Course Reader for CHEN 7100/7106. Transport Phenomena I

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1 Couse Reade fo CHEN 7100/7106 Tanspot Phenomena I Pof. W. R. Ashust Aubun Univesity Depatment of Chemical Engineeing c 2012 Name:

2 Contents Peface i 0.1 Nomenclatue i 0.2 About this Document i Rules fo Witten Student Wok on Exams Academic Honesty and Syllabus Acknowledgment Foms ii iv 1 Vecto and Tenso Mathematics The Dot Poduct The Coss Poduct The Double Dot Poduct The Opeato Catesian Coodinates Cylindical Coodinates Spheical Coodinates Cuvilinea Coodinates Cylindical:, θ, z Coodinate Convesion Diffeential Opeatos Unit Vecto Convesions Spatial Deivatives of the Unit Cylindical Coodinate Vectos Spheical:, θ, φ Coodinate Convesion Diffeential Opeatos Unit Vecto Convesions Spatial Deivatives of the Unit Spheical Coodinate Vectos The Reynolds Tanspot Theoem 5 4 Equation of Continuity Catesian Coodinates Cylindical Coodinates Spheical Coodinates

3 CONTENTS 2 5 Equation of Motion Catesian Coodinates fo a Newtonian Fluid with constant and µ Cylindical Coodinates fo a Newtonian Fluid with constant and µ Spheical Coodinates fo a Newtonian Fluid with constant and µ Catesian Coodinates in tems of τ Cylindical Coodinates in tems of τ Spheical Coodinates in tems of τ The Dissipation Function (Φ v fo Newtonian Fluids Newton s Law of Viscosity Catesian Coodinates Cylindical Coodinates Spheical Coodinates Fouie s Law of Heat Conduction Catesian Coodinates Cylindical Coodinates Spheical Coodinates Fick s Law of Binay Diffusion Catesian Coodinates Cylindical Coodinates Spheical Coodinates Equation of Enegy Catesian Coodinates Cylindical Coodinates Spheical Coodinates Equation of Continuity fo Component α in tems of j α Catesian Coodinates Cylindical Coodinates Spheical Coodinates Mathematics Abbeviated List of Odinay Diffeential Equations and Thei Solutions Leibniz Integal Rule Hypebolic Functions Taxonomy of Vecto Fields

4 Peface 0.1 Nomenclatue This couse eade follows the notation convention of Bid, Stewat and Lightfoot. Specifically, the following notation is maintained: 0.2 About this Document s = scala (nomal text v = vecto (bold τ = tenso (bold, Geek This couse eade is designed to be the only efeence document fo CHEN 7100/7106. Duing Exams, this is the only mateial you ae allowed to have in you possession. It is to be tuned in along with you exam and will be etuned to you with you gaded exam. Document vesion updated August 21, i

5 Rules fo Witten Student Wok on Exams The following ules ae in effect fo handwitten wok on exams in CHEN 7100/7106. Failue to adhee to these ules may esult in total loss of cedit on an offending poblem, even if that poblem is coectly solved! 1. All wok is to be done on pape supplied by the instucto. 2. Wite on one side only. Any makings on the back of pages will be ignoed. 3. Wite neatly and lage enough to be easily ead. 4. Obseve easonable magins. Do not wite within one inch of any edge of the pape. The next page shows an empty box fo you to use as a guide duing exams. When supeimposed on you exam wok, all you witing should fit within this box. 5. Leave oom at the top fo a staple. No witing should be coveed up by the staple. Whateve is coveed will be ignoed. 6. Put you name on each page. Numbe pages n/n, whee N is the total numbe of pages and n is the cuent page. (Same as HW fomat 7. Wok only one poblem pe page. Even if the solution is one line! 8. Clealy mak you answes. Many poblems in CHEN 7100 will have multiple components to answes. Box each component of you answe, and make sue the boxed answes ae linked with the poblem numbe. 9. Befoe handing in you wok, aange the pages in ode, face up. 10. Staple all wok togethe, with exam pape on top, in the uppe left cone. ii

