*< ° F °o.

ZV

Q NBS TECHNICAL NOTE 959

0ff£AU Of

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

libration of Time Respi

Concepts and Model Calculations

)C

LOO

J5753

w.959

1978
NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU

OF STANDARDS l 6 ° LIBRARY U.6 7^3

FIB S 1978

' Nfffi

Calibration of Time Response of Thermometers:

Concepts and Model Calculations

*. 7dC>Ai^a/ A/oftL 9d

Raymond D. Mountain

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

and

G.W. Mulholland

Institute for Applied Technology

National Bureau of Standards

Washington, D.C. 20234

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Dr. Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

U- S NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued January 1978
National Bureau of Standards Technical Note 959

Nat. Bur. Stand. (U.S.), Tech. Note 959, 34 pages (Jan. 1978) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1978

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. CI 3. 46: 959). Stock No. O03-0O3-O1878-3 Price $1.40

(Add 25 percent additional for other than U.S. mailing).
TABLE OF CONTENTS

I. Introduction 1

II. Discussion: General Features of the Time Response of a Temperature Sensor .... 4

III. Model Calculations 10

A. Derivation of Equation 1 10

Bl. Calculation of the Response Function for a Solid Sphere and a Solid

Plate by the Series Solution Method 12

B2. Calculation of the Response Function for a Solid Cylinder by the

Laplace Transform Technique 14

B3. Heat Transfer in a Hollow Cylinder 17

C. Layer Structure 19

D. Thermal Coupling of Sensor to Environment 20

E. Internal Heating Transients 23

F. Applications of Equation 1 24

References 29

XII

Calibration of Time Response of Thermometers:

Concepts and Model Calculations

Raymond D. Mountain and G. W. Mulholland

Some of the conceptual problems associated with the calibration of the

time response of a temperature sensor are examined in this report. The

discussion is in terms of a time response function which characterizes

the way a given sensor responds to changes in the temperature of its

surroundings. A series of model calculations of the response function for

idealized sensors are used to investigate the general features of the

response functions. Important features are the sensitivity of these

functions to (i) material properties of the sensor, (ii) the type of

thermal coupling of the sensor with the environment and (iii) the geometry

of the sensor. These features must be considered in the design of

procedures for calibrating the time response of thermometers.

Key Words: Calibration; response time; temperature; thermal response;

thermometer; time response function.

I. INTRODUCTION

The concept of calibration of dynamical measurements is inherently more complex

than is the concept of calibration of quasi-static (say, temperature or pressure)

measurements. This increased complexity is due to the increased number of quantities

whose characteristics must be quantitatively understood if a calibration is to be

possible. Also, it is necessary to clarify the relationship between the quantity

actually measured (e.g. electrical resistance) and the time variation of the property

inferred from that measurement.

For example, equilibrium temperatures are often determined by measuring the

electrical resistance of a platinum wire for which the temperature-resistance relation has

been determined under conditions of thermodynamic equilibrium. Two such wires, of quite

different diameter, geometrical arrangement and mounting will yield when calibrated

sensibly identical determinations of equilibrium temperatures. This is not the case for

the time response of these "thermometers" to changes in temperature. The response will

depend upon the aforementioned factors and will be especially sensitive to the properties

of the materials and to the thickness of the materials used in the mounting. Also it will

depend upon the nature of the thermal contact of the "thermometer" with the medium undergoing

thermal change involving the thermal properties of both the "thermometer" and the medium

as well as the nature of the thermal gradients over the surface of the sensor-medium

boundary. All of these must be quantitatively understood before a meaningful calibration

of the sensor can be obtained. In addition, the sensitivity of the calibration to variations

of these conditions must be known if the sensor is to be used in situations which do not exactly match those used in obtaining the calibration.

1
Some of the features involved in calibrating the time response of a temperature

sensor are examined in this report. Model calculations have been performed to indicate

how a sensor would respond to changes in the temperature of surrounding material. Some

general features of these calculations have been extracted and form the basis for the

discussion of what is involved in establishing a calibration of time response.

First let us specify what we mean by time response. Suppose that initially the

sensor and surroundings are in thermal equilibrium at an arbitrarily defined T=0. Then,

at an initial time, t=0, the surroundings undergo a change of temperature, T (t) . The

time response of the sensor is known when the temperature of the sensor T(t) is known as a

function of T

6 (t). For the situations discussed here, (those for which the temperature dependenc

of the thermodynamic and transport properties can be ignored), we show in Sec.

III-A that the relationship between T(t) and T

e

(t) takes the form

T(t) = T

e

(t) - C dy G(t-y) ^^- < (1)

If one knows G(t), one can then, using Eq. (1), predict the response of the sensor to the

external temperature T

e (t). In particular, the response function G(t) describes how the

sensor responds to a step function change in external temperature applied at t-0. If T (t) is

a unit step change as shown in Fig. 1, then

T(t) = 1 - G(t) ( 2)

as shown in Fig. 2, so G(t) has the form of Fig. 3.

It is important to keep in mind the distinction between the calibraton of the time

response of a sensor under well characterized conditions and the application of this calibration

to the mesurement of a time varying temperature. For the purposes of calibration, the external

temperature T

e

(t) is known and controlled. This greatly simplifies the analysis since the

effect the sensor has on the temperature of the medium surrounding it need not be considered.

The task of calibrating the time response of a temperature sensor is therefore one of

determining G(t) under specificied conditions. To be useful it is important to know how

sensitive this function is to changes in those conditions and how well G(t) can be determined.

Model calculations of G(t) have been made to investigate its properties and to specifically

explore the following questions:

1. How should G(t) be represented explicitly?

2
T

e

(t)

Fig. 1. A unit step change in external temperature,

T(t)

Fig. 2. The temperature response of a hypothetical thermometer,

G(t)

Fig. 3. The response function for the hypothetical thermometer of Fig. 2.

3
2. What sort of information is needed to determine G(t) from experiment?

3. What are the limitations on the T(t), T

e

(t) , G(t) relationship?

4. How sensitive is G(t) to changes in material properties, in boundary

conditions and in operating features?

