Lab 1 b Time Constant-Lab Manual-2019-Fall PDF

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Download Lab 1 b Time Constant-Lab Manual-2019-Fall PDF

Description

Lab 1 b Time Constant Lab This experiment measures the time response of a first order measurement system. The thermal time constant of stainlesssteel spheres with different diameters and masses exposed to flowing air are measured.

1

Objectives

1.1

Experimental Objectives 1. To determine the thermal time constant of a thermocouple (bare thermocouple wire, and thermocouple mounted in spherical objects) when exposed to air flowing at several velocities. 2. To determine the effect of the volume/mass of the object (in which this thermocouple is mounted) on time constant.

1.2

Learning Objectives 1. To learn how a time constant of a measurement device affects data collection. 2. To learn about the dependence of an instrument system’s response to a step-input and the influence of time constant on the response.

2

Apparatus

2.1

Equipment List 1. 2. 3. 4.

2.2

Thermocouple Rotameter and nozzle connected to building air supply Heating gun Stainless steel spheres

Measurement Tools

2.2.1 Thermocouple A thermocouple is a commonly used temperature measurement device that consists of two dissimilar metals joined together at one end in a junction as shown in Figure 1. If the two ends of the thermocouple are at different temperatures, T1 and T2, a voltage will be measured across the unjointed end at A and B; the underlying physics will be described in greater detail later in this course. The magnitude of voltage produced is a function of the difference in temperature between the two sides of the thermocouple. Through calibration with a reference standard the relationship between temperature difference and voltage has been determined for different types of thermocouple wire materials.

Figure 1: A schematic diagram of a thermocouple.

Time Constant - 1

This lab utilizes a type K thermocouple with a 100x gain op-amp for amplifying the voltage signal to a readable level. This is the most common type of thermocouple in laboratory settings; the thermocouple type describes the type of alloys employed. A schematic diagram of the circuit used for this lab is shown in Figure 2. The nominal uncertainty of type K thermocouples is 2.2°C or 0.75%, whichever is greater.

Figure 2: Schematic diagram of the thermocouple circuit used in this lab.

2.2.2 Rotameter Rotameters, or variable-area flowmeters, measure the volumetric flow rate of a gas or liquid. The rotameter reading is determined by the drag delivered to a solid object as the fluid flows past the float. For gases, this drag depends on temperature, pressure, and gas molecular weight. If measurements are carried out under conditions that differ from those used to calibrate the rotameter, a correction must be carried out. For this experiment, only nominal flowrate values are needed, so a correction will not be made.

3

Principles and Background

3.1

Definitions

3.1.1 Time Constant The time constant is the characteristic time scale required for a first order measurement system to respond to a change in signal.

3.1.2 Lumped Capacitance Model The lumped capacitance model is a common approximation in transient conduction, which may be used whenever heat conduction within an object is much faster than heat transfer across the boundary of the object (Bi < 0.1).

3.1.3 Standard Cubic Feet Per Hour (SCFH) SCFH is the molar flow rate of a gas corrected to "standardized" conditions of temperature and pressure thus representing a fixed number of moles of gas regardless of composition and actual flow conditions. The standard pressure is 14.696 psia and standard temperature is 60°F.

3.2

Possible Solution Techniques

The time constant of a system may either be determined experimentally or theoretically. In this experiment, you will determine the values both ways for a thermal body and compare the two methods.

3.3

Solution Strategy

The method used to determine the time constant depends on the type of response being measured. In all cases, a known signal is input into the system, and the output is measured. From the resulting data, the time constant is determined. For a step-change input to the body: 1. Record the initial and steady state conditions of the system, 2. Create a change in the system using the heat gun, Time Constant - 2

3. Record temperature data at consistent time increments, and 4. Use the data to determine the time constant, using the governing equation.

3.4

Governing Equations

The governing equation for a first-order system with static sensitivity, K = 1 is 𝜏

𝑑𝑋 + 𝑋 = 𝑋𝑠 𝑑𝑡

(1)

where: τ X(t) Xs

is the time constant of the system, is the output of the system, and is the input to the system.

