# Modeling Thermoelectric Systems

This post is part of a series on thermoelectric modules (TEMs), also known as Peltier devices. The prior page explored the basic function of TEMs and a method of describing their behavior mathematically based on information likely to be found in product literature. This page steps through a process of developing a thermal model for a system using a TEM to create a small beverage cooler/refrigerator, showing how different portions of the physical system are modeled, how product information can be used to derive values for the model components, and how together with the mathematical TEM model developed in the prior page, it is possible to make calculated projections of system performance and behavior.

## Notes on Thermal Modeling

In many respects electricity behaves like a fluid, particularly in that it flows when subject to pressure; we just call the pressure “voltage” when speaking of a flow of electrons. Thermal energy also flows in a fluid-like manner when subject to pressure, though in that context we call it “temperature.” So similar are these behaviors that the symbology and analysis techniques used for electrical phenomena are commonly adapted for use where actual fluid or thermal flows are the item of interest. The meaning of the symbols however, the physical quantities or processes that they are used to represent, is different depending on whether the electrical or thermal context is under consideration. Since thermoelectric devices mix both, there’s abundant opportunity for confusion. Practice and familiarity aside, suggestions for liming this confusion would include maintaining a clear mindfulness of whether one is doing an electrical or thermal calculation, and being scrupulous about carrying units throughout one’s calculations and diagrams.

Another technique for mitigating confusion involves selection of labels for model elements distinct from those conventionally used for electrical work. For example, labeling thermal resistances with the Greek letter theta as a subscript. Indicating the points in a system between which such a resistance is connected using the subscript is also common practice; for example a thermal resistance between the H ot side of a TEM and the A mbient environment might be labeled RθHA.

This resource is aimed at those interested in TEMs specifically; those who are unfamiliar with thermal modeling may find the less-complex example here a more approachable starting point. Also, the distinctions between absolute and relative temperature scales mentioned here are also concepts that are important as background to the following discussion.

Figure 1: Table showing how different physical quantities are represented when using electrical symbology for thermal modeling.

## The TEM Itself

The heat pumping characteristic of a TEM at a given drive current can be modeled as a current source in parallel with a resistance, as shown in figure 3. When the time comes to start plugging numerical values into the model, the value of the current source QMod will be set to the Qmax value for the chosen level of (electrical) drive current, and the resistance value RθMod selected to yield ΔTmax for the same (electrical) drive current when a thermal “current” of Qmax is passed through it. These values are taken directly from the equations for the device operating curves developed in the prior section of this series.

The electrical input power to the TEM is modeled as a current source connected between “ground” in the thermal model and the positive side of the current source/resistor combination. When evaluating the model, it will be assigned a value equal to the actual electrical power being applied to the TEM, which is equal to the product of the current through it and the voltage appearing across it as a result. For a given amount of current flow through a TEM, the resulting voltage depends on the temperature difference (ΔT) between its hot and cold sides. Because ΔT will show up in the expression for a model element that influences ΔT, hand calculations would be rather cumbersome and a circuit simulator is highly recommended for solving the model.

Figure 2: Physical model of CP85438 TEM

Figure 3: TEM thermal model

## Thermal Interfaces

Mechanical interfaces between physical objects inhibit the flow of thermal energy (heat) across the interface, due to imperfect contact between the objects. Use of some type of thermal interface material to improve thermal transfer is standard practice in TEM applications; the selection of such materials is covered in some depth on this page. In this example, we’ll postulate the use of Tpcm583 material and use the thermal resistance figures quoted for a 50PSI application pressure of 0.08°C-cm2/watt. The selected TEM has a surface area of 16cm2 on a side, yielding a projected interface resistance of approximately 0.005°C/W per interface to the TEM on the hot and cold side each. These are shown in the model as RθintH and RθintC respectively in figure 6.

Figure 4: TEM with thermal interface material

Figure 5: Tpcm583 datasheet excerpt highlighting thermal resistance values used.

