Predicting Temperatures of Exposure Panels: Models and Empirical

Chapter 6

Predicting Temperatures of Exposure Panels: Models and Empirical Data Daryl R. Myers

Downloaded by CORNELL UNIV on August 22, 2016 | http://pubs.acs.org Publication Date: April 15, 1999 | doi: 10.1021/bk-1999-0722.ch006

National Renewal Energy Laboratory, Center for Renewable Energy Resources, 1617 Cole Boulevard, Golden, C O 80401

This paper discusses meteorological and temperature measurements and heat transfer models for predicting panel temperatures as a function of solar irradiance, orientation, ambient temperature, windspeed, and wind direction during outdoor exposure testing. Sources of uncertainty associated with physical measurements of meteorological and temperature data, and the propagation of uncertainty into model results are briefly addressed. Equations for making estimates of panel temperatures, based on fundamental principles of time series analysis, and radiative, conductive, and convective heat transfer from empirical data will be presented. We will also discuss models and data used to predict or simulate panel temperature responses in various meteorological climates.

The thermal properties of materials and coatings and the response of these material to the thermal environment are significant components of the degradation mechanisms for materials. Two models for the rates at which reactions occur as a function of temperature are those of Arrhenius:

R =

(kT)

Ae'

and Eyring: kT

t = —e

where R is the reaction rate, t is a "nominal" lifetime , k is the Boltzman Constant, A and B are characteristic of the material or failure mechanisms, T is the absolute temperature, and E is the activation energy for the reaction. (7) It is this exponential dependence of the rate and lifetime on temperature that make it of critical interest.

© 1999 A m e r i c a n C h e m i c a l Society

Bauer and Martin; Service Life Prediction of Organic Coatings ACS Symposium Series; American Chemical Society: Washington, DC, 1999.

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72 Correlating changes in material properties with environmental stresses induced naturally (through outdoor exposure) or artificially (through indoor, possibly accelerated, testing) with a given level of confidence requires knowledge of the uncertainties associated with measurement data, and good experimental design. Predicting material responses to a variety of conditions requires estimates of the accuracy or uncertainty of the modeled environmental conditions and the material properties under investigation.(2,5)


Measurement Issues Measurements of changes in material properties and temperature data are the basis for studying the effects of thermal energy on material properties. Understanding the measurement processes and their associated uncertainties is needed when analyzing and interpreting the data. The following comments emphasize that the precision of model calculations and results will often far exceed our measurement capabilities. The physical realization of the theoretical concept of temperature, or thermal energy associated with any material has long been a scientific challenge. Today's technological tools of electronic data loggers and sensors, make its easy to assemble a temperature measurement system that can provide either relatively accurate, or highly misleading temperature data. The basic issues associated with the accuracy of temperature measurement include the physical principle involved (changes in state or electrical properties), and sources of error due to installation or instrumentation. For any temperature measurement system the temperature of the sensor., not necessarily the temperature of the device or medium the sensor is attached to is recorded (4). The International Practical Temperature Scale (5) is realized by national standardizing laboratories with an accuracy of approximately 0.0005 Kelvin (K). The scale is founded on extensive sensor characterization and international determinations of scale reference points (freezing, boiling, or melting points of various materials). Secondary temperature standards used in corporate and research metrology (calibration) laboratories can be accurate to 0.002 K . Commercial calibration standards for temperature are accurate to 0.02 K. Commercial sensor elements are quoted with accuracies of 0.2 to 2.0 K over the range of -40 °C to +100 °C. Table I summarizes the level of accuracy to be expected from this range of sensors. Table I. Temperature Sensor Accuracies -40 ° C to +150 °C Sensor Type | Typical Accuracy Standard Platinum Resistance Thermometer 0.001 K (1 milliKelvin) Platinum Resistance Thermometer 0.02 K Thermistor 0.2 K to 3.0 K Thermocouple 0.2 K to 2.0 K Radiometric methods 1.0 K to 10 K The above specifications address only the accuracy of sensors. Errors associated with self-heating by excitation currents, thermal gradients, the quality of the thermal contact of the sensor with the material to be measured, linearity, and the accuracy of electronic data collection equipment are ignored. Detailed uncertainty analysis to estimate the magnitude of such error sources is required to quantitatively evaluate the uncertainty in any measurement system.(23)


