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CÆAÆSÆEHÆIÆSÆTÆOÆRÆY
Renewal Rate of Filament Lamps: Theory and Experiment
V. J. Menon ÆD. C. Agrawal
Submitted: 21 February 2007 / in revised form: 9 September 2007 / Published online: 2 November 2007
ASM International 2007
Abstract We calculate the renewal rate of fused lamps in
an organization having large installation of filament lamps.
This is achieved by using the standard mortality curve over
the specified fractions of the half-life and setting up an
algebraic relation for the renewed fraction at successive
stages. The procedure for data analysis is described in
detail for an actual experiment performed by us.
Keywords Tungsten Filament lamps
Large installation Failure Mortality curve
Renewal rate Theory Experiment
Introduction
In a clearly written paper, Leff [1] vividly described vari-
ous operating features of incandescent filament lamps and
also emphasized the fact that their failure is history-
dependent. This statement was corroborated further by us
[2] by showing numerically that the half-life, average life,
and most probable life are almost the same for a given type
of bulb. Also, experimental mortality curves for commer-
cial bulbs are reported in an engineering handbook [3],
whose heuristic form was proposed by Leff and derived
rigorously by us [4] using the laws of mathematical sta-
tistics. While continuing our research in this area, the
following question occurred to us: ‘‘If an organization has a
large installation of fresh lamps kept under continuous
operation, then what is the mean renewal rate of fused
bulbs after a periodic (or repeated) elapse of a chosen
interval of time hereafter called the slab length?’’
We feel that a clear answer to this question is important
because of three reasons: (i) A theoretical answer to the
said question will give good training to physicists and
engineers in the use of the parameterized lamp mortality
curve [4]. (ii) Any organization (e.g., Blackpool, Lanca-
shire) would like to have the aforementioned information
because if the fused bulbs are periodically replaced by new
ones then the installation will have all the requisite space
illuminated. (iii) A knowledge of the numerical values as
well as pictorial graphs of the renewal rate at various stages
can be used by an organization to strike a balance between
the replacement cost and quality of illuminance needed.
The aim of the present paper is to address this problem
algebraically, numerically, and experimentally.
Theory
Preliminaries
It is known that many factors inherent in lamp material/
manufacture make it impossible to have each individual
bulb operate for exactly the life for which it was designed.
Rather, as the evaporation of tungsten atoms proceeds and
the hot filament gets thinner the lamp can fail in a statistical
manner in accordance with the so-called mortality curve
measured by engineers [3]. The said curve depicts the
experimental survival probability:
SðsÞKðsÞ=Kð0Þ;st=thð1aÞ
for a fresh assembly of identical lamps. Here tis the
experimental time elapsed in hours, t
h
is the rated half-life
V. J. Menon
Department of Physics, Banaras Hindu University,
Varanasi 221 005, India
D. C. Agrawal (&)
Department of Farm Engineering, Banaras Hindu University,
Varanasi 221 005, India
e-mail: dca_bhu@yahoo.com
123
J Fail. Anal. and Preven. (2007) 7:419–423
DOI 10.1007/s11668-007-9074-9
in hours, sis a dimensionless time, K(s) is the number of
surviving lamps at s, and K(0) is the initial number. We
were able to reproduce this curve quite well by using the
binomial distribution for the number of ejected tungsten
atoms yielding the formula:
SðsÞ¼0:5½1þerffYð1sÞg;Y¼3:0ð1bÞ
where erf stands for the error functions and Yis a slope
parameter. This expression will play a crucial role in the
formulation developed below.
When Replacements Are Not Done
Let us consider a large number K
0
=K(0) of identical
bulbs switched on at the instant t=0 and examine the
scenario if the fused lamps are not replaced as the time
progresses. We chose a specific time slab t
i
=it
h
where iis
a fixed positive number called the slab index. For example,
if the rated half-life t
h
=800 h and the index i=0.1 then
the chosen slab length is t
i
=0.1 9800 =80 h. Let us
look at the undisturbed lamp assembly at successive stages
of this slab. The survival chance s
j
and failure chance d
j
of
any bulb up to the end of the jth stage are given by
sj¼38;SðijÞ¼0:5½1þerf fYð1ijÞg
dj¼38;1sj;j¼1;2;3;... ð2Þ
Clearly,
sj!38;1;dj!0;if ij 1
sj!38;0;dj!1;if ij 1ð3Þ
The average number of surviving and fused bulbs read
Kj¼K0sj;Dj¼K0djð4Þ
The information in Eq 2 to 4 will be useful in the sequel.
