Thermotechnique et thermoénergétique

A NUMERICAL STUDY OF THE HEAT TRANSFER MANAGEMENT PROVIDED

BY A THERMOELECTRIC SINK-AND-FAN SYSTEM

JEAN BOGDAN DUMITRU1, ALEXANDRU MIHAIL MOREGA1,2, MIHAELA MOREGA1,3

Key words: Thermoelectric cells, Thermal management, Mathematical model, Numerical simulation.

This paper presents a mathematical model and its finite element (FEM) implementation for the thermoelectric processes that occur in a Peltier sink-and-fan device aimed at enhancing the heat exhausted by power electronics. The heat transfer management for several key parameters (the heat released by the electronic module, the ambient temperature, the convection heat transfer coefficient) is analyzed to find optimal working conditions.

1. INTRODUCTION

An important issue in the design of modern electronic systems is the heat transfer management. The heat produced by “hot” electronic devices can lead to their thermal failure, lifespan reduction of the final electronic system, or it can significantly reduce the overall performance of the device. To keep the temperature within safe working limits heat transfer methods are required. One such solution is the “Peltier” thermoelectric cooler (TEC) that can “pump” the heat exhausted by the power electronic unit.

A contemporary TEC is a solid-state device that uses the electrical current instead of the refrigerant used by the refrigerators to carry the heat released by thermal sensitive devices to the environment [1–4]. To ensure best yet economic cooling, optimal control techniques are required. Heat transfer can be controlled

1 “Politehnica” University of Bucharest, Faculty of Electrical Engineering, Splaiul Independenţei no. 313, sector 6, Bucharest, 060042, Romania, E-mail: dumitrujeanbogdan@yahoo.com.

2 “Politehnica” University of Bucharest, Faculty of Electrical Engineering, Splaiul Independenţei no. 313, sector 6, Bucharest, 060042, Romania, E-mail: amm@iem.pub.ro.

“Gheorghe Mihoc – Caius Iacob” Institute of Statistical Mathematics and Applied Mathematics, Romanian Academy, Calea 13 Septembrie no. 13, 050711, Bucharest, Romania.

3 “Politehnica” University of Bucharest, Faculty of Electrical Engineering, Splaiul Independenţei no. 313, sector 6, Bucharest, 060042, Romania, E-mail: mihaela@iem.pub.ro.

Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 58, 2, p. 205–214, Bucarest, 2013

206 Jean Bogdan Dumitru, Alexandru Mihail Morega, Mihaela Morega 2

either by adjusting the electrical current through the TEC or by controlling the heat transfer removal rate at the hot end of the Peltier cell [5].

This paper is concerned with the numerical study of the heat transfer management provided by a TEC unit with respect to several key parameters: the heat released by the electronic module, the ambient temperature, the convection heat transfer coefficient of the sink-and-fan ensemble, with the aim of finding optimal working conditions for the electronic modules.

2. THE TEC USED FOR THERMAL CONTROL

A TEC is essentially a Carnot device, and it provides for several important advantages: solid state construction, chip integrating possibility, silent operation, compact dimensions, lightweight, higher reliability, accurate temperature stability (better than +/–0.1 °C), localized cooling, short response times to temperature change, reduced power consumption, cooling below the ambient temperature, etc. [1–4].

A single-stage TEC module is composed of thermoelectric cells made of n and p-type semiconductor columns, which are sandwiched and fixed by soldering between two ceramic plates (the hot and cold ends). The couples are connected electrically in series and thermally in parallel. Multistage TEC units are also available. They “split” the hot end – cold end temperature interval in several stages, providing for higher heat pumping efficiencies.

Fig. 1 – CAD representation of a TEC module and its working principle.

