0018-151X/06/4403-0463 © 2006 Russian Academy of Sciences and Springer Science + Business Media, Inc.

High Temperature. Vol. 44, No. 3, 2006, pp. 463–470. Translated from Teplofizika Vysokikh Temperatur, Vol. 44, No. 3, 2006, pp. 465–471.

Original Russian Text Copyright © 2006 by D. V. Volosnikov, V. P. Efremov, P. V. Skripov, A. A. Starostin, and A. V. Shishkin.

HEAT AND MASS TRANSFER

AND PHYSICAL GASDYNAMICS

An Experimental Investigation of Heat Transfer

in Thermally Unstable Polymer Systems

D. V. Volosnikov1, V. P. Efremov

2, P. V. Skripov

1,

A. A. Starostin1, and A. V. Shishkin

1

1Institute of Thermophysics, Ural Division, Russian Academy of Sciences, Ekaterinburg, 620016 Russia

2Institute of High Energy Densities, Joint Institute of High Temperatures, Russian Academy of Sciences (IVTAN),

Moscow, 125412 Russia

Received July 27, 2004

Abstract—An experimental study is made into the process of heat transfer from a pulse-superheated probe

to a solidified polymer system at probe temperatures above the temperature Td∞ of the beginning of thermal

destruction of matter in a quasi-static process. Use is made of the procedures of thermal stabilization of the

pulse-superheated probe (with the characteristic time of constancy of superheated probe temperature of 1 ms)

and of shock heating (with the characteristic time of increasing the probe temperature of 1 μs and that of

monitoring its cooling-down of 1 ms). The effect of short-term thermal stability of polymers in the region of

T > Td∞ is revealed. A procedure is developed for identifying the signs of thermal destruction of polymers

in the pulsed process. The maximal values are estimated of the density of heat flux through samples without

their thermal destruction.

INTRODUCTION

The transfer under conditions of intense energy

impact on matter is studied in application to problems

involved in the simulation of thermal conditions in

rocket and space engineering, in the processing of

materials by concentrated energy fluxes, and in the

development of new working media for thermally

stressed processes [1–3]. In model experiments, com-

parison is made of the values of density of heat flux

through samples for the preassigned value of temper-

ature of the hot surface.

Experiments in Joule heating of a wire probe in

polymer melts at rates above 105 K/s are performed to

demonstrate the possibility of short-term transfer of a

polymer to the region of temperatures which exceed

significantly the temperature of the beginning of ther-

mal destruction of this polymer in a quasi-static proc-

ess Td(t → ∞) = Td∞ [4–6]. Because of the short time

of the experiment texp, no signs of thermal destruction

of matter show up until some value of T*(texp) > Td∞which is referred to as the temperature of attainable

superheat of matter. The thermal properties of poly-

mers in this region have not been studied. The prob-

lem lies in the reduction of the mean lifetime of a sys-

tem with increasing temperature T in the

experiment. The heat transfer in the region of thermal

instability of polymers T > Td∞ may be investigated

using the method of controlled pulsed heating [7] with

characteristic time . The value of texp

includes the time of development of superheated state

and the time of measurement proper with the temper-

ature preassigned in the experiment.

The known way of studying heat transfer in ther-

mally unstable polymers is based on the use of contact

methods with characteristic times of minutes and sec-

onds [3, 8]. Left outside of the capabilities of these

methods are the processes with time- and volume-con-

centrated heat release. It is such processes that make

up the subject of our investigation in which we sought

to reduce the duration of texp in order to increase the

depth of penetration into the region of thermal insta-

bility while maintaining the initial structure of matter.

The problem consisted in performing an experi-

mental investigation of the density of heat flux from a

t T Td∞–( )

texp t T( )<

HIGH TEMPERATURE Vol. 44 No. 3 2006

464 VOLOSNIKOV, EFREMOV, SKRIPOV et al.

pulse-heated probe through polymer materials and in

searching for a method of estimating the temperature-

and-time conditions of the beginning of thermal

destruction of matter by the results obtained for the

variation of heat flux in a series of pulses with the pre-

assigned heating function.