6 RULES FOR WRITTEN STUDENT WORK ON EXAMS iii

7 Academic Honesty and Syllabus Acknowledgment Foms Syllabus Acknowledgment Fom Syllabus Acknowledgment Fom: By affixing my signatue below, I cetify that I have eceived, ead and undestood the couse syllabus fo CHEN 7100/7106 pepaed by D. Radich fo the Fall semeste Signatue: Date: Pint Name: iv

8 ACADEMIC HONESTY AND SYLLABUS ACKNOWLEDGMENT FORMS v CHEN 7100/7106 Academic Honesty Statement - Exam I Academic Honesty Statement: By affixing my signatue below, I acknowledge I am awae of the Aubun Univesity policy concening academic honesty, plagiaism, and cheating. This policy is defined in the cuent Tige Cub Student Handbook, Code of Laws, Title XII, Student Academic Honesty Code, Chaptes I futhe attest that the wok I submit with exams and quizzes is solely my own and was developed duing the exam o quiz. I have used no notes, mateials, o othe aids except those pemitted by the instucto. Signatue: Date: Pint Name: Instucto Veification: Signatue: Date:

9 ACADEMIC HONESTY AND SYLLABUS ACKNOWLEDGMENT FORMS vi CHEN 7100/7106 Academic Honesty Statement - Exam II Academic Honesty Statement: By affixing my signatue below, I acknowledge I am awae of the Aubun Univesity policy concening academic honesty, plagiaism, and cheating. This policy is defined in the cuent Tige Cub Student Handbook, Code of Laws, Title XII, Student Academic Honesty Code, Chaptes I futhe attest that the wok I submit with exams and quizzes is solely my own and was developed duing the exam o quiz. I have used no notes, mateials, o othe aids except those pemitted by the instucto. Signatue: Date: Pint Name: Instucto Veification: Signatue: Date:

10 ACADEMIC HONESTY AND SYLLABUS ACKNOWLEDGMENT FORMS vii CHEN 7100/7106 Academic Honesty Statement - Final Exam Academic Honesty Statement: By affixing my signatue below, I acknowledge I am awae of the Aubun Univesity policy concening academic honesty, plagiaism, and cheating. This policy is defined in the cuent Tige Cub Student Handbook, Code of Laws, Title XII, Student Academic Honesty Code, Chaptes I futhe attest that the wok I submit with exams and quizzes is solely my own and was developed duing the exam o quiz. I have used no notes, mateials, o othe aids except those pemitted by the instucto. Signatue: Date: Pint Name: Instucto Veification: Signatue: Date:

11 Chapte 1 Vecto and Tenso Mathematics 1.1 The Dot Poduct Fo two vectos, ( a b = δ i a i i j δ j b j = i (δ i δ j a i b j = j k a k b k (1.1 Fo two tensos, σ τ = ( δ i δ j σ ij δ k δ l τ kl = i j i k l = i (δ i δ j δ k δ l σ ij τ kl δ i δ jk δ l σ ij τ kl = j i k l j ( δ i δ l m l σ im τ ml (1.2 Fo a vecto and a tenso, τ v = δ i δ j τ ij δ k v k = i j i k δ i δ jk τ ij v k = j i k δ i ( l τ il v l ( The Coss Poduct Fo two vectos, v w = i δ i v i j δ j w j = i δ i δ j v i w j = j i ɛ ijk δ k v i w j (1.4 j k Fo a vecto and a tenso, τ v = δ i δ j τ ij i j k δ k v k = i δ i ɛ jkl δ l τ ij v k j k l = i δ i δ l j l ɛ jkl τ ij v k (1.5 k 1

