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Performance and optimum design analysis of longitudinal and pin ﬁns with simultaneous heat and mass transfer: Uniﬁed and comparative investigations Balaram Kundu

*

Department of Mechanical Engineering, Jadavpur University, Raja SC Mallick Road, Kolkata 700 032, West Bengal, India Received 1 March 2006; accepted 10 August 2006 Available online 11 October 2006

Abstract In the present paper, the thermal analysis and optimization of longitudinal and pin ﬁns of uniform thickness subject to fully wet, partially wet and fully dry surface conditions are carried out analytically, and also a comparative study is made between the longitudinal and pin ﬁn for a wide range of design parameters. From the results, a signiﬁcant eﬀect on the temperature distribution in the ﬁn and the ﬁn eﬃciency with the variations in moist air psychometric conditions is noticed. For partially wet ﬁns, the length of the wet–dry interface depends on the relative humidity RH, ﬁn parameter Zd and geometry of the ﬁn. From the results, it can also be highlighted that for the same thermo-geometric and psychometric parameters, a longitudinal ﬁn gives higher eﬃciency than the corresponding pin ﬁn irrespective of surface conditions. Next, a generalized scheme for optimization has been demonstrated in such a way that either heat transfer duty or ﬁn volume can be taken as a constraint selected as per design requirement. From the optimization results, it can be pointed out that the optimum design of both the longitudinal and pin ﬁn under the partially wet surface condition is only possible for a narrow range of relative humidity whereas for the fully wet surface, a wide range is noticed. Finally, design curves have been established for a wide range of thermo-psychometric parameters, which may be helpful to a designer for estimation of the optimum design variables of a ﬁn with a minimum eﬀort. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Dehumidiﬁcation; Fin eﬃciency; Fully wet surface; Longitudinal ﬁn; Optimization; Partially wet surface; Pin ﬁn

1. Introduction For the enhancement of heat transfer, ﬁns or extended surfaces are extensively used in a variety of applications. Among the various applications, ﬁnned surfaces are widely found in air-conditioning, refrigeration, cryogenics systems and many cooling systems in industries for air cooling and dehumidifying. Since, in most cooling coils, the coil surface temperature is below the dew point temperature of the air being cooled, moisture is condensed on the ﬁn surface attached with the coil. Thus, simultaneous heat and mass transfer take place and the ﬁn surface becomes wet. Among *

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1359-4311/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2006.08.003

the factors aﬀecting the thermal performance of such wet ﬁns are the geometry, material and thermo-psychometric parameters. Depending upon the shape of a primary surface, diﬀerent types of ﬁns are used in heat exchange applications. For example, for the ﬂat basic surface, both the longitudinal and pin ﬁns are usually attached to augment the heat transfer rate from the solid surface to an adjoining ﬂuid. With regard to air, surface and the dew point temperature of the surrounding air, three diﬀerent states of a ﬁn surface are found in the literature: fully dry, fully wet and partially wet. In the case when the surface temperature is higher than the saturation temperature, condensation does not occur and the surface remains dry. The surface will be completely wet if the temperature is below the

B. Kundu / Applied Thermal Engineering 27 (2007) 976–987

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Nomenclature A

Ac B Bi Cp Cpa Cpv F g h hm hfg J k L Le m p q Q qi Qi RH s T Ta Tb Td U V

constant evaluated from the ﬁn-base and ﬁn-tip conditions of air [xb(Tt Tb) Tb(xt xb)]/ (Tt Tb) cross sectional area normal to the direction of heat conduction, m2 constant evaluated from the ﬁn-base and ﬁn-tip conditions of air, (xt xb)/(Tt Tb) Biot number hdb/k humid speciﬁc heat of surrounding air, Cpa + xCpv (J/kg of dry air/°C) speciﬁc heat of dry air (J/kg of dry air/°C) speciﬁc heat of water vapour (J/kg of dry air/°C) parameter used in Eq. (31) function deﬁned in Eq. (25) surface heat transfer coeﬃcient (W/m2/°C) mass transfer coeﬃcient (kg/m2 s) latent heat of evaporation of water (J/kg) Jacobian matrix, see Eq. (33) thermal conductivity of the ﬁn material (W/m/ °C) length of the ﬁn, see Fig. 1 (m) Lewis number (=thermal diﬀusivity/diﬀusion coeﬃcient) constant used in Eqs. (2), (4) and (5) perimeter (m) actual heat transfer rate through the ﬁn (W) dimensionless actual heat transfer, qhm1/ (2/m)pm1km(Ta Tb) ideal heat transfer rate through the ﬁn (W) dimensionless ideal heat transfer, qihm1/ (2/m)pm1km(Ta Tb) relative humidity function deﬁned in Eqs. (29) and (30) local ﬁn surface temperature (°C) dry bulb temperature of the surrounding air (°C) ﬁn-base temperature (°C) dew point temperature of surrounding air (°C) dimensionless ﬁn volume, deﬁned in Eq. (24) ﬁn volume (m3)

saturation temperature corresponding to the partial pressure of water vapour in the surrounding air. The partially wet surface occurs when the saturation temperature is less than the ﬁn-tip temperature and greater than the ﬁn-base temperature. The addition of ﬁns causes to increase the initial cost, weight, and pumping power required for the forced convection. Thus, it is essential to study the optimum design conditions. The optimization of ﬁns can be classiﬁed into two groups. The ﬁrst group of optimization problems involves the determination of the proﬁle of the ﬁn so that for a given amount of heat transfer rate, the volume of the material used is a minimum. Other group is to determine the ﬁn

x X x0 X0 Zd Zw

coordinate along the length of the ﬁn (m) non-dimensional distance, x/L distance from the ﬁn at which dry and wet surface coexisted, see Fig. 1 non-dimensional distance, x0/L pﬃﬃﬃﬃ ﬃ ﬁn parameter, Bi=w dimensionless ﬁn parameter for the wet ﬁn, Zd(1 + Bn)1/2

