A simplified model for shell-and-tubes heat exchangers:Practical application
F.Vera-García a,*,J.R.García-Cascales a ,J.Gonzálvez-Maciáb ,R.Cabello c ,R.Llopis c ,D.Sanchez c ,E.Torrella c
a
Thermal and Fluid Engineering Department,Universidad Politécnica de Cartagena,Cartagena,Murcia,Spain b
Applied Thermodynamics Department,Universidad Politécnica de Valencia,Valencia,Spain c
Department of Mechanical Engineering and Construction,Universidad Jaume I,Castellón,Spain
a r t i c l e i n f o Article history:
Received 22September 2009Accepted 7February 2010
Available online 13February 2010Keywords:
Refrigerating systems Evaporator Condenser
Shell-and-tube exchanger Modelling
Heat exchanger
a b s t r a c t
In this paper,a simplified model for the study of shell-and-tubes heat exchangers (HXs)is proposed.The model aims to agree with the HXs when they are working as condensers or evaporators.Despite its sim-plicity,the model proves to be useful to the pre-designment and correct selection of shell-and-tubes HXs working at full and complex refrigeration systems.The heat transfer coefficient and pressure drop corre-lations are specially selected and treated to implement them into the shell-and-tubes HXs presented.The model is implemented and tested in the modellization of a general refrigeration cycle and the results are compared with data obtained from a specific test bench for the analysis of shell-and-tubes HXs.
Ó2010Elsevier Ltd.All rights reserved.
1.Introduction
The modelling of refrigeration and air-conditioning systems be-gan about half a century ago.There are
several kinds of models for this type of installation.Nowadays,global models assist in the de-sign of refrigeration and/or air-conditioning systems to ensure installations are efficient.
There are many models to characterise the behaviour of the heat exchangers encountered in refrigeration cycles.There are sev-eral correlations available,so that the heat transferred and the pressure drop can be evaluated.They can be more or less complex depending on the accuracy required.
Geometrical info regarding HXs is required for these analysis.Unfortunately,this is not always available for shell-and-tubes heat exchangers due to confidentiality issues.Geometrical information is often missing and only some reference values is occasionally found in catalogues for particular experimental conditions.Although there are plenty of detailed models,they cannot be used due to this lack of information which makes the modelling process complicated.
In order to overcome this,a simple model is developed in this paper to determine the outlet conditions of shell-and-tubes heat exchangers working in a refrigeration cycle either as condensers or evaporators.
Shell-and-tube models can be classified according to the dis-cretisation detail used by different types o
f models,such as:one zone;two zone;tube by tube and cell by cell models.The model proposed here belongs to the first type,a one zone global model.One zone global models assume a global heat transfer coefficient for the whole heat exchanger that can be dependent on variables,such as air flow rate and evaporator global heat flux.Calculations of the HX uses the -NTU or LMTD solutions found for single-phase HXs.Examples of this type of approach are found in [7].
In the literature,it is possible to find several models of shell-and-tubes HXs:[19,18,26,23].The majority of these models are more complicated than the model presented in this paper.For example,Allen and Gosselin [1]present a model that estimate the total cost of shell-and-tube HXs with condensation in tubes or in shell.This model and optimization is based on a genetic algo-rithm that optimized 11variables of the HXs.Ten of them are asso-ciated with the geometry:tube pitch,tube layout patterns,baffle spacing at the center,baffle spacing at the inlet and outlet,etc.The 11th variable is the condensing fluid.The model offers the users the possibility to identify the best internal design for a given heat transfer process between two fluids,one of which is condens-ing.Where the internal geometry of the shell-and-tube HX is known,the model is highly applicable.
On the other hand,Yanik and Webb [27]present a model that predict the heat transfer and the pressure drop on a 4-pass per tube-side in a shell-and-tube HX working as evaporator.This mod-el uses detaile
d correlations to obtain the behaviour of the fluid in the tubes,and compares the results with experimental data ob-tained from the test of a single tube.In this model,the refrigerant
1359-4311/$-see front matter Ó2010Elsevier Ltd.All rights reserved.doi:10.1016/j.applthermaleng.2010.02.004
*Corresponding author.Address:Thermal and Fluid Engineering Department,Universidad Politécnica de Cartagena,30202Cartagena,Murcia,Spain.Tel.:+34968325987;fax:+34968325999.