When a voltage is applied at the electric ports of the TEC a DC current flows throughout the unit, which (by Peltier effect) conveys the heat released by the

3 Heat transfer management in a thermoelectric sink-and-fan system 207

device connected (thermally) to its cold end to the other (hot) end that is usually provided with a heat sink-and-fan device (Fig. 1) [5]. In this situation, in addition to the Joule-Lenz (electro-thermal) effect that is related to the electrical current flow through the device, Seebeck effect is a menace also because it acts into reducing the efficiency of the heat removal by a counter electromotive force, opposite to the voltage applied to the unit.

Therefore an optimal control of heat transfer should consider the complex interactions that occur in a TEC and mathematical modeling and numerical simulation may be needed in the design phase of the unit.

3. THE MATHEMATICAL AND NUMERICAL MODELS

The mathematical models that describes the heat and current flow through the TEC under steady state working conditions is [5, 8, 9]

∇ ⋅ J = 0, ∇ ⋅q = 0, (1)

where J [A/m2] is the electrical current density and q [W/m2] is the heat flux density

SeebeckOhm

TV ∇σε−∇σ−=J , (2)

PeltierFourierJoule

JJq TTV ε+∇λ−= . (3)

Here V [V] is the electrical potential, T [K] is the temperature, ε [V⋅K−1] is the Seebeck coefficient, σ [S/m] is the electrical conductivity, and λ [W⋅K−1m−1] is the thermal conductivity [2, 3, 10]. All physical properties of the thermoelectric elements are temperature dependent [5, 8–11].

The mathematical model (1)–(3) was solved numerically by implementing a general partial differential equation (PDE) mathematical model [5, 8, 9, 12]

( ) Ω∂=γ−α+∇⋅∇− in ,0uuc (4)

n ⋅ c∇u+αu− γ( )+ qu = g − hTµ, on ∂Ωhu = r, on ∂Ω

, (5)

where Ω denotes the computational domain, ∂Ω is the boundary, u is the dependent variable (T for the heat transfer problem, and V for the electrical problem), n is the outward pointing normal, µ is a Lagrange multiplier, c, α, γ, q, h, r, and g are defined such as to model the two PDEs (1).

208 Jean Bogdan Dumitru, Alexandru Mihail Morega, Mihaela Morega 4

The boundary conditions (BCs) that close the problem are set as following (Fig. 2): for the heat transfer problem, either heat flux or temperature at the cold end (the heat source), convection heat transfer on the heat sink surface, and thermal insulation for all other boundaries; for the electric field problem, either normal current density or ground (zero voltage) at the terminals and electrical insulation for all other sides.

4. NUMERICAL RESULTS AND DISCUSSIONS

A series of parametric studies were conducted on an elemental cell fitted with a plate-fin, heat sink – fan unit to study the sensitivity of the heat pumping efficiency. Fig. 2 presents the computational domain for the elemental TEC and the temperature distribution for 1 A pumping current, and α = 100 W/m2K (sink-and-fan) convection heat transfer coefficient.

Fig. 2 – The elemental TEC and plate-fin, sink-and-fan ensemble.

Temperature field (gray map) in K and heat flux (arrows).

It is assumed that the heat flow between adjacent elemental cells is negligible small, therefore the heat transfer problem can be studied at the elemental cell level.

5 Heat transfer management in a thermoelectric sink-and-fan system 209

Symmetry BCs (with respect to the coolant flow) are used to further reduce by half the computational domain.

Figure 3 presents the average temperature at the cold end versus electrical current for α = 20…180 W/m2K. The total power per elemental cell is 0.078 W and the ambient temperature is set at 293 K.

Apparently, a heat transfer coefficient less than 20 W/m2K, which means in fact natural convection, does not provide enough cooling to keep the heat source (the electronic unit) at a safe temperature.

On the other hand, increasing α beyond 100 W/m2K may not be an option since it provides for an insignificant reduction in the average temperature, for a certain electrical current. We found that an optimal control interval for the convective heat transfer coefficient is then 20…100 W/m2K.

