# Encyclopedia of Color Science and Technology

2016 Edition
| Editors: Ming Ronnier Luo

# Coefficient of Utilization, Lumen Method

Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-8071-7_117

## Definition

The lumen method is an indoor lighting calculation methodology that allows a quick assessment of the number of luminaires necessary to achieve a given average illuminance level or alternatively the average illuminance level that will be achieved for a given number of luminaires. It is valid for empty rectangular rooms with simple three-surface diffuse reflectances for ceiling, wall, and floor.

## Introduction

When an electric lamp is turned on, it emits light and it is possible to quantify the amount of light by measuring it, the result being given in the units of lumens. In itself this is a useful piece of information, but what would be more useful would be to have a method of converting this into a measure of the amount of light that would be received onto a desk from one or more luminaires or alternatively the number of luminaires necessary to achieve a given quantity of light on the desk. This calculation is known as the lumen method.

Light falling onto a surface is called illuminance and has units of lux (lx) or lumens per square meter (lm/m2). It is not possible to see illuminance as light is actually invisible, which is fortunate, else any view would be degraded by looking through a fog of light. What is actually seen is the effect light has on surfaces, the reflected light, and this is called luminance with units of candelas per square meter (cd/m2). To test this, place a sheet of white paper onto a dark surface (see Fig. 1). At the point where the edge of the paper meets the dark surface, the amount of light falling onto the two materials will be approximately the same. However, they appear completely different as the eye detects the light reflected back towards it, not the light falling onto the surfaces, and the white paper reflects more light than the dark desktop.

However, luminance is a difficult quantity to measure and changes with viewing position. Imagine a day-lit room with resultant shadows and patches of light reflected from polished surfaces. As the observer position moves within the space, the shadows and patches of light change with viewing position. As vision is essentially viewing luminance, this means that the luminance is view dependent. This makes it a difficult design quantity, so for ease, common practice is to design using illuminance which is view independent as the amount of light falling onto a surface does not change with viewing position. (There are a few exceptions, an example being traffic routes, where design may use luminance, in this case the light that is reflected from the road surface).

## Calculating the Maximum Achievable Illuminance

To be able to calculate the amount of light falling onto a surface from a given number of luminaires, it is necessary to be able to convert between the measures of lamp lumens (lm) and illuminance (lx). Remember lx is also lm/m2, so a given total quantity of lumens can be converted into illuminance by dividing it by the size of the area to be lit in m2. But it is important to consider where the area to be lit really is. Ideally luminaires should provide task lighting, that is, light the task being undertaken at any given point to the correct level of illuminance. This means that if there is a desk in a room, it would generally be lit to a level of 500 lx, the recommended level of illuminance for reading and writing tasks [4]. However, the circulation space within the room does not need this quantity of light, 200 lx being a perfectly adequate illuminance level to safely move around the space. This has two problems:
• Frequently in large spaces, it is not known where tasks such as desks will be positioned and may not be even clearly known what tasks will be performed within the space. Even if the task types and locations are known, most spaces and how they are used change through time. This means that, within reason, lighting has to be flexible enough to preserve the correct lighting conditions for changing requirements.

• Calculating the illuminance level for a particular area within a larger space is generally beyond the ability of a quick and easy calculation method. For lighting the calculation of an average illuminance across a space is relatively easy, while the calculation of the illuminance across a desk located in a room would normally require the use of a computer calculation program.

Therefore, to keep the calculation simple, it will calculate the average illuminance across a horizontal plane, called the task plane. This could either be the floor or a virtual plane at desk height depending upon the expected height of the task. So remembering the units of lm/m2, it is necessary to know the total quantity of lumens within the space. So
• For a given number of luminaires, Nlum

• Where each luminaire contains a given number of lamps, Nlamp

• And each lamp produces a given quantity of lumens, lmlamp

the total amount of lumens available, lmtotal, is equal to
$${\mathrm{lm}}_{\mathrm{total}} = {\mathrm{N}}_{\mathrm{lum}}*{\mathrm{N}}_{\mathrm{lamp}} * {\mathrm{lm}}_{\mathrm{lamp}}$$
(1)
So for a given area of task plane, Atp, and from Eq. 1, the known total amount of lumens available, lmtotal, the absolute maximum illuminance possible, lxmax, will be
$${\mathrm{lx}}_{\max} = {\mathrm{lm}}_{\mathrm{total}}/{\mathrm{A}}_{\mathrm{tp}}$$
(2)
There are two points to consider regarding the value produced by Eq. 2.
• This value assumes all the light produced by the lamps will be received onto the task, with no losses. This is generally not possible or even desirable.

• As this is the maximum illuminance theoretically (but not practically) achievable, if the illuminance is too low, at this point more luminaires and, hence more lumens, will be required.

