Lecture 6 of Agricultural instrumentation

Continuous lecture of Agricultural instrumentation

1. AE23001 Anubhab Pal North Eastern Regional Institute of Science and Technology Nirjuli – Arunachal Pradesh

2. Lecture 6 Lecture topics First order transfer functions Analogous first order elements Fluidic Electrical Mechanical

3. Transfer functions for first order element What we have already learnt The transfer function G(s) of an element is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, provided the initial conditions are zero.         First order transfer function block

4. Transfer functions for first order element For our example of temperature sensor, transfer function only relate changes in sensor temperature to the changes in its environment temperature. The overall relationship between changes in sensor output signal O and environment temperature will be steady state sensitivity times the transfer function   Steady state sensitivity

5. Transfer functions for first order element The steady state sensitivity for an ideal sensor is equal to the slope K of ideal straight line. If the temperature sensor is non-linear and subject to small temperature fluctuations, then The derivative being evaluated at the steady-state temperature T(0−) around which the fluctuations are taking place.  

6. Transfer functions for first order element Example: For a copper–constantan thermocouple measuring small fluctuations in temperature around 100 °C, ΔE/ΔT is found by evaluating dE / dT at 100 °C to give ΔE/ΔT = 35 μV °C−1. If the time constant of the thermocouple is 10s the overall dynamic relationship between changes in e.m.f . and fluid temperature is:  

7. Analogous first order elements Fluidic element

8. Analogous first order elements Fluidic element cont. Volume flow rate can be given by,         . ..(1.3) . ..(1.3 ) . ..(1.1) . ..( 1.1) Therefore, . ..(1.2) Therefore, . ..(1.2)

9. Analogous first order elements Fluidic element cont. Again, Q can be written by Now using equation (1.3) and (1.4) we can write, Therefore, . ..(1.6) . ..(1.6 ) . ..(1.4) . ..(1.4 ) . ..(1.5) . ..(1.5 )

10. Analogous first order elements Fluidic element cont. The resulting first order differential equation for the system will be, The time constant for fluidic element can be given by . ..(1.9) . ..(1.9 ) . ..(1.7) . ..(1.7 ) Or, . ..(1.8) Or, . ..(1.8 )

11. Transfer functions for first order element A simple problem, Two overhead water tanks of 1 m diameter each are connected at the bottom with a cylindrical cross section pipe of 1 cm diameter and 10 cm length. There is a valve connected in the pipe to control the flow. One of the tank is full with water level of 1.5 m above the connecting pipe centreline. If the valve is opened, what will be the water level in the second tank after 5 sec,10 sec and 15 sec. Dynamic viscosity of water at 25 C = 0.89 mPa-S

12. Transfer functions for first order element Solution, Algorithm: Step 1: Determine the governing differential equation. Step 2: Determine time constant Step 3: Solve the differential equation at t = 5 sec, 10 sec and 15 sec Step 1: From equation (1.6) we can write, Where, R F is the fluidic resistance = = = 362802.55   . ..(1.6) . ..(1.6 )

13. Transfer functions for first order element Solution, Step 2: Time constant can be given by Step 3: Solve the following differential equation     Solution T, sec h, m 5 10 15 h = 3/2 - (3* exp (-(100t/2903))/2

14. Analogous first order elements Electrical element

15. Analogous first order elements Electrical element cont. Voltage difference across the resistor is, . ..(2.1) . ..(2.1 ) Charge stored = . ..(2.2) Charge stored = . ..(2.2 ) Current = . ..(2.3) Current = . ..(2.3)

16. Analogous first order elements Electrical element cont. Now we can rewrite equation 1 as, The time constant for electrical element can be given by . ..(2.4) . ..(2.4 ) Or, . ..(2.5) Or, . ..(2.5) Or, . ..(2.6) Or, . ..(2.6) . ..(2.7) . ..(2.7)

17. Analogous first order elements Mechanical element

18. Analogous first order elements Mechanical element cont. Displacement of the system, . ..(3.1) . ..(3.1 ) Or, . ..(3.2) Or, . ..(3.2 ) Again, . ..(3.3) Again, . ..(3.3)

19. Analogous first order elements Mechanical element cont. Using equation 3.2 and 3.3, The time constant for mechanical element can be given by . ..(3.4) . ..(3.4 ) Or, . ..(3.5) Or, . ..(3.5) . ..(3.6) . ..(3.6)

20. Analogous first order elements System type Time constant Equivalent resistance Equivalent capacitance Thermal Fluidic Electrical Mechanical System type Time constant Equivalent resistance Equivalent capacitance Thermal Fluidic Electrical Mechanical

21. Thank you End of Lecture 6