Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division July 3, 2001; revision received June 4, 2003. Associate Editor: S. S. Sadhal. The coefficients most commonly used in the study and design of fins are fin efficiency (Harper and Brown 1) and fin effectiveness (Gardner 2). In their definitions, the so-called Murray-Gardner’s hypotheses or limiting assumptions are assumed 3; if these cannot be stated, the assessment of fin system performance is not correct or simply impossible. This occurs when fin-wall systems or unsteady processes are considered; in the last case, the dissipated heat does not instantaneously coincide with the heat rate through the fin base and when the efficiency is calculated using this quantity values greater than unity are obtained in the first few instants of the transient period 4. The limitations of these parameters to properly reflect the real behavior of finned systems and/or to face real world configurations have led to the definition of new coefficients. Among these are input admittance (Krause et al. 5), the augmentation factor (Heggs and Stone 6) and the enhancement factor, (Manzoor et al. 7). The last two parameters refer to fin-wall systems, and consider the fin and the wall to which it is attached as an indissoluble whole. The suitability of the different coefficients has been discussed for the steady state by Kraus 8 and Wood et al. 9. Kraus argues that the input admittance is better than efficiency and effectiveness, while Wood et al. conclude that the augmentation factor, AUG, is the only parameter that can properly represent the behavior of the fin-wall assembly. Nevertheless, AUG is unsuitable for predicting the behavior of the fin-wall assembly subjected to an increase in the heat transfer coefficient h2, since, while the heat dissipated by the fin-wall assembly increases, AUG decreases 9. This is due to the fact that an increase in h2 promotes an increase in the heat dissipated by the unfinned wall greater than the increase produced in the fin-wall assembly. In our judgment, then, none of the above coefficients fully satisfies all the requirements that an ideal performance coefficient must fulfil. After evaluating the aptitude and/or suitability of the five above-mentioned coefficients for studying fin-wall systems, we propose three new performance indicators: “thermal reverse admittance” 4, “specific reverse admittance,” and “relative reverse admittance.” They are especially suited to representing the performance of fin-wall systems under any hypotheses, including time-dependent processes. This set of indicators can be directly obtained by the Network Simulation Method (NMS), a general-purpose numerical method whose efficacy has been established for different kinds of linear and non-linear problems in the heat transfer domain 10. Briefly, an ideal performance coefficient ought to: In the search for a performance coefficient to overcome the difficulties of the coefficients defined to date, the “thermal reverse transfer admittance,” or simply “thermal reverse admittance,” Yr, is defined, whose generality means that it is not merely confined to the study of finned systems: (1)Yr=Joθiwhere Jo is the output heat rate dissipated to the surroundings and/or transferred to other elements, depending on the system considered, and θi is the excess in the input section of the same. This input section is not necessarily a physical surface but may also be an ideal limit which includes boundary conditions. In fin-wall systems (Fig. 1) θi=θ1∞ is the excess in the fluid that bathes the wall on its naked side, while Jo is the heat dissipated to fluid 2 by the external surface of the fin and the rest of the prime surface. Yr refers to the heat that is transferred by the fin or fin-wall assembly, while the Kraus input admittance, Yi, refers to the heat flux in the input section. The main reason for defining the new coefficient is that in time-dependent systems (transient process, harmonic, etc.) differences in phase exist and there is a damping of the amplitude between the flux in the thermal source and the dissipated flux, both of which depend on the excitation and on the dimensional and thermal characteristics of the system. In time-independent processes Yi=Yr.The terms “impedance” and “admittance” are widely identified as quantities of time-dependent processes, both harmonic and non-harmonic. Harmonic processes are of interest in thermal scope (day-night variations, reciprocating compressors, etc.). In these time-dependent processes, thermal admittance is not just the reverse of thermal resistance since it is defined by both modulus and phase, which normally have a marked dependence on frequency. Yi is really a “conductance,” not properly an admittance, since no reference to time-dependence is mentioned. In turn, Yr has been used with success to obtain the frequency response of fin-wall systems 4. The definition of thermal reverse admittance is general, and is not restricted to certain hypotheses: multidimensional behavior, variable thermal coefficients and properties, nonlinear boundary conditions, etc. It is defined for instantaneous heat rates and excess and their evaluation is in general made by numerical procedures, NSM being especially suitable. This coefficient is indeed useful in the calculation and design of fin systems since, once Yr, the excess θi and the number of fins per unit of length or surface are known, it is possible to calculate the overall heat transferred by the device. In turn, according to Section 2, thermal reverse admittance is a consistent parameter but cannot be considered as an ideal coefficient, for which reason we have derived new parameters from it. The reverse admittance coefficient refers neither to the quantity of material used nor to an optimum. To overcome this deficiency the “specific reverse admittance,” yr, or the relationship between the thermal reverse admittance, Yr, and the mass, m, of the fin, has been defined: (2)yr=YrW/Kmkg=JoWθiKs˙mkgA specific reverse admittance with respect to the volume is also possible. This coefficient fulfils the three first conditions in Section 2: Specific reverse admittance is a universal parameter for comparing the performance of different fin configurations and shapes, taking into account the quantity of material used. The above coefficient is dimensional and its maximum value is not unity. In order to overcome these deficiencies and to make more