Modeling of thermoelectric heat transfer in shape memory alloy actuators: Transient and multiple cycle solutions

Pergamon

Int. J. Engng Sci. Vol. 33, No. 15, pp. 2345-2364, 1995

0020-7225(95)00074--7

Copyright ~) 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225/95 $9.50+ 0.00

MODELING OF THERMOELECTRIC HEAT TRANSFER IN S H A P E M E M O R Y A L L O Y A C T U A T O R S : T R A N S I E N T AND MULTIPLE CYCLE SOLUTIONS

DIMITRIS C. LAGOUDAS and ZHONGHAI DING Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, U.S.A.

Abstract--In this paper we study the transient and multiple cycle solutions of one-dimensional symmetric thermoelectric shape memory alloy (SMA) actuators. From the model proposed by Bhattacharyya et al. [1], an approximately equivalent simpler model governed by an integrodifferential ecluation is derived first, to describe the temperature distribution in SMA. Then a numerical algorithm is proposed to solve the integro-differential equation. It is found that for the transient cool!Lng problem with constant electric current density of magnitude ~/I there is a critical value Jo of k/I .,;uch that when IJI >J0, the temperature in SMA may not be always decreasing. The heat transfer problem and the resulting phase transition of SMA, induced by a piecewise constant electric current source, are also studied. It is found that there exist restrictions on the maximum and minimum values of the current density J for a repeated complete phase transition to take place in SMA. Explicit expressions for the critical values of J are derived and their physical implications are discussed.

1. I N T R O D U C T I O N

Shape memory alloys (SMA) [2] undergo an austenitic to martensitic phase transformation upon cooling, whiLle with some hysteresis they fully recover their parent phase when heated above the austenitic finish temperature. A thermodynamic formulation of the phase transformation in SMA ihas been given by Tanaka and Nagaki [3] based on the thermodynamic approach to dissipation proposed by Edelen [4]. Many other researchers have also contributed to the thermomechanical constitutive description of SMA, some of the key references can be found in Boyd and Lagoudas [5, 6]. Shape memory alloys (SMA) have been proposed as actuators for shape and vibration control of structures, especially the Ni-Ti (Nitinol) system. The capability of producing large actuation forces during the martensitic to austenitic phase transition is the major advantage of SMA. The major disadvantage in utilizing SMA actuators is the low rate of cooling, since the time constant for heat transfer is usually large compared to the frequencies required for many engineering applications. Recently a new SMA thermoelectrically cooled or heated actuator has been proposed by Bhattacharyya et al. [1] that utilizes the thermoelectric effect for heat transfer from and to the SMA, by making the SMA actuator the cold or hot junction of a thermoelectric couple respectively. In this paper, we study the transient solutions of one-dimensional symmetric thermoelectric SMA actuators. F'rom the heat transfer model proposed by Bhattaeharyya et al. [1] which is described in Section 2, an approximately equivalent simpler model governed by an integrodifferential equation is derived in Section 3. A numerical algorithm is proposed to solve the resulting integro-differential equation for the SMA temperature in Section 4, where results are presented for both transient and multiple cycle solutions induced by a piecewise constant electric current source. 2345

2346

D.C. LAGOUDAS and Z. DING 2. PHYSICAL MODEL OF T H E R M O E L E C T R I C HEAT T R A N S F E R IN SMA A C T U A T O R S

The local law of heat conduction in a material point with position vector x at time t is given by - V " (~(x, t) + r(x, t) = Co ~O-T x( t , )

(2.1)

where Co is the heat capacity of the material and all possible heat sources/sinks per unit volume are collectively represented as r(x, t). In this paper, boldface lower case letters, or upper case letters with a vector symbol denote vectors, while boldface upper case letters denote second order tensors. The energy flux vector, (~, for material with thermoelectric properties is given by [7] Q = - K V T + ( o t T + ~/21)J

(2.2)

where K is the second order tensor of thermal conductivity, ot is the second order tensor of Seebeck coefficients, q is the particle charge, /~ is the electrochemical potential, and I is the second order identity tensor. In the absence of an electric current J, equation (2.2) reduces to the linear Fickian diffusion constitute equation. The gradient of the electrochemical potential is given by [7]

V fi, = - q ( o t r V T + Ixl)

(2.3)

where p is the second order tensor of electrical resistivity. Assuming the material is isotropic, we have K = KI,

a = al,

p = pl

(2.4)

by which equation (2.2) reduces to

O. = - K V T + ( a T + I ~ ) . L q

(2.5)

If the material is, in addition, spatially homogeneous, then VK = Va = 0, which also implies temperature independent properties for nonuniform temperature fields. Combining equations (2.3) with (2.5) and utilizing the steady state conservation of charge equation, i.e.V. J = 0, we have

v . 0 = - g r E w - p IJI2

(2.6)

where the second term on the right of equation (2.6) represents the Joule heat. Note that the assumption of steady state for the conservation of charge equations is not valid whenever the imposed electric current J is a function of time. Due to the large velocity of the electric charge carriers, however, the deviation from steady state lasts an infinitesimal time interval compared with the time period of the applied electric current, and it may be neglected. Finally, from equations (2.1) and (2.6), the heat equation for a spatially homogeneous, isotropic material with K and a independent of T, reduces to

KVET + p lJ[ 2 + r = C oOT --. Ot

(2.7)

In the above equation p and Co may still be functions of T. As seen from the above equation for a homogeneous material with temperature independent properties, the thermoelectric effect does not enter into the field equation, but it becomes

Thermoelectricheat transfer in SMAs N-semiconductor

,/! I

~ y SMA

P-semiconductor

SI /4, "~ .....