Greek symbols db half thickness, see Fig. 1 g ﬁn eﬃciency h dimensionless temperature (Ta T)/(Ta Tb) hd dimensionless dew point temperature, (Ta Td)/(Ta Tb) hp dimensionless temperature parameter, (xa BTa A)/[(Ta Tb)(1 + Bn)] w aspect ratio, db/L / temperature parameter, h + hp /0 temperature parameter, 1 + hp /d dew point temperature parameter, hd + hp x local speciﬁc humidity of air on the ﬁn surface (kg of w.v./kg of d.a.) xa speciﬁc humidity of surrounding air (kg of w.v./ kg of d.a.) xb speciﬁc humidity of air on the ﬁn surface at the base (kg of w.v./kg of d.a.) xt speciﬁc humidity of air on the ﬁn surface at the tip (kg of w.v./kg of d.a.) n dimensionless latent heat transfer parameter, hfg/CpLe2/3 Subscripts dr dry f fully wet opt optimum p partially wet w wet

dimensions for a given ﬁn shape and a desired heat dissipation rate in order that the volume of material used would be a minimum. Using the ﬁrst approach of optimization, many researchers [1–4] have concentrated their studies to obtain the optimum proﬁle shape of a ﬁn for the purely conductive and convective environments. These proﬁles are parabolic, circular or wavy in nature. The main drawback of the optimum proﬁles is their sharp tip and these conﬁgurations are diﬃcult to manufacture. In practical applications, constant thickness ﬁns are extensively found because of their simple design and ease of fabrication. The optimization of constant thickness ﬁns may be done by using the second approach of optimization.

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B. Kundu / Applied Thermal Engineering 27 (2007) 976–987

Fig. 1. Schematic diagram of the partially wet surface ﬁn of uniform thickness: (a) longitudinal ﬁn; (b) pin ﬁn.

A contemporary study of heat and mass transfer demands the development of complex mathematical models. For fully wet ﬁns, Threlkeld [5] and McQuiston [6] determined the one-dimensional ﬁn eﬃciency of a rectangular longitudinal ﬁn based on a modiﬁed dry ﬁn formula. Kilic and Onat [7] used a quasilinear one-dimensional model for vertical rectangular ﬁns. They considered constant heat and mass transfer coeﬃcients along the ﬁn. The ﬁn eﬃciency and the average temperature were lower in the case of condensation than in the case of a dry ﬁn. Toner et al. [8] have studied the rectangular and triangular ﬁns by using quasilinearization technique when condensation occurs. Wu and Bong [9] provided analytical solutions for the eﬃciency of a longitudinal ﬁn under wet surface conditions considering temperature and humidity ratio diﬀerences as the driving forces for heat and mass transfer. Kazeminejad et al. [10] calculated numerically the performance of a cooling and dehumidifying vertical rectangular ﬁn with the variation of non-uniform heat transfer coeﬃcient. Hong and Webb [11] described a quantitative evaluation of methods used to calculate the ﬁn eﬃciency for dry and wet ﬁns. They provided an empirical modiﬁcation to the Schmidt equation. Coney et al. [12] demonstrated a numerical solution for dehumidiﬁcation of air over a rectangular ﬁn taking into account the thermal resistance of the condensate ﬁlm and using a second degree polynomial to relate the humidity ratio with dry bulb temperature. Srinivasan and Shah [13] gave a summary of previous studies on condensation over rectangular ﬁns. The eﬀect of condensation on a vertical plate ﬁn of variable thickness has been numerically studied by Sharma et al. [14]. A technique to determine the performance and optimization of straight tapered longitudinal ﬁns subject to simultaneous heat and mass transfer has been established analytically by Kundu [15]. He assumed that the entire ﬁn surface temperature is below the dew point temperature of the surrounding air. Lin and Jang [16] investigated a two-dimensional ﬁn eﬃciency analysis of combined heat and mass transfer in elliptic ﬁns. The analysis of one-

dimensional ﬁn assembly heat transfer with dehumidiﬁcation has been demonstrated by Kazeminejad [17]. Liang et al. [18] have drawn a comparison between one-dimensional and two-dimensional models for wet surface ﬁn eﬃciency of a plate-ﬁn-tube heat exchanger. From the thorough literature survey summarized above, it reveals that the previous studies were concentrated mainly on the ﬁn performance of fully wet ﬁns. However, depending upon the thermo-geometric parameters and psychometric properties of air, ﬁn surfaces may be fully dry, fully wet or partially wet. No attention has been given in the literature to determine the thermal performance of cylindrical pin ﬁns in wet conditions. Nevertheless, pin ﬁns are extensively used in many heat transfer equipments. In addition, researchers have rarely concentrated on the optimization study of ﬁns under diﬀerent wet surface conditions. Thus, the analysis and optimum dimensions of a ﬁn for diﬀerent wet surface conditions is essential to carry out the exact analysis of heat exchanger under dehumidifying condition. Therefore, a combined analysis for the performance and optimization of both the fully and partially wet surfaces of longitudinal and pin ﬁns is the prime motivation of the present investigation. In the present work, an eﬀort has been made to determine analytically the performance and optimization of longitudinal and pin ﬁns of uniform thickness under fully as well as partially wet conditions. Analysis of the longitudinal and pin ﬁns has been presented in uniﬁed formulas. The eﬀects of some important parameters like relative humidity, dry bulb temperature of the surrounding air and ﬁn-base temperature on the overall ﬁn eﬃciency have been studied. From the result, it has been observed that with the increase of relative humidity the ﬁn surface temperature increases due to evolution of latent heat of condensation on the ﬁn surface. However, in comparison to a longitudinal ﬁn, this eﬀect is larger for the pin ﬁn. For all the above types of ﬁns in a partially wet surface, the eﬀect of relative humidity on the ﬁn eﬃciency does change considerably. Moreover, this result is less responsive to pin ﬁns. The range of relative humidity for keeping a partially

B. Kundu / Applied Thermal Engineering 27 (2007) 976–987

wet surface of a pin ﬁn is always larger than for the corresponding longitudinal ﬁn. The optimization of longitudinal and pin ﬁns under both the fully and partially wet surface conditions is done in a generalized form based on the calculus of variation in which either heat transfer rate or ﬁn volume can be taken as a constraint. Finally, a design curve has been drawn for a wide range of thermo-psychometric parameters. It assists to estimate the design unknown variables without the rigorous mathematical calculations.

