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Одержано функщональний зв'язок температу-ри теплопоглинаючого спаю вiд часу з врахуванням теплоeмностi комутацшних пластин i охолоджу-ючого об'екта, а також теплового навантажен-ня теплообмту на робочих спаях термоелемента з зовншнш середовищем. Показана можливють управлтня терцштстю нагрiву i охолодження термоелектричного пристрою шляхом змти вели-чини робочого струму термоелементу. Визначена ефективтсть такого управлтня у залежностi вд теплофiзичних параметрiв
Ключовi слова: термоелектричний охолоджу-вач, нестащонарний режим, перепад температур,
управлтня нрцштстю
□-□
Получена функциональная связь температуры теплопоглощающего спая от времени с учетом теплоемкости коммутационных пластин и охлаждающего объекта, а также тепловой нагрузки теплообмена на рабочих спаях термоэлемента с окружающей средой. Показана возможность управления инерционностью нагрева и охлаждения термоэлектрического устройства путем изменения величины рабочего тока термоэлемента. Определена эффективность такого управления в зависимости от теплофизических параметров
Ключевые слова: термоэлектрический охладитель, нестационарный режим, перепад температур, управление инерционностью
UDC 621.362.192
|DOI: 10.15587/1729-4061.2017.116005|
ANALYSIS OF THE POSSIBILITY TO CONTROL THE INERTIA OF THE THERMOELECTRIC COOLER
V. Zaykov
PhD, Head of Sector Research Institute «STORM» Tereshkova str., 27, Odessa, Ukraine, 65076 E-mail: [email protected] V. Mescheryakov Doctor of Technical Sciences, Professor, Head of Department Department of Informatics Odessa State Environmental University Lvivska str., 15, Odessa, Ukraine, 65016 E-mail: [email protected] Yu. Zhuravlov PhD, Associate Professor Department of Technology of Materials and Ship Repair National University «Odessa Maritime Academy» Didrikhsona str., 8, Odessa, Ukraine, 65029 E-mail: [email protected]
1. Introduction
The possibilities of using thermoelectric cooling devices (TCD) in critical systems with enhanced requirements to reliability indicators are determined not only by static, but by dynamic indicators as well. Under transient modes characteristic of the operation of thermoelectric coolers in control systems, of special importance are the temporal indicators of reaching preset thermal regimes by the cooled elements of radio-electronic equipment. Under a pulsed thermal impact even a slight excess of the permissible thermal regimes of elements can lead to failure of the thermally-loaded elements. A transient characteristic depends on the heat capacity of heat load, thermal conductivity and active resistance of thermoelement branches, spatial distribution of heat flow, etc. Accounting for the effect of given components on the transitional characteristic and, accordingly, on the possibilities to control performance efficiency of TCD, is a relevant task aimed at improving failure-free operation of the thermally-loaded components of electronic equipment.
2. Literature review and problem statement
Thermoelectric cooling devices have a significantly higher performance compared with compression systems for
maintaining thermal modes of radio-electronic equipment
[1]. This is predetermined by the low mass of a thermoelectric cooler, solid-state principles of cold and heat generation
[2], properties of the material used for thermoelements [3], and fabrication technology [4]. That is why designers of thermal mode provision systems typically confine themselves to a stationary TCD operating regime, that is, they actually neglect dynamic TCD characteristics [5-7]. However, there is a class of tasks, which are characterized by the pulse input of external heat impact, for example, laser receivers and emitters that require systems for ensuring thermal regimes [8]. For such thermally-loaded elements permissible heating temperature is limited by the values of working capacity of semiconductors, which is why it is necessary to form a cooling wave, close to the rate of rise in the heat flow of the element. This circumstance poses a problem requiring the creation of high-speed systems for ensuring thermal regimes that work in pulsed mode [9]. The issues of distribution of thermal fields in TCD are not studied enough, to say nothing of control methods. The results of research into effect of switching modes of TCD on cracking a contact between the thermoelement and the electrode [10] only partially fill this gap. This feature became the basis for the accelerated TCD test method for failure-free operation [11]. However, TCD high-speed performance issues, as well as performance efficiency control methods, were not paid due attention.
©
3. The aim and objectives of the study
The aim of present work is to study dynamic characteristics of thermoelements under a non-stationary operation mode of the thermoelectric cooling device.
