**2.2 Use of COMSOL multiphysics to model absorption in alpha brass through FEM**

COMSOL Multiphysics is a general-purpose simulation software for modeling designs, devices, and processes in all fields of engineering, manufacturing, and

**61**

**Figure 1.**

*alpha brass.*

*Analysis of the Absorption Phenomenon through the Use of Finite Element Method*

scientific research. The platform product can be used on its own or expanded with functionality from any combination of add-on modules for simulating electromagnetic fields, structural mechanics, acoustics, fluid flow, heat transfer, and chemical engineering. The add-on modules and LiveLink™ products connect seamlessly for a modeling workflow that remains the same regardless of what you

Engineers and scientists use the COMSOL Multiphysics® software to simulate designs, devices, and processes in all fields of engineering, manufacturing, and scientific research. This simulation platform encompasses all of the steps in the modeling workflow—from defining geometries, material properties, and the physics that describe specific phenomena to solving and postprocessing models for

In this section a simple model in COMSOL developed for the analysis of laser light absorption in alpha brass samples is described. The geometry of the problem is described using a concentric double cylinder, where the external cylinder represents the sample being heated, with a diameter of 40 mm, while the diameter of the internal cylinder represents the diameter of the laser beam, with a diameter of 10 mm, as shown in **Figure 1**. According to the geometry defined in the problem, a collimated beam with a constant intensity distribution is considered, in which the effects of divergence are neglected. In the general analysis, the domain is divided into two volumes, which are equivalent to the two cylinders. These volumes represent the same material, but only Beer–Lambert law is resolved in the internal domain (internal cylinder), which is the only region in which the material is heated by the laser beam. To implement Beer–Lambert law, it starts by adding the PDE interface with the dependent variables and unit

The equation itself is implemented via the general form PDE interface. Aside from

the source term, *f*, all terms within the equation are set to zero; thus, the equation being solved is *f* = 0. The source term is set to Iz-(50[1/m]\*(1 + (T-300[K])/40[K]))\*I, where the partial derivative of light intensity with respect to the z-direction is Iz, and the absorption coefficient is (50[1/m]\*(1 + (T-300[K])/40[K])), which introduces temperature dependency for illustrative purposes. This one line implements the Beer– Lambert law for a material with a temperature-dependent absorption coefficient,

*Definition of the cylindrical geometry, parameters, and initial conditions for the absorption analysis model in* 

assuming that it will also solve for the temperature field, T, in this model.

*DOI: http://dx.doi.org/10.5772/intechopen.86924*

producing accurate and trustworthy results [16].

are modeling.

settings.

scientific research. The platform product can be used on its own or expanded with functionality from any combination of add-on modules for simulating electromagnetic fields, structural mechanics, acoustics, fluid flow, heat transfer, and chemical engineering. The add-on modules and LiveLink™ products connect seamlessly for a modeling workflow that remains the same regardless of what you are modeling.

Engineers and scientists use the COMSOL Multiphysics® software to simulate designs, devices, and processes in all fields of engineering, manufacturing, and scientific research. This simulation platform encompasses all of the steps in the modeling workflow—from defining geometries, material properties, and the physics that describe specific phenomena to solving and postprocessing models for producing accurate and trustworthy results [16].

In this section a simple model in COMSOL developed for the analysis of laser light absorption in alpha brass samples is described. The geometry of the problem is described using a concentric double cylinder, where the external cylinder represents the sample being heated, with a diameter of 40 mm, while the diameter of the internal cylinder represents the diameter of the laser beam, with a diameter of 10 mm, as shown in **Figure 1**. According to the geometry defined in the problem, a collimated beam with a constant intensity distribution is considered, in which the effects of divergence are neglected. In the general analysis, the domain is divided into two volumes, which are equivalent to the two cylinders. These volumes represent the same material, but only Beer–Lambert law is resolved in the internal domain (internal cylinder), which is the only region in which the material is heated by the laser beam. To implement Beer–Lambert law, it starts by adding the PDE interface with the dependent variables and unit settings.

The equation itself is implemented via the general form PDE interface. Aside from the source term, *f*, all terms within the equation are set to zero; thus, the equation being solved is *f* = 0. The source term is set to Iz-(50[1/m]\*(1 + (T-300[K])/40[K]))\*I, where the partial derivative of light intensity with respect to the z-direction is Iz, and the absorption coefficient is (50[1/m]\*(1 + (T-300[K])/40[K])), which introduces temperature dependency for illustrative purposes. This one line implements the Beer– Lambert law for a material with a temperature-dependent absorption coefficient, assuming that it will also solve for the temperature field, T, in this model.

