Thermal design of symmetrically and asymmetrically heated channel–chimney systems in natural conv

Thermal design of symmetrically and asymmetrically heated channel–chimney systems in natural convection
Antonio Auletta,Oronzio Manca *,Marilena Musto,Sergio Nardini
Dipartimento di Ingegneria Aerospaziale e Meccanica,Seconda Universit a
degli Studi di Napoli,Real Casa dell’Annunziata,Via Roma 29,Aversa (CE)81031,Italy
Received 15May 2002;accepted 28November 2002
Abstract
In this paper,design charts for the evaluation of thermal parameters for natural convection with air in a channel–chimney system are proposed.
In the thermal analysis of natural convection in channel–chimney systems,the variables that play an important role are:the ohmic heat flux,maximum wall temperatures and geometrical parameters such as the height of the heated channel,the channel spacing and the height and spacing of unheated extensions.A simple numerical procedure to obtain the thermal design charts,a thermal optimization of the system and an uncertainty analysis due to the thermophysical properties are presented.Results are carried out for symmetrically and asymmetrically heated channels with walls at uniform heat flux and a simple estimation procedure is proposed to evaluate the error in the relevant geometrical and thermal parameters due to a different value of the reference temperature.The estimated error,however,is less than the uncertainty of the experimental data employed.Some simple examples are given to show the use of the charts.The proposed results are obtained from experimental data in the following dimensionless parameter ranges:5:06L h =b 620;1:56L =L h 64;16B =b 64;1026Ra 6106.Ó2003Elsevier Science Ltd.All rights reserved.
Keywords:Free convection;Chimney effect;Channel;Thermal design
1.Introduction
The thermal design of systems is very important for the industrial user and,for this reason,natural convection in partially open cavities such as the simple channel is widely researched.
*
Corresponding author.Tel.:+39-08-150-10217;fax:+39-08-150-10264.E-mail address:manca@unina.it (O.Manca).
1359-4311/03/$-see front matter Ó2003Elsevier Science Ltd.All rights reserved.
doi:10.1016/S1359-4311(02)00241-7
Applied Thermal Engineering 23(2003)605–621
/locate/apthermeng
This configuration is widely used in engineering applications,particularly in thermal control of components of electronic systems,as reported in [1–5]and reviewed more recently in [6,7].
The
606 A.Auletta et al./Applied Thermal Engineering 23(2003)605–621
A.Auletta et al./Applied Thermal Engineering23(2003)605–621607 present trend is also oriented toward the evaluation of optimal geometrical configurations that derive from the simple channel[6,8,9].In this context,a particularly interesting geometry is the channel–chimney system.Recently,this system has received much attention as shown in[10–24]. In[10],the chimney effect due to a parallel-walled adiabatic channel with heat sources at the channel inlet was analyzed.A similar configuration is the channel with straight adiabatic downstream extensions that was investigated in[11,14,18].In[11],a numerical study on a vertical channel with isothermal parallel walls was carried out and results for air were obtained.In[14],an investigation on a vertical isothermal or isoflux channel with straight unheated extensions located upstream or downstream of the channel was carried out numerically.In the case of the down-stream extensions,an increase of massflow rate and heat transfer was obtained.In[18]the problem treated in[14]was reformulated for an isoflux channel using the elliptic form of the governing equations in order to obtain a more realistic model.In these studies no thermal design evaluation and suggestions were given.
The effect of expansion ratio(chimney spacing/channel spacing)greater than1.0was also in-vestigated in[12,13,16,22–24].In[12],the increase of heat transfer rate in a vertical isothermal tube due to an unheated chimney attached downstream was studied numerically.An isothermal channel–chimney system was studied numerically and experimentally in[13].The effects of the geometry,i.e.the straight extensions and the abrupt or diffuse expansions,on the heat transfer characteristics in the channel were studied.A system with the symmetrically heated channel at uniform wall heatflux was studied experimentally in[16,22–24].In[16],results were derived in terms of geometric parameters and Rayleigh numbers.In[22],the experimental investigation extended the results given in[16].Optimal configurations were identified through the measured wall temperature profiles,with reference to the extension and expansion ratios of the insulated extensions.More recently,local temperature measurements of the airflow in the channel and the chimney were carried out in[23,24].In[23],the effect of the heating mode,symmetrical and asymmetrical,was pointed out.In[24],symmetrical heating was investigated more deeply.Dif-ferentfluid motion regions were observed inside the chimney and some of them confirmed the results of[23].No thermal design procedure was elaborated or proposed in[16,22–24].A periodic isothermal vertical channel expanded chimney was examined in[19].Each subsystem channel–chimney was the same as the analyzed configuration in[13].Results showed that the interaction between multiple channel–chimney systems presented an associated chimney effect stronger than in a single channel with adiabatic extensions.
