91Ó°ÊÓ

A thermocouple is a temperature-measurement device that consists of two dissimilar metal wires joined at one end. An oversimplified diagram follows. A voltage generated at the metal junction is read on a potentiometer or millivoltmeter. When certain metals are used, the voltage varies linearly with the temperature at the junction of the two metals: $$V(\mathrm{mV})=a T\left(^{\circ} \mathrm{C}\right)+b$$ An iron-constantan thermocouple (constantan is an alloy of copper and nickel) is calibrated by inserting its junction in boiling water and measuring a voltage \(V=5.27 \mathrm{mV}\), and then inserting the junction in silver chloride at its melting point and measuring \(V=24.88 \mathrm{mV}\). (a) Derivethelinear equation for \(V(\mathrm{mV})\) in terms of \(T\left(^{\circ} \mathrm{C}\right)\). Then convert it to an equation for \(T\) in terms of \(V\). (b) If the thermocouple is mounted in a chemical reactor and the voltage is observed to go from 10.0 mV to \(13.6 \mathrm{mV}\) in \(20 \mathrm{s}\), what is the average value of the rate of change of temperature, the the the terme the are a renter \(d T / d t,\) during the measurement period? (c) State the principal benefits and disadvantages of thermocouples.

Step by step solution

01

Determine the Linear Equation

By comparing the given formula \(V = aT + b\) and \(y = mx + c\), we can see that 'a' is the slope of the line that represents the change in voltage per unit temperature, and 'b' is the y-intercept, which represents the voltage at 0 degree Celsius. To determine 'a' and 'b', use the measurements provided in the question. Set up two equations using the two sets of data points, \(V = 5.27 mV\) at \(100 degrees Celsius\), the boiling point of water, and \(V = 24.88 mV\) at \(960 degrees Celsius\), the melting point of silver chloride. This creates two equations: \(5.27 = 100a + b\) and \(24.88 = 960a + b\). Subtract the first equation from the second to eliminate 'b' and find 'a'. Then substitute 'a' into the first equation to find 'b'.

02

Convert Existing Equation

To convert the existing equation in terms of temperature \(T\), we need to solve the equation for \(T\), which gives \(T = (V - b) / a\).

03

Determine Rate of Change of Temperature

Given that the voltage changes from 10.0 mV to 13.6 mV in 20 seconds, we need to find the change in temperature corresponding to this voltage change, we can substitute these values into the new equation found in step 2. The rate of change of temperature is the change in temperature divided by the change in time, i.e, \(dT/dt = ΔT/Δt\).

04

Discuss the Advantages and Disadvantages of Thermocouples

Since the question doesn't provide a list of possible advantages or disadvantages, a general response can be given. For the advantages, these could include: ease of use, wide temperature range, self-powered, and low cost. For the disadvantages, these could include: need for reference temperatures to calibrate properly, the junction voltage can be affected by parasitic thermoelectric voltages in the measurement circuits, and thermocouples can have lower accuracy compared to other types of temperature sensors.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calibration of Thermocouples

Calibration of thermocouples is an essential process to ensure accurate temperature measurements. Since thermocouples are made from dissimilar metals, they generate a voltage when heated, which correlates to the temperature at the junction point. However, for precise readings, we must establish a known relationship between the voltage generated and the actual temperature. This is where calibration comes into play.

For example, to calibrate an iron-constantan thermocouple, we would immerse the junction in environments with known temperatures, such as boiling water (100 degrees Celsius) and the melting point of silver chloride (960 degrees Celsius), and record the corresponding voltages. Using this data, we create a voltage-temperature equation specific to the thermocouple being used. This process leads to highly accurate temperature readings, which are crucial in applications where precise temperature control is vital.

Linear Voltage-Temperature Relationship

A key characteristic of many thermocouples is the linear relationship between the voltage they produce and the measured temperature, often expressed as \(V = aT + b\), with 'V' representing the voltage, 'T' representing temperature, 'a' as the slope, and 'b' as the y-intercept. This equation is reminiscent of the linear equation \(y = mx + c\), where the constants \(a\) and \(b\) must be found through calibration.

In practice, we establish this relationship by measuring voltage at two known temperatures to solve for the constants, 'a' and 'b', effectively mapping out the linear path that voltage follows as temperature increases or decreases. Once this relationship is established, we can easily convert between voltage readings and temperature, facilitating a simple yet powerful way to gauge temperature changes in various settings.

Rate of Change of Temperature

Understanding the rate of change of temperature, denoted as \( \frac{dT}{dt} \), is important in dynamic systems where temperature varies over time. It indicates how quickly or slowly the temperature is changing, which can be crucial for processes that require precise temperature control. To determine this rate, we use the linear equation derived from thermocouple calibration and apply it to the voltage measured over a specific period.

For instance, if we observe a voltage change from 10.0 mV to 13.6 mV over a span of 20 seconds, we first use our calibrated equation to convert each voltage to a corresponding temperature. The difference between these two temperatures, divided by the time interval (20 seconds), gives us the average rate at which temperature changes during that period. Monitoring and controlling this rate are paramount in many industrial processes, including chemical reactions, where proper temperature regulation is key to both safety and efficacy.