9.8 THE “CARNOT CYCLE” FOR ELASTOMERS

In elementary thermodynamics, the Carnot cycle illustrates the production ofuseful work by a gas in a heat engine. This section outlines the correspondingthermodynamic concepts for an elastomer and illustrates a demonstrationexperiment.

The conservation of energy for a system may be written

(9.41)

where the internal energy, U, is equated to as many variables as exist in thesystem. For an ideal gas (Section 9.6), P-V-T variables are selected. The cor-responding variables for an ideal elastomer are s-L-T [see equation (9.34)].Since Poisson’s ratio is nearly 0.5 for elastomers, the volume is substantiallyconstant on elongation.

By carrying a gas, elastomer, or any material through the appropriate closedloop with a high- and low-temperature portion, they may be made to performwork proportional to the area enclosed by the loop.A system undergoing sucha cycle is called a heat engine.

9.8.1 The Carnot Cycle for a Gas

In the Carnot heat engine, a gas is subjected to two isothermal steps, whichalternate with two adiabatic steps, all of which are reversible (see Figure 9.12)(48). Briefly, the gas undergoes a reversible adiabatic compression from state1 to state 2. The temperature is increased from T1 to T2. During this step thesurroundings do work |w 12| on the gas. The absolute signs are used becauseconventions require that the signs on some of the algebraic quantities hereinbe negative.

Next the gas undergoes a reversible isothermal expansion from state 2 tostate 3. While expanding, the gas does work |w 23| on the surroundings whileabsorbing heat |q2|. Then there follows a reversible adiabatic expansion of thegas from state 3 to state 4, the temperature dropping from T2 to T1. Duringthis step, the gas does work |w 34| on the surroundings.

Last, there is an isothermal compression of the gas from state 4 to state 1at T1. Work |w41| is performed on the gas, and heat |q1| flows from the gas tothe surroundings.

9.8.2 The Carnot Cycle for an Elastomer

For an elastomer, the rubber goes through a series of stress–length steps, twoadiabatically and two isothermally, as in the Carnot cycle (see Figure 9.13)(51). Beginning at length L1 and temperature T I, a stress, s, is applied stretch-ing the elastomer adiabatically to L2. The elastomer heats up to T II. The quan-tity s is related to the length by the nonlinear equation

dU V dp T dS dL= + + +s . . .

450 CROSS-LINKED POLYMERS AND RUBBER ELASTICITY

(9.42)

[see equation (9.34)]. In this step work is done on the elastomer.At T II, the elastomer is allowed to contract isothermally to L3. It absorbs

heat from its surroundings in this step and does work. As the length decreases,its entropy increases by DS (see Figure 9.13c). The elastomer then is allowedto contract adiabatically to L4, doing work, and its temperature falls to T I

again. The length of the sample is then increased isothermally from L4 to L1,work being done on the sample, and heat is given off to its surroundings. Thisstep completes the cycle.

An increase in the volume of the gas, however, corresponds to a decreasein the length of a stretched elastomer. It is important to note that at no timedoes the elastomer come to its rest length, L0. Interestingly the corresponding“rest volume” of a gas is infinitely large.

9.8.3 Work and Efficiency

The equations governing the work done during the two cycles may also becompared. For a gas,

(9.43)w g P dV= -Ú

s = - Êˈ¯

È

ÎÍ

˘

˚˙nRT

L

L

L

L0

0

2

9.8 THE “CARNOT CYCLE” FOR ELASTOMERS 451

Figure 9.12 Carnot cycle for a gas (48).

For an elastomer,

(9.44)

In both cases the cyclic integral measures the area enclosed by the foursteps in Figures 9.12 and 9.13.

The efficiencies, h̄̄ , of the two systems may also be compared. For a gas,

(9.45)

where q1 and q2 are the heat absorbed and released (opposite signs), as above.For the elastomer,

(9.46)

or in a different form,

(9.47)he

T T

T=

-II I

II

hs

e

dL

Q

T T S

Q

Q Q

Q= =

-( )=

+ÚII

II I

II

I II

II

D

hg

q q

q=

+1 2

2

w se dL= -Ú

452 CROSS-LINKED POLYMERS AND RUBBER ELASTICITY

Figure 9.13 Thermal cycle for an elastomer (51).

where QI and QII are the amounts of heat released to the low-temperaturereservoir (T I) and absorbed from the high-temperature reservoir (T II),respectively.

While the entropy change is zero for either system during the reversibleadiabatic steps (see Figures 9.12c and 9.13c), it must be emphasized that theentropy change is greater than zero for an irreversible adiabatic process. Anexample for an elastomer is “letting go” of a stretched rubber band.

9.8.4 An Elastomer Thermal Cycle Demonstration

The elastomer thermal cycle is demonstrated in Figure 9.14 (51). A bicyclewheel is mounted on a stand, with a source of heat on one side only. Stretchedrubber bands replace the spokes. On heating, the stress that the stretchedrubber bands exert is increased so that the center of gravity of the wheel isdisplaced toward 9 o’clock in the drawing. The wheel then rotates counter-clockwise (52).

Each of the steps in Figure 9.13 may be traced in Figure 9.14, although noneof the steps in Figure 9.13 are purely isothermal or adiabatic, and then ofcourse they are not strictly reversible. Steps 1 to 2 in Figure 9.13 occur at 6o’clock in Figure 9.14, where there is a (near) adiabatic length increase due togravity. At 3 o’clock, at T II, heat is absorbed (nearly) isothermally, and thelength decreases, doing work.At 12 o’clock, corresponding to steps 3 to 4, thereis an adiabatic length decrease due to gravity. Last, at 9 o’clock, steps 4 to 1,there is a (nearly) isothermal length increase, and heat is given off to the sur-roundings at T I, and work is done on the elastomer.

9.9 CONTINUUM THEORIES OF RUBBER ELASTICITY

9.9.1 The Mooney–Rivlin Equation

The statistical theory of rubber elasticity is based on the concepts of randomchain motion and the restraining power of cross-links; that is, it is a molecular

9.9 CONTINUUM THEORIES OF RUBBER ELASTICITY 453

Figure 9.14 A thermally rotated wheel, employing an elastomer as the working substance (51).