The method of controlled pulsed heating of a wire

probe (resistance thermometer) has been improved to

solve the problem [7]. A possibility is provided for

establishing the preassigned heating mode T(t) and for

monitoring the time variation of heat flux through

matter Q(t). The voltage and current in the probe sup-

ply circuit are recorded in the experiments, and the

current values of resistance and power of heating are

calculated. The rate of recording is defined by the

speed of amplification and transformation elements

and must not be lower than 106 readings per second.

A method has been found of controlling the heating

power P(t) and recording the results of successive

measurements in the computer memory. We used, in

combination, the procedures of thermal stabilization

of a pulse-heated probe and of cooling down of a

shock-heated probe. These procedures correspond to

the models of “isothermal” and “instantaneously

superheated” probes with stationary medium, which

are known in the theory of heat transfer [9, 10]. Figure

1 shows qualitatively the time dependences of temper-

ature which are realized in the experiment. The char-

acteristic times of the experiment are given for a probe

20 μm in diameter.

We investigated polymer systems prepared by

polymerization of monomer or by chemical solidifica-

tion of reactive compositions, as well as low-molecu-

lar hydrocarbons as comparison systems in thermo-

physical measurements. In the case of solidification,

the probe was immersed in the medium at the initial

stage of reaction and implanted into the bulk of sam-

ple.

EXPERIMENT

Thermal Stabilization of Pulse-Heated Probe

The first one of two procedures mentioned above

makes possible a rapid transition from the mode of

pulsed heating of the probe to the constant tempera-

ture mode T(t > t1) = Tpl(Δtpl) ≈ const (see Fig. 1)

owing to stabilizing feedbacks in the heating control

circuit. The difference in the power of heating the

probe P(Δtpl), required for maintaining the desired

temperature Tpl in different samples, reflects the dif-

ference in their thermal properties. For not too low

values of the Fourier number 0.2 < Fo ≤ 1.0, the elec-

tric quantities measured in the experiment may be cor-

related with the effective values of thermal conductiv-

ity λ(Tpl) and thermal activity b(Tpl) = (λρc)1/2

of

superheated matter within the linear model [7, 11]

(here, ρ, c is the density and specific heat of matter,

respectively).

In a series of experiments, the temperature of ther-

mal stabilization was increased, and matter made a

transition from the region of stable states (Tpl ≤ Td∞)

to the region of thermal instability (Tpl > Td∞) up to

the emergence of signs of thermal destruction. In the

case of polymer melts, the beginning of thermal

destruction was accompanied by the process of spon-

taneous boiling of highly superheated volatile prod-

ucts [4–7]. In the case of boiling, the time dependence

of power of heating the probe varied abruptly (see

Fig. 2, region Δtpl > ). The duration of the region of

thermal stabilization until the signal of boiling was

taken to be the mean lifetime of melt at the preas-

signed temperature . The beginning of thermal

T0 t1 tpk t2t

12

Tpl

Td∞

Tpk

T

Fig. 1. Qualitative time dependences of tempera-ture in experiments in thermal stabilization of asuperheated probe (subscript pl, line 1) and inshock heating of the probe (subscript pk, line 2);Tpl ~ 300–900 K, Tpk ~ 400–1300 K, Td∞ ~ 500–

650 K; t1 ~ 0.1 ms is the time required to reach

the operating conditions, (t2 – t1) = Δtpl ~ 1–10 ms

is the region of thermal stabilization, tpk is the

instant of shock heating, (t2 – tpk) = Δtm ~ 0.1–

10 ms is the region of monitoring the coolingdown of the probe.

t

t Tpl( )

HIGH TEMPERATURE Vol. 44 No. 3 2006

AN EXPERIMENTAL INVESTIGATION OF HEAT TRANSFER 465

destruction of solidified samples was not accompa-

nied by a distinct response directly at the time of

pulse. The procedure of identifying thermal destruc-

tion in solidified systems is described in the next sec-

tion.