The model calculations are of the following type. The temperature equation

3T ^

>t} X v2 T (r,t) (3)

is solved for plates, cylinders and spheres with boundary conditions of either

a) temperature continuity at the sensor surroundings interface, or

b) Newton's law of cooling which states that q, the heat flow at the surface,

is proportional to the temperature discontinuity at the surface,

q =

- A ¥-\ * = H [ T (t) - T

e (t)] . . (4) 8r I surface surface

In Eqs. 3 and 4, x = ^/(p C ) is the thermal diffusivity, p is the density, C the specific

heat at constant pressure and A is the thermal conductivity of the sensor. The heat transfer

coefficient H is a property of the material surrounding the sensor. We use the initial

condition T(r,0) = for the temperature of the sensor. We have shown in IIIA that only the

step function external temperature change case need be considered for the linear problem

defined by Eq. (3) and boundary conditions (a) or (b) . The symmetries of the plate, cylinder

and sphere have been exploited so that only one spatial dimension is involved. This greatly

simplifies the calculations and makes it possible to extract the essential features of the

problem more readily. These calculations are not new and can be found in various texts on

1-3

heat transport. For the sake of completeness we do include some of the pertinent details

of the calculations and show that the numerical evaluation of the infinite series expressions

obtained for G(t) is straightforward and accurate. Time response functions have been usefully

employed in some engineering studies .

Section II of this report contains a discussion of the factors involved in time response

determinations. The details of the calculations on which this discussion is based are found

in Section III. The third section contains a series of "appendices" to sections I and II and

can be consulted by those interested in the details of the calculations.

II. DISCUSSION: GENERAL FEATURES OF THE TIME RESPONSE OF A TEMPERATURE SENSOR

The solutions to the temperature equation, Eq. (3), for a unit step function

4
change at t=0 can be expressed in the form

oo 2/o2 n T(r,t) = 1 - V^ a.f.(r) exp [~xta. (5) z /l l

] / J J J J j-l

where the set of coefficients {a.} are determined by the initial conditions, the set of

3 coefficients {a.} are obtained from solutions of a dispersion relation which reflects

J the boundary conditions, and the set of functions (f.(r)} reflect the geometry of the

sensor. The coefficients {a.} are ordered so that a. < a.,,. The quantities v and I

3 3 J+l

are the thermal diffusivity and a characteristic length (e.g. the radius) of the sensor.

The ratio H /x is the "natural" time unit for the sensor. When the sensor is a composite

of two or more "layers", the form remains the same with the x and £ being the properties

of the active part only and with the coefficients a. and a. reflecting the complexity

introduced by the layering of materials.

A comparison of Eqs. (2) and (5) shows that the response function can be represented

as an infinite series of exponential functions with decreasing relaxation times

t. =J£ 2 /xa. . Except for very short times, only a few terms in the series are significant

and when

t > x 1 = ^/x^2

, (6)

only the first term is important. The experimental determination of G(t) can in principle

be obtained by abruptly changing the temperature of the medium in which the sensor is

embedded and then monitoring the reading of the sensor as a function of time. The

response function is then found by scaling Eq. (2) by an amount T to be

G(t) = (T6

- T(t))/T6

. (7)

Since the realization of an effective step function change in T is not always possible

we should consider the possibility of extracting G(t) from the more general result

T(t) = T

6

(t) - f

Z dy G(t-y)

dT (y)

. (1) J y

The solution of this integral equation for G(t) is possible with reasonable accuracy

if T(t) and T (t) are quite accurately known, if the zero of time is accurately known

e and if the time variation of T (t) occurs over about the same time interval as does the

variation of G(t). Some examples of this way of obtaining G(t) are discussed in Section III-F.

5
Once G(t) has been determined, the exponential series form

oo G(t) = Y] a.f.(r) exp (- x t a. 2 /£ 2 [t) = ^2 a.f.(r) exp (- X t a. ) (7) 2 /l 2

) j=l -1 J -1

extracted from Eq. (5) provides a convenient representation. The coefficients a. and

f . (r) should be computed from the geometry of the sensor and the factors in the arguments

of the exponentials adjusted to fit G(t) . Of course, the full infinite set of terms will

not be determined but a reasonable representation would involve a few terms.

One may also turn the question around to ask; if G(t) and T(t) are known, can one

infer T

6 (t)? Viewing Eq. (1) as an equation for T (t) in terms of T(t) and G(t) leads to

difficulties. It is not possible, using the procedure described in Section III-F, to

construct a function T (t) given only a set of values for G(t) and T(t) (unless G(t) = 1 -

p T(t) so that T (t) is a step function). An example illustrating the problem is discussed

p in Section III-F. It may be possible to construct solutions for T (t) in terms of specified

functions subject to some sort of least squares error condition. We have not investigated

this possibility.

e

It should not come as a surprise that solutions T (t) are difficult to obtain. As the

variation of T (t) becomes more and more rapid, less information shows up in T(t) . As

shown in the example in Section III-F, when

T

e

(t) = T sin tot (8)

for a plate, then

T(t) = A(to) sin tot + B(a>) cos tot (9)

+ decaying terms.

Both A(oj) and B(to) go to zero as to > °°. This is simply a manifestation of the inability

of the sensor to follow (or respond to) changes of sufficiently high frequency. Since any

T (t) which can be represented as a Fourier integral contains some high frequency components,

the integral equation for T (t) cannot be expected to yield solutions unless these high

frequency components are reimposed by means of constraining the solution to be expressed

in terms of some set of functions which do possess the necessary Fourier components.

The type of thermal coupling between the sensor and the surroundings can make a large

difference in the time response of a sensor. To illustrate this consider a
cylindrical sensor of radius a which is analysed in Section III-B2. At one extreme, the

cylinder is in perfect thermal contact with the surroundings and heat flow into (or out

of) the cylinder is unimpeded at the surface. This condition results in the most rapid

response. More generally, there is a surface impediment to heat flow and the heat flow at

the surface is proportional to the temperature discontinuity at the surface (Newton's law

of cooling)

;

, 3T9r

= H[Te (t) - T(a,t)] (4) r=a where the proportionality coefficient H is known as the heat transfer coefficient. This

coefficient is a measure of the ability of the surroundings to deliver heat to (or carry

it away from) the surface of the cylinder. Thus perfect thermal contact corresponds to

very large H(A/H = 0) and T (t) = T(a,t). In fluid flow problems, H can depend on the

details of the flow past the cylinder. H is smaller for laminar flow than for turbulent

flow and changes of a factor of 10 or more in H with fluid velocity can be expected.

