Transactions

The differential equations for the thermal circuit (and of the
thermal system) and those of an analogous electrical circuit
follow:
BECK—TH ERM O M ETRIC TIM E LAG 541
q — thermal current, Btu per hr
e = instantaneous voltage of angular frequency a
applied to condenser C and resistor R in series
i = electrical current
i = V —i
The instantaneous temperature t, measured by the thermometric
element (measured above the same mean as r) corresponds
to the voltage across the capacitor C. Thus
where / = unit conductance between the fluid and the thermometer
A t = difference in temperature between fluid and thermometer
at any time 0 after the sudden change in
the fluid temperature (or immersion of the thermometer
in a fluid of fixed temperature)
A t o — initial difference in temperature between thermometer
and fluid
A = area through which heat flows to immersed portion
of thermometer
V = volume of immersed portion of thermometer
y = weight per unit volume of immersed portion of
thermometer
Cp = unit heat capacity of immersed portion of thermometer
e = voltage in electrical circuit corresponding to temperature
in thermal circuit, i.e., voltage across
resistor B
Co = initial voltage drop across resistor R
By definition:
L — time constant (lag) = RC =
VyCP
fA
The quantity VyCp depends only upon the construction of the
thermometer (thermocouple) and the degree of immersion and
may be tabulated for any instrument by the manufacturer.
The unit conductance / is a function of the method of application
of the instrument, i.e., the fluid properties and the character of
the flow over the thermometric element. The unit conductance
may vary a thousandfold for different applications. The
manufacturer should tabulate typical magnitudes of the unit
conductance for common applications. Users of thermometers
can then readily compute the performance of any instrument
under transient conditions.
More complex ideal systems, involving distributed capacities
and resistances, may be devised more accurately to predict the
behavior of a thermometric element. Further, more complicated
thermometric elements may be treated analytically (the
analogous electrical circuits for which solutions are available
may be utilized).
A thermometric element may be employed to record a periodically
fluctuating temperature. Concepts developed to solve the
electrical circuit may be utilized. Again returning to Fig. 15,
the condenser of infinite capacity being replaced by a sinusoidal
generator
Thermal circuit
E l e c t r i c a l c i r c u i t
r = instantaneous temperature of the fluid (with respect
to the mean) of angular frequency a
or the ratio of the amplitudes of the thermometric element
reading t and the fluid temperature variation r
and the angular lag of t with respect to r is given by the expression
F i g . 1 5 T h e r m a l C i r c u i t f o r T h e r m o m e t e r S u d d e n l y I m ­
m e r s e d i n A F l u i d
Periodic thermal variations of complex shape may be solved by
expansion in a Fourier series. Thus, it is seen that VyCp/fA =
CR — time constant L is a fundamental constant of the thermometric
element but involves its mode of application.
The writers also wish to mention that the time lag for forcedconvection
applications (flow perpendicular to cylinders) varies
inversely as the velocity to the 0.56 to 0.6 power. Thus, the
curve illustrated as Fig. 8 by the author should not become horizontal.
W. G. Brombacher.8 D ata on two im portant classes of
instruments have been omitted from the author’s presentation
of this subject, i.e., the thermocouple and the electric-resistance
thermometer. These have been omitted for good reasons,
since a full discussion would merit a separate paper. But it
should be pointed out that these thermometers can be made with
comparatively small lag, particularly the thermocouple thermometer.
The writer is in full agreement with the author on the method
of testing which he recommends for inclusion in specifications.
However, in instruments of small lag, it m ay be preferable to
specify other readings during the test than those which give the
constant L directly; for example, a thermometer originally
at 80 F put into a bath at 211 F may be read at 111 F and 201 F.
The value of L is easily computed; thus, in the example, using
the equations in the paper, L = 0.44 Sa, where Sa is the measured
* National Bureau of Standards, Washington, D . C.