OPTIMIZATION ANALYSIS FOR LOUVER-FINNED HEAT EXCHANGERS

Transcription

1 - 1 - OPTIMIZATION ANALYSIS FOR LOUVER-FINNED HEAT EXCHANGERS Jn-Yuh Jang, Professor,Department of Mechancal Engneerng, Natonal Cheng Kung Unversty, Tanan, Tawan 70101, Bng-Ze L, graduate student,,department of Mechancal Engneerng, Natonal Cheng Kung Unversty, Tanan, Tawan 70101, Abstract: A seres of 3-D computatonal flud dynamcs analyses along wth the smplfed conugate-gradent method were carred out to study the thermal-hydraulc characterstcs for the louver-fnned heat exchanger. The effects of dfferent louver angles (θ = 0 o to 8 o ), and louver ptches (L p = 0.8 to 1. mm) on the heat transfer phenomenon and pressure drop (n terms of Colburn factor and frcton f factor) were nvestgated n detal. The optmzaton of the louver angle and louver ptch s carred out by usng the smplfed conugate-gradent method. The area reducton rato usng a louver fn relatve to the surface s the obectve functon to be maxmzed. The numercal results showed that both the f and factors are ncreased wth the ncrease of the louver angle; whle, for a gven louver angle, as the louver ptch (L p ) s creased, both the f and factors are ncreased. It s also found that, under the optmum condtons, the area reducton rato could reach up to 8 % to 37% wth Reynolds number (based on the fn ptch ) Re F p rangng from 400 to Key Words: optmzaton, Louver- fn, louver angle and louver ptch 1 INTRODUCTION Fnned-tube heat exchangers are wdely used n the ar-condtonng ndustry. To enhance the heat transfer, nterrupted surfaces lke louver fns are often used. Generally, louver fns are employed snce they can ncrease the heat transfer area and nterrupt the development of boundary layers, thus provdng hgher average heat transfer coeffcents and thermal performance. The topc of louver fns has been studed expermentally wth the frst relable data publshed by Kays and London (Kays and London, 1950). Davenport carred out smoke trace studes on a ten-tmes scale model and dscussed the effects of dfferent geometrcal parameters on the heat transfer and pressure drops (Davenport 1983). Achacha and Cowell made a comprehensve study of louvered plate fn heat exchangers and found the correlatons for heat transfer and pressure drops (Achacha and Cowell, 1988). Webb and Trauger performed flow vsualzaton tests and defned the term flow effcency as the rato of actual transverse dstance to deal transverse dstance (Webb and Trauger, 1991). They also found that the flow effcency at a gven Reynolds number s ncreased as the louverto-ptch rato s reduced. Webb and Jung presented expermental data for sx brazed alumnum heat exchangers and found that ths desgn gves a 90% hgher heat transfer coeffcent for only 5% hgher pressure drop compared wth the round tube plate fn desgn (Webb and Jung, 199). Rugh et al. expermentally studed a hgh fn densty louvered surface (Rugh et al., 199). They reported that the louver fns produce a 5% ncrease n heat transfer coeffcent and 10% ncrease n pressure drop. Chang and Wang developed the heat transfer and frcton correlatons, based on the huge data base wth 91samples of flat tube heat exchangers (Chang and Wang, 1997). Wang et al. developed heat transfer and frcton correlatons, based on the data bank wth 49 samples of round tube heat exchangers, and clamed that 90% of the expermental data were correlated wthn 15% (Wang et al., 001). The desgn of louver fn has also been studed numercally wth CFD technques. Suga et al. used a rectangular flow doman flled wth overlap Cartesan meshes to compute the flow and heat transfer over a fnte-thckness fn(suga et al. 1990). Ikuta et al. used a block-structured mesh for each louver and found good agreement between computed and measured values of overall heat transfer and pressure loss(ikuta et al., 1990). Recently, Hseh and Jang proposed successvely varable louver angles and carred out a 3-D numercal analyss on thermal-hydraulc characterstcs. Ther results showed that varyng the louver angles can effectvely enhance the heat transfer of the heat exchanges (Hseh and Jang, 006). Atknson et al. nvestgated the -D and 3-D flow and heat transfer