The time response functions for the solid cylinder for varying thermal coupling

conditions are constructed in Section III-B2. The longest relaxation time is found to be

x 1 = a

2 /312 X (10)

where 3-. is the smallest root of the transcendental equation

6 i J

i

(e) =

J (e) (11)

Here J and J, are the Bessel functions of order and order 1. The details of the

o 1 derivation of Eq. (11) are to be found in Section III-B2.

To illustrate the influence of thermal coupling on time response, we show in Fig. 4,

the inverse of x (in units of x/a 2

) as a function of A/aH. As H decreases (the surroundings

become less able to transport heat) 1/t 1 decreases rapidly. This has the result, depicted

in Fig. 5, of increasing the response time of the sensor. This indicates the importance

of specifying the boundary conditions as a part of a calibration statement of time response.
6 r

4

-

1/T, "

2 -

Fig. 4. The inverse of the longest relaxation time for a solid

cylinder as a function of the quantity, A/aH. The inverse time is

expressed in units of x/a 2

- Perfect thermal contact corresponds to A/aH = 0.

"1

PLATE

Fig. 5. Response functions for the center of Fig. 6. Comparison of the "center line"

a solid cylinder for different values of A/aH. response function for a sphere cylinder and

The time is expressed in units of a

2 /x- Perfect plate for a common value of x/^ 2

- The time

thermal contact corresponds to A/aH =0. is expressed in units of £

2/x- Here I is the

radius of the sphere or cylinder and onehalf the thickness of the plate.
Another feature which influences the response time of a sensor but which is not

likely to pose a calibration problem is the geometrical shape of the sensor. For a given

value of x/ I , where £ is the radius for the sphere and the cylinder and the half thickness

for a slab, the response time for a solid sphere is less than that for a solid cylinder

which in turn is less than that for a slab as shown in Fig. 6. Some illustrative examples

including solid and hollow sensors are contained in Sec. III-B.

Changes in the material properties for a uniform sensor change only the parameter

A/aH, which concerns the heat transfer at the sensor-surroundings interface, and the scale

factor x/£ > which is used in going from a dimensionless response time to the real response

time. For layered sensors the situation is more complicated. The addition of an electrically

insulating layer outside of the active portion of a sensor increases the response time of

the sensor in a way which depends upon both the relative thickness of the layer and the

extend to which the thermal impedances of the two materials coincide. This is examined in

Sec. III-C.

Finally, we consider the effects of internal heating on the time response. A source

term, Q/(p C ) where Q is the rate at which heat is added to the sensors of density p and

specific heat at constant pressure C , must be added to Eq. (3) with the result,

3T(

a;

t} = X V

2 T(r,t) + Q/(p C ). (12)

a t Op

If the heating rate Q is time independent, then there is no effect on the time variation

of the temperature inside the sensor although it does affect the temperature profile. If

Q does change in time and if Q does not strongly depend on the change in temperature, then

the response of the temperature of the sensor initially at a stationary temperature

condition T (r) is found in Sec. III-E to be represented as

T(r,t) = T

Q

(r) + dy G(r,t-y) Q(y)/(p

Q

Cp f dy G(r, )

The magnitude of the effect depends on the heat capacity of the sensor and may be estimated

for specific cases by means of Eq. (13)

.

In summary we reiterate the answers to the four questions posed earlier concerning

the response function G(t).

1. G(t) can usefully be represented as a series of exponential functions (Eq.(7)).
After very small times only a few terms are significant.

2. The coefficients in the exponential series representation of G(t) , Eq. (7),

can be determined by matching Eq. (1) to time temperature data. In making

this match, it is important that the zero of time be accurately known as well as

the temperatures of the sensor and of the medium.

3. Eq. (1), which relates T(t), T

e

(t) and G(t) , is not easily solved for T

e (t)

given T(t) and G(t) . The constructive procedure described in III-F is unstable

and cannot be used. The use of eq. (1) to infer "real-time" values of T (t)

will require some input as to the form of T (t) in addition to the response

function and T(t). This point requires further study.

4. G(t) is independent of how the external temperature changes and of the

nature of stationary heat sources within the sensor. The effect on G(t) of

changing the material properties or the geometry can be readily analyzed. The

effect on G(t) of layer structure in a sensor is more complicated to analyze.

Finally, the nature of the thermal contact between the medium and the sensor has

a large effect on G(t) and depends on the properties of the sensor as well as

those of the fluid.

III. MODEL CALCULATIONS

This section consists of model calculations which provide a basis for our discussion

in Sections I and II. Part A contains a derivation of Eq. (1). Part B is concerned with

the construction of response functions for plates, cylinders and spheres. The temperature

equation, Eq. (3) is solved and the response functions are deduced from the solution. The

influence of layer structure on the response function of a sphere is examined in Part C.

The effect of thermal coupling of the sensor to the environment is considered in Part D

and internal heating transients are examined in Part E. Part F is concerned with applications

of the time response equation, Eq. (1), and some limitations on how it may be used are

considered. These parts are intended to serve as appendices to the first two sections of

this report.

A. DERIVATION OF EQUATION 1. g Equation 1 describes the relation between the external temperature T (t) and the

10
temperature of the sensor T(t) in terms of the response function G(t) . The response

function indicates how the sensor responds to a unit step change in external temperature

at time t=0. The essential feature used to develop Eq . (1) is the linearity of the system.

An arbitrary variation of external temperature can be viewed as a series of step

changes. To see this, consider a plot of T (t) vs. t which is represented by a bar graph

as in Fig. A-l. The value of T at the ith increment may be written as

T

6

(i) = T

6 (i-1) + AT6

(i) (A-l)

The response of the system at a time corresponding to increment j , j > i is

e obtained by scaling Eq. (2) by AT (i) to yield

A T

e

(i) [1 - G(t.-t.)]. (A-2)

The total response of the sensor at a time t is obtained by adding up all of the individual

responses to changes occurring prior to t,

j T(t .) = ^ A T

c

(i) [1 - G(t -t

1 / j J -

1- -t.)]. (A-3)

i=0

Next we take the limit t., n -t.^O so that

l+l l ^( i,.^ dy (A-4)

and we recover Eq. (1) ,

T(t) = T

e (t) - f

dy G(t-y)^^- (1)

Fig. A-l. Decomposition of an arbitrary

temperature change into a series of step

changes

.