This differential equation can be solved for a step-input as investigated in this lab with Xs=constant, and X(t=0) = Xi yielding: −𝑡 𝑋(𝑡) − 𝑋𝑠 =𝑒 𝜏 (2) 𝑋𝑖 − 𝑋𝑠 Equation 2 is the governing equation for a step input. In this laboratory exercise, X is temperature. Often it is convenient to describe the left half of Equation 2 as the Error Fraction, Γ. This can be thought of as percentage remaining of the transition from the initial to the final value in a step-input process. At time t = 0, the error fraction is 1, or 100%. When the process is complete and X(t) = Xs, the error fraction is 0, or 0%. The error fraction is defined as: 𝑋(𝑡) − 𝑋𝑠 𝛤= (3) 𝑋𝑖 − 𝑋𝑠

3.5

Theory

The time constant or dynamic response of an instrument is an important consideration when choosing instruments. The time constant is a measure of the speed of response of an instrument to changing input conditions. Instruments with short time constants are needed for measuring rapidly fluctuating phenomena, such as the velocity of a turbulent fluid flow. Instruments with long time constants, such as the alcohol-in-glass thermometer, are used primarily for static measurements.

3.5.1 Derivation of Theoretical Time Constants To understand the origin of the time constant, consider a sphere with an embedded sensing element at temperature Ti exposed to airflow moving at velocity V and temperature Ts (Figure 3).

Figure 3: Sphere exposed to airflow

The temperature indicated by Tt, is the instantaneous temperature of the sensing element embedded in the sphere. If the temperature of the airflow, Ts, is different from Tt, heat transfer, q, between the sphere and the airflow can be expressed as (4) 𝑞 = ℎ𝐴(𝑇𝑠 − 𝑇𝑖 ) where: A is the surface area of the sphere and h is the convective heat transfer coefficient due to the motion of the airflow surrounding the object. Because of the heat transfer, the sphere’s temperature will change at a rate 𝑑𝑇𝑡 /𝑑𝑡, and this heat transfer rate is given by Time Constant - 3

where: m c t

𝑑𝑇𝑡 𝑞 = 𝑚𝑐 𝑑𝑡

(5)

is the mass of the sphere, is the specific heat of the sphere material, and is time elapsed from a reference point.

In principle, if the convective heat transfer coefficient at the surface of the object is large, it may be possible that large thermal gradients are present within the object. Consequently, the lumped capacitance model approximation will be invalid for this case. However, since in this experiment we are dealing with a metal sphere with a large thermal conductivity (k coefficient) and airflow with moderate h coefficients (see further), there will not be a large temperature gradient within the object, and the lumped model approximation is valid for our measurement conditions. Combining Equations 4 and 5 yields: 𝑚𝑐 𝑑𝑇𝑡 (6) + 𝑇𝑡 = 𝑇𝑠 [ ] ℎ𝐴 𝑑𝑡 Equation 6 is a first-order differential equation. Instruments that behave according to this type of equation are called first-order systems. The time constant of the system corresponding to Equation 6 is: 𝑚𝑐 (7) 𝜏= ℎ𝐴

3.5.2 Approximation of Time Constant using 36.8% Method It is often necessary to make a rough approximation of the time constant quickly while an experiment is being performed. This can be done rapidly by determining the time it took for the system to complete 63.2% of the transition (i.e., where 36.8% remains) as shown in Figure 4. Consider the governing equation for a step-change (Equation 2). If this equation is solved for X at one time constant, X(τ), 𝑋(𝜏) = 𝑋𝑠 + 𝑒 −1 (𝑋𝑖 − 𝑋𝑠 ) = 𝑋𝑠 + 0.368(𝑋𝑖 − 𝑋𝑠 )#( 8) If Xs and Xi for a first order system undergoing a step-change are known, one time constant of change will have occurred when the system value equals X(τ). The response of the system to a step-input is shown in Figure 2.

Figure 4: Time response of a system to a step input.

Note that at one time constant the system has decayed to 36.8% of its initial potential, or the process is 63.2% complete. In terms of error fraction, one time constant is reached when the error fraction, Γ = 0.368. Time Constant - 4

The step change percentage completed for a first-order dynamic process corresponding to two, three, etc. time constants (i.e., 𝑡 = 𝜏, 2τ, etc.) are indicated in Table 1 in dimensionless form. Table 1. Error Fraction for Time Constants

NUMBER OF TIME CONSTANTS

ERROR FRACTION, Γ(T)

0 1 2 3 4 5

1.000 0.368 0.135 0.050 0.018 0.007

At five time constants, the system is within 1% of its final value. As a result, a rule-of-thumb has developed where a system at 5τ is considered to be at steady state.

