, v

: This report has been prepared

for information and record

purposes and is not to be referenced

in any publication.

NATIONAL BUREAU OF STANDARDS REPORT

753^ /

A DIRECT ELECTRICAL-ANALOG METHOD FOR TWO-DIMENSIONAL

TRANSIENT HEAT FLOW ANALYSIS OF FIRE-EXPOSED STRUCTURES

By

D. Gross

<$B§>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March 3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and maintenance of the national standards of measurement and the provision of means and methods for making measurements consistent

with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials, devices, and structures; advisory services to government agencies on scientific and technical problems; invention and development of devices to serve special needs of the Government; and the development of standard practices, codes, and specifications. The work includes basic

and applied research, development, engineering, instrumentation, testing, evaluation, calibration services,

and various consultation and information services. Research projects are also performed for other government

agencies when the work relates to and supplements the basic program of the Bureau or when the Bureau’s

unique competence is required. The scope of activities is suggested by the listing of divisions and sections on the inside of the back cover.

Publications

The results of the Bureau’s research are published either in the Bureau’s own series of publications or

in the journals of professional and scientific societies. The Bureau itself publishes three periodicals avail- able from the Government Printing Office: The Journal of Research, published in four separate sections,

presents complete scientific and technical papers; the Technical News Bulletin presents summary and pre- liminary reports on work in progress; and Basic Radio Propagation Predictions provides data for determining

the best frequencies to use for radio communications throughout the world. There are also five series of nonperiodical publications: Monographs, Applied Mathematics Series, Handbooks, Miscellaneous Publications,

and Technical Notes. A complete listing of the Bureau’s publications can be found in National Bureau of Standards Circular

460, Publications of the National Bureau of Standards, 1901 to June 1947 (SI. 25), and the Supplement to National Bureau of Standards Circular 460, July 1947 to June 1957 ($1.50), and Miscellaneous Publication 240,

July 1957 to June 1960 (Includes Titles of Papers Published in Outside Journals 1950 to 1959) (12.25); avail- able from the Superintendent of Documents, Government Printing Office, Washington 25, D. C.
NATIONAL BUREAU OF STANDARDS REPORT

MBS PROJECT NBS REPORT

1002-12-10120 June 19, 1962 753*+

A DIRECT ELECTRICAL-ANALOG METHOD FOR TWO-DIMENSIONAL

TRANSIENT HEAT FLOW ANALYSIS OF FIRE-EXPOSED STRUCTURES

By

D. Gross

NATIONAL BUREAU OF ST

Intended for use within the

to additional evaluation and

listing of this Report, either the Office of the Director, N

however, by the Governmenl

to reproduce additional copl

IMPORTANT NOTICE

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015.

progress accounting documents

rnially published it is subjected

reproduction, or open-literature

sion is obtained in writing from Such permission is not needed,

prepared if that agency wishes

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

A DIRECT ELECTRICAL- ANALOG METHOD FOR TWO-DIMENSIONAL

TRANSIENT HEAT FLOW ANALYSIS OF FIRE-EXPOSED STRUCTURES

By

D. Gross

ABSTRACT

A continuous-resistance, continuouscapacitance analog model has been constructed

and used to simulate a fire-exposed T-beam

structure. The model was arranged to permit

study of both one and two-dimensional transient

heat flow and to allow for Newtonian heat

losses to the cool surroundings. Comparison

of the results with those of a numerical solution of the problem show agreement within 11$,

based on the value of the input signal. Considering the size limitations of the electrical

model, edge effects, and the variation in its

electrical properties, the agreement is considered reasonable.

1. Introduction

The theory, construction details and performance of an

electronic instrument which permits solution of transient heat

flow problems by direct electrical analogy has been previously

described £l7 and the instrument has been used to solve a number of one-dimensional problems. Since the method involves

the use of "lumped" resistance and capacitance elements, the

extension to two (or three) dimensions could be accomplished by

the use of discrete (i.e. lumped) circuit elements arranged in

a two- (or three-) dimensional array. Because of simplicity,

space considerations and electronic circuitry problems associated with "fast-time" instruments of this type, the use of distributed parameter elements would present distinct advantages.

Furthermore, this would eliminate the "lumping" error resulting

from the use of a limited number of discrete elements; however,

other errors associated with non-uniformity of the conductive

medium would generally be introduced.