12 CHAPTER 1. VECTOR AND TENSOR MATHEMATICS The Double Dot Poduct Fo two tensos, σ : τ = i ( δ i δ j σ ij : δ k δ l τ kl = (δ i δ j : δ k δ l σ ij τ kl j k l i j = δ il δ jk σ ij τ kl = i j m k l (σ mn τ nm (1.6 n 1.4 The Opeato = i δ i (1.7 x i Catesian Coodinates = δ x x δ y y δ z z Cylindical Coodinates Spheical Coodinates = δ δ 1 θ θ δ z z = δ δ 1 θ θ δ 1 φ sin θ φ (1.8 (1.9 (1.10

13 Chapte 2 Cuvilinea Coodinates Tanslation between Catesian (x,y,z and cuvilinea coodinates. 2.1 Cylindical:, θ, z Coodinate Convesion x = cos θ (2.1 y = sin θ (2.2 = x 2 y 2 (2.3 θ = actan(y/x (2.4 z = z ( Diffeential Opeatos x = (cos θ sin θ θ = (sin θ y cos θ θ = z z (2.6 (2.7 ( Unit Vecto Convesions δ = (cos θδ x (sin θδ y (2.9 δ θ = (sin θδ x (cos θδ y (2.10 δ x = (cos θδ (sin θδ θ (2.11 δ y = (sin θδ (cos θδ θ (2.12 δ z = δ z ( Spatial Deivatives of the Unit Cylindical Coodinate Vectos δ = 0 δ θ = 0 δ z = 0 (2.14 3

14 CHAPTER 2. CURVILINEAR COORDINATES 4 θ δ = δ θ z δ = 0 θ δ θ = δ z δ θ = 0 θ δ z = 0 (2.15 z δ z = 0 ( Spheical:, θ, φ Coodinate Convesion x = sin θ cos φ (2.17 y = sin θ sin φ (2.18 z = cos θ (2.19 = x 2 y 2 z 2 (2.20 ( x θ = actan y 2 (2.21 z φ = actan(y/x ( Diffeential Opeatos x = (sin θ cos φ cos θ cos φ θ sin φ sin θ φ = (sin θ sin φ ( cos θ sin φ cos φ y θ sin θ φ = cos θ ( sin θ z θ (2.23 (2.24 ( Unit Vecto Convesions δ = (sin θ cos φδ x (sin θ sin φδ y cos θδ z (2.26 δ θ = (cos θ cos φδ x (cos θ sin φδ y sin θδ z (2.27 δ φ = (sin φδ x (cos φδ y (2.28 δ x = (sin θ cos φδ (cos θ cos φδ θ sin φδ φ (2.29 δ y = (sin θ sin φδ (cos θ sin φδ θ cos φδ φ (2.30 δ z = cos θδ sin θδ θ ( Spatial Deivatives of the Unit Spheical Coodinate Vectos φ δ = δ φ sin θ δ = 0 θ δ = δ θ δ θ = 0 θ δ θ = δ φ δ θ = δ φ cos θ δ φ = 0 (2.32 θ δ φ = 0 (2.33 φ δ φ = δ sin θ δ θ cos θ (2.34

15 Chapte 3 The Reynolds Tanspot Theoem Basic foms: D α dv = Dt V D α dv = Dt V D α dv = Dt V V V V [ α t (αv dv [ Dα α( v dv Dt dv [αv n ds [ α t Note that α can be a scala o vecto function. S 5

16 Chapte 4 Equation of Continuity Basic foms If is constant: [ t v = 0 [ v = Catesian Coodinates 4.2 Cylindical Coodinates t x (v x y (v y z (v z = 0 (4.1 t 1 v = v x x v y y v z z = 0 (4.2 (v 1 v = Spheical Coodinates t 1 2 (2 v 1 sin θ θ (v θ z (v z = 0 (4.3 (v 1 v θ θ v z z = 0 (4.4 θ (v θ sin θ 1 sin θ φ (v φ = 0 (4.5 v = 1 2 (2 v 1 sin θ θ (v θ sin θ 1 v φ sin θ φ = 0 (4.6 6

