Thermal performance and optimization ofhyperbolic annular fins under dehumidifyingoperating conditions – analytical and numericalsolutions

S. Pashah, Abdurrahman Moinuddin, Syed M. Zubair *Mechanical Engineering Department, KFUPM Box # 1474, King Fahd University of Petroleum & Minerals,Dhahran 31261, Saudi Arabia

A R T I C L E I N F O

Article history:

Received 1 July 2015

Received in revised form 4

November 2015

Accepted 6 January 2016

Available online 3 February 2016

A B S T R A C T

The thermal performance of hyperbolic profile annular fins subjected to dehumidifying op-

erating conditions is studied. An analytical solution for completely wet fin is derived using

an approximate linear temperature–humidity relationship. A numerical solution using actual

psychrometric relationship for completely and partially wet operating conditions is then

obtained to account for the actual temperature–humidity ratio psychrometric relationship

under both partially and fully wet operating conditions. An excellent agreement is ob-

served between analytical and numerical solutions for completely wet fin.The fin optimization

is presented based on the analytical solution of completely wet fin. Finally, a finite element

formulation is used for studying the two-dimensional effects of orthotropic thermal con-

ductivity on the thermal performance of fin under partially and fully wet operating conditions.

© 2016 Elsevier Ltd and International Institute of Refrigeration. All rights reserved.

Keywords:

Annular fin

Hyperbolic profile

Analytical solution

Fin optimization

Finite element analysis

Orthotropic thermal conductivity

Mass transfer

Partially wet fin

Performance thermique et optimisation d’ailettes annulaireshyperboliques sous conditions de fonctionnementdéshumidifiant – solutions analytique et numérique

Mots clés : Ailette annulaire ; Profil hyperbolique ; Solution analytique ; Optimisation d’ailette ; Analyse d’élément fini ; Conductivité

thermique orthotrope ; Transfert de masse ; Ailette partiellement humide

* Corresponding author. Mechanical Engineering Department, KFUPM Box # 1474, King Fahd University of Petroleum & Minerals, Dhahran31261, Saudi Arabia. Tel.: +966 13 860 3135; Fax: +966 13 860 2949.

E-mail address: smzubair@kfupm.edu.sa (S.M. Zubair).http://dx.doi.org/10.1016/j.ijrefrig.2016.01.0060140-7007/© 2016 Elsevier Ltd and International Institute of Refrigeration. All rights reserved.

i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 5 ( 2 0 1 6 ) 4 2 – 5 4

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1. Introduction

Extended surfaces are widely used to enhance the rate of heattransfer between a solid and a surrounding fluid. In refrigera-tion and air conditioning equipment, if the fin surfacetemperature is lower than the dew-point temperature of in-coming moist air, then the condensation of water vapor occurson the fin surface such that heat and mass transfer occurssimultaneously.

The major variables that influence the heat and mass trans-fer include, fin geometry, fin material and operating conditions.The fin geometries that are commonly used in these heat trans-fer devices may be classified as (a) spines or pin fins (b)longitudinal or straight fins, and (c) radial or annular fins, (Krauset al., 2001). The objective of a variable profile fins is to providehigh heat transfer capability for a given additional weight ofthe fin or to provide a minimum weight for the required amountof heat to be dissipated from the finned surface. The profilesare classified as (a) rectangular, (b) triangular or trapezoidal,(c) convex parabolic and (d) concave parabolic (Yovanovich,2004). It is stated that among the whole family of annular finsof tapered cross section, the annular fin of hyperbolic profileis the foremost fin shape for usage in tubes of high perfor-mance heat exchange devices (Campo and Cui, 2008).

The performance of a fin is well described by its effi-ciency, defined as:

η = QQ

**max

(1)

where Q is heat transfer rate through the fin and Qmax is themaximum possible heat transfer rate from the fin if the entirefin surface is at the prime surface temperature and humidityratio.

High efficiency fins are desirable for effective heat trans-fer. For the case of dry operating conditions, the rate of sensibleheat loss can be increased by using force convection e.g. byusing a fan. However, for a given fin material and geometry,the maximum value of convective heat transfer under dry op-erating conditions is governed by the dimensionless parametercalled Biot number. For a circular cross-section pin fin it is givenby:

Bi = hrk

(2)

h is the convective heat transfer coefficient, k is the thermalconductivity and r is the fin radius. For metallic fin material,a fin has high efficiency (>90%) values only in very low Biotnumber range Bi � 1. For a given fin material, this conditionimplies that low h values must be used to have high effi-ciency slender pin fins. This issue can be addressed by usinga fin material with orthotropic thermal conductivity; i.e. a pinfin with different thermal conductivities kr and kz in radial and

Nomenclature

B Parameter defined in Eq. (10) [°C]

Bi Biot numberCo constant defined in Eq. (33) [kgw/kga]cp specific heat of incoming moist air stream

[J kg−1 K−1]

h convective heat transfer coefficient [W m−2 K−1]

hD mass transfer coefficient [kg m−2 s−1]

′h equivalent heat transfer coefficient defined byEq. (69) [W m−2 K−1]

ifg latent heat of evaporation for water [J kg−1]

k thermal conductivity [W m−1 K−1]L fin length [m]Le Lewis numberM parameter defined in Eq. (21) [m−3/2]M0 parameter defined in Eq. (20) [m−3/2]m0 parameter defined in Eq. (20) [m−1]Patm atmospheric pressure [Pa]Q dimensionless heat flow rate

Q * heat flow rate [W]q heat flux [W m−2]qb specified heat flux [W m−2]RH relative humidity of air

r* fin radius [m]r dimensionless fin radiusR annular fin radius ratioT temperature [°C]

′Ta equivalent ambient temperature defined by Eq.(70) [°C]

t* fin thickness [m]t dimensionless fin thickness

Greek symbolsη efficiencyψ fin aspect ratio defined after Eq. (37)θ dimensionless temperatureτ temperature difference defined by Eq. (16)τ p temperature value defined by Eq. (26)Ω parameter defined in Eq. set (32)ω humidity ratio of air [kgw kga

−1]

Finite element matrices and vectors

f{ } element load vector

k[ ] element stiffness matrix

B[ ] flux-temperature matrix

D[ ] material property matrix

N[ ] shape function matrixT{ } nodal temperature vector

Subscripts and superscriptsa ambient

b base

h convectionq conductionr radialt tipz lateralζ border between wet and dry region of a partially

wet fin

43i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 5 ( 2 0 1 6 ) 4 2 – 5 4

axial directions, respectively. The thermal conductivity of thepolymer composite materials is significantly higher in fiber axisdirection whereas the thermal conductivity in the orthogo-nal direction can be significantly lower. The thermalconductivities of some polymer composite are summarized inTable 1.