979

For the fully dry surface, the governing ﬁn equation is obtained by Eq. (1) with only consideration of zero value of mass transfer coeﬃcient hm. By applying the Chilton–Colburn analogy [19] on the relation between heat and mass transfer coeﬃcients and under the above assumptions, the governing diﬀerential equation for the longitudinal and pin ﬁns can be expressed in normalized form as follows: Fully wet surface 2

d /=dX 2 ¼ mZ 2d ð1 þ BnÞ/

2. Formulations of the mathematical model The detailed geometrical parameters of a longitudinal ﬁn and a pin ﬁn with the coordinate system taken for the present analysis are shown schematically in Fig. 1. In both ﬁns, x-axis is considered along the length of the ﬁn starting from the primary surface. Depending upon the surface temperature of a ﬁn and the dew point temperature of its surrounding air, either sensible or both sensible and latent heat transfer takes place on the ﬁn surface. Therefore, on the basis of heat transfer mechanism, the ﬁn surface can be treated as dry, fully wet or partially wet. To derive the mathematical formulations of the present theoretical model, the following assumptions are made: i. The thermal conductivity of the ﬁn material and the coeﬃcient of convective heat transfer are assumed to be constant. ii. Due to dehumidiﬁcation of air on the ﬁn surface, the thickness of the condensate water ﬁlm is very small compared to the thickness of the boundary layer. As a result the eﬀect of condensate thickness can be neglected. iii. A linear relationship between speciﬁc humidity (x) of air on the ﬁn surface and the corresponding ﬁn surface temperature (T) is taken considering a small temperature range between dew point and ﬁn-base temperature, i.e. x = A + BT, where, A and B are constants evaluated from the psychometric properties of air at the base and tip conditions. iv. Heat conduction through ﬁns is one-dimensional and steady state. v. The temperature of the surrounding air is constant. vi. Heat transfer by radiation is negligible. vii. For the analysis of longitudinal ﬁns, two extreme ﬁn surfaces parallel to the longitudinal axis are insulated.

where pﬃﬃﬃﬃﬃ Z d ¼ Bi=w; w ¼ db =L; X ¼ x=L; / ¼ h þ hp ; h ¼ ðT a T Þ=ðT a T b Þ;

ð2Þ

Bi ¼ hdb =k;

n ¼ hfg =C p Le2=3 and hp ¼ ðxa BT a AÞ=½ðT a T b Þð1 þ BnÞ

ð3Þ

and m is a constant and is equal to 1 and 2 for the longitudinal and the pin ﬁn, respectively. Partially wet surface " # d2 /=dX 2 d2 h=dX 2 0 6 X 6 X 0; 0 < X 0 < 1 mZ 2d ð1 þ BnÞ/ ¼ ð4Þ 2 X0 6 X 6 1 mZ d h Fully dry 2

d h=dX 2 ¼ mZ 2d h

ð5Þ

2.2. Boundary conditions A partially wet ﬁn is encountered when the ﬁn-base temperature is lower but the ﬁn-tip temperature is higher than the dew point temperature of the surrounding air. On the ﬁn surface there is a linear location, X = X0 as shown in Fig. 1, where the ﬁn temperature equalizes the air dew point temperature. The ﬁn is then separated into two regions: a wet region (0 6 X 6 X0) with the ﬁn surface temperature lower than the air dew point temperature, and a dry region (X0 6 X 6 1) with ﬁn surface temperature higher than the air dew point temperature. For both the surface conditions, ﬁn-base temperature is assumed to be a constant and heat exchange at the tip is considered. The boundary conditions can be expressed mathematically as follows:

2.1. Governing diﬀerential equations Fully wet surface The governing energy equation is derived by applying the law of conservation of energy to an incremental area of straight ﬁns of uniform thickness shown in Fig. 1: kAc dT =dx kAc ½dT =dx þ ðd2 T =dx2 Þdx ¼ hp dxðT a T Þ þ hm p dxðxa xÞhfg

at X ¼ 0; at X ¼ 1;

/ ¼ /0 d/=dX ¼

ð6aÞ wZ 2d ð1

þ BnÞ/

ð6bÞ

Partially wet surface ð1Þ

at X ¼ 0;

/ ¼ /0

ð7aÞ

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B. Kundu / Applied Thermal Engineering 27 (2007) 976–987

at X ¼ X 0 ; at X ¼ 1;

/ ¼ hd þ hp and h ¼ hd

ð7bÞ

wZ 2d h

ð7cÞ

dh=dX ¼

Fully dry surface at X ¼ 0; at X ¼ 1;

h¼1 dh=dX ¼

ð8aÞ wZ 2d h

ð8bÞ

2.3. Temperature distribution The temperature proﬁle in the ﬁn for the fully wet, partially wet and fully dry surfaces can be obtained by solving the governing diﬀerential equations (2), (4) and (5) with the boundary conditions (6)–(8), respectively: Fully wet surface m1=2 cosh m1=2 Z w ð1 X Þ þ w sinh m1=2 Z w ð1 X Þ / ¼ m1=2 coshðm1=2 Z w Þ þ w sinhðm1=2 Z w Þ /0 ð9Þ where Z w ¼ Z d ð1 þ BnÞ

by solving Eq. (15) using Newton–Raphson iterative method [20]. For this numerical iteration, ﬁnal results have been obtained until a convergence criterion (108) has been satisﬁed. 2.4. Heat transfer rate Actual heat transfer rate through the longitudinal and pin ﬁns has been estimated form Eqs. (9), (11) and (13) by applying the Fourier’s law of heat conduction at the ﬁn base. Heat transfer rate can be expressed in dimensionless form for diﬀerent surface conditions given below: Fully wet surface qw hm1 ð2=mÞpm1 k m ðT a T b Þ m1=2 w2m1 Z d2m2 Z w /0 m1=2 tanhðm1=2 Z w Þ þ w ¼ m1=2 þ w tanhðm1=2 Z w Þ

Qw ¼

ð16Þ

Partially wet surface 1=2

ð10Þ

Partially wet surface /0 sinh m1=2 Z w ðX 0 X Þ þ /d sinhðm1=2 Z w X Þ ; /¼ sinhðm1=2 Z w X 0 Þ 0 6 X 6 X0 ð11Þ 1=2 1=2 1=2 m cosh m Z d ð1 X Þ þ w sinh m Z d ð1 X Þ h ¼ ; hd m1=2 cosh fm1=2 Z d ð1 X 0 Þg þ w sinh fm1=2 Z d ð1 X 0 Þg ð12Þ X0 6 X 6 1 Fully dry surface m1=2 cosh m1=2 Z d ð1 X Þ þ w sinh m1=2 Z d ð1 X Þ h¼ m1=2 coshðm1=2 Z d Þ þ w sinhðm1=2 Z d Þ ð13Þ For the estimation of temperature proﬁle on the partially wet surface, the distance X0 is to be determined where the separation of the wet surface from the dry surface takes place. It can be evaluated by considering the continuity of heat conduction at the separation section. It yields ½dh=dX X ¼X 0 ¼ ½d/=dX X ¼X 0