Heat sink chimney systems were studied analytically in[15,17]and experimentally in[20,21]. The analytical solution for a vertical parallel-platefinned heat sink with a chimney was developed in[15].The solution permitted easy parameter evaluation aiming to recognize conditions of maximum heat transfer.In[17],the analysis was extended to pin-fin heat sinks.Optimal values of the pin-fin diameter and heat sink porosity were obtained for assigned thermal dissipation and system size.These two studies provided useful guidance for the thermal design of the heat sink. The experimental investigations carried out in[20,21]allowed a comparison with the results obtained in[15,17].In[20],a vertical parallel-platefinned heat sink with a chimney was inves-tigated.Results confirmed theoretical predictions for overall heat transfer and location of optima given in[15].In[21],the performance of pin-fin heat sink with a chimney was obtained.Exper-iments were compared with theoretical predictions given in[17]and a reasonable agreement was observed for overall heat transfer.
The above review shows that there is no information on thermal design for channel–chimney systems or on simple procedures to be used in order to optimize these systems.Following the procedure proposed in[25],in this study,charts allowing an easy evaluation of the significant parameters in the thermal design of a channel–chimney system are drawn up for air.The optimum spacing is also evaluated in graphical form in terms of the more significant parameters following the analysis given in[26,27].The charts are for symmetrically or asymmetrically heated channels. The charts for the thermal design and the optimization are drawn up for a given reference tem-perature in order to calculate thermophysical properties.The evaluation of the error due to the use of a different reference temperature is carried out and a method for the correction of the dependent variables on reference temperature is given.Some examples are presented to explain the graphical procedure.
2.Data reduction
In the thermal analysis of the natural convection in channel–chimney systems,the factors that play an important role(see Fig.1)are:the Ohmic heatflux q X,the wall temperatures and the geometrical parameters such as the heated channel height,L h,the channel spacing,b,the unheated extensions height and spacing,respectively L ext and B.Clearly,the channel–chimney system,in comparison to the simple channel,introduces,as dimensionless geometric ratios,not only the aspect ratio L h=b,but also the expansion ratio,B=b,and the extension ratio,L=L h,being L the height of the whole system(L¼L extþL h).The characteristic variables are the dimensionless maximum wall temperature and the channel Rayleigh number,defined as follows:
TÃ
w;max ¼
T w;maxÀT0
q c b=k
ð1
Þ
Fig.1.Sketch of the channel–chimney system.(a)Symmetrically heated channel;(b)asymmetrically heated channel. 608 A.Auletta et al./Applied Thermal Engineering23(2003)605–621
and
Ra¼Gr Pr¼g b
m2
q c b5
kL h
Prð2Þ
where q c is the mean value of the spatially averaged convective heatflux and is evaluated as follows:
q c¼
1
2L h
Z L h
q c;rðxÞd x
þ
Z L h
q c;lðxÞd x
ð3Þ
In the case of channel–chimney system,it is preferable to consider the product between the channel Rayleigh number and the expansion ratio,Ra B as a unique independent variable.This group is often related to the channel Nusselt number so defined
Nu¼
q c b
kðT wÀT0Þ
ð4Þ
The thermophysical properties of air are evaluated at the reference temperatureðT wþT0Þ=2,with
T w¼
1
2L h
Z L h
T w;rðxÞd x
þ
Z L h
T w;lðxÞd x
ð5Þ
It is worth noticing that the experimental results related to the q c values are valued by means of the relation
q c¼q XÀq rÀq kð6Þwhere q X is the heatflux due to the ohmic dissipation,(assumed to be uniform,and calculated by measuring a voltage drop and a current);q k is the conductive heatflux transferred from the plate to the ambient by conduction and calculated by a numerical procedure;q r is the radiative heatflux from the wall surface and its value is obtained,following the procedure reported in[25],by means of a numerical procedure for two-dimensional enclosure formed by the channel and the inlet and the outlet sections.The open boundary is assumed to be a black surface at ambient temperature. The channel walls are assumed to be grey.The hypothesis is that the two channel surfaces could be divided into a number of sub-surfaces and all subsurface temperatures of the enclosure are known.