Figure 6: addition of thermal interface resistances to the model

## Heat Sinks

By itself, the limited surface area of a thermoelectric device is not very efficient at transferring thermal energy to or from the surrounding environment; use of a heat sink to create a larger area for heat transfer to occur will improve system effectiveness greatly. These thermal resistances are added to the thermal model in figure 8 as RθHA­ ( H ot side to A mbient)and RθCR ( C old side to R efrigerated area) respectively. Note that on each side, they’re connected in series with the respective interface resistance.

Wakefield P/N 392-120AB is chosen as a heat sink for this example, and will be used on both hot and cold sides of the system. From the heat sink datasheet, it can be seen that changes in air flow rate can influence the resulting thermal resistance quite substantially. Under natural convection, its thermal resistance is indicated as being 0.5°C/watt, falling to roughly 0.16°C/watt if forced airflow at 100 cubic feet per minute (170 m3/hr) is provided.

As described in greater detail on this page, it’s not a trivial matter to select an air mover to provide a specified amount of airflow; fans have operating curves of their own, and identifying the point at which it will operate in conjunction with a given system usually ends up being a matter of empirical testing at some point. For this example, fan part number AFB1212HHE-TP02 will be considered. It dissipates roughly 4.8 watts of electrical power at nominal input voltage (important for thermal load considerations) and delivers approximately 120CFM at free delivery. Should this particular fan & heat sink combination produce a lesser airflow of around 50CFM, the forced-convection thermal resistance would rise to something closer to 0.22°C/watt.

Take note of the relative scale of the TEM in figure 7 at this point; in comparison to the heat sinks on either side, it’s really quite small. The 392-120AB heat sink was chosen for this example precisely because it’s much larger and more capable than those one would likely see used in practice.

Figure 7: Heat sinks (top transparent). Note their size relative to the TEM.

Figure 8: Addition of heat sinks to the thermal model.

### TEM assembly leakage

Thermal leakage between the hot and cold sides of a TEM through its own bulk is a given, but TEMs are generally not used in isolation; the assemblies into which they are integrated can introduce further losses, and these can be significant since the temperature difference between the bases of any heat sinks used is nearly that found across the TEM itself. In the thermal model of figure 9, this leakage component is represented as RθHC. ( H ot side to C old side)

One possibility for assembly-induced losses is through fasteners used to hold an assembly together. It is a common fixation method in larger assemblies with relatively massive heatsinks because it reduces the likelihood of failures due to mechanical stress. Assuming use of two #6 screws made of stainless steel, spanning the 16mm distance between the points where the fasteners contact the heat sinks in the example assembly, the thermal resistance of the path between the hot and cold sides created by the fasteners works out to roughly 110°C/W, according to the process shown in figure 11. The choice of fastener material can be quite significant as material thermal conductivities can vary widely.

Additionally, heat will tend to flow between the bases of the hot-and cold-side heat sinks. Were this gap left uninsulated, it would occur through some convection process that would be quite difficult to estimate, and likely not result in very good performance. For sake of this example, the 4mm gap will be presumed to be filled with an equal thickness of the same polystyrene foam insulation from which the enclosure walls will be theoretically constructed. Calculations for this leakage component are shown in figures 12-14, and yield an estimated thermal resistance of 10.4°C/W. When combined with the parallel leakage through the fasteners, the estimated value for RθHC works out to 9.5°C/W

Figure 9: Common leakage paths through a thermoelectric assembly.

Figure 10: Addition of TEM assembly leakage to the thermal model.

### Fastener Leakage

For purposes of this example, the fasteners used to hold the assembly together can be modeled as a cylinder having a diameter equal to the minor diameter (diameter less screw threads) of the fastener used. The indicated length is determined by TEM thickness and geometry of the fastened parts

Dimensional information for standardized fasteners and thermal properties of various materials are available from a variety of reference sources, such as Machinery’s Handbook . Stainless steels are among the least thermally conductive of the common structural metals with tabulated thermal conductivity values on the order of 14W/m°C. Together with their corrosion resistance, stainless steels are an attractive choice of fastener material for applications such as this where thermal conductivity is undesirable and presence of moisture due to condensation is likely. Use of plain carbon steel materials would be expected to result in 2-3 times greater thermal leakage, while aluminum and aluminum alloys would produce leakage some 10-15 times greater than stainless steels.

