3230 ChemE Fluid Mechanics
Spring Semester, 2009
What is a Chemical Engineer ?
a) An Engineer who manufactures chemicals. b) A Chemist who works in a factory. c) An Engineer who knows Chemistry. d) A highly-paid plumber.
Why Study Fluid Mechanics?
1. Most chemical plants convert some fluid (liquid, gas) raw material into a more desired (usually more expensive) product. Fluid mechanics provides a framework for designing and sizing equipment for (a) transporting; (b) transforming; and (c ) storing the raw materials, intermediates, and final products. Example: Chemical Plant for Transforming ethylene and benzene into ethylbenzene -----> styrene --------> polystyrene
C2H5
MCM-22 / EBZ-500
+
$0.32/lb
C
C
Alkylation
-
C2H3
C2H5
$0.30/lb C-150 / EBM-9
+
CH2 CH2
H2
Dehydrogenation Polymerization
C2H3
+
BF3 + H2 O
x
$0.42/lb
x
$0.82/lb
-
Chemical Plant for Producing Ethylbenzene (99 +%; 50,000 mton/yr) - ChemE `05
Qin
Qout
l P1 Fluid Flow in Piping: P2
!P = P2 " P1 = f (Q,l)
Design Objectives
Minimize pump (liquids, suspensions, slurries) and compressor (gases) cost. Minimize pipe acquisition and maintenance costs.
Example from ChemE 324
r z
Fluid Flow in Heat Exchangers:
! q = g(k f , !T ,ur )
Design Objectives
Maximize heat transfer coefficient and heat transfer rate. Minimize heat loss from pipes and fittings.
Example from ChemE 390
v!
Flow in Packed-bed Reactor: X = h(v! , K,"K )
Design Objectives
Maximize interfacial contact and residence time of reactant(s) in catalyst bed . Minimize entrainment of particles in exiting streams.
Example from ChemE 332
Flow in Distillation Columns: N T = !(Qg ,QL ,T , P, K L,V )
Design Objectives
Maximize contact between down-comer (liquid) and up-comer (vapor) streams. Minimize flooding.
1785-1836
ChemE 3230 Fluid Mechanics
1819-1903
C-L. Navier
G.G Stokes
Course Description
Fundamental principles and applications of fluid mechanics. This course teaches students how to apply mathematics and the physical sciences to analyze fluid flows in chemical process equipment (e.g. pipes, pumps, and packed columns), and to design fluids handling processes. Text Book
Morton M. Denn - Denn: Process Fluid Mechanics, Prentice - Hall 1980
Class Time
Class: MWF 11:15 - 12:05. (OH 155); Section: W 1:25 - 4:25 (OH 155)
Instructor
L. A. Archer, Professor 241 Weill; Phone: [phone]; Email: [e-mail] Office Hours - Tu 2-5 pm (Weill 241)
Teaching Assistants
Praveen Agarwal B40 Olin Hall Phone: [phone] email: [e-mail] Office Hours: TBA Peng Wang Olin Hall Phone: [phone] email: [e-mail] Office Hours: TBA
Course Grading
Preliminary Exams (3): 45% Weekly Homework & Quizzes: 20%; Class participation: 5% Final Exam: 30%
Exams
Three prelims are scheduled during the recitation period (1:25 - 4:25 pm) in OH 155 Exam 1: February 25; Exam 2: March 25; Exam 3: April 22 Final (comprehensive) : 7:00 - 9:30 pm on Wednesday, May 6
Homework and Quizzes
Homework will be assigned weekly (typically on Fridays) and is due at the beginning of class one week from the date assigned. Homework assignments will be completed by 3person groups. A single assignment will be submitted per group. Weekly quizzes (typically one or two problems based on the homework set due the previous week) will be assigned, solved, and graded during the recitation section.
Course Outline
TOPIC
Introduction, Basic Concepts, V&Ts, Physical Properties Dimensional Analysis & Pipe Flow External Flows - Permeation Macroscopic Balances & Applications Prelim Exam - 1 Microscopic Balances 1-D Flows Accelerating Flows & Boundary Layers Exam - 2 Converging Flows Ordering Analysis & Creeping Flow Lubrication Approximation Exam - 3 Boundary Layer Approximation Turbulence
Denn Chapter(s)
1-2 3 4 5-6 7 8 9 10 11-12 13 15 16
Dates - Approx.