The effect of orthotropic thermal conductivity on fin per-formance under dry operating conditions is presented in Fig. 1.The orthotropic property is defined by the thermal conduc-tivity ratio k k krz r z= . It is obvious that orthotropic fins havehigh efficiency in the range 0 5 1. < <Bi whereas an isotropicfin would be practically not useful in this range.

The existing work for orthotropic material is mainly cov-ering dry operating conditions (Kundu and Lee, 2011). The heatconduction in an anisotropic material is studied by Traiano et al.(1997). The analytical solution for cylindrical spines oforthotropic material is derived by Bahadur and Bar-Cohen (2007).The solution provides expressions for temperature profile andheat transfer rate. A dimensionless closed form solution fororthotropic material, cylindrical spines is derived by Zubair et al.(2010). An axisymmetric thermal non-dimensional finiteelement method has been used to study the performance oforthotropic material spines by S. Pashah et al. (2011). Mustafaet al. (2011) derived a closed-form analytical solution to study

the thermal performance of orthotropic material annular finswith a contact resistance.

There are many studies available for isotropic fin perfor-mance under combined heat and mass transfer whereas notmuch work has been done with orthotropic material. Regard-ing orthotropic fins under combined heat and mass transfer;Kundu and Lee (2011) developed a semi-analytical model forpredicting the fin efficiency of orthotropic fin-and-tube heatexchangers for square and equilateral arrays of tubes. Howeverto decrease the complexity of semi-analytical solution, a linearrelationship between the specific humidity at the fin surfaceand the fin surface temperature is considered, whereas it iswell known that a linear approximation is not valid for thisrelationship because the psychrometric correlation betweenthe two parameters is non-linear (Hyland and Wexter, 1983a,1983b; Kloppers and Kröger, 2005). The correlation also in-cludes some other parameters like wet bulb temperature;consequently, the non-linear correlation is not suitable for aclosed form solution. Thus, the polynomial (linear, quadraticand cubic) relationships between temperature (dry bulb) andhumidity ratio have commonly been used for closed form ana-lytical solutions. An excellent review of the available polynomialrelationships and limitations of some approaches are dis-cussed by Kundu (2009). An approximate cubic polynomialrelationship is given by Liang et al. (2000). The relationship isobtained by regression analysis over the temperature range 0to 30°C. The approximate cubic relationship is suitable for theclosed form solution with an acceptable range of accuracy.

The preceding discussion shows that there is a potentialof major contribution in the area for combined heat and masstransfer for orthotropic fin material.The limited amount of workavailable in open literature is approximate in nature due to sim-plifying assumptions to decrease complexity of a semi-analytical solution. However, numerical methods like finiteelement method are capable of incorporating complexitiesrelated to material properties and geometry with relative ease,particularly for two-dimensional problems.Therefore, a generalfinite element formulation that is capable of modelling com-bined heat and mass transfer can provide solution with anacceptable accuracy level. Therefore, the first objective of thepresent study is to evaluate the effect of using a lineartemperature-humidity ratio relationship on the accuracy of ahyperbolic profile annular fin by comparing its results with anumerical solution based on actual non-linear psychromet-ric relationship. The second objective is to study the effects oforthotropic thermal conductivity using a finite element for-mulation considering the non-linear psychrometric relationship.

2. Annular fin with hyperbolic profile

It is very common to make certain assumptions for simplify-ing one-dimensional analysis of the fins. The assumptionsconsidered in the present study are:

(a) The thermal conductivity of the fin, heat transfer coef-ficient and latent heat of condensation of the water vaporare constant;

(b) The heat and mass transfer are under steady-statecondition;

Table 1 – Polymer composite thermal conductivities(Bahadur and Bar-Cohen, 2007).

Filler Matrix Parallelto fiber

(W/m − K)

Normalto fiber

(W/m − K)

Continuous carbon fiber Polymer 330 3–10Discontinuous carbon fiber Polymer 10–100 3–10Graphite Epoxy 370 6.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10Bir

k* = 0.005

k* = 0.01

k* = 0.03

k* = 0.1

k* = 1

krz

krz

krz

krz

krz

L/r =10

Fig. 1 – Orthotropic pin fin efficiency under dry operatingconditions as a function of radial Biot number Bir andthermal conductivity ratio for fin aspect (length to radius)ratio = 10 (Pashah et al., 2011).

44 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 5 ( 2 0 1 6 ) 4 2 – 5 4

(c) The thermal resistance associated with the presence ofthin water film due to condensation is small and maybe neglected; and

(d) The effect of air pressure drop due to airflow is ignored.

These are some of the classical assumptions that have beencommonly used by many researchers for the analysis ofconducting-convecting finned surfaces.

2.1. Fully wet fin model

We consider an annular fin of hyperbolic profile with base thick-ness of tb* and temperature Tb . The temperature at tip is Tt ,which for fully wet situation, should be less than the dew pointtemperature of the ambient air at temperature Ta and humid-ity ratio ωa.The length of the fin is L. The fin geometry andoperating conditions are depicted in Fig. 2.

