Annals of the Academ of Romanian Scientists Series on Science and Technolog of nformation SSN 066-856 Volume 1, Number 1/008 43 POLAR CHARACTERSTC OF ENERGETC NTENSTY EMTTED BY AN ANSOTROPC THERMAL SOURCE RREGULARLY SHAPED Constantin DUMACHE 1, Corina GRUESCU, oan NCOARĂ 3 Reumat. Lucrarea preintă o metodă analitică de calcul al distribuţiei spaţiale de intensitate energetică emisă de o sursă termică aniotropă. Problema diferitelor sisteme de localiare a ţintei este calculul iradianţei produse de ţintă într-un punct situat pe o direcţie de observaţie dată. Problema poate fi soluţionată prin utiliarea caracteristicii (,). Abstract. The paper presents an analtical method to compute the spatial distribution of the energetic intensit emitted b an anisotropic thermal source. The essential problem of different target location sstems is the calculus of the irradiance produced b the target in a point on the observation direction. The problem is solvable using the characteristic (,). Ke words: locating sstem, anisotropic source, energetic intensit distribution 1. ntroduction The calculus of an locating sstem begins with the evaluation of the energetic intensit e, emitted b the target along the receiver s direction [1], []. Generall, the energetic intensit calculus is a difficult problem, because the characteristics of the radiation depend on man parameters (nature and temperature of the target, roughness, degree of corrosion, orientation in respect with the receiver). Considering a spatial polar reference sstem (fig.1), the energetic intensit is a function depending on the coordinates (,): and represents the spatial distribution of intensit. e, (1) As the graphical representation of the function, curves obtained for = ct. or = ct., is used. e is difficult, a famil of 1 Dr. Eng., Optica S.A., Timişoara. Ass. Prof., Dr. Eng., Universit Politehnica Timişoara. 3 Prof. Dr. Eng., Universit Politehnica Timişoara (inicoara@upt.ro). Copright Editura Academiei Oamenilor de Știință din România, 008
44 Constantin Dumache, Corina Gruescu, oan Nicoară observation line e (, ) Fig.1. Polar reference sstem attached to the target The curves, and, o o are defined as the polar characteristic of the intensit emitted b the object [3], [4], [5]. Some simplifing hpotheses are taken into account: the object is considered a gra bod, with a total radiance 4 R e T () where is the emission coefficient of the object surface, = 5.76.10-1 Wcm - K -4 Stefan-Boltman constant, T absolute temperature of the object surface. the irradiance at the object surface is a constant sie depending onl on the temperature T and independent on the angles and. As most targets respect these conditions on certain spectral range, the emitted radiation obes Lambert s Law. The energetic intensit along the observation direction is: cos (3) where o is the energetic intensit along the normal direction to the surface, - the angle between the normal to the surface and the observation line. f the irradiance B of a surface is constant, the energetic intensit along the normal direction is: or o o BS (4) BS cos BS a (5) Copright Editura Academiei Oamenilor de Știință din România, 008
Polar Characteristic of Energetic ntensit Emitted b an Anisotropic Thermal Source rregularl Shaped 45 where S a is the apparent surface equal to S cos. The apparent surface is the projection of the surface S on a plane normal to the observation line. The present stud proposes an analtical method to determine the angle as a function of observation direction, for an object cubic shaped.. Analtical calculus of cos The sie of cos results out of a scalar product of two vectors (fig.): R u R u n n (S) O - ur Fig.. Definition of the angle - a unit vector belonging to the observation line (OR), oriented towards the receiver. n an orthogonal reference sstem ur is defined b the relation: u R cos sin i sin sin j cos k, (6) where i, j andk are the vectors of module 1 for the reference ais. - u n - a unit vector, normal to the surface (S) in point O, oriented along the propagation direction of the radiation. un is described b the relation: u n u i u j u k, (7) where u, u and u are the components of the vector u n and depend on surface s (S) orientation. The scalar product is: u R n u cos cos sinu sin sinu cosu. (8) Copright Editura Academiei Oamenilor de Știință din România, 008