1/19-1/26 1/28 - 2/6 2/9 - 2/13 2/16 - 2/25 2/25 - 3/6 3/6 - 3/13 3/23 - 3/27 3/30 - 4/3 4/3 - 4/15 4/17- 4/22 4/22 - 4/24 4/24 - 5/1
1.1 Introduction
What is Fluid Mechanics? What is a Fluid?
A fluid is defined as a material that deforms continuously when subjected to a tangential or shear force.
FT H
z r
R "R
M
!
Planar Couette
Circular Couette
What is Fluid Mechanics?
What is a Fluid?
A fluid is defined as a material that deforms continuously when subjected to a tangential or shear force.
FT H
(t)
z r
R "R
Sir Isaac Newton
M
!
Planar Couette shear
SF CF
Circular Couette shear
SF: Simple Fluids (most gases, water, honey) CF: Complex Fluids (paints, polymers, slurries, waxes) S: Solids
S
SF & CF: (gases, liquids, quasi solids)
time
Nanoparticle Ionic Fluids
inorganic core charged oligomer corona
canopy
A- :
Cl Su: R(OCH2CH2)7O(CH2)3SO3IS: CH3(CH3)CHCH2(CH2)12CH3COO-
A:Cl
A:Su
7nm SiO2
50nm
Cores: SiO2, TiO2, Fe2O3, ZnO, SnO2, Au, Pt, Pd, CNTs, C60
What is Fluid Mechanics?
Mechanics: the branch of the physical sciences concerned with the state of
bodies that are subjected to the action of forces. Fluid Mechanics: Study of deformation and motion of fluids subject to forces. motion of
Example:
What is Fluid Mechanics?
Study of deformation and motion of fluids. Example:
Questions: What is the origin of the resistance to fluid motion in the pipe? Is a pump needed? How large should it be? What size motor? How much does it cost to operate? How are these results affected by properties of the fluid, size of the pipe, material used to construct the pipe?
Why Study Fluid Mechanics?
2. The governing (conservation) equations for momentum transport (fluid mechanics),
!("v) + # $ ("vv) = "g % #p % # $& !t
are similar in structure to equations for mass and energy transfer (ChemE 324).
!"i + # $ ("i v) = ri % # $ ji !t D1 ! ( v " v + U ) + # $ !g " v = % " (T " v) $ % " q Dt 2
Where,
D! "! = + v # $! Dt "t
Procedures developed in the course for solving these equations are applicable for other transport processes.
ChemE 3230, Spring 2009
1.3 Vector and Tensor Algebra
Vectors and tensors will be repeatedly analyzed and manipulated throughout this course. It is therefore important that you have a working knowledge of their meaning and properties. This short handout summarizes the points discussed in class and is intended to provide a basic overview of vector and tensor algebra. 1.3.9.1 Scalars and Vectors
A scalar is a quantity having only magnitude. A vector is a quantity having both direction and magnitude. The magnitude of a vector a is denoted a . A familiar example of a vector is the velocity u or the position coordinate x . Such a vector may be represented by an arrow whose length denotes its magnitude and whose orientation species its direction in space.
1.3.9.2 Vector Space, Basis vectors, Components
Let us begin with a brief review. The fundamental rules for addition of two vectors and of multiplication of a vector by a scalar form a vector space. (1) Commutative rule u+v = v+u au = ua (2) Associative rule !u + v" + w = u + !v + w" = u + v + w a ! bu " = ! ab "u (3) Distributive rule a ! u + v " = au + av ! a + b "u = au + bu (4) Existence of a zero (1.3.5) (1.3.6) (1.3.3) (1.3.4) (1.3.1) (1.3.2)
-1-
ChemE 3230, Spring 2009 (V&T Handout)
u+0 = u (4) Existence of a negative u + ! u " = 0 (5) Unit multiplication 1u = u
(1.3.7)
(1.3.8)
(1.3.9)
Suppose that a group of vectors # v $ contains n independent vectors. This group # v $ is called an n-dimensional vector space. Let us choose n such vectors e 1& e 2& %& e n to form a vector basis. By denition of linear dependence, any vector v belonging to the group of vectors # v $ may be written as a linear combination of e 1& e 2& %& e n : v = v1 e1 + v2 e2 + % + vn en (1.3.10)
The scalars v 1& v 2& %v n are called the components of v with respect to the base e 1& e 2& %& e n . When the set of base vectors is understood, we can simply display v as v = ! v 1& v 2& %& v n " The rules of operations in vector components are as follows: (1) Addition of two vectors. Let u = u 1 e 1 + u 2 e 2 + % + u n e n and v = v 1 e 1 + v 2 e 2 + % + v n e n Then w = u + v = ! u 1 + v 1 "e 1 + ! u 2 + v 2 "e 2 + % + ! u n + v n "e n w = ! u 1 + v 1& u 2 + v 2& %& u n + v n " w i = u i + v i for ! i = 1& 2& %& n " (2) Multiplication of a vector by a scalar a : v = au = au 1 e 1 + au 2 e 2 + % + au n e n v = ! au 1& au 2& %& au n " v i = au i for ! i = 1& 2& %& n " -2ChemE 3230, Spring 2009 (V&T Handout)
1.3.9.3 Scalar (Inner or Dot) Product of Two Vectors
The scalar product of two vectors is by denition a scalar, u ! v = scalar The general rules imposed on the scalar product are as follows (1) Commutative rule u!v = v!u (2) Distributive rule u ! !v + w" = u ! v + u ! w (3) In general, u ! ! av + bw " = au ! v + bu ! w (1.3.14) (1.3.13) (1.3.12) (1.3.11)
Using the above rules, the scalar product of any two vectors may be determined once the scalar products between the base vectors e 1& e 2& %& e n are specied. Several geometric concepts such as magnitude, orthogonality and directional cosine may be dened through the use of the scalar product. (1) Magnitude of a vector v= (2) Orthogonality Two vectors u and v are said to be orthogonal to each other if u ! v = 0. (3) Directional cosine If " is the angle between two vectors u and v then the directional cosine is dened by u!v cos " # ----------- . uv (1.3.17) (1.3.16) v!v (1.3.15)
-3-
ChemE 3230, Spring 2009 (V&T Handout)
1.3.9.4 Einsteins Summation Convention
The Einsteins summation convention greatly simplies things by saying repeated indices implies summation. A vector,
n
v = v1 e1 + v2 e2 + % + vn en = may be written using Einsteins convention as v = vi ei .
$ vi ei ,
i=1
The summation sign with respect to i is omitted but its presence is understood. The ith component of v may be written formally as # v $i = vi . (1.3.18)
We must make a clear distinction between two types of indices in a system of equations. For example, let v 1 = a 11 x 1 + a 12 x 2 + % + a 1n x n v 2 = a 21 x 1 + a 22 x 2 + % + a 2n x n % v n = a n1 x 1 + a n2 x 2 + % + a nn x n In Einstein notation, we have v i = a ij x j The index j is repeated implying summation and is called a dummy index because this index may be replaced by any other letter, say, k and the equation will be unaffected v i = a ik x k . The operation summing over the dummy index is called contraction. On the other hand, the index i depends on the equation we want to pick and can be chosen freely. For example, if we choose i = 2 , we get the second equation v2 = a2 j x j . -4ChemE 3230, Spring 2009 (V&T Handout)
Hence, i here a free index.
1.3.9.5 Orthonormal Basis
A basis of a n-dimensional space e 1& e 2& %& e n is called orthonormal if the n base vectors fulll the following two conditions: (1) All the n vectors e 1& e 2& %& e n are unit vectors, e1 = e2 = % = en = 1 or e i ! e i = 1 when i = j (2) The n base vectors are orthogonal to each other, e i ! e j = 0 when i % j . The above two conditions may be combined as follows e i ! e j = & where ij & ij is the Kronecker delta, which is dened by & ij = 1 for i = j & ij = 0 for i % j (1.3.23) (1.3.24) (1.3.22) (1.3.21) (1.3.20) (1.3.19)
From here on we will be dealing with 3-dimensional orthonormal vector spaces. An example is the conventional right hand cartesian coordinate system. Here the 1& 2& 3 axes correspond to the x& y& z axes, respectively, and are parallel to the unit vectors e 1& e 2& e 3 .
1.3.9.6 Vector (Cross) Product
In 3-dimensional space, the cross product of two vectors u and v is dened in the form w # u ' v is dened by the following rules: (1) w is perpendicular to both u and v , u!w = v!w = 0 (1.3.25)
-5-
ChemE 3230, Spring 2009 (V&T Handout)
(2) The magnitude of w is by denition, w = uv sin " , where " is the angle between u and v . (3) w points in the direction so that u& v& w form a right hand coordinate system. This is a convention that determines the sense of w . Hence w is a pseudo-vector (see Borisenko and Tarapov1). The following properties of the vector product are easily proven. u ' v = ! v ' u " u ' v = 0 if v = au u ' !v + w" = u ' v + u ' w u ' !v ' w" % !u ' v" ' w . 1.3.9.3.1 Orthonormal Base System Let e 1& e 2& e 3 form an orthonormal base. Then we have, e1 ' e1 = e2 ' e2 = e3 ' e3 = 0 e1 ' e2 = e3 , e2 ' e3 = e1 , e3 ' e1 = e2 . Consider the two vectors u and v , u = u 1 e 1 + u 2 e 2 + u 3 e 3 and v = v 1 e 1 + v 2 e 2 + v 3 e 3 Then, using Equation (1.3.28), u ' v = ! u2 v3 u3 v2 " ! e2 ' e3 " + ! u3 v1 u1 v3 " ! e3 ' e1 " + ! u1 v2 u2 v1 " ! e1 ' e2 " = ! u 2 v 3 u 3 v 2 "e 1 + ! u 3 v 1 u 1 v 3 "e 2 + ! u 1 v 2 u 2 v 1 "e 3 e1 e2 e3 = u1 u2 u3 . v1 v2 v3 (1.3.29) (1.3.27) (1.3.28) (1.3.26)
-6-
ChemE 3230, Spring 2009 (V&T Handout)
The above may be simplied by using Einstein notation with the Levi-Cevita or permutation tensor ' ijk , dened as follows ' ijk = 1 if ijk are even permutations of 1,2,3 ' ijk = 1 if ijk are odd permutations of 1,2,3 ' ijk = 0 if any indices are the same The ith component of u ' v can then be written as # u ' v $ i = ' ijk u j v k . (1.3.31) (1.3.30)
For example, when we set the free index i = 1 , then the non-zero cases for ' ijk are ' 123 = 1 and ' 132 = 1 . From Equation (1.3.31) # u ' v $ 1 = u 2 v 3 u 3 v 2 in agreement with Equation .
1.3.9.4 Multiple Products of Vectors
Some useful formulae involving multiplication of multiple vectors are presented below. These formulae may be proven directly by using known vectorial identities or indirectly by working through the scalar components. One important theorem in connection with the latter approach is the following statement: If a vector or tensor identity is proved true in one coordinate system, it will be true in all other coordinate systems. We can therefore prove an identity using the simple coordinate system (usually the rectangular coordinate system with orthonormal base vectors), and then rewrite it in general vector form. (1) The scalar product of three vectors: u1 u2 u3 u ! !v ' w" = v1 v2 v3 w1 w2 w3 = ' ijk u i v j w k Then applying the rules of the permutation symbol, we obtain u ! !v ' w" = v ! !w ' u" = w ! !u ' v" = u ! ! w ' v " = v ! ! u ' w " = w ! ! v ' u " . (1.3.33) (1.3.32)
-7-
ChemE 3230, Spring 2009 (V&T Handout)
The scalar product then follows a general cyclic rule. The sign of the determinant changes when any two rows in Equation (1.3.32) are interchanged. This is because interchanging two indices in the permutation symbol causes a change in sign. Note also that u ! ! v ' w " = 0 if any of the two vectors are identical. (2) Vector product of three vectors u ' !v ' w" = v!u ! w" w!u ! v" Proof A proof is obtained by using the following identity ' ijk ' klm = & il & jm & im & jl Then # u ' ! v ' w " $ i = ' ijk ' klm u j v l w m = & il & jm u j v l w m & im & jl u j v l w m = v i u j w j w i u j v j q.e.d. (3) !a ' b" ! !c ' d" = !a ! c"!b ! d" !a ! d"!b ! c" (4) Lagranges Identity u'v
2
(1.3.34)
(1.3.35)
(1.3.36)
= u v !u ! v"
2
2
2
(1.3.37)
1.3.9.5 Differentiation of a Vector Function of a Scalar Variable.
Let r ! ( " be a vector function of the scalar ( . As ( changes, the magnitude and direction of r varies, as shown in the gure. For a small change )( , r ! ( " changes by )r to r ! ( + )( " . The derivative of r is a new vector function dened by dr r(( + )() r(() # lim ------------------------------------d ( )( + * )(
(1.3.38)
-8-
ChemE 3230, Spring 2009 (V&T Handout)
)r r(() r(( + )() 0 1.3.9.3.1 The Product Rule Two vector functions u(() and v(() may be combined in various ways such as by the inner product u ! v , the cross product u ' v . Another product called the outer product uv (or u , v ) produces a higher order tensor and will be discussed later. Writing ( to denote ! or ' or , , the product rule states du dv d !u ( v" = (v+u( d( d( d(
(1.3.39)
It is important to remember here that the operations ' and , do not commute. Example: If u(() is a vector of constant magnitude then tion, u = const . Hence, du d(
2
du is perpendicular to u . By assumpd(
= 2u !
du = 0 q.e.d. d(
-9-
ChemE 3230, Spring 2009 (V&T Handout)
1.3.9 Tensors
The denition of a tensor is a natural extension of that of vectors which are sometimes referred to as 1st order tensors, while scalars are zeroth order tensors. A tensor is dened by a set of scalars and a tensor basis. In 3-dimensional space, with basis vectors e 1& e 2& e 3 , a tensor A may be written as A = A 11 e 1 e 1 + A 12 e 1 e 2 + % A 33 e 3 e 3 or in matrix form, 1 A 11 A 12 A 13 / A = / A 21 A 22 A 23 / - A 31 A 32 A 33 or in Einstein (index) notation A = A ij e i e j or # A $ ij = A ij (1.3.42) 2 0 0 0 . (1.3.40)
(1.3.41)
1.3.9.1 Dyadics
A dyad ab (or a , b ) is formed from the outer or dyad product of two vectors a and b . A dyad is thus an ordered pair of vectors and so has two directions associated with it. It is another form of a tensor A = ab . Dyads obey the following rules: (1) ! ab " ! x = a ! b ! x " The vector ! ab " ! x thus has the same direction as a . (2) a ! 3b + 4c " = 3ab + 4ac (1.3.44) (1.3.43)
Hence, ab maybe decomposed into component forms. If e 1& e 2& e 3 are the unit vectors base vectors of a cartesian coordinate system, then ab in component form reads A = ab = a 1 b 1 e 1 e 1 + a 1 b 2 e 1 e 2 + %a 3 b 3 e 3 e 3 = a i b j e i e j (1.3.45)
- 10 -
ChemE 3230, Spring 2009 (V&T Handout)
or A ij = # ab $ ij = a i b j (3) ! 3ab + 4cd " ! x = 3 ! ab " ! x + 4 ! cd " ! x (1.3.47) (1.3.46)
This means that the operation ! ab " ! x can be carried out in component form. Using Einstein notation # ! ab " ! x $ i = a i b j x j (1.3.48)
1.3.9.2 Tensor Order
The number of free indices of the tensor components is equal to the order (or rank) of the tensor. . Table 1: Rank No. of free indices 0 1 2 3 n Examples Remarks
0 1 2 3 n
T , ui vi u i , A ij u j , ' ijk u j v k 5u i ,A ,& 5 x j ij ij A ij x k , ' ijk A ijk%s
scalar vector tensor 3rd order tensor nth order tensor
If a quantity is referred to as a tensor without reference to its order, it is assumed that the quantity is a second order tensor. Operations for inner and cross products are summarized below. e i %e k e l , e m e n %e p = e i %e k e l e m e n %e p
- 11 -
ChemE 3230, Spring 2009 (V&T Handout)
e i %e k e l ! e m e n %e p = & lm ! e i %e k " , ! e n %e p " = & lm ! e i %e k e n %e p " ! e i %e k e l " ' ! e m e n %e p " = ' rlm ! e i %e k " , e r , ! e n %e p " = ' rlm ! e i %e k e r e n %e p "
1.3.9.3 Operations with Tensors
Let A and B be two tensors, then we review various operations 1.3.9.3.1 Addition # A + B $i = Ai + Bi 1.3.9.3.2 Multiplication of a Tensor by a Scalar 3 # A $ ij = 3 A ij 1.3.9.3.3 Dot Product of a Tensor and a Vector If x is a vector, the operation y = A ! x is written y = ! A ij e i e j " ! ! x k e k " = & jk A ij x k e i = A ij x j e i or y i = A ij x j 1.3.9.3.4 Dot Product of Two Tensors A ! B = ! A ij e i e j " ! ! B kl e k e l " = & jk A ij B kl e i e l = A ik B kl e i e l Often A ! A is written as A . 1.3.9.3.5 Double Dot Product of Two Tensors Two tensors may be multiplied according to the double dot operation A:B = ! A ij e i e j ": ! B kl e k e l " = A ij B kl ! e i e j ": ! e k e l " = A ij B kl ! e i :e l " ! e j :e k " = A ij B kl & il & jk = A ij B ji 1.3.9.3.6 Cross Product of a Tensor with a Vector A ' x = ! A ij e i e j " ' ! x k e k " = ' ljk A ij x k e i e l (1.3.54) (1.3.53)
2
(1.3.49)
(1.3.50)
(1.3.51)
(1.3.52)
1.3.9.4 Properties of Tensors
1.3.9.3.1 Identity Tensor The identity tensor I is dened by - 12 ChemE 3230, Spring 2009 (V&T Handout)
x = I!x In cartesian coordinates 1100 / I=/010 / -001 2 0 0 = ei ei 0 .
(1.3.55)
(1.3.56)
# I $ ij = & ij 1.3.9.3.2 Transpose The transpose of a tensor A is denoted A and is dened
T T T
(1.3.57)
A 1.3.9.3.3 Symmetry
= A ji e i e j or # A $ ij = A ji
(1.3.58)
A tensor A is symmetric if A ij = A ji . Clearly a symmetric tensor has only six independent components. Note also that ' ijk A jk = 0 6 A ij = A ji [proof] ' ijk A jk = ' ijk A kj = ' ikj A jk = ' ijk A jk = 0 (1.3.60) (1.3.59)
A tensor A is antisymmetric if A ij = A ji . Clearly an antisymmetric tensor has only three independent components, the diagonal components being zero. So it can be represented by a vector w A ij = ' ijk w k where 1 w k = -- ' kpq A pq 2
T
(1.3.61)
(1.3.62)
w is a pseudovector since it depends on the convention chosen for the cross product. We can easily show that the product, A:B = A ij B ij (this is the double dot product) of a symmetric tensor A and an the transpose of an antisymmetric tensor B is zero:
- 13 -
ChemE 3230, Spring 2009 (V&T Handout)
A ij B ij = A ji B ji = A ij B ij = 0
(1.3.63)
The second step is obtained by interchanging the indices i and j. The third step follows from the denition of symmetric and antisymmetric tensors. Any tensor A may be decomposed into a symmetric tensor A tensor A
!a" !s"
and an antisymmetric
1 1 T T !s" !a" A = -- ! A + A " + -- ! A A " = A + A 2 2 or 1 1 !s" !a" A ij = -- ! A ij + A ji " and A ij = -- ! A ij A ji ". 2 2 1.3.9.3.4 The Adjoint of a Tensor A tensor A is called the adjoint of the tensor A if for any two vectors x and y ,
(1.3.64)
(1.3.65) x!A!y = y!A!x A tensor A is called self-adjoint if A = A . A symmetric tensor satises the condition Equation (1.3.65) and is therefore self-adjoint. 1.3.9.3.5 The Trace of a Tensor The trace of a tensor A is the sum of its diagonal elements, i.e. tr ! A " = A ii = A 11 + A 22 + A 33 (1.3.66)
As will be varied later by inspection, the trace of a tensor is independent of the orientation of the coordinate system. Note also that the trace of the Kronecker delta is & ii = 3 . (1.3.67)
- 14 -
ChemE 3230, Spring 2009 (V&T Handout)