Applying the energy balance on the infinitesimal circularring of width dr* , which has an average thickness of t* , at aradius of r* from the center of the tube

q q q qr convection condensation r dr* * *= + + + (3)

where qr* is the conduction heat transfer rate in the radial di-rection. It can be expressed as,

q r t r kdTdrr* = ( )2π * *

*(4)

Note that t r*( ) shows the variable thickness of the fin alongits length, given by:

t r trrbb* ***

( ) = ⎛⎝⎜

⎞⎠⎟ (5)

also:

q qqr

drr dr rr

* * **

+ = + ∂∂ *

* (6)

therefore;

q q k r t rd Tdr

t rdTdr

rdt r

drdT

r dr r* * *+ = + ( ) + ( ) + ( )2 2 2

2

2π π π* *

**

**

** ddr

dr*

*⎡⎣⎢

⎤⎦⎥

(7)

The heat and mass transfer coefficients are related as(Chilton and Colburn, 1934):

hh

c LeD

p=23 (8)

where

h i hBD fg = (9)

and

Bi

c Le

fg

p

= 23

(10)

The parameter B can be considered as constant, becauseover the practical range of air temperature and relative hu-midity, the variations are not significant for the latent heat ofwater condensation ( i fg ), Lewis number ( Le ) and specific heatof incoming moist air ( cp). Therefore, its value is within ±1 6. %of the average value of 2433 C (Sharqawy and Zubair, 2007).Substituting all the above terms in Eq. (3) and simplifying, weget:

d Tdr r

dTdr t r

dt rdr

dTdr

hkt r

T Th

kta

D2

2

1 1 2 2* * * *

** * *

+ + ( )( ) + ( )

−( ) +rr

i fg a*( )

−( ) =ω ω 0

(11)

subject to the following boundary conditions:

at r r T T andb b b= = =, ω ω (12)

The other boundary condition, which is at the fin tip, canbe either

at for insulated tipr rdTdr

t= =, 0 (13)

or

at for convective tipr r kdTdr

h T T h it a D fg a= = −( ) + −( ), ω ω(14)

2.1.1. Analytical solution for fully wet finEquation (11) can be written in terms of temperature differ-ence as:

ddr

mrr

Bob

a

2

22τ τ ω ω

***

= + −( )[ ] (15)

where the temperature difference is,Fig. 2 – Schematic of a completely wet hyperbolic annularfin.

45i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 5 ( 2 0 1 6 ) 4 2 – 5 4

τ = −T Ta (16)

Now we need a relation between the temperature differ-ence τ and humidity ratio ω to solve the above equation. Inthis regard, we will use the same relation as that of Sharqawyand Zubair (2007), expressed as

ω = +a b T2 2 (17)

The constants a2 and b2 are defined as follows:

aT T

Tbdew b

dew bb2 = − −

−ω ω ω

(18)

bT T

dew b

dew b2 = −

−ω ω

(19)

Substituting the above expressions in Eq. (15) andintroducing.

Mmr

mh

ktb b0

2 02

02= ⎛

⎝⎜⎞⎠⎟

=* *

with (20)

M M b B20

221= +( ) (21)

we get after some manipulation,

ddr

M r M r B a

2

22

02τ τ ω ω

** *− = −( ) (22)

The boundary conditions are:

at and* *r rb b b= = =, τ τ ω ω (23)

At for insulated tip* *r rddr

t= =,τ

0 (24)

The closed-form analytical solution is expressed in termsof Airy functions, expressed as

τ ττ τ

++

=′ ⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

− ⎛⎝⎜

⎞⎠⎟

′p

b p

tAi M r Bi M r Ai M r Bi M23

23

23

23* * * rr

Ai M r Bi M r Ai M r Bi

t

t b b

*

* * *

⎛⎝⎜

⎞⎠⎟

′ ⎛⎝⎜

⎞⎠⎟

′ ⎛⎝⎜

⎞⎠⎟

− ⎛⎝⎜

⎞⎠⎟

′23

23

23 MM rt

23 *

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

(25)

where;

τ ωp

a aB a b Tb B

= − −( )+( )

2 2

21(26)

It is important to note that Airy functions are related tomodified Bessel functions of the fractional order by the fol-lowing equations (Abramowitz and Stegun, 1964).

Ai X XK X( ) = ⎛⎝

⎞⎠

1 13

23

1 33 2

π(27)

Bi X X I X I X( ) = ⎛⎝

⎞⎠ + ⎛

⎝⎞⎠

⎡⎣⎢

⎤⎦⎥−

13

23

23

1 33 2

1 33 2 (28)

Ai′ and Bi′ are their respective derivatives.The actual rate of heat transfer from the fin is given

by:

Q A kddr

A k M

Ai M r Bi M r

b br r

b b p

t b

b

**

* *

=

= +( )

×− ′ ⎛

⎝⎜⎞⎠⎟

′

=

τ

τ τ* *

2 3

23

23⎛⎛

⎝⎜⎞⎠⎟

+ ′ ⎛⎝⎜

⎞⎠⎟

′ ⎛⎝⎜

⎞⎠⎟

′ ⎛⎝⎜

⎞⎠⎟

′

Ai M r Bi M r

Ai M r Bi M

b t

t

23

23

23

2

* *

* 3323

23r Ai M r Bi M rb b t* * *

⎛⎝⎜

⎞⎠⎟

− ⎛⎝⎜

⎞⎠⎟

′ ⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

(29)

where Ab is the fin base area.The maximum heat transfer, which is the heat transfer from

the fin surface if the entire fin is at the base temperature andbase humidity ratio, is expressed as,

Q m kA b Bs b p*max = +( ) +( )02

21 τ τ (30)

where As is the fin surface area. Therefore, the fin efficiencycan be written as,

η =− ′ ⎛

⎝⎜⎞⎠⎟

′ ⎛⎝⎜

⎞⎠⎟

+ ′ ⎛⎝⎜

⎞⎠⎟−A

AM

Ai M r Bi M r Ai M r Bb

s

t b b4 3

23

23

23* * * ′′ ⎛

⎝⎜⎞⎠⎟

′ ⎛⎝⎜

⎞⎠⎟

′ ⎛⎝⎜

⎞⎠⎟

− ⎛⎝⎜

⎞

i M r

Ai M r Bi M r Ai M r

t

t b b

23

23

23

23

*

* * *⎠⎠⎟

′ ⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥Bi M rt

23 *

(31)

2.1.2. Numerical solution for fully wet finAlthough the assumption of linear relationship (cf. Eq. (17)) sim-plifies the analytical solution of the differential Eq. (34), it doesnot account for the actual non-linear psychrometric correla-tions of an air-water vapor mixture. Therefore, we will obtaina numerical solution using the actual non-linear psycrometricrelationship.

