Kiran Joy, International Journal of Advanced Engineering Technology E-ISSN

Size: px

Start display at page:

Download "Kiran Joy, International Journal of Advanced Engineering Technology E-ISSN"

Grant Lester
5 years ago
Views:

Kran oy, nternatonal ournal of Advanced Engneerng Technology E-SS 0976-3945 nt Adv Engg Tech/Vol.

1 Kran oy, nternatonal ournal of Advanced Engneerng Technology E-SS nt Adv Engg Tech/Vol. V/ssue /Aprl-une,04/9-95 Research Paper DETERMATO O RADATVE VEW ACTOR WTOUT COSDERG TE SADOWG EECT Kran oy Address for Correspondence Department of mechancal engneerng, Vmal yoth Engneerng College, Kannur, nda ABSTRACT Vew factor refers to the fracton of heat energy leavng a radatng surface whch s ntercepted on another surface. Ths vew factor can be determned wth or wthout consderng the shadow effect. Shadow effect refers to the reducton n net radatve transfer between two surfaces, due to the obstructon of radaton by a thrd surface present n between them. Ths paper presents the detals of a numercal ntegraton technque for evaluatng vew factors wthout consderng the shadow effect and a comparson of the results obed wth analytcal results. A ORTRA code was developed for vew factor computaton usng double area ntegraton method. KEWORDS Cranes, 3 axs accelerometer, sensor based, wreless transmsson, rado frequency (R Transcever).. TRODUCTO n engneerng applcatons where radaton s the prmary mode of heat transfer, the accurate determnaton of net radatve transfer between surfaces s a necessty. n complcated geometres where shadow effect comes nto pcture, vew factors can be determned n such a way that t ncorporates the reducton n radaton heat transfer due to shadowng. n the absence of a partcpatng medum, radaton heat exchange between two surfaces depends upon ther temperatures, thermooptcal propertes lke emssvty (spectral or drectonal) and on the relatve locatons and orentatons between these surfaces. Apart from these, the shadowng effect of other surfaces (.e. other surfaces obstructng the drect vew between two surfaces) must also be consdered for computng radatve heat exchange and hence thermal response of the surfaces. n the absence of an absorbng medum, radatve heat exchange between surfaces s a functon of the surface condtons and the optcal vew that each surface has of the others [3]. The net radant heat transferred n the wavelength range, λ to λ+ dλ, between two dffuse surfaces that form an enclosure and whose respectve emssve powers are const over the surface, s gven by ( e e ) Q ε ε + + ε A ε A A whereε s the surface emssvty and e s the emssve power n a gven spectral range. f the surfaces are sothermal and gray e can be replaced 4 by σt to determne the net total radant heat transfer. The term s commonly referred to as the vew factor, shape factor or confguraton factor whch s defned as cosθ cosθ A A A π S and represents the fracton of radant energy leavng surface that s ntercepted by surface. depends on surface only. Also relatonshp. The sum of A A satsfes the recprocty s unty for a surface. The exstng methods of evaluatng vew factors, double area ntegraton and contour double ntegraton, are used. n both methods surface s dvded nto number of fnte dfferental areas upon whch the ntegraton s carred out. n contour double ntegraton method, the double ntegraton over the area of nvolved surfaces s converted nto contour ntegraton by applyng Stokes theorem [].Durng ths process the ntegrand becomes a functon of logarthmc terms. g.vew factor computaton between fnte surfaces The advantage clamed wth the contour method over the area ntegraton method s that the ntegrand s a smple functon of geometrcal data whch results n fast and accurate computaton. The present work descrbes about the double area ntegraton method to determne the vew factors between radatng surfaces. n the present work, vew factors are calculated for the followng two geometres of practcal nterest. Two dentcal parallel square surfaces drectly opposed to each other. Two dentcal square surfaces perpendcular to each other wth common edge. 3. VEW ACTOR ESTMATO WT DOUBLE AREA TEGRATO METOD A computer code s developed n ORTRA language based on the present method for the computaton process. The dmensons of the square geometry as shown n g, taken for computatons are,,, L, where L s the vertcal dsce between two square surfaces. 3.Methodology 3.. Descretzaton of the source and target surfaces. Descretzaton s defned as dvdng the source/target surfaces nto a number of small dfferental sub surfaces called as elements of equal/unequal dmensons. Each element s surrounded by four neghborng nodes. Dsce between each node wll be same for each element. Dsce between each node represents the sze of an element, say dx and dy. Descretzaton s done for both source and target surfaces. Descretzaton of the surface s mport n order to reduce the error n the computed value of

