www.elsevier.com/locate/enbuild

Energy and Buildings 37 (2005) 485–492

A design tool for predicting the performances of light pipes

David Jenkins*, Tariq Muneer, Jorge Kubie

School of Engineering, Napier University, Edinburgh EH105DT, UK

Received 18 May 2004; received in revised form 11 August 2004; accepted 14 September 2004

Abstract

Light pipes are simple means of directing daylight (diffuse and direct light) into interior spaces. Previous work by the authors described the

initial work on a luminous flux and illuminance predictive model for straight light pipes, using a basic equation for illuminance distribution as

a function of horizontal distance. Further work has now produced a model that uses the cosine law of illuminance to describe the distribution of

light from the light pipe diffuser as well as takes into account pipe elbow pieces or bends. The resulting illuminance model can be described as

a quartic cosine model. By producing a ‘‘luxplot’’ prediction for any given light pipe application, it is possible to maximise the potential of

these daylight providers and design their configuration to suit any given need. As part of this study, wide-ranging illuminance and luminous

flux data were collected both for the formulation of this model (as the formula is semi-empirical) and its validation.

# 2004 Elsevier B.V. All rights reserved.

Keywords: Light pipes; Daylight; Modelling; Lighting

1. Introduction

Much literature has been published that highlights the

problems associated with a lack of daylight illumination in

places such as office spaces and schools [1]. Additionally, a

large amount of energy is consumed within the UK that is a

result of electric lighting. For example, recent statistics

published by the Department for Trade and Industry [2],

show that approximately 53.5 million tonnes of oil

equivalent is associated with electrical lighting/appliances

in the domestic and industrial sectors.

One method for optimising the available daylight would

be to use light pipes. These are simple light guides, varying

in size depending on the application, that reflect the sunlight

and diffuse skylight into a given interior space. By using

devices such as light pipes, the daytime energy consumption

can be significantly reduced in buildings throughout the

country. This obviously requires a reasonably accurate

design tool so that it is possible to put forward a specific

* Corresponding author.

E-mail addresses: davepjenkins@hotmail.com,

djenkins@napier.ac.uk (D. Jenkins).

0378-7788/$ – see front matter # 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.enbuild.2004.09.014

configuration for any situation that might require a minimum

illuminance at a specific level.

The basic passive light pipe design considered for this

study is a collector on the roof of a building attached to

reflective tubing (forming the pipe itself) that reached a

diffuser/emitter at ceiling level. There are however different

types of light pipes that vary in pipe transmittance, collector

transmittance and diffuser transmittance. The pipe con-

sidered for this study has a reflectivity of 95% and is made

from polished aluminium sheets. The light is therefore

transported via multiple specular reflection through the pipe

itself. The collector is a hemispherical polycarbonate top

dome and has, in previous literature [3], been given a

transmission of 88%. The diffuser is also made from clear

polycarbonate material and has a stippled finish that, while

letting a significant amount of light through, diffuses the

light throughout the room. The non-uniform nature of this

diffuser can, for short pipes under clear sky conditions,

produce asymmetric light distributions that are difficult (or

impossible) to predict. The detail of this problem is

discussed in previous modelling work by the authors [4]

but the model is justified by using an overcast sky

approximation. Clear sky predictions can still be calculated

but the actual distribution of light may be inaccurate for

D. Jenkins et al. / Energy and Buildings 37 (2005) 485–492486

List of symbols

A pipe aspect ratio

Eex external illuminance (lux)

Ein internal illuminance (lux)

Etheory internal illuminance determined from theory

alone (lux)

lu luminous intensity distribution (candela)

r pipe radius (m)

Tpipe transmission of pipe material per unit aspect

ratio

V vertical distance of pipe diffuser to point of

measurement/prediction (m)

Greek letters

b diffuser area (m2)

x elbow reduction factor

f luminous flux of pipe (=fSP for straight pipe)

(lumen)

fSP luminous flux of straight pipe (lumen)

’ angle of pipe elbow (8)u angle between line from diffuser to point of

measurement and line normal to diffuser centre

(8)t total pipe transmission

tdiffuser transmission of diffuser

tdome transmission of top dome

v solid angle of light distribution from diffuser

(steradian)

certain scenarios; however, the predictions can still be a

good indicator of the general light level in a given room.

The actual transmission of the diffuser alone has not been

measured but, from mathematical procedures, the combined

transmission of the dome and diffuser is approximated in

Section 2.1.