Figure 11: Estimating the thermal resistance of the selected fasteners.

### Insulation Leakage

The insulation value of construction materials is commonly specified in terms of an R-value, which indicates the thermal resistance of one unit area of a given insulation product. Here, as elsewhere, the U.S. customary systems of measure are at odds with those adopted in other regions. In the U.S., R-values are typically quoted in units of hrft2°F/Btu, which yields numbers about 5.7 times larger than the correlating metric units of °C*m2/W. Quoted R-values typically reference the thermal properties of an insulation product as-sold, so a product variant that is twice as thick as another will carry an R-value roughly double that of the thinner product, though the thermal conductivity of the material itself per unit of thickness is (to a first approximation) constant and independent of the final product thickness. In order to calculate a thermal resistance in a specific situation, a quoted R-value must be divided by the area across which heat transfer occurs. If using quoted R-vales from an insulation product to estimate thermal resistance for the same material in a different thickness, as is done here, quoted R-values must also be adjusted accordingly.

Polystyrene foam insulation materials are commonly cited as providing an R-value of about 5 hrft2°F/Btu or 0.88 °Cm2/W per inch (25.4mm) of thickness. A 4 mm thickness would offer roughly a sixth of that figure, and the area enclosed by the heat sink bases in this example, less that covered by the TEM, is about 0.0134m2. The thermal resistance to leakage between the bases of the heat sinks then, can be estimated as about 10.4°C/W through the process shown in figures 12-14.

Note that R-values relate to thermal resistivity, which is essentially thermal conductivity upside down. Whether a larger number is “better” in any given case depends entirely on which way one chooses to flip the numbers, and whether one’s goal is to transmit thermal energy or to prevent it’s transmission. Both can be encountered both ways in context of thermoelectric applications, and it’s another opportunity for confusion to arise; retaining units of measure throughout one’s calculations is quite helpful in avoiding errors from such.

Figure 12: Geometry of insulation between hot-and cold-side heat sinks.

Figure 13: Excerpt from Owens Corning® Foamular®150 datasheet. This material is a rigid polystyrene foam insulation material commonly used in home construction. Values shown are per inch of material thickness.

Figure 14: Estimation of thermal leakage resistance through insulation in the modeled TEM assembly.

### Enclosure leakage

The enclosure around the refrigerated space will be modeled as a cube approximately 12 inches (0.3m) on a side, formed from 1-inch thick polystyrene foam insulation, deducting the area occupied by the TEM apparatus. The thickness of the insulation will also be treated as negligible in terms of determining heat-transfer area. At roughly 1/6th of the enclosure’s overall dimensions in this case, it’s a simplification that’s likely starting to become significant in terms of affecting the accuracy of the final result, compared to a situation where the material thickness is smaller in relation to the enclosure’s overall dimensions. Other error sources in the model being developed also exist however, and it should be appreciated that the results ultimately obtained from analyses such as this are merely informed estimates, rather than a precise calculation. Figure 15 shows dimensions of the enclosure for the refrigerated space, and calculations estimating the thermal resistance through its walls as 1.63°C/W. This is shown added to the model as RθAR in the thermal model of figure 16.

Figure 15: Dimensions of enclosure and estimation of its thermal resistance.

Figure 16: System model with thermal resistance of case added

### Internal Dissipation

Electrical power applied to some device within the refrigerated region, a fan in this example, should be regarded as being converted entirely to thermal form unless it can be accounted for as leaving the refrigerated area through some other means, such as an electrical or optical signal of some kind. Represented as current source QInt on the thermal model, its value is set equal to whatever electrical power is applied to the cold-side fan, taken to be 4.8W in this case per the specifications of the chosen fan. Though perhaps less common, other processes such as chemical reactions that occur within the refrigerated area may also release (or absorb) heat and need to be accounted for.