11
Another way of deriving Eq (1) is to use the Laplace transform technique to solve Eq.

(3). The solution to the temperature equation, Eq. (3), is

f(z) = r(z) T

e

(z) (A-5)

where T(z) is the Laplace transform of the solution, T (z) is the Laplace transform of the

temperature on the boundary and T(z) (the Green function for the problem) depends on geometrical

factors and material properties. The formal solution is found by using the convolution

theorem for Laplace transforms with the result

„t = J

dy r(t-3

*

' T(t) = / dy r(t-y) T

c (y). (A-6)

Unfortunately lim r(x) is infinite so it is desirable to integrate by parts to remove this

x^O

feature. The result is

.t

r,e,

T(t) = T

S

(t) - / dy G(t-y) d

^y

(y)

(1)

with T^CO) = so that the divergence disappears. Now we see that

G(z) - A - I<*1 < A" 7 > z z

or

8(«) - 1 -i"

1 {^} (A-8)

If the pole at z=0 in T(z)/z is excluded, then G(t) can be obtained as

G(t) - - 1 / dze

Zt 1^2. , ( A_ 9 ) ZTTl C Z

where C is the usual contour for inverting Laplace transforms except that the pole at z=0

is not enclosed by C.

Bl. CALCULATION OF THE RESPONSE FUNCTION FOR A SOLID SPHERE AND A SOLID PLATE BY THE

SERIES SOLUTION METHOD

The series solution method is used here to derive G(t) for a solid sphere. The problem

is formulated as follows. A solid sphere of radius R is located at the origin of a coordinate

system. The initial temperature distribution is

T(r,0) =0 r < R. (Bl-1)

The temperature for r > R is fixed at T . The temperature equation is

12
3T(r,t) = X

3t

|

"3 2T(r,t) + 2 3T(r,t)

L 9r 2 r 3r (Bl-2)

because of spherical symmetry.

We postulate a solution (r < R) of the form

T(r,t) = V a.R.(r) T.(t)

j=0

(Bl-3)

Substitution of this into Eq. (Dl-1) yields

d

2 R,(r) dR, (r)

r

dT (t)

XT.(t) dt

Solving for T.(t) we find

R.(r) dr2

-+ - S— c

:

T (t) = T (0) exp(-C xt) .

The equation for R. (r) is

, d2R

i (r) + 2r dR

i (r) + r

2 C. R.(r) = . r ^r^- —dr J J

The solution of this equation which is finite at r = is

R^r) =J (C. 1/2D.

(Bl-4)

(Bl-5)

(Bl-6)

(Bl-7)

Using the orthogonality relations for spherical Bessel functions (j ) and the initial

conditions, we find

T(r,t) = T

e |l - 2 ^(-l) k+1

j Q (^) exp[- X t(^k/R) 2

]l . (Bl-8)

The response function for a sphere is thus

G(r,t) = 2 ) (-D

k+1

j (^) exp[-X t(7Tk/R) 2

] (Bl-9)

The response function for a plate is

£

"1 r t

, N 4 V* (-D cos (22 + 1) itx exp -xt / 21 + 1 \

( x ' c ) = 7 2^ 2* + 1 flL J L

' 2L V

(Bl-10)

where x is measured from the center of the plate and 2L is the width of the plate,

The same technique used in deriving Eq. (Bl-9) can be used to derive Eq. (Bl-10).

These expressions were evaluated for x=0 and r=0 and the results are shown

in Fig. 6 (located in Sect. II).

13
For short times these series converge very slowly and many terms are needed.

However, for longer times, say xt/R2

> .1, the convergence is quite good and only a few

terms are required. This is important when systems with more than one layer are involved

since the C.'s are no longer derivable from simple transcendental equations with closed

I form solutions. Instead numerical solutions of transcendental equations are required.

This is illustrated in Section III-C for the case of a two layer spherical sensor.

B2: CALCULATION OF THE RESPONSE FUNCTION FOR A SOLID CYLINDER BY THE LAPLACE TRANSFORM

TECHNIQUE

The Laplace transform technique is widely used for solving heat conduction problems

in solids. We illustrate the technique by solving heat conduction problems for an

infinite solid cylinder with a "radiation" boundary condition.

Suppose that initially the sensor and surroundings are in thermal equilibrium at

an arbitrarily defined T=0. Then, at an initial time, t=0, the surroundings undergo a

e step change of temperature, T . The heat conduction equation of interest and the

associated "radiative" boundary condition are given by

|£ = XV 2 T r < a (B2-1)

, 3T

9n + H (T(r=a) - T

e

) = (B2-2) r=a Making use of cylindrical symmetry, we obtain for Eqs. (B2-1) and (B2-2)

3T 3 2 T , 1 3T _ ,_. _. -KZ = X T

-7 + _ T- r < a (B2-3)

3t 3r^ r 9r

, 3T

3r + H(T (r=a) - T

e

) = . (B2-4) r=a Taking the Laplace transform of Eqs. (B2-3) and (B2-4) yields

fl + ± f ~ £ f = (B2-5)

dr"1 r dr x

X -^ + H(T - —) = at r=a. (B2-6) dr p

00 where T = I e *""Tdt Jo

14
By changing variables, to = */7" r, we see that Eq. (D2-5) is the modified Bessel

differential equation for the case of zero eigenvalue.

fj + l f _ T = o

dto to dto (B2-7)

The solution to (B2-7) that satisfies the boundary condition and that remains finite at

the origin must be of the form

f = F(p) I Q

( Jy r) (B2-8)

where I is the modified Bessel function of the first kind,

o The function F(p) is obtained by substituting Eq. (B2-8) into Eq. (B2-6) . We thus

find

r) T = HT I Q

( £

*'[va#v^ vvH

(B2-9)

The quantity of interest T can be obtained from T by taking the inverse Laplace

transform. Complex variable theory tells us that the inverse Laplace transform may be

obtained by the following prescription:

T = 27i /

ePt T(p) dp ' (B2-10)

Y-i°°

where y is to the right of the poles of T(p) . The only singularities for the integrand

are simple poles along the negative p axis including the point p=0. We thus evaluate

the integral above by performing the integration around the contour shown in Fig. B-l. The

Fig. B-l. Contour used to evaluate Eq. (B2-10)

15
contour integral can be readily performed by using the residue theorem which states

that the integral around a closed loop is equal to 2iri times the sum of the residues.

