Derivation of Keitz Model Using Conventional Definitions

The derivation found in

Ultraviolet Lamp Output Measurement: A Concise Derivation of the Keitz Equation by Michael Sasges, Jim Robinson, and Farnaz Daynouri Ozone: Science & Engineering, 34: 306–309

uses some symbols defined differently from the usual optic definitions and, importantly, treats radiance from the source in a different manner.  Therefore, the derivation done below follows convention and tries to more clearly describe the ray optics at play.  It is then discussed in comparison to paper cited.

Conventional Approach for a Flat Diffuse Source

In the conventional approach one defines radiance of a ray from a source in terms of L_o = power per area of the source per solid angle and this value can vary with the angle at which it leaves the flat surface of the source.  Additionally, rays with this radiance will leave the flat source surface in a beam at an angle  and Lambert’s law will reduce the cross-sectional area of this beam with the angle with respect to the  normal to the surface according to the cosine law.  Therefore the power in a ray bundle or beam passing through the flat surface into a solid angle in a direction off-normal to the surface is:

where L_o is in power per area per steradian of solid angle and dA_s is the source area.

For a diffuse source L_o is taken as isotropic (not a function of θ) and therefore is a constant.  However, the beam cross-section is still a function of θ and therein arises the cos θ. One may visualize the radiance pattern as depicted in Figure 1.

A small flat diffuse area sits at the origin with its normal pointing in the +y direction.  The yellow sphere represents the radiance pattern described by equation (1).  A point on its surface represents the direction of a ray beam and its distance from the origin represents the magnitude of its power = L_o cosθ dA_s . The radiance L_o is essentially the brightness of the source.  One normally wants to relate its value to the total power that passes through this small flat area into the half-space into which it is radiating.  Therefore one begins with an expression for the power in a beam into a solid angle.

dω is the solid angle subtended by a small (imaginary) detector on the surface of the sphere of radius r and the detector has area  dA_d  with its normal at an angle ς relative to the normal of the spherical surface at the point at which the detector is positioned.  The surface of the hemisphere is capturing all of the rays emitting from the source and therefore one makes the sphere be the detector by setting  ς = 0  and one then integrates over the hemisphere.

At this point one can make use of the symmetry of the radiance pattern around the y-axis.  Shown in Figure 2 is an anullar ring of thickness r dθ for which cos θ is constant. 

Therefore the integration becomes:

which is the total power per unit area passing through and out of the surface. This is defined as the exitance of the source.

Conventional Approach for a Diffuse Line Source

In Keitz a diffuse flat source is not assumed and instead a line source is assumed so the problem changes from three dimensions to two dimensions as far as the radiance is concerned even though the line source still radiates in all three dimensions.  If one aligns the line source at the origin with its axis coincident with the x-axis, the earlier equation (1) applies in any plane coincident with the x-axis and L_o takes on the significance of power per line length per steradian and dA_s → dl_s .  The source is diffuse so L_o is constant.  One then writes:

where dl_s  is the width of the line source.

Since this equation is true in any plane coincident with the x-axis one can visualize the three dimensional  radiance pattern being a torus as depicted below.  The “hole” in the torus has zero radius.

The distance from the origin to any point on the surface of the torus represents the magnitude of power in a beam that is flowing in the direction of the point.  The cross-sectional view below shows this along with a few additional rays being drawn.

 As before one wants to relate the radiance L_o to the total power being sent out of the differential element of line source dl_s.  Since the line source sends out power in all directions in three dimensional space a sphere is drawn in the above figure to be the receiving surface of this power.  The two figures above demonstrate the symmetry of the radiance pattern about the x-axis so that for constant cos θ the solid angle subtended is a ring coincident with the x-axis of thickness r dθ  and circumference 2πr cos θ.  This is seen in Figure 4a.

When one puts this together in the same manner as was done for the flat area source where one makes the surface of the sphere the receiving surface of which all elements of its surface point directly at the line source one writes:

where the factor of 2 accounts for both halves of the sphere. The evaluated integral results in π/4  thus yielding the following expression:

where the exitance in this case is power per unit length of line source. In the Sasges, et al paper mentioned above the authors set M=P/L where P is the total power in the lamp with length L since every source element dl_s has the same exitance.

