Topics

TopicsSubject topics: - introduction (this lecture)- fitting- turbine machinery- dimensional analysis, use, selection of variables, preprocessing of variables- measuring stations, connection with experimental fluid dynamics, sensing equipment, sampling- design of the experiment, choice of variables- regression and statistical analysis- labview?- cavitation, examples of experimental modeling (Matevž Dular)

Topics

We will use few equations, but this is often impossible ...If possible, we will ese examples.

Seminars, I invite you to participate and prepare them, we agree on the prize. You can select the topic from the subject area or close to this subject.

The exams are written (within the deadlines set out or agreed by agreement) or oral (in agreement with me after e-mail a few days in advance)

Exercises and lectures are assessed separately. This means that you can do first lectures and then practice. When you pass lectures or exercises, the grade is valid as long as I take care of this subject.

The possibility of student work and later the master's thesis and/or praktikum.

3

Topics

Topics2017/18:- introduction 2 hours- fitting 2 hours- turbine machines, 4 hours- dimensional analysis, 4 hours- measuring stations, 4 hours- regression, probably 6 hours- clothes drying machine, example, probably 2 hours- stone or glass wool, example, probably 2 hours- experimental modeling - cavitation (Matevž), probably 4 hours

Fitting

Fitting- this is ... fitting- fitting is the process of determining the mathematical function (curves) that best fits the data points- the fit may contain interpolation (exact fit is required) or smoothing (an approximate function that moves smoothly between values) or extrapolation (matching outside the interval of the measured data)- fitting (in engineering) is part of the regression analysis (statistics, later chapters)- the engineer tries to understand the phenomenon and adjust the fit function to the process- example: first laboratory exercise, cavitation, flow channel resistance, dp as a function of flowUsually we fit the function in the form

The line passes through two points:

The square function fits precisely to three points:

Figure: interpolation (black line) and fitting (red line)

𝑦 = 𝑓 𝑥

𝑦 = 𝛽0 + 𝛽1𝑥

𝑦 = 𝛽0 + 𝛽1𝑥 + 𝛽2𝑥2

Fitting

- the fit of a cubic function means more generally that the function is determined by several conditions, the conditions being points, inclination, curvature- the slope and curvature are prescribed at both edges of the interval (eg two points and a slope to fit the cubic function)- splines (laboratory exercise 1)- if we have more than n + 1 conditions, the polynomial curve can still pass through, but this is not certain- generally, if we have more than n + 1 conditions (mostly points), we need a certain method that will carry out an approximation. This is mostly the least squares method (matlab eg polyfit).

matlab: p=polyfit (x,y,n)pyton: numpy.polyfitexcel : right mouse click, add trendline

- good understanding of engineering process helps ...

Figure: fitting withfunctions of different

Fitting

There are several reasons to use the approximation (least squares method) instead of increasing the order of the polynomial fit:- measurement uncertainty,- accurate solution may be difficult to calculate and the approximate solution is easier to solve,- oscillating behavior of polynomial at high order (between adjacent points through which the curve goes exactly, we expect that the fitted curve takes place "somewhere in between"),- reducing the number of inflection points (for high-orderpolynoms), Trigonometric functions, rational and exponential functions, are rarely used for fitting. In power and process engineeringwe often use power functions.We prescribe the function we fit in advance (n-th orderpolynomial, exponential function, or whatever).

Figure: ordinary least squares vs total least squares

Fitting

What is experimental modeling?pressure in a closed vesseldepending on temperature

The experimental modeling of the variation of the selected variable consists of:

- planning of the experiment,- making of the experiment,- pre-processing of measurement results,- modeling, where the model consists of a deterministic part and noise with a certain distribution,- post-processing of results,- validation of the model,- use of modeled data (estimation, forecast, calibration, process optimization)

𝑦 = 𝑓 Ԧ𝑥

𝑦 = 𝑓 Ԧ𝑥; Ԧ𝛽 + 휀

Figure: left - measurementsin the middle - fitted function fright - epsilon of values, histogram and distribution

Fitting

We drive car with constant speed- equation

- fitting

Accelerating the car with ICE, does force F change over time? - ICE power depends on rotational frequency- drag (airflow)- shifting of gears

Braking distance of the car d depending on the initialspeed, what physical equation would I use?- drag- coefficient of friction between the ground and the wheels- heating the brake discs