When Fused Lamps are Replaced
Next, we decide to keep on renewing the fused bulbs at the
end of successive stages (each of slab length t
i
) and shall
label the corresponding symbols by a superscript star. At
the end of the first stage the starred surviving and fused
numbers are simply
K
1¼K1¼K0s1;D
1¼D1¼K0d1ð5Þ
These D
1fused bulbs are replaced by new ones so that the
total number of lamps is restored to the value K
0
. Next,
after the elapse of jstages the number of surviving bulbs
will be
K
j¼KjþD
1sj1þD
2sj2þ D
j1s1ð6Þ
whose physical explanation is clear; namely, K
j
lamps have
survived undisturbed over jslabs, D
1renewed lot has
survived over j– 1 slabs, D
2renewed lot have survived
over j– 2 slabs, and so on and finally the last D
j1renewed
lot have survived over one slab. The number of renewals
done over the jth stage alone is, of course, D
j=K0K
j.
Often it is more convenient to get rid of the K
0
dependence
by working with the functions
s
jK
j.K0¼sjþd
1sj1þ d
j1s1ð7aÞ
d
jD
j.K0¼djd
1sj1d
j1s1ð7bÞ
In practical applications the quantity
R
j100 d
jð8Þ
may be called the percentage renewal rate corresponding to
the jth stage.
Explicit Examples for j=1, 2, 3, ...
From Eq 7b the fused fractions over the first, second,
third,…stages are seen to be
d
1¼d1¼1s1ð9aÞ
d
2¼d2d
1s1¼1s1s2þs2
1ð9bÞ
d
3¼d3d
1s2d
2s1
¼1s1s2s3þs2
1s3
1þ2s1s2ð9cÞ
and so on. If the slab index i[1 then s
2
,s
3
,…can be
neglected compared with s2
1,s3
1,.... Also, if the renewal
stage j[10 (say) the series for d
jbecomes geometric
namely
d
j1s1þs2
1s3
1þffi1=1þs1
ðÞ ð10Þ
Hence the renewal rate should saturate at an estimated
value
R
jestðÞffi100=1þs1
ðÞif i[1;j[10 ð11Þ
which is a very interesting result.
We illustrate the above theory for the following choices
of the index
i¼0:1;0:2;0:3;0:4;0:5;0:6;0:7;0:8;0:9;1:0;1:5;2:0
ð12Þ
As mentioned already the index i=0.1 implies that
renewals are to be done after every 80 h if the rated half-
life is 800 hr. The renewal rate R
jdefined by Eq 8 was
computed numerically from the series (Eq 7b) for
successive stages j=1, 2, 3,…, and the results are
summarized in Table 1.
420 J Fail. Anal. and Preven. (2007) 7:419–423
123
For every index iit was found that R
jfluctuates with the
renewal stage j. This is understandable because the suc-
cessive terms of the series (Eq 7b) exhibit increments due
to the addition of new bulbs that also undergo subsequent
decays stage by stage. The up-down feature of R
jversus j
was found to persist up to a large stage J(say) beyond
which it saturated. For example, if i=0.4 a saturated
value namely 33.3% occurred for J=20 onward. Of
course, this phenomenon also happened with other choices
of index ias summarized in Table 1.
In Eq 11 an analytical estimate for the saturated value of
renewal rate R
J(est) was obtained provided that the slab
index iis more than 1. It is satisfying to note that this
estimate given in column 4 of Table 1matches reasonably
with the computed results down to iC0.7. The fact that
R
J¼1becomes essentially 100% for iC1.5 is physically
expected because by the time the first renewal is made the
originally installed bulbs have all failed.
Experimental Work
Since the actual data on the mortality curve as well as the
renewal rates are not usually revealed by the bulb manu-
facturers, we decided to perform our own experiment in the
Department of Farm Engineering, Banaras Hindu Univer-
sity. For this purpose, an assembly of 50 new Philips
(India) lamps with the rating 40 W, 220 V (AC) was taken
and installed in the horizontal orientation and uniformly
distributed over a lab area 11 m 97 m. The experiment
was performed in two parts.
