Dynamic Response of a First Order system

Introduction

Thermocouples are used in many capacities in a lab setting as they are a simple and effective way to find flow temperatures.  This experiment was conducted to explore how a thermocouple would respond in flow with an oscillating temperature.  While the goal was to incorporate the thermocouple in a 292.5±7.5 K flow that oscillates as 0.5 Hz, this was a simplified experiment used to estimate the response time of the selected thermocouple and determine if it was capable of giving an accurate reading in the desired environment.

Background

The response of a thermocouple is a very good approximation of a first order system where energy is only stored by thermal capacitance, and possesses some thermal resistance.  A first order system is characterized by a static sensitivity K, in this experiment it was considered to be 1, and a time constant τ.  This experiment focused on determining the time constant of a supplied thermocouple.  Physically τ is a determination of how fast the sensor of a thermocouple can reach equilibrium with the source around it. This can be affected by diameter, material, and how exposed the bead is. [1] gives an equation for τ of a thermocouple time constant. Mathematically τ determines the response of the system to change.  In an oscillating system this determines the phase delay:  and attenuation:  , of the sensor to the environment.  In the case of a step or impulse response, τ merely dictates how fast the system returns to equilibrium.  This was used to calculate the response of a thermocouple, by moving the sensor from one heat source to another and knowing at time t=τ the sensors value would be 63.3% of the way to the final value.

Procedure

The experimental set-up is depicted in Figure 1. The test involved two cups of water; one heated to approximately 63°C, and the other left to equilibrate with the ambient temperature of approximately 23°C.  A T-type thermocouple was used to read voltage data into a CPU via a DAQ module, where it was converted to meaningful temperature values and printed out real-time on the display.The procedure for collecting temperature data was as follows: the thermocouple was submerged in the hot water until it reached equilibrium with the water. Equilibrium was determined by watching the temperature readout on the display.  The thermocouple was then taken out of the hot water cup and submerged in the cold water, and again allowed to reach equilibrium with the water. Finally the thermocouple was moved back to the hot water, and the entire process was repeated several times for repeatability. 

Figure 1. Schematic of Experimental Setup

Analysis of Results:

The data was collected at a frequency of 4Hz over a period of 290 seconds.  The maximum temperature of the hot water was recorded as 630C and the minimum temperature of the cold water was recorded as 230C. Figure 2 shows four complete cycles of temperature readings. Some of the initial data are truncated due to abrupt fluctuation. The data recording with the thermocouple was done manually. Therefore the time span of each cycle was different. The temperature distribution seemed very repeatable in all four cycles. Table 1 shows the extreme values of each cycle. It is important to note that the temperature of the hot water was not constant throughout the whole experiment which was recorded in the range of 63-590C.

Figure 3 shows a complete cycle of the temperature measurement. It took approximately 40 sec to complete one cycle. During this time the temperature reached the maximum temperature of 630C within the first 20 sec and then the temperature dropped to 230C as the thermocouple was inserted into the cold water. No abrupt fluctuation was observed in the data. The time constant was calculated for both hot temperature measurement (step response) and cold temperature measurement (impulse response). The calculated time constant was 1.92 sec for the step response. The time constant was calculated based on the time it took to reach the instantaneous temperature value of 63% of the maximum temperature.  It was observed that the time constant of the impulse response varies within 1% of the value of step response. Therefore the time constant was considered to be 1.92 sec for both the step response and impulse response cases. Figure 4 shows a step response temperature data normalized by the time constant.

The measured data was compared with an analytical solution of a first order system. Figure 5 shows the comparison of the step response data with the analytical solution. It was observed that the measured temperature agreed with the analytical solution. There was some deviation observed in the temperature data when the temperature value approached the maximum temperature. Similar trends have been observed for impulse the response case which is shown in Figure 6.

Discussion and Conclusion:

The temperature response of a thermocouple has been observed in this experiment where two different temperature sources (hot water, cold water) have been used. A  T-type thermocouple was used during the measurement with a sampling frequency of 4Hz. The estimated time constant was 1.92 sec. Although the estimated time constant for the impulse response was slightly higher (~1%) than the step response, it was considered to be 1.92 for both cases. The experimental uncertainty of the time constant is less than 1%. The calculated time constant was used to approximate an expected measured value for the expected actual value of the oscillating temperature experiment that the thermocouple was to be used in. The attenuation of the measured temperature was 83.6% of the magnitude of the actual temperature fluctuation. Thus it was determined that the thermocouple would not be acceptable for use in this experiment. It was determined that this thermocouple would only be acceptable in applications oscillating at a frequency of no more than 0.027Hz. Where acceptability is defined as less than 5% attenuation from the actual temperature.A possible source of error in this experiment was the manual change of the thermocouple from the hot water to the cold water. Because it is not possible for a person to move the thermocouple infinitely quickly, it was not possible to record a true step response going from the hot water to the cold water. Although having the cold water at ambient allowed the transfer through ambient air to better approximate an infinitely fast transition, any air currents that may have been present would also have had an effect on the data. The experiment was limited in its accuracy by this transition time.

References:

[1] Wheeler, Anthony J., Ahmad Reza Ganji, Vaidyanadhan Venkata Krishnan, and Brian S. Thurow. Introduction to engineering experimentation. New Jersey: Prentice Hall, 1996.

[2] Tavoularis, Stavros. Measurement in fluid mechanics. Cambridge University Press, 2005.

Cite this article: M.A. Hossain, 'Dynamic Response of a first order system',Department of Mechanical and Aerospace Engineering, The Ohio State University.