ð14Þ

Diﬀerentiating Eqs. (11) and (12) and substituting them in Eq. (14), one can get the following equation:

Z w /0 /d cosh m1=2 Z w X 0

m1=2 þ w tanh m1=2 Z d ð1 X 0 Þ Z d hd sinh m1=2 Z w X 0

m1=2 tanh m1=2 Z d ð1 X 0 Þ þ w ¼ 0 ð15Þ From Eq. (15), it follows that the linear distance X0 is a function of thermo-geometric and psychometric parameters. For a given design condition, X0 can be calculated

qp hm1 ð2=mÞpm1 k m ðT a T b Þ m1=2 w2m1 Z d2m2 Z w /0 coshðm1=2 Z w X 0 Þ /d ¼ sinhðm1=2 Z w X 0 Þ

Qp ¼

ð17Þ

Fully dry surface qdr hm1 ð2=mÞpm1 k m ðT a T b Þ m1=2 w2m1 Z d2m1 m1=2 tanhðm1=2 Z d Þ þ w ¼ m1=2 þ w tanhðm1=2 Z d Þ

Qdr ¼

ð18Þ

2.5. Fin eﬃciency The overall ﬁn eﬃciency is deﬁned as the ratio of the rate of actual total heat transfer to the rate of ideal total heat transfer. The latter is deﬁned as the heat rate through a ﬁn that would be transferred if the entire ﬁn surface were maintained at its base temperature. For the calculation of actual heat transfer rate through a partially wet ﬁn, analysis of two separate regions is essential. The actual total heat transfer must include both the sensible heat transfer and the latent heat transfer originated by mass transfer (dehumidiﬁcation). The sensible heat transfer is due to the convection from the air to the ﬁn surface because of their temperature diﬀerence and the latent heat transfer is caused by humidity ratio diﬀerence between the air and the ﬁn. The maximum heat transfer rate for both the partially and fully wet surface corresponds to an ideal ﬁn whose surface temperature equals the temperature at the ﬁn base under a fully wet condition. Therefore, the ideal heat transfer rate can be written in non-dimensional form given below:

B. Kundu / Applied Thermal Engineering 27 (2007) 976–987

can be obtained from the expression given below in dimensionless form as

Fully wet surface m1

ðQi Þj ¼

ðqi Þj h ð2=mÞpm1 k m ðT a T b Þ

¼ w2m1 Z 2m2 Z 2w ðm þ wÞð1 þ BnÞ/0 ; d

U ¼ Vhmþ1 =k mþ1 ¼ ð2=mÞpm1 Z 2mþ2 w2mþ1 d j ¼ w; p

ð19Þ

ðqi Þdr hm1 ¼ w2m1 Z 2m d ðm þ wÞ ð2=mÞpm1 k m ðT a T b Þ

ð20Þ

Fully dry surface ðQi Þdr ¼

From the deﬁnition of ﬁn eﬃciency, it can be expressed as follows: Fully wet surface m1=2 m1=2 tanhðm1=2 Z w Þ þ w ð21Þ gw ¼ Z w ðm þ wÞfm1=2 þ w tanhðm1=2 Z w Þg Partially wet surface m1=2 /0 coshðm1=2 Z w X 0 Þ /d gp ¼ Z w ðm þ wÞ/0 coshðm1=2 Z w X 0 Þ Fully dry surface m1=2 m1=2 tanhðm1=2 Z d Þ þ w gdr ¼ Z d ðm þ wÞfm1=2 þ w tanhðm1=2 Z d Þg

981

ð22Þ

ð23Þ

ð24Þ

For the fully wet surface ﬁn with the thermo-psychometric parameters being constant, the heat transfer rate (Q) and the ﬁn volume (U) are functions of Zd and w only. However, in the case of partially wet surface ﬁns, heat transfer rate depends not only on the Zd and w but also on the linear location separating the dry and wet surface, X0. Moreover, the length X0 is also function of the Zd, w and the psychometric condition. For a given thermo-psychometric parameter, it is two variables and one constraint optimization problem where either heat transfer or ﬁn volume can be treated as a constraint. Here, it is noteworthy that the deﬁnition of a local, global, or an inﬂection point remains the same as that of a single variable functions, but the optimality criteria for multivariate functions are diﬀerent. The optimality criteria for the present optimization problem can be derived using Lagrange multiplier technique. The following optimality function is obtained after eliminating the Lagrange multiplier from the Euler’s equations: " # oQj =oZ d ½gðZ d ; wÞ ¼ ½ oU =ow oU =oZ d ¼ ½0 oQj =ow j¼w;p;dr

It may be noted that the calculation of ﬁn eﬃciency for the partially wet surfaces can be done by using the weighted average of dry and wet ﬁn eﬃciencies [16,21].

ð25Þ Using Eqs. (16)–(18) and (24) and Eq. (25) can be written after simpliﬁcation for the fully wet, partially wet and fully dry surface separately as follows:

2.6. Optimization Fully wet surface The ﬁn optimization is classiﬁed on the basis of two approaches. The ﬁrst approach to optimization is for a given volume or desired heat transfer rate, and thermo-psychometric parameters the shape of the ﬁn (parabola or circular) that may be obtained would maximize the heat transfer rate or minimize the ﬁn volume. The second approach is for a desired heat transfer rate or given volume, proﬁle (rectangular, triangular, etc.) and thermo-psychometric parameters, it seeks the dimensions of the ﬁn that will minimize the ﬁn volume or maximize the heat transfer rate. Although the ﬁrst approach of optimization is superior to the second approach with respect to the heat transfer rate per unit volume, the second kind is extensively used, because the resulting ﬁn proﬁle obtained from the ﬁrst kind is diﬃcult to manufacture. Nevertheless, from the past literature, the superiority of the optimum proﬁle is measured, based on the volume, which is marginally lower than the optimum triangular ﬁn. For the present problem, the second kind of approach of optimization is adopted to establish the optimality criteria. In this section, an optimization scheme has been developed for both ﬁns of uniform thickness with combined heat and mass transfer using uniﬁed formulas. The volume (V) of the two types of ﬁns, namely, longitudinal and pin ﬁn

gðZ d ; wÞ ¼ m1=2 ½wð4m2 þ 2m þ 1Þ þ Z w ðw2 mÞ þ ð4m2 1Þðm þ w2 Þ tanhðm1=2 Z w Þ þ m1=2 ½wð4m2 2m 3Þ þ Z w ðm w2 Þtanh2 ðm1=2 Z w Þ ¼ 0