It can be shown that the integral along the semicircle goes to zero as the radius

approaches <*>. Thus the contour integral is equal to the integral in Eq. (B2-10) and

T = 2lrf t ePt f (P>] • (B2-11)

Res.

In order to evaluate the residues, we first must find the poles of Eq. (B2-9) , or,

equivalently, the roots of the equation

hjr "V^-iM^" -°. (B2- 12)

where we have made use of the relation I~(z) = I (z) . The roots of Eq. (B2-12) are

along the negative axis so that the argument of the modified Bessel function is imaginary,

By making use of the relationship between the Bessel function J and the modified Bessel

function I, we can express Eq. (B2-12) in terms of J_ and J with real arguments.

a J. (a a) - - J_(ot a) = (B2-13) n 1 n A (J n

(B2-14)

The quantity a is the n root of Eq. (B2-13) with a corresponding to the smallest

root in absolute magnitude. Carslaw and Jasger's book contains the first five roots to

Eq. (B2-13) for a wide range of values for the parameter H/X.

f-Vi Since all the poles are simple, the residue at the n pole is given by the formula

R

n

= [(p - Pn

) ept f(P )] (B2-15) P=Pn

We treat the case Pn=0 separately from the other values of p . For this case we obtain

from Eqs. (B2-9) and (B2-15) plus the properties of the modified Bessel functions at

zero argument

R = T

e (B2-16)

For the other residues, we must factor out the n root from the denominator in

Eq. B2-9. Let

D(p) = (p - pn

) Dl (p) = I ± {$ a)^|\ f

I Q

(^Jpa). (B2-17)

16
By differentiation we find

W= a? D(P)

We make use of Eq. B2-12 plus the identity, n

z I'^z) + I 1 (z) = zI

Q

(z) ,

to obtain the following expression for D (p ) w= ^

a)

2ap

2 PT /Ha^

(B2-18)

(B2-19)

(B2-20)

From this equation plus Eq. (B2-9) and Eq. (B2-15) we obtain the residue,

w=

2 T H a exp(p'> J o

(V^ r)

JIo# a)

(B2-21)

i 2 ^n_ _ / X \

The solution for T can now be obtained from Eqs. (B2-21) , (B2-16) , and the relation

between I and J_.

T = T

e _

2 T Ha

OO E

exp(-a

n

2 xt)J (an

r)

n=] J

Q

(an

a) >V+F<

(B2-22)

The response function (1- T/T ) is thus found to be

G(t) = ^E exp(-a

n

2 xt) J

Q

(anr)

n=l J

n

( a.»•) [^dfj

(B2-23)

B3. HEAT TRANSFER IN A HOLLOW CYLINDER

The problem of heat transfer in a hollow cylinder is of interest to us for two

reasons. First.it has an additional parameter, the ratio of the outer to inner radii

b/a, which is not found in the solid cylinder. Second, it is expected that a hollow

cylinder can be made to more nearly approximate a real platinum-resistance thermometer

than can a solid cylinder.

We consider the case of heat transfer in the hollow cylinder with "radiation"

boundary condition at the outer surface and with no heat flux across the inner surface.

Again, we are interested in the case of a step change in temperature T of the surroundings

17
at t=0. The heat equation and the boundary conditions are given below:

|£ = XV

2 T b >r> a (B3-1)

_3T 3n + H[T(r=b) - T ] = r=b

(B3-2)

_9T 9n

= r=a

(B3-3)

This problem is solved by the Laplace transform technique described in Section B2,

The response function at r=a is given by

G(t) = L D

*a) (B3-3)

n=l n

D(a ) = ~ a

2 {J.(a ) [Y.(a b/a) - a mY 1 (a b/a)] + Y_(a ) [-J_(a b/a) +

n 4 n (J n U n nin Un On

anin mJ, (a b/a)] - —alnJ. (a ) [Y n (a b/a) + ma Yn (a b/a)] + In nUn

Y n (a ) [J, (a b/a) + ma Jn In In (a b/a)]} , n n

(B3-4)

where m = —— and Y is the solution to the Bessel differential equation that is infinite

an

at the origin. The a are the roots to the transcendental equation

ma [J (a b/a) Y, (a ) - Y. (a b/a) J. (a )] + [J, (a ) Yn nin (a b/a) - Y. (a ) J_(a b/a)] = 0. In In In InUn InUn

We show in Fig. B-2 how the response time increases for a hollow cylinder as the

outer diameter b increases with the inner diameter fixed. The response function is

evaluated at the inner surface and the dimensionless ratio X/aH equals 3.33 for each

curve. The time is expressed in units of a /x18
1 2

Fig. B-2. Response functions for hollows cylinders for different values of

b/a, the ratio of the outer diameter to the inner diameter. The ratio A/aH is

3.33 for each case and time is expressed in units of a

2 /y.

C. LAYER STRUCTURE

The effect of layers on the time response of a sensor can be illustrated by considering

the case of a two-layer, spherical sensor. An idealized version of this object is sketched in

Fig. C-l. The inner, active region 1 is surrounded by an outer protective/electrically

insulating region 2. The temperature at the center can be related to the temperature at

surface 2 quite conveniently by using the Laplace transform method.

The result is

T

Q = T

2 /D(x1 ,y,R,A) (C-l)

Here x.. = q-,R,

X9 = Yxi =

<l9 £

(C-2)

and

A = X ± /X

2

; R = R

1 /R2 Fig. C-l. Sketch of a two layer spherical

sensor. Region 1 is assumed to be the active

region and region 2 is a protective layer.

sinh x,

D(x r

y, R, A) =

R cosh (yx

1

) + (1 - A) (1 - R)

sinh (yx

1 )

YX!