Note that the fan on the exterior of the assembly is not represented in the model of figure 18. This is because it is situated in the ambient environment into which all heat being extracted from the refrigerated area and applied in electrical form is ultimately being discarded. When designing devices like a small refrigerator, the ambient environment is commonly approximated as an infinite thermal reservoir; a body at some known or assumed temperature to or from which any amount of thermal energy can be transferred without causing a change in that temperature. The ambient environment is shown as voltage source TA in the completed model of figure 20. It should be recognized that approximating the ambient environment in which a device is situated as an infinite thermal reservoir may or may not be a reasonable simplification to make in context of a larger system; were the assembly being modeled placed in an enclosed space such as a small closet, it’s likely that the electrical power being applied would cause the ambient temperature to increase.

Figure 17: TEM assembly mounted to enclosure.

Figure 18: Thermal model with internal power dissipation added.

## Thermal Storage

While not necessary for estimating steady-state performance, accounting for the thermal storage capacity of system components and contents allows insights to be gained about how a system might behave in a dynamic manner. For situations where no change of physical phase is involved (such as freezing of water) an estimate can be made simply by considering the mass of the component and the specific heat value of the material from which it is made; tabulations of such values can be found in material property tables and many thermodynamics textbooks. Thermal reservoirs of this sort can be modeled as capacitors, connected between the node in the thermal model in which they are present and the model’s “ground” chosen in this example as absolute zero temperature. The values of thermal capacitances in such models will commonly be much larger than those of their electrical kin; that’s entirely normal, though perhaps unsettling to folks familiar with electronics, where kilofarad capacitor values are decidedly uncommon.

A standard U.S. sized 12-pack (12 fluid ounce, 355ml) of Favorite Aqueous Beverage will be used for this example, and modeled as pure water. Given their mass the heat sinks merit consideration also, and will be modeled as pure aluminum and assumed to maintain a constant temperature throughout. Such an assumption is of course false, but not greatly more so than ignoring their influence altogether.

Using these assumptions/simplifications, the hot- and cold-side heat sinks, based on their weight as listed in the datasheet, offer a thermal capacitance of about 1.8kJ/°C each, represented as CθHot and CθCold respectively in figure 20. The 12-pack, roughly 18kJ/°C and indicated as CθBev.

Also added to the model at this stage is the ambient temperature of the system’s operating environment, represented as voltage source TA, connected between “ground” and the thermal resistance of the hot-side heat sink. An ideal voltage source is capable of sourcing or sinking any amount of current with no change in voltage across it; the thermal-model analogy is the assumption that any amount of thermal energy can be transferred to or from a system’s surroundings with no resulting change in temperature.

Figure 19: Completed system model with calculations estimating thermal capacitance of heat sinks and cooler contents.

Figure 20: The completed model. Added capacitors represent thermal storage capacity of cooler contents and heat sinks, voltage source represents the temperature of the ambient environment in which the cooler is operating.

## Simulation Results

The thermal model developed above was solved using LTspice, a circuit simulation program downloadable at no cost from Analog Devices Inc. A few different scenarios were examined, including different airflow rates, addition of a 0.5" (12.7cm) thick aluminum spacer between the TEM and the cold-side heat sink, use of different fastener materials, a thermal interface resistance higher than that indicated by the product’s datasheet, and omission of the fan on the cold side of the assembly. The same basic circuit structure shown in figure 22 was used for each, adjusting component values to represent the different situations. A summary of the different model element values used under different scenarios studied is shown in figure 21.

Figure 21: Summary of model component values used in simulation

Figure 22: Circuit model in LTspice.

### Standard assembly with varying airflow and drive current, with and without cold-side spacer

Figure 24 shows the predicted steady-state temperatures for the refrigerated space, for several different scenarios. The horizontal line indicates the ambient temperature of the room in which the device is situated. The solid lines indicate results for the system as developed above, dashed lines with the addition of an aluminum spacer block between the cold side of the TEM and its respective heat sink, as shown in figure 23. These values represent the lowest refrigerated area temperature achievable by the system for the scenario being modeled. The length of time it takes to reach these temperatures after the system is started is a different matter, discussed shortly.