The requirement of constant temperature of the

probe during the time of measurement, which is dic-

tated by the adopted model of heat transfer [9],

restricts the operating speed and, as a consequence,

the depth of penetration into the region of thermal

instability on the level Tpl – Td∞ = (200±100) K. In

order to increase the value of superheat while retain-

ing the reproducibility of the results, it was necessary

to reduce the duration of impact texp.

Cooling Down of Shock-Heated Probe

In this procedure, the characteristic time of

increase in the probe temperature is 10–6

s. The curves

of cooling down of the shock-heated probe are record-

ed in the experiment. This procedure employed quali-

tatively is an adaptation of the method of rapid heating

of thin metal samples under conditions of discharge of

a high-voltage capacitor [12, 13] to the conditions of

our investigation. The general requirements apply to

the operating speed of the power switch and to the val-

ue of the time constant of the capacitor discharge cir-

cuit (less than 1 μs). The first condition is met by

using IGBT transistors with low connection energy.

The second condition required a reduction of the wire

probe resistance. The discharge currents in the probe

circuit reach 100 A. The value of attainable superheat

(Tpk – Td∞) exceeds the respective value in the proce-

dure of thermal stabilization by a factor of two or

three.

Following is the description of the experimental

scheme. A shock-heating pulse is delivered to a ther-

mally stabilized probe at a temperature T(t < tpk) =

= Tpl(t) at a preassigned instant of time t = tpk. After

the shock pulse, the low measuring current I(t > tpk) =

= Im provides for monitoring of the probe temperature

T(t, Im) which is the main quantity measured in the

experiment. Figure 3 gives the form of the current

pulse and the behavior of the probe temperature in two

substances for a preassigned value of Tpl. The initial

rate of cooling down is defined by the density of heat

flux through matter and serves as a characteristic of

heat transfer under conditions of thermal shock.

The resolution of the method increases with the

measuring current Im; however, the time of sample

Ppl, W

3.2

3.0

2.8

2.6

0.6

1.3

2.0

2.7

Δtpl, ms850

860870

880890

t(Tpl)

Tpl, K

Fig. 2. The power of heating the probe in PPG-2000 polypropylene glycol (Td∞ ≈ 550 K) as a function of time

Δtpl with the increase in temperature Tpl with a step of 2.5 K (atmospheric pressure). Solid line indicates the

boundary of spontaneous boiling . t Tpl( )

HIGH TEMPERATURE Vol. 44 No. 3 2006

466 VOLOSNIKOV, EFREMOV, SKRIPOV et al.

residence in the region of thermal instability increases

with the same value of maximal probe temperature

Tpk. The heat release from the measuring current

introduces a perturbation into the mode of cooling

down of the probe,

, (1)

where H and Q denote the probe enthalpy and the heat

flux through matter, and RT (t) is the probe resistance.

In writing Eq. (1), we ignore the thermal effect of

physicochemical transformations and end corrections.

If the probe is taken to be an ideal cylinder with a vol-

ume-uniform temperature field, the density of heat

flux q(t) through the side surface of the probe of

length l, radius r, and heat capacity per unit volume

Cp(T)ρ(T) in the process of cooling down may be

determined by the relation

. (2)

In order to correlate the effective thermal proper-

ties of shock-superheated matter with the rate of cool-

ing down of the probe, we used the model of heat

transfer between an infinite cylinder instantaneously

heated to a preassigned temperature and the ambient

medium [10]. On the assumption that the temperature

is uniform over the cylinder cross section and the ther-

mal properties of the ambient medium are constant,

the model gives the following correlation between the

rate of cooling down of the cylinder and the properties

of the medium:

where ΔT0 is the temperature difference (Tpk – T0), ΔT

is the current temperature difference, t is the time of

cooling down of the probe, α = (2πrb)2C–2

, β =

= 2πr2ρcC

–1, r is the probe radius, and C is the heat

capacity of the probe per unit length.