Fig. 3 – Average temperature at the cold end, Tav,cold, versus the electrical current

for different convection cooling conditions.

Figure 4 shows the sensitivity of the average temperature at the cold end of the TEC, Tav,cold, with the electrical current.

210 Jean Bogdan Dumitru, Alexandru Mihail Morega, Mihaela Morega 6

Fig. 4 – The average temperature at the cold end, Tav,cold, for a TEC with 127 elements.

The total heat power per elemental TEC is 0.078 W and ambient temperature is set to 293 K. Qualitatively three different heat transfer regions are observable.

For a specific current, a steep reduction of Tav,cold for increasing α is initially (α < ~50 W/m2K) noticeable. However this region is to be avoided since the too high temperatures suggest that the electronic device would be damaged.

Then, for higher α (> 80 W/m2K) Tav,cold reaches a plateau where the heat pumping efficiency of the TEC is significantly reduced. For these two (asymptotic) regions linear control may be conveniently implemented to manage the heat exhaust. These regions merge through a nonlinear, knee-like section, for α = 40…80 W/m2K. It is more likely to use the TEC in the moderate convection heat transfer regime of the plate-fin, sink-and-fan system.

Comparing the results presented in Figs. 3, 4, it may be conjectured that the adjustment of the current and of the sink-and-fan convection performance (i.e., the mass flow rate of the coolant (air), or the fan speed) is advisable for optimal control.

A third parametric study, summarized in Fig. 5, was conducted to outline the influence of the heat source power, Q [W], upon the temperature at the cold end of the TEC for different values of the electrical current. The convection heat transfer coefficient is 100 W/m2K, and the ambient temperature is assumed 293 K.

7 Heat transfer management in a thermoelectric sink-and-fan system 211

Fig. 5 – Average temperature at the cold end, Tav,cold, vs. electrical current.

The heat influx at the cold end is a parameter.

The coefficient of performance (COP), defined as ratio between the heat exhausted at the hot end through the electrical power of the TEC, for different hot end – cold end temperature differences was studied too. Fig. 6 shows the COP as a function of the electrical current. It may be noticed that the TEC reaches a unique COP(I) asymptotic behavior for electrical currents exceeding 1 A.

These numerical simulation results are consistent with the available datasheet for the TEC device [6], prompting that such an analysis may evidence optimal working conditions for the TEC and can stand as base point for development an optimal control technique. Apparently, for each specific ∆Th-c the optimal COP is associated with a certain value of the electrical current. It is then for the heat transfer designer to select appropriate working conditions (electrical current, exhausted power) for a maximum COP.

Finally, Fig. 7 shows the maximum, reachable COPs for the TEC pin-fin, sink-and-fan TEC. Comparing it with the similar curve for the pin-fin, sink-and-fan TEC it may be inferred that the plate-fin, sink-and-fan TEC provides for superior heat transfer management [13]. It is still to ascertain the validity of this conclusion when the assumed convection heat transfer coefficient is replaced by numerical simulation results obtained for the fully, combined, fluid flow and heat transfer problem.

212 Jean Bogdan Dumitru, Alexandru Mihail Morega, Mihaela Morega 8

Fig. 6 – The COP of the TEC. The hot to cold end temperature drop, ∆Th-c, is a parameter.

Fig. 7 – The COP of the TEC. The hot to cold end temperature drop, ∆Th-c, is a parameter.

9 Heat transfer management in a thermoelectric sink-and-fan system 213

5. CONCLUSIONS

Modern power electronics require efficient means to exhaust the heat produced by operating at ever increasing heat transfer rates, to provide for safe thermal operating conditions. To this end, solid-state semiconductor devices may be added to the sink-and-fan systems providing for precise, controllable thermal management. This paper presents a mathematical model and numerical simulations for the thermoelectric processes that occur in a TEC under steady state working conditions, with the aim to analyze the heat transfer management provided by a TEC provided with a sink-and-fan system.

