Callendar Van Dusen
Callendar Van Dusen
In the field of industrial process control, the accuracy of temperature measurement is a cornerstone of operational efficiency and safety. While many sensors provide a rough estimate of thermal conditions, high-precision applications—particularly those involving the monitoring of volatile liquids or complex chemical reactions—require a more sophisticated mathematical approach. The Callendar Van Dusen equation serves as the standard for defining the relationship between the resistance of a Platinum Resistance Thermometer (PRT) and its temperature.
For engineers and technicians working with advanced instrumentation, understanding this equation is essential for calibrating sensors, ensuring system interoperability, and achieving the low-uncertainty measurements required in modern industrial automation. This guide explores the principles, mathematical foundations, and practical applications of the Callendar Van Dusen equation within the context of industrial measurement.
The Fundamental Physics of RTDs
Before examining the specific mathematics of the Callendar Van Dusen equation, it is necessary to understand the hardware it describes: the Resistance Temperature Detector (RTD). RTDs operate on the principle that the electrical resistance of a metal increases as its temperature rises. This phenomenon, known as the temperature coefficient of resistance, is highly predictable in noble metals like platinum.
Platinum is the preferred material for high-precision RTDs (often called Pt100 or Pt1000 sensors) because of its chemical stability, wide temperature range, and high resistance to contamination. However, the relationship between temperature and resistance is not perfectly linear. At a fundamental level, as the metal atoms vibrate more vigorously with increasing heat, they interfere more frequently with the flow of electrons, increasing resistance. While a simple linear approximation might suffice for a narrow range (e.g., $0^\circ \text{C}$ to $100^\circ \text{C}$), broader industrial ranges require a polynomial equation to account for the slight curvature in the resistance-temperature characteristic.
Mathematical Structure of the Callendar Van Dusen Equation
The Callendar Van Dusen equation was developed in the early 20th century. Hugh Longbourne Callendar established the basic quadratic equation for temperatures above $0^\circ \text{C}$, and Milton S. Van Dusen later added a fourth-order term to account for the behavior of platinum at sub-zero temperatures.
The equation is typically expressed in two parts, depending on the temperature range being measured:
For Temperatures Above $0^\circ \text{C}$ ($t \geq 0^\circ \text{C}$):
$$R_t = R_0 [1 + At + Bt^2]$$
For Temperatures Below $0^\circ \text{C}$ ($t < 0^\circ \text{C}$):
$$R_t = R_0 [1 + At + Bt^2 + C(t – 100)t^3]$$
Where:
* $R_t$: The resistance of the sensor at temperature $t$.
* $R_0$: The resistance of the sensor at $0^\circ \text{C}$ (e.g., $100\, \Omega$ for a Pt100 sensor).
* $t$: The temperature in degrees Celsius ($^\circ \text{C}$).
* $A, B, C$: The Callendar Van Dusen constants derived from calibration.
In most industrial applications following the IEC 60751 standard, the constants for a standard platinum sensor with an Alpha ($\alpha$) of 0.003850 are:
* $A = 3.9083 \times 10^{-3} \, ^\circ \text{C}^{-1}$
* $B = -5.775 \times 10^{-7} \, ^\circ \text{C}^{-2}$
* $C = -4.183 \times 10^{-12} \, ^\circ \text{C}^{-4}$
Understanding the Constants: A, B, C, and Alpha
The accuracy of a temperature measurement system depends on how well these constants match the specific sensor in use. While standard coefficients are used for "interchangeable" industrial RTDs, high-precision laboratory or critical process sensors undergo individual calibration to determine their unique Callendar Van Dusen constants.
The Alpha ($\alpha$) Constant
The Alpha constant represents the average change in resistance per degree Celsius between $0^\circ \text{C}$ and $100^\circ \text{C}$. It is calculated as:
$$\alpha = \frac{R_{100} – R_0}{100 \cdot R_0}$$
The most common industrial standard is the DIN/IEC 60751 standard, which specifies an Alpha of 0.003850. Other standards, such as the older American standard, used 0.003920. It is critical to ensure that the transmitter or controller is configured for the correct Alpha value, as a mismatch can lead to significant measurement errors.
The Delta ($\delta$) and Beta ($\beta$) Parameters
Historically, the equation was also written using $\delta$ (Delta) and $\beta$ (Beta) parameters, which describe the deviation from linearity. Delta describes the curvature at positive temperatures, while Beta describes the curvature below $0^\circ \text{C}$. Modern digital transmitters usually allow the direct input of $A, B,$ and $C$ constants, which are mathematically derived from $\alpha, \delta,$ and $\beta$.
Integration with Industrial Level Measurement Systems
While the Callendar Van Dusen equation is a temperature-specific formula, its importance extends to level measurement technologies. Many level sensors rely on temperature data to maintain accuracy. For those interested in the broader scope of industrial instrumentation, the Main Page of specialized manufacturers offers a comprehensive look at how these technologies converge.