E-mail address:francisco.vera@upct.es (F.Vera-García).Applied Thermal Engineering 30(2010)
1231–1241
Contents lists available at ScienceDirect
Applied Thermal Engineering
j o u r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /a p t h e r m e n
g
and the water were in baffled crossflow.The tube was divided into 1.0ft incremental lengths along the refrigerantflow for each pass, thus the tube was divided into24axial increments.The calcula-tions started at the refrigerant inlet and proceeded through incre-mental steps to the refrigerant exit.The analysis used the LMTD method to calculate the heat transfer in each step.In this case, the internal geometry of the shell-and-tube HX is needed to model the HX correctly.
The usefulness of the model presented here is restricted to cases where minimum information on the performance of heat exchang-ers is provided by the manufacturer.As detailed,the objective of the model is to provide the outlet conditions,where the inlet con-ditions are known as part of the modelling procedure.
In this paper,the model is presented and the correlations used to estimate the heat transfer coefficients are introduced.These coefficients depend on the type of HX considered in the modelling process(condensers or evaporators).There is then a description of the testing facilities,test measures and measuring process.All components of the compressed refrigeration cycle are modelled with the proposed shell-and-tubes HX model.There is a compari-son between the measured and modelled results andfinally,the conclusions are presented,as well as some applications of the pro-posed model.
2.Model description
Global models assist in the design of refrigeration and/or air-conditioning systems to ensure installations are efficient.Individ-ual models of each component are required to ensure the accuracy of the global model.The connection between the models of indi-vidual components yields a system of equations which can be solved by means of a Newton–Raphson algorithm.This is the case for the glob
al model used in this study which yields the steady state solution to model a global installation.This global model is implemented in the ARTÓcode environment,which has been widely tested,used and cited by several authors[11,16,15,25].
While each component could be modelled with a different level of approximation,this article is focused on the modellisation of heat exchanger components.When HXs are working as condensers and/or evaporators,it is common to know the inlet conditions:the fluids’inlet temperaturesðT0
1i
and T0
2i
Þand the massflow ratesð_m0
1 and_m0
2
Þ.It should be noted that variables with a prime symbolð0Þrefer to the design point.
In general,the manufacturer’s catalogue contains several tested parameters.These will be know as the Catalogued Working Condi-tions(CWC).The following CWC are often known:condensation temperature or evaporation temperature,depending on the case ðT cond or T e v apÞ;secondaryfluid massflow rateð_m2Þ;secondaryfluid inlet temperatureðT2iÞand the heat exchanged in the HX(Q). Where this information is know,the rest of the CWC parameters can be easily calculated as follows:
T2o¼T2iÀ
Q
_m
2
c p2
ðevaporation caseÞ;
T2o¼T2iþ
Q
_m
2
c p2
ðcondensation caseÞ;
ð1Þ
where the log mean temperature difference,D T lm,and the overall heat transfer coefficient per area,UA,are calculated in CWC,
UA¼
Q
D T lm
:ð2Þ
It is assumed that the parametersððT cond or T e v ap,_m2;T2i and QÞare known.They all are in most catalogues used during the develop-
Nomenclature
A areaðm2Þ
c p specific heat at constant pressure(J/kg K)
C g Correlation coefficients depending on geometry
C f Correlation coefficients depending onfluid and/orflux CWC Catalogued Working Conditions
D diameter(m)
f friction factor
g gravityðm=s2Þ
HX heat exchanger
h enthalpy(J/kg)
h fg latent heat(J/kg)
_mfluid massflow rateðkg=sÞ
N number of baffles in the shell side
N cw number of effective cross-flow rows(shell side)
N c number of cross-flow rows(shell side)
NTU Number of Transfer Unit
p pressure(Pa)
Pr Prandtl number
q heatfluxðW=m2Þ
Q heat exchanged(W)
Re Reynolds number
R l correction factor on pressure drop(tube–baffle and shell-to-baffle)
R b correction factor on pressure drop(bypass offlow)
S surfaceðm2Þ
T temperature(K,°C)
U global coefficient of heat transfer
Greeks
a heat transfer coefficientðW=m2KÞ
b heat transfer coefficient ratio
D increment
effectiveness
g dynamic viscosityðN s=m2Þ
k thermal conductivityðW=m2KÞq densityðkg=m3Þ
r surface tension(N/m)
Subscripts and superscripts
1refrigerantfluid
2secondaryfluid
c cross section
cond condenser
e external
evap evaporator
ffluid
fgfluid to gas
g gas or vapour
geom geometry
guess guess value
i internal or input conditions
lm logarithm mean
max maximum value
new new calculation
o output conditions
s shell side
sat saturation
w wall or window
0modelling/design point
1232 F.Vera-García et al./Applied Thermal Engineering30(2010)1231–1241
ment of this work.If the catalogue does not have these parameters, other equivalent ones are available,and the UA in CWC could be cal-culated in a similar way.
