Appunti Advanced Manufacturing B, Sbobinature di Tecnologia Meccanica

Scarica Appunti Advanced Manufacturing B e più Sbobinature in PDF di Tecnologia Meccanica solo su Docsity!

ADVANCED MANUFACTURING PROCESS B

October 8, 2024

4 CONTENTS

Chapter 1

Resume Termodynamics

1.0.2 Fourier law

q′′ n = −k ·

∂T

∂n

Where k is the thermal conductivity of the medium. Introducing the Fourier Law in the 1 st^ thermodynamic law, we have

−k · ∂T∂x

∂x

−k · ∂T∂y

∂y

−k · ∂T∂z

∂z

• ˙q = ρ · cp ·

∂T

∂t

Assuming k = const. (thermal conductivity of the medium)

∂^2 T

∂x^2

∂^2 T

∂y^2

∂^2 T

∂z^2

q˙ k

ρ · cp k

∂T

∂t

α : Thermal diffusivity

α =

k ρ · cp

[

m^2 s

]

The heat diffusion equation becomes:

∂^2 T

∂x^2

∂^2 T

∂y^2

∂^2 T

∂z^2

q˙ k

α

∂T

∂t

From the Heat Diffusion Equation to the extended planar heat source model. Let's assume that a laser beam irradiates the surface of a material in order to harden it. The material properties are as follows:

• No internal heat sources.
• No phase changes.
• Assuming only conduction.

Let's introduce the Heat Diffusion Equation.

∂T

∂t = α ·

∂^2 T

∂x^2

∂^2 T

∂y^2

∂^2 T

∂z^2

Where T is the temperature, t is the time, x, y and z are the spatial coordinates and α is the thermal diffusivity of the material. Simplification to one-dimensional problem: assuming temperature variation only along one direction, say x. The Heat Diffusion Equation becomes

∂T

∂t = α ·

∂^2 T

∂x^2 Bundary conditions : they allow to find a unique solution for the temperature distribution in the material over time.

      

T (x, 0) = T 0

− k · ∂T∂x

x=0 =^ q

′′ 0 (const.)

T (∞, t) = T 0

8 CHAPTER 1. RESUME TERMODYNAMICS

• The first condition states that at the initial time instant, the temperature T at any position x in the model is equal to T 0 , which is the initial uniform temperature throughout the domain.
• The second condition specifies that, at the surface, a constant heat flux is spreading by conduction.(if q''= 0 -> adiabatic surface)
• The third condition indicates that the temperature remains unchanged and equal to the initial temperature T 0 far away from the heat source, suggesting that the heat source's impact does not extend infinitely.

10 CHAPTER 2. THERMAL MODEL

2.1 Thermal model

Define the principal models for a generic heat source. First of all, the sum of the dimensions of the source and the thermal field should be 3. Hence:

• 1D model: the planar source. It has 2 dimensions acting on the surface and the temperature varies in depth.
• 2D model: the line source. It has one dimension (normal to the material surface) and the temperature varies in the other 2 dimensions.
• 3D model: the point source. It has no dimensions and the temperature should vary in all the dimensions.

For the advanced manufacturing processes, we are mainly interested in the motionless 1 D, moving 2 D and 3 D thermal models.

2.1.1 Laser process phases:

1. At t < 0 , T (x, 0) = T 0
2. At t = 0 the laser is turned on.
3. At t = τ the laser is turned off (cooling stage).

The solution to the heat equation has been given by Carslaw and Jaeger:

T (x, t) = T 0 + q 0 ′′ ·

4 αt k

· ierfc

x √ 4 αt

we need to model the temperature raise in time and space for a motionless heat source. The modelling of the heating phase will allow us to understand the operational parameters to reach a desired temperature in space and time. The equation expresses the temperature in one dimension (depth) and time as a function of q0 ”, duration of the interaction and material properties

2.1.2 Error function

Let's focus more in detail on the ierfc(x) function, starting from the Normalized Gaussian function.