2347

A[

t---.~~-------~--~

I

_

I, L - I-~----d ~ 1 . L .I Fig. 1. One-dimensionalthree phase N-SMA-P thermoelectricsystem. important when interface conditions are evaluated, across interfaces separating dissimilar materials. Assuming a coherent interface without any distributed sources, the interface flux condition is obtained from (2.5), by requiring that the energy flux Q . N be continuous across the interface with normal vector N'. In addition, continuity of temperature across the interface is assumed. For the 1-D heat conduction problem in which a thin SMA material is confined between two semiconductors, an N-type and a P-type, forming the junction of a thermoelectric element, as shown in Fig. 1, equation (2.7) reduces to [1] "

OETN"

J~,N-~x2 ~X, t) + pNj2(t) --

HP C NdTN x A (TN(x, t) -- To) = v --~-( , t),

d d 7
t>0,

(2.8)

02T~ P C s dT~ (x, t), Ks ~ (x, t) + pj2(t) - H-~ (T~(x, t) - To) = v dt

d d -7
t>0,

(2.9)

Kp 02TP (x, t) + peJZ(t) - n P e dTp -Ox -T A (Tp(x, t) - To) = Cv - ~ ( x , t), d -L-~
d < --~,

t>0.

(2.10)

Note that the electric current density vector is given in 1-D case by J(t) = J(t)ix where ix is the unit vector in the x-direction. In deriving the above equations, it has been assumed that I H ~ (TN- T0), -HA(T~-- - To) and - H P ( T e - To) in N-element, SMA and P-element r= A A respectively, where H is the heat convection coefficient, P is the perimeter and A is the area of the cross section, which are assumed to be independent of x. Convective heat transfer does occur around the sides of the respective phases, and it has been included approximately as a source term in the heat conduction equation for the constituent phases. The interface conditions for the 1-D case are

T~(~, t ) = TN(~, t),

T~( - d , t ) = Te( - d , t),

(2.11)

[-K~OT~( d't = --KN -~xOTN[dt7't ) + a N T N ( d ' t ) J(t)'

t

0x

:

7'

where the interface conditions (2.12) follow from the continuity of the energy flux Q • N with

2348

D. C. L A G O U D A S and Z. D I N G Table 1. Material parameters [1] Ni-Ti SMA Thermal conductivity K Heat capacity Co Electrical resistivity p Seebeck coefficient a

P-Element

2.2 x 10 -2 W / ( m m K) 2.12 × 10 -3 J / ( m m 3 K) 6.32 x 10 -4 f~ mm 1.2 x 10 -5 V / K

1.63 4.35 1.15 2.15

× × x x

10 -3 W / ( m m K) 10 -3 J / ( m m 3 K) 10 -2 f2 mm 10 -4 V / K

/V = ix and the assumption that /2s =/~i and as = 0 (the latter condition is justified as the Seebeck coefficient of an SMA is very low compared to that of the N or P semiconductor as shown in Table 1). Moreover, the end boundary conditions are

TN(L +d, t) : To, Te(-L -d

2'

t) = To,

(2.13)

while the initial conditions are stated as

d d -2< - x < - L + ~ ,

TN(x, O) = T°(x), T~(x, O) = T°(x),

d d - ~<-x <-~,

Te(x,O) = T°(x),

-L-~<-x

d

d

- -~, <

(2.14)

where T°(x), T°(x) and T°(x) are the initial temperature distributions in N-element, SMA and P-element respectively. The heat capacities, C~ and CoP, of N- and P-type semiconductors are taken to be independent of temperature, while the heat capacity C[ of the SMA material varies significantly with T due to the temperature induced phase transformation. During the forward transformation (austenite to martesite), the heat capacity of a typical Ni-Ti SMA has been experimentally determined [8] and may be represented by the function [1]

C~= C ° + [Ln

_

In 100

C°(M, - My)]. IMs - M:I

e-(2

In

,O0/,M~ M:),r (M,+M:a),Mf<_r <--Ms, -

--

(2.15)

and the reverse transformation (martesite to austenite) is represented by the function CSo = C ° + [LH --

C°(A/-As)] " -

In 100 -

IA, -,4:1

e-(2In1OO/IA"-A:I) IT-(A"+A:/2)IAf <-- T<-As,

(2.16)

with Co = C O at all other temperatures. Equations (2.15) and (2.16) are plotted in Fig. 6 for hI/=278K, M s = 2 9 6 K , A s = 3 0 2 K and A/=324K, L H = 0 . 1 J / m m 3 and Cv= o 2.12 x 10-3j/(mm-K), which are typical values for Nitinol SMA. The parameter, Ln in equations (2.15) and (2.16) is the latent heat of transformation and is equivalent to the area under the curve in Fig. 6 above the horizontal line C[ = C~ between Ms and M/or As and ,4. Even though the electrical resistivity p changes with temperature during the phase transformation of SMA, lack of precise experimental data limits us in assuming that p is constant. Dependence of p on temperature with possible hysteresis can be accounted for mathematically by varying the magnitude [J[ of current density vector J.