+ A(l - R)

cosh X-. sinh (yx, )

Yx x

(C-3)

The zeros of D occur when x = R^lz/x-, is purely imaginary. By excluding the pole at x =0,

G(t) can be constructed using the technique outlined in Section III-A. Fig. C-2 shows

X

l G(t) for the case A = 0.4, — = 2.25 for several values of R. As expected, the

19
presence of an outer layer slows down the response of the sensor. Time Is expressed in units

of R

1

2 /x1

.

Fig. C-2. Response functions for a two-layer spherical sensor for several values

Of Ri/Rz - For this calculation \\/\2 =0.4 and xi/X2 = 2 - 2 5. The time is expressed

in units of R\ z /x\-

D. THERMAL COUPLING OF SENSOR TO ENVIRONMENT

As pointed out earlier, the type of thermal coupling between the sensor and the surroundings

can make a large difference in the response time of a sensor. Below we shall consider two

types of boundary conditions for the heat transfer at the interface between a sensor and its

surroundings: (1) Prescribed surface temperature or good thermal contact and (2) "radiation"

boundary condition. In the first case the temperature at the surface is a specified function

of time. Mathematically this is the easiest boundary condition to study, though in

practice it is difficult to achieve. The "radiation" boundary condition, also referred

to as "Newton's law", states that the heat flux at the surface is proportional to the

temperature difference between the surface of the sensor and the medium. Actual conditions

more often conform to the "radiation" boundary condition rather than the prescribed

surface temperature condition.

Suppose that initially the sensor and surroundings are in thermal equilibrium at

e t=0. Then, at an initial time, t=0, the surroundings undergo a step change to T . The

heat equation and the prescribed surface temperature boundary condition for a solid

cylinder of radius a_ are given by: — = XV 2 T

at x for r < a

20

(D-l)
T(r=a) = T

e

t > (D-2)

Eq. (D-2) means that the temperature at the outer surface of the cylinder changes instantly to

T at t=0 (good thermal contact). The response function G(t) for this problem, which can be

obtained using the techniques outlined in Sec. III-B, is

„a2

00 en xt T , N J.(a r)

r(t) = 7 V - n

^ KZJ *". aa J. (a a) (D-3) n=l n 1 n

where the J's are Bessel functions. The a are the roots to the equation

J

Q

(an

a) = (D-4)

A plot of the quantity G(t) versus t (time in units of a

2 /x) is given in Fig. 5 (labeled 0)

for T evaluated at the center of the cylinder.

For the case of the "radiative" boundary conditions, Eq. (D-2) is replaced by

3T

X to + H(T(r=a) - T

e

) = 0. (D-5)

The first term is the heat flux from conduction with the 3T/3n being the normal derivative of

the temperature at the surface of the cylindrical sensor. The symbol H stands for the heat

transfer coefficient. The response function for this problem with the boundary condition Eq.

(D-5) and with a unit step change in temperature is given by Eq. (B2-23) , «o - as y

7<-">>Vv>

(B2_ 23)

n=l ( ^2 j/HaVl J n

(a a)

_(aan

} X\)j ° n Here the a are the roots of the transcendental equation

an HJ

l (an

a) =

J (an

a) ' (D" 6)

For the case of the "radiative" boundary condition we have one more parameter, H, than

for the case of the prescribed surface temperature. In the limit X/H approaches zero, the

radiative boundary condition becomes the same as the prescribed surface temperature boundary

condition. As the ratio increases the response time of the sensor decreases as shown in Fig.

5, where the plots are parametrized by the dimensionless ratio X/aH. The increase in the

response time with an increase in X/aH is also indicated by the plot of the inverse

of the longest relaxation time, 1/t , vs. X/aH in Fig. 4 where t is expressed in

1 1 units of a

2 /x- The longest relaxation time is defined as the time at which the

21
argument of the exponent for the first term in Eq. (B2-23) is unity.

In designing a thermometer, one would like to minimize the response time and this

can be done by maximizing the heat transfer coefficient H. Below we give an empirical

2 equation for H in terms of the Reynolds number N , the Prandtl number N , the radius

of the sensor a, and the thermal conductivity of the fluid A .

c n Npr

= ^

(D_ 9)

In these equations n is the fluid viscosity, p the fluid density, and C the constant

pressure heat capacity and v is the fluid velocity. Eq. (D-7) is valid for N between

50 and 10,000.

We illustrate the use of the equations in this section by predicting the response

time of a model platinum-resistance thermometer to a step change in temperature when

immersed in H„0 moving 1 m/s. The model consists of an infinite cylinder of powdered

A1„0_, a material used in many commercial platinum resistance thermometers, with a

platinum wire of infinitesimal thickness in the center. The temperature of such a

thermometer is the temperature at the center of the cylinder. We take the radius of

the cylinder J to be .001 m. From the data in Table D-l we calculate NT1 , N_, , and

Re Pr

-| from Eqs. (D-8) , (D-9) , and (D-7) respectively: N

Rg

= 2xl0 3

, Npr

= 7.0, and -| =

2.4. Then we find from Fig. 4 that the longest relaxation time t = 1.33, which from

the thermal properties of Al 0, corresponds to 0.1 s.

Table D-l Approximate Values of the Transport Properties

for Al and HO at 20°C

-k)] p[kg/m 3

] Cp Substance A[J/( [J/(kg-K)] n[Pa s]

A1

2 3 30

a

H

2 .59

a

3800

a 753b

1000

a 4200

a .001

a

Handbook of Chemistry and Physics, 56th Edition 1975-1976 (CRC Press, Cleveland, 1975),

D. C. Ginnings and R. J. Corruccini, J. Res. NBS 38, 593 (1947).

22
E. INTERNAL HEATING TRANSIENTS

The value of knowing G(t) is further illustrated by studying the time response of a

spherical sensor to changes in internal heating. Suppose the sensor is embedded in a

reservoir at T=0 and is in equilibrium with it for t < 0. At t=0, an internal heat

source is turned on. This problem may be posed as follows:

T(r, t<0) = (E-l)

3t X ur ' t; +

P c r < a (E-2)

P where

Q(t) = (0 t <

r < a (E-3)

(Q(t) t >

and T(r > a,t) = . (E-4)

This problem may be conveniently treated using the Laplace transform method. In

terms of V(r,t) = rT(r,t), the equation in transform variables becomes

V( r > 2 > _ z j, s m -rQ(z) -rQ(z) , . 3r2 X V(r ' Z)

XPC

p

A • (E5)

The solution finite at the origin is

V(r,z) = Asinh r \z/x H xQ(z)

Xz (E-6)

so

f (r>z) , Asinh r Vz7? , _M^) _ (E_ 7)

This solution is a superposition of the homogeneous solution and a particular solution

appropriate to the heating pattern. By invoking the boundary condition

T(a,t) = (E-8)

we obtain

g(z) T(r,z) =

zp Cp

1 _ a. sinh r yz/x (E-9)

r sinh a «/z/x !