2 - - phenomenon of louvered fnned-tube heat exchangers (Atknsona et al., 1998). They found that the 3- D models gve predctons of overall heat transfer n better agreement wth expermental observatons. Recently, Hseh and Jang numercally nvestgated the optmal desgn of a louver fnned-tube heat exchanger usng the Taguch method (Hseh and Jang, 007). The optmal desgn values for each parameter were all presented. Jang and Tsa used the smplfed conugate-gradent method (SCGM) to fnd the optmal louver angle of a louver fn heat exchanger (Jang and Tsa., 007) for a gven louver ptch. The search for optmum louver angles rangng from θ = 15 o to 45 o were carred. However, they dd not serach the optmum combnaton of louver anale and louve ptch smulatneously. Ths has motvated the present nvestgaon. In the frst part of ths study, the effects of dfferent louver angles (θ = 0 o ~8 o ), and louver ptches (L p = 0.8~1. mm) on the heat transfer phenomenon and pressure drop s nvestgated usng a commercal CFD code CFD-RC. The results are presented n terms of Colburn factor and frcton f factor.in the second part, the optmzaton of the louver angle and louver ptch s carred out by usng the smplfed conugate-gradent method. The area reducton rato usng a louver fn relatve to the surface s the obectve functon to be maxmzed. MATHEMATICAL ANALYSIS.1 Governng Equaton Fgure 1 descrbes the physcal model and the computatonal doman. Fgure 1 Schematc dagram of computatonal doman and geometrcal parameters

3 - 3 - Table 1 The geometry of the physcal model Parameter fn ptch, F p fn thckness,δ louver ptch, L p louver wdth, L w louver angle,θ Value.7 mm (# 9/nch) 0.1 mm 0.8 ~ 1. mm 6.4 mm 0 o ~ 8 o The relevant geometrcal dmensons of the louver fn are also shown n Table 1. The louver angle θ and louver ptch L p are the man operatng parameters n the present study. The relevant numercal results were acheved n the range of 400 <Re Fp < 1600, 0< θ < 8, 0.8 mm < Lp < 1. mm. Frst, we run the the prsent phscal model wth 3-D steady lamnar and turbulent modes, respectvely. The numercal results were compared wth those of the expermentas by prevous mvestogator (Davenport 1983) as shown n Fg.. It s found that the flud may be modelled as a turbulent flow wth κ ε model wth reasonably accuracy. Equatons for contnuty, momentum and energy may be expressed n tensor form: u = 0 p ρ ( u u ) = + μ u ρ u ' u ' () ( u ' T ') ρ c p ( u T ) = k f T ρc (3) p In the above equatons, ρ s flud densty, μ s dynamc vscosty, c p s heat capacty, and k f s thermal conductvty of the flud.the turbulent model adopted here s the ρ ρ u κ t ( u κ ) = u + + ρ( P ε ) u l σ r k ut ε ε + + ρ [ c1pr cε σ ε κ ( u ε ) = ] l κ ε model: In the above equatons, κ andε are turbulent knetc energy and turbulent energy dsspaton rate, respectvely, defned as κ = 1 ( ) u u (6) u u (7) ε =ν ( ) The local Nusselt number s defned as, T hf ( ) w Fp p Nu n (8) = = k Tw Tb where F p s the fn ptch, T w s the fn temperature, and T b s the bulk temperature, whch s expressed as v = TdA T b (9) vda The average Nusselt number s gven by (1) (4) (5)

4 - 4 - NudA Nu = (10) da The Colburn factor ( ) and frcton factor(f) are defned as = Nu 1/ 3 Re Fp Pr (11) p F n p p f = 1 (1) ρ V 4L n where Re Fp s defned as ρvnfp Re = (13) Fp μ and P n s the nlet pressure, Pr s the Prandtl number, L s the flow length.. Boundary Condton At the nlet, the flow velocty V n s assumed to be unform, and the temperature T n s taken to be 5C. At the fn surfaces, no-slp condtons and constant wall temperature T w (75C) are specfed. Cyclc boundary condtons are set on both the upper and lower boundares, whle symmetrc boundary condtons are set on both sdes of the doman..3 Performance Evaluaton Crtera (PEC) Many performance evaluaton crtera (PEC) have been developed for evaluatng the performance of heat exchangers. The VG-1 (varable geometry) performance crtera, as descrbed by Webb(webb, 1994), represents the possblty of surface area reducton by usng enhanced surfaces havng fxed heat transfer and pumpng power. ha A G = (14) h A A G where the subscrpt ref refers to reference plate fn, and G s the mass velocty. The pumpng power s calculated as fa G GAc P = ( )( ) (15) Ac ρ ρ where A c s the flow area at mnmum cross secton. The pumpng power rato relatve to the reference plane fn can be obtaned by 3 P A G = (16) P A G and by the elmnaton of the term ha h A = (17) P 1/ 3 A / 3 f 1/3 ( ) ( ) ( ) P A f Under the followng condtons Q/Q = 1 ha/h A = 1 P/P = 1 we may obtan the area reducton rato relatve to the reference plane fn as (18)

