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A. Solve the block diagram below using Skogestad’s method for approximating transfer functions (FOPTD). Using Simulink, simulate unit step responses of the original model and the approximate FOPTD model and plot them for purposes of comparison. Estimate values for any unknown terms if applicable. Comment on the accuracy of the approximate model.

- A beaker of water is to be heated using an immersed electrical heater. There is no inflow or outflow.

- At time zero, the heater is turned on, using 110 Watts of power that is converted to heat. The following temperature data are collected:

Temperature, °C |
Time, s |

24 | 0 |

31 | 500 |

37 | 1000 |

43 | 1500 |

If the dynamic model is described by

(1)

estimate the term *MC* using either numerical integration or an analytical solution of Eq. (1).

- After the temperature of the fluid reaches 45 °C, the heater is turned off and the fluid is allowed to cool. The following data are collected during the cool-down period.

Temperature, °C |
Time, s |

45 | 0 |

39 | 5000 |

34 | 15000 |

31 | 20000 |

The model for the cool-down period is

(2)

Using the estimated value of MC from part (a), estimate the parameter group *UA* based on the ambient temperature . Use numerical or analytical solutions to Eq. (2) to match the predicted value of *T* vs. the experimental data.

- Note that in part (a), the heat loss to the surroundings was ignored. Using the estimated value of
*UA*in (b), modify the model in (a) to include heat loss. Then examine the error in the model with and without ambient heat loss and plot it as a function of time (*0≤t≤1500*). What do you conclude about the nature of the model?

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