Normalizing Eq. (11) using the following relations:

rrL

T TT T

a

a b

a

a b

= = −−

= −−

*, ,θ ω ω

ω ωΩ (32)

And substituting:

Co BT T

a b

a b

= −−

⎛⎝⎜

⎞⎠⎟

ω ω(33)

We get

ddr

m Lrr

Coob

2

22θ θ= ( ) +[ ]Ω (34)

The boundary conditions are:

at andr rb= = =, θ 1 1Ω (35)

and

at for insulated tipr rddr

t= =,θ

0 (36)

or

46 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 5 ( 2 0 1 6 ) 4 2 – 5 4

r rddr

m L Cot o= = − ( ) +( ),θ ψ θ2 Ω (37)

where ψ = ( )t Lb* 2 is the fin aspect ratio.The heat transfer to the fin is the summation of heat trans-

ferred by convection and condensation, which can be estimatedin the dimensionless form as,

QQ

r k T Tm L R Co rdrfin

b a b r

r

b

t

=−( )

= ( ) −( ) +( )∫*

22 10

2

πψ θ

*Ω (38)

R r rt b= * * denotes the radius ratio of tip to base of the fin.The maximum dimensionless heat transfer rate Qmax wouldexist if the entire fin surface were at the fin base tempera-ture and humidity ratio. It can be written as,

QQ

r k T Tm L R Co

b a bmax

max*=−( )

= ( ) +( ) +( )2

1 102

πψ

*(39)

Therefore, the fin efficiency can be expressed as,

ηθ

=−( ) +( )

+( ) +( )

∫2 1

1 1

R Co rdr

R Cor

r

b

t

Ω(40)

Note that for a fin subjected to combined heat and masstransfer, the total heat transfer comprises both the sensibleas well as the latent heat transfer caused by the temperaturedifference and mass transfer, respectively. Therefore, the ef-ficiency of a fully wet fin depends on the distribution oftemperature as well as humidity ratio on the fin surface.

2.2. Partially wet fin model

A partially wet fin condition exists when the fin base tem-perature is lower, but the fin tip temperature is higher thanthe dew point of air. Under such situation, there is a radius,r r= ζ , where the surface temperature equals the dew point ofthe air, i.e. T r Tdewζ( ) = . The fin is then divided into two regions:a wet region for r r rb ≤ ≤ ζ , with the surface temperature lowerthan Tdew and a dry region from r r rtζ ≤ ≤ , with the surface tem-perature higher than Tdew (see Fig. 3). In this regard, separategoverning differential equations must be written for each region.

For r r rb ≤ ≤ ζ

ddr

m Lrr

Coob

2

22θ θ= ( ) +[ ]Ω (41)

and for r r rtζ ≤ ≤

ddr

m Lrr

ob

2

22θ θ= ( ) [ ] (42)

The boundary conditions that differ from completely wetcase are as follows,

At the section separating the fully wet and fully dry regions

at r r dew= =ζ θ θ, (43)

For the convective tip:

at r rddr

m Lt o= = −( ),θ ψθ2 (44)

The actual dimensionless heat transferred to the fin surface,is the summation of the heat transferred in the wet and dryregions that can be calculated by,

Q m L R Co rdr rdrr

r

r

r

b

t

= ( ) −( ) +( ) +⎡

⎣⎢⎢

⎤

⎦⎥⎥

∫ ∫2 102ψ θ θ

ζ

ζ

Ω (45)

Therefore, the fin efficiency can be expressed as,

η

θ θζ

ζ=

−( ) +( ) +⎡

⎣⎢⎢

⎤

⎦⎥⎥

+( ) +[ ]

∫ ∫2 1

1 1

R Co rdr rdr

R Cor

r

r

r

b

t

Ω(46)

3. Results and discussion

The one-dimensional mathematical models, developed in theprevious section, are solved numerically for both completelywet and partially wet fins with insulated tip boundary condi-tions. The governing differential equations and thecorresponding boundary conditions are written in finite-difference form and are solved using either successive over-relaxation (SOR) or under relaxation method as per requirement(Patrick, 1998). In order to calculate the humidity ratio corre-sponding to wet fin surface temperature using actual

Fig. 3 – Schematic of a partially wet hyperbolic annular fin.

47i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 5 ( 2 0 1 6 ) 4 2 – 5 4

psychrometric relationships, the method is implemented in EES(Klein, 2015) that has built-in psychrometric property calcu-lator. The solution has been obtained in an iterative mannerwith the convergence criterion of 1 10 06× − .

Grid independence is described as the improvement in nu-merical results upon successive reduction in the cell size. Asthe grid is refined, grid cells become smaller, the number ofcells in the domain grows, and the spatial discretization errorasymptotically approaches zero (excluding the round-off error).For our case, we have discretized the one-dimensional findomain into 51 nodes as the starting point, thereafter increas-ing it to 111 nodes in the step interval of 10 nodes. The gridindependence test results for fin efficiency are shown in Table 2.Looking at the table, we can say that 101 nodes provide sat-isfactory accurate results without consuming unacceptableamount of time.