The sensitivity of the heat pumping efficiency was evaluated for different control parameters: the electrical current and the convective heat transfer of the sink-and-fan system (e.g., by adjusting the mass flow rate of the coolant – air – through the fan speed).

The COP shows off an extremum (maximum) that is sensitive to the cold to hot end temperature drop, and a unique asymptotic behavior for electrical currents exceeding 1 A. These results are consistent with the available datasheet for the TEC device. The hot end temperature vs. the electrical current characteristic is linear while the hot end temperature vs. h characteristic is nonlinear.

For a certain electrical current, the increase of the convective heat transfer coefficient α above 100 W/m2K results in an insignificant reduction of the average hot and temperature.

The plate-fin, sink-and-fan TEC provides for superior heat transfer management (higher COP for same ∆Th-c) than the pin-fin, sink-and-fan TEC.

It may be conjectured that the study may evidence the optimal working conditions for the TEC and can stand as base point for development an optimal control technique.

ACKNOWLEDGMENTS

The work was conducted in the Laboratory for Multiphysics Modeling at UPB. The first author acknowledges the support offered by the Sectorial Operational Program Human Resources Development 2007–2013 of the Romanian Ministry of Labour, Family and Social Protection through the Financial Agreement POSDRU/107/1.5/S/76903.

Received on January 8, 2013

REFERENCES

1. G. Jeffrey Snyder, M. Soto, R. Alley, D. Koester, B. Conner, Hot Spot Cooling using Embedded Thermoelectric Coolers, Extreme Thermal Solutions, Research Triangle Park, NC 27709, The 22nd IEEE SEMI-THERM Symposium, 2006.

214 Jean Bogdan Dumitru, Alexandru Mihail Morega, Mihaela Morega 10

2. I. Sauciuc, H. Erturk, G. Chrysler, V. Bala, R. Mahajan, Thermal devices integrated with thermoelectric modules with applications to CPU cooling, Proc. InterPACK 2005, 73243, 2005.

3. K. Fukutani, A. Shakouri, Optimization of bulk thermoelectric modules for chip cooling applications, Proc. InterPACK 2005, 73410, 2005.

4. J. Bierschenk, D. Johnson, Extending the limits of air-cooling with thermoelectrically enhanced heat sinks, The 9th Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems (IEEE Cat. No. 04CH37543), 1 (2004) pp. 679–684.

5. J.B. Dumitru, A.M. Morega, D.G. Chamorro, M. Morega, Numerical Simulation of Thermoelectric Heat Transfer Management, The 11th International Conference on Applied and Theoretical Electricity, University of Craiova, 2012.

6. * * *, Thermal Solutions, http://thermal.ferrotec.com/thermal-home/. 7. C.J.M. Lasance, R.E. Simons, Advances in high-performance cooling for electronics, Electronics

Cooling, Nov. 1, 2005. 8. A.M. Morega, M. Morega, A FEM model for thermoelectric and thermomagnetic effects, Rev.

Roum. Sci. Techn. – Électrotechn. et Énerg., 48, 2–3, pp. 187–197, 2003. 9. A.M. Morega, M. Morega, M.A. Panait, Structural optimization of a thermoelectric generator by

numerical simulation, Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 55, 1, pp. 3–12, 2010.

10. M. Jaegle, Multiphysics simulation of thermoelectric systems – modeling of Peltier-cooling and thermoelectric generation, Proc. of the COMSOL Conference, Hannover, 2008.

11. * * *, A. Bejan, Heat Transfer, New York, Wiley, 1993. 12. Comsol A.B., v.3.5a (2010), v.4.2a (2012), Sweden. 13. J.B. Dumitru, A.M. Morega, M. Morega, Heat transfer management in a pin-fin, sink-and-fan

thermoelectric system, Simpozionul Naţional de Electrotehnică Teoretică SNET '12; http://snet.elth.pub.ro/snet2012/.