Transactions

The Bureau of Standards paper2 gives the integration and solution
of this equation for several special conditions, of which only
two are of interest to us: The behavior of a body (bulb) immersed
into another body (bath) of constant temperature, and of a
temperature rising at a uniform rate.
The equation for the constant temperature is
Written in logarithmic form, Equation |2] becomes
BECK—TH ERM O M ETRIC T IM E LAG 537
which is the form used in the main paper. Solved for L, this
formula permits the determination of L from any two timed
temperature readings
I t will be noted that with —-----— = e the formula becomes
T — 2 2
L = s, which is the relation on which the recommended method
of measuring time lag is based.
The equations for the steady-rising bath are as follows
S E C O N D S
F i g . 1 0 C o m p a r i s 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 S u d d e n
B u l b T r a n s f e r (A ) a n d U n i f o r m R a t e o f T e m p e r a t u r e R i s e
(B)
The condition shown by the curve, Fig. 2, is based on T0 =
Th in which case Equation [6] becomes
The term rse S/ L becomes increasingly smaller with time so
that the thermometer finally approaches the condition
In equation [8], the temperature lag T — T* eventually will
equal the rate of temperature change times the time lag. Fig. 2
shows rL as a dash line and illustrates the gradual approach of
the bulb temperature to this line.
Appendix— 2
In the main part of the paper there has been no discussion of
the events which occur immediately after transferring a thermometer
bulb from one temperature medium into another one at
higher or lower temperature. The author felt that they were of
no concern to the practical engineer for whom the paper is intended.
The methods proposed for determining the time lag were
deliberately arranged to eliminate the effects caused by the sudden
contact of the bulb with a medium at a considerably different
temperature.
In a preliminary discussion3 of the paper, the question was
brought up whether in the theory of temperature regulators the
time lag as discussed in the main paper, or that for “sudden bulb
contact” would have to be considered. Fig. 1 shows the tem ­
perature curve without the “sudden contact” effect; Fig 10, curve
A, shows an actual temperature-response type with a drop at the
beginning, which is due to the sudden contact of the bulb with
hot water. The reason for the drop is as follows: The heat
proceeds like a wave from the bulb surface to the interior. During
1 Annual Meeting, Philadelphia, Pa., December 4 - 8 , 1 9 3 9 , o f T h e
A m e r i c a n S o c i e t y o p M e c h a n i c a l E n g i n e e r s .
F i g . 11
D i p E f f e c t f o r S u d d e n B u l b T r a n s f e r
the time of traverse through the bulb wall, the latter expands,
whereas, the mercury has not expanded as yet. Therefore, there
is an increase in bulb volume without an increase in mercury
volume, which causes the temperature indication to drop.
To clear up the effect of this phenomenon on the response of
thermometers, the author made some tests with apparatus which
happened to be available. Due to lack of time and equipment
these tests are by no means complete, but they yielded sufficient
quantitative information to give an idea of the true effects, and
may serve as a basis for further tests where more accurate information
may be required.
The tests were made by means of a recording thermometer with
a circular chart revolving once in 10 sec. One of the tests, transferred
to cross-section paper, is shown in Fig. 10, curve A . This
was made with a mercury-in-steel thermometer system, the bulb
consisting of KA2 stainless steel. Fig. 11 shows the start of the
curve from four different tests. Each test was made with a
different temperature interval between the bulb and the water
bath in which the bulb was suddenly inserted. The result is interesting
in several respects.
1 The time required for the reading to come back to the
original bulb temperature is practically the same in each case.
The variations shown are not greater than might be expected due
to imperfections in the experimental equipment.