ONE IMENSIONAL TRIANGULAR FIN EXPERIMENT Techncal Advsor: r..c. Look, Jr. Verson: /3/
7. GENERAL OJECTIVES a) To understand a one-dmensonal epermental appromaton. b) To understand the art of epermental measurement; n partcular, the judcous use of data. c) To learn a practcal method of measurng a convectve heat transfer coeffcent for a trangular fn usng a statstcal analyss. 7. INTROUCTION Whle fns are used everyday, t s often beleved that the nformaton presented n basc tetbooks s not a real world descrpton; t cannot be observed n the laboratory. Further, the parameters lsted n the back of the book that are used n an analyss appear to be gven by some mysterous means. Ths eperment s desgned to demonstrate that the fn concept s straghtforward and accurate and there s no mystery to thermophyscal property evaluaton. 7.3 THEORY Consder a secton of the one-dmensonal trangular fn as n Fgure. Note that the root s at temperature T(wall), T W, and that the ambent s at a temperature T(nfnty), T. Heat s lost by the fn through the convecton coeffcent, h of the slanted surface. Snce the fn s consdered one-dmensonal (.e., the z dmenson s effectvely nfnte and the permeter can be appromated as z snce l <<z), the Frst Law of Thermodynamcs, descrbng the temperature, T( ), varaton wthn the fn, takes the form d ka d dt d ( ) h p( ) ( T T ) ds. () d 63
Let θ T T, L, L L l, h l k, and assume h and k to be constants. Thus, Eqn. () reduces to d d dθ L θ d L. () The soluton of Eqn. () s obtaned by usng the followng boundary condtons:, θ fnte (3), θ θ. () The soluton s θ θ I I ( ) ( ) (5) where θ T, I s the modfed essel Functon of the frst knd and T W zeroth order and hl L. (6) k L At ths pont, f we know the value of all the parameters (h, l, k, L, A(), p(), T W and T ), we could calculate a temperature profle nsde the fn as a functon of. Of the parameters lsted, h (convecton heat transfer coeffcent) s the most dffcult to measure epermentally. In order to estmate a magntude for h, an epermental method may be used. Ths method s based upon a statstcal analyss of poston,, and temperature, T (or θ) data usng equaton (5). Ths method s referred to as the Method of Least Squares (MLS) [3,], a form of lnear regresson. ased upon the assumpton that the errors n the epermental measurements follow a Gaussan 6
dstrbuton, the MLS produces a unque value for the determned constants. That s, the magntudes of the constants that are determned gve the most probable form of the gven equaton that fts the data. The MLS s based upon the dfferences between the ndependent values of the epermental data and the equaton from the theory. For convenence, let θ θ θ, then from the theory at each poston, θ ( ) I (7) where s a constant to be determned so as to ft that epermental data.. For the sake of clarty, let I ( ) In ths case, the resdual for each data pont s I. (8) θ I. (9) Note that θ and (or I ) are the nput data. Then the sum of the resduals squared s θ θi S I () The necessary condton [] for ths sum to be mnmum wth respect to s that S θ I I () Or, ( ) ( ) θ I θi () I I That s when an ntal guess of s made, s calculated usng Eqn (). A correcton to s calculated by usng (3) 65
66 where the correcton s calculated as follows: If we let, then epandng the essels functon by Taylor seres epanson and truncatng the hgher order terms we get ( ) ( ) ( ) ( )... I ( ) ( ) I. () So, ( ) ( ) I θ ( ) I θ (5) Then, [] { } [] [] [ ] S (6) Lettng S, we get 3 3 (7)
67 Further smplfcaton yelds 3 (8) Ths s a quadratc n and ts soluton s A A C ± (9) where, 3 A () () C () and, ( ) I θ. (3) The teratve procedure used s based on the ntal guess for, producng a, whch then requres a new, then a new and so on. Ths procedure s repeated untl the varance defned by Eqn () s mnmum. ( ) /n Var () where, /n (5) and (6)