5 - 5 - A A = ( f f ) 1/ ( ) 3 / (19) 3 NUMERICAL METHOD The governng equatons are solved numercally usng a commercal CFD code (CFD-RC, 01). The numercal methodology s brefly descrbed here. A thrd-order upwnd TVD (total varable dmnshng) scheme s used to model the convectve terms of the governng equatons, whle secondorder central dfference schemes are employed for the vscous and source terms. A grd system of 19,000 grd ponts s typcally adopted n the computaton doman. A careful check for the grdndependence of the numercal solutons s made to ensure the accuracy and valdty of the numercal results. For ths purpose, four grd systems, 61,440, 1,400, 19,000 and 83,360, are tested. The relatve error between the solutons for the 19,000 and 83,360 grd systems n the frcton factor s wthn 3%, whle the computaton tme of the 83,360 grd systems s tmes that of the 19,000 grd system. Consderng the computng tme, the 19,000 grd system s adopted here. The convergence crteron s satsfed when the resduals of all varables are less than 1.0x OPTIMIZATION In the present study, the smplfed conugate-gradent method (Jang and Tsa, 011) s combned wth a fnte dfferental method code ( CFD-RC 01) as an optmzer to search the optmum louver angle (θ) and louver ptch (Lp). The obectve functon J ob s defned as the maxmum area reducton rato (1-A/A ref ). Above all, the SCGM method evaluates the gradent of the obectve functon, and then t sets up a new conugate drecton for the updated desgn varables wth the help of a drect numercal senstvty analyss. The ntal guess for the value of each search varable s made, and n the successve steps, the conugate-gradent coeffcents and the search drectons are evaluated to estmate the new search varables. The solutons obtaned from the fnte dfference method are then used to calculate the value of the obectve functon, whch s further transmtted back to the optmzer for the purpose of calculatng the consecutve searchng drectons. The procedure for applyng ths method s descrbed n the followng: (1) Generate an ntal guess for two desgn varables (x 1,x ) louver angle (θ) and louver ptch (Lp). () Adopt the fnte dfference method to predct the velocty and temperature felds assocated wth the latest θ and Lp, and then calculate the obectve functon J ob (x 1,x ). (3) When the value of J ob (x 1,x ) reaches a maxmum, the optmzaton process s termnated. Otherwse, proceed to step (4). (4) Determne the gradent functons, ( J ob / 1 ) (k) and ( Job / ) (k), by applyng a small perturbaton (Δx 1,Δx ) to each value of x 1 and x, and calculate the correspondng change n obectve functon (ΔJ ob ). Then, the gradent functon wth respect to each value of the desgn varables (x 1,x ) can be calculated by the drect numercal dfferentaton. (k) ( k+ 1) ( k+ 1) (5) Calculate the conugate-gradent coeffcentsγ, and the search drectons, ξ1 andξ, for each search varable. (6) Assgn values to the coeffcents of descent drecton (β 1, β ) for both values of the desgn varables (x 1, x ). Specfcally, those values are chosen by a tral-and-error process. In general, the coeffcents of descent drecton (β 1, β ) are wthn a range of 0.01 to (7) Update the desgn varables wth ( k+ 1) ( k ) ( k) ( k+ 1) ( k) ( k ) x = x + β ξ and x = x + β ξ (3) RESULTS AND DISCUSSION The present study manly evaluated the nfluences of louver angle (θ) and louver ptch (Lp) on the local and overall flow and heat transfer characterstcs of louver fn. Furthermore, optmzaton analyses were utlzed n order to search the optmum combnaton of (θ, Lp) and maxmum obectve functon (1 A/A ref ). Fgure shows the comparsons of and f factors between the present numercal