The pole at z=0 has zero residue so using the convolution theorem we obtain

• c) = *k £

T(r,t) = —i- / dy G(r,t-y) Q(y) (E-10)

°0C

p ^0

where

23
G(r,t) = 2 2_j ^ -1 ^ exP

~ xt ("T^ a n=l (mr r/a). (E-ll)

is the response function for a sphere.

Equation 13 is obtained by changing the initial condition T(r,t) = to

T(r,0) = T

Q

(r) . (E-12)

F. APPLICATIONS OF EQUATION 1.

For numerical work, the discrete form of Eq. (1)

i T

6 - T6 T = T

e

- 5 V £ G i+1 i- 1

is used. The time increment is 6, Gn = 1 and e. . is 1 except if i = or i = i when it

1,2

is 1/2. If we set T -11= T - , we can solve for G. in terms of the quantities {T.}

j i and {T . }. The first step, j = 1, is found to be

G

1 = 2(1^ - T

1 )/Tei - (Te2 - T

e

Q

)/2T

e

1

. (F-2)

for j >2 the result is

G

j = 2 (T

&

. - T.)/Te

i - (T 6

. +1

- T

e

j _ 1 )/2T6

1

j-1 ( F- 3 >

i=l

To illustrate the use of Eqs. (F-2) and (F-3) consider the following model calculation.

The temperature, T(t), at the center of a plate is determined using Eq. (Bl-3) and the

postulated external temperature

T

e

(t) = 1 - exp(-0.6T) (F-4)

in Eq. (1). The calculated values of T(t) and the values of T

e

(t) obtained from Eq. (F4) are then used to estimate G(t) using Eqs. (F-2) and (F-3). Although it does not

explicitly appear in the equations, a time increment of 6 = 0.05 was used. The results

ar« displayed in Fig. Fl. The actual error

AG = G . - G (F-5) calc exact

i» shown in the upper curve. This is to be compared with G computed using Eq. Bl2 -1

3. The time interval ^t <.l is in the usual reduced units of (2L/tt) x where 2L is

24
t Fig. Fl. Lower curve: The response function G(t) for a slab sensor from Eq . Bl-3.

Upper curve: the error in G(t) determined using Eqs. F-2 and F-3.

the thickness and x is the thermal diffusivity of the plate. We see that if T(t) and

T (t) are accurately known, then G(t) can be extracted with reasonable accuracy using

the discrete version of Eq. 1 embodied in Eqs. (F-2) and (F-3). If T (t) changes slowly

compared with G(t), then it's procedure will not be accurate. To see this consider Eq.

e e (F-2); slow changes in T (t) mean that the difference T - T

1 will be small with a

resulting loss in precision which will lead to an inaccurate estimation of G(t) . This

is a point which should be checked with model calculations of the type described here

before this procedure is applied to other problems.

A further note of caution is in order. The algebraic solution to Eq. (F-l) for T.

in terms of the {T.} and {G.} is readily found to be

T

i+1

= 4(T^

j-l

r. - t.) + Vl -2 £ Vi (T i+i * Ti-i} j " X (F" 6)

where G =1, TL,=0 and the value of T. is specificied. The quantities {T.} and {G.} 1 1

M

J J are assumed to be known. Unfortunately, this way of estimating T is numerically unstable.

To illustrate this we use the example considered above. The exact values of the set

{G.} for a plate of thickness 2L and of the set {T.} when Eq. (F-4) applies have been

used infer a set of {T.} values using Eq. (F-6) and a time increment of 6 = 0.05. The

25
results are displayed in Fig. B-2. The dashed curves are the estimates of T (t) obtained

using Eq. (F-6) with T

e and with T^, T^ and T

p specificied by Eq. (F-4)

0.5 1.0

Fig. F-2. An illustration of the unsuitability of Eq . F-6 for inferring external

temperatures. The solid curve is the correct result for the example considered

and the dashed curves are obtained using F-6 with different numbers of points

constrained to be on the correct curve.

These results indicate that T

e (t) cannot be constructed from a knowledge of G(t)

and T(t) by this procedure. As suggested in Section I, some additional constraints on

the form of T (t) are needed.

The response of a sensor to various frequency components is a complicated function

of the frequency. This can be seen by considering the case of a sinusoidally varying

external temperature

T (t) = T sin oot (F-7)

applied to a plate. The temperature T(t) at the center of a sheet of thickness 2L is

found by using Eq. (1) and G(t) for the plate which is derived in Section III-B1. The

result is

26
TOO

T = A(w) sin wt + B(co) cos lot + <f>(t) (F-8)

where <j>(t) is a transient. Written out explicitly

Fig. F-3. The frequency dependent Fig. F-4 . The steady state amplitude y and

amplitudes A and B in Eq. F-8 in terms of phase shif t n of Eq . F-12 as functions of the

dimensionless frequency ft=wL 2 /x;2L is the dimensionless frequency Q=wL 2 /x;2L is the

plate. thickness of the plate.

A(u) = i - *f z

1=0 —(21 + 1) {to 2 F^WFTTT +

(F-9)

.M - - m V

^F^f

(-1)" (24 + 1)

21 +

and

( t) = « £ 1

£=0j ,

-')

J

}

(F-10)

xtl

2

"

-1) (2£ + 1) e ^[fr H^m (F-ll)

The frequency dependance for A(a>) and B (w) are shown in Fig. B-3 in terms of the

reduced frequency Q = wL2 /x for 0.2 < 9. <2.0. The large oj limits of A and B are both

zero. The steady state response can be expressed as

T(t) = y(w)sin(ojt + n (co)) (F-12)

The quantities y(a>) and n(w) are shown in Fig. (F-4) as functions of ft. For 9. = 2, the

steady state amplitude has been reduced by over 20% of the input amplitude and the phase

27
has been shifted by 50°.

