A heat flux sensor typically consists of a thermopile or sometimes just a pair of thermocouples in which the elements are separated by a thin layer of thermal resistance material. Under a temperature gradient, the two thermopile junction layers will be at different temperatures and will therefore register a voltage. The heat flux is proportional to this differential voltage. Note that there must be a temperature gradient; if there is none, both thermocouple junction layers will be at the same temperature and hence register no voltage. The thermal resistance layer is usually made as thin as possible to improve the sensor's response time (see Photo 1). To help ensure a proper thermal gradient, heat flux sensors should be designed to have a high thermal conductivity (see Figure 1).

Figure 1. Heat flux through a thermal resistance layer will create a temperature gradient. The differential temperature between the top and bottom of the resistance layer is proportional to the heat flux through the material. In its simplest configuration, a heat flux sensor requires only two thermocouples on either side of a thermal resistance layer (A). Heat flux sensors typically use a thermopile, an array of thermocouples, to increase the sensitivity of the sensor (B). The thermopile in a heat flux sensor is usually very thin so that heat flow will be perpendicular to the surface rather than through the thermocouple metals.

Measuring the Three Modes of
Heat Transfer
  Radiation. Most heat flux sensors are calibrated using radiation heat sources, which are the most consistently repeatable sources for this purpose. However, the fraction of the radiation

Figure 2. By conservation of energy, all radiation incident on a surface must be either absorbed or reflected. The percent absorbed is dictated by the emissivity of the object for a graybody. Most objects can be reasonably approximated as graybodies.

absorbed by the sensor, emissivity , is never 100%, so the absorbed heat flux differs from the incident heat flux. (The assumption here—and it is usually a good one—is that the object in question is a graybody whose absorptivity and emissivity are equal.) All heat flux sensors can measure only the absorbed heat flux, regardless of its source or the mode of heat transfer. Sensors are typically coated black to improve emissivity so that the absorbed radiation is nearly equal to the incident radiation (see Figure 2). The relation of incident and absorbed heat flux for a radiation source is given by:

(1)

  Conduction. When the heat flux is not from a radiation source, emissivity is not an issue. For a conductive heat flux, where the sensor is in direct contact with a heated material, the governing equation at the material surface is:

(2)

where:

k	= thermal conductivity of the sensor
T/n	= thermal gradient with n as the unit vector perpendicular to the surface through which the heat flux is being measured

Because the incident and absorbed heat flux are the same for a purely conductive heat flux, a heat flux sensor will read the actual incident heat flux. The caveat here is that the sensor must have good thermal contact. If the contact is poor, there will effectively be a high thermal resistance between the sensor and the material of interest, which can seriously alter the sensor reading. This is discussed in more detail below.

  Convection. For convective heat flux, the heat flux equation is:

(3)

where:

h	= heat transfer coefficient of the sensor
T	= temperature difference between the sensor and the fluid

The heat transfer coefficient is a function of the fluid's thermal conductivity and the fluid flow characteristics. Unfortunately, fluid flow is extremely complex and difficult to model; the heat transfer coefficient is therefore usually determined only by measuring the surface heat flux. This procedure assumes that the heat transfer coefficient for the heat flux sensor and the surrounding system are the same, so that the incident and absorbed heat flux are equal. The accuracy of this assumption will vary with different system configurations and materials.

All three modes of heat transfer can be measured as described above. When radiation is mixed with the other modes, however, the question arises as to what fraction of the heat flux must be corrected for emissivity and what fraction need not be. Ideally, the different modes can be isolated by, for example, using a heat flux sensor in a radiometer configuration to view only the radiation sources. If the modes cannot be differentiated experimentally, it becomes necessary to make some intelligent estimates of the relative fractions of the heat flux each mode contributes. In these cases the emissivity of the heat flux sensor should be as high as possible to minimize error. Some sensors are restricted to the mode of heat transfer for which they can be used; a Gardon gauge, for instance, should be used only for radiation detection. Other sensors such as the Vatell HFM or Episensor can measure heat transfer in any mode.

Photo 2. A water-cooled housing allows this Gardon gauge to withstand much higher heat flux levels. Because actively cooled sensors can create a "cold spot" in the system, they are best used in radiation measurements, where a possible cold-spot disturbance will have negligible effect on heat flux measurements.

Heat flux sensors take two basic shapes—a flat, surface-attached, layered wafer or an insert-style cylinder. Because of its greater surface area, the surface-attached configuration is usually more sensitive than cylindrical designs. On the other hand, cylindrical sensors generally can withstand higher operating temperatures and are more easily water cooled (see Photo 2).