6 - 6 - results and the prevous mvestogator (Davenport 1983). The present results showed good agreements wthn a maxmum of 16% dscrepancy. [] Fgure Comparson of the and f factors for the present study and prevous lterature Fgure 3 shows the comparson of streamlne dstrbutons for dfferent louver angles at Re Fp =400, L p =1.00 mm. It can be seen that, at θ = 0 o, the streamlne of the flud s qute smooth and much of the flud flows through the channels between the fns rather than through the louvers. On the other hand, at θ = 8 o, the flow s dverted through the louver passages, whch s so called louver drected flow. (a) θ= 0 o (b) θ= 4 o (c) θ= 8 o Fgure 3 Streamlne dstrbutons at Re Fp = 30 and L p = 1.00 mm: (a)θ= 0 o, (b)θ= 4 o, (c)θ= 8 o

7 - 7 - Temperature (K) (a) θ= 0 o (b) θ= 4 o (c) θ= 8 o Fgure 4 Isotherms at Re Fp = 30 and L p = 1.00 mm:(a)θ= 0 o, (b)θ= 4 o, (c)θ= 8 o Fgure 4 shows the comparson of temperature dstrbutons for dfferent louver angles. At θ = 8 o, the temperature at outlet s hgher because of the ntensfed turbulence, whch means the performance of heat transfer s better.the frcton factor f and the Colburn factor versus Re Fp wth varous louver angles are shown n Fgure 5. As the louver angle s ncreased, both the and f are ncreased. For the present study, when the louver angle s ncreased from 0 o to 8 o, the values of f s ncreased by 36.80%, whle the values of s ncreased by 8.66%, across the Re Fp range ( ). (a)frcton factor

8 - 8 - (b) Colburn factor Fgure 5 Varatons of f and factors versus Re Fp wth dfferent louver angles Fgure 6 The area reducton rato versus Re Fp for dfferent louver angles The area reducton ratos relatve to the fn surfaces, 1- A/A ref, for dfferent louver angles are presented n Fgure 6. It s shown that the area reducton rato s ncreased as the louver angle s ncreased. For example, at Re Fp = 800, the area reducton ratos for θ = 0 o to 4 o and 8 o are 9.60%, 19.6% and 7.43%, respectvely. The frcton factor f and the Colburn factor versus Re Fp wth varous louver ptches are shown n Fgure 7. As the louver ptch s ncreased, both and f are ncreased. For the present study, when the louver ptch s ncreased from 0.8 mm to 1.mm, the values of f s ncreased by.33%, whle the values of s ncreased by 4.0%, across the Re Fp range ( ). The area reducton ratos relatve to the fn surfaces, 1- A/A ref, for dfferent louver ptches are presented n Fgure 8. It s shown that the area reducton rato s ncreased as L p s ncreased.

9 - 9 - (a) frcton factor (b) Colburn factor Fgure 7 Varatons of f and factors versus Re Fp wth dfferent louver ptches

10 Fgure 8 The area reducton rato versus Re Fp for dfferent louver ptches Fgure 9 dsplays the teraton process used to search the optmum combnaton of louver angle (θ) and louver ptch (Lp) for the maxmzaton of obectve functon(.e. area reducton rato, 1-A/Aref) at Re Fp = 400 and The constant area reducton rato contours are plotted as a functon of θ and Lp, where the dark red area represents the maxmum area reducton rato. It s seen that, wth the ntal values (θ = 35,Lp = 1.mm) by usng the smple conugated gradent method (SCGM), the optmal θ and Lp combnaton s obtaned for around 33 and 57 teratons, respectvely. Thus, the current optmzaton method provdes a tremendous savngs n regard to computatonal tme.for Re Fp = 400, the optmal θ and Lp combnaton are obtaned (θ = 33.55,Lp = 0.84mm) and the area reducton rato s 7.64%. For Re Fp = 1600, the optmal θ and Lp combnaton are (θ =.3,Lp = 1.0 mm) and the area reducton rato s 36.76%. The searched optmum combnaton of θ and Lp for louver fn wth specfc values of Re Fp = 400, 600, 800, 1000, 100 and 1600 are tabulated n Table. It s seen that, the area reducton rato of 7.64% to 36.76% s acheved across the range of Re Fp.= 400 to (a) Re Fp =400