Kran oy, nternatonal ournal of Advanced Engneerng Technology E-SS 0976-3945 vew factor. Error decreases when the descretzaton s large.

2 Kran oy, nternatonal ournal of Advanced Engneerng Technology E-SS vew factor. Error decreases when the descretzaton s large. Descretzaton s defned as dvdng the source/target surfaces nto a number of small dfferental sub surfaces called as elements of equal/unequal dmensons. Each element s surrounded by four neghborng nodes. Dsce between each node wll be same for each element. Dsce between each node represents the sze of an element, say dx and dy. Descretzaton s done for both source and target surfaces. Descretzaton of the surface s mport n order to reduce the error n the computed value of vew factor. Error decreases when the descretzaton s large. 3.. Element connectvty and drecton cosnes of normal to actve area. Element connectvty s defned as the nformaton about a gven element and ts surroundng nodes. Element s bascally a plane sub-surface havng neghborng nodes. odes surroundng an element are numbered n counter-clockwse drecton Computaton of drecton cosnes element normals or computng the drecton cosnes of normals to the elements two vector lnes (say A and B ) connectng opposte nodes on corners are consdered as shown n fgure. The vector or cross products of these two vectors are found. The drecton cosnes are computed by usng followng relatons Elem Elem Elem (A B) (A B) Z(A B) l, m, n A B A B A B 3..4 Computaton of element centrods. An element s centrod s defned as the average of the co-ordnates (say x, y, z) of the all surroundng nodes. n the present case, each element s surrounded by four neghborng nodes. Element centrod of an element s computed by usng the formula E neghb neghb x y z x Ey Ez neghb,, neghb neghb neghb 3..5 Computaton of dsce between source and target elements, S. n order to compute the dsce between source and target elements, a ray or vector lne (say A or B ) s drawn from source elements to target elements connectng ther centrods. Then the value or magntude of ether of these two vector lnes gves the dsce between source and target elements. The dsce S can be computed by usng the formula S s ( _ T _ S) + ( _ T _ S) + ( Z_ T Z_ S) 3..6 Computaton of vew factor from source to target surface. Vew factor s calculated from each element n source surface to every element n target surface usng the double area ntegraton method. The double area ntegraton formula to determne vew factor between two surfaces s gven by cosθ cosθ A π S A A where A and A are fnte areas of surface and. n the present problem ntegraton of s replaced by summaton of. s s t cos θ cos θ π S 3..7 Valdaton of the code Valdaton of the code s necessary n order to fnd the percentage of error exstng between the computed vew factor value and the vew factor obed from the analytcal soluton. computedv analytcal V % error *00 analytcal V or all cases number elements per edge () s consdered to be 00, e; RESULTS AD DSCUSSO Case Two dentcal parallel square surfaces drectly opposed to each other. A computer code s developed n ORTRA language based on the presentmethod for the computaton process.the dmensons of the square geometryas shown n g, taken for computatons are,,, L, where L s the vertcal dsce between two square surfaces. g. Two dentcal parallel square surfaces drectly opposed to each other The vew factor from the program s compared wth the value from the analytcal expressons gven n reference [4] and are shown n Table.The comparson s shown n g.3.a graph, shown n g 4 s plotted for vew factor aganst /L rato. The analytcal value of vew factor for the above confguraton can be determned from followng equaton []. π + (+ ln + + )(+ + ) + / nt Adv Engg Tech/Vol. V/ssue /Aprl-une,04/9-95 Where, and L L

Kran oy, nternatonal ournal of Advanced Engneerng Technology E-SS 0976-3945 Table g.3 Comparson of vew factor for parallel square surfaces g.

of elements per edge Case : Two dentcal square surfaces geometry as shown n g.7, taken for computatons perpendcular to each other wth common edge.