The modelling approach that is taken is semi-empirical so

measurements were taken as part of this study. This appears

to be the most effective method, mainly because light pipe

transmissions (particularly for aluminium pipes that may

have imperfections along their lengths) can be difficult to

ascertain from pure theory. The aim of the model is to be

able to predict both the luminous flux and illuminance of any

size light pipe (with or without elbow sections), for any

situation. To produce this kind of versatility, measurements

are taken over the range of pipe diameters 0.3 m–0.53 m and

pipe lengths of 0.6 m–5.4 m.

2. Modelling procedure

It has already been stated that parts of this modelling

procedure has been discussed elsewhere [4]. However, rather

than just mention the new areas that have been developed,

the whole procedure will be summarised to clarify the step-

by-step approach taken (although much of this is subtly

different to the previous article referenced).

2.1. Luminous flux predictions

The theory used for producing the luminous flux model is

as follows. Consider a light pipe of cross-sectional radius r,

for an external illuminance Eex, and flux transmission t. The

cross-sectional area of the pipe will therefore be pr2. The

flux of light at the top of the pipe, f0, will be a product of the

external illuminance and light pipe area, or:

f0 ¼ Eexpr2 (1)

The total luminous flux output for a straight pipe, fSP, will

therefore be:

fSP ¼ tEexpr2 (2)

This is the basic equation behind the luminous flux

model. Eex and r are measurables for any given situation, so

it is just t that needs to be ascertained. While various papers

have sought to predict the transmission of pipes through

theory alone [5], the problem that often arises is that there

are a large number of variables to account for, some of which

are difficult to quantify. As a result, it was decided to

use data to approximate the transmission of a pipe as a

function of aspect ratio (the ratio of the pipe length to pipe

diameter).

The necessary data was obtained from a test-shed at

Nottingham University. A 0.3 m diameter pipe was

measured with lengths ranging from 0.6 m to 5.4 m and,

using a photometric integrator to measure the pipe luminous

flux at the diffuser and a global illuminance sensor to

measure the external illuminance, a relationship between

aspect ratio and pipe transmission was ascertained.

Specifically it was found that:

t ¼ 0:82e�0:11A (3)

where A is the aspect ratio of the pipe. It can be seen that, for

zero length, the transmission is still not unity; this is due to

the collector and diffuser absorbing a fraction of the light.

The coefficient of 0.82 implies a combined transmission of

collector and diffuser of 0.82. The results therefore imply

that, if the dome were attached directly to the diffuser, 82%

of the light would be transmitted from outside. This value is

somewhat larger than might be expected for the materials

involved, and would ideally require further investigation.

Incidentally, if this value was inaccurate, it would not affect

the illuminance predictions (Eq. (13)) due to the use of an

empirical constant (that would accommodate any inaccuracy

in this respect).

Eq. (3) can now be substituted into Eq. (2) to give:

fSP ¼ 0:82Eex e�0:11Apr2 (4)

and this is the semi-empirical model to predict the luminous

flux from a straight light pipe.

D. Jenkins et al. / Energy and Buildings 37 (2005) 485–492 487

Although this procedure was applied to a specific light

pipe design (as defined in introduction), it can easily be

modified for other pipe designs. Firstly, if the combined

transmission of the collector and diffuser is known (or can be

measured), then this can be substituted into Eq. (4) instead of

the ‘‘0.82’’ value. If another pipe has a different reflectivity,

this can also be accounted for. The exponential term in Eq.

(4) is directly related to the pipe reflectivity. In actual fact,

e�0.11 gives the transmission of the pipe material per unit

aspect ratio so this can easily be altered for any other pipe.

This is especially important as films now exist that, through

total internal reflection, can produce reflectivities of 98%

and above, so any model should ideally be able to

accommodate this. Hence, Eq. (4) can be generalised to:

fSP ¼ ðtdometdiffuserÞTpipeEexpr2 (5)

where tdome and tdiffuser are the transmissions of the dome

and diffuser, respectively, and Tpipe is the transmission of a

piece of pipe of unit aspect ratio.

2.2. Theoretical light distribution from a point source

The ideal way of modelling the horizontal distribution

would be to use a theoretical form that is validated by actual

data. This would remove the need for using pure measure-

ments to construct the distribution of light from the diffuser.

Data can be used to optimise any empirical coefficients

involved or that are introduced for accuracy. The approach

taken is to use modified theory, and then through data apply

an empirical factor to produce the required results.

Fig. 1 shows a point source of light illuminating a

horizontal plane. From the inverse square law of illumi-

nance, the illuminance (E) directly below the a light source

is given by:

E ¼ I0

V2(6)

where I0 is the luminous intensity (in candela) at the point

directly below the point source.