Equation (10) is the power emitted through each line source element along the length of the lamp. At this point Figure 5 taken from the referenced paper can be used to find the irradiance falling on a flat sensor in a plane normal to the lamp axis and bi-secting the lamp.

The integral can be looked up in any of a number of Tables of Integrals and is

evaluated at the limits stated.  At the lower limit both terms are zero and at the upper limit one gets:

At this point one can put this result in terms of the angle α found in most presentations of the Keitz equation where

So the final result is:

Discussion

There are two complaints of a substantive nature regarding the derivation in the paper by Sasges, et al. The first is the authors treat differently what herein is L_o cos θ dl_s .  They define

and call I_o the flux per unit area in the direction normal to the line element. Of course the line element does not have an area but only a length, so this is confusing.  Mathematically I_o is equivalent to L_o. There is no mention whether their “area” is Lambertian.  Their power I(θ) in rays that flow in a direction at angle θ varies while conventionally in a diffuse source it does not. The power in the ray is constant for a diffuse source.   Only beam cross-section varies as angle changes, and this is Lambert’s Cosine Law.  Only the power in ray bundles (or beams) varies with angle. But, mathematically there is no consequence that the authors treat things this way but the physics is wrong .

Next the authors set up the integral to get the total power that flows to the enclosing sphere.  They do not describe properly the set up, but in the end mathematically the result is the same as if they set it up properly.  They define a receiving area on the sphere surface using the symmetry of the source about the x-axis.

Then the power received by all of the surface of the sphere is:

This is somewhat nonsensical because the total power flowing out of the line source element cannot be a function of the size of the sphere!  The authors have failed to take into account power flowing into a solid angle.  They have not set this up properly in that the dA should be dA/r² where the latter is the solid angle subtended by dA.  However, as before, this has no mathematical consequence because this error is compensated for when the result is used in calculating radiance received by a detector in a plane bi-secting a long linear lamp. However, in the paper the quantity I_o π² is not singled out and properly identified as the exitance M of the line source element.

Appendix

Because the starting point in the Keitz model derivation is a source that is one- dimensional, that is, a line with a differential  length, equation (7) was the starting point.  There the radiance L_o took on the significance of power per unit length per steradian.  The radiance pattern in Figure 3 resulted and the equation for exitance M resulted in equation (9).  Its unit for a line source is power per unit length.

One instead could have started at an earlier point in regards to applying the conventional physics by beginning with the flat diffuse source found in equation (1) that has radiance pattern seen in Figure 1. One imagines a cylinder of length dl_scentered at the origin with axis coincident with the x-axis and having a small radius R.  It is essentially a ring because it is so short.  The surface of this ring can be constructed of flat diffuse source elements of area dA_s with their normal pointing in the radial direction.  These source elements are described by equation (1) where L_o takes on the usual significance of power per unit area per steradian. The radiance pattern of each of these elements is the same as in Figure 1 except each element is displaced from the origin by radius R. Each of these flat source elements have exitance M defined in equation (6) and its units are power per unit area.  One such element is shown in Figure 6 with its spherical radiance pattern and its integrating hemisphere used to calculate M.

Now one wants to find the total power emitting through the surface of the ring per unit length of ring. One can visualize the radiance pattern from the surface of ring by replicating the spherical pattern seen in Figure 1 for each and every surface element comprising the surface of the ring.  Thus the spheres sweep out a toroidal radiance pattern similar to Figure 3 but that it now has an inner radius R.  This toroidal radiance pattern is seen in the video and in Figure 7 along with its integrating sphere that is used to calculate the total power emitting from the surface of the ring.

Therefore to get the total power through the surface of the ring one needs to sum all of the surface elements.  One starts by re-writing equation (5).

The differential length R dθ is the arc length on the ring representing one side of a flat surface element.  When one sums up all of the surface elements one sees that these arc lengths just add up to the circumference of the ring.

One now recognizes that 2R is the diameter of this cylindrical surface element and that its product with L_o defines a radiance with dimensions power per unit length per steradians, which is the radiance defined in equation (7).   At this point one can identify that equation (17) is really the same as equation (9).  Keeping in mind that Φ is really a differential element of power since the ring has a differentially small length one writes:

where L_o and M take on the meaning of power per unit length per steradian and power per unit length, respectively.  The radius R can be taken to be very small approaching zero. Consequently one arrives at the same starting point as in equation (7) yet the light source is built with flat diffuse sources.