A new performance evaluation method and its application in fin-tube surface design of small diameter tube

Abstract

In this paper, a simple yet efficient performance comparison method is proposed based on the assumptions of constant properties and identical frontal area. For this method, no correlations are required, and a small number of discrete data are sufficient. To illustrate the feasibility of the proposed approach, a new slotted fin with 4 mm tubes is designed to replace the original louvered fin with tubes of 7 mm. The orthogonal design method is adopted in the fin design to reduce the number of computational cases significantly, and yet a nearly optimum combination of major geometric factors can still be obtained. The reasonable parametric combination of 3 global parameters is obtained by analyzing the numerical results of 16 plain plate fins. Based on this result, 3 new slotted fins with different fin pitches are studied. The slotted fin with a fin pitch of 1.4 mm is recommended after considering the heat transfer, comprehensive performance, and cost of material and operation. The result shows that compared with the original louvered fin, the recommended fin not only increases the heat transfer rate by 2.2%, 22.5%, and 13.7% under an identical flow rate, identical pressure drop, and identical pumping power constraint, respectively, but also saves approximately 36% of the copper tube materials.

Keywords

performance evaluation / orthogonal design / small-diameter tube

Introduction

Heat exchangers are widely used in various engineering fields. To improve the overall performance of a heat exchanger effectively, including the reduction of its size and saving of energy for its operation [1], many augmentation techniques have been reported and extensive data on the basic heat transfer and friction characteristics have been published [2-5]. An effective performance comparison method of these enhanced techniques for a specified purpose at some given operating conditions is very useful for users to select a required technique appropriately.

Different performance comparison methods might lead to different criteria for designers and users to select a preferred type or the most suitable one. Webb and Eckert [6] first developed equations to compare the performance between the roughened tubes and smooth tubes. Thereafter, this issue have become a matter of concern for more and more researchers, and several evaluation methods have been proposed. Generally, these assessment methods can be classified into following categories: the evaluation criteria based on the first law of thermodynamics, such as the 12 combinations for the single-phase flow heat transfer [1,7], the criteria based on identical pumping power/identical pressure drop [8-13], the goodness factor [14,15] used in plate-fin-tube surfaces, and the criterion based on correlating the heat transfer coefficient and the dissipation energy in fluid [16]; and the assessment methods based on the second law of thermodynamics by determining the entropy generation or exergy generation in the enhancement process [17-24]. In 2007 a new concept called entransy was introduced by Guo et al. [25] for comparison purposes. Among many comparison methods mentioned above, the criteria based on identical pumping power/identical pressure drop are simpler and more straightforward.

With the emergence of worldwide energy shortage, researchers of enhanced heat transfer techniques now give more attention to energy-saving purposes. Thus, the performance comparison between enhanced and reference surfaces based on identical pressure drop or identical pumping power seem more attractive, apart from being simple. To visualize the performance comparisons, different kinds of plots have been suggested [26], among which none could be used to indicate the energy-saving effect. Therefore, Fan et al. [27] presented a performance evaluation plot for the energy-saving effect based on the method proposed by Webb and Bergles [1, 7]. In this plot, users can compare the performances between the enhanced and reference surfaces and also select the best operating condition for a given surface intuitively and quickly. This evaluation method is based on the important assumptions that the heat transfer and friction factor correlations for the reference surface being studied are available and the characteristic dimension used for calculating the dimensionless characteristic number of enhanced surface is the same as that of the referenced one. In developing a newly enhanced structure, designers are always faced with the problem of how to evaluate the performance of an in-designing temporary structure. For a temporary in-designing structure, its geometric size and shape are subject to change; hence, meeting the above assumptions can be very difficult. As a complement to this method, a performance evaluation approach for in-designing structures is highly needed. In this paper, a simple yet efficient performance comparison method is proposed where no correlations are required, and only several discrete data are sufficient.