17 Chapte 5 Equation of Motion Basic foms [ Dv Dt = p µ 2 v g [ Dv = p [ τ g Dt 5.1 Catesian Coodinates fo a Newtonian Fluid with constant and µ ( vx t v v x x x v v x y y v z ( vy t v v y x x v v y y y v z v x z v y z = p [ 2 x µ v x x 2 2 v x y 2 2 v x z 2 = p [ 2 y µ v y x 2 2 v y y 2 2 v y z 2 g x (5.1 g y (5.2 ( vz t v v z x x v v z y y v z v z z = p z [ 2 v z µ x 2 2 v z y 2 2 v z z 2 g z ( Cylindical Coodinates fo a Newtonian Fluid with constant and µ ( v t v v v θ v θ v v z ( vθ t v v θ v θ v θ θ v v θ z z v2 θ = p [ ( 1 µ (v z v v θ = 1 p θ [ ( 1 µ (v θ v θ v θ θ 2 2 v z 2 2 v θ z 2 2 v θ 2 g (5.4 θ 2 v 2 g θ (5.5 θ 7

18 CHAPTER 5. EQUATION OF MOTION 8 ( vz t v v z v θ v z θ v z v z = p [ 1 z z µ ( v z 1 2 v z 2 θ 2 2 v z z 2 g z ( Spheical Coodinates fo a Newtonian Fluid with constant and µ ( v t v v v θ v θ µ v φ v φ v2 θ v2 φ sin θ [ (2 v 1 2 sin θ = p θ ( sin θ v θ 1 2 sin 2 θ 2 v φ 2 g (5.7 ( v θ t v v θ v θ µ [ 1 2 v θ θ v φ v θ sin θ φ v v θ vφ 2 cot θ = 1 p θ ( 2 v θ 1 ( 1 2 θ sin θ θ (v θ sin θ 1 2 v θ 2 sin 2 θ φ 2 2 v 2 θ v φ φ 2 cot θ 2 sin θ g θ (5.8 ( vφ t v µ v φ v θ v φ θ [ ( v φ v φ v φ sin θ 1 2 θ φ v φv v θ v φ cot θ ( 1 sin θ θ (v φ sin θ 5.4 Catesian Coodinates in tems of τ ( vx t v v x x x v v x y y v z ( vy t v v y x x v v y y y v z ( vz t v v z x x v v z y y v z v x z = p x = 1 p sin θ φ 1 2 v φ 2 sin 2 θ φ cot θ 2 sin θ [ x τ xx y τ yx z τ zx 2 sin θ v θ φ v φ g φ (5.9 g x (5.10 v y = p [ z y x τ xy y τ yy z τ zy g y (5.11 v z = p [ z z x τ xz y τ yz z τ zz g z (5.12

19 CHAPTER 5. EQUATION OF MOTION Cylindical Coodinates in tems of τ ( v t v v v θ v θ v v z z v2 θ = p [ 1 (τ 1 θ τ θ z τ z τ θθ g (5.13 ( vθ t v v θ v θ v θ θ v v θ z z v v θ = 1 p θ [ 1 2 (2 τ θ 1 θ τ θθ z τ zθ τ θ τ θ g θ (5.14 ( vz t v v z v θ v z θ v z v z z = p [ 1 z (τ z 1 θ τ θz z τ zz g z ( Spheical Coodinates in tems of τ ( v t v v v θ v θ v φ v sin θ φ v2 θ v2 φ [ 1 2 (2 τ 1 sin θ = p θ (τ θ sin θ 1 sin θ φ τ φ τ θθ τ φφ g (5.16 ( v θ t v v θ v θ v θ θ v φ v θ sin θ φ v v θ vφ 2 cot θ [ 1 3 (3 τ θ 1 sin θ = 1 p θ θ (τ θθ sin θ 1 φ τ φθ sin θ (τ θ τ θ τ φφ cot θ g θ (5.17 ( vφ t v v φ v θ v φ θ v φ v φ sin θ φ v φv v θ v φ cot θ = 1 p sin θ φ [ 1 3 (3 τ φ 1 sin θ θ (τ θφ sin θ 1 φ τ φφ sin θ (τ φ τ φ τ φθ cot θ g φ (5.18