3.Uncertainty analysis
The uncertainty in the calculated quantities was determined according to the standard single sample analysis recommended by[28,29].Accordingly,the uncertainty of a dependent variable U as a function of the uncertainties in the independent variables X i is given by the relation
d U¼
o U
o X1
d X1
2
"
þ
o U
o X2
d X2
2
þÁÁÁþ
o U
o X n
d X n
2#1=2
ð7Þ
A.Auletta et al./Applied Thermal Engineering23(2003)605–621609
The uncertainty in the values of the air thermophysical properties to be assumed negligible.On the basis of Eqs.(1),(2),(4)and (6)and of the maximum percentage uncertainties in the values of the independent variables,which are reported in Table 1,the maximum uncertainty in Ra ranged from 6%to 8%whereas the maximum uncertainty in Nu turned out to be 4–6%and approxi-mately the same in T Ã
w ;max .
4.Analysis and procedure 4.1.Maximum wall temperature
Dimensionless maximum wall temperatures were correlated to channel Rayleigh number,the extension ratio and the expansion ratio for both cases of symmetrically and asymmetrically heated configurations.The proposed correlation equations are in the form
T Ãw ;max ¼a ðRa B =b Þp
ðL =L h Þ
q
ð8Þ
whose coefficients a ,p and q were evaluated by means of the least squares method and are re-ported,together with the values of the regression coefficients,in Table 2.The coefficients in Eq.(8)are valid in the following intervals:1:56L =L h 64,16B =b 64,1026Ra 6106.
According to what was proposed in [25]and starting from Eq.(8),correlations between di-mensional quantities (maximum wall temperature rise,channel and chimney spacing and length,convective heat flux)can be derived.In this way dimensional graphical instruments (charts )can also be drawn.4.2.Design charts
In the following,the steps of the analytical procedure by means of which it is possible to furnish
dimensional relationships that relate the T Ã
w ;max to the project parameters are shown.
The procedure to obtain this dimensional relationship depends on the choice of the input and the output parameters.The construction of a graphic tool that allows the drawing of the channel
Table 1
Maximum percent uncertainties Variable D T max b q X q r q k q c Uncertainty
1.1%
1.2%
2%
5%
4%
3%
Table 2
Coefficients for Eq.(8)
a
p q r 2Symmetrical 1.35)0.191)0.03170.975Asymmetrical
1.69
)0.155
0.0288
0.955
610 A.Auletta et al./Applied Thermal Engineering 23(2003)605–621
spacing,b,with respect to a constrain on the maximum wall temperature,D T w;max,is reported, when the other geometrical dimensions of the system are known.
By manipulating Eq.(8)and transforming it into a dimensional form on the grounds of Eqs.(1) and(2)the following expression can be obtained
D T w;max¼c1q1þp
c b1þ4p
L h
b
Àp B
b
p L
L h
q
ð9Þ
where
c1¼a
k
g b
m2k
Pr
p
ð10Þ
is assumed‘‘constant’’.
Then,it is possible to proceed sequentially to the elimination of one variable at a time getting, for instance,the following dimensional groups
f1¼D T w;max
L h
b
p
¼c1q1þp
c
b1þ4p
B
b
p L
L h
q
ð11Þ
f2¼D T w;max
L h
b
p
qÀð1þpÞ
c
¼c1b1þ4p
B
b
p L
L h
q
ð12Þ
f3¼D T w;max
L h
b
p
qÀð1þpÞ
c
L
L h
Àq
¼c1b1þ4p
B
b
p
ð13Þ
f4¼D T w;max
L h
b
p
qÀð1þpÞ
c
L
L h
Àq B
b
Àp
¼c1b1þ4pð14Þ
through the evaluation of which it is possible,in the end,to reach the value of the channel spacing, b,that allows the maximum wall temperature to be maintained at afixed value.It is useful to observe that the technique under examination can be used both in the design phase and in the phase of the verification of the performance of an existing channel.
4.3.Optimization of the channel spacing
The evaluation of the optimum value of the spacing between the plates,b opt,was obtained by following the procedure suggested in[26,27].A result of this evaluation is a value for the distance between the plates which minimizes either the difference between the average wall temperature and the entering air temperature,T wÀT0,for a determined total heat transfer rate,Q T,or maximizes the total heat transfer rate for an established value of the difference between the average wall temperature and the entering air temperature.Hence,the distance value b,which minimizes the ðT wÀT0Þ=Q T ratio,is the optimum spacing value.In the following the difference T wÀT0is set as D T.
