3: Mathematics Review

Transcription

1 3: Mathematics Review B. Rouben McMaster University Course EP 4D03/6D03 Nuclear Reactor Analysis (Reactor Physics) 015 Sept.-Dec. 015 September 1

2 Review of: Table of Contents Co-ordinate systems (Cartesian, Cylindrical, Spherical) Differential lengths, surface areas, volumes Integrating over solid angle Gradient, Divergence, Laplacian Leakage out of a volume and Gauss theorem 015 September

3 Co-Ordinate Systems A point in 3-dimensional space can be identified in a number of different co-ordinate systems. The ones which will be most useful to us are: Cartesian: here the co-ordinates are the familiar x, y, z Cylindrical: here the co-ordinates are r,, z (see diagram on next slide) Spherical: here the co-ordinates are r,, (or ) (see diagram on nd next slide). Sometimes it is convenient to use =cos as variable instead of. 015 September 3

4 Cylindrical Co-Ordinates z is the height of the point above the x-y plane; r is the distance from the origin to the point s projection on the x- y plane; (I sometimes use ) is the angle of rotation from the x- axis to the point s projection on the x- y plane 015 September 4

5 Polar Co-Ordinate System for Solid Angle 015 September 5

6 Spherical Co-Ordinates r is the distance of the point from the origin; is the polar angle ( latitude of point from z-axis, 0 to 180 deg); (I use ) is the angle of rotation from the x-axis to the point s projection on the x-y plane (similar to in cylindrical coordinates) 015 September 6

7 Differential Lengths (Distances) Differential lengths ds corresponding to differentials in the various co-ordinates are needed for use in 1-d integrals in the various systems. They can be easily evaluated as the distances corresponding to changes in the various directions. In Cartesian co-ordinates, ds = dx, dy, dz In cylindrical co-ordinates, ds = dr, rd, dz (Note: I sometimes use instead of ) In spherical co-ordinates, ds= dr, rd, rsin d (Note: I sometimes use instead of ). 015 September 7

8 Differential Surface Areas Differential surface areas perpendicular to the various directions (see some in next slide) are: In Cartesian co-ordinates, perpendicular to directions along changes in x, y, z: dydz, dzdx, dxdy In cylindrical co-ordinates, perpendicular to directions along changes in r,, z: rddz, drdz, rdrd In spherical co-ordinates, perpendicular to directions along changes in r,, (or ): r sindd, rsindrd, rdrd 015 September 8

9 Differential Surface Areas 015 September 9

10 Differential Volumes Differential volumes dv are needed in 3-d integrals in the various co-ordinate systems. They are (see next slide for drawings): In Cartesian co-ordinates, dv = dxdydz In cylindrical co-ordinates, dv= rdrddz (Note: I sometimes use instead of ) In spherical co-ordinates, dv= rsindrddr = r sin d drd = (-) r drdd (Note: I sometimes use instead of ) 015 September 10

11 Differential Volumes 015 September 11

12 Spherical Co-Ordinates: Instead of Note: In spherical co-ordinates, it is often very useful, especially when evaluating integrals, to use cos as the variable, instead of the ( latitude ) variable. This is so because d = - sind, so that the more complicated quantity on the right can easily be replaced by the simpler differential d. The range = 0 to is replaced by the range = -1 to +1 (when expressed in this order instead of +1 to -1, it removes the minus sign). 015 September 1

13 Integrating Over Solid Angle Integrating over a solid angle is essentially equivalent to integrating over the surface of a unit sphere. We must be aware of the range of solid angle (and therefore, of and ) that we need to consider. f ( ) d rangeof rangeof rangeof rangeof rangeof f f sindd dd [ We must not forget the sin!!] 015 September 13

14 Integrating Over All Solid Angles In this case d d sind d d 015 September 14

15 Neutrons Crossing Unit Area of a Plane For neutrons crossing the plane from below, is between 0 and / ( = 0 to 1). For neutrons crossing from above to below, is between / and ( = -1 to 0). In this case the number will be negative. 015 September 15

16 Derivative of a Function The derivative of a function f of a single variable (x) is the rate of change of f with respect to x, The change f in the value of f when x changes by a small amount x can then be approximated by: x df f * x dx df dx 015 September 16

17 Gradient If we now consider a function f of 3 variables (x, y, z), then the value of f will (in general) change if there is a change in any of the 3 variables. We can define the directional rates of change with respect to each variable separately: And the change in the value of f when there are small changes in all the variables can be approximated by 015 September 17 z f y f x f,, z z f y y f x x f f