ð26Þ

Partially wet surface gðZ d ; wÞ ¼ Z w ½/0 ð4m2 1Þ sinhðm1=2 Z w X 0 Þ þ m1=2 Z w /d f2wðm þ 1ÞoX 0 =ow Z d oX 0 =oZ d X 0 g coshðm1=2 Z w X 0 Þ /d ð4m2 1Þ sinhðm1=2 Z w X 0 Þ m1=2 Z w /0 f2wðm þ 1ÞoX 0 =ow X 0 Z d oX 0 =oZ d g

ð27Þ

Fully dry surface gðZ d ; wÞ ¼ m1=2 ½wð4m2 þ 2m þ 1Þ þ Z d ðw2 mÞ þ ð4m2 1Þðm þ w2 Þ tanhðm1=2 Z d Þ þ m1=2 ½wð4m2 2m 3Þ þ Z d ðm w2 Þtanh2 ðm1 =2Z d Þ ¼ 0

ð28Þ

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B. Kundu / Applied Thermal Engineering 27 (2007) 976–987

Eqs. (26)–(28) can be solved with a constraint equation. According to the speciﬁcation of a design, either the ﬁn volume U or the heat transfer rate Q, may be taken as a constraint. Thus the constraint equation can be constructed as follows:

where ogðZ d ; wÞ osðZ d ; wÞ D1 ¼ Z d þ sðZ d ; wÞ gðZ d ; wÞ det J ðZ d ; wÞ ow ow

ð35Þ

and Volume constraint sðZ d ; wÞ ¼ ð2=mÞpm1 Z d2mþ2 w2mþ1 U ¼ 0

ð29Þ

osðZ d ; wÞ ogðZ d ; wÞ D2 ¼ w þ gðZ d ; wÞ sðZ d ; wÞ det J ðZ d ; wÞ oZ d oZ d

ð36Þ

Heat transfer rate constraint 2

3 sðZ d ; wÞ 6 7 4 sðZ d ; wÞ 5 sðZ d ; wÞ 2 3 Z w /0 m1=2 tanhðm1=2 Z w Þ þ w = m1=2 þ w tanhðm1=2 Z w Þ Qw 6 7 Z w /0 coshðm1=2 Z w X 0 Þ /d = coshðm1=2 Z w X 0 Þ Qp ¼ F4 5 1=2 Z d m tanhðm1=2 Z d Þ þ w = m1=2 þ w tanhðm1=2 Z d Þ Qdr 2 3 0 6 7 ¼ 405 ð30Þ 0

where F ¼ m1=2 w2m1 Z d2m2

ð31Þ

Eqs. (26)–(29) or (30) being non-linear and non-homogenous algebraic equations, they can be solved numerically to determine the optimum values of Zd and w. The generalized Newton–Rhapson method [20] has been suggested for the solution. For the partially wet surface ﬁns, one extra algorithm is required for calculating the values of oX0/ oZd, oX0/ow, o2 X 0 =oZ 2d , o2X0/ow2 and o2X0/oZdow used for the optimization algorithm. These derivatives can be obtained by diﬀerentiating Eq. (15) and solving them using Newton–Raphson iterative method [20]. To satisfy the convergence criterion (suﬃcient condition), selection of initial guess values for the roots are to be made cautiously. A brief outline of the Newton–Rapshon iteration scheme and convergence criteria (necessary condition) for the present problem is given below: " # " # " # gfðZ d Þj ; wj g ðZ d Þjþ1 ðZ d Þj 1 ¼ ½J fðZ d Þj ; wj g sfðZ d Þj ; wj g wjþ1 wj

The above process may be repeated to obtain the optimum values of Bi and w until a desired accuracy (108 in the present case) has been achieved. 3. Results and discussion Based on the above analysis, results for the present investigation are taken for a wide range of thermo-psychometric parameters and a constant ambient absolute pressure of 1.01325 bar. The dry bulb temperature (surrounding air temperature), ﬁn-base temperature and relative humidity have been taken as the psychometric parameters. Dew point temperature is calculated from the air properties such as relative humidity and dry bulb temperature. However, the present analysis has been established as a function of speciﬁc humidity. So, for the ﬁrst step to obtain the results of the present work, the conversion of a relative humidity into a speciﬁc humidity is provided. The detailed analysis for this conversion is available elsewhere [5]. In Fig. 2, the temperature distribution in the longitudinal ﬁn is plotted against the dimensionless distance X measured from the ﬁn base, for the wet surface of the relative humidities of 70% and 100%, and for the dry surface. The ﬁn parameter Zd, aspect ratio w, dry bulb temperature Ta and ﬁn-base temperature Tb are chosen arbitrarily as 1.5, 0.05, 30 °C and 5 °C, respectively. By comparison, temperature distribution in the pin ﬁn is also depicted in the same ﬁgure. The graph reveals that the dimensionless temperature h in both ﬁns decreases gradually with the increase 1.0

ð32Þ

Longitudinal fin Spine

0.8

j

Dry Surface

0.6 0.4

Zd=1.5 ψ=0.05

0.2

Ta=30 C

j

The suﬃx ‘‘j’’ denotes the value at the jth iteration. At each step of this iteration the convergence criteria is selected as follows: ( ) oD2 oD1 oD2 oD1 <1 þ ; þ Max ð34Þ oZ d j oZ d j ow j ow j

RH=70% (Partially wet)

θ

Here, J denotes the Jacobian matrix, which is expressed as ih i 3 2h o o gðZ ; wÞ gðZ ; wÞ h n oi d d ow j j7 6 oZ d ih i 5 J ðZ d Þj ; wj ¼ 4 h ð33Þ o o sðZ ; wÞ sðZ ; wÞ d d oZ d ow

o

o

Tb=5 C

0.0 0.0

0.2

RH=100%

0.4

0.6

0.8

1.0

X

Fig. 2. Eﬀects of relative humidity of air on the temperature distribution in longitudinal and pin ﬁns.