Thermoelectric heat transfer in SMAs

2349

0.020

0.018 0.016 0.014 0.012 0.010 L" 0.008 0.006 U 0.004 0.002 0.000

I I I I

.

•

260

.

.

.

i

.

270

.

.

.

J

. . . .

280

i

.

290

.

.

.

,

.

300

.

.

.

°

t I t t

.

3 I0

.

.

.

i

.

.

.

.

320

i

.

.

.

.

330

340

T [°K] Fig. 2. The phase transformation of Ni-Ti SMA.

3. A N A L Y T I C A L M O D E L L I N G

3.1 Two-phase model To be able to derive a closed-form solution for the 1-D three-phase model described in the previous section, this model needs to be further modified. Due to the relatively large heat conduction coefficient of the SMA compared with the heat conduction coefficient of the P and N semiconductors,, the temperature distribution in the SMA material is expected to be almost constant for small ratios d/L. It is, therefore, possible to simplify the three phase model into an approximate two phase model, by incorporating the SMA into the P-N interface. Assuming that T~(x, t) is con:stant along the x-axis, the interface condition (2.11) becomes

The flux interface condition (2.12) can be combined with the field equation (2.9) by first d d integrating equation (2.9) from x = - 2 to x = ~ , which results in the following equation

K OT$(~,t)-K ax $ Ox

+ dpj2(/) -

HP fan j_aa(T(x,t) - To)dx fa/2 dT~ (x, t) dx,

= C~ J-ale dt

(3.2)

and then substituting (2.12) in the above equation to obtain

KN.~x ~,OTN/t)d -Ke--~x ~- d t)-ctNTN( d, t)J(t)+ ctt,Tt,(-d2, t)J(t)+ dpj2(t) HP f~,cz r a/z dZ A e,2(T~(x't)-T°)dx=Cv~-a/2 dt (x,t)dx. $

[$ ~I°I$.P

w

.

$

(3.3)

2350

D . C . L A G O U D A S and Z. DING

By utilizing the assumption that T~(x, t) = Te - d, t = TN ~, t for x e

2'

, the inter-

face condition given by equation (3.3) is reduced to the following form, which contains only the temperature at N or P elements:

OTN/d t) - Kp OTP d t) + p~dj2(t) HdP KN--~X ~2' --~X (--2" - ' - - A - ( T N ( d ' t ) -T°) =aNTN(d,t)J(t)_ a pT[ s dTN (~, d t). e ~ - ~d, t)J(t) + C~d--~-

(3.4)

Since the heat conduction equation (2.9) for SMA can be replaced by the interface equation (3.4), for simplicity of notation, we shift the domains of the remaining field equations (2.8) and

(2.10) forN-andP-typesemiconductors,[-L

d2 '

d] and [d , L + ~d] , to [ - L , 0] and [0, L]

respectively. Thus, the two phase model which is a simplification of the three phase model defined by equations (2.8)-(2.14) is described by the following equations for N- and P-type semiconductors:

C~

(x, t) = KN ~x 2 (x, t) + pNfl(t) --

[ ~ C~

(TN(X,t) -- To), 0 < x < L, t > 0

.02Tp, He (x,t)=t~e-~x2(X,t)+pej2(t)---~-(Tp(x,t)-To),

-L
t>0.

(3.5)

The boundary conditions are given by

TN(L, t) = To, Tp(-L, t) = To,

t >-O, t >-O,

(3.6)

and the initial conditions are

TN(x,O)= To, Te(x,O) To,

O<-x<-L, - L <-x<-O,

(3.7)

while the interface conditions (3.1) and (3.4) become

TN(O, t) = Tp(O, t),

t >IO,

(3.8)

K N-~X OTN(0 ' t) -- Kp ~-~ (0, t) + psdJ2(t) ---H--~-(TN(O, t) - To) =TN(O,t)(otN-aP)J(t)+C~ud~(O,t),

t>0.

(3.9)

3.2 Symmetric two-phase model If C~ - Cv - Cv, K N = Kp = K, Ol N = - - O r P and ON = O P = P, then the temperature distributions TN(X, t), and Tp(x, t) of (3.5)-(3.8) are symmetric, namely, __

P__

TN(X,t)= Tt,(-x,t),

O<-x<-L,

t>_O.

Thermoelectricheat transferin SMAs

2351

The boundary value problem (3.5)-(3.9) is then reduced to

C o °r(x, 02T (x, t) + N 2 ( t ) - ~ - ( r ( x , t ) o-T t ) = K Tx T(x, 0) =-- TO,

O<-x<-L,

T(L, t) = To,

t -> 0,

-

To),

O
t>0

2K-~xT(0, t) + p, dj2(t) --H--~ (T(O, t) - To)

= r(o t)(t~N - t~e)J(t) + C~od~-~(O, t),

t > 0.