To achieve the set aim, the following tasks must be solved:
- to design a model of the thermoelectric cooler under a non-stationary mode connecting temperature difference between the electrodes and parameters of the thermoelement, load, and working current;
- to determine a possibility to control the rate of change in the heating and cooling of the thermoelectric device.
- branches of thermoelements possess identical thermo-physical parameters;
- lateral surfaces of thermoelement branches are adia-batically insulated.
Then the distribution of temperature T as a function of the x coordinate, directed along the length of thermoelements from a heat-absorbing junction to the heat-generating one, and time t, is described by the equation of thermal conductivity
4. Design of the model of a thermoelectric cooling device under a non-stationary mode
The operation of thermoelectric cooling devices is determined by both stationary and non-stationary cooling process. Stationary regime is characteristic of the systems for ensuring thermal regimes of thermally-loaded elements whose heat load changes insignificantly. For the non-stationary regime, important is the time for reaching the specified level of cooling time, as well as the ability to control over time a given level of temperature on the heat-absorbing layer.
In this case, of interest are the transient processes that occur when electric current of different form passes the thermoelement.
Heat conduction equation in a homogeneous medium can be represented in a general form [12]:
at dt
dx2 k
dy2
aV
dz2
T dt '
,9T 9x2'
where 1 is the thermal conductivity of thermoelement branches; c is the volumetric heat capacity of TCD; p is the specific electric resistance of a thermoelement branch; j is the density of supply current.
Physical parameters are related to the area of the cooled surface. At the initial point of time temperature is constant throughout the entire volume of the thermoelement, while the temperature of a heat-generating junction is constant over time due to the intensive heat exchange T0.
At the heat-absorbing junction, one takes into account a Peltier effect, heat capacity of the switching plates and the cooled object, heat losses into the environment and a heat load.
The initial and boundary conditions for equation (1) can be written in the form:
n=o=T ;
xdL
dx
ejT + g ^ +ai (T-To )-
(2) (3)
T , = t,
I x=l 0>
where a =— ; k is the coefficient of thermal conductivity; \cp
c is the specific heat of the substance; p is the density. In this case, the initial condition that produces initial temperature distribution at t=0:
TL=fCx У, z).
In the case of distribution of heat in a solid body with linear dimensions, for example in a homogeneous rod, which is located along the x axis, the equation can be written
Given such a form of the equation, one does not take into account heat exchange between the ends and the side surface of the rod and the surrounding space.
The issues of non-stationary thermoelectric cooling were previously considered for a quite simplified thermal model of TCD [10].
In a general case, taking account of various factors ultimately comes down to controlling the temperature of thermoelement junction in line with the assigned law.
Of greatest interest is the study of thermal processes on a heat-absorbing thermoelement junction taking into consideration heat load.
The thermal model is built under the following assumptions:
where e is the thermal emf coefficient; g1 is the heat capacity of the switching plate and the cooled object; a1 is the heat transfer coefficient; q is the thermal load; l is the length of a thermoelement branch.
In a general case, power qt consists of the power of heat sources and Joule heat power, dissipated at the heat-absorbing junction (j2pc, where pc is the contact electric resistance of a thermoelement branch).
We shall subsequently operate with the following dimen-sionless parameters:
®=zT; x=x/l; F0=at/l2; Bl1=a1l/X;
ni=gi/(cl); K= qt (zlA); ^ =pL(K, = Kgv%n),
where a is the coefficient of thermal diffusivity of thermo-_ e2
electric material; 2 = — is the thermoelectric efficiency of pA
material.
Equation (1) with initial and boundary conditions in di-mensionless units can be represented in the following form:
d20 90 2
d© =W "V' ^ =0 =0o;
a©
dXX=0
®L=©o.
d©
= v© + n ^^ + BlX (0-00 ) + KH;
(4)
(5)
Using the operational method, it is easy to find the Laplace transform for the temperature of a heat-absorbing junction:
©( p)\ =Q0 | (C^P - 1) - P( VQ° - ShJP (6)
x=° P p2 [-JpchJp + (v + Bi + h1p)] sh^p
Transition to the temporal domain is carried out in line with the Riemann-Mellin formula and a deduction theorem:
Q(Fo) = ©o -
v©0 - 0,5v2 - K l 1 + v+B,
+ ±Ak exp(-S2kFo), (7)
where
2
[Sk (vQ0 - K )sin 8,-v2 (1 - cos 8, )]
8k [8k (2hi +1)sin 8k + (hi82 -1 -v- Bi )cos 8k ] '
The magnitudes Sk that are included here are the positive roots of transcendental equation
lated values of the two lowest roots 51 and S2 for a wide range of change in the parameters v+Ba and Calculation results are given in Table 1.