For the calculation of the outlet conditions at the design point, the -NTU method could be used in such a way that if
NTU0¼
UA0
m
2
c p2
;ð3Þ
and the effectiveness is
0¼1ÀeðÀNTU0Þ;ð4Þthe heat exchanged in the condenser or in the evaporator can be re-calculated as
j Q0
new
j¼ 0j Q0max j;ð5Þ
where j Q0
max
j is the maximum heat exchanged in the HX and is given by
Q0
max ¼m0
2
c p2ðT0
2i
ÀT0
1i
Þðevaporation caseÞ;
Q0
max ¼m0
2
c p2ðT0
1i
ÀT0
2i
Þðcondensation caseÞ:
ð6Þ
The outlet conditions can be determined with this new Q0
new
.It is necessary to know the product UA0of the design’s working condi-tions and UA0depends on Q0and/or the outlet conditions of the de-sign point,as it is shown in Eqs.(3)–(6).An iteration procedure will be thus necessary,and so repeated until the convergence based on
the heat,Q0
new ¼Q0
guess
,is reached.To begin the calculations afirst
guess is needed,in this case,it is proposed that the HX effectiveness is 0¼0:6at design’s working conditions.With this value of 0,the first outlet conditions guessed at design point are given by
h0 1o ¼h0
1i
þ 0Q0maxðevaporation caseÞ;
T0 2o ¼T2iÀ
0Q0
max
_m
2
c p2
;
ð7Þ
and/or
h0 1o ¼h0
1i
À 0Q0maxðcondensation caseÞ;
T0 2o ¼T2iþ
0Q0
max
_m
2
c p2
:
ð8Þ
In the other hand,for the calculations of the product UA0it is necessary to calculate both heat transfer coefficients,the internal tubeða0iÞand the external oneða0eÞat the design’s working condi-tions,as follows:
U0¼
1
1
a0
e
þR foþD x w
k w
þR ft A e
A i
þ1a0
i
A e
A i
;ð9Þ
where R fo and R fi are the outside and inside fouling resistances,D x w and k w are the wall thickness and thermal conductivity,and a0
i
:internal tubes heat transfer coefficient at design’s working conditions,
A i:internal heat transfer surface,
a0
e
:external tubes heat transfer coefficient at design’s working conditions,
A e:external heat transfer surface.
If fouling and thickness resistance are neglected,the mean wall heat transfer area is defining,A,and regrouping the last equation,it is possible to write
1 UA0¼
1
a0
i
A0
i
þ
1
a0
e
A0
e
¼
a0
i
A0
i
þa0e A0e
a0
i
A0
i
a0
e
A0
e
:ð10Þ
The heat transfer coefficients mostly depend on thefluids,the HX geometry and theflow conditions in the shell-tubes HX at the design’s working conditions.
