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# Engineering Unit Operations for Biomass Conversion WPS 760

NCS

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This 24 page Class Notes was uploaded by Emiliano Gutkowski on Thursday October 15, 2015. The Class Notes belongs to WPS 760 at North Carolina State University taught by Orlando Rojas in Fall. Since its upload, it has received 15 views. For similar materials see /class/223793/wps-760-north-carolina-state-university in Paper Science And Engineering at North Carolina State University.

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Date Created: 10/15/15

Summary of Some Math Operations Relevant to WPS 760 Scalars vectors and tensors Scalars One number Mass volume density Vectors Numberdirection or several numbers Position force velocity Tensors Dependence of vectors upon vectors motion Many numbers simplest case a table Deformation stress Vector a quantity with direction Examples Position velocity acceleration Notation gt 739 Z r Coordinate representation a column r1 Vzvxivyjvzk r 2 r2 rlel r2e2 r3e3 or notations r3 Vu1VJwk Length 2 2 2 lrl wlrl r2 r3 Vectors and transformations Is any set of three numbers a vector If you don t change your coordinate systemyes If you change your coordinate systemno Example Simple rotation l rl r1os rZ sin l rZ 41 sin rzcos l 72 General romu39on Euler angles Conclusion anul 39 Vector operations Sums and di 39erences Element by element cum uuucl Mlltiplicztin Four as as 1 Productofvectorandscalar Ml la laz la 3 2 Scalar dot product oftwo vectors a b ab cos alll azbz 513123 Other notation a 3 Vector cross product of tWo vectors 11L aabLb ab absin In coordinate representation 61 62 63 at b a1 a2 a3 bl 52 53 What happens if We change signs Other notation a X b 4 Tensor product oftWo vectors Rules of vector operations All standard rules for algebra are valid but one exception vector product is anti commutative nub ill all Some rules for products 1 If a L b their scalar product is zero ab 0 2 If a b their vector product is zero 11 0 3 Cyclic rule duel bvlcvall 4 bac minus cab rule for any three vectors a c ltacgticltmgt Description of motion Lagrange coordinates I sit on the bank and look as the river ows r gt 39 Use il for hydrodynamics Euler coordinates I sit on a raft owing With the river Ar gt Ax Use il for elasticity theory Examples of motion Tr anslatjon sra AsAr Stretch S1 23971 52 72 33 r3 Rotation s1rlcos rzsin s2 r1 sm r2 cos S3 2 7393 Note the difference between this equation and coordinate change equation 0 coordinate change the vector is stationary the coordinate system moves 0 rotation the vector moves the coordinate system is stationary Slam Tensors and matrices Description of motion tensors We see that motion is a transformation of a vector How to describe this uniformly For scalar variables we could write x gtyx Ayy39Ax We want to write for vectors something like r gt sr As 21 Ar What is 121 It is not a scalar x is parallel to x It is not a vector a x is a scalar and a x is normal to x It is a new object It is called tensor a tensor of second rank Tensors of the second rank correspond to linear transformations vector gt vector Notation 721 Elementary tensor operations Multiplication of tensor and vector by de nition Multiplication of tensor and scalar isotropic stretch mammals Addition and subtraction Ai3xzzlxil x Multiplication inner product partial convolution let then 3121quot t Note the order Rules of tensor operations Unit tensor 5 no motion r r 521 21E 21 All usual algebraic rules are valid in tensor algebra but commutativity Coordinate representation After coordinate system is chosen vectors can be represented by columns in 3D How can we represent tensors The linear transformation rules are 51 mil a12r2 aura 52 a21r1 azzrz 23 S3 a31r1 aazrz 33 We need 3 X 3 9 numbers Matrix all all 13 A all all 6123 a31 a a Language simpli cation Matrix of rotation instead of matrix representing the tensor of rotation Coordinate change if we change coordinate system tensors do not change but their matrices do Tensor equation s Tr Matrix equation sTr