(3.10)

Since the interface boundary condition in equation (3.10) is complicated, it is hard to derive theoretically and even numerically the temperature distribution by classical methods such as separation of va:riables, the integral transform method, or the finite element method. Furthermore, because of the temperature dependence of the heat capacity of SMA, the boundary value problem (3.10) becomes extremely challenging to study theoretically and numerically. In the following, the boundary value problem (3.10) is transformed into an integro-differential equation that describes the temperature at the interface x = 0. Then by analysing and solving this integro-differential equation, the temperature distribution on the SMA is evaluated. Let us first look: at the solution to the homogeneous boundary value problem derived from (3.10), i.e.

HP "Co~ot (x, t) = K ~22 (x, t) + pj2(t) - ~ (tp(x, t) - To), q~(x, O) = O,

O <-x <- L,

~p(0, t) = 0,

t -->0,

~(L, t) =- 0,

t->0,

O
t>O,

(3.11)

where J(t) is a bounded function. It can be shown that the solution of (3.11) can be expressed by the following convergent Fourier series

~o(x, t) = ~ a,(t)sin((2n -1) L ) , n=l

where

an(t) =

'

K

A.CoL

and

(2n - 1)7r L

2352

D.C. LAGOUDAS and Z. DING

Let T(x, t ) = T(x, t ) - ~o(x, t), it follows from (3.10) and (3.11) that T(x, t) satisfies t h e following equation and boundary and initial data: 0 2 f~ HP ~ - [ ( ,t)=K-~x2(X,t)-----~- (x,t),

C 07" X

T(x, 0) = To,

O<-x<-L,

T(L, t) = To,

t -> 0,

O
2K 0:--T(0'oxt) + pflJZ(t) - - - ~ (T(O, t) - To) + ~ L

= T(O, t)(a N - aP)J(t) + c~dd-:-T(o, t), Ot

t>0

F(t - v ) ~

+ p IJ(v)l 2 dT

t>0,

(3.12)

where

F(t) =

exp -

h, +

t .

(3.13)

n=l

If the interface temperature is known, then to solve the above problem reduces to solving the problem given below:

Co~tt (x, t)=Ka2---~-~(X,ax 2 t) -----:-~(x,t), HP A

~O(x, O) -- To,

O<-x<-L,

,/,(L, t) = To,

t -> O,

t~(O, t) = g(t),

-

-

0
t>0

t >--O,

(3.14)

where g(t) is a bounded function with bounded derivative g'(t) and g(0) = To. Let ~s(x, t) = qJ(x, t) -

[g(t) + -~x (To - g(t)) ] , then ~(x, t) satisfies

.Co~t (x,t)=KO2qJ Oxa (x,t) - ~HP ~J(x, t) _(Cog,(t)+Tg(t)_HPTol L-x A ] L d~(x, 0) = 0,

O<-x<-L,

~ ( L , t) = 0,

t -> 0,

2(0, t) = 0,

t -->0.

HPTo A '

O
t>0

(3.15)

The solution of (3.15) is given by t~(x, t ) =

b,(t)sin T

'

n=l

where '

K

C--'~ g(r)

CoA / n'-~+~oA ( 1 - ( - 1 ) " ) - ~ z ] dr'

(3.16)

Thermoelectric heat transfer in SMAs n~

and

ix,=--. L

By

the

assumption

on

g(t),

it

is

easy to

2353

check

that

~(x, t) =

n~

~n=l

b.(t)sin--L-- is uniformly convergent on [0, L]. Hence d/(x, t) = ~ b , ( t ) s i n ~- ~ n=l

[g(t) + x (To- g(t)) ]. L

(3.17)

If T(t) denotes the temperature at the SMA interface x = 0, then

T(t) = T(O, t) = T(O, t). Let g(t) = T(t) in (3.14), consequently in (3.16) and (3.17), we obtain

T(x, t) = ~ b.(t)sin---~-nmc [ T(t) + Zx ( T o - T(t)) ] .

(3.18)

n=l

From equation (3.18), we can evaluate the following derivative, __

01"(0, t)= ~~ b.(t)tz. -t 0X

To-

T(t)

L

n=l

2 ~ G(t _ r)(d_7_T(z) + HP

: - -L

\ clt

_ HPTo]

~ A T (z )

CvA /

dr

4

- -L

~"F ( t - z ) HPTo + To- T(t) ~ . A dr L

where =

exp - ~ / x 2 + n = 1

t .

(3.19)

C v

Substituting the expression of 07"/Ox(O, t) into the interface equation in (3.12), we finally obtain the following SMA interface temperature integro-differential equation G(t - ~)

(~) + v,

r(~) d~ + M r ) d_:__r(t) + vz(t)r(t) = ~(t) Ol

[ T ( 0 ) = To,

(3.20)

where

-C~dL -.

~(T):= 4K '

vl

=

HP • CvA'

v2(/)-

L ( a N - ote)J(t) + _1+ _PHLd _. 4K 2 4KA'

and

o~(t)=

fo

psdL =2-HPTo 2 f t F(t - z)I/(z)l 2 dz + To + PHLdTo _ G ( t - z ) ~ , A dZ + -~t~.lo -2 _ 4KA + 4K 3 (t),

with G(t) given by (3.19) and F(t) given by (3.13).