#### **Figure 1.**

*Definition of the cylindrical geometry, parameters, and initial conditions for the absorption analysis model in alpha brass.*

*Modern Spectroscopic Techniques and Applications*

\_\_\_ <sup>∂</sup>*<sup>I</sup>*

ρ*Cp* \_\_\_ <sup>∂</sup>*<sup>T</sup>*

**2.1 Beer–Lambert law deviations**

*2.1.1 Light scattering*

from Beer–Lambert law are the following:

scattering of light are proportional to 1/λ<sup>4</sup>

the emission fluorescence is collected.

at a wavelength of 3300 cm<sup>−</sup><sup>1</sup>

wavelength that is 3300 cm<sup>−</sup><sup>1</sup>

**brass through FEM**

*2.1.2 Fluorescence*

be written in differential form for the light intensity **I** as

tion for temperature distribution within the material:

such as laser light, and we despise losses by reflection and dispersion in the material, considering a minimum refraction and scattering, the Beer–Lambert law can

where z is the coordinate along the beam direction and α(*T*) is the temperaturedependent absorption coefficient of the material. Because the temperature can vary in space and time, we must also solve the next governing partial differential equa-

where Q is the heat source term equals the absorbed light. Eqs. (2) and (3) present a bidirectionally coupled multiphysics problem that is possible to solve within the core architecture of COMSOL Multiphysics program based in the finite element method [10].

Beer–Lambert law states that the optical density is directly proportional to the concentration of the species they absorb. However, deviations from this law can occur due to instrumental and intrinsic causes [11–13]. Among the main deviations

There are two dispersion phenomena, one depends on the size of the solute particle or any suspended material. Biological samples are usually cloudy because macromolecules or other large aggregates scatter light. The optical densities resulting from the

recognized as an absorption background which increases rapidly with the decrease in wavelength [14]. The second type of dispersion is known as the Raman scattering. In this phenomenon, part of the excitation energy of light is abstracted by vibrational modes of the solvent molecules. In the case of water or hydroxyl solvents, the most dominant vibrations that absorb this energy are the OH groups, whose vibration energy is observed

If the optical density of the sample is high and if the absorbing species are fluorescent, the emitted light can reach the detector. This process will result in derivations of Beer**–**Lambert law. The effect can be minimized by maintaining the distance of the detector from the sample and decreasing the efficiency with which

COMSOL Multiphysics is a general-purpose simulation software for modeling designs, devices, and processes in all fields of engineering, manufacturing, and

length of the Raman scattering (λRA) can be calculated as λ RA−<sup>1</sup>

**2.2 Use of COMSOL multiphysics to model absorption in alpha** 

<sup>∂</sup>*<sup>z</sup>* <sup>=</sup> <sup>α</sup>(*T*)*<sup>I</sup>* (2)

<sup>∂</sup>*<sup>t</sup>* <sup>−</sup> <sup>∇</sup>∙(*<sup>k</sup>* <sup>∇</sup> T) <sup>=</sup> <sup>Q</sup> <sup>=</sup> <sup>α</sup>(*T*)*<sup>I</sup>* (3)

(Rayleigh scattering) and can therefore be

= λex−<sup>1</sup>

–0.00033 [15].

. The Raman signal of the solvents will be observed at a

less in energy than the excitation wavelength. The wave-

**60**

Since this equation is linear and stationary, the initial values do not affect the solution for the intensity variable. The zero-flux boundary condition is the natural boundary condition and does not impose a constraint or loading term. It is appropriate on most faces, with the exception of the illuminated face. We will assume that the incident laser light intensity follows a Gaussian distribution with respect to distance from the origin. At the origin, and immediately above the material, the incident intensity is 0.3 W/mm2 . Some of the laser light will be reflected at the dielectric interface, so the intensity of light at the surface of the material is reduced to 0.95 of the incident intensity. This condition is implemented with a Dirichlet boundary condition. At the face opposite to the incident face, the default zero-flux boundary condition can be physically interpreted as meaning that any light reaching that boundary will leave the domain.

With these settings described above, the problem of temperature-dependent light absorption governed by the Beer–Lambert law has been implemented. It is, of course, also necessary to solve for the temperature variation in the material over time. In this case thermal conductivity of alpha brass of 93.32 W/m/K is considered, with a density of 2000 kg/m3 and a specific heat of 1000 J/kg/K that is initially at 300 K with a volumetric heat source.

The heat source itself is simply the absorption coefficient times the intensity or, equivalently, the derivative of the intensity with respect to the propagation direction. Most other boundaries can be left at the default thermal insulation, which will also be appropriate for implementing the symmetry of the temperature field. However, at the illuminated boundary, the temperature will rise significantly, and radiative heat loss can occur. This can be modeled with the diffuse surface boundary condition, which takes the ambient temperature of the surroundings and the surface emissivity as inputs.

It is worth noting that using the diffuse surface boundary condition implies that the object radiates as a gray body. However, the gray body assumption would imply that this material is opaque. So how is it possible to use Beer–Lambert law, which is appropriate for semitransparent materials? It is possible to solve this apparent discrepancy by observing that the absorption capacity of the material is highly dependent on the wavelength. The depth of penetration is relatively large. However, when the part is heated, it is reradiated mainly in the long infrared regime. At long infrared wavelengths, it is possible to assume that the depth of penetration is very small and, therefore, the assumption that the volume of the material is opaque for radiation is valid.

**63**

**Figure 4.**

*method.*

**Figure 3.**

*alpha brass.*

*Analysis of the Absorption Phenomenon through the Use of Finite Element Method*

It is possible to solve this model either for the steady-state solution or for the transient response. In **Figure 2** the propagation of heat in the alpha brass material is shown. In **Figure 3** the changes of temperature as a function of depth along the centerline over a time of 40 seconds for six different kinds of alpha brass are shown. In **Figure 4** the mesh used to solve the equations through finite element is shown. Although it is not necessary to use a swept mesh in the absorption direction, applying this feature provides a smoot solution for the light intensity with relatively fewer elements than a tetrahedral mesh. The plot of temperature with respect to

*Creation of the mesh for the calculation of the heat distribution in alpha brass by means of the finite element* 

*Temperature as a function of depth along the centerline over 40 seconds of time, for six different samples of* 

*DOI: http://dx.doi.org/10.5772/intechopen.86924*

**Figure 2.** *Propagation of heat in the alpha brass material, (a) surface profile and (b) penetration profile.*

*Analysis of the Absorption Phenomenon through the Use of Finite Element Method DOI: http://dx.doi.org/10.5772/intechopen.86924*

**Figure 3.**

*Modern Spectroscopic Techniques and Applications*

the incident intensity is 0.3 W/mm2

ing that boundary will leave the domain.

with a density of 2000 kg/m3

surface emissivity as inputs.

300 K with a volumetric heat source.

Since this equation is linear and stationary, the initial values do not affect the solution for the intensity variable. The zero-flux boundary condition is the natural boundary condition and does not impose a constraint or loading term. It is appropriate on most faces, with the exception of the illuminated face. We will assume that the incident laser light intensity follows a Gaussian distribution with respect to distance from the origin. At the origin, and immediately above the material,

dielectric interface, so the intensity of light at the surface of the material is reduced to 0.95 of the incident intensity. This condition is implemented with a Dirichlet boundary condition. At the face opposite to the incident face, the default zero-flux boundary condition can be physically interpreted as meaning that any light reach-

With these settings described above, the problem of temperature-dependent light absorption governed by the Beer–Lambert law has been implemented. It is, of course, also necessary to solve for the temperature variation in the material over time. In this case thermal conductivity of alpha brass of 93.32 W/m/K is considered,

The heat source itself is simply the absorption coefficient times the intensity or, equivalently, the derivative of the intensity with respect to the propagation direction. Most other boundaries can be left at the default thermal insulation, which will also be appropriate for implementing the symmetry of the temperature field. However, at the illuminated boundary, the temperature will rise significantly, and radiative heat loss can occur. This can be modeled with the diffuse surface boundary condition, which takes the ambient temperature of the surroundings and the

It is worth noting that using the diffuse surface boundary condition implies that the object radiates as a gray body. However, the gray body assumption would imply that this material is opaque. So how is it possible to use Beer–Lambert law, which is appropriate for semitransparent materials? It is possible to solve this apparent discrepancy by observing that the absorption capacity of the material is highly dependent on the wavelength. The depth of penetration is relatively large. However, when the part is heated, it is reradiated mainly in the long infrared regime. At long infrared wavelengths, it is possible to assume that the depth of penetration is very small and, therefore, the

. Some of the laser light will be reflected at the

and a specific heat of 1000 J/kg/K that is initially at

**62**

**Figure 2.**

*Propagation of heat in the alpha brass material, (a) surface profile and (b) penetration profile.*

assumption that the volume of the material is opaque for radiation is valid.

*Temperature as a function of depth along the centerline over 40 seconds of time, for six different samples of alpha brass.*

*Creation of the mesh for the calculation of the heat distribution in alpha brass by means of the finite element method.*

It is possible to solve this model either for the steady-state solution or for the transient response. In **Figure 2** the propagation of heat in the alpha brass material is shown. In **Figure 3** the changes of temperature as a function of depth along the centerline over a time of 40 seconds for six different kinds of alpha brass are shown.

In **Figure 4** the mesh used to solve the equations through finite element is shown. Although it is not necessary to use a swept mesh in the absorption direction, applying this feature provides a smoot solution for the light intensity with relatively fewer elements than a tetrahedral mesh. The plot of temperature with respect to

depth at the centerline illustrates the effect of the varying absorption coefficient due to its increase.