11 (b)re Fp =1,600 Fgure 9 Iteraton process to search the optmum combnaton of θ and Lp 6 CONCLUSION Table The searched optmum combnaton of θ and Lp for dfferent Re Fp Re Fp L p (mm) θ( o Area reducton rato ) (%) Flud flow and hear transfer for louver-fnned heat exchanger were studed numercally. The effects of dfferent louver angles (θ =0 o ~ 8 o ), and louver ptches (L p = 0.8 ~1. mm) on the heat transfer phenomenon and pressure drop were studed n detal. The optmzaton of the louver angle (θ) and louver ptch (L p ) was performed by usng a smplfed conugate-gradent method. A searchng procedure for the optmum louver angle (θ) and louver ptch (L p ), rangng from 0 o < θ < 35 o and 0.5 mm < L p < 1. mm, respectvely, was executed. Based on the numercal results, the followng conclusons can be drawn: 1. At a fxed louver ptch, as the louver angle (θ) s ncreased both the and f factors are ncreased. For example, as the louver angle s ncreased from 0 o to 8 o, the and f factors are ncreased by 8.66% and 36.80%, respectvely.. At a fxed louver angler, as the louver ptch (L p ) s ncreased, both the and f factors are ncreased. 3. The searched optmum obectve functon assocated wth an optmal combnaton of θ and Lp for dfferent Re Fp are obtaned for less than 60 teratons. Ths demonstrates that the current optmzaton method provdes a tremendous savngs n regard to computatonal tme for the present physcal model. Under the optmum condtons, the area reducton rato could reach up to 8 % to 37% wth Reynolds number rangng from 400 to 1600.

12 - 1-7 ACKNOWLEDGEMENT Fnancal support for ths work was provded by the Natonal Scence Councl of Tawan, under contract NSC E MY 8 REFERENCES Achacha, A., T.A. Cowell "Heat transfer and pressure drop characterstcs of flat tube and louvered plate fn surfaces," Expermental Thermal and Flud Scence, vol. 1, pp Atknsona, K.N., R. Drakulc, M.R. Hekal, T.A. Cowell "Two- and three-dmensonal numercal models of flow and heat transfer over louvred fn arrays n compact heat exchangers," Internatonal Journal of Heat and Mass Transfer, vol. 41, pp Chang, Y.J., C.C. Wang "A generalzed heat transfer correlaton for Iouver fn geometry," Internatonal Journal of Heat and Mass Transfer, vol. 40, pp Davenport, C.L "Correlatons for heat transfer and flow frcton characterstcs of louvred fn," AIChE Symposum Seres, vol. 79, pp Hseh, C.T., J.Y. Jang "3-D thermal-hydraulc analyss for louver fn heat exchangers wth varable louver angle," Appled Thermal Engneerng, vol. 6, pp Hseh, C.T., J.Y. Jang. 01. "Parametrc study and optmzaton of louver fnned-tube heat exchangers by Taguch method," Appled Thermal Engneerng, vol. 4, pp Ikuta, S., Y. Sasak, K. Tanaka, M. Takag, R. Hmeno "Numercal analyss of heat transfer around louver assembles," SAE paper No Jang, J.Y., Y.C. Tsa "Optmum louver angle desgn for a louvered fn heat exchanger," Internatonal Journal of the Physcal Scences vol. 6, pp , 011. Kays, W.M., A.L. London "Heat transfer and flow frcton characterstcs of some compact heat exchanger surfaces-part I: Test system and procedure," ASME Journal of Heat Transfer, vol. 7, pp Rugh, J.P., J.T. Pearson, S. Ramadhyan "A Study of a Very Compact Heat Exchanger Used for Passenger Compartment Heatng n Automobles," ASME Symposum Seres, vol. 01, pp Suga, K., H. Aok, T. Shnagawa "Numercal analyss on two-dmensonal flow and heat transfer of louvered fns usng overlad grds," JSME nternatonal ournal. Ser., Fluds engneerng, heat transfer, power, combuston, thermophyscal propertes, vol. 33, pp Wang, C.C., W.S. Lee, W.J. Sheu. 001 "A comparatve study of compact enhanced fn-and-tube heat exchangers," Internatonal Journal of Heat and Mass Transfer, vol. 44, pp Webb, R.L., P. Trauger "Flow structure n the louvered fn heat exchanger geometry," Expermental Thermal and Flud Scence, vol. 4, pp Webb, R.L., S.H. Jung "Ar sde performance of enhanced brazed alumnum heat exchangers," ASHRAE Transactons,, vol. 98, Pt., pp Webb RL, 1994, Prncples of Enhanced Heat Transfer, New York, John Wley & Sons