4. NUMERICAL SOLUTIONS A numerical algorithm will be developed in this section to solve equation (3.20) by using the finite difference scheme. This is necessary due to the complexity of the integro-differential equation, especially the nonlinearity of the function/x(T) and the presence of the derivative dT/dt in the integral. Let h be the size of the time step. The time derivative is approximated by

dT T(t) - T(t - h) -~(t) ~h

2354

D.C. LAGOUDAS and 7_.. DING

Based on the above approximation of the derivative, equation (3.20) is discretized for n = 1, 2, 3 . . . . . as follows:

~. G. ,., [7", hTk , + v, Tk ] + tt(T, i) T " - T~-l + v2(nh)T, = F~, h

(4.1)

k=l

where nh

7", = T(nh);

G, =

f{

G(t) dt;

F,, = F(nh).

(4.2)

rl - | ) h

Simplifying equation (4.1), we obtain the following formula, n

I

(1 + vth)GiT, + ~, ((1 + v,h)G,-k+l - G, k)Tk - G, To k-1

+ (/.t(T,_,)+ v2(nh)h)T . -I.tT,-1 =F,.

(4.3)

From equation (4.3), the following explicit formula for the SMA temperature at the interface is finally derived: T, =

(I + v,h)G, + l~(To) + v2(h)h

[ (To)To +

F, + G, Tol,

(4.4)

1

(1 + v,h)G, + #(T,_,) + v2(nh)h

x ~(L-,)L-,+~((L

,-(1.v,h)G._,.,)r~+F.+G.ro,

n=2.3,4

.....

(4.5)

k-!

To solve the symmetric two phase model (3.10), the finite difference scheme will be used, by taking advantage of the explicit form of the interface temperature (4.4) and (4.5). Since the temperature in SMA interface can be evaluated by using (4.4) and (4.5), the original problem is reduced to solving a heat equation with Dirichlet boundary conditions, for which the finite difference scheme is straightforward. The above algorithm has been tested with data taken from Table 1, and it is found to be efficient and insensitive to the time stepsize, as Table 2 shows. The material parameters of the Ni-Ti SMA and the N-semiconductor element at room temperature To = 300K shown in Table 1 are adopted from [1]. Since the SMA Seebeck coefficient is substantially smaller compared to that of the semiconductor element, we have assumed a, =0. In the following we assume that H = 2.5 x 10-~W/mm2K, L = 4 . 0 r a m , d=2.0mm. P=4smm and A = s Z m m 2 where s is the side length of the tetragonai cross-section. Here we set s = 1.0 ram. Table 2 displays the comparison between temperatures in SMA at time (in seconds) t = 5, 10, 15, 20, 25, 30 corresponding to different time stepsizes for constant current density J = - 2 . 0 A / m m 2. A stepsize of time h = 0.1 will be used for all subsequent calculations. Note also that for this example C'o and thus ~ has been assumed constant for simplicity. The variable C'o case will be treated in detail later.

Table 2. Comparison of the temperature corresponding to different $tel~iZe of time Stepstzc

T(5)

T(10)

T(15)

T(20)

T(25)

T(30)

h = 0.I

260.37

261.86

264.20

265.65

266.49

266.97

h = 0.2 h - 0.5 h = 1.0

260.78 261.98 263.91

26Z05 262.61 263.55

264.30 26,1.59 265.08

265.71 265.86 266.13

266.52 266.61 266.75

266.98 267.03 267.11

T h e r m o e l e c t r i c h e a t t r a n s f e r in S M A s

2355

310

J ---3.1 [A/mm~] 300 290

t

280

J =-2.4

270

J =-0.6 260

J =-L8 250

J ---1A

240

.... 0

' .... 5

' .... 10

~ .... 15

J .... 20

t .... 25

30

t [second] Fig. 3. T e m p e r a t u r e

in S M A c o r r e s p o n d i n g t o d i f f e r e n t c u r r e n t d e n s i t i e s f o r c o n s t a n t C~.

4.1 Transient solutions Based on the numerical algorithm described above, the temperature in the SMA interface as a function of time is shown in Fig. 3 for different constant current densities, corresponding to cooling. Notice that only equations (4.4) and (4.5) need to be used for the evaluation of SMA temperature. For the constant C~ case, we find from Fig. 3 that the temperature profiles in SMA are quite different corresponding to different current densities. For J = - 0 . 6 and -1.4 A/mm 2, the corresponding temperature in SMA always decreases asymptotically to the steady state temperature. For J = - 2 . 4 and -3.1 A/mm 2, the corresponding temperature in SMA decreases first, it reaches its lowest value at some finite time and then asymptotically increases to the steady state temperature. The material constants used to obtain the results of Fig. 3 are taken from Table 1. The interesting behavior shown in Fig. 3 can be letter understood by the following theorem. THEOREM 4.1 Let the current density J(t) be a constant function and the heat capacity of SMA, C~, be a constant in equation (3.20). Let T(t) be the solution of the integro-differential equation (3.20) with J(t) = J <-O. Then there exists a critical value Jo for ~ll, the magnitude of electric current density, such that when 0 <- IJI <- Jo, T(t) is always decreasing to its stable temperature T~.