N (x) =

π

2 · σ

· e−^

x^2 2 ·σ^2

The error function erf(x) is the integral of the Normalized Gaussian function. Calling

2.1. THERMAL MODEL 11

t^2 =

x^2 2 · σ^2

The error function is

erf(x) =

∫ (^) x

−x

N (x) =

π

∫ (^) x

−x

e−t

2 · dt

Since it is symmetric (odd function), it's equal to

erf(x) =

π

∫ (^) x

0

e−t

2 · dt

Complementary error function.

erfc(x) = 1 − erf(x)

Integral of the complementary error function.

ierfc(x) =

∫ (^) x

−x

erfc(x) = − erfc(x) · x + e−x

2 √ π

Where x represents the depth of the material we are heating, hence x > 0. [ierfc(0) = 0.56; ierfc(1) = 0.053] The functions are plotted below:

heat equation

Taking back the solution obtained for the heat equation

T (x, t) = T 0 + q 0 ′′ ·

4 αt k · ierfc

x √ 4 αt

We can clarify the following points:

• The temperature at any location within the material, denoted by x, and at any moment in time t, is calculated using the formula above. This formula takes into account the material's thermal conductivity k, the supplied heat flux q′′ 0 , and the thermal diffusivity α of the material.
• The term ierfc

√x 4 αt

represents the integral of the complementary error function, which helps to map out how heat is distributed through the material as time progresses.

• The idea of a 'thermal length', represented by D, is introduced to gauge the extent of heat spread within the material over time, defined as

4 αt. At a depth equal to this thermal length, the temperature increase is roughly 10 % of the increase observed at the surface, indicating that most of the heat has dissipated beyond this depth, supporting the assumption of a semi-infinite material. At twice the thermal length, the temperature

2.1. THERMAL MODEL 13

The procedure will be the following:

T 1 f = Tamb +

q 0 ′′ √ πk

[√

4 αttot −

4 α (ttot − τ )

]

= Tamb + q′′ 0 · B

T 2 f = T 1 f + q′′ 0 · B = Tamb + q 0 ′′ · B + q′′ 0 · B = Tamb + 2q′′ 0 · B

Tf = T 2 f +

q 0 ′′

4 ατ √ πk

= Tamb + 2q′′ 0 · B + q′′ 0 · C

14 CHAPTER 2. THERMAL MODEL

16 CHAPTER 3. LASER

3.1 Laser modeling

Laser is an acronym for Light Amplification by Stimulated Emission of Radiation. When we talk about Laser we mean the device, while when we talk about the "laser beam", we mean the coherent, collimated, and monochromatic beam of electro-magnetic radiation with wavelength ranging from ultraviolet to infrared.

The main definition of an electro-magnetic radiation is a propagating waves associated with the oscillating electric field E and, orthogonal to it, the magnetic field B. Ligth as electromagnetic field

∂^2 Ey ∂z^2

= ε 0 μ 0 ∂^2 Ey ∂t^2 E · B = 0 E × B = v

Maxwell law:

E = A sin

2 π λ

(vt − x) + φ

[ε 0 : Permittivity; φ :Wave phase ; μ 0 : Permeability; v: Wave velocity] Laser light is a type of electromagnetic wave that moves in a direction perpendicular to the electric and magnetic fields. Electromagnetic radiation, like laser light, consists of waves where an electric field (E) and a magnetic field (B) constantly change. Since the magnetic field is always at a right angle to the electric field, we usually focus on how the electric field changes to describe the wave.

In this case, the electric field moves up and down (along the y-axis), changing with both space and time. For waves that vary in a smooth (sinusoidal) pattern, A is the amplitude (height of the wave), λ is the wavelength (distance between wave peaks), and v is the speed of light.

When working with lasers in manufacturing, it’s important to remember that the electric field part of the wave interacts with the material. The λ of the laser determines how the light affects the material during processing.

We can analyse the electric part of the electromagnetic field with

• A fixed position in space, where we observe the passage of the wave in time with variable amplitude and the peak to peak distance is the wave frequency
• A fixed position in time, where we observe that the wave has variable amplitude in space and the peak to peak distance is the wavelength.