B. Kundu / Applied Thermal Engineering 27 (2007) 976–987

in X value and a minimum temperature attains at the ﬁn tip (X = 1). The decrease in dimensionless temperature h indicates the increase in dimensional temperature T. Therefore, the maximum temperature in the ﬁn is obtained at the ﬁn tip. For the above thermo-geometric and psychometric parameters, the tip temperature for the longitudinal ﬁn is lower than the dew point and as a result the entire ﬁn surface is fully wet. However, in the case of pin ﬁn for the relative humidity of 70%, dew point temperature lies in between tip and base temperature and, therefore, the ﬁn becomes partially wet (X0 = 0.623). Normally, for all types of wet ﬁns, the ﬁn surface temperature increases with the increase in relative humidity of air. This can be explained from the condensation point of view. With the increase in relative humidity, condensation of moisture on the ﬁn surface enhances. This releases more latent heat of condensation resulting in the increase in surface temperature. In general, surface temperature of a pin ﬁn is higher than that of the longitudinal ﬁn for the same thermo-physical parameters. In Fig. 3, the eﬃciency g for both the longitudinal and pin ﬁns is plotted against the relative humidity RH over the whole range of relative humidities from RH = 0% to RH = 100%. For both the longitudinal and pin ﬁns, the partially wet surface begins at the same humidity of 20% owing to the same base temperature Tb = 5 °C. The range of relative humidity for the partially wet pin ﬁn is more in comparison with the longitudinal ﬁn. This fact can be explained on the basis of higher ﬁn-tip temperature observed for the pin ﬁn. From the ﬁgure, it can be established that, in general, the ﬁn eﬃciency for the partially wet surface depends signiﬁcantly on the relative humidity. However, for the pin ﬁn, this eﬀect is relatively less important. In nature, the ﬁn eﬃciency for the dry ﬁn is independent of the relative humidity of air whereas the ﬁn eﬃciency for the fully wet ﬁn depends weakly on the relative humidity for all types of ﬁns. It can further be noted from the ﬁgure that the eﬃciency for the longitudinal ﬁn is always greater than for the pin ﬁn for identical thermo-geometric and psychometric parameters.

983

The distance from the ﬁn base to the section of separation between the dry and wet region X0 for a partially wet ﬁn as a function of the relative humidity and the ﬁn parameter Zd for both the longitudinal and pin ﬁn with the constant values of w = 0.05, Ta = 30 °C and Tb = 5 °C, are depicted in Fig. 4. For the partially wet surface, the value of X0 increases with the increase in relative humidity up to a certain value, then, there is a sharp rise in the X0 value until it attains the fully wet surface (X0 = 1). From this ﬁgure, it can be found that, for the partially wet ﬁn, the range of relative humidity increases with the increase of Zd. It can be explained in the following way: From Eqs. (11) and (12), it is clear that X0 is function of Zd, aspect ratio w and psychometric properties of air. For a constant value of w and psychometric properties, the ﬁn parameter Zd is a function of Bi only. The higher value of Zd signiﬁes the higher value of Biot number. This may give the higher conductive resistance in the ﬁn and consequently, it may raise the tip temperature. In comparison with the longitudinal ﬁn, a larger range of relative humidity of air is satisﬁed in partial wet conditions in case of pin ﬁns. Fig. 5 depicts the variation of the ﬁn eﬃciency of longitudinal as well as pin ﬁn as a function of ﬁn-base temperature and relative humidity. For the relative humidity of 70% and 100%, the wet ﬁn eﬃciency gradually decreases with a constant rate as the ﬁn-base temperature is increased. Although the temperature variation in the ﬁn decreases with the increase of the ﬁn-base temperature, the ﬁn eﬃciency decreases due to the higher value of latent heat of condensation. In the case of the partially wet surface of relative humidity 40%, a reversed trend is observed. The ﬁn eﬃciency increases with the increase in ﬁn-base temperature because the dry region increases rapidly. However, this increase in ﬁn eﬃciency is slower for the smaller value of ﬁn-base temperature. Fig. 6 delivers the idea about the variation of the ﬁn eﬃciency for both the longitudinal ﬁn and pin ﬁn as a function of ﬁn parameter Zd. The results are taken by keeping a constant ﬁn-base temperature, dry bulb temperature and

1.0 o

o

Zd=1.5, Ψ =0.05, Ta=30 C, Tb=5 C Fully dry

0.8 Zd=1.0 0.6

Longitudinal fin Spine

0.5

η

Fully wet

Partially wet

Zd=1.5 T =5oC b 0.2

Longitudinal fin Spine

Fully dry Partially wet

0.2 0

o

Ta=30 C

0.4

0.4 0.3

Ψ =0.05

X0

0.6

20

40

60

Fully wet

80

100

RH (%) Fig. 3. Eﬃciency of longitudinal and pin ﬁns as a function of relative humidity of air.

0.0 20

30

40

50

60

70

80

90

RH (%) Fig. 4. Location separating the dry and wet regions as a function of the surrounding air relative humidity and the ﬁn parameter Zd.