Both internal and external heat transfer surfaces depend on the HX geometry.As commented before,the geometrical information is missing in the majority of the cases in this type of HX,and to avoid this problem,the following strategy is carried out: let us consider that the global heat transfer coefficient,UA at Catalogued Working Conditions(CWC),is given as the following equation
1
UA
¼
1
a i A iþ
1
a e A e;ð11Þwhereða iÞis the internal heat transfer coefficients andða eÞis the external one working at the CWC.Grouping the different terms of the last equations it becomes
1
UA
¼
a i A iþa e A e
a i A i a e A e;ð12Þdividing Eqs.(12)and(10)
UA0
¼
a0
i
a0
e
a i a e
a i A iþa e A e
a0
i
A0
i
þa0e A0e
;ð13Þand defining the ratios b e;b i as
b e¼
a0
e
a e;
b i¼
a0
i
a i:
ð14Þ
According to Bell and Muller[4]an approximately optimum HXs design can be obtained if the resistances of the two sensible heat transfer processes(internal and external)are approximately equal.The internal and external thermal resistance are normally balanced in shell-and-tubes HXs because this class of HXs are tech-nology very mature and with high experience in the design,there-fore,the modifications in the design of this HXs are naturally arrived to have shell-and-tubes HXs with the thermal resistances balanced.This requirement can be stated as:
1
a i A i%
1
a e A e;ð15Þor
A e
A i
%
a i
a e:ð16Þ
If shell-and-tubes evaporators and condensers are designed in a way that the thermal resistances are balanced
a i A i¼a e A e;ð17Þwhich is a common practice according to Bell and Muller[4]and Kays and London[20]books.Then,expression(13)can be written as:
UA0
UA
¼b i b e
2a i A i
a i b
i
A iþa e b e A e
¼
2b i b e
b iþb e
;ð18Þ
where the expressions of the coefficients b i and b e depend on the heat transfer correlations which chosen to calculate the heat trans-fer coefficient.
Therefore,UA0is determined by means of
UA0¼UA
2b i b e
b iþb e
:ð19Þ
In case that the thermal resistances are not balanced,Eq.(17)is not valid,an study carried out by the authors of this paper shows that the relative error obtained between approximation Eq.(18) and the correct Eq.(13)is less than10%when there are differences of100%between the internal and external thermal resistances,
F.Vera-García et al./Applied Thermal Engineering30(2010)1231–12411233
j a i A iÀa e A e j
a i A i%100%)j Eq:ð13ÞÀEq:ð18Þj
Eq:ð13Þ
<10%:ð20Þ
Therefore,the approximation used by the authors and suggested by Bell and Muller[4]and Kays and London[20]books is valid.
In certain cases,an iterative procedure based on the -NTU method is needed as the heat transfer depends on the heat ex-changed.A summarised version of this algorithm is shown in Fig.1.
The amount of available correlations in the existing literature is fairly extensive.It is possible to choose the most appropriate cor-relation whether a condenser or an evaporator is being studied and whether the refrigerantflows through or outside the tubes. The correlations utilised for each case and the estimation of heat transfer coefficient ratiosðbÞare described in Section3.
Regarding pressure drop,a simple method for its evaluation is proposed in the secondaryfluid side(fluid without a phase change).The refrigerant side has not been considered since the lack of information makes its evaluation difficult.No information is provided by the manufacturer,contrary to what happens with the secondaryfluid.In the section,several expressions have been derived for the ratio
D p0
D p
;ð21Þ
which is calculated as a function of thefluid properties and the massflow rate at manufacturer and design‘s working conditions. In the hand,the most important pressure drop comes from second-aryfluid because this will determine the needed energy for the cir-culating pump of thisfluid.
3.Estimation of heat transfer coefficient ratios
In this section,it is going to be explained and justified the cho-sen correlations in order to calculate correctly the heat transfer coefficient ratiosðbÞ.
Due to the methodology,it is necessary that the correlation be-tween heat transfer coefficientsðaÞcan be separated in two func-tional terms.One must include all geometrical terms and the other the type offluid and its working conditions.The correlation of heat transfer coefficients can be written in this way
a¼C gðgeometryÞÂC fðfluid;fluxÞ;ð22Þwhere C gðgeometryÞis a mathematical functional which inclu
des all geometrical parameters of the heat transfer coefficient correlation and,C fðfluid;fluxÞis a functional which includes type offluid and working conditions parameters.
Moreover,it is necessary to choose the most appropriate corre-lation for condensers or evaporators,and refrigerantflows through or outside the tubes.The following configurations are studied in the next sections:
Condensers in which the refrigerantflows through the tubes. Condensers in which the refrigerantflows outside the tubes (through the shell).
Evaporators in which the refrigerantflows inside the tubes.
Evaporators in which the refrigerantflows outside the tubes (through the shell).