Figure 4 gives comparison of the experimental data

for a solidified EDT-10 epoxy-dian composition with

the results of calculation involving the substitution

into Eq. (3) of approximations for thermal properties

at T(t) > Td∞ obtained in [14]. The example demon-

strates qualitative agreement between model and

experiment.

Composition and Operation of the Experimental

Setup

The time variation of the probe temperature is

formed by computer-controlled software-hardware

0 100 800 1000 1200 1400 1600t, μs

200

400

600

800

1000

1200T, K

0.5

1.0

1.5

2.0

2.5

3.0

I, A

Tpk

1

2

2

1

Tpl

Ipl

tpk

Im

Fig. 3. Time variation of current in the probe andprobe temperatures in experiments with shockheating of the probe in (1) elastomer and (2) vit-reous polymer (the current pulse associated withthe capacitor discharge is not shown because ofthe adopted scale).

ΔH t tpk–( ) Q t( ) td

tpk

t

∫ Im2

RT t( ) td

tpk

t

∫–=

q t( ) Cp T( )ρ T( )r/2 dT/dt–( ) +=

+ Im2RT t( )/ 2πrl( )

ΔT

ΔT0

---------4

π2β---------×

3( )

=

×αty2–( )exp

y yJ0 βy( ) J1 βy( )–[ ]2 yY0 βy( ) Y1 βy( )–[ ]2+{ }----------------------------------------------------------------------------------------------------------------- y ,d

0

∞

∫

0 0.1 0.2 0.3 0.4 0.5t–tpk, ms

0.4

0.5

0.6

0.7

0.8

0.9

1.0ΔT/ΔT0

1

3

2

Fig. 4. Time variation of relative superheating ofthe probe in experiment with solidified EDT-10epoxy-dian composition (curve 1) and in calcula-

tions by relation (3) at β = 3, α = 1.5×103 (curve

2), and α = 2×103 (curve 3).

HIGH TEMPERATURE Vol. 44 No. 3 2006

AN EXPERIMENTAL INVESTIGATION OF HEAT TRANSFER 467

means (Fig. 5). The heating of the probe (Tpl – T0) is

initiated on command via control circuits CCC and

CC1. The capacitor of power supply PS1 is discharged

to the probe resistor PR by a switch S1. The PS1 pow-

er supply provides for controlled and stable voltage on

the capacitor prior to the pulse delivery. At the same

time, a heating current function generator HFG starts

operating along with a heating power regulator HPR

for maintaining the probe temperature on the attained

level T(t < tpk) = Tpl. The HFG includes a memory unit

in which an array of numerical values of the heating

function is pre-stored. The sampling of values is per-

formed at a frequency defined by a programmable

timer PT. The HFG is loaded by the results of prelim-

inary experiments with fitting of the desired heating

function [15].

The shock-heating pulse is shaped by a switch S2

with delay D using the control circuit CC2. The switch

structure provides for the transfer of energy of the

charge of capacitor of power supply PS2 to heat the

probe in 1 μs [14]. The use of high-power IGBT tran-

sistors of the IRG4PC50W type with low effective

connection energy and residual voltage of 2–2.5 V

makes it possible to transfer 50% and more of energy

directly for heating the probe. Figure 6 gives the tem-

perature Tpk as a function of capacitor voltage.

The measuring part of the experimental setup

employs high-speed operational amplifiers with cir-

cuits of input overload protection. A scaling amplifier

SA and a current meter CM provide for the shaping of

signals proportional to the voltage drop and to the

probe supply current at the input of recording circuit

RC with an error of 0.1–0.2%. The RC includes a

12-bit ADC with a conversion time of 0.8 μs. The

probe temperature is computer-calculated by the val-

ues of probe resistance at each instant of recording the

results [11]. The entire temperature measuring circuit

is calibrated against R4830/1 standard bridge resist-

ance by direct substitution of the probe resistance.