Table 1
Values of roots 5, and S2
Xv+B.i ni 0 0.5 1.0 2.0 3.0 5.0 10 15 20
0 1.57 1.84 2.03 2.29 2.46 2.65 2.86 2.95 3.0
4.71 4.82 4.91 5.1 5.23 5.45 5.76 5.91 6.0
0.10 1.43 1.70 1.90 2.18 2.37 2.60 2.84 2.94 2.98
4.31 4.40 4.49 4.67 4.85 5.14 5.60 5.82 5.94
0.25 1.27 1.52 1.72 2.02 2.24 2.51 2.81 2.92 2.98
3.94 4.0 4.06 4.19 4.34 4.63 5.25 5.62 5.82
0.50 1.08 1.31 1.49 1.79 2.02 2.34 2.73 2.88 2.95
3.64 3.67 3.71 3.78 3.86 4.05 4.59 5.092 5.46
1.00 0.86 1.05 1.21 1.47 1.68 2.01 2.53 2.78 2.90
3.43 3.44 3.45 3.47 3.50 3.57 3.83 4.18 4.55
8
h482 -v-Bii
(8)
It follows from ratio (7) that temperature difference A©= =©0-© between the thermoelement junctions can be represented in the form
A© = ©stat-/(Fo),
where
f (F0) Aexp(-8kF).
(9)
(10)
Series (10) determines the speed at which a heat-absorbing junction temperature approaches the stationary value. In this case
A0stat =
_v©0 (0,5)v2 - K
1 +v+ B,
(11)
The maximum temperature difference is reached at optimum current density v^, which can be determined from ratio
„ _[B2 + 2QoB + 2K(1 + 2£,k)]7 -B v _ 1 + 2^k
(12)
The maximum temperature difference can be determined from relation
1
A©™- = ©o + B - [ B2 + 2©0B + 2K (1 + 2^)]2, (13) where
5=(1+5a)(1+2 Zk).
The case for v+Bi1=0 and n1=0 is the limiting. Since the magnitudes of 5k are localized in the respective intervals 5ke((k-1)n, kn), series (10) quickly converges at large enough values of F0.
Data from Table 1 show that an increase in heat capacity n1 leads to a decrease in the values of 5k, that is, to an increase in the inertia of thermoelements. An increase in heat transfer Biv as well as in current density v, exerts an opposite effect on the magnitude of 5k.
Pre-exponential factors Ak are also dependent on parameters n1, v and Bi1. In addition, the magnitudes Ak are affected by parameters ©0 and Kn. Fig. 1-4 show dependences of coefficients A1 and A2, as well as 82 and 82, on the ratio of current density v to the optimum value of v^ for various combinations of parameters Bi1 and n1. Dependence of A1 on current density v has a maximum, after which A1 mono-tonically decreases, passing a zero to the negative region. Coefficient A2 monotonically increases with an increase in current v. With an increase in heat transfer B^ coefficients of the series decrease, which corresponds to a reduction of the stationary temperature difference (Fig. 1, 2). Values S42 and 82 increase monotonically with an increase in current power. In the cases where parameters Bi1 and n1 accept limiting values close to zero, or sufficiently large, it is easy to determine analytically relevant values of 5k and Ak, specifically:
- at n1^0 v+Bs1^0 K=0
8K _-(2k-1); Ak_
8 n(2 k -1) v©0 + 2(-1)k
(2k -1)3 n3
at v+Ba >>0 K=0
2v2 [(-1)k -1]
8k_ kn; A_ 2 ,L2-] ;
k k rc2k2(v + Bt1)
- at n1>>0 K=0
5. Analysis of the model of a thermoelectric cooler under a non-stationary mode
Calculating temperature difference © from formula (7) implies a knowledge of the roots of equation (8). We calcu-
81 1+v+Bi1; 8k _ (k- 1)n, (k _ 2,3...), v© -05v2 2v2 il+(-1 )k]
A _ ©0„ 0,„v ; A_ v4L V ] (k_ 2,3...).