𝑦 = 𝛽0 + 𝛽1𝑥

𝑎 =𝐹

𝑚

𝑠 = 𝑣 ∙ 𝑡

𝑑 = 𝛽1 ∙ 𝑣 + 𝛽2 ∙ 𝑣2

Fitting

Examples of fitting

Tensile strength of concrete- we choose the appropriate function(rational, polynomial)

Power number of Rushton mixer- logarithm- fittingt with logarithmic values- antilogarithm

𝑦𝑚𝑜𝑑𝑒𝑙1 =𝑏0 + 𝑏1 ∙ 𝑥

1 + 𝑏2 ∙ 𝑥

𝑦𝑚𝑜𝑑𝑒𝑙2 = 𝑏0 + 𝑏1 ∙ 𝑥 + 𝑏2 ∙ 𝑥2 + 𝑏3 ∙ 𝑥4

𝑁𝑝 =𝑃

𝜌 ∙ 𝑛3 𝑑5

Turbine machinery

Turbine machineryDo not fall into the contents of this subject ... kkhhmm. Principle of operation: through velocity change.Turbine and piston machines perform a similar task, while the pistons exerta force on the pistonIn turbines, guide vanes increase velocity of the fluid in tangential direction (continuity), and then the runner reduces this velocity. The change in energy from the liquid (to mechanical) is through the change in the velocity of the liquid. For turbine:- the fluid has a pressure at the inlet and a little of kinetic energy in the direction of flow,- in guide vanes, the pressure changes in a kinetic energy - tangential- in the runner, the kinetic energy in the tengential direction changes to the shaft power - in the draft tube, the remaining kinetic energy changes to the pressureThe Bernoulli equation is approximately valid.

𝐹p = 𝑝𝑆

Turbine machinery

Types?- open / closed- according to the flow direction- according to the direction of energy conversion- depending on the type of application- according to physical action

Turbine machinery

The first law of thermodynamics for turbine machines

Q is the heat going over the walls of the machine, but also in E is the heat hidden in the enthalpy, it enters and exits the machine with a flow of liquid (specific per unit mass)

the specific internal energy and the term pv are enthalpy

we also know the total enthalpy, with which we write the first law of thermodynamics for a turbine machine as follows

𝐸 = 𝑄 − 𝐴

𝐸 = 𝑄 − 𝐴 = 𝐸𝑓𝑙𝑢𝑖𝑑 𝑝𝑜𝑤𝑒𝑟 𝑜𝑢𝑡 2 − 𝐸𝑓𝑙𝑢𝑖𝑑 𝑝𝑜𝑤𝑒𝑟 𝑖𝑛 1

ሶ𝑚 𝑢1 + 𝑝1𝑣1 +1

2𝑐12 + 𝑔𝑧1 + ሶ𝑄 = ሶ𝑚 𝑢2 + 𝑝2𝑣2 +

1

2𝑐22 + 𝑔𝑧2 + ሶ𝐴

ℎ = 𝑢 + 𝑝𝑣

ℎ01 + 𝑔𝑧1 + 𝑞 = ℎ02 + 𝑔𝑧2 + 𝑎

Turbine machinery

There are many companies in Slovenia that produce fans, pumps and water turbines. There are no gas turbine producers, so we can ignore the compressibility and heat flow.General example:

fans:

pumps:

𝑒 =𝐸

𝑚= 𝑌 = 𝑔𝐻 =

𝑝2𝜌+1

2𝑣22 + 𝑔𝑧2 −

𝑝1𝜌−1

2𝑣12 − 𝑔𝑧1

𝐸

𝑉= 𝑝2 +

1

2𝜌𝑣2

2 + 𝜌𝑔𝑧2 − 𝑝1 −1

2𝜌𝑣1

2 − 𝜌𝑔𝑧1

𝐸

𝑚𝑔=𝑌

𝑔= 𝐻 =

𝑝2𝜌𝑔

+1

2

𝑣22

𝑔+ 𝑧2 −

𝑝1𝜌𝑔

−1

2

𝑣12

𝑔− 𝑧1

Turbine machinery

Change of enthalpy,velocity and pressure

centrifugal compressor

axial compressor

ℎ0 = ℎ +1

2𝑐2 = 𝑢 + 𝑝𝑣 +

1

2𝑐2

Turbine machinery

Triangles of velocityc - absolute velocityw - relative velocityu - tangential velocity

turbine equation: Euler equation, first form

second form

Ԧ𝑐 = 𝑤 + 𝑢

−𝑃𝑠ℎ𝑎𝑓𝑡= ሶ𝑚𝜔 𝑟2𝑐𝑢2 − 𝑟1𝑐𝑢1

−𝑃𝑠ℎ𝑎𝑓𝑡= ሶ𝑚 𝑢2𝑐𝑢2 − 𝑢1𝑐𝑢1

−𝑃𝑠ℎ𝑎𝑓𝑡= ሶ𝑚 ℎ02 − ℎ01

−𝑃𝑠ℎ𝑎𝑓𝑡

ሶ𝑚=1

2𝑐22 − 𝑐1

2 + 𝑢22 − 𝑢1

2 − 𝑤22 − 𝑤1

2

Turbine machinery

Triangles of velocity:radial turbine (Francis)

