CONVECTIVE DIDACTIC THERMOELECTRIC DEVICES

J.D. Tellez 1, 2, J.M. Sanchez* 2, J.M. Redondo* 1 , T. Vila 1

1 Department of Applied Physics, Universitat Politecnica de Catalunya, 08034 Barcelona,Spain2 Department of Research, BEROTZA, Pol. Noain-Eskirotz, Nave 1, 31191 Eskirotz,Navarra, Spain (e-mails: * berotza@berotza.com, * redondo@fa.upc.edu )

Abstract

Design and operation of thermoelectric coolers and heaters that may be used in detailled lab-oratory experiments of buoyancy driven trurbulence, as well as in the general fluid mechanicsof convection is important, there seems to be a lack of comparison between numerical modelsof turbulent flows ( Including Kinematic Simulation and DNS) and non-homogeneous exper-iments. [1-3] We present the results of a university-industrial colaboration that developed aFluid Dynamic Didactic Apparatus able to model steady and transient Thermoelectric drivenconvective models based on the control of thermal boundary conditions and also optimized toperform flow measurements inside a closed enclosure. The Thermoelectric Convection DidacticDevice (TCDD) presented here is basically designed for a range of didactic uses, but a widerange of innovative research options are available, both in the small 4x4 device shown in figure1 and in larger and higher power equipments. We present here both the thermoelectric andthe fluid flow description of the TCDD. The coupling of heat transfer and electric conductionwithin the semiconductor Thermal assemblies is important and takes into account the localthermoelectric effects, including Joule and Seebeck heating, Thomson effect, Peltier effect andFouriers heat conduction. The macroscale heating cooling equations are presented togetherwith the calibration and flow characteristics of the thermolectric convective devices shownonly for an example configuration, but many other are possible.

Keywords: Thermoelectric, heating- cooling, convection, turbuence.

1 Introduction

We calculate the velocity and vorticity patterns and discuss the scaling of the velocity and vorticityin a thermoelectric driven heating and cooling experimental device in order to understand and mapthe different transitions between two and three dimensional convection in an enclosure as shownin the TCDD (Figures 1 and 2), this is an example of convective driven and boundary conditionegulated complex flows, with enough thermal power the resulting flow is fully turbulent and thusa wide range of flow regimes can be studied. The size of the water tank of the TCDD is , of 0.2x 0.2 x 0.1 m and the heat sources or sinks can be regulated both in power and sign [1-3] usingThermoelectric cells. The thermal convective driven flows are generated by Seebeck and Peltiereffects in 4 wall extended positions of 0.05 x 0.05 cm each. The parameter range of convective cellarray varies strongly with the Topology of the boundary conditions. At present side heat fluxes areand estimated from empirical correlations as functions of Rayleigh, Peclet and Nusselt numbers[4-6].

Figure 1: Set up of the Thermoelectric Device (TCDD) prototipe with its temperature measure-ment and regulating circuit designed by BEROTZA.

Figure 2: Set up of a Thermoelectric driven flow with PIV, and the resulting velocity/vorticityplot using program DigiFlow.

We present here a description of the basic Thermoelectric driven heating and cooling experi-mental device in order to regulate the heat fluxes in the TCDD. It is also quite interesting froma didactic point of view to be able to alter theglobal geometry from a 2D flow to the full 3 Dcomplex flows. The size of the device shown here is of 0.2 x 0.2 x 0.1 m and the heat sources. ALarger Prototipe is available of 1 x 1 x 0.2 m, where the buoyancy driven flows can be generated bySeebeck and Peltier effects in between 4 to 40 wall positions. The parameter range of convectivecell array varies strongly with the topology of the boundary conditions and buoyancy [4]. Thetilting possibilities of the BEROTZA built experimental device(TCDD) also allow to heat/coolat any angle [5,6] (Redondo 1995, Redondo et al.2013). Fluid visualizations shown here are justperformed by PI and Particle tracking, future research will include Pearlescence tracers and shad-owgraph [4,12]. The equipment can be set up with an array of temperature ( thermistor probes)to be places at different positions as seen in figure 3.

Figure 3: Probe distribution in the Thermoelectric Device designed by BEROTZA.

When the heat source on the wall is switched on it is assumed that a turbulent natural convec-tion boundary layer forms adjacent to the wall. The flow in the boundary layer per se is complex.Indeed there is still a great deal we do not know about the behaviour of high Rayleigh numbernatural convection boundary layers despite the fact that they play a very important part in thethermal behaviour of buildings and other structures. (Note: a Rayleigh number of order 1011is typical of a large space in a building with a heated wall of height H=5 m and temperaturedifference of 10C between wall and the air). However, at high boundary layer Rayleigh numberone might reasonably assume that effects of molecular viscosity are small compared to those ofturbulent mixing and that the ”entrainment assumption” applies. The entrainment assumption isused frequently in the analysis of buoyant plumes and models the observation that the horizontalvelocity of fluid entrained into the plume is directly proportional to the vertical velocity.[10-13].We describe next the application of the thermoelectric equations on thermoelectric efficiency andcooling, and we finally show some examples of complex convective flow patterns using PIV.