Conductive-sheet, conductive-solid and conductive-liquid

analogs find considerable use in simulating two-and three-
dimensional fields governed by Laplace's equation,, eo go steadystate heat flow For transient heat flow involving potential energy storing elements, however, previous use appears to

have been limited primarily to continuous-resistance, lumpedcapacitance analog So Apparently, the first published description of a method utilizing a distributed resistance and capacitance analog was described by Fatt £~37°

This report presents data obtained by use of a continuousresistance, continuous-capacitance analog simulating a fireexposed T-beam structure and compares the results with corresponding solutions obtained by means of finite difference computations Z* 47; it is intended to demonstrate the feasibility of

using such a model for the approximate solution of one- and

two-dimensional transient heat flow problems

»

2o Analog Model System

An ideal two-dimensional analog consists of a layer of

dielectric material of uniform capacity, to one surface of which

is applied a layer of conductive material of uniform resistivity,

the other surface being a perfectly conducting layer connected

to grounds Ideally both the dielectric and the resistance layer

should exhibit the same degree of homogeneity and isotropicity as

the thermal body being simulated 0 Of the more readily available

dielectric materials, barium titanate was chosen even though it

is known to be nonisotropic to some extent . A 4 by 4 by 1/8 in0 barium titanate sheet containing a painted resistance layer on

one surface and silver foil on the other surface was specially

fabricated by the Centralab Division of Globe Union Mfg 0 Go 0 in

February 1958o This was the largest size piece which could be

conveniently produced at the time; however, the thickness of the

dielectric was maintained to a tolerance of + ChOOl in0 Direct measurements were made of the resistance at several

positions across the entire surface and these are listed, in Table

1. The non-uniformity of the resistance layer at the measuring

point locations was estimated, by means of voltage measurements,

to be about ± 10 percents The time constant (RC) measurements

listed in Table 1 were obtained using the circuit shown in Figo 1,

A time constant of 7*+°7 msec, is well suited for measurements by

conventional cathode ray oseilloscopeso From these measurements,

the capacitance is 0 o 000h6h mfd/cnr which corresponds to a

dielectric constant, neglecting edge effects, of approximately

l660 o
- 3

-

Table 1. Measured Resistance and RC Product

R RC

X-direction meg sec

1 in. 1.60 .0752

2 in. 1.40

3 in. 1.40 .074-7

4 in. 1.80 .0770

Y-direction

1 in.

2 in. 1.60 .0760

3 in. .0741

4 in. .0712

Averages 1. 56 .0747

The analog model was arranged so that probes could be

positioned on its conductive surface through vertically

aligned hole's in two sheets of clear plastic placed as shown

in Fig. 2. Although a measuring grid of this type was selected

for convenience, the system is basically continuous and does not

restrict measurements to any finite number of space points., Because of symmetry, one-half of the T-beam may be represented by

the inverted L-section of Figure 3° A series of 19 "fixed"

probes were placed at selected points and were spring-loaded to

ensure good contact with the surface . The tips of the 0.075 in.

dia. probes were beveled to provide a contact area of approximately 0.03 in. dia. Each of the probes was connected to an individual discharge circuit and, through a manual selector switch, . to the high impedance measuring circuit. One additional "movable"

probe was used to make measurements at any other grid point desired. The line representing the fire-exposed surface was
outlined on the resistive paint surface using highly conductive

silver paint and then covered with a sheet of aluminum foil

formed to the same outline » The lower plastic sheet was fastened down firmly assuring good contact at the aluminum foilsilver paint interface o A lead connected to the aluminum foil

electrode was fed a voltage-time signal derived from a photoformer-type of signal generator which simulated the shape of the

prescribed temperature-time curve . Another lead from the silver

foil on the lower surface of the model was connected to the instrument ground

.

The problem chosen for study was one for which numerical

solutions by means of a digital computer were available £47

„

The lower surfaces (ceiling) of a long horizontal slab and beam

structure (T-beam) are subjected to the standard fire exposure

defined for fire endurance tests £57 while the upper surface

(floor) loses heat by radiation and Newtonian cooling to an

ambient maintained at the initial temperature of the structure

.

Although consideration was given in ref* 4 to a wide range of

physical and thermal parameters, this study was concerned only

with the following:

3 ° Two-dimensional T-beam Problem

3°1 Thermal

Thermal conductivity:

Thermal diffusivity:

Concrete density:

Beam width:

Beam to slab ratio:

Slab thickness:

Specific heat:

4o2

4 in* (10ol6 cm)

8 in* (20*32 cm)

140 lb/ft^(2<.24 g/cm^)

0*577 Btu/lb F (2®4i

watt sec/g C)

0*905 Btu/hr ft F

(0*0156 watt/cm C)

0*0112 f

t

2/hr (0o 00289

cm^/sec

)
- 5

-

The loss of heat from the unexposed upper surface by radiation and convection was assumed to take place according to the

relation

V3

q- = 0 o 275 t + 0.173 rr**4 -i

o

-p1

oo

1

—

1

oo n

1

—

1

p where q = rate of heat loss, Btu/hr ft

t = temperature rise above initial, deg F

tQ = initial temperature, 68 F

3o2 Electrical

Analogous thermal and electrical elements for both the onedimensional and two-dimensional cases are listed in Table 2.