explicit the reference to a standard, relative reverse admittance, yrel, is defined as: (3)yrel=yryr,optwhere yr,opt is the specific admittance in the optimum fin for a given type (longitudinal, annular, rectangular, etc.) and/or material. The determination of this coefficient requires the determination of the optimum, which is specific to every fin system. yrel is a dimensionless coefficient with a maximum value of unity. Thus, it may be considered as an ideal coefficient because it fulfils all the requirements (Section 2). Yrel indicates the margin of enhancement that a fin design can obtain for a given shape. This information is not provided by any of the fin performance indicators defined to date. Table 1 shows the extent to which the different extended surface performance indicators fulfil the required characteristics. It is clear that only reverse admittances are always consistent and that relative reverse admittance is the coefficient that accomplishes all the characteristics of an ideal indicator. As an example, we shall calculate the reverse admittances Yr,yr,yrel of the straight fin-wall assembly of rectangular profile of Fig. 1. Classical Murray-Gardner’s hypotheses are assumed for simplicity, except that unsteady flux and diabatic extreme of the fin are also considered; θb excess is a priori unknown. Fin and wall are of pure aluminum k=229W/mK,ρ=2700kg/m3,c=945J/kgK;θ1=1K and the heat transfer coefficients on both sides of the assembly are h1=100W/m2K,h2=10W/m2K. The dimensions are Lw=5s˙10−3m,Lf=5s˙10−2m,Ly=10−2m,e=10−3m,b=1m; the fin with these dimensions is called “initial” fin. Due to symmetry, only the section between y=0 and y=Ly needs to be analyzed. The mathematical model consists of the following set of equations: 0<x<Lw:∂jw/∂x+ρc∂T/∂t=0;jw=−k∂T/∂xLw<x<Lw+Lf:∂jf/∂x+ρc∂T/∂t+h2P/SfT−T2∞=0;jf=−k∂T/∂xx=0:h1T1∞−T=−k∂T/∂x(4)x=Lw:−Swk∂T/∂x=−Sfk∂T/∂x+Sw−Sfh2T−T2∞x=Lw+Lf:−k∂T/∂x=h2Tf−T2∞t=0;0<x<Lw+Lf:T=T0,T2∞=T0=0In the steady state the simulation of the fin-wall system gives values of Yr=0.733W/K and yr=1.358W/kgK. The efficiency of the fin in these conditions is η=0.964. To determine the value of the relative admittance yrel, the “optimum” of the system must be found, i.e., the dimensions of the assembly for a given volume (or mass) for which the dissipated heat is maximum. If the restrictions of the problem are the volume of the fin Vf=10−4m3,Ly=0.005m, and 0.01m<Lf<0.2m, the optimum fin length, Lf,opt, as determined by means of NSM 11, is 0.13 m, which corresponds to half the fin thickness, eopt=3.846×10−4m. The coefficients of performance in the optimum are ηopt=0.635,Yr,opt=0.954W/K, and yr,opt=1.767W/kgK. Finally, the relative reverse admittance, yrel, of the initial fin is 0.768, indicating that the fin design can be substantially improved. As has been demonstrated, efficiency behaves inconsistently and so comparison of the efficiency of the two fins provides confusing information: the efficiency of the initial fin is greater than that of the fin of optimum dimensions. On the other hand, the admittances of the two fin-wall systems reflect the real situation: either the reverse admittance or the specific reverse admittance is bigger in the fin of optimum dimensions and the relative admittance of the studied fin provides a value that points to the distance from the optimum. Figure 2 shows the instantaneous input admittance Yi and reverse admittance Yr of the fin-wall assembly of the fin of the initial dimensions (i) and the fin of optimum dimensions (o). The difference between both coefficients is clear. Both values coincide in the steady state but not in the transient time. Table 2 shows the values of the performance indicators for the initial and optimum fin-wall assemblies. Efficiency, effectiveness, augmentation, and enhancement factors have been computed directly from their definitions 12567 and NSM simulation results. After a study of the conceptual and application characteristics of currently used extended surfaces indicators, in a general analysis of fin-wall systems, the characteristics that must be fulfilled by an ideal coefficient have been detailed. With these in mind, thermal reverse admittance, specific reverse admittance and relative reverse admittance are proposed as new performance indicators for the analysis of extended surfaces under any hypothesis, especially time-dependent processes. These new coefficients, for their generality and precision, outmatch the currently used performance coefficients; the last two are also linked to the concept of system optimization. Thermal reverse admittance, which allows us to know the dissipated heat once the temperature gradient between fluids is known, is an absolute scaling factor of fin-wall assemblies; specific reverse admittance is a measure of efficient use of material for heat transfer task; and relative reverse admittance provides information as to the margin of improvement in fin design. Finally, a straight fin-wall system has been analyzed as an example of how the new coefficients can be applied. Their superiority over other coefficients is demonstrated. Network Simulation Method is especially suitable for the direct and accurate numerical determination of the proposed coefficient without using special mathematical developments. Although initially developed for extended surfaces, the thermal reverse admittance and derived coefficients could be applied to the analysis of other thermal systems. AUG= augmentation factor b= with of the fin (m) c= specific heat (kJ kg−1K−1) e= half fin thickness (m) h= heat transfer coefficient (Wm−2K−1) j= heat rate density (Wm−2) J= heat rate (W), J=js˙Sk= thermal conductivity of the material (Wm−1K−1) L= length (m) Ly= half fin pitch (m) m= mass of the fin or fin-wall assembly (kg) P= perimeter of the fin (m), P=22e+bS= cross-sectional area, external surface (of the fin) (m2) t= time (s) T= temperature of the material at point x (K) V= volume of the fin or fin-wall assembly (m3) x= space co-ordinate (m) Y= admittance (W/K) y= specific admittance (W/kg K) Greek Symbolsη = fin efficiency ρ = density (kg m−3) θ = temperature excess, i.e., T−T2∞KSubscriptsb= base of the fin f= fin i= input o= output opt = refers to the optimum fin r= reverse rel = refers to relative reverse admittance w= wall 0 = initial condition 1 = refers to the unfinned side of the wall 2 = refers to the finned side of the wall ∞ = refers to the external media far from the surfaces