fluid dynamics

CAMS in the School of Computing, Engineering and Physical Sciences

Introductory fluid dynamics by Dr J. Whitty 1  m 2  m 3 m 1

Lessons structure • The lessons will in general be subdivided in to eight number of parts, viz.: 1) 2) 3) 4) 5) 6) 7) 8) 2

Statement of learning objectives Points of orders Introductory material (Types of flow) Concept introduction (The conservation of mass) Development of related principles (flow continuity) Concrete principle examples via – reinforcement examination type exercises Summary and feedback Formative assessment, via homework task

Learning Objectives After the session the students should be able to: – State and use the basic thermodynamic laws – Derive the conservation of mass – Describe the differences between flow regimes – Calculate simple fluid flow mechanisms – Evaluate volumetric flow rates in fluid simple systems

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Recap: Laws of thermodynamics • These are quite simply the 4 axioms (self evident truths) of all modern Physics, they are known as the four Laws of Thermodynamics and relate to the quantities of

– Zeroth: Temperature – First: Energy – Second: Disorder (Entropy) – Third: Balance of them all

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Consequences of the first law: Flow Processes • If we consider the 1. The mass flow is first law based on constant and equal to the outlet mass flow some fluid passing 2. The cross-section through a control properties of the inlet volume above a and outlet are constant datum (at sea-level) for consentience. Conservation of Mass Application of the Mass cannot be destroyed or created first law, with the following assumption: 5

Conservation of mass Both Heat and fluid flow must adhere to the principal of the flow of mass and energy. Here we can consider a system (sometimes referred to as a control volume) with fluid flow (or heat) in and out of the system

 in  m 1  m 2 m

i.e. 6

3  m  out m

 in  m  out m

The unit of mass flow the kg per second (kg/s). Because speed has magnitude and direction, it vector quantity.

Consequence??

The Consequence of to Conservation of Mass

1. The mass (and sometimes volume) flow rate of a in-viscid, incompressible fluid (like water or oil) is constant. 2. This principle is one of probably the fundamental assumption in the field of Fluid Mechanics, this will now be explored! Class Examples Time: Think of some process which adhere to the above 7

Fluids in motion As an example of this principle we will investigate the concept of a fluid (say water) in motion. There is still a little terminology that is required before we proceed, these being: 1.

Assumptions regarding the fluid in motion, namely:

a)

Viscid

b) In-viscid 2.

Assumptions regarding the type of flow regime’

a)

Laminar

b) Transition c)

3.

8

Turbulent

Assumptions regarding Compressibility: 1.

Compressible, or

2.

In-compressible

1. Viscosity •

•

The viscosity of a fluid is the internal resistance to a change in the shape. Typically viscous fluids are treacle like: glycerine and thick oils. All fluids have some type of viscosity, however some fluids have such small viscosities have (e.g. water, air) can be considered in-viscid i.e. the viscosity of the fluid can be ignored! It is these type of fluids we considered here. Hence we have: 1. Viscid fluids (includes fluid viscosity effects) 2. In-viscid fluids (neglects fluid viscosity effects)

Since the math is considerably reduced when inviscid fluids are concerned it is these types we consider! 9

2. Flow regime’ • Laminar

• Turbulent

• Transition flow Class Exercise: Use the internet to find defientions of the above! 10

3. Compressibility • Incompressible fluid: Where the density of the fluid remains constant! (This course)

• Incompressible fluid: Where the density of the fluid changes during the flow process! (Not this course)

• When the Compressibility (Bulk) Modulus is? K

p

v

p  x y z

Class Question: What? 11

K

p

v

p   x y z

Continuity of flow • For the system shown, given that the flow is laminar, in-viscid and incompressible, find the flow rate at the outlet. A1

v1 m/s

v2 m/s A2

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Continuity of flow; Solution: • Here we could just apply the conservation of mass, as we know it is a consequence of the first law of thermodynamics, thus: A1 1 xt  A2  2 xt  A3  3 xt which implies 1 t

 A11 x   1t  A2  2 x   1t  A3 3 x 

and gives:

1  m 2  m 13

As density and the volume of then control volume are  3 constant! m

The Continuity Equation: Using the fact that. The flow is in-compressible:

A1 xt   A2 xt   A3 xt   A1v1  A2v2  A3v3

• We have now we’ve proved the continuity equitation (I wonder why I have spent so many slides on it?) The Continuity Equation: :

A1v1  A2 v2  A3v3 14

Example #2 • Evaluate the velocity of the fluid exiting the barrel of beer: 20mm DIA 6 m/s

1 m/s 30mm DIA

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Example #2; solution: • Apply the continuity equation, thus:  D12   D22  v1    4   4 20 2 6  30 2 1 



 D32  v2     4 20 2 v3

   





 v3  

  

6 20 2  30 2 -1  8 . 25 ms Hence: v3  20 2

Can you drink BEER that quickly?

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Class Problems 3. A system has two inlet rates of 3m3/s & 2m3/s what is the approximate output velocity [2]; and what assumptions did you make [3]? 4. For the system shown, determine the volumetric flow rate and velocity at the outlet. Given the large diameter pipe is 1.25 that of the smaller. 3.2m/s

vout m/s 1.6m/s 17

Class problem; solution #4: • Here were are given the volumetric flow rate, hence by continuity we have: A1v1  A2v2  A3v3

M1A1 3 -1

Q1  Q2  Q3  3  2  5m s

• There are three assumptions in place here: – The flow regime is laminarB1 – The fluid is incompressibleB1 – The fluid is in-viscidB1 18

Class problem; solution #2: •

Apply the continuity equation taking D and 1.25D along as parameter, thus: 

D 2  3.2 



(1.5D) 2 1.6  Q3

4 4 D 2 3.2  1.52 1.6  Q3 4  Q3  5.537D 2





M1

M2 A1

The required velocity can be found from the flow rate thus: Q3  A3v3   D 2v3 M1 4

5.537 D 2 

19

v3 

4





4

M1

D 2 v3

5.537  7.05ms1

A1

Examination type questions 1. Explain, using cogent examples: three laws of thermodynamics [6]. a) Use formulae to describe three mechanisms of heat transfer [6]. b) Find the total heat lost an asbestos (thermal conductivity 0.15W/mK) reinforced steel wall (thermal conductivity 50W/mK), given that the concrete is twice the thickness of the steel. [8]

150oC

20

25oC

Examination type questions 2. State three states of matter. [3] a) Explain the meaning of incompressible flow [2]. b) Given that the large pipe is 1.4 times the diameter of the small pipe evaluate the velocity at the output [12], 3.4m/s 2.1m/s

c) Clearly state the assumptions of the modelling process [3]. 21

Summary • Have we met our learning objectives: specifically, are you now able to do: – State and use the basic thermodynamic laws – Derive the conservation of mass – Describe the differences between flow regimes – Calculate simple fluid flow mechanisms – Evaluate volumetric flow rates in fluid simple systems 22