Table 2. Analogous Thermal and Electrica]L Element So

One- Dimensional

Case

Two-Dimensional Case

(Conductive Sheet

Analog

)

Element Symbol Unit Symbol Unit

Thermal Resistance cm

2

C i 1 cmC

k watt k watt

Electrical Resistance R megohm R megohm/

square

Thermal Capacitance psxt watt-sec psA

t

watt-sec

cm^C cm C

Electrical Capacitance C cAe pfd

In addition to the capacity, resistivity and time-scaling factors

in the one-dimensional case, a linear scale factor X = L+./L

e is introduced in the two-dimensional case to account for the use

of a small-scale electrical model o Based on the dimensions and

the measured electrical values of the analog model, the values

of the scaling factors (in Cog„s. units) were as follows:
- 6

m = . 1 = 1 - 4l.l

kR (.0156) (1.56)

n = ps , L

t x 2 = (2.24) (2.41) (5)

2 = 2.91 x 10^

c

V L

; (.000464)

9 " 1

x = L

t = 4 = 5

L 0.8

e

For a three-hour fire exposure duration, each time division

on the read-out oscilloscope was chosen to represent 4 minutes

of thermal time and the oscillator was therefore set for a frequency of k

mn cb (4l . 1) (2. 91 x 10 p )(l) ^ kCe

f = t

t = 4 x 60

' " '

To simulate convective and radiative heat losses, it is

common to use non-linear resistive elements [1,6]. To maintain

the continuous, geometrical nature of a two-dimensional analog

would require the use of a distributed voltage-sensitive resistive film. In the absence of such a material, it becomes necessary to employ discrete elements, so that a combined geometric

and network analog results. The size of the connector contact

is important since, if too large, there is the tendency to form

a short circuit between adjacent contacts, whereas if too small,

the flow lines tend to crowd [71. Because of a lack of nonlinear elements of the desired properties and since simple resistors may approximate non-linear resistors over a limited

voltage range, conventional resistors were used. The contact

size was approximately 0.03 in. dia. and eight heat loss resistors were arranged 0.4 in. apart as shown in Fig. 3* The values

of the individual resistors were determined by linear approximations to the electrical equivalent of the heat loss curve over

the approximate temperature ranges (derived from the numerical

solution). The appropriate current-voltage curve was obtained

by use of the relation

q _ 106 At

i m A E

where A t = l44 F/volt, as measured.

A E
- 7

-

These resistor values check very closely (approximately 6%)

the values obtained by employing the "eff ective" surface heat

transfer coefficient of Fig. 4 (Ref. 4), and using the relation

1 . “hff

4. Discussion of Results

Temperature rise data on the two-dimensional analog model

of the reinforced concrete T-beam problem were obtained at- several

locations as indicated in Fig. 3*. For direct comparison with the

numerical results given in ref. [4], all data were taken at a thermal time of 2 hours after application of the standard temperature-time fire exposure along the surface designated F. Section

A-A is sufficiently removed from two-dimensional effects and hence

the data in Fig. 4 represents the one-dimensional temperature gradient for a horizontal slab whose lower surface is fire-exposed

and which loses heat from its upper ("unexposed") surface by

Newtonian cooling. The analog results are within 7 %<> based on

the value of the input signal , of the corresponding results obtained by numerical solution of the finite difference equations.

In Figures 5 and 6 are shown the temperature gradients along

Sections B-B and C-C, where two-dimensional heat flow patterns

are evident. Here, the analog and numerical values do not differ

by more than 11%.

Higher analog values along part of the B-B axis of symmetry

may be due to difficulties associated with measurements along an

edge surface. Although this problem also exists along the other

edge, its effect is partially masked by the Influence of the

heat loss resistors at the "unexposed" surface. Some of the factors involved in edge measurements include:

( 1 ) the conventional edge effect for a parallel plate

capacitor which results in increase in capacitance

close to the edge [81.