The total heat transfer rate from the walls of a channel,Q T,is given by
Q T¼2Sqð15Þ
A.Auletta et al./Applied Thermal Engineering23(2003)605–621611
where S is the single plate surface.From the definition of the average Nusselt number
q¼h D T¼Nuk
b
D Tð16Þ
hence
D T Q T ¼
b
2Sk Nu
ð17Þ
The average Nusselt number is obtained by the following composite correlation as reported in[22] for a symmetrically heated channel–chimney system and,in[23],for the asymmetrically heated configuration
Nu¼
L
L h
m
c1Ra
B
b
n
q
þc2Ra
B
b
p
q 1=q
ð18Þ
in terms of the asymptotic limits in the case of fully developedflow and isolated plate.The co-efficients values are reported in Table3.These coefficients are valid in the following intervals: 1:56L=L h64;16B=b64;1026Ra6106
From the definition of Rayleigh number,Eq.(2),and from relation(18),the Eq.(17)is found to reduce to
D T Q T ¼
b L
h
Àm
c1c0q X
h
B b5
n
h i q
þc2c0q X
h
B b5
p
h i q
n oÀ1=q
2kS
ð19Þ
where
c0¼Pr g b
m2k
ð20Þ
By differentiating Eq.(19)with respect to b and setting the derivative to zero,it is possible tofind the optimal value for the channel spacing,b opt,that minimizes D T=Q T ratio
b opt¼K
B
b
q X
L h
À1=5
ð21Þ
where
K¼c0
c1 c2
"
8
<
:À5nÀ1
5pÀ1
1=q#1=ðnÀpÞ9=
;
À1=5
ð22Þ
Table3
Coefficients for Eq.(18)
c1c2m n p q r2 Symmetrical0.259 1.420.02680.3990.150)2.020.965 Asymmetrical 1.64 1.63)0.02870.7480.144)0.5780.979 612 A.Auletta et al./Applied Thermal Engineering23(2003)605–621
Hence,the optimum value of the channel spacing is
b opt ¼6:79R
B
b À0:2
ð23Þ
for the symmetrical heating,and
b opt ¼3:69R
B
b À0:2
ð24Þ
for the asymmetrical heating,with
R ¼
g b m 2k q X L h Pr ¼c 0
q X
L h
ð25Þ
according to the procedure suggested by Bar-Cohen and Rohsenow [26].
5.Evaluation of the error due to the reference temperature
The charts proposed in this paper are in terms of dimensional variables.A reference temper-ature has to be introduced in order to obtain dimensional variables from dimensionless ones.This
means that a number of charts should be introduced for each reference temperature.In the fol-lowing,a procedure to correct an error due to a reference temperature different from the chosen one is proposed.This allows the introduction of a unique group of charts at an assigned reference temperature,which can be employed for every condition.A sensitivity analysis of the parameters with respect to the reference temperature is carried out according to the procedure suggested in [28,29].
The significant variables for thermal design are D T max ,q X ,L h ,L ext ,B and b .From Eq.(9)each variable U can be written as follows:
U ¼const m a k b b c
ð26Þ
where a ,b and c are reported in Table 2.Hence
d U U 2¼a d m m 2þb d k k 2þc d b
b
2ð27Þ
From [30]the dependence of k and m on reference temperature is expressed by
k ¼k 0ðT r =T 0Þ
0:814
ð28Þm ¼m 0ðT r =T 0Þ1:733
ð29Þ
For T 0¼273:15K one has k 0¼0:0241W/m K and m 0¼13:3Â10À6m 2/s,T r is in Kelvin.Since b ¼1=T r Eq.(27)becomes d U U ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3:00a 2þ0:633b 2þc 2
p d T r T r ¼f d T r
T r ð30ÞThe values of the coefficient f for the significant variables are presented in Table 4.
A.Auletta et al./Applied Thermal Engineering 23(2003)605–621613
The Eq.(26)can be rewritten taking into account the Eqs.(28)and (29)as follows:U ¼const T n
r ð31Þ
with n ¼1:733a þ0:814b Àc ,where a ,b and c are those reported in Table 2.In addition,the values of n are reported in Table 4for both cases of heating.If U r0is the value of the variable U at T r0¼40°C ¼313K,one has
U U r0¼T r
313 n
ð32ÞThe reference temperature is defined as follows:
T r ¼
T w þT 0
2
ð33Þ
where T w is the average temperature of the channel walls.