1 +v+ B,
n4( k -l)4 hl
(nvQ0 - 2v2 ) ( n2
A0 = AQstat - -^- -expI -—F,
n \ 4
- at v+B. >>1 K =0
4v2
A0 = A0 stat + 8-^-r
stat n2 (v + Bti)
exp (-rc 2F0 );
- at h1>>1 AQ = AQ
. | 1 + v + Bi1 r 1 - exp I--± F0
(14)
(15)
(16)
§1 (q0 - K- J sin § - v(l-cos §1 ) = 0.
(17)
Determining current density that satisfies this condition (v0) should be carried out numerically, in combination with solving equation (8). The values of v0, obtained in this case, can be compared with the optimal under a stationary mode. The v0 calculation data for some values of parameters n and Bn (at @0=0.6 and K=0) are given in Table 2.
Fig. 3. Dependence of magnitude 5^ on current density at Q0=0.6 and 7=300 K for various values of Bn and h
Fig. 1. Dependence of coefficient A1 on current density v at Q0=0.6 and 7=300 K for various values of Bn and h
Accordingly, a temperature change at large enough values of F0 is determined from the following relations: - at n1^0 v+B.^0 K, =0
Fig. 4. Dependence of magnitude §2 on current density at Q0=0.6 and 7=300 K for various values of Bn and h1
The value of v0 can be derived analytically, at K^0, in this case
Q„ arccos -
v=
Q
1+a
+ 2Qo ' Since v°T = J 1 + 2Q0 -1, then
1 +
Jl + 2Q0
arccos -
Q
l + Q,
(18)
(19)
Fig. 2. Dependence of coefficient A2 on current density v at Q0=0.6 and 7=300 K for various values of Bn and h1
Temperature of the heat-absorbing junction, depending on the current strength, can approach stationary from the side of larger or smaller values, that is, either monotonically or passing through a minimum. The condition that differentiates these two cases implies that coefficient A is equal to zero:
Ratio (19) for any ©0 exceeds n/2, that is, it is larger than
unity. An increase in heat load n1 contributes to the growth of
current v0 while heat transfer Bn reduces magnitude v0 (Table 2). v0
In a general case, ratio vo can be both larger and lower
vT
than unity. Thus, an extremum at the dependence curve 0(F0), occurring at v=v0, may emerge at the current values that are less than optimal (for sufficiently large values of Bn) and which exceed the optimal under a stationary mode. Because 82>81, the mode, which is transitional from monotonic fall in temperature over time to a change in temperature with an extremum, is the "fastest". Current density v0 corresponds to the least time of the transition process, while mode v=v0
appears appropriate under condition that current v0 ensures sufficient reduction in temperature under a stationary state.
Table 2
Ratio between v0 and vT at different thermal load on a heat-absorbing junction
ni 0 0.1 0.5 1.0 œ 0 0 0
Bi 0 0 0 0 0 0 1.0 5.0
V°r 0.483 0.483 0.483 0.483 0.483 0.483 0.53 0.573
V0 0.79 0.83 0.97 1.05 1.20 0.79 0.66 0.37
V = Vo/Vr 1.64 1.74 2.0 2.17 2.48 1.64 1.25 0.65
ni 0 0 0 0.1 0.5 1.0 0 -
Bi 10 œ 5.0 5.0 5.0 5.0 5.0 -
V°T 0.584 0.60 0.573 0.573 0.573 0.573 0.573 -
V0 0.24 0 0.37 0.41 0.55 0.72 1.20 -
V = Vo/Vr 0.41 0 0.65 0.71 0.97 1.25 2.09 -
vQq -K + 4 vQQ -K +-V -F +--
0(FQ)=0Q----Fq + ^ h2
hi
(vqq -K)[1--! (v + Bî1)] + -1v2
F 3/2 -
2-3
-F2 +
| 8 (vQq-K)[1-2-I(V+B,i)]]+-iV2[1--I(V+/!,)]
hi5jn -4
fq5/2 +... (20)
leads to a decrease in the cooling rate during initial period. The first two terms of decomposition (21) coincide with the corresponding decomposition terms for the case of cooling the plates from a source of constant cooling capacity equal to v©0—K;.
