The Monte Carlo Ray-Trace Method in Radiation Heat Transfer and Applied Optics (eBook)

J. Robert Mahan (Autor)

eBook Download: EPUB

2018
John Wiley & Sons (Verlag)
978-1-119-51850-1 (ISBN)

Infos

A groundbreaking guide dedicated exclusively to the MCRT method in radiation heat transfer and applied optics 

The Monte Carlo Ray-Trace Method in Radiation Heat Transfer and Applied Optics offers the most modern and up-to-date approach to radiation heat transfer modelling and performance evaluation of optical instruments. The Monte Carlo ray-trace (MCRT) method is based on the statistically predictable behavior of entities, called rays, which describe the paths followed by energy bundles as they are emitted, reflected, scattered, refracted, diffracted and ultimately absorbed.

The author - a noted expert on the subject - covers a wide variety of topics including the mathematics and statistics of ray tracing, the physics of thermal radiation, basic principles of geometrical and physical optics, radiant heat exchange among surfaces and within participating media, and the statistical evaluation of uncertainty of results obtained using the method. The book is a guide to help formulate and solve models that accurately describe the distribution of radiant energy in thermal and optical systems of practical engineering interest. This important guide:

• Combines radiation heat transfer and applied optics into a single discipline
• Covers the MCRT method, which has emerged as the dominant tool for radiation heat transfer modelling
• Helps readers to formulate and solve models that describe the distribution of radiant energy
• Features pages of color images and a wealth of line drawings

Written for faculty and graduate students in mechanical and aerospace engineering and applied optics professionals, The Monte Carlo Ray-Trace Method in Radiation Heat Transfer and Applied Optics is the first book dedicated exclusively to the MCRT method. 

J. Robert Mahan is Professor Emeritus of Mechanical Engineering at Virginia Polytechnic Institute and State University, where he leads the NASA-funded Thermal Radiation Group.

Textauszug

1
Fundamentals of Ray Tracing

A ray is defined as the collection of straight‐line path segments followed by an energy bundle from its point of emission to its point of absorption. The definition includes the possibility of intermediate reflection, scattering, refraction, and even diffraction events. Ray tracing involves the application of basic mathematics to the process of identifying the intersection of ray segments with surfaces. Most engineering and science students acquire the required mathematical tools long before they enter university. The current chapter provides a review of the mathematical principles governing ray tracing and the related issues of meshing and indexing.

1.1 Rays and Ray Segments

A ray is defined here as the continuous sequence of straight‐line paths connecting a point on one surface, from which an energy bundle is emitted, to a point on a second surface – or perhaps even on the same surface – where it is ultimately absorbed. One or several reflections from intervening surfaces may occur between emission and absorption of the energy bundle. The path followed by the energy bundle between reflections is referred to as a ray segment. Two situations are normally considered: either (i) the power of the emitted energy bundle does not change as it is reflected from one surface to the next until it reaches the surface where all its power is ultimately absorbed; or (ii) a fraction of the emitted power is left behind with each reflection until the remaining power is deemed to have dropped below a threshold value, at which point the ray trace is terminated. Both approaches have their adherents and are in common use, and both are developed in detail in this book.

1.2 The Enclosure

The enclosure is an essential concept in all approaches to radiation heat transfer analysis. We define the enclosure as an ensemble of surfaces, both real and imaginary, arranged in such a manner that a ray emitted into the interior of the enclosure cannot escape. Energy is conserved within the enclosure under this definition. If a ray does leave the enclosure through an opening, represented by an imaginary surface, the energy it carries is deducted from the overall energy balance.

1.3 Mathematical Preliminaries

Consider two points, P0 and P1, in three‐dimensional space, as illustrated in Figure 1.1. Let the Cartesian coordinates of points P0 and P1 be (x0, y0, z0) and (x1, y1, z1), respectively. Then the vector directed from P0 to P1 is

(1.1)

and its magnitude is

(1.2)

In Eq. 1.1 i, j, and k are the unit vectors directed along the x‐, y‐, and z‐axes, respectively. Note that the distance t from P0 to P1 must always be real and positive.

Figure 1.1 Relationships among the quantities introduced in Section 1.3.

The unit vector in the direction of V is

(1.3)

where L, M, and N are the direction cosines illustrated in Figure 1.1. The direction cosines are defined

(1.4)

where α, β, and γ are the angles between the unit vector v and the x‐, y‐, and z‐axes, respectively. Equations 1.1 and 1.3 can be combined to define the equations for the line segment connecting point P0 to point P1

(1.5)

The three equations embodied in Eq. 1.5 are arguably the most important relationships in geometrical optics, because they form the basis for navigation of rays within an enclosure.

The general equation for a surface in Cartesian coordinates is

(1.6)

The simplest, and perhaps most common, surface used in fabricating an enclosure is the plane, illustrated in Figure 1.2. In order to derive the equation for a plane, we must know the unit normal vector n at a point (x′, y′, z′) in the plane and the coordinates of a second point (x1, y1, z1) in the plane. Then, because n and U are in quadrature, it must be true that

(1.7)

Figure 1.2 Definition of a plane surface.

or

(1.8)

To find the intersection of the ray segment V = (x1 − x0) i + (y1 – y0) j + (z1 – z0) k with the plane, we introduce Eq. 1.5 into Eq. 1.8, obtaining

(1.9)

Finally, solving Eq. 1.9 for t we obtain

(1.10)

or

(1.11)

Note that if n and v are in quadrature, n · v = 0, in which case t is undefined. The interpretation is that the ray passes parallel to the plane and so can never intersect it. We must anticipate this eventuality when programming. This is perhaps an appropriate juncture to emphasize the natural compatibility of Cartesian coordinates with the vector nature of ray tracing.

A more instructive example is the intersection of a ray segment with a sphere of radius R whose center is located at (xC, yC, zC); that is,

(1.12)

Suppose a ray is emitted from point (x0, y0, z0) in the direction (L, M, N) and we want to find its point of intersection (x1, y1, z1) with this sphere. As in the previous example, this may be accomplished by finding the point (x1, y1, z1) that simultaneously satisfies the three equations for the straight line connecting the two points, Eq. 1.5, and the equation for the sphere, Eq. 1.12; that is,

(1.13)

Happily, the solution of Eq. 1.13 for the distance t is just about the most challenging mathematical operation we encounter in ray tracing.

It is convenient to use the symbolic solver feature of Matlab to solve quadratic equations (see Problems 1.2–1.7). However, solution of Eq. 1.13 is relatively straightforward and provides an opportunity to point out certain useful properties of the quadratic coefficients. Upon carrying out the indicated operations and rearranging the result, we have

(1.14)

or

(1.15)

where

(1.16)

(1.17)

and

(1.18)

The coefficients A, B, and C are defined in terms of known quantities and, thus, are themselves known. Equation 1.14 can now be solved for t, yielding

(1.19)

We define a quadratic surface as any surface whose algebraic equation S(x, y, z) = 0 is second‐order. It turns out that, in addition to plane surfaces, essentially all enclosures of practical engineering interest include such surfaces or surfaces that can be subdivided into such surfaces. Listed in Table 1.1 are some of the quadratic surfaces commonly encountered in radiation heat transfer and applied optics. Equation 1.19, with generally different expressions for the coefficients A, B, and C, yields the candidate values of the distance t for all quadratic surfaces.

Table 1.1 Quadratic surfaces commonly encountered in radiation heat transfer and applied optics modeling.

Name

S(x, y, z) = 0

Notes

Sphere

(x − xC)2 + (y − yC)2 + (z − zC)2 − R2 = 0

R = radius, center at (xC, yC, zC)

Tri‐axial ellipsoid

a, b, c = semi‐axes, center at (xC, yC, zC)

Spheroid

Prolate if c > a, oblate if c < a, center at (xC, yC, zC)

Elliptic paraboloid

Upward‐opening, origin at (xC, yC, zC)

Hyperbolic paraboloid (“Potato chip”)

(x/a)2 − (y/b)2 − z = 0

Opens up along x‐axis, down along y‐axis

Two‐sheet hyperboloid

(x/a)2 + (y/a)2 − (z/c)2 + 1 = 0

Rotationally symmetric about...

Produkt-Details

• Erscheint lt. Verlag: 26.11.2018
• Reihe/Serie: Wiley-ASME Press Series
• Reihe/Serie: Wiley-ASME Press Series
• Reihe/Serie: Wiley-ASME Press Series
• Sprache: englisch
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: binning of rays on a surface element • blackbody radiation distribution function • directional spectral emissive power • fundamentals of thermal radiation • hemispherical total emissive power • ideal models for emission, reflection, and absorption in radiation heat transfer • intensity of radiation • <p>A guide to the Monte Carlo Ray-Trace Method in Radiation Heat Transfer and Applied Optics • Maschinenbau • mechanical engineering • Physics • Physik • radiation distribution factor for diffuse-specular exchange • Resource to The Monte Carlo Ray-Trace Method in Radiation Heat Transfer and Applied Optics • text on The Monte Carlo Ray-Trace Method in Radiation Heat Transfer and Applied Optics • thermal and optical analysis of radiometric instruments</p> • Thermal Physics & Statistical Mechanics • thermodynamics • Thermodynamik • Understanding The Monte Carlo Ray-Trace Method in Radiation Heat Transfer and Applied Optics • Wärmelehre • Wärmelehre u. Statistische Mechanik
• ISBN-10: 1-119-51850-4 / 1119518504
• ISBN-13: 978-1-119-51850-1 / 9781119518501