T= and Jo are given by the following formulas: 1

Cv

p,

2pr/2~j 2

T.=

(Vl ~h

1 Cvvll~) L(t~N- ae) J +2 + - ~ + 4K

where 1"ll =

2

C~,\L: and J0 =

- b + X/'~ + 4ac 2a

2356

D . C . LAGOUDAS and Z. DING

where a-

V3L2pl a N

- ael

2KTr2

V'3pL+ V'3pL 4 b=Psd+ C--

la ~

-

2~t

~

[

1 + PHLd Cs 2KA ( 1 ___e Cv)] ,

a~'l To 4

The proof of this theorem is possible by using the Laplace transform and theory of complex variables. A detailed proof is given in a forthcoming paper. REMARK 4.1 An essential assumption for the validity of the above theorem is that C~ is constant. The results in Theorem 4.1 cease to hold whenever C~ is a function of temperature, which is the case during the phase transformation of SMA. Even though we have not proved such a theorem yet for the variable C~, case, a similar theorem should be valid for the variable C~ case. In fact, this has been numerically verified as shown in Fig. 4 for C~(T) given by (2.15) and (2.16). Corresponding to the data given in Table 1 and constant C~, we obtain that Jo ~ 1.8415 A/mm 2, 0.77617To + 26.875J 2 T(oo) - 0.77617 - 0.2638J K.

4.2 Multiple cycle solutions When the applied current J(t) is a function of time, it is possible to generate multiple cycle solutions for temperature which correspond to repeated forward and reverse phase transformation in SMA. The requirement for alternating phase transformation poses an interesting control problem, in which the target maximum and minimum temperatures are given and the optimum current J(t), corresponding to minimum time, needs to be evaluated. 310 J =-3.1 [A/ram~] 300 29O 280 J =-2.4

270 J ---0.6

260

J =-1.8 250

J =-IA

240

''

0

5

10

15 t

20

25

30

[second]

Fig. 4. Temperature in SMA corresponding to different current densities for variable C~.

Thermoelectric heat transfer in S M A s 330

1.5,

320

1.0 ,.7-~

310

0.S

300

0.0

29O

-0.5

280

-1.0

270

5

IO

(a)

15

20

25

30

35

2357

-1.5

40

o

5

10

(b)

t [second]

15

20

25

30

35

40

t [second]

Fig. 5. (a) S M A temperature for constant C~; (b) current density applied (J = J÷).

A representative example is shown in Figs 5, 6 for the constant C~ case and Figs 7, 8 for the variable C~ case, where a current density J(t) of constant magnitude [J(t)l = 1 A/mm 2 is applied and its direction is alternating (square pulse) based on the requirement that the forward and reverse phase transformation in SMA are completed. Figure 5 shows the temperature of the SMA and the cun'ent density applied while Fig. 6 shows the temperature distribution along the semiconductor for the constant C~ case. Figure 7 shows the temperature of the SMA and the current density applied while Fig. 8 shows the temperature distribution along the semiconductor for the variable C~ case. As seen from Fig. 7 for the variable C~ case, the temperature of SMA changes substantially differently from the constant C~ case shown in Fig. 5 during the phase transformation.

320 ~

310"

o

300"

290"

280"

30"

2( +

% 400 300 -

200

t [second x 10l ] Fig. 6. T e m p e r a t u r e distribution in the semiconductor.

2358

D.C.

LAGOUDAS

and Z. DING

330

1.5

320

1.0

310

0.5

o

300

.~

0.0

290

•~.

-0.5

2110

-I.0

27O

$

(a)

I0

15

20

25

30

35

-I.5

40

t [second]

)

,,,,i,,,.. 5

10

......... 15

, .................. 20 25 ~10

L 35

40

t [second]

(b)

Fig. 7. ( a ) S M A t e m p e r a t u r e for variable C~; ( b ) current d e n s i t y applied ( J _ = J + ) .

Another representative example is shown in Figs 9, 10 for the constant C~ case and Figs 11, 12 for the variable C~ case, where a piecewise constant current density J(t) is applied and its direction is alternating (square pulse) based on the requirement that the forward and reverse phase transformation in SMA are completed. In this case, the maximum value J÷ and minimum value - J _ of J(t) are not of the same magnitude, i.e. J÷ ~ J_. Figure 9 shows the temperature of the SMA and the current density applied while Fig. 10 shows the temperature distribution along the semiconductor for the constant C~ case. Figure 11 shows the temperature of the SMA and the current density applied while Fig. 12 shows the temperature distribution along the semiconductor for the variable C~ case. By comparing the temperature of SMA Fig. 5 and Fig. 9 for the constant C~, case with the temperature of SMA in Fig. 7 and Fig. 11, we found that the time intervals of Figs 9 and 11 are significantly greater than that of Figs 5 and 7, respectively. Also, since during heating, both the Joule heating effect and thermoelectric effect are both contributing to the increase in temperature, while during cooling, the Joule effect opposes

310 300 2902RN4~

F 400 300 2OO

100

t[secon6 ~<~0"~]

0 Fig. 8. T e m p e r a t u r e distribution in the s e m i c o n d u c t o r .

Thermoelectric heat transfer in SMAs

2359

1.5

330

1.0

O

310

~

o.5

30O

,~

0.0

29O

-0.5

28O

-1.0

!

2"/0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

o

(a)

5

~o

~s

2o

2s 3o

3s

4o

4s

-1.5

,,,,ll,,i

so

ss +o

t [second]

i

...i...

