Transactions

532 TRANSACTIONS OF THE A.S.M.E. AUGUST, 1941
definition provides a convenient method of measuring L. The
time L in seconds is the same in both cases; the mathematical
relationship is shown in the Harper reference.2 Note also that
L is independent of the temperature scale used.
Fig. 1 shows the typical behavior of a thermometer bulb. The
points have been determined directly by experiment; the curve
has been calculated from L , determined by the simplified test
method described later on.
In the following sections, we will first discuss the use of L for
predicting the behavior of a thermometer under various conditions;
list various factors affecting L; then describe methods of
determining L and, finally, give actual values of L for different
types of thermometer systems, sizes of bulbs, etc.
U s e s o p T im e - L a g C o n s t a n t L
In the first definition, L represents the time the thermometer
will lag behind the actual temperature. In the case of a constant
rate of temperature change, this time is the same for any rate of
temperature change. Let us suppose L is 10 sec, the temperature
rising a t a rate of 2 deg per sec, and the thermometer reads 100
deg a t a certain instant. By definition, the actual temperature
was 100 deg 10 sec previously and, as it is rising at a rate of 2 deg
per sec, it is now 100 + (10 X 2) = 120 deg. In other words,
the thermometer is lagging 20 deg. Generally, we can say that,
if the temperature changes a t a rate of n deg per sec, the thermometer
will at any definite moment show n X L deg either less
or more than the actual temperature, depending upon whether
the temperature is rising or falling. Rate of change n may be expressed
in any standard temperature scale, as Fahrenheit or
centigrade; the result, of course, must be taken in the same scale.
The foregoing relation is true only if the rate of change n has
been maintained for a certain length of time, which, for practical
purposes, can be taken as 3 to 4 L. This will be explained later.
The second definition states that L represents the time it takes
a thermometer, if transferred to a higher temperature T, to move
from one indication Ti to another one T 2, which is J/e of the
original temperature difference T — Ti below the final temperature
T. In other words, L is the time for the thermometer indication
to proceed from Ti to Ti, if T t = T — 1/e (T — Ti) = T —
0.368 (T — Ti). Generally, it will be desired to determine the
time required to come within smaller percentages of the final
temperature. The factor 1/e brings us to within 36.8 per cent
(roughly 40 per cent) of the final temperature. The general
formula brought into a convenient form is
To give a yet clearer picture of the thermometer response,
actual values are inserted in the following table for a thermometer
having a time lag L — 10 sec, which has been suddenly transferred
from a 100-deg bath into a 200-deg bath:
Thermometer reading, F .
Time elapsed, sec.........
100 150 1 6 3 .2 180 190 195 199 1 9 9 .9
0 7 10 16 23 30 46 69
It is possible to reduce practically all thermometer-response
problems either to the case of a uniform rate of change, or of a
sudden change of temperature at the bulb, or to a combination of
both. For practical purposes, we can say that, in case of a sudden
change, it requires 3L to get fairly close to the final temperature,
or 5L if we want to be very accurate. For a temperature
changing at a rate of n deg per sec, the thermometer will lag n
X L deg. If we start from a steady temperature at which the
thermometer shows the true temperature, and then change at a
F i g . 2 T h e r m o m e t e r R e s p o n s e f o r S t e a d y T e m p e r a t u r e
C h a n g i n g I n t o a U N ir o R M T e m p e r a t u r e R i s e
where Sa = total number of seconds to come within a times the
difference of initial and final temperature, and
The following table shows t for various values of a, and gives
a fairly good picture of the behavior of a thermometer:
Suppose we transfer a thermometer suddenly from 100 deg to
200 deg, and want to know when the thermometer indication will
have reached 199 F. In this case, T = 200, rl \ = 100, Tt = 199,
200__199
and a = -------------- = 0.01. We will, therefore, take the value
200 — 100
4.6 L for o = 0.01. If the change were from 190 to 200 deg, and
we again want to know the time of reaching the 199-deg mark, we
will use the value 2.3L for a = 0.1.
F i g . 3
D e t e r m i n a t i o n o f T h e r m o m e t e r R e s p o n s e f o r A r b i t r a r y
T e m p e r a t u r e C h a n g e s
rate of n deg per sec, we will not develop the full temperature
lag of n X L deg until after a time of about 3L. This type of behavior
is shown in Fig. 2.
For cases of a more complicated nature, a graphic method can
be used. This method is illustrated in Fig. 3. I t is based on a
reversal of the formula for temperature lag. We found that n
X L = Ti represents the temperature lag 7\ for a rate of change
of n deg. Inversely, if the temperature difference T, between bath
and bulb is known, we can draw the conclusion that the thermometer
indication will change at the rate of n deg per sec. The