On the other side of the refrigerant exchangers there is a sec-ondaryfluid whose state does not change.To calculate the heat transfer coefficient ratio it is not necessary to take into account if the HX is a condenser or an evaporator.However,as the secondary fluid canflow through the shell side or through the tubes,a differ-ent correlation will be necessary for each case.
Section3.1shows the correlations for the secondaryfluid which flows through the shell and through tube
s.The cases concerning the refrigerant side are presented in the following sections.
3.1.Heat transfer coefficient ratios for secondaryfluids
In this section,the chosen correlations for the secondaryfluid whichflows both through the shell and through tubes are going to be explained.In the shell-and-tubes HXs there is a brine,whose state does not change,whichflows in the shell side or in the tubes. Thus,it is only necessary to decide appropriate correlations for both cases:single phaseflows through the shell(outside the tubes),b e,and;single phaseflows in the tubes,b i.
3.1.1.Secondaryfluidflowing through the shell.Heat transfer
coefficient and pressure drop
In this case,the heat transfer coefficient used was defined by Kern(according to Hewitt[17]).The expression of the heat transfer coefficient is given by
a e¼0:36k
e
D e_m s
g
0:55c
p
g
1=3g
g
w
:ð23Þ
Thus,the expression obtained for the heat transfer coefficient ratio b e is
b
e
¼
a0
e
a e¼
c0
p
p
1=3k0 2=3g
g0
0:217_m0
_m
0:55
;ð24Þwhere there is no influence on the geometry of HXs.
1234 F.Vera-García et al./Applied Thermal Engineering30(2010)1231–1241
On the other hand,in single phase flow,the pressure drop across the shell side (excluding the nozzles)can be calculated by means of Bell–Delaware correlation,and by:
D p ¼ðN À1ÞD p c R b R t R l þN D p w R l þ2D p c R b
1þN cw
N c
;ð25Þ
where the first term of the right hand side is the pressure drop in cross flow between baffle tips,the second is the pressure drop in the window flow and the last one corresponds to the pressure drop in the two end zones of the HX.The meaning of each parameter of the equation is defined at nomenclature section.
A detailed description and justification of these parameters can be found in Bell and Mueller’s digital publication [4].Grouping Eq.(25)the pressure drop can be written as
D p ¼ðN À1ÞR b þ2R b
1þ
N cw c
!
D p c þNR l D p w ;
ð26Þ
using the definition of pressure ratio,Eq.(21),as an approximation,
the following expression for the pressure ratio can be assumed
D p 0D p ¼
D p 0w D p w D p 0c
D p c
:
ð27Þ
With Eq.(27)is possible to calculate the pressure drop in the de-sign’s conditions if the CWC are known,whenever the new ratios D p 0w w and D p 0c
c
are defined.The first one can be derived by considering the original definition of D p 0w ,which is basically a geometrical func-tion that multiplies the coefficient _m
2
q ,corrected by the viscosity
function if the Reynolds number is lower than or equal to 100,therefore,after these considerations
D p 0w
w ¼q q 0_
m 0_m ÀÁ21þg 0g
Re 6100;
q q 0_m 0_m
ÀÁ
2
Re >100:24ð28Þ
For the second term,
D p 0c
c
,the expression proposed by Hewitt in [17]
is employed
D p c ¼N c K f _m
S m
;
ð29Þ
where S m is the window flow area and K f is defined graphically in the same reference [17],which depends on the pitch-diameter and the Reynolds number.Thus,this pressure ratio is
D p 0c D p c ¼K 0f
K f _m 0_m
:
ð30Þ
Taking into account the influence of the geometrical variables on
the K f plots,the expression of Zakauskas and Ulinskas (according to [17])has been used for the estimation of the ratio K 0f
f
.