Turbine machinery

Triangles of velocity:radial compressor

Turbine machinery

influence of blade angle on characteristics, radial fan:

Turbine machinery

triangles of velocity: axial turbine (left) and axial compressor (right)

Turbine machinery

axial turbine machines:- aircraft engines, ventilation fans, gas and water turbines- the flow of enters and exits on the same radius, tangential speeds do not change- for fans

for axial entry 𝑐𝑢1 = 04 main examples:- a stator followed by a runner (the stator increases, runner decrease tangential velocity, losses are generated in the driver)

- a runner followed by a stator (the most common version)

- a stator followed by a runner and another stator (the input and output velocities are the same; the runner produces static pressure only

- a stator followed by a runner, both of which get flow at a certain angle (for multi-stage versions, the speeds are lower than in the above cases, losses are the smallest, no equation)

𝑃𝑠ℎ𝑎𝑓𝑡

ሶ𝑚=∆𝑝

𝜌= 𝑢2𝑐𝑢2 − 𝑢1𝑐𝑢1 𝑢1 = 𝑢2 = 𝑢

∆𝑝

𝜌= 𝑢𝑐𝑢2

∆𝑝

𝜌= 𝑢𝑐𝑢1

∆𝑝

𝜌= 𝑢𝑐𝑢2

∆𝑝

𝜌= 2𝑢𝑐𝑢1 = 2𝑢𝑐𝑢2

Turbine machinery

turbine engine characteristics (and NPSH):- for a water turbine, but what about the other- unregulated- single regulated- AvčeIf instead of a flow or pressure a flow and pressure number is used, characteristic or hill diagram for different sizes are ... the same.

Turbine machinery

Similarity theory:Turbine machinery- geometric similarity- kinematic similarity- dynamic similarityReynolds, Euler, Thoma, Froude, Weber numbersflow, pressure numberso

change from model to prototype (density constant)

water turbines efficiency

𝜑 =ሶ𝑉

𝑛 𝑑3 𝜓 =𝑔𝐻

𝑛2 𝑑2

variable flow rate pressure hydraulic power

n n n 2 n 3

d d 3 d 2 d 5

1

𝜑m =ሶ𝑉m

𝑛m 𝑑m3 = 𝜑p=

ሶ𝑉p

𝑛p 𝑑p3

𝜓m =𝑔 𝐻m

𝑛m2 𝑑m

2 = 𝜓p=𝑔 𝐻p

𝑛p2 𝑑p

2

∆𝜂 = 1 − 𝜂m 𝑉 1 −𝑅𝑒m𝑅𝑒p

𝛼

𝜆 =𝑃

𝜌 𝑛3 𝑑5𝜓 =

𝑔𝐻

𝜌 𝑛2 𝑑2

𝜓 =𝐻

𝑛2 𝑑2

Turbine machinery

𝑛s =𝑛 ሶ𝑉

𝐸 ൗ3 4=

𝑛 ሶ𝑉

𝑔 𝐻 ൗ3 4

𝑛s =3.652 𝑛 ሶ𝑉

𝐻 ൗ3 4

𝑛s =576 𝜑

𝜓 ൗ3 4

𝛿 =𝜓14

𝜑12

=𝑑 ∙ 𝑔𝐻

14

𝜌14 ∙ ሶ𝑉

14

Turbine machinery

The importance of similarity theory or experimental modeling?The theory of similarity helps us decide which function to use for experimental modeling

Examples:- we have an existing mixer, but we need to figure out the power to mix the substance with a different density- we have an air mill, and we would like to grind instead of 2 t / h now 3 t / h of product- the fan consumes too much electricity in cold days and turns off the circuit breaker; can this be repaired?- we develop a drying machine, but we are interested in how the process variables affect the energy consumption and the drying time- there is too much dust in the textile production hall, and the power of the suction system needs to be increased- how much costs the dissolved oxygen concentration increase in treatment plant reactors by 0.1 g / l- etc.