1. Identify Sources of Uncertainty

• Furnace Temperature Uniformity: The temperature may not be exactly the same throughout the furnace chamber. Variations in temperature across the space where the thermocouple is placed can lead to uncertainty in the measured temperature.

• Furnace Temperature Stability: The furnace temperature may fluctuate over time. These short - term and long - term drifts in temperature can affect the accuracy of the calibration.

• Reference Thermometer Uncertainty: If a reference thermometer is used to calibrate the thermocouple, its own accuracy and precision contribute to the overall uncertainty. This includes uncertainties in its calibration, resolution, and repeatability.

• Thermocouple Characteristics: The properties of the thermocouple itself, such as its non - linearity, hysteresis, and aging effects, can introduce uncertainty. Different types of thermocouples have different levels of inherent uncertainty.

• Measurement System Resolution: The resolution of the measurement system used to record the temperature (e.g., the digital display on the furnace controller or the data logger) can limit the accuracy of the measurement. If the resolution is too low, small changes in temperature may not be accurately captured.

• Environmental Factors: Changes in the surrounding environment, such as ambient temperature, humidity, and air drafts, can affect the heat transfer in the furnace and the performance of the thermocouple.

2. Quantify Each Source of Uncertainty

• Furnace Temperature Uniformity: This can be determined by measuring the temperature at different points within the furnace chamber using a set of calibrated thermocouples. The standard deviation of these measurements can be used as a measure of the temperature uniformity uncertainty.

• Furnace Temperature Stability: Monitor the furnace temperature over an extended period and calculate the standard deviation of the temperature fluctuations. This value represents the uncertainty due to temperature stability.

• Reference Thermometer Uncertainty: The manufacturer's specifications for the reference thermometer usually provide information about its calibration uncertainty. This value can be used directly or adjusted based on the actual calibration procedures and frequency. Additionally, the repeatability and resolution of the reference thermometer can be quantified through repeated measurements and used to calculate the associated uncertainties.

• Thermocouple Characteristics: Manufacturers often provide data on the typical non - linearity, hysteresis, and aging effects of thermocouples. These values can be used as a starting point for estimating the uncertainty. Experimental tests can also be conducted to measure these characteristics for the specific thermocouples being used.

• Measurement System Resolution: The uncertainty due to the measurement system resolution is typically taken as half of the resolution of the instrument. For example, if a digital thermometer has a resolution of 0.1°C, the uncertainty due to resolution is ±0.05°C.

• Environmental Factors: The effect of environmental factors can be estimated by conducting experiments in which the environmental conditions are varied and the resulting changes in the thermocouple readings are measured. Statistical analysis can then be used to quantify the uncertainty associated with these environmental effects.

3. Calculate the Combined Uncertainty

• The combined uncertainty is calculated by combining the individual uncertainties from each source using the root - sum - of - squares (RSS) method. If $u_1,u_2,u_3,\cdots ,u_n$ are the uncertainties from different sources, the combined uncertainty $u_c$ is given by the formula:

4. Expand the Uncertainty to a Coverage Interval

• To provide a more comprehensive measure of the uncertainty, the combined uncertainty is often expanded to a coverage interval. This is done by multiplying the combined uncertainty by a coverage factor $k$. The most commonly used coverage factor is $k=2$ for a 95% confidence level. The expanded uncertainty $U$ is then given by:

5. Document and Review the Uncertainty Budget

• The entire uncertainty budget analysis should be carefully documented, including the identification of sources of uncertainty, the methods used to quantify them, and the calculations performed. This documentation is essential for traceability and for demonstrating compliance with relevant standards and regulations. Regular reviews of the uncertainty budget should be conducted to ensure its accuracy and relevance. Changes in the calibration process, equipment, or environmental conditions may require updates to the uncertainty budget analysis.