Examples of matrices Identity matrix no motion or deformation 1 0 0 E 0 1 0 0 0 Stretch A 0 A 0 1 0 0 1 Rotation cos sin 0 A sin cos 0 0 0 1 Slant 11 0 0 A 0 0 0 0 0 1 Matrix operations Coordinate representation of tensor operations Multiplication of matrix and vector by de nition 1 1 sfr s Za r Example Multiplication of matrix and scalar BxiA b11224 Example 2 3 4 10 15 20 5 0 1 2 0 5 10 0 4 5 0 20 25 Addition and subtraction CAiB cljaljblj Example 2 3 4 1 2 3 3 5 7 0 1 2 4 5 6 4 6 8 0 4 5 7 8 9 7 12 14 Transposing re ection with respect to diagonal A gtAT fl61 Multiplication row by column Note the order C AB cl 2611ka alk bkj 16 Example stretch and rotation First stretch then rotate cos sin 0 l 0 0 lcos sin 0 sin cos 0 0 1 0 lsin cos 0 0 0 1 0 0 1 0 0 1 First rotate then stretch it 0 0 cos sin 0 leos lsin 0 0 l 0 sin cos 0 sin cos 0 0 0 l K 0 0 l 0 0 l Scalars are tensors of zero rank vectors are tensor of rst rank The general rule 2 Tensor 2 The rank of a tensor is the number of its subscripts Not so elementary tensor operations Tensor product just glue them together a b In matrix notation cl all b J 611 b1 611 b2 611 b3 C 6sz1 czle a b a3b1 alibi alibi The general case Tensor of kth rank 9 Tensor of lth rank Tensor of klth rank Operator v and its applications Differentiating a scalar Let aw m be a scalar How ean we differentiate m Three numbers They are components ofavecmr Bx grada E 3y E a This is gradient ane difference x xy5yzamp xyzi 3f x g Vector language agtr Jame r av grad Example 2D functl ons Example Electn eld 7 potential Ezgraday Directinnal derivative etus dAfferenuate m the duecnon n 1 hm n grad as M When s largesw as Answer when n grad garadw Vector v m x n mu n y symbah vector grada Va Simple properties oi V V wV Vw V lIW VUI Differentiation of a vector Question How can we dilfereniiele avector a An wer Multiply it by V Three kinds ofproducts b by a 1 Dotproductub 2 Crossproducluxb 3 Tensorproductu b Divergence De nition Dot product of V and u is called dzvergence ofthe Vector 11 6a diviiVu aax w 6x 8 ay 32 This is ascalar Meaning oi divergence 1D problem incompressible liquid in apipe p1 In ux va 5A In ux per unit volume In ux 6vx 11m river 6A ampc 6x 3D problem sum all contributions In ux per unit volume is 6v 6Vy 6v Z le v 6x 6y 62 Example Continuity equation in hydrodynamics Consider a small volume 5V The total mass in this volume p5V Change of this mass Direct estimation 16th 6t In ux estimation div pv639V6t Result 6 d1v 0 at IN TWO ways of multiplication by V Divergence 6 divaVaiaa 6x 6x 6y 62 z This is a scalar Directional derivative 6 6 a V ax a 6 y aZ 6x 6y 62 This is a scalar operator Examples of directinnal derivative Dxfferenuauon ofa scalar a V a grad d Dxfferenuauon of a vector a Vl7 Thxs rs a vecmr Simple properties of v VaVl7 V 6 W MW 5 Curl De nition cross product of V and a is called m1 of the vector a El 22 23 curlaVgtlta a i a 8x at 82 ax dy a fa dl curla S Simple properties of v Vxabanbe Vx aV xa an Rules about zeros curl grad V X V 0 diV curlaVVgtlta 0 Tensor derivative Question What is V 9 a Answer Tensor with coordinates 6a 6x J i Laplacian Question What is V2 Answer A scalar operator De nition The scalar operator 62 62 62 62 VZA Bxl xl 6x2 Byz 622 is called Laplacian Example Electrostatics diV E 47273 E grad Therefore V2 47rp Other applications Hydrodynamics of ideal liquid V2 0 Elasticity theory V27 0 20 Curvilinear orthogonal coordinates Why curvilinear coordinates Because the world is not rectangular How can you calculate Diffusion from a cylinder Flow aroun a Sphele Field ofa charged ellipsoid Many omel exciting things quot 39 39 llmln39 al a 39 39 39 439 Examples of curvilinear coordinates Circular lindrical coordinates x pcos y pslud 22 Spherical coordinates xrsint9cos yrsint95in zrcost9 Divergence Cylindrical coordinates Spherical coordinates Laplaman Cylmdncal coordmates Spherical coordmates curl a Cylmdncal coordmates Spherical coordmates Homework Gradient and divergence Consider the function 1 Calculate g rad a 2 Prove that am gmd p Vza and calculate ms quanmy

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