System Identification of RTD Dynamics (Tom Co, 11/26/2006; 11/25/2007, 11/15/2009)

Transcription

1 System Identification of RTD Dynamics (Tom Co, 11/26/26; 11/25/27, 11/15/29) Laboratory Objective: To obtain a mathematical model of RTD dynamics General procedures for mathematical modeling of dynamic processes: 1. Propose differential equation a) using physical/chemical laws b) empirical ( curve fitting ) c) mixed 2. Gather data 3. Estimate model parameters 4. Test model and modify if necessary Solution of first order ordinary differential equation: Then the solution is given by: dt τ + T = A τ, A constant dt initial condition T() = T (1) t T ( t) = A ( A T ) exp (2) τ Check: 1. At t =, T ( ) = A ( A T ) exp() = T 2. Substitute equation (2) into equation (1), d τ A dt τ Steady State Temperature t τ ( A T ) exp + A ( A T ) 1 τ t τ ( A T ) exp + A ( A T ) 1. Set time derivatives to zero in equation (1) τ 2. Note that limt ( t) = A = T steady state t + steady state T = A T = A t exp = A τ t exp = A τ A = A 1

2 Time Constant, (τ ): Question: What happens when t = τ? Answer: T ( τ ) will have a value 63.2% between T and A. Proof: Take equation (2), set Thermocouple Dynamics: t = τ, subtract T from both sides, then rearrange : T ( τ ) T T ( τ ) T T ( τ ) = A ( A T ) T ( τ ) T = = A ( A T ) ( A T ) 1 ( A T )( 1 e ) = 1 e 1 e e 1 1 =.632 rate of change rate of rate of flow of flow of of energy in = + energy + energy energy in energy out thermocouple generated transferred dt mc p = + + ha( Tsurrounding T ) dt mc p dt + T = Tsurrounding ha dt T Thus the model is given by a first order differential equation in which the time mc p constant is given by: τ = ha Complications: Additional circuitry and signal filtering will introduce lags. Quick fix: Modify the model by introducing time FOPTD (First Order Plus Time Delay) Model: Let the surrounding temperature be a step function, FOPTD model. 2

3 T surrounding T = T new when t < t when t t Then the response is a first-order response but ed by an amount of τ (seconds or minutes) : T ( t) T = T new ( T T ) new ( t τ ) t + exp τ when when t < t t t + τ + τ time =.5, time constant = 4 Temperature (deg C) T surround T Time (secs) 3

4 Experimental Procedure: Part I. Obtain RTD data 1. Prepare a beaker of warm water. Warm it to around 5-6 o C. 2. Load the program: two_rtd. 3. Dip both RTDs into another beaker until it is at equilibrium with room temperature. 4. At a chosen elapsed time, dip both RTDs into the warm water. Record the value of the elapsed time when the RTDs were dipped. 5. Wait a few minutes until a new steady state is obtained (about 3 to 5 minutes), then click on the [STOP] button. Part II. Obtain a first-order-plus-time- (FOPTD) model 1. Open an Excel spreadsheet file and set-up initial parameters. (see Figure 4). (Note: Except for the time, the other values are initial approximations, they will be modified later by SOLVER.) Figure 4. Intial parameter setup for spreadsheet. 2. Set-up headings for Time, Temperature data and Estimated Temperature columns. Insert data from file generated from Part I: First select the cell for inserting the data. Then select [Data] [Get External Data] [ From Text] menu items (see Figure 5). Next, open the labview data file ( recall that files might have been save with *.lvm extension). 4

5 Figure 5. Importing data. 3. After importing the data, generate the column for temperature estimates based on the following formula (see Figure 6): where, T ( t E T ) = T initial, ss final, ss if < t ( T T ) final, ss t E initial, ss + τ t exp t E Elapsed time T final, ss Final steady state temperature T, initial steady state temperature initial ss E t t time ( when RTDs were dipped) τ time τ Time constant τ τ 4. Calculate a column of squared errors (see Figure 6). 5. Set a cell for the root means squared error (RMS). (see Figure 7) 5

6 Figure 6. Setup columns for estimated temperature and the squared error Figure 7. Calculate the RMS value. 6. Access the Solver window by selecting [Data] [Solver...] menu item to minimize RMS cell by changing the corresponding cells of the parameters (see Figure 8 ), then click [SOLVE] button. 6

7 Figure 8. Minimize RMS cell by adjusting cells for, time constant, initial and final steady temperature. 7. Plot the estimated temperature together with the raw temperature data to check how good the fit is (see Figure 9). Figure 9. Plot the model prediction together with data. 7

8 8. Report the plot together with the dynamic model (with estimated values for τ and τ inserted into the following formula) : τ dt + T = Tsurrounding t dt ( τ ) 8