Before the cooler or heat sink can be selected, the cooling requirements must be defined. This includes determining the amount of heat to be pumped. Minimizing the heat load allows the cooler to achieve colder temperatures or reduces the power required to reach the defined cooling level. The following describes the techniques used to estimate active and passive heat loads, and applies only to steady state heat loads. If the heat load is of a transient nature or involves more complex factors such as air or fluid flow, we suggest that you call one of our applications engineers for assistance.

Heat load

The heat load may consist of two types; active or passive, or a combination of the two. An active load is the heat dissipated by the device being cooled. It is generally equal to the input power to the device. Passive heat loads are parasitic in nature and may consist of radiation, convection or conduction.

Active Heat Load

The general equation for active heat load power dissipation is:

• Q_active = V²/R = VI = I²R

where:

• Q_active = active heat load (watts)
• V = voltage applied to the device being cooled (volts)
• R = device resistance (ohms)
• I = current through the device (amps)

For example, a typical lead selenide (PbSe) infrared detector is operated at a bias voltage of 50 volts and a resistance of 0.5 megohms. The active load therefore, is .005 watts.

Radiation

When two objects at different temperatures come within proximity of each other, heat is exchanged between them. This occurs through electromagnetic radiation emitted from one object and absorbed by the other. The hot object will experience a net heat loss and the cold object a net heat gain as a result of the temperature difference. This is called thermal radiation.

Radiation heat loads are usually considered insignificant when the system is operated in a gaseous environment since the other passive heat loads are typically much greater in magnitude. Radiation loading is usually significant in systems with small active loads and large temperature differences, especially when operating in a vacuum environment.

The fundamental equation which describes radiation loading is:

• Q_rad = F e s A (T_amb⁴ - T_c⁴)

where:

• Q_rad = radiation heat load (watts)
• F = shape factor (worst case value = 1)
• e = emissivity (worst case value = 1)
• s = Stefan-Boltzman constant (5.667 X 10^-8w/m²K⁴)
• A = area of cooled surface (m²)
• T_amb = Ambient temperature (Kelvin)
• T_c = TEC cold ceramic temperature (Kelvin)

Example Calculation: A Charge Coupled Device is being cooled from an ambient temperature of 27°C (300K) to -50°C (223K). The known parameters are:

The detector surface area (includes 4 edges + top surface) is 8.54 X 10 -4 m2 and has an emissivity of 1. Assume the shape factor = 1

From the equation above:
  Qrad = (1)(1) (5.66X10-8 W/m2K4) (8.54 X 10-4 m2) [(300 K)4 - (223 K)4] = 0.272 W

Convection

When the temperature of a fluid (in this case, a gas) flowing over an object differs from that of the object, heat transfer occurs. The amount of heat transfer varies depending on the fluid flow rate. Convective heat loads on TECs are generally a result of natural (or free) convection. This is the case when gas flow is not artificially induced as with a fan or pump, but rather occurs naturally from the varying density in the gas caused by the temperature difference between the object being cooled and the gas.

The convective loading on a system is a function of the exposed area and the difference in temperature between this area and the surrounding gas. Convective loading is usually most significant in systems operating in a gaseous environment with small active loads or large temperature differences.

The fundamental equation which describes convective loading is:

• Q_conv = h A (T_air - T_c)

where:

• Q_conv = convective heat load (watts)
• h = convective heat transfer coefficient (w/m²C)

(typical value 21.7 for a flat, horizontal plate in air at 1 atm)

• A = exposed surface area (m²)
• T_air = temperature of surrounding air(C)
• T_c = temperature of cold surface (C)

Example Calculation: A square plate is being cooled from 25°C to 5°C. The top and four sides are exposed surfaces. The plate is 0.006 meters thick and each side is 0.1 meters long.

From the Convection equation:

• Q_conv = (21.7 w/m²C (0.0124 m²)(25°C - 5°C)

Q_conv= 5.4 watts It is very important to avoid allowing condensation to form when cooling below the dew point. This problem may be avoided by enclosing the cooling system in a dry gas or a vacuum environment.

Conduction

Conductive heat transfer occurs when energy exchange takes place by direct impact of molecules from a high temperature region to a low temperature region.

Conductive heat loading on a system may occur through lead wires, mounting screws, etc., which form a thermal path from the device being cooled to the heat sink or ambient environment.

The fundamental equation which describes conductive loading is:

Q_cond= k A/L DT

where:

Q_cond = conductive heat load (watts) k = thermal conductivity of the material (w/m C) A = cross-sectional area of the material (m²) L = length of the heat path (m) DT = temperature difference across the heat path(C)

(usually ambient or heat sink temperature minus cold side temperature).

Example Calculation : A TEC is used as a black body reference. A temperature sensor is attached to the cold surface of the TEC. It has two platinum leads which have a diameter of 25mm and are 12 mm long. These leads are attached to pins on the heat sink. The cold plate is at -20°C while the heat sink is at 30°C.

The known parameters are:

1. k = 70.9 w/mC, from Table I
2. DT = [30 - (-20)] = 50°C
3. A = pi d² / 4 = 3.14159 (25 m^-6)² / 4

A = 4.91 X 10 ^-10 m² A(2 wires)= (2)(4.91 X 10 ^-10m²) = 9.82 X 10 ^-10 m²

4. L = 12mm = .012m

From the equation above:

Q_cond = [(70.9 w/mC)(9.82 X 10^-10 m²)] (50°C) / (.012m) Q_cond= 0.0003 watts

Since the conductive load is inversely proportional to the length of the wire, the conductive load can be reduced by using longer wires.

Combine Convection and Conduction

The following equation can be used for estimating heat losses due to convection and conduction of an enclosure.

• Q _passive = (A x DT)/(x/k + 1/h)

where:

• Q = Heat load (watts)
• A = Total external surface area of enclosure (m²)
• x = Thickness of insulation (m)
• k = Thermal conductivity of insulation (w/m C)
• h = Convective heat transfer coefficient (w/m² C)
• DT = Temperature change (C)

Transient

Some designs require a set amount of time to reach the desired temperature. The following equation may be used to estimate the time required:

• t = [(rho) (V) (C_p) (T₁ - T₂)]/Q

where:

• t = Time (seconds)
• rho = Density (g/cm³)
• V = Volume (cm³)
• C_p= Specific heat (J/g C)
• T₁-T₂ = Temperature change (C)

• Q = (Qt_o + Qt_t) / 2 (J/s, J/s = watts)

Qt_o is the initial heat pumping capacity when the temperature difference across the cooler is zero. Qt_t is the heat pumping capacity when the desired temperature difference is reached and heat pumping capacity is decreased. Qt_o and Qt_t are used to obtain average values.

Heat loading may occur through one or more of four modes: active, radiation, convection or conduction. By utilizing these equations you can estimate your heat loads. The resulting information can then be used to select a suitable TEC for your application (see section IV).