EXPERIMENTAL RESULTS

AND DISCUSSION

The experiments involved samples of vitreous pol-

ymers (such as a solidified composition on the basis of

EDT-10 oligomer and PMMA polymethyl methacr-

ylate obtained by polymerization of monomer using

CCC

Com

pute

r

D

CC1

PS1

PS2

CC2

S1

S2

PR

HPR

PS3

SA

CM

HFG

PT

RC

Fig. 5. The scheme of controlling the wire probe temperature in experiments in pulsed heating of solidifiedpolymers (see the text for explanation).

40 60 80 100 120 140 160 180U, V

400

500

600

700

800

900

1000Tpk, K

Fig. 6. The maximal temperature in the pulse as afunction of capacitor voltage.

HIGH TEMPERATURE Vol. 44 No. 3 2006

468 VOLOSNIKOV, EFREMOV, SKRIPOV et al.

benzoyl peroxide), “styrosyl” elastomers [16], and

low-molecular hydrocarbons (heptane and glycerol).

The probes were made of platinum wire 20 μm in di-

ameter. The initial probe resistance R0 = R(T0 =

= 293 K) was (3±0.25) Ohm in experiments with ther-

mal stabilization and (1±0.05) Ohm in experiments

with shock heating. Cells with ten probes implanted in

each cell were used in experiments with solidified sys-

tems. These experiments were performed at atmospher-

ic pressure, and experiments with liquids – at supercrit-

ical pressures p ~ 10 to 20 MPa. The sensitivity of the

setup was checked in experiments with heptane under

conditions of pressure increased with a step of 1 MPa.

The experiments were performed in series of sev-

eral pulses with stepped increase in the probe temper-

ature (Tpl or Tpk with preassigned values of Δtpl or Im)

from series to series. Recorded during the experiments

was the variation of temperature T(t), of power P(Δtpl)

required for maintaining the selected value of Tpl(thermal stabilization procedure), or of power

(shock heating procedure). The

upper limit of increasing the probe temperature was

provided by the values of and , the reproduc-

tion of which resulted in irreversible changes of ther-

mal resistance of the initial sample. These changes

were traced by the values of integral mean power in

the region of thermal stabilization Pav(Δtpl) or by the

values of integral mean temperature in the region of

cooling down of the probe Tav(Δtm). A sign of thermal

destruction was provided by a systematic increase in

thermal resistance from pulse to pulse (increase in

temperature Tav(Δtm) or decrease in power Pav(Δtpl) in

a series) caused by the disturbance of continuity of

contact between the probe and matter (see Fig. 7).

Figure 8 gives comparison of the estimates of temper-

ature corresponding to the beginning of thermal

destruction of EDT-10 sample in two modes of pulsed

heating.

The data on the density of heat flux from a shock-

heated probe q(t = ti) were calculated by the arrays of

values of T(ti). Here, ti = tpk + iτ, where i is the number

of measurement point after a shock-heating pulse, and

τ is the interval between measurement points. The cal-

culation was performed at iτ > 10 μs for putting aside

the region with transient phenomena in the amplifier.

The results are generalized in Fig. 9. These data were

800 1000 1200 1400t, μs

400

500

600

700

800

900

1000

T, K1100

1

10

Fig. 7. Time variation of the probe temperature inexperiments with shock heating of solidified EDT-

10 composition at Td∞ ≈ 550 K and

(series of ten pulses each were performed for eachvalue of Tpk); arrows in the top inset indicate

curves which correspond to the first and tenthpulses.

Tpk* 1000 K≈

Im2 RT Δtm t tpk–=( )

Tpl* Tpk

*

600 800 1000 T, K

0

5

10

5

0

10

–ΔPav, arbitrary units –ΔTav, K

Tpl Tpk**

Fig. 8. Estimation of the temperature of the begin-ning of thermal destruction of EDT-10 in series often pulses by the difference of integral mean val-ues of temperature in the region of cooling downof the probe Δtm = 1 ms (right-hand axis, solid

points) and of power in the region of thermal sta-bilization of the probe Δtpl = 1 ms (left-hand axis,

hollow points) between the tenth and first pulsesas functions of temperature Tpk and Tpl, respec-

tively. Dotted lines define the corridor of scatterof the data which are not associated with thermaldestruction.