In the design of a newly enhanced structure, the designers are often puzzled by the problem of how to compare the performance among different design schemes to obtain an optimum or near optimum combination for a selected structure. A direct but very time-consuming way is to list the cases of all possible combinations of the major influencing factors. Performance prediction is then conducted for every case to find the best one. In this paper, the orthogonal design method [28-32] proposed by Taguchi is adopted to reduce the number of computational cases significantly but still be able to obtain a nearly optimum combination of major geometric factors.

First, the equations for constructing a plot for the comparison of in-designing surfaces are derived. The Taguchi method is then implemented to find a nearly optimum structure for a plain plate fin-tube surface. Based on the result, a new slotted fin with tubes of 4 mm is proposed to replace the original louvered fin with tubes of 7 mm at an inlet velocity range of 0.5-3.0 m/s.

Basic equations for the performance evaluation method

Based on our own research experiences, both the fluid temperature difference between the inlet and the outlet, ΔT, and the pressure drop of the fluid, Δp, are obtained the easiest in numerical simulations or experimental measurements. The heat transfer medium is specified for a given study. Hence, the temperature difference and pressure drop correspond to a certain amount of heat transfer rate and pumping power at a given fluid velocity, respectively. Thus, the temperature difference and the pressure drop can be selected as representatives of the heat transfer and flow characteristics at a given velocity to compare the performance among different configurations.

To facilitate the comparison process, the performance evaluation is based on the following assumptions: the thermophysical properties of fluid are constant [1]; the air flow is three-dimensional, incompressible, laminar, and steady; the tube wall temperature is constant; and the total frontal area for all configurations is the same.

Based on the definition of the heat transfer rate and pumping power, the ratios of the heat transfer rate and the pumping power between two configurations can be expressed as

(1){\displaystyle \frac{{\mathrm{\Phi }}_{\mathrm{2}}}{{\mathrm{\Phi }}_{\mathrm{1}}}}={\displaystyle \frac{{(\rho ·V·A_{\mathrm{c}}·c_p·\mathrm{\Delta }T)}_{\mathrm{2}}}{{(\rho ·V·A_{\mathrm{c}}·c_p·\mathrm{\Delta }T)}_{\mathrm{1}}}},

(2){\displaystyle \frac{P_{\mathrm{2}}}{P_{\mathrm{1}}}}={\displaystyle \frac{{(A_{\mathrm{c}}·V·\mathrm{\Delta }p)}_{\mathrm{2}}}{{(A_{\mathrm{c}}·V·\mathrm{\Delta }p)}_{\mathrm{1}}}}.

The following discussion is presented for the three comparison indices: identical flow rate, identical pressure drop, and identical pumping power.

For identical flow rate, the equations are as follows:

(3){\displaystyle \frac{{\mathrm{\Phi }}_{\mathrm{2}}}{{\mathrm{\Phi }}_{\mathrm{1}}}}={\displaystyle \frac{\mathrm{\Delta }{T}_{\mathrm{2}}}{\mathrm{\Delta }{T}_{\mathrm{1}}}},

(4){\displaystyle \frac{P_{\mathrm{2}}}{P_{\mathrm{1}}}}={\displaystyle \frac{\mathrm{\Delta }{p}_{\mathrm{2}}}{\mathrm{\Delta }{p}_{\mathrm{1}}}},

(5){\displaystyle \frac{{\mathrm{\Phi }}_{\mathrm{2}}/P_{\mathrm{2}}}{{\mathrm{\Phi }}_{\mathrm{1}}/P_{\mathrm{1}}}}={\displaystyle \frac{{(\mathrm{\Delta }T/\mathrm{\Delta }p)}_{\mathrm{2}}}{{(\mathrm{\Delta }T/\mathrm{\Delta }p)}_{\mathrm{1}}}}.