0

5

i0

15

2O

(b)

25

30

35

t [second]

Fig. 9. (a) SMA temperature for constant C~.; (b) current density applied (J_ # J ÷ ) .

opposes the thermoelectric, unequal J÷ and J_ can result in equal time intervals (compare Figs 5 and 7 with 9 and 11, respectively). Because of the existence of the heat convection and Joule terms in equation (3.10), there is a range of applied current densities, J(t), beyond which a repeatable complete phase transformation is not possible. For example, for a very small J÷ and J_, the SMA might not be able to cool down or heat up enough for a complete forward or reverse transformation respectively. Thus, the direction of or(t) may never be switched, since the target temperature that triggers the change in the cur~ent direction, may never be achieved. The same is also true for a very large current J_, where the Joule effect overcomes the thermoelectric effect and the necessary cooling may not be realized. This interesting phenomenon is shown in Figs 13 and 14 for the

320 i

310 o

300 290"

280" 4(

300 250 150 I00 50

t -second ~ 2 ~

xO'~'~

Fig. 10. Temperature distribution in the semiconductor.

2360

.o.

D . C . L A G O U D A S and Z. DING

3311

1.5

32O

1.0

310

0.5

30O

0.0

29O

%

28O

270

.0.5

-i.0 0

5

to

(a)

,5

2o

25

3o

35

40

4s 50 55 ~o

-1.5

5

tO

15

20

(b)

t [second]

25

30

35

40

45

50

55

60

t [second]

Fig. 11. (a) SMA temperature for variable C~; (b) current density applied (J # J÷).

constant C[ and variable C[ cases, respectively, where IJ(t)], the magnitude of J(t), is assumed to be constant. For the constant C~ case, the following theorem explains the results shown in Fig. 13. THEOREM 4.2 Given M / a n d A f such that Mf <- To <-Af, let C~(T) = C O and J(t) be a piecewise constant function whose value is either J÷ or -J_, J÷ >- 0 and J_ >-O. The transition from - J _ to J÷ is based on the requirement that the forward phase transformation in SMA is completed, while the transition from J÷ to - J _ is based on the requirement that the reverse phase transformation in SMA is completed. There exist a critical value Jl for J÷ such that Ay is reachable and a critical

320310"

~' o . 300. 290"

280'

+

.\

150 50

l "second x 2 x lO'~

Fig. 12. Temperature distribution in the semiconductor.

Thermoelectric heat transfer in SMAs

2361

330

tl I~0-~

320

[A/mm~]

310 300 290

LI I=IA

280

IJ 1=2.48

270 0

5

'

-

10

~

15

20

~ 25

30

35

40

t [second] Fig. 13. SMA temperature corresponding to different IJ(t)l = J for constant C~o.

range (J2, J3) for J__, such that My is reachable. The critical values of J~, J2 and J3 are given by the following formulas. •/1 = m i n { x / ~ A } + 4~l~2(A'f- T o ) - ~3Af El} 2~2 ,~ , 12 = ~3Mf

- ~/ ~2M~ --

4~ ~2(To -

M:)

2~:2 J3

=

(4.6)

'

(4.7)

~3M: + V'~M~ - 4~:1~2(To - M/) 2~:2 '

(4.8)

where l+Cv -CS - u

~1 = VlT]I -[-~

~2 = ~ vl I~ + 2pn'--'-~2

V1/.I.,

L(aN - ae)

Co '

~:3 =

4K

(4.9) '

'I/2=

'01 = --

+ V1

C~

(4.10) Cv

+

VI

330

U I~J.

320 310 °,2.., 300 290 280 270

li 0

5

I0

15

[A/mm~]

IJ I=1.4

IJ I=2.48

20

25

30

35

40

t [second] Fig. 14. SMA temperature corresponding to different IJ(t)l = J for variable C'~,.

2362

D . C . L A G O U D A S and Z. DING

PROOF. From Theorem 4.1, the system (3.20) is stable for any J(t) = -J_ <- O. It is easy to verify that the system is also stable for those J(t) -- J+ > 0 and J+ e (0, ~1/¢3) where the expressions of ¢~ and ¢3 are given in (4.9). From Theorem 4.1, the stable temperature T~ of the system is given by T~

¢1 To + ¢2J2 ~1 + ~3J

where ~1, ~2 and ~3 are given in (4.9). When J(t) =J÷ E (0, ~1/~3), which corresponds to the heating problem, and when T~-< At, the temperature T(t) will not reach A r or reach it at infinite time. By solving T~ - A r and noticing that J e (0, ~1/~3), we obtain that when

O
(4.11)

A s is reachable. When J(t) = -J_ <- O, which corresponds to the cooling problem, and when T~-> My, My is not reachable. By solving T~-> My, we obtain

0 <--J_ <--~3Mf

-

X/¢32M}- 4¢~ ¢2(To - Mr)

(4.12)

2¢~ or 2

2

j_ _> ~:3MI + X/~3M~ - 4~:1~:2(T0- Mr)

26

(4.13)