Hydrostatic Level Measurement
Hydrostatic level transmitters calculate the height of a liquid based on the pressure exerted by the liquid column. However, the density of a liquid changes with temperature. To provide an accurate level reading in units of length (meters) or volume (liters), the system must compensate for these density changes. High-precision RTDs, characterized using the Callendar Van Dusen equation, provide the thermal data necessary for the PLC or transmitter to perform real-time density correction.
Ultrasonic and Radar Level Meters
In ultrasonic level measurement, the sensor calculates distance based on the time it takes for a sound wave to travel to the surface and back. The speed of sound in air or gas is highly dependent on temperature. A $1^\circ \text{C}$ change in temperature can result in a significant error in level measurement if not compensated. Similarly, while radar (microwave) signals are less affected by temperature than ultrasonic waves, the mechanical components and the dielectric constant of certain vapors can be temperature-sensitive. In these cases, precise temperature monitoring ensures the integrity of the level data.
Selection and Calibration Criteria for Industrial Sensors
When selecting an RTD for an application requiring Callendar Van Dusen characterization, engineers must balance cost with the required level of uncertainty. The following table summarizes common RTD classes as defined by IEC 60751:
| Tolerance Class | Tolerance at $0^\circ \text{C}$ | Temperature Range | Application Suitability |
| :— | :— | :— | :— |
| Class B | $\pm 0.30^\circ \text{C}$ | $-196^\circ \text{C}$ to $+600^\circ \text{C}$ | General industrial monitoring |
| Class A | $\pm 0.15^\circ \text{C}$ | $-100^\circ \text{C}$ to $+450^\circ \text{C}$ | Critical process control |
| 1/3 DIN | $\pm 0.10^\circ \text{C}$ | $0^\circ \text{C}$ to $+150^\circ \text{C}$ | High-precision food/pharma |
| 1/10 DIN | $\pm 0.03^\circ \text{C}$ | $0^\circ \text{C}$ to $+100^\circ \text{C}$ | Calibration and lab reference |
Individual Calibration
For the highest accuracy, a sensor is calibrated at several fixed points (such as the triple point of water or the freezing points of various metals). The resulting resistance-temperature data pairs are used to calculate custom $A, B,$ and $C$ constants for that specific probe. This process eliminates the "interchangeability error" inherent in standard Class A or B sensors.
Installation and Maintenance Best Practices
To maintain the accuracy provided by the Callendar Van Dusen equation, proper physical installation is paramount. Even the most precisely calibrated sensor will yield incorrect data if installed poorly.
1. Wiring Configurations: Use 3-wire or 4-wire configurations to eliminate lead wire resistance. In a 2-wire setup, the resistance of the cables is added to the sensor resistance, creating a positive temperature bias that the CVD equation cannot correct.
2. Immersion Depth: Ensure the sensor is immersed deeply enough into the process to prevent "stem conduction," where heat travels along the sensor sheath to the ambient environment, cooling the sensing element.
3. Thermowell Selection: While thermowells protect the sensor from pressure and corrosion, they introduce a thermal lag. Using a thermal transfer compound can improve response times.
4. Self-Heating: RTDs require an excitation current to measure resistance. If the current is too high, the sensor will heat up internally, leading to an error. Most industrial transmitters limit this current to less than $1\, \text{mA}$.
Limitations and Frequently Asked Questions
What is the difference between Callendar Van Dusen and ITS-90?
The International Temperature Scale of 1990 (ITS-90) is a more complex set of equations used primarily in national metrology labs and for primary standard thermometers. While ITS-90 is more accurate over very wide ranges, the Callendar Van Dusen equation remains the industrial standard due to its relative simplicity and sufficient accuracy for process control.
Can I use CVD constants from one sensor for another?
No. If you are using custom-calibrated constants ($A, B, C$), they are unique to that specific physical probe. If the probe is replaced, the transmitter must be updated with the constants of the new probe. If using standard Class A or B sensors, standard constants can be used across all sensors of that class.
Why is the 'C' constant often zero?
The 'C' constant in the Callendar Van Dusen equation is only used for temperatures below $0^\circ \text{C}$. If your process always operates above the freezing point of water, the $C$ term is omitted, and the equation simplifies to the Callendar quadratic form.
How often should sensors be re-calibrated?
RTDs are known for their stability, but mechanical shock, vibration, and extreme thermal cycling can cause "drift." In critical B2B environments, annual calibration is standard. If the drift exceeds the required tolerance, new Callendar Van Dusen constants should be calculated and programmed into the measurement system.
By leveraging the precision of the Callendar Van Dusen equation, industrial operators can ensure that their temperature measurements—and the level measurements that depend on them—remain accurate, reliable, and compliant with international standards.