In the first part, no replacements were made. The
assembly was monitored at regular intervals of 12 h to
look for failures. The instants of recorded failures were
called t
‘
and a total of 32 data points were obtained such
that even the last bulb failed. Table 2shows these raw
data along with the surviving fraction S=K/50. Inspec-
tion of these raw data revealed that 25 lamps survived up
to 1416 h, which is good guess value for the half-life.
Next, a chi-square search was made on these Svalues
based on the theoretical formula (Eq 1b) using a range for
the unknown slope parameter Yand half-life t
h
as
2.5 \Y\3.5 and 1350 \t
h
\1450, respectively. For
every trial choice of Yand t
h
the svalues were defined as
Table 1 Slab index i, the stage Jat which the renewal rate has sat-
urated, and the percentage saturation value R
Jusing Eq 8 and 7(b).
The fourth column shows the algebraically estimated value of R
J(est)
using Eq 11
iJR
JR
J(est)
0.1 53 9.5
0.2 21 18.2
0.3 22 26.1
0.4 20 33.3
0.5 14 39.9
0.6 14 46.4
0.7 30 51.4 52.9
0.8 21 55.3 55.6
0.9 10 60.1 60.1
1.0 7 66.7 66.7
1.5 1 98.3 99.5
2.0 1 100.0 100.0
Table 2 Experimental data corresponding to the mortality curve
shown in Fig. 1. The theoretical fit was obtained with Y=3.0 and
t
h
=1400 h
Serial ‘Time
t
‘
,h
Failed
number
Surviving
number, K
Surviving
fraction
S=K/50
Error bar
from
(Eq A1)
1 840 2 48 0.96 0.03
2 852 1 47 0.94 0.03
3 936 1 46 0.92 0.04
4 960 1 45 0.90 0.04
5 972 1 44 0.88 0.05
6 996 1 43 0.86 0.05
7 1008 1 42 0.84 0.05
8 1032 2 40 0.80 0.06
9 1104 1 39 0.78 0.06
10 1176 5 34 0.68 0.07
11 1200 1 33 0.66 0.07
12 1260 1 32 0.64 0.07
13 1296 1 31 0.62 0.07
14 1308 1 30 0.60 0.07
15 1332 2 28 0.56 0.07
16 1356 1 27 0.54 0.07
17 1380 1 26 0.52 0.07
18 1416 1 25 0.50 0.07
19 1476 2 23 0.46 0.07
20 1524 3 20 0.40 0.07
21 1548 3 17 0.34 0.07
22 1560 1 16 0.32 0.07
23 1632 1 15 0.30 0.06
24 1644 2 13 0.26 0.06
25 1656 2 11 0.22 0.06
26 1716 5 6 0.12 0.05
27 1764 1 5 0.10 0.04
28 1812 1 4 0.08 0.04
29 1836 1 3 0.06 0.03
30 1860 1 2 0.04 0.03
31 1980 1 1 0.02 0.02
32 2568 1 0 0 0
J Fail. Anal. and Preven. (2007) 7:419–423 421
123
s=t
‘
/t
h
(1 B‘B32). The final best fit parameters
obtained were
Y¼3:0;th¼1400 h ð13Þ
Figure 1depicts our results obtained on mortality sta-
tistics showing the experimental data, theoretical fit along
with error bars calculated in accordance with Appendix A.
The agreement between experiment and theory is quite
satisfactory.
In the second part replacements were made starting
again from an assembly of 50 new lamps. Repeated
interval for replacement was chosen as 560 h correspond-
ing to the slab index i=0.4, and five stages were covered.
The replacement values obtained experimentally and cal-
culated theoretically (Eq. 7b, 8) are displayed in Table 3
and plotted in Fig. 2. The error bars were computed as
described in the Appendix A. The matching between theory
and experiment is once again satisfying.
Conclusions
Our theoretical formulation and experimental work
described above can be of great benefit because:
(i) The data presented in Tables 1to 3and shown in
Fig. 1and 2can serve as model data in this area of
research. The procedure to use the mortality curve can
be adopted by physicists and engineers for doing their
own replacement experiments extended to much
higher stages with the bulbs kept in other orientations.
