Spine Fin Efficiency A Three Sided Pyramidal Fin of Equilateral Triangular Cross-Sectional Area

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1 Proceedings of the 006 WSEAS/IASME International Conference on Heat and Mass Transfer, Miai, Florida, USA, January 18-0, 006 (pp13-18) Spine Fin Efficiency A Three Sided Pyraidal Fin of Equilateral Triangular Cross-Sectional Area RICHARD G. CARRANZA Consultant 9315 Vanwood Houston, TX USA Abstract: A odel is presented for deterining the fin efficiency of a spine fin with the geoetry of a three sided pyraid of equilateral triangular cross-sectional area (fro here on referred to as TSPECA). It is assued that heat transfer is one-diensional. The differential equation describing the syste is solved using finite differences. The fin efficiency is then calculated using nuerical integration, the trapezoidal rule. The results are copared against an equivalent fin of constant cross-sectional area. It is found that the errors are considerable when approxiating the fin geoetry as that of constant crosssection - as high as 19%. It is recoended that the spine fin be odeled as variable in cross-section throughout its length. Key Words: fin efficiency, finite differences, spine fin, convection, conduction, heat transfer. 1 Introduction The interest in deterining the fin efficiency for TSPECA is based on a previous work by Carranza [1]. Carranza odels the spine fin as a three sided pyraid, but deterines fin efficiencies for geoetries that are not equilateral triangular in cross-section. Each side of the triangle has a different length. Moreover, the fins analyzed are of fixed diensions. In total, only two fins with different sets of physical paraeters are studied. For this reason, a general ethod for deterining the fin efficiency is presented. A robust odel is outlined that calculates the fin efficiency of a TSPECA for a variety of spine lengths, widths, aterials and teperatures. Any physical or theral paraeter can be varied. The proble is siply restricted to a pyraid with equilateral triangular crosssection. Before beginning the analysis, it is necessary to review why the fin efficiency is required. In heat transfer studies, h o is easured experientally for finned surfaces through the Fourier heat transfer equation, where the overall heat transfer coefficient is defined relative to the pipe inner radius: U i 1 = 1 ri ln ro / ri Ai + + h k h ( A + A η) i o b f (1) As one can see, h o is deterined for a finned pipe through the use of Equation 1. All of the physical paraeters in Equation 1 are easured. The h i is easily calculated fro established correlations. The proble with Equation 1 is that η is a function of h o. That is, the h o ust be estiated to deterine η. Once η is quantified, then h o is calculated fro Equation 1 and checked against the initial estiate. For this reason, a systeatic ethod for deterining η is required. η is a function of fin geoetry, fin aterial, teperature profile

2 Proceedings of the 006 WSEAS/IASME International Conference on Heat and Mass Transfer, Miai, Florida, USA, January 18-0, 006 (pp13-18) and uch ore. All of these issues are addressed in this work. Geoetry The geoetry of the spine in question is that of a three sided pyraid (see Figure 1). The footprint of the spine, at the base, is that of an equilateral triangle. It is then assued that the sides slowly decrease in size as they approach the tip of the spine. The rate of decrease is assued to be a linear function of the spine height. Thus, the cross-sectional area of the spine is always that of an equilateral triangle for any cross-section throughout the length of the spine. It ust be noted that the cross-sectional area of the spine is changing throughout its length. The cross-sectional area is zero at the spine tip. Thus, ost odels for deterining the spine efficiency cannot be used for this spine geoetry since they assue constant cross-sectional area. Therefore, a set of special geoetrical equations are established for this spine geoetry - TSPECA. The first step is to assign a spatial coordinate syste to the odel (see Figure ). At the base, the crosssectional area of the spine is shaped like an equilateral triangle. Each side of the triangle has a length a. The height of the spine is assigned the ter b. As was stated, the sides of the triangle decrease linearly as they approach the tip of the spine. The length of the sides of the triangle are zero at the spine tip. The geoetry ust be addressed atheatically. To do so, a paraeter is required that deterines the triangle s side length as a function of x, l(x). This is done by aking l(x) a dependent variable of the independent variables a and b. Hence, the following equation is proposed for l(x): TOP VIEW SIDE VIEW a Figure 1. Scheatic of spine geoetrical paraeters. b l ( a x ) = a b x = a β x () Knowing the length of the sides of the equilateral triangle as a function of x, the perieter of the spine is known as a function of x : das P = 3l( x) = = 3( a x) dx β (3) Lastly, the cross-sectional area of any equilateral triangle is deterined fro siple geoetry:

