UVC Lamps, Part 1: Comparison of Models for Irradiance

• Introduction
• Keitz model
• Beggs model
• Modest model
• Comparison of models
• Conclusion
• References

Introduction

When one deploys a UVC germicidal lamp for the purpose of disinfection, there are two fundamental relationships that need to be known. Such lamps are low pressure mercury discharge lamps with UV transparent envelopes and are in the form of long tubular shapes. The fluency (dose) of UVC energy that is needed to substantially deactivate the target pathogen is one. The other is the UVC irradiance as a function of distance from a lamp of a given rated power. This latter information allows one to set up distance and time of lamp operation so that sufficient dose is delivered to the objects to be disinfected. The topic presented here deals with this second aspect, namely the models used to calculate irradiance.

When one searches the literature quite a number of research papers are found that use a model to calculate irradiance. This can seem confusing because they are not all the same. A bit of study shows that there appears to be three different mathematical models. These can be referred to by citing their origins: the Keitz model (1971)[1], the Modest model (1993)[2], and the Beggs model (1998)[3]. The Keitz model is embraced in the handbook[4] of the American Society of Heating, Refrigerating and Air-Conditioning Engineers while the Modest model is specified in the lighting handbook[5] of the Illuminating Engineering Society of North America.

Since these three models are mathematically different from each other, the question arises: do they all predict the same behavior, as they should? The answer to this question is explored in this post.

Keitz model

The model for the lamp is depicted in the following diagram:

The detector is placed at point P on the perpendicular to the lamp axis (A-B) that bisects the lamp length L. The distance to the detector is a and alpha is the angle shown in Figure 1. The Keitz model assumes the detector is a Lambertian surface with normal perpendicular to the lamp axis and therefore obeys the cosine law. The expression for irradiance is shown in equation (1).

E = \frac{\phi}{2\pi^2La} (2\alpha + \sin{2\alpha}) \; (1) \\ \;

E is the irradiance measured at the detector, that is, power per unit area incident on the detector. The value of phi is the total UVC power output of the lamp (which is typically 1/3 of the rated lamp power for low pressure mercury discharge lamps).

Beggs model

The Beggs model is mathematically similar to the Keitz model but yet different, which immediately raises the question “why?” when one first encounters these models. Beggs developed the model shown in Figure 2.

This a bit more general than what Keitz developed in that the point where irradiance is calculated is not on the line bisecting the lamp. Also, the detector is a pathogen, which means the detector is essentially a small sphere and the cosine law does not apply to it. The expression is shown in equation (2).

E = \frac{\phi}{((l_{1} + l_{2})h}(\sin{\alpha_{1}} + \sin{\alpha_{2}}) \;(2) \\ \:

The symbols take on the same meaning except here the distance from the lamp is h and the lamp length is broken into two segments. One can immediately see that when the pathogen is on the centerline bisecting the lamp that the angles become equal and equation (2) does NOT reduce to equation (1).

Modest model

The geometry in this model is essentially the same as in the previous two models except the lamp is broken into two segments and the irradiance is calculated for each segment, then added together. Furthermore, the lamp is explicitly treated as a cylinder whereas the previous two models treat the lamp as a line source. In Figure 3 is shown the geometry for a lamp segment.

The detector is aligned at the end of the lamp segment and is a Lambertian surface obeying the cosine law. The lamp segment length is l, the lamp radius is r, and the distance to the detector is x. The expression for the irradiance at the detector is considerably more complicated than the previous two models.

E = \frac{\phi}{2\pi rl}F \; \;(3)\\ \;\\where \;F = \frac{L}{\pi H}\left[\frac{1}{L} \arctan{\left(\frac{L}{\sqrt[2]{H^2 - 1} }\right)} - \arctan{\left(M\right)} + \frac{X-2H}{\sqrt[2]{XY}}\arctan{\left(M\sqrt[2]{\frac{X}{Y}}\right)}\right] \\ \;\\and \; H = \frac{x}{r}, \; L = \frac{l}{r}, \; X = (1+H)^2 + L^2, \; Y = (1-H)^2 + L^2, \\ \;\\and\; M = \sqrt[2]{\frac{H-1}{H+1}}\\ \;

Again phi is the total lamp UVC power as before. It is noted that F is a dimensionless value (called the view factor) and the factor it multiplies is recognized to be the exitance, which is the surface intensity of the diffuse radiant power emitting from the lamp. Since equation (3) applies only for one lamp segment, the expression needs to be evaluated for both lamp segments comprising a complete lamp and the results added. Note that this model is similar to the Beggs model in that the detector need not be on the centerline bisecting the lamp.