To evaluate the effect of approximate linear relationshipbetween temperature and humidity ratio; the analytical solu-tion is compared with the numerical solution (based on actualpsychrometric relation) in Fig. 4. Since the analytical solutionis valid for completely wet conditions, therefore, the geomet-ric and operating conditions are selected to ensure completelywet operating conditions. The graph depicts that there is verygood agreement between two results. It can be noted that thenumerical solution gives slightly higher efficiency than the ana-lytical solution, nevertheless the maximum difference inefficiency values is 2% that corresponds to m L0 0 5= . . The dif-ference of surface temperature over the dimensionless fin length

is presented in Fig. 5; that shows that the maximum tempera-ture difference is approximately 0.7°C (at fin tip) thatcorresponds to m L0 1= . For practicality, the fin will be oper-ating in the range m L0 0 6≤ . , and the maximum temperaturedifference is 0.2°C. So this shows that the approximate linearrelation given by Sharqawy and Zubair (2007) has provided areasonable accurate non-iterative analytical solution for thecase of hyperbolic fin under considered operating conditions.

Now the fin performance is studied using a numerical ap-proach because it covers both fully wet and partially wetoperating conditions. The results for different radii ratio arepresented in Fig. 6. The fin base and the ambient tempera-ture values are the same for all cases whereas the RH valueshave been varied from 0.2 to 1. The fin is fully dry for RH = 0.2over the considered range of fin parameter m L0 .The fins arefully wet for high RH and low m L0 values. Moreover, the finefficiency decreases with increasing wet portion of the fin, thusthe completely dry fin condition has the highest efficiency andthe highest relative humidity has always the lowest efficiency.

The effect of radius ratio on the fin efficiency for RH = 0.6is shown in Fig. 7. It shows that the lower radius ratio has higherfin efficiency.This can be explained by the fact that a low radiusratio fin can be obtained by truncating tip portion of a highradius ratio fin, since the tip portion is obviously less effec-tive than the fin base portion due to low temperature differencebetween the fin surface and ambient air.

4. Fin optimization

The optimum fin dimensions can be obtained by using an op-timization techniques, for example in a recent paper Huangand Chung (2014) developed an inverse design algorithm usingthe conjugate gradient method (CGM) to estimate the optimumshape for fully wet annular fins. Optimization of fins can bedescribed as a process through which we find the optimumdimensions of a fin for a required amount of heat transfer, ordetermining the maximum possible heat transfer if the

Table 2 – Grid independence test data for fin efficiency.

Case Number of nodes Fin efficiency

A 51 0.8257B 61 0.8253C 71 0.8251D 81 0.8249E 91 0.8247F 101 0.8246G 111 0.8245

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4m0L

AnalyticalNumerical

Tb = 7 °CTa = 27 °CRH = 0.6R = 4

Fig. 4 – A comparison of completely wet fin efficiency: fromanalytical and numerical solutions.

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1x

m L=0.2m L=0.6m L=1

Tb = 7 °CTa = 27 °CRH = 0.6R = 4

0

0

0

Fig. 5 – Fin surface temperature difference betweenanalytical and numerical solutions.

48 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 5 ( 2 0 1 6 ) 4 2 – 5 4

dimensions of the fin are known (Ullmann and Kalman, 1989).In the following, we will follow the latter approach for theoptimization.

In this regard, we define following new dimensionless pa-rameters (Sharqawy and Zubair, 2007):

uhrk

b= 2 * (47)

vVr

t r r rrb

b b t b

b

= = −( )π

ππ*

* * * **3 3

2(48)

wrt

b

b

= **

(49)

where u is the fin heat transfer parameter i.e. modifiedversion of m Lo , v is the dimensionless fin volume and wisthe dimensionless fin-base radius. We observe thefollowing:

vw

rr

t

b

= −⎡⎣⎢

⎤⎦⎥

21

**

(50)

ψ =−( )

12 1w R

(51)

m L u R w0 1= −( ) (52)

rRb =

−1

1(53)

(a) (b)

(c) (d)

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2 2.5 3 3.5 4m0L

RH=0.2 (Fully dry)RH=0.4 (P.W. for m L ≥1.0)RH=0.6 (P.W. for m L ≥1.5)RH=1.0 (P.W. for m L ≥3.5)

Tb = 7 °CTa = 27 °CR = 2

0

0

0

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4m0L

RH=0.2 (Fully dry)

RH=0.4 (P.W. for m L ≥1.0)

RH=0.6 (P.W. for m L ≥1.5)

RH=1.0 (P.W. for m L ≥3.0)

Tb = 7 °CTa = 27 °CR = 3

0

0

0

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4m0L

RH=0.2 (Fully dry)

RH=0.4 (P.W. for m L ≥0.5)

RH=0.6 (P.W. for m L ≥1.0)

RH=1.0 (P.W. for m L ≥3.0)

Tb = 7 °CTa = 27 °CR = 4

0

0

0

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4m0L

RH=0.2 (Fully dry)

RH=0.4 (P.W. for m L ≥0.5)

RH=0.6 (P.W. for m L ≥1.0)

RH=1.0 (P.W. for m L ≥2.5)

Tb = 7 °CTa = 27 °CR = 5

0

0

0

Fig. 6 – Effect of relative humidity on fin efficiency. (a) R = 2, (b) R = 3, (c) R = 4, and (d) R = 5.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4m0L

R=2 (P.W. for m L ≥1.5)

R=3 (P.W. for m L ≥1.5)

R=4 (P.W. for m L ≥1.0)

R=5 (P.W. for m L ≥1.0)

Tb = 7 °CTa = 27 °CRH = 0.6

0

0

0

0

Fig. 7 – Effect of radius ratio on the fin efficiency for ahyperbolic annular fin.