From Eqs. (3-5), the heat transfer rate, pumping power, and comprehensive performance are proportional to the temperature difference of the two heat transfer medium, pressure drop, and ratio of temperature difference to pressure drop, respectively. Thus, there are two possible ways to construct the performance evaluation plot for identical flow rate. One is assigning Δp₂/Δp₁ and ΔT₂/ΔT₁ as the abscissa and the ordinate, respectively, and the other is assigning Δp and ΔT for the same positions. These two expressions are essentially the same. In such plot, the abscissa represents the flow, whereas the ordinate represents the heat transfer characteristics, and the slope represents the comprehensive performance. The former is more intuitive when the comparison is made at a given inlet velocity, and the latter is more convenient when given an inlet velocity range.

For identical pressure drop, the equations are:

(6){\displaystyle \frac{{\mathrm{\Phi }}_{\mathrm{2}}}{{\mathrm{\Phi }}_{\mathrm{1}}}}={\displaystyle \frac{{(V·\mathrm{\Delta }T)}_{\mathrm{2}}}{{(V·\mathrm{\Delta }T)}_{\mathrm{1}}}},

(7)\mathrm{\Delta }{p}_{\mathrm{1}}(V_{\mathrm{1}})=\mathrm{\Delta }{p}_{\mathrm{2}}(V_{\mathrm{2}}).

The evaluation plot for identical pressure drop can be constructed by taking Δp and V·ΔT as the abscissa and the ordinate, respectively. Certainly, V·ΔT and Δp can also be taken as the abscissa and V as the ordinate to obtain the inlet velocity under the identical pressure drop constraint. Of the two, the former is simpler, but the latter is more intuitive and presents more information, including the velocity and the heat transfer rate of the compared structure corresponding to the reference under identical pressure drop.

Similarly, for identical pumping power, the equations are

(8){\displaystyle \frac{{\mathrm{\Phi }}_{\mathrm{2}}}{{\mathrm{\Phi }}_{\mathrm{1}}}}={\displaystyle \frac{{(V·\mathrm{\Delta }T)}_{\mathrm{2}}}{{(V·\mathrm{\Delta }T)}_{\mathrm{1}}}},

(9){\displaystyle \frac{P_{\mathrm{2}}}{P_{\mathrm{1}}}}={\displaystyle \frac{{(V·\mathrm{\Delta }p)}_{\mathrm{2}}}{{(V·\mathrm{\Delta }p)}_{\mathrm{1}}}}=\mathrm{1.}

The evaluation plot for identical pumping power can be constructed by taking V·Δp and V·ΔT as the abscissa and the ordinate, respectively, or taking V·ΔT and V·Δp as the abscissa and the inlet velocity V as the ordinate.

Application to fin design

A new slotted fin with tubes of 4 mm is designed using the proposed method. The original fin-and-tube heat exchanger is a condenser of an air-conditioning system that uses a copper tube 7 mm in diameter with a louvered fin surface. To investigate the applicability of the tube with a smaller diameter while still satisfying the original heat transfer and pressure drop requirements, the air-side heat transfer and pressure drop, tube-side heat transfer and pressure drop, and circuit arrangement should be simultaneously studied.

Physical and mathematical models

Figure 1 is a pictorial view of a fin-and-tube heat exchanger with two-row tubes arranged in staggered manner. All kinds of fin-tube heat exchanger are modeled in this figure. Apart from the streamwise tube number, the only difference is in the details of the fin surface structure. Due to the periodic character of the tube arrangement in a span-wise direction and of the fin arrangement in an axial direction, two representative units for the selection of modeling are presented in Fig. 1: practice A and practice B [33]. To ensure the reasonability of the uniform inlet and the fully developed outlet boundary conditions, one time and six times of the fin length are extended in both upstream (pre-extended) and downstream (after-extended) parts of the computational domain, respectively. By neglecting the details of the fin surface, the major features of the computational domain are represented in Fig. 2 [33]. To assure the accuracy of the numerical results, fin thickness and thermal conduction are considered; the fin collar outside diameter is adopted as the tube outside diameter in the numerical simulation.