7. EXPERIMENTAL SETUP The confguraton and schematc of the epermental setup (- trangular fn) s presented n Fgs. and respectvely. The sdes and the bottom are all well nsulated. Ths setup represents the upper half of a symmetrc nfntely long fn from a thermal pont of vew. Fgure 3 presents the actual epermental fn nformaton. The temperature ndcators are thermocouples embedded wthn the stanless steel fn as ndcated n Fgure. 7.5 ATA ACQUISITION PROCEURE a. Famlarze yourself completely wth the epermental setup. o NOT, under any crcumstances, touch the stanless steel surfaces because the setup wll be turned on before class. Fgure 3 s ncluded so that you may vew the setup wthout takng t apart and subjectng yourself to harm. b. Confrm that the temperature profle s at a steady state (not a functon of tme). o not adjust the heater temperature settng. You should have data to prove that steady state ests. c. Take the data as per the ncluded data sheet, ncludng T. Once agan confrm the repeatablty of these temperatures. 7.6 ATA REUCTION PROCEURE a. Frst plot T (or θ) versus n order to check the data for the approprate shape as well as to check ts one-dmensonal character. 68
b. If the shape s approprate, enter T W, T nf and the data sets (T and X ) nto the data reducton program TRIFIN. Note that Tw s the arthmetc average of the temperature readngs at the fn root. c. Plot Varance, Var, versus the convecton coeffcent, h. Ths may requre several dfferent runs of the program to get a complete curve. d. At the mnmum of ths curve, pck off the value of h. 7.7 POINTS OF INTEREST In your techncal memorandum you should nclude a dscusson of the followng: ) The physcal sgnfcance of the boundary condtons lsted by Equatons (3) and (). ) Eplan clearly how nsulaton on the two sdes smulates a fn whch s nfntely long n the z drecton. 3) Prove that your data was taken whle the fn was at steady state. ) Plots of T versus and Var versus h and the determned the value of the best h. 5) As an append, consder a dfferental element of the length d of the epermental fn and obtan the governng equaton for the fn. Apply the boundary condtons and hence obtan Equaton 5. State the assumptons you made n the process of obtanng Equaton 5. Are they all justfed for the epermental setup used? Eplan. 6) As an append, you should nclude a step-by-step development of the statstcal analyss (Eqns. 9 to ). 69
REFERENCES. Chapman, A.J., Heat Transfer, Fourth ed., Macmllan Publshng Company, New York, Coller Macmllan Publshers, London, 98, pp. 67-89.. Hldebrand, F.., Advanced Calculus for Applcatons, second ed., Prentce-Hall, Inc., Englewood Clffs, New Jersey, 976, pp. -53. 3. an, L.J., and M.Engelhardt, Introducton to Probablty and Mathematcal Statstcs, PWS Publshers, 987, pp. 3-37.. Lpson, C., and N.J. Sheth, Statstcal esgn of Engneerng Eperments, McGraw- Hll ook Company, 973, pp. 37-387. 7
NOMENCLATURE fn surface ot number ( hl /k ) for a trangular fn h surface convecton coeffcent for a trangular fn ( W/m C ) k thermal conductvty ( W/m C ) l one half fn thckness at the root (m) L fn length (m) s slant length (m) T fn temperature ( C ) T W fn root temperature ( C ) T ambent temperature ( C ) coordnate along fn length / L y coordnate along fn heght z coordnate along fn wdth L L / l A( ) cross-sectonal area parallel to the wall and enclosed by the permeter (.e. z /L ) p( ) permeter of the cross sectonal area of the trangular fn (.e. (zy) z, as z>>y from a thermodynamc pont of vew ) ds ((d ) (dy) ) / θ (T - T ) θ T W - T 7
One mensonal Trangular Fn Epermental ata Sheet ate: Twall Name: T TC# 3 5 6 7 8 9 3 5 6 epth from Fn ottom (n) stance from root (n)...5.8.3...6..5.9..5.5.5.5.5 3.75.5.5 3.75.5.5 3.75 Temperature (F) TC# tme 3 5 6 7 8 9 3 5 6 7
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