The illuminance received at point P, a direct distance D

from the point source, will be related to the luminous

intensity at angle u(Iu) by:

E ¼ Iucos u

D2(7)

Fig. 1. Point source of light illuminating horizontal plane.

and by simple trigonometry,

D ¼ V

cos u(8)

giving:

Etheory ¼ Iucos3 u

V2(9)

This expression is the ‘‘cos3 law of illuminance’’ [6] and

can be used to describe the horizontal distribution of light

from a point source.

2.3. Deriving relationship for actual light

distribution from light pipe

The source of light in question, i.e. the light pipe diffuser,

is not actually a point source (although it can be approxi-

mated as one for large distances away from the diffuser). If

the source is approximated to a flat disc shape, the area of the

diffuser ‘‘seen’’ by any given point (b0) is a factor of cos u

smaller than the actual surface area (where u is defined in

Fig. 1) or, for diffuser surface area b:

b0 ¼ bcos u (10)

The quantities that were measured as part of this work

were the luminous flux leaving the diffuser (fSP for straight

pipe) and illuminance at some point below the diff-

user,so it is desirable to have Eq. (9) expressed in these

terms.

Luminous flux and intensity can be related by the solid

angle, v, in the equation:

Iu ¼fSPcos u

v(11)

where the cos u term comes from Eq. (10) and accounts for

the difference in horizontal distribution between a point

source and a source of given surface area (for a point source,

emitting equalling in all directions, the intensity distribution

would just be the ratio of the luminous flux over the solid

angle without the cos u factor, as discussed in literature [6]).

Substituting this into Eq. (9) gives:

Etheory ¼ fSPcos4 u

vV2(12)

There are still slight modifications that have to be made to

this. Firstly, the empirical constant, g, must be placed in

front of this equation so that, when comparing data with

predictions, the optimum match can be found. It follows

from this that Eq. (12) will not be equal to the actual

predicted illuminance. This is probably due to the slightly

non-Lambertian distribution of light from the pipe diffuser. g

will just be a number and will include the solid angle term v

(=2p) for simplicity. If a suitable empirical term is found that

is accurate for a wide range of data, then this task is

relatively straightforward. If the term is largely different for

a wide range of datasets, then some variable (i.e. a factor that

varies between each set of measurements taken) has not been

accounted for. In fact, over the measurements that were

D. Jenkins et al. / Energy and Buildings 37 (2005) 485–492488

taken for this study, the empirical term is found to be

approximately constant at 0.494 (in that this value achieves

the best comparison with all data). This includes all constant

terms identified. The final equation for the internal

illuminance at any point is then:

Ei ¼ 0:494fSPcos4 u

V2(13)

This is proposed as being suitable for all straight light

pipes. It should be noted that this does not include

contribution from internally reflected components, although

the data collected would have had a small reflected

component anyway (as the illuminance data was collected

from real-life scenarios rather than test environments). It is

reasoned that, although the internally reflected component

from light pipes should be an area of investigation, the fact

that the light from light pipes is directly vertically

downwards (as opposed to windows which tend to project

light onto walls) means that this component will not be as

significant as with windows. Also, the floor of a given room

is typically not very reflective and so the internally reflected

component would be reduced as a result.

The next step of the modelling procedure is to apply a

factor for elbow pieces.

2.4. Ascertaining effects of pipe elbows

Measurements of the losses due to pipe elbows were

carried out, with the help of Nottingham University, on

bends ranging from 58 to 758. From this data the following

expression was derived:

x ¼ e�0:0052’ (14)where ’ is the angle of the pipe elbow in degrees and x is the

ratio of a pipe with an elbow to that of a straight pipe of equal

length.

This expression is purely derived from measurements but

its ‘‘form’’ was deduced partly from theory. Fig. 2 refers to

Fig. 2. Measurements (and ‘‘theory’’) with trendline taken at Nottingham Uni

the theoretical trend line. This is deduced from the following

calculations.

The simplest way of approaching this is to consider a

single elbow of known angle (for example, 158). For a given

percentage loss, say a, it is proposed that the losses of other

elbows can be estimated by working in multiples of the

original angle (this is shown to be a suitable approximation

in Fig. 1).

Working on this principle, the following process shows

the theoretical loss for a number of bends, n.