As the comparison reference, the performance of the original louvered fin with 7 mm tubes is also simulated. Figure 3 shows the schematic diagram of the original (reference) fin. Table 1 lists its geometrical parameters.

According to [32], for the fin-and-tube heat transfer surface, the most important geometric parameters are the fin pitch, transverse tube spacing, and longitudinal tube spacing, apart from the structure of the fin surface. The global parametric sensitivity of these three factors are first analyzed for the plain plate fin-and-tube surface using the Taguchi method, and four levels of every factor are selected based on orthogonal array L₁₆(4³) [34], where L is the code for orthogonal array, 16 is the experiment number completed in orthogonal design, 4 is the number of levels for every factor, and 3 is the number of factors arranged in orthogonal array. Table 2 presents the factors and levels used in this study. In total, 16 kinds of plain plate fin models are made by compounding the levels on each factor (Table 3).

Governing equations and boundary conditions

The governing equations can be expressed as follows:

The continuity equation is

(10){\displaystyle \frac{\partial }{\partial x_i}}(\rho u_i)=\mathrm{0.}

The momentum equation is

(11){\displaystyle \frac{\partial }{\partial x_i}}(\rho u_i{u}_k)={\displaystyle \frac{\partial }{\partial x_i}}\left(\eta {\displaystyle \frac{\partial u_k}{\partial x_i}}\right)-{\displaystyle \frac{\partial p}{\partial x_k}}.

The energy equation is

(12){\displaystyle \frac{\partial }{\partial x_i}}(\rho u_{i}T)={\displaystyle \frac{\partial }{\partial x_i}}\left({\displaystyle \frac{\lambda }{c_p}}{\displaystyle \frac{\partial T}{\partial x_i}}\right).

The fin surfaces are regarded as a part of the solution domain; hence, no boundary conditions are required. All the boundary conditions of the computation domain are as follows:

In the x-coordinate direction:

At the inlet:

(13)u=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t},v=w=\mathrm{0},T_{\mathrm{i}\mathrm{n}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.

At the outlet:

(14){\displaystyle \frac{\partial u}{\partial x}}={\displaystyle \frac{\partial v}{\partial x}}={\displaystyle \frac{\partial w}{\partial x}}={\displaystyle \frac{\partial T}{\partial x}}=\mathrm{0.}

In the y coordinate direction:

The fin surface and fluid region is

(15){\displaystyle \frac{\partial u}{\partial y}}={\displaystyle \frac{\partial w}{\partial y}}=\mathrm{0},v=\mathrm{0},{\displaystyle \frac{\partial T}{\partial y}}=\mathrm{0.}

The tube region is

(16)u=v=w=\mathrm{0},T_w=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.

In the z coordinate direction:

In the pre-extended region: symmetry condition

In the fin coil region and after-extended region: symmetry condition for the plain plate fin model and periodic condition for louvered fin model

Numerical methods

The grid systems are generated by the commercial software GAMBIT. The discretized equations are solved by the full-field computational method, where the solid part of the fin is regarded as a special fluid with very large viscosity. The second-order upwind and central difference are used to discretize the convective terms and diffusive terms, respectively. The coupling between velocities and pressure is treated with the SIMPLEC algorithm. The problem is solved by FLUENT, and parallel computing is adopted to shorten the computing time. In the preliminary study, the influence of grid density on the computational results is investigated, and all numerical results presented below can be regarded as grid-independent. The convergence criteria are as follows: the reduction of the residuals is below the order of 10^-4-10^-5 for the continuity equation, 10^-6-10^-7 for the momentum equations, and 10^-7-10^-8 for the energy equation.

In the numerical simulation, the calculation parameters are as follows: the inlet velocity of air is 0.25-3.0 m/s, the inlet temperature of air is 308 K, and the wall temperature of the tubes is 318 K.