984

B. Kundu / Applied Thermal Engineering 27 (2007) 976–987

0.34

0.44

X0=0.309

RH=40%

RH=40%

X0=0.209

0.42 X =0.498 0 Partially wet

0.32

X0=0.136

Zd=1.5

Zd=1.5

o

η

Ta=30 C Ψ=0.05

0.38

RH=

0.36

RH=

0.30

η

0.40 70%

o

Ta=30 C Ψ=0.05

X0=0.705 RH=

70%

0.28

X0=0.518

RH=

100%

100%

0.26

Fully wet

0.34

2

4

6

8

2

10

4

6

8

10

Tb

Tb

Fig. 5. Fin eﬃciency for diﬀerent wet surface conditions as a function of ﬁn-base temperature: (a) longitudinal ﬁn; (b) pin ﬁn.

aspect ratio w. In general, the ﬁn eﬃciency decreases with the increasing value of Zd. The eﬃciency of the longitudinal ﬁn is greater than that of the pin ﬁn. The dry ﬁn is more eﬃcient than the wet ﬁn. The eﬃciency of wet ﬁns gradually decreases with an increase in relative humidity due to increment in temperature gradient. All the above trends have also been noticed in the previous ﬁgures. Another important feature to be noticed here is the partially wet region for the longitudinal ﬁn starts at a higher value of ﬁn parameter Zd than for the pin ﬁn due to lower conductive resistance. From the graph, it may be noted that with the increase in Zd, the ﬁn eﬃciency of the partially wet ﬁn decreases although the dry region increases with Zd. It may further be highlighted that the ﬁn eﬃciency for the wet ﬁn varies insigniﬁcantly with the variation of relative humidity than the variation of ﬁn parameter Zd. Optimum dimension of a ﬁn may be obtained either by maximizing the rate of heat transfer for a ﬁxed ﬁn volume or by minimizing the ﬁn volume for a given rate of heat transfer duty. Depending upon the surface condition (fully wet, partially wet or fully dry) and an objective function selected among any one from Eqs. (26)–(28), it has to be

1.0 0.9

Longitudinal Fin Spine

0.8

Fully Dry

RH=50% X0=0.382

η

0.7 0.6 RH=100%

t in ar St

0.5

Ψ=0.05

0.4

o

et

Tb=5 C

0.2

yw all rti pa

o

Ta=30 C

0.3

0.5

0.0

X0=0.257

1.0

1.5

2.0

Zd Fig. 6. Eﬃciency of wet ﬁns as a function of ﬁn parameter Zd for longitudinal and pin ﬁns.

solved simultaneously with the constraint equation (29) or (30) in which either the heat transfer rate or the ﬁn volPartially wet Surface Fully Wet

0.14

C 100%

U=0.001

0.00042 U=0.000001

o

0.12

o

Ta=30 C

Ta=30 C

80%

o

o

0.00035 Tb=5 C

Tb=5 C 60%

0.08 0.06

A Fully Dry

B

RH=40%

0.02 0.003

0.006

Bi

60%

0.00028

40%

0.00021

Partially wet Surface Fully Wet

0.04

Q

Q

0.10

0.009 0.012

C

0% 10 = RH % 80

B 0.00014

A Fully Dry Surface 0.0018 0.0027 0.0036 0.0045

Bi

Fig. 7. Heat transfer rate for the wet surface ﬁn as a function of Bi and relative humidity of air: (a) longitudinal ﬁn; (b) pin ﬁn.

B. Kundu / Applied Thermal Engineering 27 (2007) 976–987

ume can be taken as a constraint according to the requirement of a particular design. The inﬂuence of Biot number on the heat transfer rate through the longitudinal ﬁn has been depicted in Fig. 7a for the diﬀerent relative humidities and an identical ﬁn volume. A salient feature to be noticed in the ﬁgure is that with the increment of Biot number heat transfer rate increases and ultimately reaches a maximum value for a particular relative humidity. Further with increase in Biot number, a reversed trend is observed. Here A–B–C is the locus of points having highest heat transfer rate. It is necessary to note that from C–B the maximum heat transfer rate occurs when the ﬁn is in fully wet condition. But from A–B partially wet condition is of prime importance since maximum heat transfer rate occurs under the partially wet conditions. The locus of the values of Biot number for the maximum heat transfer rate of fully wet ﬁn is almost linear. Similar features are observed for the pin ﬁn as shown in Fig. 7b. Finally, design curves are drawn to obtain the optimal design variables for the longitudinal ﬁn as shown in

985

Fig. 8. This ﬁgure can be reliably used to get optimum ﬁn dimensions within the base temperature range of 2– 10 °C and ambient temperature range of 25–35 °C. The variation of the optimum aspect ratio w is depicted in Fig. 8b as a function of ﬁn volume. It is evident that for a ﬁxed value of ﬁn volume, optimum value of w increases with the increase of relative humidity. Fig. 8 may be used together as the design curve for the longitudinal ﬁn. Now for a particular case, if heat transfer rate Q and relative humidity are prior speciﬁed design constants along with the base and ambient temperature, the ﬁn volume can found from Fig. 8a. After getting the ﬁn volume at the optimum condition, the geometrical parameter w is determined, and the optimum Bi can subsequently be obtained from the expression of ﬁn volume (Eq. (24)) by substituting the known design parameters calculated from the previous steps. Thus, the design curves give all the optimum design unknowns. Therefore, with the help of design curves, tedious numerical calculations can be avoided. For the optimum pin ﬁn, a similar nature of design curves has been obtained as shown in Fig. 9.

0.20 % 00 =1 8 0% RH = RH

0.16

o

R

0%

0 10 H=

%

RH

0.0006 0.08 Fully dry surface o

0.04

=8

RH

0%

= 60

RH =

0.0004

o

Ta=25 C, Tb=2 C o

o

Ta=35 C, Tb=10 C

0.0008

60 %

RH=4

0.12

o

Qopt

Qopt

RH=

o

Ta=25 C, Tb=2 C;

%

40%

o

Ta=35 C, Tb=10 C 0.00 0.000

Fully dry surface 0.001

0.002

0.003

0.0002 0.000001

0.004

U o

o

o

0.25

0.30

0.20

0.25 RH=40% RH=60% RH=80%

0.10

o

o

o

Ta=35 C, Tb=10 C

RH=60% RH=40%

0.20 RH=80% 0.15 RH=100%

RH=100% 0.05

Fully dry surface

Fully dry surface

0.10 0.00 0.000

0.000004

0.35

Ψopt

Ψopt

o

Ta=25 C, Tb=2 C;

o

Ta=35 C, Tb=10 C

0.15

0.000003

U

Ta=25 C, Tb=2 C 0.30

0.000002

0.001

0.002

0.003

0.004

U Fig. 8. Optimum design variables for the longitudinal ﬁn of wet surface conditions as a function of ﬁn volume: (a) optimum heat transfer rate; (b) optimum aspect ratio.

0.000001

0.000002

0.000003

0.000004

U Fig. 9. Optimum design variables for pin ﬁns of wet surface conditions as a function of ﬁn volume: (a) optimum heat transfer rate; (b) optimum aspect ratio.