Combine (4.11)-(4.13), we obtain that when J+ > J l and J2 < J - Jl and J_ ~ (J2, J3)- From the numerical solutions, the sequence of time intervals seems to converge to some constant value, i.e. J(t) is almost periodic in time. However, this case of temperature control is different from the one when a periodic current J(t) is applied, because in that case the final range of operating temperature (Tmin, Tmax) , after the transient response of the system has been eliminated, will in general be different from (My, AI). REMARK 4.3 Similar to Remark 4.1, a corresponding theorem for the variable C~, case should hold. Even though we have not proved such a theorem yet for the variable C[ case, similar results, in fact, have been numerically verified as shown in Fig. 14. Often in applications, a partial phase transformation is also of interest. In this case, the target range (Tmi,, Tmax) is specified such that (Tmi., Tmax)c (MI, At) ' thus a partial phase transformation between Tmi, and Tmax is available, provided that the target temperatures are reachable for applied J(t). The control problem is then similar to that for complete phase transformation with My and A r replaced by T m i n and Tmaxrespectively. Such an example is shown in Fig. 15, where is also seen that the time interval for partial transformation is much smaller than for the complete transformation case, shown in Fig. 7. A final numerical result for comparison purposes, is shown in Figs 16 and 17, where [J(t)[ = 1.0A/mm 2, Ms = A I and My =As, and both full and partial transformations are considered. This case corresponds to lack of hysteresis in the phase transformation of SMA and, for given phase transformation range, it represents the shortest time interval, provided that all other parameters remain constant. To further reduce the time interval and increase the frequency response, one needs to change either material parameters of the system, e.g. the Seebeck coefficient a, or geometric parameters, e.g. the thickness d of the SMA.

Thermoelectric heat transfer in SMAs

2363

292

29O

288 ¢> 286 284 282

o

~

,'o

'

,5

t [secondl S M A temperature corresponding to variable C~, and

Fig. 15.

T.~i. =

(Mr + M~)/2 a n d

Tm.~ =

(Af + A,)/2. 300

290 .o. 280

270

, 0

i

5

,

,

t Fig. 16.

SMA

temperaturc

corresponding

,

.

n I0

,

,

15

[second]

to variable

C~,, a n d

Tmi,, = M r

and

Tm,~ = M~.

320

31o

~"

300

290

280 0

,

,

,

5n

,

,

,

,

! i0

15

t [second] Fig.

17. S M A

temperature corresponding to variable C'~, and

Tmi. =

(3Mr + M ~ ) / 4

and

Tm~=

(Mr + 3 M s ) / 4 . 5. C O N C L U S I O N

In this paper we studied the transient and multiple cycle solutions of one-dimensional symmetric thermoelectric shape memory alloy (SMA) actuator. From the model proposed by Bhattacharyya et al. [1], an approximately equivalent simpler model governed by an integral differential equation was derived first, to describe the temperature distribution in SMA. Then the existence and uniqueness of transient solutions were established and a numerical algorithm was proposed to solve the integro-differential equation. It was found that for the transient cooling problem with constant current density J there is a critical value J0 of IJI, the magnitude of current density, such that when IJI > J0, the temperature in SMA may be not decreasing. The heat transfer problem and the resulting phase transition of SMA, induced by a piecewise constant current source was also studied. It was found that there exist restrictions on the maximum and minimum values of the piecewise constant current density J for a complete phase transition in SMA. Explicit expressions for the critical values of J were derived.

2364

D.C. L A G O U D A S and Z. DING

Due to the relatively large heat conduction coefficient of the SMA compared with the heat conduction coefficient of the P and N semiconductors, the temperature distribution in the SMA material is expected to be almost constant for small ratio d/L, and the results derived in this paper are valid. When the ratio d/L is not small and the physical domain of SMA can be no longer described by one-dimensional heat conduction equation, a proper two-dimensional model of SMA actuator should be introduced and studied. The two-dimensional model of SMA actuator is being currently investigated by the authors. The results will be presented in a separate subsequent paper. Acknowledgments--This work was supported by the Grant No. N00014-94-1-0268 from the Office of Naval Research to Texas A&M University, College Station. The continuing interest by Dr Roshdy Barsoum in this work is gratefully acknowledged. The authors acknowledge many useful discussions with Professor V. K. Kinra and Dr A. Bhattacharyya on thermoelectric cooling problems.

REFERENCES [1] A. BHATTACHARYYA, D. C. LAGOUDAS, Y. WANG and V. K. KINRA, Thermoelectrical cooling of shape memory alloy actuators: theoretical modelling and experiment. In Active Materials and Smart Structures (Edited by G. L. ANDERSON and D. C. LAGOUDAS), SPIE Vol. 2427 (1994). [2] W. J. BUEHLER and R. C. WILEY, Nickel-base alloys. US Patent 3174 (1965). [3] K. TANAKA and S. NAGAKI, Ing. Arch. 51, 287 (1982). [4] D. C. EDELEN, Int. J. Engng Sci. 12, 397 (1974). [5] J. G. BOYD and D. C. LAGOUDAS, J. lntell. Mater. Structures 5, 333 (1994). [6] J. G. BOYD and D. C. LAGOUDAS, A thermodynamical constitutive model for shape memory materials. Part I. The monolithic shape memory alloy. Submitted to Int. J. Plasticity. [7] C. A. DOMENCIALI, Rev. Modern Phys. 26, 237 (1954). [8] C. M. JACKSON, H. J. WAGNER and WASILEWSKI, A report on 55-Nitinol the alloy with a memory: its physical metallurgy, properties and applications. Technology Utilization Office, NASA, Washington, DC (1972).