46 Constantin Dumache, Corina Gruescu, oan Nicoară The relations (5) and (8) lead to the epression for the energetic intensit:, BS cos sinu sin sinu cosu. (9) f k surfaces of the object are visible from the point R, the polar characteristic is: k, B S cos sinu sin sinu cosu i1 Where S i represents the area of the order i surface, B i the correspondent irradiance. 3. Characteristic (,) for a cube i i i i i, (10) Assume a cube of side a (fig.3), at temperature T, each face radiating the value B of irradiance. All faces, ecept the one contained in the plane, radiate. R u R a Fig.3. Cub shaped object For 090 o and 090 o, u =u =u =1 and the relation (9) becomes:, a Bcos sin sin sin cos. (11) The relation (11) can be written in terms of intensities:, cos sin sin sin cos o o o, (1) where o = o = o =a B is the energetic intensit along the directions O, O and O. For =90 o results the characteristic in the plane : Where cos sin cos o o o o, (13) Copright Editura Academiei Oamenilor de Știință din România, 008
Polar Characteristic of Energetic ntensit Emitted b an Anisotropic Thermal Source rregularl Shaped 47 o arctg. o 4 Relation (13) represents the equation of a circle, passing through the point O, of 1 radius r= o o a B and center C a B, a B, belonging to the line =/4. The function () written in an orthogonal reference sstem becomes (fig. 4): 0. (14) o o (0, ) o ( ) ( )= + 4 o C( o, o ) o O (0, ) o Fig.4. Graphic representation of the function (). Similarl, the characteristics (,) in planes (=/) and (=0) are also circles, described b the equations: o o 0, (15) o o 0, identical to equation (14). The three circles belong to a sphere centered in the origin of the reference sstem. ts epression is: The sie of the sphere radius is: 0. (16) o 1 3 R o o o a B. (17) The center of the sphere situated on the main diagonal of the cube has the following coordinates: o o Copright Editura Academiei Oamenilor de Știință din România, 008
48 Constantin Dumache, Corina Gruescu, oan Nicoară o o o C,C,C. (18) The main diagonal of the cube follows the direction on which the energetic intensit maimum is ( ma 3a B ). n order to demonstrate this assumption, the angles and are determined from the condition (,)=ma. The results are:,, 0 for 4 and meaning the main diagonal direction. 0 4 for arctg, (19) Similarl, for the angles /<<, the characteristic is a sphere passing through point O and having the center situated on the main diagonal. n order to design the characteristic (,), vector radii are traced at lengths proportional to e computed with relation (11), at different values of and. The evolut of radii end provides the characteristic. n figure 5 is shown a characteristic drawn for =45 o and =0 o, 15 o, 30 o, 45 o, 60 o, 75 o and 90 o. As the energetic intensit is known, the total energetic flow, emitted b the cube into a hemisphere can be calculated: e or e, d, / 0 0 sindd 3 cos sin sin sin cos sindd a B 3 a B (0) C( a B, a B) a B Fig. 5. Plot of characteristic (,). Copright Editura Academiei Oamenilor de Știință din România, 008
Polar Characteristic of Energetic ntensit Emitted b an Anisotropic Thermal Source rregularl Shaped 49 The radiance produced b the cube at the surface of a receiver which is placed on the observation line, at the distance d, is:, E o o R, (1) d where ( o, o ) are the angles which define the observation line. Conclusions For the objects on the field, situated at a long distance to the receiver, is difficult to find the characteristic (,), from several reasons: the object surfaces have various shapes and unknown orientation in relation to the receiver the temperature gradient and emission coefficient of these surfaces are also unknown. The solving of the problem is possible b means of a simplified object model. The model is shared in regular geometrical shapes. Onl significant surfaces of the object should be taken into account. Each surface is treated as a gra bod, at known, constant temperature. Using equation (10) results the function (,). Relation (1) allows determination of radiance onto the pupil entrance of the locating sstem, which represents the basic element for the energetic calculus of the sstem. The entire demonstration above refers to a cube shape. An other geometrical shape can be studied in a similar manner in order to get the analtical equation in energetic intensit. Copright Editura Academiei Oamenilor de Știință din România, 008
50 Constantin Dumache, Corina Gruescu, oan Nicoară R E F E R E N C E S [1] M. Baas, Handbook of Optics, vol.,, McGraw Hill nc., NY, 1988. [] E. Hecht, Optics, 3 rd Edition, A. W. Longman, 1998. [3] Constantin Dumache, Contribuţii privind determinarea poiţiei unei surse de radiaţie în infraroşu folosind modulaţia fluului radiant incident, PhD Thesis, Universitatea Politehnica Timişoara, 005. [4] C. Dumache,. Nicoară, Utiliarea transformatelor Fourier la determinarea caracteristicii statice a traductorului optic de poiţie utiliat la măsurarea mărimilor reglate, a XXX-a Sesiune de Comunicări Ştiinţifice cu participare internaţională, Academia Tehnică Militară, Bucureşti, 001. [5] C. Gruescu,. Nicoară, Aparate optice. Analia şi sintea sistemelor optice lenticulare, Editura Politehnica, Timişoara, 004. Copright Editura Academiei Oamenilor de Știință din România, 008