They may also verify that replacement rate saturates
after number of Jstages mentioned in Table 1.
(ii) It is well recognized that a lamp failure decreases the
local illuminance and thus affects lighting unifor-
mity. Suppose an organization predecides that a net
failure of 33% bulbs will not affect the normal
working of its personnel. Then the appropriate
choice of slab index becomes i=0.4 because, as
seen from Table 1, a saturation value of R
J¼33:3%
is reached at the J=20th stage. At intermediate
stages the value of R
jcan also be precalculated from
Eq 7b, and 8. This information can be conveyed to
the supplier of the lamps.
(iii) As defined in the IESNA Lighting Handbook [5]
‘‘Group relamping entails replacing all of the lamps
Fig. 1 Plot of our experimental data corresponding to the mortality
curve as mentioned in Table 2. The theoretical fit was obtained with
Y=3.0 and t
h
= 1400 h (see Eq 1b)
Fig. 2 Plot of our experimental and theoretical values of renewal rate
versus replacement stage jfor the slab index i=0.4, i.e., slab length
560 h as mentioned in Table 3
Table 3 Experimental and theoretical values of renewal rate for the
slab index i=0.4, i.e., slab length 560 h (see Fig. 2)
Stage jCalculated R
jExperimental R
jError bar from Eq A3
1 0.6 0 1
2 19.3 22 6
3 60.6 58 8
4 23.6 24 6
5 24.1 26 5
422 J Fail. Anal. and Preven. (2007) 7:419–423
123
in a system together after a fixed interval, called the
economic group relamping interval. Group relam-
ping can reduce the cost of operating a lighting
system while keeping illuminance levels close to the
design value. Typically, the most economical time to
group relamp is between 70 and 80% of rated life.’’
Therefore, an economically viable choice of the slab
index is i&0.8 for which the saturated replacement
rate after 21 stages would become R
J¼55:3%.
Acknowledgment VJM thanks the University Grants Commission
for financial support.
Appendix A. Error Bar Calculation
If replacements are not done then, under the assumption of
a binomial distribution, the number K(s) of surviving lamps
at s=t
‘
/t
h
will have mean K(0) S(s) and variance K(0) S(s)
(1 – S(s)). Here t
‘
(1 B‘B32) are the instants of time at
which raw data for the mortality curve were recorded.
Hence the variance of surviving fraction at sis given by
ðdSðsÞÞ2
DE
¼SðsÞð1SðsÞÞ=Kð0ÞðA1Þ
whose square root yields the error bars shown on the mortality
curve of Fig. 1. However, if similar formula is to be applied to
the case when a slab length t
i
and proposed replacement stage
jhave been chosen then the relevant surviving fractions are
called s
j
(see Eq 2) whose variance reads
ðdsjÞ2
DE
¼sjdj=Kð0ÞðA2Þ
If the periodic replacements are done, then attention
must be focused on the starred decayed fraction d
jwritten
in Eq 9(a) to (c). Taking their calculus differentials we find
dd
1¼38;ds1
dd
2¼38;(1 2s1)ds1ds2
dd
3¼38;(1 2s12s2þ3s2
1)ds1(1 2s1Þds2ds3
ðA3Þ
and so on. Their variances hdd
j
2iare obtained by squaring,
taking expectation values with the help of (Eq A2), and
dropping cross variances. Final square roots of the resulting
expressions give the error bars shown in Fig. 2(after
multiplication by 100).
References
1. Leff, H.S.: Illuminating physics with light bulbs, Phys. Teach. 28,
30–35 (Jan 1990).
2. Menon, V.J., Agrawal, D.C.: Lifetimes of incandescent bulbs,
Phys. Teach. 41, 100–101 (Feb 2003).
3. Rea, M.S. (ed.): Chapter 6. In: IESNA Lighting Handbook,
pp. 14–15. IESNA, New York, (2000).
4. Menon, V.J., Agrawal, D.C.: A theory of filament lamp’s failure
statistics. Euro. Phys. J. Appl. Phys. 34, 117–121 (2006); A theory
for the mortality curve of filament lamps, J. Mater. Eng. Perf. 16,
1–6 (2007).
5. Rea, M.S. (ed.): Chapter 28. In: IESNA Lighting Handbook, p. 2.
IESNA, New York, (2000).
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