For straight pipe, output = fSP

For one elbow with percentage loss ‘‘a’’, output = fSP � afSP

For two identical elbows, output = fSP � afSP � a(fSP -

� afSP)

Continuing this produces the general formula:

x ¼ ð1 � aÞn (15)

where x is previously defined.

Mathematically, this is equivalent to the formula:

x ¼ enlnð1�aÞ (16)

where n is the ‘‘number of bends’’. So, for example, if the

loss associated with a 158 bend is 8.5%, then a = 0.085 and n

will be a multiple of 158 (so, to produce an estimate for 308,n will be 2).

The above equations are not meant as cast-iron solutions

to the elbow loss problem. They do however show that an

exponential form between elbow angle and percentage loss

would be expected and so they are useful in finding the

correct form of equation to use to fit a trend line to the data

points (as has been done in Fig. 2). They can be directly

compared to Eq. (14) to calculate the properties a and n.

Using the measurements for the 158 elbow produces the

trend line shown in Fig. 2. This is actually quite close to the

trend line derived purely from measurements (as would be

expected), with the small difference probably being due to

versity (January 13–17, 2003) for elbow pieces of 0.3 m diameter pipe.

D. Jenkins et al. / Energy and Buildings 37 (2005) 485–492 489

Fig. 3. Examples of different configuration of same pipe components.

the fact that, for example, a 308 elbow will not have precisely

the same effect as two 158 elbows (due the nature of the

elbow pieces).

3. Proposed light pipe predictive model

The described mathematical treatment and the use of

data have enabled an equation for any configuration of light

pipe (straight or elbowed) to be derived. This will now

be expressed explicitly and validated against collected

data.

3.1. Final expression for light pipe predictions

To incorporate Eq. (14) into the expressions for the

straight pipe models is relatively straightforward. For a pipe

with a single elbow piece, Eq. (14) is multiplied by the

predicted output for a straight pipe (Eq. (13)). For multiple

elbow pieces, this operation is repeated for the second, third,

etc. elbows. The assumption is made that the position of each

Fig. 4. Luminous flux measurements (from April 5 to 14, 2002, Nottingh

elbow is not important; only the angle of the bend in each

case. For example, it is assumed that, both pipes in Fig. 3

would produce the same outputs.

The internal luminous flux from a pipe with n number of

elbows, fi, is thus given by:

fi ¼Yn

1

xnfSP (17)

or more explicitly:

fi ¼ 0:822Yn

1

xnEex e�0111Apr2 (18)

whereQ

is the product sign (read as ‘‘the product of . . .’’).Therefore, for a pipe with no elbow pieces, fi will just equal

fSP.

Finally, we have an expression that can be proposed for

the illuminance at any point below a light pipe, for any size

light pipe, and with any number of elbows. Using Eq. (13)

for a straight pipe gives Eq. (19) for n pipe elbows:

Ei ¼ 0:494Yn

1

xn

fSPcos4 u

V2(19)

where x is given in Eq. (14). It should be stressed that the

straight pipe luminous flux has been used here and the

effects of pipe elbows applied after this. An alternative

procedure would have been to apply the pipe elbow effects

to the luminous flux prediction (of Eq. (4)) and then use this

for finding illuminances. Eq. (19) is the final equation that all

internal illuminance predictions are based on in this study.

Expanding this further so that it is in terms of inputs only

(where it is assumed that luminous flux will not be known by

the user of the equation) gives:

Ei ¼ 0:406Y’n

’1

e�0:0052’ Eex e�0:111Apr2cos4 u

V2(20)

The value of 0.406 is an empirical constant that is a

product of all derived constants from the collected data. Its

am University) and predictions for straight 0.3 m pipe, 1.2 m long.

D. Jenkins et al. / Energy and Buildings 37 (2005) 485–492490

Fig. 5. Illuminance measurements (July 17, 2002, Nottingham University) and predictions for straight 0.45 m diameter pipe, 3 m long. Taken at vertical

distance of 2 m and horizontal distance of 1 m from pipe diffuser.

value is partly dependent on the transmissions of the diffuser

and top dome; as these are always the same for this study, the

value is assumed (and proven within the limits of the data

collected) to be constant.

3.2. Validation with data

Several pipe configurations were measured in various

sites ranging from 0.3 m to 0.53 m in diameter and 0.6 m to

6 m in length (as well as elbows of varying angles). For

conciseness, a small selection of this data will be shown as a

brief validation of the model.

Fig. 4 shows luminous flux measurements compared to

the model (of Eq. (4)) for data over a period of ten days.