( 2 ) the finite dimension of the probe tip and its proximity to the edge,

and ( 3 ) the imperfect registration of the grounding plate,

which is actually recessed approximately 0.03 in.

from the edge.
- 8 -

To avoid the edge effect at the axis of symmetry, it would

be preferable to utilize the complete T-section. This would also

provide twice the measuring capabilities (and averaging- advantages), but would probably require a larger area model for reasonable accuracy.

In most experimental work, the difference or error between

the measured and true potentials is expressed as a percent of the

true potential. However, In analog-system measurements, this

expression is not suitable since the true potential may be very

small or even zero. A more feasible method for gaging the performance of an analog system is to express the difference between

the measured and true potentials as a percent of the largest potential difference existing in the field, generally the voltage

difference between boundary electrodes [2]. This method has been

employed here to compare the analog and the finite difference

results, neither of which can be considered true values.

It should be borne in mind that solutions of transient heat

flow problems by finite difference computations are only approximations and involve errors which depend upon the space and time

grids chosen and the associated problems of instability and convergence. It is generally difficult to estimate the magnitude

of such errors. Because of a lack of knowledge of the accuracy

of the numerical computations, analog measurements may be as

much a check on finite difference calculations as the reverse.

In fire endurance tests, the performance of the test construction is measured by the time period during which it withstands the standard fire exposure without failure according to

prescribed physical and thermal criteria. For wall or floor

constructions, thermal failure generally involves a temperature elevation on the "unexposed" surface of 2^0 degrees F.

Structural failure of stressed structural steel members usually

occurs at a temperature of about 1000 F and therefore particular

attention was paid to the lower portion of the beam where reinforcing members would be most subject to high temperature.

Studies of this kind are useful both for estimating relative

fire endurance times and as an aid in selecting appropriate steel

and concrete combinations for particular applications.
- 9 -

The feasibility of using a continuous-resistance, continuous-capacitance analog of the type described for two-dimensional

transient heat flow analysis has been demonstrated. Considering

the size limitations of the electrical model and the variations

in its electrical properties, the comparison with the numerical

solutions, at least for the locations and time period' measured,

were considered reasonable. The exploratory nature of the analog model described in this report leave a number of questions

regarding experimental technique, measurement error and overall

accuracy as yet unevaluated. In particular, the magnitude'* of

possible errors introduced by edge effects, by contact resistance and by capacitive coupling between probes and lead wires

should be evaluated. For situations requiring increased pre-cision, a more thorough exploration should be made of the dielectric properties of barium titanate (or alternate dielectrics),

especially with regard to piezoelectric, temperature, hysteresis

nonisotropic and dielectric loss effects.

5* Acknowledgments

The probes and mounting assembly for the analog model

were made by William H. Bailey. Two Antioch College cooperative students, Glenn Crout and Peter Kleban, worked out experimental details and made some of the early measurements.

USCOMM-NBS-DC
10 -

6. References

[ll A. F. Robertson and D. Gross, An Electrical-Analog Method

for Transient Heat-Flow Analysis, NBS J. Res. 61, 105

(Aug. 1958).

C 2 1 W. J. Karplus, Analog Simulation - Solution of Field

Problems (McGraw-Hill Book Co., Inc., New York, N.Y. , 1958)

.

[3"1 I. Fatt, A New Electric Analogue Model for Nonsteady State

Flow Problems, A. I. Ch.E. Journal, 4, 49 (March 1958)

[41 A. F. Robertson and S. M. Genensky, Theoretical Heat

Transfer Relationships for Fire Endurance of Reinforced

Concrete, NBS Report No. 4854 (Sept. 1956).

[51 American Society for Testing Materials, Standard Methods

of Fire Tests of Building Construction and Materials,

ASTM Designation E 119-58 (1958).

[61 D, I. Lawson and J. H. McGuire, The Solution of Transient

Heat Flow Problems by Analogous Electrical Networks,

Inst. Mech. Engr. Proc. [A 1 167, 275 (1952).

[71 V. Paschkis, Combined Geometric and Network Analog Computer

for Transient Heat Flow, Journal of Heat Transfer, Trans.

ASME, Vol. 81 C, No. 2, l44 (May 1959).

D. K. Reitan, Accurate Determination of the Capacitance of

Rectangular Parallel Plate Capacitors, J. Appl. Phys.

30, 172 (Feb. 1959).

[81
hovable

Figure l.\; Schematic diagram for measurement of the time

constant of a continuous-res is tance, continuous -capacitance

analog.