In a channel the value of D T w ¼T w ÀT 0can be related to the value of D T w ;max ¼T w ;max ÀT 0.From experimental data this relation can be written as
D T w ¼f 1þf 2D T w ;max
ð34Þ
where,for instance,in the symmetrical case the values of f 1and f 2are )0.0952and 0.899,re-spectively.
Set T r as the value of the reference temperature at the design condition,one has
T r ¼
T w þT 02
¼1
2D T þT 0
ð35Þ
hence,for the Eq.(34)
T r ¼
f 1þf 2D T w ;max
2
þT 0
ð36Þ
Finally the Eq.(32)can be rewritten as follows:
U U r0¼f 1þf 2D T w ;max 2:313 þT 0
313
n
ð37Þ
When D T w ;max is known,the Eq.(37)provides the correction of one of the thermal or geometrical variables directly.When D T w ;max is unknown,one can proceed by means of an iterative method.
Table 4
Coefficients for Eqs.(34)and (35)
a
b c f n Symmetrical 0.382)0.809)0.1910.9430.194Asymmetrical
0.440
)0.780
0.220
1.01
0.348
614 A.Auletta et al./Applied Thermal Engineering 23(2003)605–621
6.Results and discussion
6.1.Design charts
The dimensional terms D T w;max,q X and b together with the non-dimensional ratios L h=b,B=b and L=L h are related by Eq.(9)whose coefficients are given in Table2.So,charts for thermal design are drawn up for symmetrical and asymmetrical heated channel–chimney systems.These charts allow the evaluation of any one parameter among the previously cited dimensional or non-dimensional terms if the other ones are known.Charts are drawn up by assuming thermophysical properties evaluated at40°C.
Figs.2and3show charts for both symmetrical and asymmetrical configurations.In Fig.2a steep variation of the value assumed by b is noticeable.Moreover,this curve reaches lower values
Fig.2.Design chart for the symmetrically heated configuration.
in correspondence with high values of heat flux.It can be interpreted as a consequence of the stronger driving force necessary to overcome a bigger drag related to the presence of a narrower channel.On the other hand,with constant values for driving force and channel spacing,if the expansion ratio increases maximum wall temperature decreases,for an assigned aspect ratio.Charts show that the influence of L =L h with respect to the other non-dimensional ratios is very little.The same can also be said of the asymmetrical configuration (Fig.3).
6.2.Example of application
Figs.4and 5show an application example of this graphical instrument for the symmetrical and the asymmetrical configurations,respectively.It consists in evaluating the value of channel spacing b when the other geometrical and thermal variables are known.The following values are
assumed:
Fig.3.Design chart for the asymmetrically heated configuration.
D T w ;max ¼15°C ;L h =b ¼10;q X ¼300W =m 2;L =L h ¼2:0;B =b ¼4:0:
The results obtained were b ¼0:0087and 0.0021m for the symmetrical and asymmetrical con-figurations,respectively.
It can be observed that channel spacing evaluated for the symmetrical system is bigger than the one obtained for the asymmetrically heated channel.Analogously,a reduction of b causes a re-duction in D T w ;max .
6.3.Optimization of the channel spacing
In Fig.6a,D T =Q T ratio values,Eq.(19),related to the symmetrical configuration are reported in terms of channel spacing b ,for B =b ¼2:0and q X =L h between 500and 5000W/m 3.It
is
Fig.4.Example of use for a symmetric configuration design.
noticeable that the diagrams decrease dramatically for the low values of b .They reach a minimum depending on q X =L h ratio,then increase with a negative concavity.This means that,for very narrow channels,a small increase in spacing value gives rise to a big improvement in the chimney effect and,therefore,of thermal performance.After reaching minimum,an increase of b value does not give rise to appreciable variations in the thermal performance of the system and the ratio D T =Q T tends to the single plate asymptotic value.
A similar path is observed in Fig.6b where values of D T =Q T ratio related to the symmetrical configuration are reported,in terms of channel spacing b ,for q X =L h ¼2000W/m 3and
B =b varying between 1.0and 4.0.