3 Proceedings of the 006 WSEAS/IASME International Conference on Heat and Mass Transfer, Miai, Florida, USA, January 18-0, 006 (pp13-18) A c = l 3 4 (4) Substituting Equation into Equation 4, the final equation for the cross-sectional area of the spine fin is derived. ( ) 3 Ac ( x) = a aβx + β x (5) 4 d Θ dθ ( Χ 1) + + α Θ = 0 (8) dχ dχ where Θ(0) = 1 and Θ(1) = 0. The diensionless variables in Equation 8 are defined in Equations 9 and 10. T T Θ = T T b (9) In heat transfer studies, an iportant quantity is the rate of change of crosssectional area as a function of fin length. This is a rate ter used later when the heat balance is conducted around the fin. For the sake of unifority, the equation is presented here. Y Χ = x b (10) da c 3 = ( aβ + β x) (6) dx SPINE FIN X 3 Theory 3.1 Transport Equations Prior to calculating the fin efficiency, it is necessary to find the teperature profile through the fin. This is done with the aid of an energy balance []: Acd T dac dt ho das + ( T( x) T 0 ) = (7) dx dx dx k dx The boundary conditions for the spine fin are T(0)=T b and T(b)= T. Equation 7 is best solved if done via diensionless variables. Diensionless variables ake the solution ore general and applicable. Furtherore, note that Equation 7 is a universal equation that applies to any different types of fins. Equations - 6 are substituted into Equation 7 aking it applicable to a TSPECA. b TUBE WALL Figure. Spatial Coordinate syste used in odel developent. 3. Nuerical Solution A nuerical solution is proposed for Equation 8. The Equation is solved using finite differences. The central difference equations are used; and when the boundary conditions are applied, a square, tridiagonal atrix is produced. The atrix is then solved using the Thoas Algorith. The Thoas Algorith is useful for tridiagonal systes because it does not store the zeros of the atrix which lie outside the band. Only the values inside the band are stored. This saves eory and decreases coputation tie. After closer inspection, one notices that the only quantity not specifically deterined in

4 Proceedings of the 006 WSEAS/IASME International Conference on Heat and Mass Transfer, Miai, Florida, USA, January 18-0, 006 (pp13-18) Equation 8 is α. Nevertheless, values for α are arbitrarily specified. Values for α are selected between 0 and 10. Thus, a copletely different teperature profile is calculated for each value of α. 4 Results 4.1 Teperature Profile The teperature profile throughout the fin is obtained fro Equation 8 and presented in Figure 3. It is clear that as α increases in agnitude, the heat transfer through the spine increases. That is, as heat transfer is enhanced, ost of the heat is exchanged before reaching the tip of the spine. The fin acts as a channel that sends heat to the tip fro the base. As α increases, the heat is lost before reaching the tip. Therefore, the spine s teperature (especially near the tip) begins to approach the teperature of the convective bulk fluid. This phenoena is ost easily seen by exaining the definition of α: hob hob α = 4 3 = 4 3 βk ak (11) As α gets larger, the following heat transfer echaniss are occurring within the spine fin: h o is increasing. As h o increases, the rate of heat lost by convection increases. Heat does not reach the fin tip; and for this reason, the teperature at the tip is closer to the teperature of the convective fluid. b is increasing. As b increases, the heat ust travel farther to reach the tip. More theral resistance is experienced and this increases the likelihood that heat is transferred by convection rather than by conduction. a is decreasing. As a decreases, the crosssectional area for heat flow reduces. Thus, heat does not have a suitable avenue for reaching the fin tip. k is decreasing. As k decreases, theral resistance to heat transfer increases. And once again, heat is inhibited fro reaching the tip. It is iportant to note that α is siply the ratio of the theral conductance due to convection divided by the theral conductance due to conduction for a straight fin with constant equilateral triangular crosssectional area. To see this, it is necessary to restate the definition of theral conductance: Kconv ho As ho ab T = 1 R = = = 3 (1) conv k Ac ka Kcond T = 1 R = = L = 3 4b cond (13) Now, by siply dividing Equation 1 by Equation 13, Equation 11 is derived. Thus, it is clear that as α increases, the heat transfer due to convection increases. Less heat reaches the spine tip and causes the teperature of the spine to ore closely approxiate the teperature of the convective bulk fluid. Figure 3. Θ versus Χ for specified values of α.

