Table of Contents

Heat Flux Sensors

Temperature control systems, explosion suppression, and measurement of
heat transfer coefficients are just some of the applications
that can benefit from heat flux measurement.

Adam Barnes, Vatell Corp.

Part 1 of this article, a discussion of the theory of heat flux sensors, appeared in the January 1999 issue of Sensors. The equations and figures in Part 2 are numbered consecutively from those in Part 1.

As technology advances, industrial, military, and commercial systems are becoming more dependent on feedback from sensors for intelligent characterization and monitoring. Cutting-edge materials and systems require higher standards of performance that can be achieved only with more precise data. Temperature control systems, explosion suppression, and measurement of heat transfer coefficients are just some of the applications that can benefit from heat flux measurement.

Heat Flux vs. Temperature
Heat flux is often confused with temperature. Although they are related, a heat flux measurement
This 3x scaled-up model of a high-pressure gas turbine blade designed by General Electric incorporates an array of heat flux microsensors along its surface to measure the convective heat flux of the blade in a wind tunnel. The measurements can then be used to determine the heat transfer coefficient of the blade at set points along its curvature.
contains more information than a temperature measurement, and consequently is often more useful. Temperature is an indication of the internal energy of matter, whereas heat flux is the rate of energy transfer per unit area. To some extent, heat flux can be considered an indication of where the temperature is headed. Temperature is dependent on the material present; heat flux measures the energy crossing a boundary and therefore is not restricted by the thermal mass of the system.

For a mechanical analogy, consider moving a heavy tank on a wheeled cart. As the tank is pushed, it gains some velocity. A temperature measurement would be analogous to measuring the velocity of the tank; a heat flux measurement would be analogous to measuring the force acting on the tank. Mathematically, the relationship can be viewed as:



= force
= mass
a = accerelation
v = velocity
t = time

For heat flux, the equation takes the form:



q"  = heat flux
k = thermal conductivity of the material
T = temperature
x = a spatial coordinate

The correspondence between the two equations is easily discerned.

Continuing the tank example, if the cart with the tank begins to roll down an incline and its velocity becomes too great, its considerable momentum makes slowing it down a difficult exercise. If the only feedback you have is the tank's velocity, you could easily lose control of it. But if you are monitoring the force acting on the tank, it is readily apparent at the beginning of the incline that the tank does not need much, if any, additional pushing and therefore can be kept under control before it begins moving too fast.

From this analogy, one obvious use of heat flux sensors is in control systems that rely on temperature measurements in a feedback loop. Independence from the thermal mass of the heat flux sensor can also benefit other applications such as those involving short bursts of energy—explosions, for example. In other cases, such as determining heat transfer coefficients, there is no direct alternative to making a heat flux measurement. In this instance, the heat flux sensor is a uniquely valuable tool.

When a control system uses temperature for feedback, the response time is inherently slow because of the system's thermal mass. Heat flux provides a much more rapid response since it
measures energy transfer instead of the resultant temperature shift, and so is not dependent on the thermal mass. The standard proportional-integral control (PI), while functional, tends to be sluggish. To improve response time, a differential temperature term is frequently added to make a proportional-integral-differential (PID) control.

One typical example is a temperature-controlled PID furnace that maintains the temperature within a preset band by controlling the power to the heating coils. Although the differential temperature term improves response time, the tradeoff is that it also makes PID controllers inherently less stable. For many slowly changing temperature systems, PID control works admirably. Rapid change, however, whether from a quick ramp time or a sudden perturbation to the system, poses a difficult challenge to PID control. This is especially true of big systems because of their large thermal mass, which adds a considerable time delay to the system response.

PID control is further complicated by the fact that the parameters generally need to be optimized for different temperature values because the more rapid heat loss at higher temperatures usually requires faster response times. If the PID parameters are set for a fast response time, the system will tend to be unstable, with large amplitude overshoot in the transient response and wide oscillations in steady state. Even if the control band around the set point is small, say only a degree or two, the temperature can still fluctuate by many degrees. Because of the potential for rapid cooling, this can be the case even when there is simply a large temperature difference between the system being regulated and the ambient temperature. Smoother, more stable control is possible by reducing the effects of the differential term, but then the system becomes slow, unresponsive, and only slightly better than it would be under PI control only.