HIGH TEMPERATURE Vol. 44 No. 3 2006

AN EXPERIMENTAL INVESTIGATION OF HEAT TRANSFER 469

obtained by formula (2) using the measured values of

resistance RT(t) of a probe of given size, as well as the

data on heat capacity and density of platinum from

[17]. The resultant estimates for heat flux were veri-

fied in the following experiment. The heat loss by a

shock-heated probe over some time interval after a

shock pulse t – tpk ~ 10–5

s in duration was compen-

sated by fitting the measuring current function Im(t).

The power required for maintaining constant the

probe temperature T(t – tpk) ≈ Tpk was recorded in the

experiment. Its value in a first approximation corre-

sponded to the heat flux through matter.

In accordance with relation (3), the initial rate of

cooling down of a shock-heated probe is defined by

the thermal activity of matter. The linear model of

unsteady-state heat transfer between an “instantane-

ously” thermostatted probe and the medium [7, 11]

and the data of pulsed experiment with constant super-

heat of the probe P(Δtpl) were used to compare the

effective values of thermal activity of the investigated

samples obtained in this model. The probe tempera-

ture was increased to (Δtpl = 2.5 ms). The

results of comparison are given in Fig. 10. Referenc-

ing was made by the known data for the thermal activ-

ity of heptane [18]. It was found that the thermal

activity varies from sample to sample in the same

manner as the heat flux density in experiments with

shock heating (Fig. 9). This confirms the consistency

of the results for different ways of penetration into the

region of thermal instability of matter.

CONCLUSIONS

A study is begun into a subject that is new for the

physics of rapid probe measurements, namely, the

microvolume of solidified polymer which is pulse-

heated relative to the temperature of the beginning of

thermal destruction under quasi-static conditions. An

approach is suggested to the investigation of heat

transfer in superheated polymers, including heat

transfer against the background of physicochemical

transformations in the system. Characteristic values of

density of heat flux through matter have been deter-

mined with the duration of heating the probe of 1 μs

and the depth of penetration into the region of thermal

instability of up to 500 K. The possibility of compar-

ing the effective values of thermal activity of sub-

stances in this region has been demonstrated. The sur-

vivability of polymer systems under extreme temper-

ature-and-time conditions preassigned by the suggest-

ed procedures has been assessed.

600 800 1000T(t20), K

15

20

25

5

10

q, MW/m2

400

30

35

40

45

501 2

3 4

5

Fig. 9. The heat flux density at the twentiethmicrosecond (t20) as a function of the probe tem-

perature T(t20) in experiments in shock heating of

the probe in (1) glycerol, (2) EDT-10, (3) PMMA,(4) styrosyl, (5) heptane.

Tpl Tpl*>

400 600 800 1000T, K

300

400

500

600

700

800

900

1000

1100b, W s1/2 m–2 K–1

1

2

3

4

5

Fig. 10. The effective thermal activity of solidi-fied samples (2–4) as a function of temperatureaccording to the data obtained using the methodof thermal stabilization of the probe. Designationsof the materials are the same as in Fig. 9. Arrows

indicate the values of , and dotted lines indi-

cate the values of Td∞. Curves 1 and 5 for liquids

are constructed by the data of [18].

Tpl*

HIGH TEMPERATURE Vol. 44 No. 3 2006

470 VOLOSNIKOV, EFREMOV, SKRIPOV et al.

ACKNOWLEDGMENTS

This study was supported by the program of the

Presidium of the Russian Academy of Sciences on

Thermophysics and Mechanics of Intense Energy

Impacts, by the Russian Foundation for Basic

Research, and by the Foundation for Promotion of

Domestic Science.

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