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White Paper: A solution to compensate for the smallest thermal influences
A solution to compensate for the
smallest thermal influences
Introduction
Nowadays, precision instruments of all types (e.g. metrology instrumentation, precision machining tools, etc.) have
reached remarkably high levels of accuracy. Thermal influences are present in all systems and limit the achieved
precision e.g. by inducing thermal expansion. In order to compensate for these effects, it is important to measure the
thermal influences.
The current approach to control thermal effects is based on multi-parameter models. These models are derived
empirically, and are used to measure and predict thermal influences. The most common parameters in such models
are temperatures at various locations in the system and situational information (e.g. power consumption of motor).
A certain compensation of thermal effects is accomplished with this method. However, in some applications, higher
measurement precision and robustness towards external factors like changing ambient temperature is required to
obtain a satisfactory compensation.
Reaching higher precision levels with the presently applied multi-parameter models is challenging due to three
reasons:
1. Non-linear temperature profile: A large number of temperature sensors are necessary to reconstruct the
temperature profile with sufficient accuracy.
2. Temperature resolution: Standard temperature sensors have a limited temperature resolution, which
limits the measurement accuracy.
3. Heat flow dynamics: The change in incoming or outgoing heat flow is unknown since it is currently not
determined experimentally. An exact statement about whether the system is heating up or cooling down is
therefore difficult.
Extending the system with additional temperature sensors addresses only the first of the issues mentioned here.
Further, the marginal improvement of precision decreases with each added temperature sensor.
This document suggests using “heat flux” as an additional parameter, and shows its benefits with regards to these
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three reasons. Heat flux (in W/m ) can be directly measured with a dedicated heat flux sensor (HFS). Systems that
might benefit from heat flux sensors:
 Dosing systems,
 Positioning systems
 Lithography
 Bonding systems
 Metrology systems
To illustrate the potential benefits, Figure 1 shows an example of the improvements gained by adding a single heat
flux sensor to a high-precision metrology instrument.
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White Paper: A solution to compensate for the smallest thermal influences
Figure 1: Uncertainty in the prediction of mechanical deformations in a high precision metrology instrument. T =
temperature, HFS = Heat Flux Sensor.
Figure 1 shows the measured and predicted mechanical deformations based on a temperature-only model and a
model with one heat flux sensor added. Whereas the method applying only temperature sensors has a rather large
uncertainty interval, adding one heat flux sensor to the system, improves the precision of the determined
deformation value by a factor of 4.
In the following, we explain this finding by addressing the impact of a heat flux measurements on each of the above
mentioned issues. For the sake of illustration we base the explanations on a simplified machine element. However,
the same arguments can be applied to more complex geometries and systems.
The block considered here is of uniform composition, W = 100 mm wide, T = 100 mm thick and L = 300 mm long. It is
connected to a heat source on one end, which could for example represent a spindle in a milling machine. Due to the
heat load, the block is heated with respect to its initial temperature T=20 °C and a thermal expansion is induced
which leads to a displacement of the tool center point (TCP).
By placing a heat flux sensor at the heated end of the beam, the amount of heat exchanged via this surface is known
and available for the analysis of the system’s state.
Reason 1 – Simplified simulation of non-linear temperature profiles
Before reaching steady state, the temperature profile along any structural element is non-linear. This is especially
true for materials with high thermal resistances as is shown in Figure 2 for a mineral cast beam. Using heat flux
information together with geometry and material properties, enables a good estimation of the temperature profile
by applying a neural network simulation. Hence, complex thermal FEM simulations based on temperature-only
information, can be simplified to manageable neural network models.
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White Paper: A solution to compensate for the smallest thermal influences
Figure 2: Left: Momentary temperature profile and thermal expansion in a block of mineral cast induced by a heat flux of 100
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W/m . (Color scale shows temperature distribution along beam; thermal expansion is shown by exaggerated geometric
deformation). Right: Temperature profile along center axis of beam. Due to the large thermal resistance of the material the
temperature gradient is highly non-linear. Hence, thermal expansion (here approx. 12 µm) is mainly concentrated at the hot
end of the beam.
Reason 2 – Improved temperature resolution
In order to determine the thermal expansion of a specific machine element, its mean temperature elevation ΔTmean
above room temperature needs to be known. This temperature increase can be measured directly with temperature
sensors placed along the beam. A typical resolution of a PT100 sensor is ±0.1 K. This temperature resolution can be
insufficient for high-precision compensation.
Knowing how much heat has entered the beam, ΔTmean can be determined from material properties and the
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geometry of the device. Typical heat flux sensors have a resolution of ± 1 W/m . Figure 3 shows a simple example of
a copper block where the heat flux resolution is used to calculate the mean temperature. By solely considering the
HFS resolution as a source of error, a resolution of ± 10 mK for the determined ΔTmean is obtained. Radiation losses
are considered in this example, which lead to an outgoing heat flow Qout.
Material parameter
Copper
ρ [kg/m3]
8700
Cp [J/(kg*K)]
385
α [µm/K]
17
λ [W/(m*K)]
400
Results
ΔTmean [K]
0.38
Resolution ΔTmean of a temperature sensor
± 0.1 K
Resolution ΔTmean of a heat flux sensor*
± 10 mK
* typical heat flux sensor resolution of ± 1 W/m
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Figure 3: Copper block with an incoming heat flux of 50 W/m . ΔTmean is the temperature difference between the initial and
the steady state of the simulation.
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White Paper: A solution to compensate for the smallest thermal influences
Reason 3 – Dynamic measurement
Measuring the heat flux inside a system directly
shows how much heat flows in which direction at
the measurement location. Figure 4 shows an
example based on the simple beam introduced
above. Here, the heat source is a motor, which is
running at different power levels in irregular
intervals. The motor creates heat, which is
partially absorbed by the beam and leads to a
mechanical deformation. There are two methods
to measure this effect. First, a temperature
sensor mounted at the beam surface can
measure the temperature change ΔTmean induced
by the heat flux. Or second, it can be measured
directly using a heat flux sensor. While both
methods give valuable information, the heat flux
methods provides additional insight into the
state of the system. When negative heat flows
occur (i.e. heat is actually flowing back into the
motor cooling), the measured temperature value
ΔTmean might show the same values as during heat
up although the beam is cooling down. This effect
occurs due to a non-linear temperature profile.
Therefore, heat flux provides a better
understanding of dynamic effects and as a
consequence allows for anticipatory
compensation.
Figure 4: Top: Typical operation profile for a spindle motor. It is
running at different speeds and switched on and off for several
time periods. Part of the heat is dissipated into the supporting
beam. Bottom: Heat flux and temperature curves caused by the
operation profile shown in the top graph. Heat flux can even be
negative (i.e. heat is actually flowing back into the motor
cooling). The two encircled points in the temperature curve
show, that although the temperature is the same, the heat flux
in the system can differ greatly.
Conclusion
It has been shown that the application of heat flux as an additional measurement parameter allows to gain valuable
information about a system under thermal influences. While this document shows the advantage of heat flux sensors
by means of simple examples, heat flux sensors provide benefits mainly in more complex systems.
As the integration of heat flux sensors can be challenging, we are happy to assist you.
Revision History
Date
Revision
Changes
27. November 2014
v.1.6
Initial revision
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