Fig. 5 is a comparison of illuminance data and the straight

Fig. 6. Illuminance measurements (September 14–15, 2002, office in High Wyc

elbows. Taken 1.5 m directly below pipe diffuser.

pipe illuminance model (of Eq. (13)) and finally Fig. 6 uses

Eq. (20) to compare predictions of the elbowed (i.e. final)

model with data. All predictions matched the collected data

(for the dimensions described in Section 1) satisfactorily and

suggested the models to be valid design tools for light pipe

illuminance and luminous flux predictions. The next stage is

to display predictions, particularly illuminance predictions,

in an accessible way.

3.3. Displaying predictions

3.3.1. Luminous flux predictions

Luminous flux predictions can easily be displayed in

tabular form, as they do not describe information regard–

ing the distribution of light. As a result, in the current

ombe) and predictions for 0.53 m diameter pipe, 1.2 m long, with two 308

D. Jenkins et al. / Energy and Buildings 37 (2005) 485–492 491

Fig. 7. Spreadsheet for luminous flux model.

framework, a simple spreadsheet can be used which requires

simple inputs and the output can be displayed on the

same page (see Fig. 7). The final luminous flux outputs of the

light pipes can then be compared to the luminous flux

associated with different electric lighting fixtures of given

efficacies.

3.3.2. Illuminance predictions

A slightly more detailed form for displaying illuminance

predictions is recommended so as to appreciate the total

light that is received by a given room. Several MS-Excel

spreadsheets can be used together to produce a ‘‘luxplot’’

that provides a colour representation of the illuminance

distribution across a two-dimensional horizontal plane. Such

a luxplot can be produced for any size room, with any

Fig. 8. Input page for illuminance model (using smallest version

number of pipes, of any dimension or configuration (and

with any number of elbows). In the current framework, an

input page (Fig. 8) provides all the information needed by

the internal calculations of the model, and defines a grid co-

ordinate system across the area in question (the grid is

usually split into squares of 0.5 m � 0.5 m) This informa-

tion is used to calculate, for each pipe in the room, the

illuminance at every point within the grid. Then the

illuminances at each point form each pipe are simply added

to produce a grid of illuminance values across this area. A

simple VBA macro is then run to convert these values into

colours. The final output (see Fig. 9), which can be produced

in seconds, is then a good representation of the light

distribution predicted in that room for the configuration

defined.

for a maximum of 10 m � 10 m room and for nine pipes).

D. Jenkins et al. / Energy and Buildings 37 (2005) 485–492492

Fig. 9. Final luxplot output (for values inputted in Fig. 8).

4. Conclusions

Through the use measurements and refined theory, a

model has been formulated that predicts the luminous flux

and illuminance (below diffuser) of any number and any size

light pipes (with any number of elbows) in a room of any

dimensions. The predictions for single pipes have been

tested against data and (as seen in Figs. 4–6), the results are

satisfactory.

The illuminance model should be used with caution when

considering clear sky predictions, as the actual illuminance

distribution predicted (particularly for short pipes) may not

be accurate. However, in such a case it is probable that the

magnitude of the values given would still be indicative of the

performance of that pipe (or pipes) involved in the

prediction. Nevertheless, the performance of the model

will be more satisfactory when compared to data from

overcast skies, as is the case for many daylight-associated

models.

The main purpose of the model is to accurately calculate

the light pipe(s) needed to maintain a specific light level in a

room. Therefore, for any given room, an estimate can be

formulated for the time period throughout the year (based on

external illuminance approximations) that electrical lighting

can be switched off (or reduced). From this, it will then be

possible to quantify the energy savings (and, using published

statistics relating energy consumption with pollutant

production [7,8], the reduction in pollutant emissions)

associated with the reduction of daytime electrical lighting.

References

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[2] DTI UK Energy Sector Indicators Report, 2003.

[3] D.J. Carter, The measured and predicted performance of passive solar

light pipe systems, Lighting Research and Technology 34 (1) (2002)

39–52.

[4] D. Jenkins, T. Muneer, Modelling light pipe performances—a natural

daylighting solution, Building and Environment 38 (2003) 965–972.

[5] L. Shao, A.A. Elmualim, I. Yohannes, Mirror light pipes: daylighting

performance in real buildings, Lighting Research and Technology 30

(1) (1998) 37–44.

[6] D.C. Pritchard, Lighting, Addison Wesley Longman Ltd., 1999.

[7] G. Weir, T. Muneer, Life Cycle Analysis of double-glazed windows,

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[8] E.S. Rubin, C.I. Davidson, Introduction to Engineering and the

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