So how is this result converted into a more practical value? The losses in light come from three main effects:
• Losses within the light fitting

• Losses through aging and dirt

• Losses through light not going directly to the task plane but via reflection from room surfaces

## Accounting for Losses Within the Light Fitting

A luminaire is a housing containing a light source with associated control gear, optics, wiring, and electrical connections, both to the light source and to the external power supply. This creates a number of problems:
• The luminaire has its own microclimate, the temperature inside the fitting generally being different to that of the air outside. Designing a fitting so that the internal temperature is the optimum for the light source it contains requires skilled design, and the larger the difference between the optimum light source conditions and the actual luminaire conditions results in an increasing reduction in lumen output from the light source and/or reduced operating life expectancy for the source.

• An optic, no matter how well designed, has an element of inefficiency. No surface will reflect 100 % of any light incident upon it and losses are cumulative. So, for example, if a metal reflector has a reflectance of 92 %, then light that bounces once off the reflector before exiting the luminaire loses 8 % of its initial lumens. If it requires two bounces to exit the fitting, only 0.92 * 0.92 = 0.85 or 85 % of the initial lumens exit the fitting, 15 % being lost. Similarly optics that rely on transmission of light, such as diffusers or prismatic controllers, absorb some of the light, no material being 100 % transmissive.

• Components within the luminaire will obstruct light, and this light may become trapped and never exit the fixture. For example, light emitted from the light source upwards into a ceiling recessed luminaire will need to be reflected around the light source, and some light will be lost in this process, trapped behind the source. (This can cause extra problems if the light is absorbed by the source, causing it to heat up further). Many reflectors trap light in their back which may be open and shaped as the inside of a V with low reflectance.

Therefore, a measure is needed to quantify how all of these situations affect the lumen output of the luminaire. This is the light output ratio (LOR). Essentially the LOR is the ratio of light emitted from the luminaire to the light emitted by the light source, and the value of LOR is luminaire specific.

So to account for these effects, Eq. 2 is modified as shown in Eq. 3 below:
$${\mathrm{lx}}_{\mathrm{luminaires}} = {\mathrm{lm}}_{\mathrm{total}} * \mathrm{L}\mathrm{O}\mathrm{R}/{\mathrm{A}}_{\mathrm{tp}}$$
(3)

## Accounting for Losses Through Aging and Dirt

When a lighting system is first installed, the lamps are new and all functioning, the luminaires are clean, and generally the room surfaces (floor, walls, and ceiling) are clean. However, through time the condition of the installation will deteriorate. As light sources age, their lumen output reduces (lumen depreciation) and some lamps will fail completely. Dust and dirt will gather on the reflecting surfaces of the luminaires, reducing how efficiently light is directed from the light source out of the fitting, and room surfaces will become dirty and marked through everyday wear and tear. All of these will reduce the amount of light reaching the task plane.

A suitable maintenance routine, such as renewing aging lamps and cleaning the luminaires, will help minimize these impacts, but the amount of light received onto the task plane will still vary through life, reducing through time between maintenance cycles, increasing back to close to the original lighting levels immediately after a maintenance cycle (see Fig. 2).
However, a lighting design should be designed to produce a level of maintained illuminance, so that even at the point of maximum reduction in light just before maintenance is performed, the required light level is still achieved. To account for this, a maintenance factor (mf) is used, which is the amount of light lost when the light sources are at their oldest and the luminaires and room surfaces are at their dirtiest. So modifying Eq. 3 produces
$${\mathrm{lx}}_{\mathrm{mf}} = {\mathrm{lm}}_{\mathrm{total}} * \mathrm{L}\mathrm{O}\mathrm{R} * \mathrm{m}\mathrm{f}/{\mathrm{A}}_{\mathrm{tp}}$$
(4)
Further advice on the determination of maintenance factors is available from the Commission Internationale de L’Eclaraige [3].

## Accounting for Reflection Losses from Room Surfaces

Equation 4 still makes one major assumption that all of the light from the luminaires goes directly onto the working plane.