In order to estimate an error in temperature calculations by analytical formulae using the first terms of the corresponding series, it is possible to apply data from the numerical solution to equation (4). Fig. 5 shows temperature difference dependences on time for various values of ratio v = v0/n and parameters n1 and BlX (©0=0.6; Kl =0).
©o-0 0,10
0,08
0,06 -
0,04 -
0,02
6i
y ___
r-T*'
' 1
-» —
When solving heat equation (4) in the form (7), we observe a fast convergence for sufficiently large values of time F0. To build a transition process for small values of time F0, it is necessary to apply a solution to equation (4) as a series for powers of F0. It is possible to obtain a decomposition for powers p-1/2 from formula (6) ©(p):
e( p)=Q0-^zK J_ +2vf;2 - K -
p hi P hi p;
(ve0 - k )[i-h (v+Bn )]+htv2 - h?P3 +
(vQo - K) [i - 2hi (v + BiX)] + hiv2 [i - 2hi (v + BiX)]
h4p7/2 ""
which corresponds to the following series in a temporal domain
0 0,02 0,04 0,06 0,08 0,1 0,5 0,9 F0 Fig. 5. Temperature difference dependence at the thermoelement on time F0 at e0=0.6; 7=300 K; n1=0 and B;1=0 for different values of current density v: 1 - 0.5; 2 - 0.75; 3 - 1.0; 4 - 1.25; 5 - 1.5; 6 - 2.0
For the case n1=0; Bl1=0 (Fig. 5), high current values are matched by curves with a larger lifting angle at the coordinate origin. The curves for currents that are smaller than v0 (for the examined case v0=1.64 v°T) are monotonic in character and do not intersect at v0 <v°T, while the curves that correspond to currents exceeding values v°T, start to intersect with the preceding ones. When current exceeds magnitude v0, the temperature course has an extremum. The highest speed of establishing a stationary differential for the represented dependences is demonstrated by curve (5) (v = 1,5v°T and close to value v0).
Graphic dependences, shown in Fig. 6 (n1=1; Bl1=0), allow us to observe the effect of heat capacity of the load on the cooling progress. An extremum on curve A©0=f(F0) is not observed even at v = 2 (in this case, v0 = 2,17v°T). The highest rate of establishing a stationary state corresponds to v = 2.
Similarly, it is possible to find decomposition ©(F0) for powers F0 for the case of negligible heat capacity n1:
e(Fo)=e 0--=( ve0 - k )Jf0+
Vtc
+ [(vQo -K)(v + Bn) + v2]Fo -
- 34n[(veo - K )(v+BlX )+v2 ](v+Bn) Fo3/2 +
+2[(vQo - K )(v + BiX) + v2 ](v + BiX)2 Fo2 -... (21)
Comparing formulae (20) and (21) reveals that the presence of heat capacity on a heat-absorbing junction
0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 F0 Fig. 6. Temperature difference dependence at the thermoelement on time F0 at Q0=0.6; 7=300 K; n1=0 and B=0 for different values of current density V: 1 - 0.5; 2 - 1.0; 3 - 1.5
For large values of heat transfer coefficient (Fig. 7) (r)i=0; 5^=5), the value of V, at which an extremum of temperature progress appears, is less than unity, while curve A©(F0) for v = v°t has an extremum. The magnitude of temperature
difference at the extremum point of curve A© =f(F0) is larger than the stationary. At Bi1=5; n1=1 (Fig. 8) is characterized by a mutual compensation of the effect of temperature and heat capacity of the load, resulting in an extremum occurring at v = 1,25.
Fig. 7. Temperature difference dependence at the thermoelement on time F0 at Q0=0.6; 7=300 K; n1=0 and B;1=5 for different values of current density v: 1 - 0.5; 2 - 1.0; 3 - 1.5
To obtain expression A© in a time function, it is convenient to use a model of the thermoelement with an infinite length of branches. In this case, it should be taken into consideration that at a sufficiently short duration of the cooling process, after switching on current, the temperature field at a heat-generating junction does not affect temperature of the heat-absorbing junction. While a Fourier criterion F0=at/P is much smaller than unity, the progress of temperature at a heat-absorbing junction does not depend on the length of thermoelement branches, which is why it can be considered infinite.