Figure 2 0 Mounting details of analog model
R 1 s 24.4 M

/ R Z : 17.4 M

Hfc'AT Loss ft £ S l S To f? S ^ "R4 - R© - • 3 M

A

OUTLINE

4x4 iM. tvAODE t

Figure 3. Grid pattern for analog model

of fire-exposed T- beam

FfiACTtov

or

Sept-h
Tff/Uf

fs

G'SE

,

400 -

FlQE - EKPoS&b

Surface

'

•

?0o

. -

7"!' ;

j-

:

0 i

r . ; .

i

! i

j

'

• j

‘ f

'' ’

i c

'•2£ .50 .75 1.0 FftACriONi OF bePTTH

U. S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and

Boulder, Colorado, is suggested in the following listing of the divisions and sections engaged in technical work.

In general, each section carries out specialized research, development, and engineering in the field indicated by

its title. A brief description of the activities, and of the resultant publications, appears on the inside of the

front cover. WASHINGTON, D. C.

Electricity. Resistance and Reactance. Electrochemistry. Electrical Instruments. Magnetic Measurements

Dielectrics. High Voltage.

Metrology. Photometry and Colorimetry. Refractometry. Photographic Research. Length. Engineering Metrology.

Mass and Scale. Volumetry and Densimetry.

Heat. Temperature Physics. Heat Measurements. Cryogenic Physics. Equation of State. Statistical Physics.

Radiation Physics. X-ray. Radioactivity. Radiation Theory. High Energy Radiation. Radiological Equipment.

Nucleonic Instrumentation. Neutron Physics.

Analytical and Inorganic Chemistry. Pure Substances. Spectrochemistry. Solution Chemistry. Standard Refer- ence Materials. Applied Analytical Research. Crystal Chemistry.

Mechanics. Sound. Pressure and Vacuum. Fluid Mechanics. Engineering Mechanics. Rheology. Combustion

Controls. Polymers. Macromolecules: Synthesis and Structure. Polymer Chemistry. Polymer Physics. Polymer Characterization. Polymer Evaluation and Testing. Applied Polymer Standards and Research. Dental Research.

Metallurgy. Engineering Metallurgy. Microscopy and Diffraction. Metal Reactions. Metal Physics. Electrolysis

and Metal Deposition.

inorganic Solids. Engineering Ceramics. Glass. Solid State Chemistry. Crystal Growth. Physical Properties.

Crystallography.

Building Research. Structural Engineering. Fire Research. Mechanical Systems. Organic Building Materials. Codes and Safety Standards. Heat Transfer. Inorganic Building Materials. Metallic Building Materials. Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics. Operations Research. Data Itocessing Systems. Components and Techniques. Computer Technology. Measurements Automation.

Engineering Applications. Systems Analysis.

Atomic Physics. Spectroscopy. Infrared Spectroscopy. Solid State Physics. Electron Physics. Atomic Physics.

Instrumentation. Engineering Electronics. Electron Devices. Electronic Instrumentation. Mechanical Instru- ments. Basic Instrumentation. Physical Chemistry. Thermochemistry. Surface Chemistry. Organic Chemistry. Molecular Spectroscopy. Molecular Kinetics. Mass Spectrometry.

Office of Weights and Measures. BOULDER, COLO.

Cryogenic Engineering Laboratory. Cryogenic Equipment. Cryogenic Processes. Properties of Materials. Cryogenic Technical Services. CENTRAL RADIO PROPAGATION LABORATORY

Ionosphere Research and Propagation. Low Frequency and Very Low Frequency Research. Ionosphere Research. Prediction Services. Sun-Earth Relationships. Field Engineering. Radio Warning Services. Vertical Soundings Research.

Radio Propagation Engineering. Data Reduction Instrumentation. Radio Noise. Tropospheric Measurements.

Tropospheric Analysis. Propagation-Terrain Effects. Radio-Meteorology. Lower Atmosphere Physics.

Radio Systems. Applied Electromagnetic Theory. High Frequency and Very High Frequency Research. Modulalation Research. Antenna Research. Navigation Systems.

Upper Atmosphere and S^ace Physics. Upper Atmosphere and Plasma Physics. Ionosphere and Exosphere Scatter. Airglow and Aurora. Ionospheric Radio Astronomy.

RADIO STANDARDS LABORATORY

Radio Physics. Radio Broadcast Service. Radio and Microwave Materials. Atomic Frequency and Time-Interval

Standards. Millimeter-Wave Research.

Circuit Standards. High Frequency Electrical Standards. Microwave Circuit Standards. Electronic Calibration

Center.