For in-line arrays,
D p 0w p w ¼}0:272þ0:207Â103Re þ0:102Â103Re 2À0:286Â10
3
Re 3
;for 3<Re <2Â103
;
}0:267þ0:249Â104À0:927Â107Re 2þ0:1Â1011Re 3
;for 2Â103<Re <2Â106:
2
6
666
64ð31Þ
For triangular arrays,
D p 0
w D p w ¼}0:795þ0:247Â103Re þ0:335Â103Re 2À0:155Â104Re 3þ0:241Â10
4
Re 4
;for 3<Re <2Â103;
}}0:245þ0:339Â104Re À0:984Â107Re 2þ0:133Â1011Re 3À0:599Â1013Re 4
;for 2Â103<Re <2Â106:
2
6
666
64ð32Þ
The only geometrical information necessary is the position of the
tube inside of the shell.In case that this information is not able,it is possible to use Eq.(32)as an approximation.
3.1.2.Secondary fluid flowing through the tubes
In this case,the heat transfer coefficient of the secondary fluid flowing through the tubes can be calculated by means of the Dittus–Boelter [14]correlations,which are given by:
a i ¼k D
characteriseRe 0:8Pr 0:4;
ð33Þ
taking into account that,in this case,the Reynolds number is
Re ¼
_mD c g
:The heat transfer coefficient can be written as:
a i ¼
C
A c D
c 0:4p k
0:6
g
_m 0:8;ð34Þ
where the first factor on the right hand side,C A 0:8c D
0:2,depends on the HX geometry,the second factor depends on the fluid and the third is a function of the mass flow rate.Therefore,b i is calculated as:
b i ¼a 0i
a i ¼
c 0p g c p g 0
0:4k 0k 0:6_m
0_m
0:8
;ð35Þ
where the properties must be evaluated at the mean temperature,
between the inlet and the outlet of the HX.
The estimation of the pressure drop of secondary fluids flowing through the tubes of the HXs has been calculated as the pressure drop for friction of single phase fluid through a single tube (exclud-ing the connection nozzles or the distributor shell).It is important to consider that the internal geometrical information of the tubes are not available for the shell-and-tubes HXs model.
The friction pressure drop in a pipe with length L and an inter-nal diameter D is given by Eq.(36)
D p ¼2f L D _m
2q S :
ð36Þ
Fanning friction factor,f ,can be calculated by means of the Darcy–Weisbach expression when fully developed laminar regime flow is encountered,and for the turbulent regime the Blasius equation is used:
f ¼16Re
;for Re 62300;0:079Re 1
4;for 2300<Re <2Â104;
0:046Re 15;for Re >2Â104:
8>><>>:
ð37Þ
Then,Eq.(36)can be re-grouped as follows:
D p ¼2L DS
2 f _m
2q
;
ð38Þ
where first factor clearly depends on the geometrical characteristics
of the tube and the second one depends on the mass flow rate and the fluid properties.With these considerations and using the defini-tion of pressure ratio,Eq.(21),the following expression results:
D p 0D p ¼
f 0f q q 0_m
0_m
2
;ð39Þ
where the friction factor ratio,f 0
f ,can be calculated by means of
f 0¼
g 0g _m
0_m
for Re 62300;g 0g _m 0_
1
4
for 2300<Re <2Â104;g 0g _m 0_ 15for Re >2Â104:8>>>>><>>>>>:ð40Þ
In this case,the only geometrical information necessary is to decide
if the regime flow is laminar or turbulent.
F.Vera-García et al./Applied Thermal Engineering 30(2010)1231–12411235
3.2.Condensers with refrigerant inside the tubes
Most of the available correlations for the evaluation of the con-densation heat transfer coefficient are local.The average correla-tion suggested by Bell in [4]and developed by Boyko and Kruzhilin [5]has been used in this case.Thus,the heat transfer coefficient is given by:
a i ¼0:024k f i Re 0:8Pr 0:431
1þ
q f q g
"#
;ð41Þ
where the Re number is the liquid only Reynolds number.
Re ¼
D i _m g f i
:Substituting the Reynolds number expression in Eq.(41)and rearranging it
a i ¼
0:024
D i S i
"
#
k
0:57
g f c 0:43p 1
1þ
q f q g
"#
_m 0:8;ð42Þ
a geometry-dependent factor is obtained,½0:024
D 0:2i
S 0:8i
,that multiplies an-other one depending on the fluid and the mass flow rate.
The definition of b i in Eq.(14)is applied by dividing the heat transfer coefficient at design’s working conditions by the coeffi-cient at CWC The heat transfer coefficient ratio for a condenser with the refrigerant flux inside the tubes ðb i Þis
b i ¼a 0i a i ¼k 0f f !0:57g f g f !0:37
c pf
pf !0:431þq 0f =q 0g q f q g "#_m 0_ !