3 Kran oy, nternatonal ournal of Advanced Engneerng Technology E-SS Table g.3 Comparson of vew factor for parallel square surfaces g.4 Comparson of analytcal vew factors g.5 Error comparson for parallel surfaces g.6 Comparson of computng tme for dfferent no.of elements per edge Case : Two dentcal square surfaces geometry as shown n g.7, taken for computatons perpendcular to each other wth common edge. are,,, L0, havng a common edge on - A computer code s developed n ORTRA axs. language based on the present method for the computaton process. The dmensons of the square nt Adv Engg Tech/Vol. V/ssue /Aprl-une,04/9-95

Kran oy, nternatonal ournal of Advanced Engneerng Technology E-SS 0976-3945 g.

4 Kran oy, nternatonal ournal of Advanced Engneerng Technology E-SS g.6 Two dentcal square surfaces perpendcular to each other wth common edge W + W (+ W )(+ + ln πw 4 + W + ( + ) W ( (+ W Z / The vew factor from the program s compared wth the value from the analytcal expressons gven n reference [4] and are shown n Table.The analytcal value of vew factor for the above confguraton can be determned from followng equaton []. / W ) / ( + W ) W + W + ) (+ + W ) )( W + ) (+ )( + W ) W / and Table g.8 Comparson of vew factor for parallel square surfaces g.9 Comparson of analytcal vew factors nt Adv Engg Tech/Vol. V/ssue /Aprl-une,04/9-95 g.0 Error comparson for parallel surfaces

5 Kran oy, nternatonal ournal of Advanced Engneerng Technology E-SS g. Comparson of computng tme for dfferent no.of elements per edge COCLUSO The work explans the use of a new numercal scheme for the computaton of vew factor wthout consderng shadow effect. n the case of square geometry the result s found to vary from the analytcal value by only percentage n Case and 0.676percentage n Case. The comparson of vew factor s done by plottng a graph and s found to be n agreement. Another graph drawn wth vew factor along -axs and /L rato or Z/ rato along -axs can be used to fnd out the vew factor wthout shadow effect of a smlar geometry wth any dmensons. The present method can also be used for calculatng vew factor wth shadowng effect created by multple obstructons. The studes carred out were for smpler geometres where structured grds could easly be made. owever practcal problems are more complex whch are not easly amenable to structured grds and hence tme consumng. REERECES. Segel, R, owell.r, Thermal Radaton eat Transfer, McGraw-ll, 97.. rank P. ncropera, undamentals of eat and Mass Transfer, 3. A.. Emery, 0. ohansson, M. Lobo, A. Abrous, A Comparatve Study of Methods for Computng the Dffuse Radaton Vew factors for Complex Structures, ournal of eat Transfer, MA 99, Vol. 3/43 4. V. Rammohan Rao and V.M.K.Sastr, Effcent evaluaton of dffuse vew factors for radaton, ournal of eat and Mass Transfer, Vol.39, o.6, pp.8-86, 996 nt Adv Engg Tech/Vol. V/ssue /Aprl-une,04/9-95

2x x l. Module 3: Element Properties Lecture 4: Lagrange and Serendipity Elements

2x x l. Module 3: Element Properties Lecture 4: Lagrange and Serendipity Elements Module 3: Element Propertes Lecture : Lagrange and Serendpty Elements 5 In last lecture note, the nterpolaton functons are derved on the bass of assumed polynomal from Pascal s trangle for the fled varable.