Boundary conditions for heat equation (1) - (8) in this case are: - at x=0
x|X = ejT-al(T0-T ) + g |T-y2p,-q ;
- at X^œ
* = 0.
3x
(22)
(23)
00-0
0,02
0,01
— " V /
3 V
/ A
/ V
« —
0 0,02 0,04 0,06 0,(
0,10 0,5 0,9 F0
Fig. 8. Temperature difference dependence at the thermoelement on time F0 at Q0=0.6; 7=300 K; n1=0 and B;1=5 for different values of current density v: 1 - 0.5; 2 - 1.0; 3 - 2.0
In Fig. 5-7, circles denote data on the estimation of function A©(F0) according to formula (7); triangles - using formulae (20) and (21). Comparison with the curves, built based on the results of calculating function A©(F0) by numerical methods, shows that at F0 e(0;0,05) a good approximation accuracy is ensured by only three terms of series (11) or (12). At F0>0.04, sufficient accuracy is achieved when considering two terms of series (7).
The obtained ratios for calculating the cooling process could be used for a heating mode (in the case when the temperature at a heat-absorbing junction is maintained constant), after replacing magnitude v in the relevant formulae with -v. Parameters Sk will correspond to the roots of equation
tg § =
§
hS +v-Bi1
The values of Sk for the case of a heating mode are less than during cooling at identical values of v, n1 and Bi1. To determine the magnitudes of S1 and S2, it is possible to employ data from Table 1; in this case, the input parameter should be the difference Bi1- v.
Expression for a non-stationary temperature progress at stepwise current form can generally be represented only in the form of a series. This circumstance greatly complicates its analysis.
The heat-absorbing junction boundary condition takes into consideration the Peltier heat, heat transfer into the surrounding environment, heat capacity of the cooled object, Joule heat released on contacts, and the power of permanent sources of heat.
We shall consider the case when a heat-absorbing junction is insulated from the surrounding environment; then at x =0
i dT 'T
X—= ejT. ox
We shall reduce equation (1) to the dimensionless form; in this case, we shall employ ratio le=X/(ej) as the equivalent length. In this case
32Q . = 3Q
3x2 + = a( ^ );
QL =Q
= Q in x = 0;
3Q .. — = 0 in x ^
(24)
(25)
(26)
where
= x = eX- F = — =
X i ^ - F0e = 12 =
dt e2 y 2at
I X
I
X2
Equation (24) is easily solved using an operational method, which is why a heat-absorbing junction temperature change over time can be written in the form
AQ( F0 ) = (1 + Q0 )(1 - exp F^ezfc^ ) - . (27)
Thus, a heat-absorbing junction temperature change depends only on parameter F0e, that is, on product /1. Expression (27) has a maximum at a specific value of argument F0e = F0*e, derived from equation
yfrfL exp Ke ■e
1+ Q
(28)
Thus, at ©0=0.6 F^ = 0,081, A©max=0.0895 and an increase in current j causes a corresponding decrease in the time of reaching a maximum, and vice versa. The magnitude of maximum A© does not depend on current. The largest temperature difference at the thermoelement is achieved under a stationary mode:
DQmax = 1 + ©0 -V1 + 2©0 at ©0=0.6, A©max = 0,117.
The largest temperature difference under a non-stationary mode is less than the maximum temperature difference under a stationary mode. Thus, at ©0=0.6, the ratio between these magnitudes is equal to 0.766. These data are also correct for a thermoelement with ultimate branch length l if the Fourier criterion at/l2<<\.
The maximum drop of temperature at a heat-absorbing junction for a semi-infinite thermoelement occurs at
e2j2at*
that is, at
aL_(Al2 F * l2 "Iejl) K*
Thus., condition at/l2<<i is reduced to inequality v>>^F*e. As demonstrated earlier (Table 2, formula (18)), the minimum value of current v0, at which there appears an extremum, is 0.791, and -JF _ 0,28.