3.5.3 Linearization of the Step-Input Governing Equation Often, the time constant for a system needs to be determined from a dataset of X(t) values. Linear regression is preferred as it performs a least-squares fit of the data to minimize error in parameter estimation. Also, it uses all data points in the estimation. However, the governing equation for a step input (Equation 2) is not linear. To linearize this equation, the natural log is taken of both sides to yield Equation 9: 𝑋𝑠 − 𝑋(𝑡) 𝑡 𝑙𝑛 𝐥 𝐧 ( (9) )=− 𝜏 𝑋𝑠 − 𝑋𝑖 If the quantity on the left, the natural log of the error fraction, is plotted as a function of time, t, a straight line will be obtained with a slope equal to -1/τ. The uncertainty of the slope can then be used to find the uncertainty in the time constant, a random uncertainty representing the uncertainty in regression fit.

3.5.4 The Time Constant Compared with Other Response Values The time constant is only one way to define the response time of a first-order system. The rise time is the time which it takes a system to go from 10% to 90% of its final value, or 0.1 ≤ 𝛤 ≤ 0.9. The half-life is the time which it takes a system to reach 50% of its final value, or Γ = 0.5. This occurs at 0.7 time constants.

3.5.5 Theoretical Calculation of the Time Constant of the Stainless-Steel Sphere As discussed earlier, Equation 7 can be used to determine the thermal time constant of a first order system to a step input. While the calculation of the time constant for a bare thermocouple is difficult, reasonable estimates could be made for the sphere. The properties of the spheres required to determine the time constant are given in Table 2. Table 2. Properties of the stainless-steel spheres [1]

Property Diameter (in.) Thermal conductivity (W m-1 K-1) @ 25°C Heat capacity (J kg-1 K-1) Density (kg m-3)

Value 0.375 or 1.0 16 500 8030

Convective heat transfer coefficient depends on the fluid, and the flow parameters. The governing equations are also different for forced and natural convection. Typically, these coefficients are estimated from the Nusselt number. The correlation for a Nusselt number for a sphere in an external flow is [2]: Time Constant - 5

where:

𝑁𝑢𝑠𝑝ℎ𝑒𝑟𝑒 =

ℎ𝐷

= 2 + [0.4𝑅𝑒

12

+

4 2 𝜇 ∞ 1 0.06𝑅𝑒 3 ] 𝑃𝑟0.4 (

(10)

) 𝑘 𝜇𝑠 h: convective heat transfer coefficient of the fluid D: diameter of the sphere k: thermal conductivity of the fluid Re: Reynolds number Pr: Prandtl number 𝜇∞ : dynamic viscosity of the fluid in the bulk flow (in this case at room temperature) 𝜇𝑠 : dynamic viscosity of the fluid at the surface In this particular case, Pr =0.715 (@ 25 ºC). Also, we assume that the dynamic viscosity of the fluid remains constant, so: µ∞/µs=1.

The Reynolds number is:

𝑅𝑒 =

where:

𝜌𝑉𝐷 𝜇

(11)

ρ: density of the fluid V: velocity of the fluid flow D: diameter of the sphere µ: dynamic viscosity of the fluid One can estimate the velocity at the nozzle by dividing the volumetric flow rate by the nozzle area (diameter of the nozzle is 1.25 in). All parameters can be evaluated at room temperature air. The required properties of air are shown in Table 3. Table 3. Properties of air at room temperature

Property -1

Value -1

µ (kg m s )

1.8444x10-5

K (W m-1 K-1) Pr ρ (kg m-3)

0.025969 0.715 1.1845

It should be noted that the accuracy of the correlation expressed in Equation 10 is approximately 30%. Additionally, the correlation assumes that the velocity is determined in the far undisturbed flow field. However, in this experiment, the outlet of the nozzle is in the transitional to turbulent flow regime and it is relatively close to the sphere, thus increasing the uncertainty significantly. Reducing the flow rate is a possibility although it will increase the measurement time significantly! In addition, a wire is attached to the ball that will influence the flow. The overall difference between the theoretical and experimental time constant results could vary by a factor of 2 in the current lab. While the absolute values might differ, trends as a function of flow velocity and volume can be easily confirmed by considering the ratio of h or time constants.

Time Constant - 6

4 4.1

Procedure Note keeping 1. Record the date and name of the experiment in your laboratory notebook. 2. Record your objective in your laboratory notebook. 3. Make a sketch of the apparatus. 4. Make a table of the equipment, including corresponding model number and uncertainties. 5. Write down the procedure you’ll likely follow in this experiment. Do not get too detailed here. Only list the basic steps that will let you determine the time constants. 6. Make sure to record all your results. This may be done by printing and taping graphs of your results in your notebook, but all results from your experiment should be represented.