Sensitivity analysis of the three global parameters

The numerical data are first analyzed to evaluate the effect of the three parameters through the proposed optimization criterion. The design aim is to obtain the maximum heat transfer rate (ΔT) and the best comprehensive performance (ΔT/Δp). Table 3 presents the numerical results of the 16 plain fins at an inlet velocity of 2.0 m/s. Based on the table, Fin 9 has the highest temperature difference, and Fin 16 has the maximum ratio of temperature difference to pressure drop. That is, Fin 9 has the largest heat transfer capability, and Fin 16 possesses the best comprehensive performance at an inlet velocity of 2.0 m/s.

Figure 4 presents the heat transfer characteristics of the three factors at an inlet velocity of 2.0 m/s. The steeper slope of the response graph indicates a stronger influence of the factor on the target characteristics, and the largest temperature difference of all levels of each factor has the best heat transfer performance. Therefore, the fin pitch has a great influence, the longitudinal tube spacing has a relatively large effect, and the transverse tube spacing has the smallest effect. The heat transfer rate (ΔT) increases with the decrease in fin pitch and the transverse tube spacing or the increase in longitudinal tube spacing. The nearly optimum parameters can be obtained by a combination of levels showing the largest heat transfer rate (ΔT) in each control factor, which is A₁B₁C₃ (Table 2). That is, the approximate optimum values for the three parameters are 17 mm for the transverse tube spacing, 12.4 mm for the longitudinal tube spacing, and 1.2 mm for the fin pitch. These results are taken as the bases for the design of a new slotted fin.

We now turn to the performance comparison between the two plain plate fins with tubes of 4 mm (Fins 9 and 16) and the original louvered fin with tubes of 7 mm under an identical flow rate inlet velocity of 2.0 m/s. Figure 5 illustrates the comparison results. At this inlet velocity, the original louver fin has the highest heat transfer capability with the associated highest pressure drop, followed by Fin 9; Fin 16 has the lowest. However, the comprehensive performances of the three fins, i.e., \mathrm{\Delta }T/\mathrm{\Delta }p, are the opposite. In other words, Fin 9 is better than Fin 16, which is superior to the original louvered fin under the given conditions.

Figure 6 demonstrates the performance assessment of the three fins at an inlet velocity range of 0.25-3.0 m/s. In the figure, 10 points for each fin are presented, corresponding to the inlet velocity of 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0, 2.5, and 3.0 m/s from left to right. The value of ΔT_max is obtained by adopting the tube wall temperature and subtracting the inlet one of air, which corresponds to the maximum theoretical heat transfer capability. The difference between ΔT_max and the ordinate value of the point represent the ultimate capacity of potential heat transfer augmentation. The greater the difference is, the greater the extent of the potential heat transfer enhancement. In Fig. 6, the heat transfer rates of the three fins are nearly the same when the maximum theoretical heat transfer rate is at an inlet velocity of 0.25 m/s. That is, at this inlet velocity, the adoption of an enhanced structure can only increase the pressure drop but does not contribute to increasing the heat transfer capacity. For the potential extent of enhanced heat transfer, the sequence of the enhanced extent is Fin 16, Fin 9, and the original fin at the same inlet velocity; the extent increases with the increase in inlet velocity for a given fin structure. Moreover, at inlet velocities of 0.25-0.75 m/s, the heat transfer rate of Fin 9 is close to that of the original fin, whereas its pressure drop is obviously less than that of the original one. Thus, the plain plate fin (Fin 9) is selected for substituting the original louvered fin when the inlet velocity is less than 1.0 m/s. However, the heat transfer rate of the plate fins is much lower than that of the original louvered fin when the inlet velocity is larger than 1.0 m/s. Therefore, in the higher inlet velocity range, the heat transfer rate is selected as the design target to ensure that the heat transfer capability of the new slotted fin can meet the requirement of the original thermal load. The following results are presented for the inlet velocity greater than 1.0 m/s.