2 Thermoelectric Device Control

we describe the basic betwen the thermel fluxes and the thermoelectric design parameters If thedefinitions of the electric definitions below correspond to a single ( one cell) thermoelectric device,we can derive the:

Expression of the termoelectric Seebek, Peltier and Thomson coefficients

The potential, V, between sides 1 (hot) and 2 (cold) of a thermoelectric appliance, may bewritten as

V =

∫ 2

1

E dx (1)

and using the electric field as, E = qV , may be written as:

V =1

q

∫ 2

1

∂φ

∂xdx−

∫ 2

1

J(ρ1 + ρ2) dx+1

q

∫ Tc

Tf

(S∗c − S∗

f ) dT (2)

This may be simplified as sides 1 and 2 are at the same temperature as:

E =1

q

∫ Tc

Tf

(S∗c − S∗

f ) dT (3)

And the Seebek coefficient is defined as:

α =S∗

q.

The heat absorbed per unit volume by the circuit when it is transversed by an electric currentof density J , and

w = (TS∗ + Φ)Je − k∇T

is given by Q = dwdx , and may be written as:

Q = −T Jq

∂S∗

∂x− J2

σ− ∂

∂x

(k∂T

∂x

).

The second term corresponds to heat by Jule effect and the third one is due to thermal con-duction, the first term includes Peltier and Thomson effects.

π =Q

I

Integrating the first term over a couple, we have

dQ = Q1Sdx = −T Jsq

(S∗B − S∗

A)

and thus

πAB =T

q(S ∗A −S∗

B).

Thomson effect shows that a gradient of temperature applied to a material where a current Iflows provoques an absortion or an emision of heat, and is defined as:

τ =dQdx

I ∂T∂x

Differenciating the heat fluxdQ

dx= −I ∂T

∂x

(T

q

∂S∗

∂T

)we can write

τ = −Tq

∂S∗

∂T.

Using these definitions Kelvin relations are found:

αAB =πABT

and

T.∂αAB∂T

= τB − τA.

2.1 Calculations for TEC design and Coeficient of Performance

In a practical thermoelectric design, after the thermal loads have been calculated, the optimumIntensities and voltages may be found using the local heat balances between both sides of the TEC.The heat produced on the hot side p is

Qp1 = S T1 I

where T1 is the hot side temperature , S the Seebeck coefficient and I is the intensity of theelectrical current.

Conversely the heat absorbed at the cold side is:

Q2 = S T2 I,

where T2 is the cold side temperature.Assuming that Joule heating flows equally to both sides of the TEC, it can be written as:

Qj = 0.5 I2 R,

where R is the resistence of the semiconductors.The loss of heat by conduction across the TEC is:

Qc = κ (T1 − T2),

where κ is the thermal conductivity of the n and p materials taking account of the length andarea.

The absorbed heat at the cold side is then:

Qnet = Qp2 −Qj −Qcand the hot side produceded heat is:

Qd = Qp1 +Qj −Qc

The electrical power, by the 1st Thermodynamic law must be then:

P = Qd −Qnet = S(T1 − T2) + I2R

There are two basic ways of TEC behaviour, whithin a system, a) maximum heat transfer andb) maximum COP (Coeficient of Performance). The first way may be found by derivating withrespect to I and equating to zero, thus

Imax =S TcR

The maximum temperature diference may be found substituting Imax in the expression of thenet heat production.

∆Tmax = (T1 − T2)max =0.5S2T 2

c

R κ.

The figure of merit is defined as:

Z =S2

R κ.

For a system to work at maximum Coeficient of Performance, COP, it is necessari to maximizethe rate between the net transported heat and the electric power.

COP =QnetP

which may be written as:

COP =ST2I − 0.5RI2 − κ(T1 − T2)

RI2 + S(T1 − T2).

We find COPmax, diferenciating with respect to the intensity, and equating to zero, this max-imum corresponds to the intensity Icop that may be expressed as

Icop =κ(T1 − T2)

0.5S(T1 + T2).[(1 + 0.5Z(T1 + T2))1/2 + 1].

Using Icop, in the expresion of the COP, we find

COPmax =[1 + 0.5Z(T1 + T2)]1/2 − T1/T2

[1 + 0.5Z(T1 + T2)]1/2 + 1.

T2T1 − T2

.

This calculations are important for the correct design of a thermoelectric system, which may beajusted to work either at maximum COP or at maximum cooling. The adjustment of workingintensity depends on the use of the application. see the references [4-6]

Figure 4: Description of a single element of Thermoelectric Device and external Temperatureregulating circuit designed by BEROTZA.