5 Proceedings of the 006 WSEAS/IASME International Conference on Heat and Mass Transfer, Miai, Florida, USA, January 18-0, 006 (pp13-18) 4. Fin Efficiency The fin efficiency is now deterined for TSPECA. This is done by using the strict definition of the fin efficiency: η = = ax A= A A= s ho ( T( x) T ) da 0 (14) h A ( T T ) o s b Forally, the fin efficiency is defined as the actual heat transferred by the fin divided by the heat the fin would transfer if it were at one hoogenous teperature, T b. Equation 14 is not yet in the proper for for developing useful results. Therefore, it is transfored through the use of Equations 3, 9, and 10 into a function of diensionless variables. Equations 15, 16, and 17. However, when Equations, 3, and 4 are substituted into Equation 17, a convenient basis is found (note, l(x)=a, P(x)=3a and A c =a /4). b = α (18) Thus, the efficiencies for TSPECA and a fin of constant cross-sectional area are both plotted as a function of the sae independent variable, α Χ = 1 η = Θ ( 1 Χ )d Χ (15) Χ= 0 In this for, Equation 15 is specifically setup for deterining the fin efficiency of a TSPECA. Equation 15 is solved nuerically using the Trapezoidal rule and the data generated via Equation 8. The results are presented in Figure 4 as a function of α. Note that the efficiency for a fin with constant cross-sectional area is also presented in Figure 4. The fin has the sae geoetrical diensions as TSPECA, given in Figure 1. The relationships for calculating the efficiency of a fin with constant cross-sectional area are well established and given in Equations 16 and 17 [3]: η = tanh( b) b (16) h P o = (17) k A Fro siple observation, it is difficult to see a coon basis on which to copare c Figure 4. η versus α: a coparison of fin efficiencies for TSPECA and an equivalent fin of constant cross-sectional area. 5 Conclusion A odel is presented for deterining the fin efficiency of a TSPECA. The results are presented in Figure 4. It is iportant to note that in any heat transfer applications the fin efficiency is approxiated using Equations 16 and 17. The equations are applied often to helical fins, plate fins, and spine fins in HVAC applications - despite they are valid only for straight fins of constant cross-sectional area. See Carranza for a coplete overview and description [4]. This work, however, shows that there is a large discrepancy between the efficiency for a fin of constant cross-sectional area and a TSPECA. In Figure 4, the error between the

6 Proceedings of the 006 WSEAS/IASME International Conference on Heat and Mass Transfer, Miai, Florida, USA, January 18-0, 006 (pp13-18) two ethods is 1.1% and 19.% for values of α at 1 and, respectively. Therefore, it is recoended that a detailed attept be ade at odeling the physical geoetry of any fin in question with as uch accuracy as possible; and that siplistic assuptions, such as constant cross-sectional area, be kept to a iniu. Otherwise, significant errors in the fin efficiency are ade. For this reason, a odel for a TSPECA is presented. The fin efficiencies are deterined fro a atheatical syste that is detailed and precise. 6 Noenclature A= area K= theral conductance L= length of conductance P= perieter = heat transfer rate R= theral resistance T= teperature U= overall heat transfer coefficient a= fin paraeter specified in Figure 1 b= fin paraeter specified in Figure h= individual heat transfer coefficient k= theral conductivity l= fin geoetrical paraeter specified in Equation = fin efficiency paraeter specified in Equation 17 r= pipe radius x= spatial coordinate y= spatial coordinate Θ= diensionless teperature Χ= diensionless length α= ratio of conductance due to convection per conductance due to conduction β= ratio of a to b η= fin efficiency c= cross-sectional f= fin i= inside = etal ax= axiu o= outside s= surface = bulk fluid conditions References: [1] R. G. Carranza, Spine fin efficiency of spined pipe heat exchangers, National Heat Transfer Conference Proceedings, Altanta, [] F. P. Incropera and D. P. DeWitt, Fundaentals of Heat Transfer, John Wiley and Sons, New York, [3] Mcuiston, F. C., and Parker, J. D., Heating, Ventilating, and Air Conditioning, 3 rd ed., John Wiley and Sons, New York, [4] R. G. Carranza, A survey of iportant auxiliary equations for selected copact heat exchangers, ASME 005 Suer Heat Transfer Conference Proceedings, San Francisco, Subscripts b= base