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

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

Applied Thermal Engineering 27 (2007) 976–987 www.elsevier.com/locate/apthermeng Performance and optimum design analysis of longitudinal and pin fins ...

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Applied Thermal Engineering 27 (2007) 976–987 www.elsevier.com/locate/apthermeng

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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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.

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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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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.

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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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