Starting from value v=v0, a temperature difference magnitude at the point of extremum, calculated for models with branches of finite and semi-infinite length, is almost identical. Fig. 9 shows dependence of temperature difference A© on v (Curve 1) and value of A© at the point of extremum of function A©(F0) (Curve 2) for the thermoelement with a finite length of branches. At v=v0 the maximum occurs when Wœ, with an increase in v, the value of yJF* is reduced and at v^œ JF0* ^0.
Fig. 9. Temperature difference dependence at the thermoelement with a thermally-insulated heat-absorbing junction on current density v at 0o=O.6 and 7=300 K under a stationary mode (1) and at the extremum point of function A©(F0) (2)
In a general case, when the boundary condition at a heat-absorbing junction is determined from ratio (22), it is easy to obtain an analytical expression that would yield dependence A©0 on time. However, in this case, A©0 is a function not only of dimensionless parameters ©0 and F0, but also of dimensionless complexes:
B _a. h _gL_gj•
ie X ej 1 cl cX
^ = PL=M. K = qzL=qz
k pie P^ ' 1 ^ ej '
The influence of thermal resistances at a junction decreases with increasing current while the effect of heat capacity and electrical resistance increases.
Fig. 10 shows, for the case Bn=5 (^=0; ^k=0; K =0; ©0=0.6), dependence of A©stat (Curve 1) and A© at the point of maximum (Curve 2) on current v. Minimum value of current v0, at which there appears a maximum, is 0.37. In this case, v0 < vT.
In the case under consideration, temperature difference at the point of maximum slightly exceeds the stationary. At v—the magnitude of A© coincides with a temperature difference at B=0. Curve 2 is built based on the results of computer calculations. Results of calculation by formula for the thermoelement with semi-infinite branches produce almost identical results, starting from v~1.
Fig. 10. Temperature difference dependence at the thermoelement on current density v taking into consideration
heat transfer at a heat-absorbing junction at 7=300 K and B;1=5 (ni=0; ^k=0; K=0; ©0=0.6) under a stationary mode (1) and at the extremum point of function A©(F0) (2)
The presence of heat capacity and contact resistance results in the fact the value of A© at the point of maximum falls with increasing current. In the case of the presence of resistance, the extremum occurs only at small enough currents at ^k<©0 or j < P^©°, when the Peltier heat magnitude
exceeds, during initial point of time, the Joule heat at the contacts of a heat-absorbing junction.
6. Discussion of results of an analysis of the possibility to control the inertia of a thermoelectric cooler
The obtained ratios allow us to argue about the following:
- with a growth in heat transfer coefficient B^=5 and n1=0 dependence A© =/(v) acquires an extremum; a temperature difference at the extremum point surpasses the stationary (v>1.0);
- in order to obtain a simpler ratio A© =/(F0), it is convenient to use a model of the thermoelement with a semi-infinite branch length, at which the Fourier criterion F0= =at/l2<<1. This ensures the possibility of computer implementation of the ratio in real time, as well as enables control over a temperature difference at a heat-absorbing junction in a time function;
- a growth of current leads to a decrease in the effect of thermal resistances at a heat-absorbing junction, while the effect of heat capacity and electric resistance increases;
- an increase in heat capacity at a heat-absorbing junction slows the rate of cooling.
Thus, when heat capacity of an object far exceeds the heat capacity of switching plates, additional effect of non-stationary cooling is negligible and comprises 1-2 % of the stationary.
The advantage of the proposed approach is enabling the possibility to improve response rate of the thermoelectric cooler on fluctuating thermal impact. This makes it possible to create thermoelectric systems for ensuring thermal regimes of thermally-loaded elements of enhanced energy stability. The effectiveness of such systems will be determined not only by the limitations, accepted when designing a model of the non-stationary thermoelectric cooler, but by control system as well. Input indicators of control system, the sensors applied, data processing algorithms should influence the inertia and reliability indicators of thermoelectric cooling system, which requires further research.
7. Conclusions
1. We designed a model of non-stationary working mode of the cooling thermoelement, in the framework of which we determined functional dependence of temperature of the heat-absorbing junction on time. The model takes into account heat capacity of the switching plates and the cooled object, heat load on a working junction, heat transfer to the environment.
2. We substantiated the possibility to control the rate of cooling and heating a thermoelement when changing the device's working current at different thermal loads. Thus, when current density increases by four times, the same temperature difference at the thermoelement during cooling is reached 8-10 times faster.
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