A number of observations can be made from these results that are worthy of note. First and perhaps most importantly, operation at the maximum rated drive current of 8.5A does not yield optimal cooling results under any of the scenarios covered. Beyond about 5.1A, increasing the electrical drive level to the TEM would be expected to produce an increase in the temperature of the refrigerated area, relative to operation at lower drive levels. In the no-fan (natural convection) case, operation at maximum drive levels would be expected to produce no cooling benefit at all, or even a warming of the “refrigerated” area relative to the ambient atmosphere. It should be borne in mind that the heat sinks chosen for this model are quite massive and generously sized, weighing roughly 2kg (5lb.) each.

A second observation that can be made is the benefit of adding the aluminum spacer to the cold side of the assembly; doing so involves lengthening the fasteners and thickening the insulation layer between the hot- and cold-side heat sinks, reducing the thermal leakage that occurs within the assembly itself. That modification improves the achievable cooling benefit by 1-2°C across the board, roughly a 5% performance improvement in each of the scenarios examined.

Figure 23: Cross section of TEM assembly including 0.5" (12.7mm) cold-side spacer

Figure 24: Simulation results for 5 drive current levels with 3 airflow levels, with and without 0.5" cold side spacer.

### Effect of Fastener Material Selection

Simulation results showing projected steady-state temperatures in the refrigerated area are shown in figure 25 for 3 different fastener materials, with and without use of an aluminum cold-side spacer block. Stainless steel, (carbon) steel, and aluminum fasteners are modeled, taking the value of carbon steel’s thermal conductivity as 50W/m°C.

In the scenario under consideration, the difference between use of carbon and stainless steels is not very pronounced; less so than that of using a spacer block. Because fastener leakage isn’t the dominant source of thermal load on the TEM in these scenarios, changing it even by a factor of roughly 3 doesn’t produce a very noticeable effect on achievable cooling benefit. Use of an aluminum fastener however would bring leakage through that path into significance compared to the other thermal loads, costing something on the order of 2-3°C of achievable cooling effect, and comparable to the use of steel fasteners without a spacer block. It should be noted that these results are specific to the scenario being studied; in situations where the refrigerated area is better insulated and fastener leakage is more pronounced in comparison to the other load sources, the influence of material selection will be more pronounced.

Figure 25: Simulation results showing the influence of fastener material selection, with and without use of cold-side spacer.

### Effect of Interface Resistance

Figure 26 shows simulation results comparing the model as developed above (100CFM airflow, SS fasteners, no spacer) if the thermal interface resistance is increased from 0.005°C/W to 0.05°C/W. Such variations can potentially be caused by a variety of factors, including excessive surface roughness or non-flatness of mating surfaces, insufficient clamping pressure, or inappropriate application of the interface material. At the 5.1A drive current level, such a change is projected to cost approximately 3°C worth of net cooling benefit, or about 10% of the total achievable difference between the refrigerated area and the surrounding environment.

Further information on thermal interface material selection and use can be found on this page.

Figure 26: Effect of thermal interface resistance on achievable cooling effect. Air flow at 100CFM, without spacer.

### Effect of Lesser Heat Sinks

Many of the various informal online resources that discuss use of thermoelectric modules make only passing references, if any at all to the thermal resistances of the heat sinks chosen for demonstration. In these cases, the selection of heat sinks seems often to be based on devices having an appealing size, being conveniently available from a drawer of spare parts, or improvised from available materials. In these cases, one might see something such as a 655-53AB epoxied to the cold side of a device, and a 394-1AB glued to the other. Such an assembly would appear something like figure 27, which arguably looks better-proportioned than the assembly in figure 7. What sort of performance might these seemingly more “right-sized” heat sinks offer?