Figure 5: Set up of a Thermoelectric Device and temperature regulating circuit designed byBEROTZA.

3 Analysis of Transients in Thermoelectric Devices

A detailled study of the transients is difficult, even in simplified models of the TEC, constant aver-age properties of the modules, have to be obtained from the manufacturers, such as conductivitiesand resistences, they also vary with temperature. If we assume a constant temperature throughoutthe TEC, the basic diferential equation is:

mcdT

dt= Q−Qnet (49)

where T is temperature, t is time Q external heat and QC the heat pumped by the TEC, m themass of the TEC and c the average especific heat of the TEC. Then

mcdT

dt= Q− (STI − 1

2I2R− κ∆T )

Simplifying and separating the variables we may express the equation as

dT

T − Tc=SI + κ

mcdt

and integrating obtain

T = (T0 − Tc) e−SI+κmc t + Tc

The term mcSI+κ is often called temporal constant, but we have to note that all terms are

temperature dependent and will have fixed values only at fixed points at selected temperatures.The equations for the hot side temperature may be obtained in the same way. The temperatureat estacionary state Te is given by:

Te =−Qh +Qj + κ∆T

SI + κ+ Tc

and the transient temperature is

T = Te(Te − Tc) e−SI+κmc t

All these ecuations are programed in a TEC optimisation subrutine that may be used afterthe external heat balance has been calculated, The final electric set up depends on the heat fluxesdesired at each of the 4 positions shown in figures 1-3.

3.1 Discussion and Conclusions

Several convective flows were set up but here we will show es in figure 2, a configuration withthe 4 thermoelectric cells placed at the sides, with the two lower ones being cooled while the twoupper ones are heaeted with the same power as to mantain the enclosure at constant temperature,this is a convenient configuration because the molecular propieties of the fluid, such as viscosity orthermal diffusivity may considered as constant during the flow transients that set up the convectiveregimes. [?] presented the experiments nd numerical models in non-linear thermoelectric systems.

Figure 6: Velocity Patterns after seting the Thermoelectric Heat and .

Thermoelectric coolers offer the potential to better control and enhance the cooling of electronicmodules to regulate operating temperatures or to allow higher power. Thermoelectric coolers arelimited by the maximum heat fluxes and have lower coefficient of performance (COP) but offer awide range of boundary fluid control.

Environmental and Engineering Fluid Mechanics laboratories at university or professionalschools may incorporate student practical work in several of many fields that need understandingsuch as: ENVIRONMENTAL FLOWS: Convection in the Atmosphere; Thermal plumes in the

ABL; Sea and Mountain Breezes; Inversion layers; Thermohaline convection; Thermal and SolutalMixing; Diffusion; Turbulence scaling...FLUID DYNAMICS AND HEAT TRANSFER: NaturalConvection, Laminar and Turbulent Convection, Enclosed Flows, Bottom and Side wall ThermalBoundary layers. Rayleigh and Nusselt number evaluation. Turbulent correlation,NUCLEAR,CIVIL AND INDUSTRIAL ENGINEERING: Ventilation, Nuclear Reactor Cooling, ThermalStratification, Buoyant Mixing, Wall thermal correlations, Chemical Thermal Reactions. OtherSelf-similarity techniques can be applied to the velocity and vorticity PIV obtained maps [14-16],The advantage of a flexible convective flow generator is the versatility in obtaining different 2D or3D turbulent flows.

One of many practical aplications, where the angle between the enclosure and gravity definescomplex, but realistic conditions, includes naturally ventilated rooms or greenhouses containinga heated vertical surface (or at different angles). By assuming the boundary layer to be fullyturbulent and that the effects of viscosity are therefore negligible it is then possible to model theboundary layer as a buoyant plume developing as a result of a vertically distributed source ofbuoyancy. The governing differential equations governing the flow in both the boundary layer andin the room itself have been solved using a numerical technique for the transient cases of a sealedroom and a ventilated room. For example in Figure 6 a Velocity (2D) Map at the centerline of aside cooled convective pattern shows quite clearly the production of baroclinic vorticity due to theconvection near the centre of the tank. Similar descriptors of the velocity and vorticity fields canbe seen in figures 2 and 5. In a similar way in figure 7 only one convective cell of 1/4 the area ofthe 2D 20x20 cm pattern is generated by only heating one of the side thermoelectric devices andcooling the other three.

Figure 7: Velocity Patterns producing a single cell in the bottom left corner of the TCDD .

Acknowledgment

Partial financial support for the present project was provided by the European Science Foundation - Thermo

Nanovea and BEROTZA under the Grant No. UPC-BEROTZA-2014-0880 .

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Figure 8: Side view of the Thermoelectric Device designed by BEROTZA.