In Fig.7optimal channel spacing values are reported as a function of expansion ratio B =b and for several values of q X =L h and also for both symmetrical (Fig.7a)and asymmetrical configu-rations (Fig.7b).It is noticeable that,under the same conditions,for the symmetrical case b
opt
Fig.5.Example of use for an asymmetric configuration design.
value is almost twice the b opt value for the asymmetrical case.Moreover,for assigned B =b ,b opt decreases when D T =Q T increases and this indicates that when driving force decreases the channel has to be enlarged in order to compensate for the pressure drop.
Eq.(21)together with Eq.(22)show the relation between the optimal channel spacing,ther-mophysical properties,the expansion ratio B =b and the q X =L h ratio.Evaluating b opt for assigned B =b ,q X and L h values,means finding the optimal channel spacing b for a given configuration that leads to a minimum value of D T ,as reported in [25].Moreover,for an assigned b value,the value of the chimney spacing,B ,that thermally optimizes the channel–chimney configuration is also determined from Eq.(21).Finally,for an assigned B value constrained,for instance,by space limitations,Eq.(21)directly permits the evaluation of the optimal channel
gap.
Fig.6.D T =Q T ratio behavior for symmetrically heated configuration:(a)vs.channel spacing b ,for different q X =L h ratio values and with B =b ¼2:0;(b)vs.channel spacing b ,for different B =b ratio values and with q X =L h ¼2000W/m 3
.Fig.7.Optimal channel spacing behavior vs.B =b spacing,for different q X =L h ratio values:(a)symmetrically heated configuration;(b)asymmetrically heated configuration.
7.Conclusions
Correlation equations between dimensionless maximum wall temperature,Rayleigh number and geometrical parameters for natural convection of air in channel–chimney systems with symmetrically or asymmetrically heated walls are proposed.The validity ranges are:5:06L h= b620;1:56L=L h64,16B=b64,1026Ra6106.
They are very simply expressed and allow the design of channel–chimney systems by means of an easy graphical procedure.The charts are obtained for a reference temperature equal to40°C and a procedure is proposed for the correction of the variable values calculated with a different reference temperature,as suggested in[25].The estimated error,however,is less than the un-certainty of the experimental data employed.In order to aid understanding,examples of appli-cations are given.The spacing value,which optimises the thermal performance of configurations, is obtained from composite correlation equations between average Nusselt and Rayleigh numbers as functions of geometrical parameters.Results are presented in both analytical and graphical form.For single channel–chimney systems useful indications for isothermal channel walls can be found in[13,15,20].Several suggestions for extending the proposed design procedure to systems of parallel chimneys–channels can be obtained from[19].Particularly,in a system of parallel vertical chimneys–channels,the interaction between chimney and channel increases with the Grashof number and decreases with the channel spacing[19].This recommendation can be considered valid up to channel inclination angles of45°from the vertical[31],whereas for higher angles the flow exiting from a channel strongly interferes with theflow exiting from the above channel.It should be noted that studies on isoflux channels for different angles of inclination are needed in order to obtain more accurate information for these configurations,since studies[19,31]are in-vestigations into isothermal systems.
Acknowledgements
This research was supported by CNR under grant Bilateral research no.99.01943.CT07and MURST under2001grant research program.
References
[1]F.P.Incropera,Convection heat transfer in electronic equipment cooling,ASME Journal of Heat Transfer110(4b)
(1988)1097–1111.
[2]R.J.Moffat,A.Ortega,in:A.Bar-Cohen,A.D.Kraus(Eds.),Advances in Thermal Modelling of Electronic
Components and Systems,vol.1,Hemisphere,Washington,DC,1988,pp.129–282.
[3]G.P.Peterson,A.Ortega,in:Advances in Heat Transfer,vol.20,Academic Press,New York,1990,pp.181–314.
[4]T.Aihara,in:Cooling Techniques for Computers,Hemisphere,Washington,DC,1991,pp.1–45.
[5]A.Bar-Cohen,A.D.Kraus,in:Advances in Thermal Modelling of Electronic Components and Systems,vol.4,
ASME PRESS,New York,1998,pp.1–469.
[6]O.Manca,B.Morrone,S.Nardini,V.Naso,Natural convection in open channels,in:Computational Analysis of
Convection Heat Transfer,WIT Press,Southampton,2000,pp.235–278.
[7]N.Bianco,B.Morrone,S.Nardini,V.Naso,Air natural convection between inclined parallel plates with uniform
heatflux at the walls,International Journal of Heat and Technology18(2)(2000)27–36.。