4.2

Time Constant of a Sphere with Incorporated Thermocouple Subjected to a Gas Flow

In this section, you will collect data to calculate the time constant of a bare thermocouple and two stainless steel spheres with a diameter of 0.375 in. and 1 in., exposed to air flowing at different speeds. You will first find the time constant of the bare thermocouple, and 0.375 in. sphere in moving air at three different volumetric flow rates. The proposed flow rates are 200, 400, and 600 standard cubic feet per hour (SCFH). For the stainless-steel ball with a diameter of 1 in, only perform the measurement at 600 SCFH. (Note: We apologize for the use of English units in the experimental system! Please convert this to metric in completing the analysis) Use the provided LabVIEW program and perform the following steps: 1. Place the object (bare thermocouple or spheres) 1 in. away from the nozzle. 2. Start recording the temperature before you switch on the heat gun. Record the steady state temperature (this temperature is equal to the temperature of the cooling air). 3. Pick an initial temperature where you will begin recording temperature values. It will benefit you to use the largest temperature range you have available but do not exceed in any case a temperature of 120 °C to prevent melting of the connection between the thermocouple wire and stainless-steel sphere. 4. Heat the body up past the initial temperature you picked from which you start the time constant measurement. 5. Adjust the airflow to the desired condition using the rotameter. 6. Record the time-dependent temperature values for the bare thermocouple and spheres at the requested volumetric flowrates. Stop recording data when steady state is reached. 7. After you collect a set of data, quickly use the 36.8% method to approximate the time constant. The time constant should decrease as the convective heat transfer increases and increase with increasing mass. 8. You may need to adjust the sampling rate and integration time of your LabVIEW program for different objects to ensure collecting enough data points. In general, the temperature decay should be recorded for more than 5 time constants. You should have rough approximations of all the time constants. Afterwards you will calculate time constants by the theoretical method and use linear regression to find more exact values. Your data collection is now complete. Time Constant - 7

5

Data Analysis

5.1

Time Constant of a Sphere with Incorporated Thermocouple Subjected to a Gas Flow

Here you will find the time constant of the body subjected to a gas flow. To do this, you will use three different methods: linear regression, the 36.8% method, and theoretical correlation. Note that your analysis should only use the cool-down region and therefore should place the maximum temperature at t = 0 s and remove everything before that point. First, determine the time constants for each flowrate using the 36.8% method. 1. Create a plot of non-dimensional temperature (error fraction) as a function of time for all your cases. Estimate the time constant for each of the cases on the plot by estimating when the response was 63.2% complete and indicating this on the plot. If the data are too cluttered, you might consider separating the plots into two to aide clarity. Plot should begin at the chosen initial temperature. Now, determine the time constants using linear regression. 2. Make a plot of 𝑙𝑛 𝑙𝑛 ((𝑇(𝑡) − 𝑇𝑠 )/(𝑇𝑖 − 𝑇𝑠 )) as a function of time for each measurement individually. Add a linear trend line to each of the data sets, showing the fit equation and R2 value for each. It is important to assess the range of data you will be using to fit the linear trend line. A common approach is the slope between 10 and 90% of the difference between the maximum and the steady state temperature. If the data are too cluttered, you might consider separating the plots into two to aide clarity. 3. Perform a linear regression of the data in Excel for each of the data sets. From it, obtain uncertainty values for the slope. Using those, propagate the error to find the uncertainty in the time constant. Use the results to add confidence intervals to your plot from step 2. Confidence intervals should be dashed lines, and the data points should be symbols of different shape for different each series. If the plot becomes too cluttered, break the data into more parts. Finally, find the time constant using theoretical correlations for the experiments with the spheres. 4. Find the Reynolds number (Re) using Equation 11. 5. Find the Nusselt number (Nu) using the right portion of Equation 10. 6. Find the convective heat transfer coefficient (h) using the left portion of Equation 10. 7. Find the time constant using Equation 7 as well as its uncertainty using the 30% uncertainty in the Nusselt number. Finally, tabulate the time constants found for each data set (including the ones not plotted) using each of the three techniques, along with uncertainties where applicable. Comment on the results. If the results vary across the different techniques, discuss possible reasons. Discuss the quantitative variation of time constant within each technique of estimation, using theory as a guide. Are each of the trends internally consistent? Your analysis...