An elegant alternative to the differential term is the use of a heat flux measurement. In this configuration, the system is set to the desired temperature. Then the controller adjusts to match the input power to the output heat flux so that the energy going into the system is equal to the energy escaping from it. Because the controller is looking at the energy flow in the system, the physical parameters of the system, such as its thermal mass, are circumvented. The benefit is much tighter control. Different temperature set points do not require different parameters in a heat flux controller since the controller is being set to balance heat flux instead of tracking temperature. In a similar fashion, the heat flux sensor can give a better indication of heat input required for an efficient temperature ramp without overshooting the set point temperature.

From a mathematical viewpoint, the PID control works to push the error e(t) to zero:

   e(t) = T(t) - T0


T(t)  = measured temperature input
T0 = set point temperature

The control voltage vc(t) takes the form:



kp, ki, kd  = scale factors for the proportional, integral, and differential terms, respectively

The controller's use of heat flux essentially replaces the differential term with the summation of the heat flux:



khf  = scale factor for the heat flux summation term

By eliminating the differential term, the system becomes inherently more stable. The heat flux term effectively predicts where the temperature is going and allows the controller to make corrections before the temperature actually has a chance to appreciably deviate. This kind of predictive control process, known as feedforward control, provides a much faster response to perturbations in a system and allows for much more robust control of the system than is provided by feedback control (see Figure 4).

click to enlarge this image click to enlarge this image Figure 4. Furnace control systems can be based on feedback (A) or feedback plus feedforward (B). The advantages of the latter are faster response of perturbations and more robust control of the system.

Of course, the use of heat flux sensors for control system instrumentation can be applied to automotive climate control, temperature regulation in spacecraft, and other systems characterized by both active heating and cooling because the technique measures both the magnitude and the direction of heat flow.

Consider, for example, an automotive air conditioning system that needs to be set to a constant temperature. Here, the system would simply be maintained with a negative heat flux instead of the positive flux described in the furnace example. Because the heat flux sensor is not dependent on the thermal mass of the system, it can respond more quickly than a temperature sensor to environmental changes. If the car is being heated primarily by the sun, and the sun goes behind a cloud, a PID temperature-controlled system must wait until the car cools off before it knows to turn down the air conditioning. The car becomes cooler than its set point and will remain so for some time. A heat flux­based system responds instantly to the decrease in incoming heat and turns down the air conditioning before the car has a chance to become too cold, allowing the passengers to enjoy maximum comfort.

Another instance in which heat flux provides more information than temperature is in short-duration, high-energy events such as explosions or ultrashort laser pulses. When a large thermal mass is irradiated with a short, high-energy pulse, the overall rise in temperature will be small, making measurement difficult. One might want to keep the thermal mass associated with the thermocouple small to solve this problem. Unfortunately, doing so would mean dissociating the thermocouple from the thermal mass of the system.

The temperature measurement thus obtained, however, would be indicative not of the system, but rather only of the small mass to which the thermocouple has been attached. Because a heat flux sensor measures the energy flow, however, it can reliably measure the energy in the pulse without concern for the thermal mass and provide a more accurate picture of the system.

Figure 5. Heat flux sensors can be embedded in an airfoil to measure the heat transfer coefficient in laminar, transition, and turbulent flow regions.
Heat flux sensors are also used to determine the heat transfer coefficient when convection is present, such as in aerospace components (see Figure 5). Applications dealing with convection often involve complex fluid mechanics, and hence are difficult to model theoretically. This is particularly true for conditions involving turbulent flow. Measured values are therefore important. Because the heat transfer coefficient is directly dependent on heat flux, the only way to achieve a direct measurement is to use a heat flux sensor. Although indirect methods for determining the heat transfer coefficient do exist, the heat flux sensor offers the advantages of being a simple, direct measurement and it allows transient data to be captured.

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

Adam Barnes is an Electrical Engineer at Vatell Corp., 2001 S. Main St., Blacksburg, VA 24060; 540-951-4004, fax 540-953-3010,

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