Taking their thermal resistances with forced convection straight from the datasheets as 2°C/W and 0.9°C/W respectively, neglecting any leakage between the hot- and cold-side heat sinks whatsoever (disconnecting RHC in the model) and also neglecting any power applied to a cold-side fan, simulation returns the performance curves shown in figure 28. Even with these generous assumptions, the “refrigerated” area of the model ends up warmer than the surrounding ambient environment at drive currents a bit above 5.1A.

Figure 27: Model of TEM assembly using smaller heat sinks

Experience suggests that many persons approaching the use of TEMs from an informal standpoint are often surprised by such results; this seems to stem from a from an expectation that these devices “make cold.” To understand them instead as creating a temperature difference is perhaps more accurate, particularly in conjunction with the notion of the electrical input causing the hot side to become hot; the colder that this “hot” can be kept relative to the ambient temperature, the better the potential for obtaining a below-ambient temperature on the cold side of the assembly. If a TEM can generate a maximum 50°C temperature difference for a given electrical input, but that electrical input causes the hot side to warm up by 60° over ambient for a given assembly, guess what? The “cold” side of the TEM will end up 10°C warmer than it would have been otherwise. If the hot side is instead held to 20°C above ambient through the use of suitable thermal management, it would then be possible to create a cold-side temperature as much as 30°C below ambient.

Figure 28: Projected steady-state temperatures in the refrigerated area for the modeled system, using heat sink part numbers 655-53AB and 394-1AB.

### Dynamic Behavior

To this point, discussion of simulation results has focused on steady-state temperatures, those obtained once everything has settled down and reached a stable operating point; little has been mentioned so far as to how much time is required to reach such a point. This issue is illustrated in figure . Adding an arbitrary voltage source with a value that toggles between 0 and 1 to the simulation model and including its value in the expressions for the current sources allows (rough) simulation of turning the electrical power to the TEM and fans on & off. The scenario modeled is for airflow at 100CFM using a cold-side spacer, drive current of 5.1A, and assuming a good thermal interface. Voltage values in the simulation circuit represent temperatures on the Kelvin scale; the 273.15° offset to convert to Celsius is incorporated in the relevant plots, allowing them to be read (perhaps) more intuitively in terms of °C.

Ultimately, the temperature in the refrigerated area for this scenario is predicted to end up at about -4°C. Getting most of the way there from a 22°C (295K) starting point requires something on the order of 20,000 seconds, approximately five and a half hours. Though perhaps not totally unserviceable, there are certainly more expedient means for chilling a case of one’s Favorite Aqueous Beverage. Approaching the predicted steady-state temperature closely (within a fractional degree or so) takes a substantially longer period, on the order of about 60,000 seconds, or about 16-17 hours.

This particular model does have a few problems however. For one, it’s pessimistic with how quickly the refrigerated Beverage will resume a tepid and unappealing near-room temperature state when the cooling system is shut off, because the convection coefficients associated with the heat sinks are not changed to reflect the change in their performance in the absence of airflow. Secondly, though it predicts a below-freezing end temperature, it does not account for extraction of the (significant) extra thermal energy required to change water from a liquid to a solid. If one’s Favorite Aqueous Beverage happens to be some form of slushie then, the model under-predicts the time from starting the chill cycle to the attainment of satisfaction.

Finally, the transfer of heat from the case of FAB is assumed here to be associated with a negligible thermal resistance. While this is decidedly not the case, over the time scales of several hours involved it’s likely to be a reasonable simplification for purposes of estimation, at least in the forced-convection cases where the air volume in the refrigerated space is being circulated on the order of once per second. In cases where natural convection is used interior to the box, chill times may be extended significantly.

Figure 29: Transient simulation model and results for forced convection at 100CFM, without spacer.

## Summary

It’s appreciated that some readers may find many of the concepts and techniques presented here inaccessible or difficult to grasp for various reasons. Though taking the time to figure out such tools is highly recommended, two key ideas stand out as holding special importance for anyone desiring to use a TEM for cooling purposes:

1. Keep the hot side as cool as possible, recognizing that doing so might require more drastic measures than one might expect.
2. Best results are commonly achieved using 20-40% of maximum rated electrical input.