49i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 5 ( 2 0 1 6 ) 4 2 – 5 4

rR

Rt =

− 1(54)

R vw= + 1 (55)

where ψ , m L0 , rb, rt and R are all functions of u, v and w.So by keeping u and v constant, and considering w as the onlyindependent variable, the maximum heat dissipated from thefin can be obtained. This basically would mean that we havea finite quantity of material, which implies that we knowthermal properties of the material. Moreover, we have priorknowledge of the working environment in which we are goingto employ the fin, thus completely defining u. Finally, we findthe dimensions of the fin that will give us maximum heat trans-fer rate for the chosen thermal-geometric conditions.

The heat transfer given by Eq. (29) can be written in the fol-lowing non-dimensional form:

Qu b B

wAi C Bi C Ai C Bi CAi C Bb = +( )⎛

⎝⎜⎞⎠⎟

− ′ ( ) ′ ( ) + ′ ( ) ′ ( )′ ( )

1 2

2 3

1 2 2 1

1 ii C Ai C Bi C2 2 1( ) − ( ) ′ ( )⎡⎣⎢

⎤⎦⎥

(56)

where

C u w b Bvw

1 2

2 31 1

2= +( )( ) +⎛

⎝⎞⎠ (57)

C u w b B2 2

2 31= +( )( ) (58)

So by differentiating Eq. (56) with respect to w and equat-ing to zero, for fixed values of u and v we can find out thatparticular value of w, which will give us the maximum Q .The resulting analytical expression is rather complex there-fore the outcome is presented in a graphical form in Fig. 8,where the numerical results are presented in terms of dimen-sionless fin-base radius ( w) and heat transfer rate ( Q ) as afunction of dimensionless volume ( v) and fin heat transfer pa-rameter ( u). The results are also compared with the numericalsolution. It can be seen that the numerical results are in goodagreement with the analytical solutions such that the numeri-cal solution gives higher Q than analytical solution but lesservalue of w. It is worth noting that the analytical results arebased on an approximate linear temperature–humidity ratiorelationship given by Sharqawy and Zubair (2007) whereas thenumerical solution is based on the corresponding actual psy-chrometric relationship. A salient feature of presenting theresults in a graphical format is: although we started with aknown volume of fin material and found the maximum heattransfer rate, now we can go in the reverse direction, i.e. for agiven heat transfer rate we can find the minimum required finmaterial.

The above optimization results are also presented in termsof fitted regression equations (within ±1% to the actual nu-merical data) as:

w c vn= ⋅1 1 (59)

Q c vn= ⋅2 2 (60)

c u1 1 78863 0 599547= ⋅ −. . (61)

c u2 0 78741 1 2158= ⋅. . (62)

and the exponents’ n1 and n2 are

n u u u1 0 681613 0 0169324 0 00385546= − + ⋅ ( ) − ⋅ ⋅ ( ). . ln . ln (63)

n u2 0 376657 0 0186014= − ⋅ ( ). . ln (64)

It is important to emphasize that the regression equationsshown above are applicable only in the range of numerical datapresented in these figures. However, by including a correc-tion factor similar to Sharqawy and Zubair (2007), thedimensionless fin parameter u and heat transfer parameter Q,these regression equations can be used to a wide range of op-erating conditions provided the fin is completely wet.

5. Finite element formulation

The objective of this section is to develop a generalized finiteelement formulation with combined heat and mass transfer.

0.01 0.1 1 1010-1

100

101

102

v [-]

w [

-]

Analytical SolutionAnalytical SolutionNumerical SolutionNumerical Solution

u =0.25

0.51.02.0

0.01 0.1 1 1010-2

10-1

100

101

v [-]

Q [

-]

Analytical SolutionAnalytical SolutionNumerical SolutionNumerical Solution

u =2.0

u =1.0

u =0.25

u =0.5

Fig. 8 – Dimensionless optimal dimensions and heattransfer for a hyperbolic profile annular fin; comparisonwith numerical solution (a) Fin base versus fin volume (b)Heat loss versus fin volume.

50 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 5 ( 2 0 1 6 ) 4 2 – 5 4

The element should be capable of modeling conduction phe-nomena with convection and condensation boundary conditionsfor a two-dimensional orthotropic fin material. The main ad-vantage of this formulation is that it can be used for anyarbitrary fin shape, specified heat flux boundary condition,specified temperature boundary condition and composite finmaterials (e.g. a coating layer), without any additional math-ematical complexity. Some of the advantages of the finiteelement method over finite difference method are (Peiró andSherwin, 2005):

a. The finite difference (FDM) uses the differential form of thegoverning equations whereas the finite element method(FEM) is based on integral formulation. The use of integralform in FEM provides a more natural treatment of Neumannboundary condition (i.e. imposed heat flux in case of finproblem) than the FDM.

b. FEM is better suited than FDM to deal with the complex ge-ometries in two-dimensional problems because the integralform does not depend on the special mesh structure.

Based on the above discussion, the finite element formu-lation can be used to extend the present study for other finshapes, boundary conditions and composite fins.

The governing differential equation for steady state heatconduction with no internal heat generation in an orthotropicmaterial can be expressed as (Lewis et al., 2004).

kT

xk

Ty

kT

zx y z

∂∂

+ ∂∂

+ ∂∂

=2

2

2

2

2

20 (65)

The boundary conditions are:

T T S

kTx

l kTy

m kTz

n q S

kTx

l kTy

b

x y z b

x y

=∂∂

+ ∂∂

+ ∂∂

+ =

∂∂

+ ∂∂

on

on

1

20ˆ ˆ ˆ

ˆ ˆ̂ ˆm kTz

n h T T h i Sz a D fg a+ ∂∂

+ −( ) + −( ) =ω ω 0 3on

(66)

where, k kx y, , and kz are the thermal conductivities in orthogo-

nal directions, qb is the specified heat flux and ˆ, ˆl m , and n̂ are

the surface normals.The boundary conditions are very commonin heat transfer problems and are called boundary condi-tions of first, second and third kinds, respectively (Lienhard andLienhard, 2011). In the context of fin application, if the primesurface is at constant temperature, the surface S1 will be thefin base. Alternatively, a fin can receive a constant heat fluxfrom a prime surface to its base. For such cases, the surfaceS2 would be the fin base. It is worth mentioning that bothboundary conditions cannot occur simultaneously at the finbase.The surface S3 would be the remaining fin surface throughwhich heat transfer occurs between the fin and the surround-ing environment.