Design of a new slotted fin with 4 mm tubes

To determine reasonable global parameters, aside from heat transfer and pressure drop, other factors such as fin material strength and manufacturing convenience should also be considered. The following global parameters are selected for the new slotted fin design: the transverse tube spacing is 17 mm and the longitudinal tube spacing is adjusted to 13 mm. The abovementioned studies on the three global parameters on the heat transfer and pressure drop effects are conducted for the plain plate fin surface. Considering that for the slotted or louvered fin surface, the existence of slots may require a higher fin pitch; hence, fin pitches of 1.2, 1.3, and 1.4 mm are adopted. Figure 7 shows the details of the new slotted fin configuration, where f_p represents the fin pitch and the slit height, and L_s is half of the fin pitch f_p. The details of the slots geometry can be found in our previous studies [12, 14, 26, 32].

Results and discussion

Comparisons are made to show the advantages of this new design. First, the performance comparisons between the new slotted fins and the original louvered fin are made under the identical flow rate constraint. Figures 8 and 9 present the heat transfer and pressure drop characteristics of the new and the original fins, respectively. All of the three slotted fins with 4 mm tubes meet the requirement of the original louvered fin with 7 mm tubes under the same frontal area, and the heat transfer rate (∆T) and the pressure drop of the slotted fin increase with the reduction in the fin pitch. Figure 10 shows the comprehensive performance of the new slotted fins. The slotted fins with fin pitches of 1.4 and 1.3 mm give better comprehensive performances than the original fin, and the comprehensive performance of the slotted fins is improved with the increase in fin pitch. Thus, the slotted fin with a fin pitch of 1.4 mm is recommended to replace the original louvered fin. The heat transfer rate of the recommended fin (8.961 K) is 1.022 times higher than that of the original louvered fin (8.766 K) at a velocity of 2.0 m/s, whereas the pressure drop of the new fin (28.86 Pa) is only 67.77% that of the original fin (42.58 Pa).

Figure 11 shows the comparison of heat transfer capability (v · ∆T) between the recommended fin and the original fin under an identical pressure drop. Taking the louvered fin (referenced fin) at an inlet velocity of 2 m/s as an example, Points 0 and 1 are the crossing points of the line of v = 2.0 m/s and the heat transfer and pressure drop characteristic curves, respectively. The left ordinate value of Point 0 presents the heat transfer rate (17.53 K·m/s), and the right ordinate value of Point 1 shows the pressure drops of the original louvered fin. The horizontal line L_a is drawn through Point 1, and crosses the curve of the slotted fin pressure drop at Point 2. The abscissa value of Point 2 presents the velocity of the recommended fin (2.67 m/s) at which its pressure drop is equal to that of the louvered fin. The crossing point (Point 3) of the vertical line L_b and the heat transfer characteristic line for the recommended slotted fin can be obtained. The ordinate value of Point 3 presents the heat transfer rate of the recommended slotted fin (21.48 K·m/s), which possesses the same pressure drop as the reference fin at a velocity of 2 m/s. Thus, the ratios of the heat transfer rate between the recommended fin (21.48 K·m/s) and the original fin (17.53 K·m/s) is 1.225 under the identical pressure drop constraint. This means that compared with the original louvered fin, the heat transfer rate of the recommended slotted fin is increased by 22.5%.

Figure 12 shows the comparison under the identical pumping power constraint. At a velocity of 2.36 m/s, the new fin consumes the same pumping power as that of the original louvered fin at a velocity of 2.0 m/s, and the ratio of the heat transfer rate between the slotted fin (19.94 K·m/s) and the original louvered fin (17.53 K·m/s) is 1.137. Thus, relative to the original louvered fin, the heat transfer rate of the recommended slotted fin increases by 13.7% under the identical pumping power constraint.

The volume and material of the aluminum fin of the new heat exchanger with 4 mm tubes are increased by 2.4%, and the material of the copper tube is decreased to 64% under the condition of the same tube thickness. Therefore, the new fin may be a promising fin substitute for the original fin with 7 mm tubes.