20 CHAPTER 5. EQUATION OF MOTION The Dissipation Function (Φ v fo Newtonian Fluids Catesian Coodinates [ (vx 2 Φ v = 2 x Cylindical Coodinates [ (v 2 Φ v = 2 Spheical Coodinates [ (v 2 ( 1 Φ v = 2 [ sin θ ( 2 vy y ( 1 v θ θ v 2 ( 2 [ vz vy z x v x y ( 2 vz z [ vx z v z x [ [ v z v z v θ θ v 2 ( 1 v φ sin θ ( vφ 1 θ sin θ [ vz y v y z ( vθ [ [ vx x v y y v z z v 2 θ (v 1 φ v v θ cot θ 2 2 [ 1 v sin θ φ v θ φ sin θ [ 1 ( v sin θ [ ( vφ [ 1 v z 2 ( θ v θ z v θ θ v 2 z (5.20 z 2 θ (v θ sin θ 1 sin θ ( vθ 1 v 2 θ v 2 φ (5.21 φ

21 Chapte 6 Newton s Law of Viscosity Basic Fom: [ ( τ = µ v ( v ( 3 2 µ κ ( v δ 6.1 Catesian Coodinates 6.2 Cylindical Coodinates [ τ xx = µ 2 v ( x 2µ x 3 κ ( v (6.1 [ τ yy = µ 2 v ( y 2µ y 3 κ ( v (6.2 [ τ zz = µ 2 v ( z 2µ z 3 κ ( v (6.3 [ vy τ xy = τ yx = µ x v x (6.4 y [ vy τ yz = τ zy = µ z v z (6.5 y [ vz τ zx = τ xz = µ x v x (6.6 z [ τ = µ 2 v ( 2µ 3 κ ( v (6.7 [ ( 1 v θ τ θθ = µ 2 θ v ( 2µ 3 κ ( v (6.8 [ τ zz = µ 2 v ( z 2µ z 3 κ ( v (6.9 [ τ θ = τ θ = µ ( vθ 1 v (6.10 θ [ 1 v z τ θz = τ zθ = µ θ v θ (6.11 z [ vz τ z = τ z = µ v (6.12 z 11

22 CHAPTER 6. NEWTON S LAW OF VISCOSITY Spheical Coodinates [ τ = µ 2 v ( 2µ 3 κ ( v (6.13 [ ( 1 v θ τ θθ = µ 2 θ v ( 2µ 3 κ ( v (6.14 [ ( 1 v φ τ φφ = µ 2 sin θ φ v ( v θ cot θ 2µ 3 κ ( v (6.15 [ τ θ = τ θ = µ ( vθ 1 v (6.16 θ [ sin θ ( vφ τ θφ = τ φθ = µ 1 v θ (6.17 θ sin θ sin θ φ [ 1 v τ φ = τ φ = µ sin θ φ ( vφ (6.18

23 Chapte 7 Fouie s Law of Heat Conduction Basic Fom: [q = k T 7.1 Catesian Coodinates q x = k T x q y = k T y q z = k T z (7.1 (7.2 ( Cylindical Coodinates q = k T q θ = k T θ q z = k T z (7.4 (7.5 ( Spheical Coodinates q = k T q θ = k T θ 1 T q φ = k sin θ φ (7.7 (7.8 (7.9 13