18 Gradient (cont d) The gradient of f can be thought of as a vector. It of f where, in can different the be denoted ways, f f f f f f,, i j k x y z x y z in the nd notation, i, j, and k are x, y, and z f, and directions it can be written respectively. in the a couple unit vectors 015 September 18

19 Gradient (cont d) If we write the increments in the independent variables of f also as a vector in the general direction R, R, x, y z xi yj zk then the increment in f can be written as a dot product of the two vectors: f f x x f y y f z z f R 015 September 19

20 Physical Meaning of Gradient If we imagine the increment in the independent variables to be a unit change in the direction R, i. e., R 1 then we can see that the increment in f, f is simply the projection of f in the direction (see next slide). This projection will be largest if the direction of R is the same as that of the vector f. This tells us that f is the rate of change of f in the direction in which it increases most rapidly! 015 September 0 f R R

21 The Increment in f as a Projection AB f R Projection of f on R B ( R is defined as a unit vector ) R A f 015 September 1

22 The Gradient Operator In the previous slides, we defined the gradient of a function f. We can think of that gradient as the action of a gradient operator on the function f. In this interpretation, the gradient operator in Cartesian co-ordinates is written as We can see that the gradient operator is a vector operator. x, y, z 015 September

23 Gradient Operator in Other Co-Ordinates From our knowledge of the differential lengths corresponding to the various variables (see prior slide), we can write the components of the gradient operator in other co-ordinates: In In cylindrica l spherical co co ordinates, ordinates, r r, 1,, r z 1, r r 1 sin 015 September 3

24 The Divergence of a Vector In the previous slides, we saw that the gradient operator could be defined as a vector operator. We saw the action of the gradient operator on a scalar function f. But we can also define the action of the gradient operator can also operate on a vector function,. This action is defined as the divergence of the function, and is a dot product, written as. F In Cartesian co-ordinates, the divergence is: ( div F or) F Fx x Fy y Fz z 015 September 4 F

25 Physical Meaning of Divergence The physical meaning of the divergence is that it is the leakage of the vector function out of a an infinitesimal volume around the point where the divergence is calculated, divided by the infinitesimal volume. See proof of this divergence theorem in next slide. Note: The proof is given in Cartesian coordinates, but holds for any shape of the infinitesimal volume. 015 September 5

26 Divergence Theorem for Vector (Current) J x x Infinitesimal volume; sides perpendicular to paper are dy and dz x+dx J x dx Net leakage in x direction Leakage in x direction Volume J x x a similar exp ression for leakage in Leakage out of Volume Volume J xx dx J xx x dx J x* 015 September 6 J x J x x x dx* dy * dz J y y * dy * dz dy * dz y and z. J z J z

27 Divergence in Other Co-Ordinates If we write the vector function F in terms of its components in other co-ordinates, the divergence operator becomes: In cylindrica l co ordinates, 1 1 F Fz div F rfr r r r z In spherical co ordinates, 1 1 div F r r r r sin r F sin F 1 F r sin 015 September 7

28 Leakage out of a Finite Volume Subdivide a finite volume into infinitesimal subvolumes; apply the divergence theorem in each subvolume, and add all (i.e., integrate). The internal leakages (across internal surfaces out of one subvolume and into a neighbouring subvolume) obviously cancel out, leaving only the leakage out of the external surface. Therefore the net leakage out of the finite volume = the volume integral of the divergence of the current. 015 September 8

29 Leakage out of a Finite Volume We have just proved Gauss s famous Theorem: Leakage out of Volume V Surface int egral J dsˆ Volume int egral External Surface J dv V 015 September 9

30 Laplacian The Laplacian of a function f, denoted, is defined as the divergence of the gradient of f : f f The Laplacian of the flux is useful in reactor physics, because of the Divergence Theorem and an approximation (Fick s Law), J E E, which says that the net current is proportional to the negative of the gradient of the flux, i.e., that the net neutron current flows from areas of high flux to areas of low flux. 015 September 30 f

31 Laplacian The Fick s Law approximation allows us to simplify to D, where D is the constant of proportionality between the flux gradient and the current. J In a homogeneous reactor, i.e., if D is uniform in space, D becomes D, and in this case we need the formulation of the Laplacian of the flux in the various co-ordinate systems. These are given in their most general form in the next slide. 015 September 31

32 Laplacian of the Flux 015 September 3 sin 1 sin sin 1 1 : 1 1 : : r f r r f r r r ordinates co spherical In z r r r r r ordinates co l cylindrica In z y x ordinates co Cartesian In

33 END 015 September 33