Comparison of models

Now that all three models are expressed mathematically, and all three expressions are considerably different from one another, it begs to understand if they all predict the same irradiance for a given lamp and a given detector geometry respect to the lamp. The easiest way to do this is simply numerically evaluate them. For this purpose a particular UVC lamp of the twin tube type that was used in measurements described in Part 2 of this particular post is chosen. It is shown in Figure 4.

The lamp is driven by a self-resonating electronic ballast at 32 KHz frequency and is incorporated in the base seen in the figure. A plastic frame that fixtures the lamp at the top is removed so that the full arc length is exposed. The parameters of this lamp are the following:

1. Lamp is rated at 36 watts.
2. Wall plug efficiency is taken as 33 percent. Therefore, total UVC power is 12 watts.
3. The tubes are T5. Therefore, tube diameter is taken to be 5/8 inches.
4. The arc length of each tube is 15 inches.
5. The irradiance along the bisected centerline of a single tube as a function of distance from the tube is calculated. A single tube is taken to emit 6 watts of UVC.
6. The result is multiplied by two to account for two tubes.

MathCad was the tool used to do the calculations. The evaluation results in the comparison seen in Figure 5.

Remarkably it is seen that the Keitz model and the Modest model agree precisely. The Beggs model predicts a substantially different behavior. One difference is that Beggs assumes a non-Lambertian detector. However, the literature reveals that investigators have shown this renders a very small difference. If one divides the Beggs result by the square of pi, the comparison becomes that shown in Figure 6.

Remarkably the Beggs model comes into substantial agreement with the others. It seems clear that the Beggs model should have a pi squared in the denominator. This can be verified by consulting the paper by Sasges, et al [6], which is a precise derivation of the Keitz model. In that paper one may follow the derivation leading to the following equation:

I_{L}\left[r,\theta,\gamma \right] = \frac{dP\cos{\theta}}{\pi^2r^2}\cos{\gamma} \\ \;

Since the detector is a spherical, non-Lambertian particle, the cosine of theta does not apply and is replaced by 1. Then one may continue with the derivation and the integral becomes:

I\left[P,L,D\right] = \int_{\frac{-L}{2}}^{\frac{L}{2}} \frac{\frac{P}{L}}{\pi^2\left(D^2+x^2 \right)} \frac{D}{\sqrt[2]{D^2+x^2}} dx \\ \;

The integral is a definite integral and can be looked up in any book of mathematical tables[7]. Clearly there is a pi squared in the denominator. The result is:

\frac{2P}{\pi^2L}\sin(\alpha) \\ \; \\ where\; \alpha = \arcsin{\frac{\frac{L}{2}}{\sqrt[2]{D^2+(\frac{L}{2}^2)}}} \\ \;

It now can be recognized this is the same as equation (2) evaluated on the bisecting centerline but appropriately corrected with the missing pi squared.

Conclusion

The Keitz and Modest models agree over the complete range of distances, including close to the lamp. The Beggs model assumes spherical particles as the detector and therefore deviates somewhat from the others close to the lamp in the manner expected, which is to intercept a bit more irradiance. Needless to say this is true only after correcting the Beggs model. In my search of the literature I have not come across any paper that subsequently used the Beggs model and has recognized this error.

References

[1] Keitz, H. A. E. 1971. Light Calculations and Measurements. 2nd Ed. Eindhoven, Netherlands: N. V. Phillips.

[2] M. F. Modest, Radiative Heat Transfer(McGraw-Hill, New York, 1993)

[3] C. B. Beggs, et al, “An engineering approach to the control of Mycobacterium tuberculosis and other airborne pathogens: a UK hospital based pilot study”, TRANSACTIONS OF THE ROYAL SOCIETY OF TROPICAL MEDICINE AND HYGIENE (2000) 94,141-146.

[4] Chapter 17, “Ultraviolet Lamp Systems”, 2016 ASHRAE Handbook—HVAC Systems and Equipment

[5] The IESNA LIGHTING HANDBOOK, Ninth Edition, Illuminating Engineering Society of North America, Figure 9-2.

[6] Michael Sasges, et al, “Ultraviolet Lamp Output Measurement: A Concise Derivation of the Keitz Equation”, Ozone: Science & Engineering, 34: 306–309

[7] Dwight, H. B., Tables of Integrals and Other Mathematical Data, (The MacMillan Company, 1947), 200.3, p.45.