This example illustrates the difficulties involved in attempting to infer T

e

(t)

directly from T(t) and G(t). The nerp^arv n'nfn^,^^ • K J i ' ine necessary information is simply not available in those

data.

28
REFERENCES

1. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford University

Press, London, 1947).

2. W. H. McAdams, Heat Transmission (McGraw-Hill Book Company, Inc., New York,

1954).

3. B. Gebhart, Heat Transfer 2nd edition (Mc-Graw-Hill Book Company, Inc., New York,

1971).

4. T. Kusuda, "Thermal Response Factors for Multilayer Structures of Various Heat

Conduction Systems", ASHRAE Trans. 75, 246 (1969).

ft U.S. GOVERNMENT PRINTING OFFICE: 1978 —261-238/438

29
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-959

2. Gov't Accession

No.

4. TITLE AND SUBTITLE

Calibration of Time Response of Thermometers:

Concepts and Model Calculations

3. Recipient's Accession No.

5. Publication Date January 1978

6. Performing Organization Code 221.05

7. AUTHOR(S)

Raymond D. Mountain and G. W. Mulholland

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP) 13. Type of Report & Period Covered Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less (actual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) Some of the conceptual problems associated with the calibration of the time

response of a temperature sensor are examined in this report. The discussion is

in terms of a time response function which characterizes the way a given sensor responds to changes in the temperature of its surroundings. A series of model

calculations of the response function for idealized sensors are used to investigate

the general features of the response functions. Important features are the

sensitivity of these functions to (i) material properties of the sensor, (ii) the

type of thermal coupling of the sensor with the environment and (iii) the geometry

of the sensor. These features must be considered in the design of procedures for

calibrating the time response of thermometers.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons) Calibration; response time; temperature; thermal response;

thermometer; time response function.

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NBS TECHNICAL PUBLICATIONS

PERIODICALS

JOURNAL OF RESEARCH—The Journal of Research

of the National Bureau of Standards reports NBS research and development in those disciplines of the physical and

engineering sciences in which the Bureau is active. These

include physics, chemistry, engineering, mathematics, and computer sciences. Papers cover a broad range of subjects, with major emphasis on measurement methodology, and

the basic technology underlying standardization. Also in- cluded from time to time are survey articles on topics closely

related to the Bureau's technical and scientific programs. As

a special service to subscribers each issue contains complete

citations to all recent NBS publications in NBS and non- NBS media. Issued six times a year. Annual subscription: domestic $17.00; foreign $21.25. Single copy, $3.00 domestic;

$3.75 foreign. Note: The Journal was formerly published in two sections: Section A "Physics and Chemistry" and Section B "Mathematical Sciences." DIMENSIONS/NBS

This monthly magazine is published to inform scientists, engineers, businessmen, industry, teachers, students, and

consumers of the latest advances in science and technology, with primary emphasis on the work at NBS. The magazine

highlights and reviews such issues as energy research, fire protection, building technology, metric conversion, pollution

abatement, health and safety, and consumer product per- formance. In addition, it reports the results of Bureau pro- grams in measurement standards and techniques, properties of matter and materials, engineering standards and services, instrumentation, and automatic data processing.

Annual subscription: Domestic, $12.50; Foreign $15.65.

NONPERIODICALS

Monographs—Major contributions to the technical liter- ature on various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with interested industries, professional organizations,

and regulatory bodies. Special Publications—Include proceedings of conferences sponsored by NBS, NBS annual reports, and other special

publications appropriate to this grouping such as wall charts, pocket cards, and bibliographies.

Applied Mathematics Series—Mathematical tables, manuals, and studies of special interest to physicists, engineers,

chemists, biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties of

materials, compiled from the world's literature and critically evaluated. Developed under a world-wide program co- ordinated by NBS. Program under authority of National

Standard Data Act (Public Law 90-396).

NOTE: At present the principal publication outlet for these data is the Journal of Physical and Chemical Reference Data (JPCRD) published quarterly for NBS by the American Chemical Society (ACS) and the American Institute of Physics (AIP). Subscriptions, reprints, and supplements

available from ACS, 1155 Sixteenth St. N.W., Wash., D.C.

20056.

Building Science Series—Disseminates technical information developed at the Bureau on building materials, components,

systems, and whole structures. The series presents research

results, test methods, and performance criteria related to the

structural and environmental functions and the durability and safety characteristics of building elements and systems. Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a subject.

Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often

serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures

published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations. The purpose

of the standards is to establish nationally recognized require- ments for products, and to provide all concerned interests with a basis for common understanding of the characteristics of the products. NBS administers this program as a supple- ment to the activities of the private sector standardizing

organizations.

Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest

to the consumer. Easily understandable language and

illustrations provide useful background knowledge for shopping in today's technological marketplace.

Order above NBS publications from: Superintendent of Documents, Government Printing Office, Washington, D.C.

20402. Order following NBS publications—NBSIR's and FIPS from

the National Technical Information Services, Springfield, Va. 22161.

Federal Information Processing Standards Publications (FIPS PUB)—Publications in this series collectively constitute the Federal Information Processing Standards Register.

Register serves as the official source of information in the Federal Government regarding standards issued by NBS

pursuant to the Federal Property and Administrative Services Act of 1949 as amended, Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order 11717

(38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal Regulations).

NBS Interagency Reports (NBSIR)—A special series of

interim or final reports on work performed by NBS for outside sponsors (both government and non-government).

In general, initial distribution is handled by the sponsor;

public distribution is by the National Technical Information

Services (Springfield, Va. 22161) in paper copy or microfiche form.

BIBLIOGRAPHIC SUBSCRIPTION SERVICES

The following current-awareness and literature-survey bibli- ographies are issued periodically by the Bureau: Cryogenic Data Center Current Awareness Service. A litera- ture survey issued biweekly. Annual subscription: Domestic, $25.00; Foreign, $30.00.

Liquified Natural Gas. A literature survey issued quarterly.

Annual subscription: $20.00.

Superconducting Devices and Materials. A literature survey

issued quarterly. Annual subscription: $30.00. Send subscription orders and remittances for the preceding bibliographic

services to National Bureau of Standards, Cryogenic Data

Center (275.02) Boulder, Colorado 80302.
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