To solve Eq. (65) with boundary conditions (66), an addi-tional equation for ω is required.The psychrometric correlationbetween ω and T is non-linear, therefore following linear re-lationship is used:

ω = +a bT (67)

The details about the justification of using the linear rela-tionship and corresponding values of coefficient a and b arepresented in Appendix.

Therefore, by using the relationships in Eq. (67), the lastboundary condition in Eq. (66) becomes:

kTx

l kTy

m kTz

n h T Tx y z a∂∂

+ ∂∂

+ ∂∂

+ ′ ′ −( ) =ˆ ˆ ˆ 0 (68)

where ′ ′ −( )h T Ta is the combined heat transfer due to convec-tion and condensation. Therefore ′h and ′Ta are thecorresponding equivalent heat transfer coefficient and equiva-lent ambient temperature given by:

′ = +( )h h Bb1 (69)

′ = + −( )+

TT B a

Bba

a aω1

(70)

5.1. The variational principle

Variational principle is used here for deriving the finite elementformulation. The variational principle specifies a scalar quan-tity (functional Π), defined by an integral form for a continuumproblem. The solution of the continuum problem is a func-tion that makes Π stationary with respect to arbitrary changesin it (Zienkiewicz and Taylor, 2000).

For no internal heat generation, the functional for heat trans-fer problem is (Logan, 2007):

Π Π Π= + +U q h (71)

The functional has three terms that are associated with in-ternal energy U( ), heat conduction Πq( ) and heat convectionΠh( ) . For the governing equation (65) with associated bound-

ary conditions (66), the terms are (Logan, 2007):

U kTx

kTy

kTz

dVx y z

V

q

= ∂∂

⎛⎝

⎞⎠ + ∂

∂⎛⎝⎜

⎞⎠⎟

+ ∂∂

⎛⎝

⎞⎠

⎡

⎣⎢⎢

⎤

⎦⎥⎥

=

∫∫∫12

2 2 2

Π −−

= −( )

∫∫

∫∫

q TdS

h T T dS

b

S

h a

S

2

3

12

2Π

(72)

Therefore, for the case of equivalent heat transfer ′ ′ −( )h T Ta

(cf. Eq. (68)) the functional becomes:

Π = ∂∂

⎛⎝

⎞⎠ + ∂

∂⎛⎝⎜

⎞⎠⎟

+ ∂∂

⎛⎝

⎞⎠

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−∫∫∫12

2 2 2

kTx

kTy

kTz

dV qx y z

V

bTTdS

h T T dS

S

a

S

2

3

12

2

∫∫

∫∫+ ′ ′ −( )(73)

5.2. The finite element formulation

Minimization of Eq. (73) with respect to T yields:

B D B N N T[ ] [ ][ ] + [ ] [ ]⎡

⎣⎢

⎤

⎦⎥{ }

= [ ] +

∫∫∫ ∫∫

∫∫

T

V

aT

S

Tb

S

dV hT dS

N q dS

Χ

ϒ3

2

hhT dSaT

S

N[ ]∫∫3

(74)

51i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 5 ( 2 0 1 6 ) 4 2 – 5 4

where,

X and= + = + ( ) −( )+

11

12

2

2

b BB T a

b Ba aϒ ω

(75)

The equation (74) is of the form:

k T f⎡⎣ ⎤⎦{ } = { } (76)

Therefore, the element stiffness matrix and load vectors maybe deduced as follows:

k k k

k D B k N N

q h

q h

[ ] = [ ] + [ ][ ] = [ ] [ ][ ] ⎡⎣ ⎤⎦ = [ ] [ ]∫∫∫

with

and XB dV h dT

V

T SSS3

∫∫ (77)

Similarly,

f f f

f N f N

q h

q h

{ } = { } + { }

{ } = [ ] { } = [ ]∫∫ ∫∫, :with

andTb

S

aT

S

q dS hT dS2 3

ϒ (78)

It is worth mentioning that the factors X and ϒ are intro-duced due to the condensation heat transfer. For the unit valuesof both parameters, the formulation will be transformed intostandard formulation (for sensible heat transfer only) avail-able in many textbooks on finite element method.This situationwould occur for B = 0 (cf. Eq. (75)). The physical interpreta-tion is that the latent heat of water condensation ( i fg ) (cf. Eq.(10)) under dry operating conditions is zero.

5.3. Finite element model validation

The two dimensional axisymmetric finite element formula-tion has been implemented using commercial software MATLAB(2015). Four-node quadrilateral isoperimetric element has beenused for meshing the two-dimensional computational domainof a hyperbolic profile annular fin. As before, the grid inde-pendence was verified by obtaining a series of solutions withdifferent mesh sizes. The mesh size was then selected afterwhich the grid independence was observed. It is worth men-tioning that according to boundary conditions specified by Eq.(66), the fin base corresponds to surface S1 (i.e. specified tem-perature), the fin surface corresponds to surface S3 (i.e.convection and condensation) and the insulated tip corre-sponds to surface S2 (i.e. zero heat flux). The results of finiteelement model for isotropic fin case are validated against thenumerical solution presented in section 2.1.2. The results areshown in Fig. 9. A closed agreement is observed between thetwo solutions. This shows that the presented finite elementformulation is capable of accounting for the non-linear rela-tionship between temperature and humidity ratio withacceptable engineering accuracy because the numerical solu-tion of section 2.1.2 is based on actual psychrometric data.