0:8:
ð43Þ
In this equation all the parameters depend on the type and condi-tions of the refrigerant fluid.As they d
o not depend on the geometry
of HXs it is not necessary to know the inside geometry of the cell-and-tubes HXs to model them.
Eq.(24)is used to calculate b e and to complete the model in this case.
3.3.Condensers with refrigerant through the shell
In this case,the heat transfer ratio that has to be calculated is exterior,b e ,because the refrigerant is flowing through the shell.The external heat transfer coefficient can be calculated by means of a Nusselt-like correlation (according to Hewitt’s book [17])as in:
a e ¼C k 3f h fg q f ðq f Àq g Þg g f sat w "#1f geom :
ð44Þ
In this equation,the values of C ¼0:725and f geom ¼1are consid-ered for plain tubes.Different values of C and f geom are available in the existing literature [17].They are usually function of the internal surface of the tube.If the HX has low-finned in the Beatty and Katz correlation [2],f geom depends on the fin efficiency,the total fin area and so on.These correlations only consider gravitational
ef-fects.Surface tension effects could be calculated by using a factor which is a function of both surface tension and the HXs’geometry (like in the correlation explained by Belghazi et al.in [3]).
The correlation can be rearranged by grouping the geometrical variables and invariants in the first term (parameter C g in Eq.(46)),and the fluid parameters and those depending on the heat flux in the second term (parameter C f in Eq.(46)),in a way that the heat transfer coefficient becomes
a e ¼C
g D
0:25f ðN 2Þk 3f h gf q f ðq f Àq g Þ
g f
"#0:251
ðT sat ÀT w Þ0:25
"#;
ð45Þa e ¼C g C f
1
ðT sat ÀT w Þ:
ð46Þ
If the heat flux is defined as q 00¼Q ¼a e ðT sat ÀT w Þthe heat transfer coefficient,a e ,is given by
a e ¼C 43
g C 43
f Q A
À13
;
ð47Þ
and,according to the definition,the coefficient b e is calculated by
b e ¼a
0e
a e ¼C 0f C f !4
3Q 0Q
1
;
ð48Þ
where the surface tension effect has been considered similar to the
CWC and to the design’s working conditions.This means
C f ¼
k 3f h fg q f ðq f Àq g Þ
g f
!1
4
:
ð49Þ
Eq.(35)is used to calculate the b i on the other side of the condensers.
3.4.Evaporators with refrigerant flowing inside the tubes
In the case that the refrigerant flows through the tubes and it goes trough boiling process,the correlation chosen for the heat transfer coefficient is the Cooper correlation [10].This is widely ac-cepted by researchers (as the references [16,25,24,15]show)to characterise pool and nucleate boiling but it is not very useful for the case of boiling inside tubes.However,as this correlation de-pends on the fluid properties,fluid pressure and heat flux:
a i ¼55p 0:12
r
Àlog 10p r ðÞÀ0:55M À0:5q 0:67;ð50Þ
it is possible to obtain easily the ratio b i ,which is given by
b i ¼
log 10p 0r
log 10p r
!À0:55Q
0Q
!0:67
;ð51Þ
unlike other correlations which could be considered more appropri-ate for this case.
The definition of b e ,Eq.(24),is exactly the same as in the con-denser case with refrigerant through the tubes.3.5.Evaporators with refrigerant through the shell
The same equations defined in the previous section will be used for the calculations of b e parameter in the case that the refrigerant flows through the shell.In case temperature differences between refrigerant and surface are high,a continuous vapour film covers the surface and a correlation of film boiling could be used to calcu-late b e parameter.The expression of the heat transfer coefficient used is the one proposed by Bromley for film evaporation [6]
a e ¼C k 3g h fg q g ðq f Àq g Þg D ðT w ÀT sat Þ
"#14
f geom ;
ð52Þ
to which a geometrical factor has been added,f geom ,following the same trend found in Section 3.3.
The heat transfer coefficient can be re-written as
1236 F.Vera-García et al./Applied Thermal Engineering 30(2010)1231–1241
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