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For heat transfer from the body core to skin, equations are a combination of series convection, conduction, and radiation transfers at various boundaries.

HF = h∆T + k∆T/x [W/m²]

Write down equations for junctions and two-port elements.

For heat flux transducer transformer-element,

HFamb*Tamb = HFsk*Tsk =>

HFamb = (Tsk/Tamb) * HFsk * (1 + D)

D = sensor introduced deviance

HFrad + HFeva + HFcon + HFamb = 0

Trad = Teva = Tcon = Tamb

HFsk + HFbc = 0

Tsk = Tbc

Replace variables until the right sides of the equations are expressed in terms of states and system parameters.

HFbc = (h∆T + k∆T/x + ε(T)· σ ·T^4 – Hfeva)·(Tsk/Tamb)·(1+D)

More complex models can be solved using MATLAB or a variety of simulation tools designed specifically for Bondgraph models.

Measuring device

Heat flux sensor defined

Heat flux is defined as the flow per a unit of area per unit of time. For this paper, we intend to utilize [W/m²] (or Joule/second/meter²).

The majority of heat flux transducers, including the one visualized for this project, are based as its beginning element the thermocouple. A thermocouple consists of a bond between to materials (typically copper and constantan) that behave differently to heat. This bond will form an impedance directly proportional to the temperature.

Numerous thermocouples connected in series create a thermopile, with the increased number of joints improving the signal. The thermopile is then embedded in a filling material which should provide a uniform distribution and a thermal impedance, at which point it may be considered a transducer. As heat flows from one side of the transducer to the, thermocouple joints create a voltage differential representing heat flux.

Figure 4: Heat Flux Transducer:

There are several sources of errors related to heat flux sensors, including dynamic effects, lateral fluxes, distortion/deflection, and additional data acquisition elements.

Dynamic effects errors are introduced when the heat fluxes change at a rate faster than the response time of the sensor. When specifying a sensor, it is therefore preferable to utilize sensors with a low mass. In the practice of a heat flux transducer for measuring heat flow to and from the human body, minimizing the size and mass of the sensor also improves the comfort and therefore utility of the device.

The heat flux of interested in body-ambient measurements is the flux perpendicular to the skin. If a sensor is sensitive to lateral fluxes, i.e. flux in the direction parallel to the surface of the skin that will introduce an error in the measurement. Lateral fluxes are minimized in the construction of sensors by maintaining uniformity in the filling material.

Distortion and deflection of heat flow occurs when the thermal impedance of the sensor varies significantly from the characteristics of the material being measured. Distortion is minimized by limiting the thermal impedance of the filling material, increasing the size of the sensor surface area, or correcting for this error in sensor calibration. Deflection is minimized by adding a guard around the edge of the transducer, composed of the same material as the filling. Heat flow will be irregular in the guard region, but as it is not populated with thermocouples, heat flux measurements are not affected.

Self calibrating sensor

A self calibrating heat flux sensor is achieved by combining the heat flux transducer described above with a film heating element on one side. At regular intervals, the film heater is activated with a known generated heat flux, 50% of which will pass through the heat flux transducer. In the case of non-matching conductivities, a deviation occurs, which is then used to generate a new calibration factor to compensate for sensor errors.

Specification of selected device

With the above knowledge and some assumptions on how one would want to proceed with our theoretical application, one can generate a performance based specification for the design of a heat flux transducer:

Parameter Specification and Justification
Sensitivity 125 µV/(W/m²)
Range -2000 to 2000 W/m²
Response time <30s
Sensitive area 12.5cm² (.00125m²); Equivalent to a 5cm x 2.5cm patch
Thermal conductivity 0.209 W/m·K; equivalent to human epidermis
Guard area 0.5cm distance on each edge
Non stability 5% (maximum change in sensitivity per calendar year under calibration conditions)
Operating temperature 0 to 70ºC
Impedance 5 Ω
Accuracy ±5%
Self Calibration Yes


The methodology proposed for using the above sensor and model in the prevention of heat stress illness incorporates several steps. A heat flux transducer is affixed to the human body for the purpose of measuring heat flux during activity. Measurements are taken and the heat transfer is tracked over a period of time. System equations are utilized to develop an algorithm to recognize decrease of body thermoregulatory response and the onset of heat stress illness, particularly in comparison to the environmental conditions present. Finally, once normal thermoregulatory response has been characterized and monitoring algorithms have been developed, heat transfer is monitored for deviations, which may be accounted for by a decrease in the body thermoregulatory response.

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Source:  OpenStax, Body ambient bondgraph model using heat flux transducer. OpenStax CNX. May 15, 2008 Download for free at http://cnx.org/content/col10530/1.1
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