The first issue to take into account is the thermal gradient across the sensor. As previously noted, if there is none, no heat flux will be measured. This is especially important in long-duration tests in which a sensor may heat up to a uniform temperature and need to be actively cooled. Because of the thermal gradient requirement, heat flux sensors do not function well if they are not mounted because without some way to dissipate absorbed heat they will quickly come to a uniform or near-uniform temperature. Care must be taken when mounting a heat flux sensor in a substrate such as copper or aluminum, which are characterized by a high thermal conductivity. These materials will have little or no thermal gradient because heat distributes itself quickly. As a general guideline for good heat flux measurements, the sensor's thermal conductivity should be the same as or larger than the material in which it is mounted. In a well-designed sensor, thermal conductivity will be high and response will be rapid. Many heat flux sensors will therefore function well in almost any substrate but it is nevertheless important to be aware of the problems inherent in materials with high thermal conductivity.

The next factor to consider is thermal contact resistance. If a heat flux sensor does not make good thermal contact with the material to be measured, it will cause a local hot spot to form (or a cold spot in the case where the heat flux is negative). This hot spot will alter thermal gradients and change the convective and conductive heat transfer coefficients. For this reason, cylindrical sensors are usually pressed into a substrate or held tightly in place with a mounting nut. Flat, layered sensors are usually mounted with a thermally conductive adhesive to minimize contact resistance. Simply butting a sensor against a surface may still result in a heat flux reading, but the contact resistance will keep the reading from being particularly meaningful. In a similar vein, water-cooling a sensor

Figure 3. By protruding into the stream, a sensor can affect the fluid flow in a convection measurement. The flow is not only altered physically, but is also influenced by the thermal "bump" created by the sensor. The degree of the disturbance depends on the extent of the protrusion and the fluid velocity.

must be done carefully because a temperature mismatch between the sensor and the substrate will occur and skew the measurements.

Fluid flow, whether gas or liquid, must be examined as well. This convection can be forced (e.g., a jet of gas or liquid in a pipe) or natural (e.g., hot air rising). A heat flux sensor can disturb the convection in a system in two ways: physically and thermally (see Figure 3). Physically, even a flush-mounted sensor creates a discontinuity in the surface—the greater the protrusion, the greater the disruption. Thermally, the sensor alters the local temperature gradient due to its spatial protrusion. The impact of the disruption caused by the sensor will depend on the speed of the fluid flow. Disruption is greater for a laminar than for a turbulent flow because of the rapidly changing, chaotic nature of the latter. The system can be considered effectively undisturbed when:

(4)

where:

	= thickness of the sensor
KSENSOR and KSUBSTRATE	= thermal conductivities for sensor and substrate, respectively
R	= radius of the sensor

When dealing with a radiation source, two factors in particular must be considered. The first is the emissivity of the sensor, as discussed earlier. The second is the amount of radiation from the source that actually impinges on the sensor, a quantity that is dependent on the sensor's position relative to the source and other nearby objects. Because heat flux drops with the square of the distance from the source, the sensor must be located properly to ensure accurate measurements. That is to say, if the surface of interest is 10 cm from the radiation source, the sensor should take measurements 10 cm from the source. Additionally, objects in the vicinity of the heat flux sensor can block, reflect, or reradiate heat. If it is close enough to the heat source, the sensor itself can reradiate heat back at the source, raising the temperature of the source and interfering with measurements. If the sensor is too far away, however, its field of view may increase to the point where it includes objects other than the heat source of interest. Finally, if the radiation source does not emit in a spatially uniform pattern, the sensor's position relative to the source becomes important. For example, an LED does not emit in a spatially uniform pattern, but rather emits more radiation forward than it does toward the sides.

Acknowledgment
The author wishes to thank Dr. Tom Diller of Virginia Tech and Lawrence Langley, president of Vatell Corp., for their assistance with this article.

For Further Reading
Diller, T.E. 1993. "Advances in Heat Flux Measurements," Advances in Heat Transfer, Vol. 23, Academic Press:279-368.

Lartz, D.J. et al. 1994. "Heat Flux Measurement Used for Feedforward Temperature Control," Proc 10th International Heat Transfer Conference, Vol. 2, Brighton, UK.

Schmidt, F.W. et al. 1984. Introduction to Thermal Sciences—Thermodynamics, Fluid Dynamics, Heat Transfer, John Wiley & Sons, 1984.

Wesley, D. A. 1979. "Thin disk on a convectively cooled plate—application to heat flux measurement errors," ASME Journal of Heat Transfer, Vol. 82:341-348.

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