h W
À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1À
h S
h W
2
À
4f C3
dh W
s
2
4
3
5ð9aÞ
R max¼
d
C1
1À
h S
h W
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1À
h S
h W
2
À
4f C3
dh W
s
2
4
3
5ð9bÞ
For subcooledflow boiling,h W=T WÀT L=T WÀT S+T SÀT L=D T W+D T sub,h S=T SÀT L=D T sub, and D T W is wall superheat(T WÀT S).So the minimum active cavity radius R min and the maximum active cav-ity radius R max can be predicted from Eq.(9)for subcooledflow boiling as
R min¼
d
C1
1À
D T sub
D T WþD T sub
À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1À
D T sub
D T WþD T sub
2
À
4f C3
dðD T WþD T subÞ
s
2
4
3
5ð10aÞ
R max¼
d
C1
1À
车的档位D T sub
D T WþD T sub
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1À
D T sub
D T WþD T sub
2
À
4f C3
dðD T WþD T subÞ
s
2
4
3
5ð10bÞ
0.00.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0服务承诺制
1
2
3
4
5
6
7
8
9
10
d
f
=-slope=1.79
L
n
(
N
a
)
Ln (D c)
B.Xiao,B.Yu/International Journal of Multipha Flow33(2007)1126–11391129
最小乮最大乯有效腔半径
预计
where f ¼2r T S
G L
,C 1¼ð1þcos /Þ
and C 3=1+cos /,with /being the contact angle of the fluid and the heater material,and r is surface tension of fluid.d is the thermal boundary layer thickness which can be usually ex-presd as d ¼
k L
ð11Þ
where h is the single-pha heat transfer coefficient for forced convection,which is calculated by Eq.(6).For a boiling system,the fractal dimension d f of nucleation sites is given by Yu and Cheng (2002b)as
贴美甲纸d f ¼ln 1
D c ;max
c ;min
大米粒
2
ln D
c ;max
c ;min
ð12Þwhere D c ;max is the averaged value over all the maximum active cavities as
D c ;max ¼1ðT W ÀT S ÞZ T W T S D c ;max ðT W Þd T W ¼1D T W X m j ¼1D c ;max ðT W j Þd T W ¼1m X
m
j ¼1
D c ;max ðT W j Þ
ð13Þ
where m =D T W /d T W ,and a constant d T W is assumed.In the above equation,T W j ¼T S þj ðd T W Þwith j =1,
2,...,m .For example,if we choo d T W =0.2°C then m =5for D T W =1°C,and m =50for D T W =10°C.
0.00.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1
2
34567
8910d f =-slope=1.80
L n (N a )
Ln (D c )
1130 B.Xiao,B.Yu /International Journal of Multipha Flow 33(2007)1126–1139
流体集热材料的常数角度
流体表面张力
边界层热量厚度
强制对流下单相传热系数
最大有效腔平均值
被假定
Figs.1a,2a and 3a are three photo images from the reference by Chang et al.(2002)for nucleation sites in
subcooled flow boiling experiments.A vertical,one-side heated rectangular channel was ud as test ction.The experimental conditions were the following:p =1.13·105Pa,G =2000kg/m 2s (mass flux),T S =103°C,D T sub =42.7°C,and q W =6.1MW/m 2for Fig.1a;T S =103°C,D T sub =43.1°C,and
q W =5.8MW/m 2for Fig.2a;T S =103°C,D T sub =41.6°C,and q W =7.0MW/m 2for Fig.3a.A linear rela-tionship on the log–log coordinate can be obtained as shown in Figs.1b,2b and 3b after the box-counting
Table 1
A comparison on the fractal dimensions between the prent model predictions by Eq.(12)and the Box-counting method applied to the experimental data by Chang et al.(2002)at pressure p =1.13·105Pa and mass flux G =2000(kg/m 2s)Ca no.q W (MW/m 2)D T sub (°C)Method and model predictions d f 1 6.142.7Box-counting method for Fig.1a 1.81Prediction by Eq.(12)
1.842 5.843.1Box-counting method for Fig.2a 1.79Prediction by Eq.(12)
1.793
7.0
41.6
Box-counting method for Fig.3a 1.80Prediction by Eq.(12)
1.81
野荠菜
B.Xiao,B.Yu /International Journal of Multipha Flow 33(2007)1126–11391131
壁热流量也液体过冷温度参考
垂直矩形通道
实验截面线性关系