Conclusion

In this paper, a simple yet efficient performance comparison method is proposed based on the assumptions of constant properties and identical frontal areas. To illustrate the feasibility of the proposed approach, a new slotted fin with 4 mm tubes is designed using the orthogonal design method. The conclusions are as follows:

1) The nearly optimum combination of global parameters is obtained based on the analysis of numerical results of 16 plain plate fins using the orthogonal design method.

2) Three slotted fin with 4 mm tubes are designed based on the reasonable combination of global parameters. The slotted fin with a fin pitch of 1.4 mm is recommended based on the overall consideration of heat transfer capability, comprehensive performance, and cost of material and operation.

3) Compared with that of the original louvered fin, the heat transfer rates of the recommended fin increase by 2.2%, 22.5% and 13.7% under identical flow rate, identical pressure drop, and identical pumping power constraint, respectively; the copper tube materials saved are approximately 36%. Therefore, the new fin may be a promising fin substitute for the original fin with 7 mm tubes.

References

[1] Webb R L, Kim N H. Principles of Enhanced Heat Transfer. 2nd ed. Boca Raton: Taylor &Francis, 2005

[2] Bergles A E. Heat transfer enhancement-The maturing of second-generation heat transfer technology. Heat Transfer Engineering, 1997, 18(1): 47-55

[3] Wang C C. Technology review-A survey of recent patents of fin-and-tube heat exchangers. Enhanced Heat Transfer, 2000, 7(5): 333-345

[4] Shah R K, Pekulic D P. Fundamentals of Heat Exchanger Design. Hobokin, New Jersey: John Wiley & Sons, 2003, 696

[5] Wang L B, Tao W Q. Heat transfer and fluid flow characteristics of plate array aligned at angles to the flow direction. International Journal of Heat and Mass Transfer, 1995, 38(16): 3053-3063

[6] Webb R L, Eckert E R. Application of rough surfaces to heat exchanger design. International Journal of Heat and Mass transfer, 1972, 15(9): 1647-1658

[7] Bergles A E. Techniques to enhance heat transfer. In: Rohsenow W M, Hartnett J P, Cho Y L, eds. Handbook of Heat Transfer. 3rd ed. New York: McGraw-Hill, 1998, Chapter 11

[8] Sparrow E M, Tao W Q. Symmetric vs asymmetric periodic disturbances at the walls of a heated flow passage. International Journal of Heat and Mass transfer, 1984, 27(11): 2133-2144

[9] Huang H Z, Tao W Q. An experimental study on heat/mass transfer and pressure drop characteristics for arrays of nonuniform plate length positioned obliquely to the flow direction. ASME Journal of Heat Transfer, 1993, 115(3): 568-575

[10] Lue S S, Huang H Z, Tao W Q. Experimental study on heat transfer and pressure drop characteristics in the developing region for arrays of obliquely positioned plates of nonuniform length. Experimental Thermal and Fluid Science, 1993, 7(1): 30-38

[11] Yuan Z X, Tao W Q, Wang Q W. Numerical prediction for laminar forced convection heat transfer in parallel plate channels with streamwise-periodic rod disturbances. International Journal for Numerical Methods in Fluids, 1998, 28(9): 1371-1387

[12] Yu B, Nie J H, Wang Q W, Tao W Q. Experimental study on the pressure drop and heat transfer characteristics of tubes with internal wave-like longitudinal fins. Heat and Mass Transfer, 1999, 35(1): 65-73

[13] Cheng Y P, Qu Z G, Tao W Q, He Y L. Numerical design of efficient slotted fin surface based on the field synergy principle. Numerical Heat Transfer, Part A, 2004, 45(6): 517-538

[14] Shah R K, Afimiwala K A, Mayne R W. Heat exchanger optimization. In: 6th International Heat Transfer Conference. Washington, DC: Hemisphere Publishing Corp, 1978, 4: 185-191