6. Effects of orthotropic thermal conductivity

The advantage of finite element formulation compared to stan-dard finite difference approach is that it can be used for any

irregular geometry without any additional mathematical com-plexities. The objective of this section is to study the effectsof orthotropic thermal conductivity on thermal performanceof hyperbolic profile annular fins under dehumidifying oper-ating conditions. The fin analyses presented in precedingsections are one-dimensional, whereas the heat transfer in anorthotropic annular fin is certainly a two-dimensional problem.Thus, the axisymmetric form of the derived finite element for-mulation will be used in this section.

The study under dry operating conditions (Pashah et al.,2011) has shown that the thermal conductivity has consider-able effects when the thermal conductivity in fin longitudinaldirection is higher than the conductivity in lateral direction.For the values given in Table 1, the thermal conductivity ratiok k krz r z= ranges between 1 and 110, therefore the effect ofthermal conductivity ratio has been studied over the range

1 300≤ ≤krz , the reason to add the value 300 will be dis-cussed in the following.

The results are presented in Fig. 10. It is obvious that forpractical reasons, the isotropic fin (i.e., krz = 1) will be useful inthe range m L0 0 5≤ . because after that, the fin efficiency dropsexponentially. This implies that for a fixed fin geometry andmaterial, the h values must be small (cf. Eq. (20)), because forhigher h values only base portion of the fin would be effec-tive and tip portion would be at the same temperature as theincoming ambient air stream. This means that a slender finoperating under forced convection would have low efficiency.However, increasing the thermal conductivity ratio starts in-creasing the fin efficiency for higher values of m L0 . For exampleif the objective is to attain a minimum fin efficiency of 60%then krz = 50 would give the same efficiency for m L0 3 5= . (thecorresponding efficiency for an isotropic case ( krz = 1) wouldbe only 9%). Another important observation is that the fin ispartially wet for the isotropic fin case; increase in thermal con-ductivity ratio decreases the dry portion of the fin and increasesthe fin efficiency simultaneously such that the fin becomescompletely wet over the considered range of m L0 for krz ≥ 20.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

m0L

FEANumerical

Tb = 7 °CTa = 27 °CRH = 0.6R = 4

Fig. 9 – Comparison of results for an isotropic materialhyperbolic fin obtained through finite element andnumerical solution based on finite difference method.

52 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 5 ( 2 0 1 6 ) 4 2 – 5 4

It is also interesting to note that the gain in thermal per-formance is not linearly related to increase in thermalconductivity ratio. The gain in fin efficiency against krz valuesis presented in Fig. 11. It can be seen that with respect to iso-tropic fin at m L0 4= ; the gain in fin efficiency is 47% for krz = 50,compared to 60% and 76% for krz = 100 and krz = 300, respec-tively. It means that the gain in efficiency is much higher inthe range 2 50≤ ≤krz than it is in the range 50 300≤ ≤krz .

7. Concluding remarks

The conclusions from the present work may be summarizedas,

1. The closed-form analytical solution for completely wet hy-perbolic fin; based on an approximate linear temperature–humidity ratio provided excellent results when comparedto numerical solution.The results show a maximum 2% errorin fin efficiency value with respect to the numerical solu-tion based on actual psychrometric relationship, for theconsidered set of geometric and operating parameters.

2. A numerical solution is more convenient for partially wethyperbolic fin operating conditions due to non-linear actualpsychrometric relationship between the air temperature andhumidity ratio.

3. For completely wet fin conditions, regression equations(based on analytical solution) have been developed to cal-culate the optimum fin geometry when the fin volume orthe heat transfer rate is specified.

4. The orthotropic thermal conductivity effects are studiedusing a finite element formulation under fully and par-tially wet fin conditions. The results showed that the higherthermal conductivity ratio results in better thermal perfor-mance of fins at higher values of fin parameter m L0 andfully wet condition.

Acknowledgements

The authors would like to acknowledge the financial supportgiven by the King Fahd University of Petroleum and Mineralsfor this research work from Budget Head SB131005. Syed Zubairalso acknowledges support from the King Fahd University ofPetroleum and Minerals through the project IN121042.

Appendix: Piecewise linear relationship betweentemperature and humidity ratio

The approximate cubic polynomial relationship between tem-perature and humidity ratio is (Liang et al., 2000):

ω = + + +( ) × ≤ ≤ °−3 7444 0 3078 0 0046 0 0004 10 0 302 3 3. . . .T T T T C

(A1)

In order to get a linear finite element formulation, the abovepolynomial relationship has been approximated by thirty piece-wise linear polynomials (each over a temperature interval of1°C) of the form:

ω = +a bT (A2)

The coefficients a and b are obtained by a regression analy-sis on 100 data points, each over 1°C temperature interval. Thecoefficient values are considered with 20 decimal places. Thedifference between temperature values obtained through Eq.(A1) and Eq. (A2) are within 0.005°C.

The piecewise linear polynomial is implemented by ob-taining the finite element solution in iterative manner. A firstsolution is obtained by assuming dry operating conditions andthen following iterative solution uses appropriate a and b valuesbased on preceding iteration temperature results. The solu-tion is assumed to be converged when the maximum

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4m0L

Tb = 7 °CTa = 27 °CRH = 0.6R = 4

krz =300

krz =100

krz =50krz =30

krz =20krz =10krz =5krz =2

PW

Fig. 10 – Effect of orthotropic thermal conductivity on finefficiency operating under dehumidifying conditions. Thepartially wet region is below the curve PW.

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2 2.5 3 3.5 4m0L

Tb = 7 °CTa = 27 °CRH = 0.6R = 4

krz =300

krz =100

krz =50krz =30

krz =20krz =10

krz =5

krz =2

Fig. 11 – Gain in fin efficiency with respect to isotropicmaterial fin; for different values of thermal conductivityratios.

53i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 5 ( 2 0 1 6 ) 4 2 – 5 4

temperature difference at any point on the fin surface for thetwo successive iterations is less than 0.01°C.

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54 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 6 5 ( 2 0 1 6 ) 4 2 – 5 4