However, this is rarely the case, light being directed onto the room surfaces (and it is rarely desirable for all light to go directly to the task as this would result in a pool of light within a dark room which would be an uncomfortable work environment). So some of the light hits the room surfaces (see Fig. 3), and some of this light will be reflected back onto the task (see Fig. 4). However, it should be remembered that no surface will have 100 % reflectance. The quantity of light reflected will depend upon the material properties of a surface, a light-colored wall typically having a reflectance in the region of 60 %, so a quantity of light will be lost with each reflection from a surface. To account for this, a measure called the utilization factor (UF) is used. This is a measure of the total amount of light from a luminaire that reaches the task, both directly from the luminaire and indirectly through reflection from room surfaces. Manufacturers publish tables of utilization factors, and these vary as values are dependent upon the luminaire distribution, the room reflectance, and also upon the ratio of wall surface area to ceiling/floor surface area. (Given a ceiling with a high reflectance within a room that is tall and thin proportionally, there is a large amount of wall surface area to ceiling surface area, so the high reflectance ceiling will have less effect than for a large open plan office with a large ceiling surface area compared with the wall surface area).
To calculate the ratio of wall to ceiling/floor surface area, the room index (K) is used. This is defined as
$$\mathrm{K} = \left(\mathrm{area}\ \mathrm{of}\ \mathrm{ceiling} + \mathrm{area}\ \mathrm{of}\ \mathrm{floor}\right) /\mathrm{total}\ \mathrm{area}\ \mathrm{of}\ \mathrm{wall}$$
(5)
For a room of length L, width W, and height H (where the height is the distance between the task plane and the luminaire plane, see Fig. 5), Eq. 5 becomes
$$\mathrm{K} = \left(\left(\mathrm{L}\ *\ \mathrm{W}\right)+\left(\mathrm{L}\ *\ \mathrm{W}\right)\right)/\left(2\left(\mathrm{L}\ *\ \mathrm{H}\right)+2\left(\mathrm{W}\ *\ \mathrm{H}\right)\right)$$
Therefore,
$$\mathrm{K} = 2\left(\mathrm{L}\ *\ \mathrm{W}\right)/2\mathrm{H}\left(\mathrm{L} + \mathrm{W}\right)$$
Therefore,
$$\mathrm{K} = \mathrm{L}\ *\ \mathrm{W}/\left(\mathrm{L}+\mathrm{W}\right)\mathrm{H}$$
(6)
So given the example table of utilization factors shown in Table 1, it can be seen that values of utilization factor are supplied for a variety of room indices and surface reflectance’s. From the table, it can be seen that for a room with reflectance’s of
Coefficient of Utilization, Lumen Method, Table 1

An example utilization factor table

• Ceiling 70 %

• Walls 50 %

• Floor 20 %

and with dimensions
• Length 12 m

• Width 12 m

• Height 3 m

which give a room index of
$$\mathrm{K} = \left(12\ *\ 12\right)/\left(\left(12 + 12\right)*3\right)=2$$
the utilization factor would be 0.69.

(Values used in the calculation are always percentages, so in this case, 69 % = 0.69. Some tables may already show the values as fractions, in this case 0.69).

Therefore, using Eq. 4 adjusted by the utilization factor gives
$${\mathrm{lx}}_{\mathrm{final}} = {\mathrm{lm}}_{\mathrm{total}}*\ \mathrm{L}\mathrm{O}\mathrm{R}\ *\ \mathrm{m}\mathrm{f}\ *\ \mathrm{U}\mathrm{F}/{\mathrm{A}}_{\mathrm{tp}}$$
(7)
So with knowledge of the type and number of luminaires in a space, it is possible to calculate approximately how much illumination the space will have.

Information on the calculation of utilization factor tables is available from the Commission Internationale de L’Eclaraige [1, 2].

## Calculating the Number of Luminaires Required in a Room

Equation 7 may also be rearranged to allow the calculation of the required number of luminaires to achieve a given illumination level.
$${\mathrm{lm}}_{\mathrm{total}}={\mathrm{lx}}_{\mathrm{final}} * {\mathrm{A}}_{\mathrm{tp}}/\mathrm{L}\mathrm{O}\mathrm{R} * \mathrm{m}\mathrm{f} * \mathrm{U}\mathrm{F}$$
(8)
And using Eq. 1 gives
$${\mathrm{N}}_{\mathrm{lum}} * {\mathrm{N}}_{\mathrm{lamp}} * {\mathrm{lm}}_{\mathrm{lamp}}={\mathrm{lx}}_{\mathrm{final}} * {\mathrm{A}}_{\mathrm{tp}}/\mathrm{L}\mathrm{O}\mathrm{R} * \mathrm{m}\mathrm{f} * \mathrm{U}\mathrm{F}$$
So the number of luminaires required to achieve a given level of illumination is
$${\mathrm{N}}_{\mathrm{lum}}={\mathrm{lx}}_{\mathrm{final}}* {\mathrm{A}}_{\mathrm{tp}}/\mathrm{L}\mathrm{O}\mathrm{R} * \mathrm{m}\mathrm{f} * \mathrm{U}\mathrm{F} * {\mathrm{N}}_{\mathrm{lamp}} * {\mathrm{lm}}_{\mathrm{lamp}}$$
(9)

## References

1. 1.
CIE Publication 40: Calculations for interior lighting – basic method (1978)Google Scholar
2. 2.
CIE Publication 52: Calculations for interior lighting – applied method (1982)Google Scholar
3. 3.
CIE Publication 97: Guide on the maintenance of indoor electric lighting systems (2005)Google Scholar
4. 4.
ISO 8995-1:2002(E)/CIE S 008/E: 2001: lighting of work places part 1: indoorGoogle Scholar