[15] Qu Z G, Tao W Q, He Y L. Three dimensional numerical simulation on laminar heat transfer and fluid flow characteristics of strip fin surface with X-array arrangement of strips. ASME Journal of Heat Transfer, 2004, 126(5): 697-707

[16] Sano Y, Usui H. Evaluation of heat transfer promoters by the fluid dissipation energy. Texas: Scripta Publishing Co., 1982, 91-96

[17] Bejan A. General criterion for rating heat exchanger performance. International Journal of Heat and Mass transfer, 1978, 21(5): 655-658

[18] Bejan A. Second law analysis in heat transfer. Energy, 1980, 5(8,9): 721-732

[19] William R O, Bejan A. Conservation of available work (exergy) by using promoters of swirl flow in forced convection heat transfer. Energy, 1980, 5(7): 587-596

[20] Chen B H, Huang W H, Second-law analysis for heat transfer enhancement on a rib-type turbulence promoter. Energy, 1988, 13(2): 167-175

[21] Chen B H, Huang W H. Performance evaluation criteria for enhanced heat transfer surface. Int Comm Heat Mass transfer, 1988, 15(1): 59-72

[22] Zimparov V D, Vulchanov N L. Performance evaluation criteria for enhanced heat transfer surfaces. International Communications in Heat and Mass Transfer, 1994, 37(12): 1807-1816

[23] Prasad R C, Shen J H. Performance evaluation of convective heat transfer enhancement devices using exergy analysis. International Journal of Heat and Mass Transfer, 1993, 36(17): 4193-4197

[24] Prasad R C, Shen J H, Performance evaluation using exergy analysis-application to wire-coil inserts in forced convection heat transfer. International Journal of Heat and Mass Transfer, 1994, 37(15): 2297-2303

[25] Guo Z Y, Zhu H Y, Liang X G. Entransy--A physical quantity describing heat transfer ability. International Journal of Heat and Mass Transfer, 2007, 50(13,14): 2545-2556

[26] Manglik R M. Heat Transfer Enhancement. In: Bejan A, Kraus A, eds. Heat Transfer Handbook. Hobokin, New Jersey: John Wiley & Son, 2003

[27] Fan J F, Ding W K, Zhang J F, He Y L, Tao W Q. A performance evaluation plot of enhanced heat transfer techniques oriented for energy-saving. International Journal of Heat and Mass Transfer, 2009, 52(1, 2): 33-44

[28] Yun J Y, Lee K S. Influence of design parameters on the heat transfer and flow friction characteristics of the heat exchanger with slit fins. International Journal of Heat and Mass Transfer, 2000, 43(14): 2529-2539

[29] Bilen K, Yapici S, Celik C. A Taguchi approach for investigation of heat transfer from a surface equipped with rectangular blocks. Energy Conversion and Management, 2001, 42(8): 951-961

[30] Yakut K, Alemdaroglu N, Sahin B, Celik C. Optimum design-parameters of a heat exchanger having exchanger having hexagonal fins. Applied Energy, 2006, 83(2): 82-98

[31] Sanders P A, Thole K A. Effects of winglets to augment tube wall heat transfer in louvered fin heat exchangers. International Journal of Heat and Mass Transfer, 2006, 49(21,22): 4058-4069

[32] Fan J F, He Y L, Tao W Q. The flow and heat transfer characteristics of an air side louvered fin-and-tube heat transfer surface with tubes of 4 mm. In: 2nd Asian Symposium on Computational Heat Transfer and Fluid Flow, Jeju, Korea, 2009

[33] Tao W Q, Cheng Y P, Lee T S. The Influence of strip location on the pressure drop and heat transfer performance of a slotted fin. Numerical Heat Transfer, Part A: Applications, 2007, 52(5): 463-480

[34] Li Y Y, Xu C R. Experimental Design and Data Processing. Beijing: Chemical Industry Press, 2005

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg