Computational Fluid Dynamics (CFD)

Momentum Transport Lecture 1Francisco J. Trujillo1. Introduction

3/76What is fluid mechanics?Movement of fluids.Well-developed body of theory.Describes the flow of product through equipmentSo, how can we use fluid mechanics in the chemical industry?

What is Computational Fluid Dynamics (CFD)?CFD is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyse problems that involve fluid flows.

The equations that represent the detailed movement of fluids and how heat and mass moves within those fluids are very complex. Those are the transport equations of momentum, energy and mass.

CFD uses computers to perform the calculations required to solve the equations of transport which due to their complexity cannot be solved analytically. 5/76Two types of problemsMicro - focus on specific details in flow field, e.g. in boundary layers.Macro overall characterisation of flow behaviour, e.g.Flow through pipe to determine required pump size.Temperature profiles of food cans in retort.Flow through an extruder energy requirement.Packed bed reactor for determining overall pressure drop or temperature profile.

Example 1: Flow in a tubeFor simplicity, in engineering and applications it is said that the velocity of a fluid in a pipeline has a certain value v. But in reality this is a an averaged value. Close inspection shows that for instance a laminar fluid flowing in a pipeline exhibits a parabolic velocity profile.

Transport Phenomena (Fluid dynamics ) allows calculating the precise velocity profile in tubes.ddxyExample 2: heat exchangersHeat exchangers are calculated and designed in a simplified manner by :Using a log mean temperature difference ignoring the temperature and velocity profiles inside the exchanger. Estimating overall heat transfer coefficients based on empirical equations

Example 2: heat exchangersTransport phenomena allows determining precise temperature and velocity profiles, and without the need of using empirical equations to calculate heat transfer coefficients (Heat Transfer was already explained in the first part of the curse by A. Professor Tuan Pham). This allows designing better exchangers while optimizing its operation.

2. Vector Analysis

Why reviewing vector algebra?The equations of transport of momentum, energy, mass and chemical species are very long. To simplify them vectorial symbols have been created. Then those symbols need to be understood. Vectors vs. scalars If you walk 4 kilometres due north and then 3 kilometres due east, you will have gone a total of 7 kilometres, but you are not 7 kilometres from where you set out.

Vectors describe quantities like this , which evidently do not add in the ordinary way.

Vectors have direction as well as magnitude (length) and is essential to take both into account when you combine them.

Velocity, acceleration, force and momentum are examples of vectors

Scalars are quantities that have magnitude but no direction. Examples of scalars are mass, density, temperature.

4 k3 k5 k12/76VectorsBold type or a bar over a variable indicates a vector.

v means velocity vector.

All of the above are representations of vectors

13/76Example velocity field of a river

Magnitude of a VectorsA vector has magnitude or norm: which is the length of the arrow

The magnitude or norm of the vector can be calculate as:

In the example:

4 k3 k5 k4 k3 k5 k

abc

Direction of a VectorsThe direction of a vector can be calculated as:

The magnitude of the vector can be calculate as:

In the example:4 k3 k5 k

abc

Unit vectorThe direction of the vector can also be given as a unit vector, which is a vector whose length or dimension is 1.

The unit vector is obtained by dividing each component of the vector by its norm :

Vector:

Norm:

Unit vector:

Unit vector exampleCalculate the unit vector for the vector below

4 k3 k5 k

Verify if the norm of the unit vector is 1:

Instead of calculating angles, whichIs more difficult on a 3D space,The unit vector gives you the Direction of the vectorMultiplication of a vector and a scalarCan we multiply a scalar with a vector?YES

When can this be needed?

For instance lets assume that v is the velocity vector:

And then assume that the velocity is increased 5 times,The multiplication is done as:Verify the norm of the vectors:This confirms that the norm v2 is 5 times the norm of v118Multiplication of a vector and a scalar

scalarVector19Multiplication of two vectors:Dot productSometimes you need to multiply two vectors and you need the result to be a scalar.

For instance the mass flow trough a pipeline can be calculated as:

This example is one dimensional as the velocity is perpendicular to the area.

But what happens in a 3D example?flow

areavelocitydensity20Multiplication of two vectors:Dot productAssume that the yellow frame is a window and the red arrow is the velocity vector of air.The black arrow is the unit vector that indicates the direction of the window (perpendicular direction to the plane):

On which arrangement there will be a bigger mass flow entering the window?On a) then b) and finally on c) where air cannot pass through the window

airairaira)b)c)21Multiplication of two vectors:Dot product

airairaira)b)c)If the velocity of the air has a different angle compared to the area unit vector the mass flow will be different.The area can be represented as the product of the Area (scalar) and the area unit vector:

The dot product which is defined as:

The dot product can be used to give the correct mass flow (scalar) passing through the window

22Multiplication of two vectors:Dot productLets assume that that area is 1 m2 and that the density of the air is 1kg/m3 :

airairaira)b)c)

The velocity has changes direction but the norm is the same.The dot product is given the right mass flow entering the window23Multiplication of two vectors:Dot productThe dot product multiply two vectors and the result is a scalar:

The dot product of two vectors u and v can also be calculated as

Dot productSymbol

So the dot product is:zero for perpendicular vectors Maximum for parallel vectors24Multiplication of two vectors:cross productThe cross product multiply two vectors and the result is a vector:

Right hand rule25Multiplication of two vectors:cross productThe magnitude of the cross product can be interpreted as the positive area of the parallelogram having a and b as sides (figure below)

So the magnitude of the cross product vector is:Zero for parallel vectorsMaximum for perpendicular vectors

26Kronecker delta i,j and the permutation symbol i,j,kMany formulas can be expressed compactly in terms of Kronecker delta i,j and the permutation symbol i,j,k

Unit vector in the direction of the axisLet 1, 2 and 3 be unit vectors (Vectors of unit magnitude) in the direction of the 1, 2, 3 axis:

This is the notation of the Transport phenomena textbook instead of

29/76TensorsTensors have 9 components (e.g. stress):

First index = rowSecond index = columnThe shear for a Newtonian fluid

When extended to 3D is represented with a tensorNomenclatureIn addiction boldface Greek letters with one subscript (such as i) are vectors.For simplicity 0 stands as for the zero scalar, zero vector or zero tensorFor vectors and tensors, several kinds of multiplications are possible such as the single dot the double dot and the cross . Those special multiplications can produce a scalar, a vector or a tensor, so for clarification we enclose these special multiplications in different kinds of parenthesis to indicate the type of result:

Scalar (lightface, Italic)Vector (boldface, Roman)Tensor (second order tensor, Boldface Greek)

Scalar VectorSecond order tensor NomenclatureNo special significance is attached to the kind of parenthesis if the only operation enclosed are addition and subtraction, or a multiplication in which , and do not appear.

Hence:

Scalar VectorSecond order tensor VectorScalar Scalars can be regarded as zero-order tensorsVectors as first-order tensorsThe multiplication signs may be interpreted as:

In which represents the sum of the order of the quantities being multipliedNomenclatureMultiplication signOrder of the resultnone

Is order 0 + 2 = 2 Is order 1 + 1 = 2 Is order 1 + 1 = 2 Is order 1 + 1 -1 = 1 Is order 2 + 2 - 4 = 0 Is order 2 + 2 - 2 = 2 Appendix A of the textbook show all the possible operations between scalars vectors and tensors

NomenclatureMultiplication of a tensor by a scalarMultiplication of a tensor by a scalar corresponds to multiplying each component of the tensor by the scalar

Addition of tensorsThe sum of two tensors is a tensor whose components are the sum of the corresponding components of the two tensors

36/76Dyadic product - making a tensor from the product of two vectorsFor a long time we only considered that two vectors could only be multiplied as a dot product to yield a scalar, or a cross product to yield a vector. Having been introduced to a tensor, the third way to multiply the two vectors is to obtain a tensor - a dyadic product. Having,

This is called the dyadic product which yields a tensor. One way to remember the product is to notice that the first vector provides the same row elements and the second vector the same column elements.

The tensor single dot productThis operation multiplies two tensors and the result is a tensor.

38/76Tensor double dot productThis multiplication between two tensors results in a scalar .

39/76Dot product of a tensor with a vectorThis is a dot product between a tensor and a vector and results in a vector

40/76Cross product of a tensor with a vectorThis is a cross product between a tensor and a vector and results in a tensor

Useful identitiesUseful identities numbered as in the Appendix A of the textbook:

is a unit vector

(A.3-21)(A.3-20)(A.3-22)(A.3-23)(A.3-24)(A.3-25)The derivative df/dx tell us how rapidly the function f(x) varies when we change the argument x by a tiny amount dx:

In other words, if we change x by an amount dx, then f changes by an amount df.

42/76Differential calculusOrdinary derivatives:

Geometrical interpretation: the derivative df/dx is the slope of the graph of f versus x.

43/76The operator is called del:

It looks like a vector but gives the instruction to differentiate what follows

The operator

44/76Suppose now that we have a function of 3 variables. For instance the temperature T(x, y, z) in a room. We want to generalize the notion of derivative to functions like T, which depends not on one but three spatial variables. Gradient:A derivative is supposed to tell us how fast the function varies. What in which direction?

45/76The gradient, has the formal appearance of a vector , multiplying a scalar T:

Gradient

46/76The gradient, which is defined as:

It is a vector quantity, with 3 components. It is the generalized derivative that we were looking for.

The gradient has a magnitude and direction as any vector.

The gradient points in the direction of maximum increase of the function T.

The magnitude of the gradient gives the slope (rate of increase) along this maximal directionGradient:

47/76Gradient - utilitySo when we see a collection of terms like this:

it is quick and easy to instead write:

with the same meaning.

48/76Gradient Example:

From the definition of the divergence can be constructed as:

The divergence has the appearance of dot product between and the vector v. Observe that the divergence of a vector is a scalar.

49/76Divergence

50/76The name divergence is well chosen. The divergence of a vector is a measure of how much the vector spreads out (diverges) from the point in question. It is a type of derivative of a vector.

Figure (a) has a large positive divergence (if the arrows point in, it will be a large negative divergence)Figure (b) has zero divergenceFigure (c) has a positive divergence

Divergence Geometrical interpretation

51/76Notice how the divergence is not just a change of direction, but as you can see on figures a) and c) the vectors are becoming bigger as if there were source that augments the vector field.

So the vector will have a non zero divergence if the vector field is spreading out in a expansive or diminish way.The divergence is a type of measure of change of the vector field.

Divergence Geometrical interpretation

52/76Divergence - utilitySo when we see a collection of terms like this:

it is quick and easy to instead write:

with the same meaning.

53/76Divergence - Example

54/76From the definition of the curl is defined as :

It has the appearance of a cross product between and v.like any cross product, the curl is a vector.

The Curl

55/76The name curl is also well chosen, because measures how much the vector v curls around a point in space. Thus, the tree functions below have zero curl.

Whereas the function in figures below has a substantial curl, pointing in the k direction.

The Curl geometrical interpretation

56/76Curl - example

The gradient of a Vector fieldIs the dyadic product of and v. It yields a tensor

The divergence of a tensor fieldThis is a dot product between and a tensor yielding a vector

The laplacian of a scalarIt is the divergence of the gradient of the scalar s

Where is the laplacian operator

The laplacian operator

61/76SummaryGRAD: (rate of change of a scalar with direction and magnitude)

DIV: (spread of the vector function Measure of change of the vector field As it is expanding (source) or Diminishing (sink) )

CURL: (rotation, curl of the function)

62/76Summarygrad v (tensor produced of the dyadic product ofDel operator and the vector v)

Divergence of a tensor (This is a dot product between del operatorand a tensor yielding a vector )

Laplacian: (It is the divergence of the gradient of the scalar s)

Useful differential relations - IUseful identities numbered as in the Appendix A of the textbook:

is a unit vector

(A.4-19)(A.4-18)(A.4-20)(A.4-21)(A.4-22)(A.4-23)Useful differential relations -IIUseful identities numbered as in the Appendix A of the textbook:

for the symmetric ( is symmetric for Newtonian fluids ij =ji)

is a unit vector

(A.4-25)(A.4-24)(A.4-26)(A.4-27)(A.4-28)

(A.4-29)65/76Surface integralA surface integral is an expression of the form:

Where v is a vector function and da is an infinitesimal patch of area perpendicular to the surface area. If the surface is close :

66/76Volume integralA volume integral is an expression of the form:

Where is a scalar function and d is an infinitesimal volume element.

For instance if is density of a substance (which might vary from point to point), then the volume integral would give the total mass.

67/76The fundamental theorem for divergenceThe fundamental theorem for divergence states that:

It says that the integral of a divergence of vector function over a volume is equal to the value of the function integrated over the surface that encloses the volume.

This theorem will be used when explaining the finite volume method , which is the numerical method used to solve the transport equations.

3. Viscosity Most important concept in fluids flowHigher viscosityLower viscosity

Viscosity Viscosity is a measure of a fluids resistance to flow or to be deform

Higher viscosityLower viscosityHigher viscosityLower viscosityViscosity is essentially a fluids friction.Viscosity transforms kinetic energy of motion into heatFor that reason pumping viscous fluids generates heat

70/76Resistance to flowFluid initially at rest:yxt