986

B. Kundu / Applied Thermal Engineering 27 (2007) 976–987

4. Conclusions In the present study, thermal performance and optimization of longitudinal and pin ﬁns with uniform thickness under fully as well as partially wet surface conditions have been investigated analytically. Also, a comparative study on the ﬁn performance between the longitudinal and pin ﬁns has been demonstrated. The optimization analysis has been presented in a generalized form such that either heat transfer duty or ﬁn volume can be treated as a constraint. The optimality criteria have been derived using the Lagrange multiplier technique. In addition, the method for optimization of partially wet surface ﬁns has also been established. Design curves are plotted for a wide range of thermo-psychometric parameters. The following conclusions may be drawn from the results obtained by using the present analysis: 1. Whether the ﬁn surface is dry, partially or fully wet, the main deciding factors are the ﬁn parameter Zd and the psychometric properties of the surrounding air like dry bulb temperature, relative humidity, etc. 2. The overall ﬁn eﬃciency of longitudinal and pin ﬁns for partially wet surface conditions is inﬂuenced strongly by the relative humidity. But this eﬀect is less sensitive for the pin ﬁn. However, for the fully wet surface of all types of ﬁns, ﬁn eﬃciency does not have a noteworthy change with the relative humidity. 3. Among the various surface conditions, ﬁn eﬃciency for the dry surface becomes a maximum and its value for the wet surface gradually decreases with the increase in relative humidity because of the greater variation of temperature in the ﬁn. 4. Fin eﬃciency for the wet surface ﬁn decreases with both the increase of dry bulb and ﬁn-base temperature. But for a partially wet surface with the relative humidity of 40%, ﬁn eﬃciency increases with the base temperature. 5. The range of relative humidity for maintaining the partially wet surface of pin ﬁns is always greater than for the longitudinal ﬁn. 6. The ﬁn parameter Zd is one of the main factors to determine the wet surface condition of any ﬁn. For maintaining a partially wet surface of a pin ﬁn, Zd value is always lesser in comparison with that of a longitudinal ﬁn. 7. The ﬁn eﬃciency of pin ﬁns shows a low value in comparison with that of longitudinal ﬁn irrespective of surface conditions. 8. Whether the ﬁn surface is either partially or fully wet at the optimum design condition, the main responsible factor is air relative humidity. A very small range of relative humidity is found for the optimum ﬁn under partially wet surface conditions. 9. For diﬀerent constraint ﬁn volumes, the locus of optimum Bi calculated on the basis of maximum heat

transfer rate in partially wet ﬁn is found to be less steep than for the fully wet ﬁn. 10. With the help of the design charts, a designer can easily estimate the unknown design variables for the optimum design of ﬁns under fully dry, partially wet and fully wet conditions without prior knowledge of its mathematical analysis. Acknowledgements This research is supported by the Jadavpur University (Research Project Grant P-1/1057/05 under Seed Support of Potential for Excellence Scheme) whose assistant is hereby gratefully acknowledged. References [1] E. Schmidt, Die Wa´rmeu´bertragung dutch Rippen, Zeitschrift des Vereines Deutscher Ingenieure 70 (1926) 885–889. [2] R.J. Duﬃn, A variational problem relating to cooling ﬁns, J. Math. Mech. 8 (1959) 47–56. [3] D.Q. Kern, A.D. Kraus, Extended Surface Heat Transfer, McGraw Hill, New York, 1972. [4] G. Fabbri, A genetic algorithm for ﬁn proﬁle optimization, Int. J. Heat Mass Transfer 40 (1997) 2165–2172. [5] J.L. Threlkeld, Thermal Environment Engineering, Prentice-Hall, New York, 1970. [6] F.C. McQuiston, Fin eﬃciency with combined heat and mass transfer, ASHRAE Trans. 71 (1975) 350–355. [7] A. Kilic, K. Onat, The optimum shape for convecting rectangular ﬁns when condensation occurs, Warme Stoﬀubertragung 15 (1981) 125– 133. [8] M. Toner, A. Kilic, K. Onat, Comparison of rectangular and triangular ﬁns when condensation occurs, Warme Stoﬀubertragung 17 (1983) 65–72. [9] G. Wu, T.Y. Bong, Overall eﬃciency of a straight ﬁn with combined heat and mass transfer, ASHRAE Trans.: Res. (1994) 367–374. [10] H. Kazeminejad, M.A. Yaghoubi, F. Bahri, Conjugate forced convection–conduction analysis of the performance of a cooling and dehumidifying vertical rectangular ﬁn, Int. J. Heat Mass Transfer 36 (1993) 3625–3631. [11] K.T. Hong, R.L. Webb, Calculation of ﬁn eﬃciency for wet and dry ﬁns, HVAC&R Res. 2 (1) (1996) 27–40. [12] J.E.R. Coney, C.G.W. Sheppard, E.A.M. Elshafei, Fin performance with condensation from humid air: a numerical investigation, Int. J. Heat Fluid Flow 10 (3) (1989) 224–231. [13] V. Srinivasan, R.K. Shah, Fin eﬃciency of extended surface in twophase ﬂow, Int. J. Heat Fluid Flow 18 (4) (1997) 419–429. [14] P.K. Sarma, S.P. Chary, V. Dharma Rao, Condensation on a vertical plate ﬁn of variable thickness, Int. J. Heat Mass Transfer 31 (1988) 1941–1944. [15] B. Kundu, Analytical study of the eﬀect of dehumidiﬁcation of air on the performance and optimization of straight tapered ﬁns, Int. Commun. Heat Mass Transfer 29 (2002) 269–278. [16] C. Lin, J. Jang, A two-dimensional ﬁn eﬃciency analysis of combined heat and mass transfer in elliptic ﬁns, Int. J. Heat Mass Transfer 45 (2002) 3839–3847. [17] H. Kazeminejad, Analysis of one-dimensional ﬁn assembly heat transfer with dehumidiﬁcation, Int. J. Heat Mass Transfer 38 (3) (1995) 455–462. [18] S.Y. Liang, T.N. Wong, J.K. Nathan, Comparison of onedimensional and two-dimensional models for wet surface ﬁn eﬃciency of a plat-ﬁn-tube heat exchanger, Appl. Therm. Eng. 20 (2000) 941–962.

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