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Uncertainties quoted in Table I may increase by up to a factor of two, even in well characterized measurement systems, in the presence of these additional sources of error. Thus the accuracy of real world temperature measurements can estimated as approximately 0.5 °C to 1.0 °C. The World Meteorological Organization (WMO) specifies the accuracy requirements for synoptic temperature observations of ambient temperature at 0.5 °C (6). The combined uncertainty in very good meteorological and device temperature measurements would then be 1.0 °C to 2.0 °C. Uncertainties of this magnitude need to be kept in mind when comparing thermal model computations with measured data. Modem computer software and data analysis products often display an inordinate number of significant digits which cannot be taken at face value. An appreciation of measurement accuracy combined with engineering judgment should be used in evaluating model results. Heat Transfer: Theoretical Considerations The three fundamental means of exchanging thermal energy, or heat transfer, are conductive, convective, and radiative transfer (7,8). The first two involve physical media, namely solids and fluids (either liquids or gases). The third is based on the transmission, absorption, and reflection of energy in the form of photons. We outline the basic principles of each form of energy exchange to further the understanding of the models discussed later. Heat Transfer Mechanisms. When there is a temperature gradient within a solid body, the transfer rate, q , or conduction, of energy through the body is function of the thermal conductivity, k, the area, A, and the temperature gradient normal to the area. This is Fourier's Law, expressed as: q = -kA— dX where the minus sign reflects the second law of thermodynamics requiring heat transfer to be from warmer to cooler regions. When the temperature within a body is changing with time, the rate at which energy is transferred Qk, is a function of the mass, m, the heat capacity, Cp, and the rate of change of temperature, 3T, as a function of position within the material, 9 X , namely dT 'dX Convective heat transfer occurs when a fluid (liquid or gas) receives or provides thermal energy to a system. Newton's Law of Cooling states that the heat, q, transferred between two a body immersed in a fluid at different temperatures, is proportional to the area of contact and the difference in temperature: q=

h.A'(T,-T ) f



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where h is the convective heat transfer coefficient, A is the area, and Ts and Tf are the temperatures of the surface and fluid, respectively. Thermal conductivity of solids varies as a function of temperature, generally as a quadratic in the difference between a reference temperature and the body temperature. For "low" temperatures (within one hundred degrees of ambient), the relationship is generally linear: k = ko (1 + b (T-Tref)) where ko is the thermal conductivity at a reference temperature. In applications around ambient, tabulated values of thermal conductivity are adequate. The transfer of energy through substrates to coatings, or in the reverse direction, requires application of the principles of conductivity. When several layers of material separate two different temperature regimes (in the simplest case, at steady state conditions), superposition holds, and a sequential application of Fourier's law is appropriate. Integration of Fourier's law over the thickness of a material layer results in an expression of the heat as proportional to the difference in temperature (thermal 'potential difference') and inversely proportional to the thermal conductivity ( thermal 'resistance'); namely T -T AT q = -kA— = -kA . x -x Ax 1A

2

X

JA

L

2

x

For layers of materials in 'series', where the heat energy flowing out of one layer enters the adjacent layer the continuity of the heat flow allows summation of thermal 'resistances' analogous to Ohm's law for electrical resistances in series. Li a two-layer sheet of material situated in an environment with temperature T l at side one of layer one, and T3 on side two of layer two, assuming thermal conductivities are k l and k2, and thickness x l and x2, at the boundary between the layers, r -7;

T -T

2

3

Predicting Temperatures of Exposure Panels: Models and Empirical

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