24 Chapte 8 Fick s Law of Binay Diffusion Basic Fom: [j A = D AB ω A 8.1 Catesian Coodinates j Ax = D AB ω A x j Ay = D AB ω A y j Az = D AB ω A z (8.1 (8.2 ( Cylindical Coodinates ω A j A = D AB j Aθ = D ω A AB θ j Az = D AB ω A z (8.4 (8.5 ( Spheical Coodinates ω A j A = D AB j Aθ = D ω A AB θ j Aφ = sin θ D AB ω A φ (8.7 (8.8 (8.9 14

25 Chapte 9 Equation of Enegy Basic Foms [ Ĉp DT Dt = q [ Ĉp 9.1 Catesian Coodinates Ĉp ( ln Dp ln T p DT Dt = k 2 T µφ v Dt τ : v ( [ T t v T x x v T y y v T qx z = z x q y y q ( z ln Dp (τ : v (9.1 z ln T p Dt Ĉp ( [ T t v T x x v T y y v T 2 T z = k z x 2 2 T y 2 2 T z 2 µφ v ( Cylindical Coodinates ( T Ĉp t v T v [ θ T θ v T 1 z = z (q 1 q θ θ q z z ( ln ln T Ĉp ( T t v T v [ θ T θ v T 1 z = k z 9.3 Spheical Coodinates ( T Ĉp t v T v θ = T θ [ 1 2 (2 q 1 sin θ v φ T sin θ φ θ (q θ sin θ 1 sin θ p Dp (τ : v (9.3 Dt ( T 1 2 T 2 θ 2 2 T z 2 µφ v (9.4 q φ φ ( ln Dp (τ : v (9.5 ln T p Dt 15

26 CHAPTER 9. EQUATION OF ENERGY 16 Ĉp ( T t v T v θ T = k θ v φ T sin θ φ [ ( T 1 2 sin θ ( sin θ T θ θ 1 2 T 2 sin 2 θ φ 2 µφ v (9.6

27 Chapte 10 Equation of Continuity fo Component α in tems of j α Basic Foms [ Dω α Dt 10.1 Catesian Coodinates ( ωα t ω α v x x v ω α y y v z = ( j α α [ ω α jαx = z x j αy y j αz α (10.1 z 10.2 Cylindical Coodinates ( ωα t ω α v v θ ω α θ v z ω α z [ 1 = (j α 1 j αθ θ j αz α (10.2 z 10.3 Spheical Coodinates ( ωα ω α v t v θ ω α θ = v φ sin θ ω α φ [ 1 2 (2 j α 1 sin θ θ (j αθ sin θ 1 sin θ j αφ α (10.3 φ 17

28 Chapte 11 Mathematics 11.1 Abbeviated List of Odinay Diffeential Equations and Thei Solutions 1 x 2 1 x 2 ( d dx d dx Equation dy dx = f(x g(y dy f(xy = g(x dx e ( Solution gdy = ( fdx fdx C 1 (11.1 e ( fdx gdx C 1 (11.2 d 2 y dx 2 a2 y = 0 C 1 cos(ax C 2 sin(ax (11.3 d 2 y dx 2 a2 y = 0 C 1 e ax C 2 e ax (11.4 x 2 dy a 2 y = 0 dx ( x 2 dy a 2 y = 0 dx 11.2 Leibniz Integal Rule C 1 x cos(ax C 2 sin(ax x (11.5 C 1 x eax C 2 x e ax (11.6 Let then di(t dt = d dt β(t α(t I(t = f(x, tdx = β(t α(t β(t α(t f(x, tdx (11.7 f(x, tdx f(β, tdβ f(α, tdα t dt dt ( Hypebolic Functions cosh(x = ex e x 2 sinh(x = ex e x 2 (11.9 (

29 CHAPTER 11. MATHEMATICS Taxonomy of Vecto Fields Assume a vecto a. tanh(x = sinh(x cosh(x = ex e x e x e x (11.11 Name o Designation Defining o Chaacteistic Popety Solenoidal a = 0 Iotational a = 0 Laplacian a = a = 0 Beltami a ( a = 0 Consevative a = φ, whee φ is a scala field.