The product of wave frequency and wavelength is the speed of light. c = λ f

However, for some purposes, the electro-magnetic radiations can be treated in 2 different ways:

3.1. LASER MODELING 17

Waves

⇒ Particle approach : this perspective assumes the electromagnetic radiation as a stream of particles which are called photons, and it is more appropriate to describe the process of stimulated emission.

The energy of a photon is the product of Planck's constant

h = 6. 63 · 10 −^34 Js

with the wave frequency

e = h · f = h ·

c λ

Therefore

⇒ Waves approach : it facilitates the application of concepts like absorption, transmission, and reflection, which are fundamental to wave behavior, hence it is more suitable to delineate the propagation of a laser beam.

This distinction is key in comprehensively understanding both the generation and propagation of laser light.

3.1.1 Laser generation

At the core of laser beam generation lies the concept of atom excitation. To bring an atom to an "excited" state, energy must be provided to it. This energy should be equal to the difference between the atom's initial energy level e 1 (the ground state) and the destination energy level e 4 (the excited state, for hypothesis equal to 4 ).

There are two primary methods to excite an atom, i.e., to move electrons to a higher energy level (further from the nucleus), known as "pumping process".

1. The optical method , represented below, where an external light source is used to energize an active medium. The external lamp could be: - Xenium or Krypton flashlamp. Here, some wavelength of the flash (emission spectrum of flashlamp) matches with the absorption characteristics of the active laser medium facilitating population inversion. - Diode laser of suitable wavelength. This kind of laser are suitable for pumping solid-state lasers, like Nd:YAG.

3.1. LASER MODELING 19

3.1.3 Light amplification

This amplification is realized through a particular arrangement of mirrors, forming what is known as a resonator.

Mirrors can be made of Zinc selenide ( ZnSe ) or Gallium arsenide (GaAs). Typically, these mirrors are gently curved to help focus the light within the active medium, often a rod. The res- onator's design allows the stimulated photons to reflect repeatedly between the mirrors, traversing the rod multiple times. Each pass through the rod triggers additional emissions from electrons, thereby amplifying the light.

The mirrors in a laser system could be partially or totally coated, depending on their goal: one mirror is fully reflective (100%), ensuring maximum light reflection back into the rod, while the other is partially transmissive. This partial transmission allows some of the light to exit, forming the laser's output beam. The extent to which the output mirror transmits light can vary with different types of lasers, generally ranging from 1 % to 50 %. This mirrored mechanism, in conjunction with the processes of population inversion and stimulated emission, plays a crucial role in creating a laser beam. The mirrors do more than just amplify the light; they are key in preserving the laser's monochromatic and coherent characteristics, ensuring the beam is both powerful and sharply focused. Therefore, this final step of optical amplification is essential in transforming the initial photon emissions into a functional and effective laser beam.

3.1.4 Properties of laser radiation

• Monochromaticity : this refers to the laser's ability to emit light at a single wavelength (or color). Different materials absorb light differently at various wavelengths, which can influence the efficiency and effectiveness of the laser process.
• Beam Coherence : coherence relates to waves with same phase in time and space. Coherent light allows for precise focusing and control over the micromachining process, leading to higher precision and accuracy in the machined products. Example:

20 CHAPTER 3. LASER

• Space and Time coherence
• Incoherent light
• Collimation : this is the laser beam's ability to remain parallel over long distances. As can be seen from the image below, high collimated beam (the upper one) can be focused on a very small area even at longer distances, and consequently the beam intensity is preserved even at long distances.

Instead, if the beam is not well collimated (like the lower one in the image), after passing through a focusing lens it will not be focused to the smallest beam size possible, so the intensity isn't preserved.

Hence, collimation is crucial for maintaining the focus and intensity of the laser beam even at longer distances between the laser source and the workpiece.

• Focusability : it refers to the ability to concentrate the laser beam to a small spot. The smaller the focal spot, the higher the intensity and the finer the details that can be machined.
• Brightness or Radiance : this property defines how well the energy of a beam is confined into a unit area (what we call intensity of the laser beam). Normally, the beam energy is defined as

E = P · τ

Where τ is the pulse duration. The energy density (for PW emission)