Development of FO-ODEs for Selected Thermofluid Systems

1. Development of FO-ODEs for Selected Thermofluid Systems P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Understand the Role of Time in Thermofluids….

2. Three Selected problems fit for FO-ODE • Development of a design model for thermometers. • Development of Travel model for minute particles in a fluid : Principle of Momentum Transfer. • Development of primitive models for Space Vehicles: Principles of Chemical Reaction & Momentum Transfer.

3. Separable First Order Differential Equations • A first-order ODE is separable if it can be written in the form where the function v(y) is independent of x and u(x) is independent of y. Homogeneous first order ODE: Inhomogeneous first order ODE:

4. Basic Ideas for Development of FO-ODE Model • Collection of all possible or known integrals for both left and right hand side. • Collect all known laws of nature for the thermofluid system under study. • Develop suitable assumptions or approximations to fit for a solvable FO-ODE. • Find the solution. • Carry out a parametric analysis.

5. Design of Thermometers in Thermofluids Zeroth Law of Thermodynamics • Every system in this universe spontaneously move towards equilibrium with its surroundings. • If A and C are in thermal equilibrium with B, then A is in thermal equilibrium with C. Maxwell [1872] • Design Rule: • Every thermal instrument must be designed to reach thermal equilibrium with the system as fast as possible. • Independent variable : time t • Dependent variable : Temperature T • Engineering Mathematics: Obtain a suitable ODE for this application.

6. Options for Thermometers Liquid in Glass Thermometer Thermocouple Thermometer

7. Industrial Temperature Sensors

8. Construction of A Thermometer

9. Liquid in Glass Thermometer

10. Final Measurable Output

11. The bulb of Thermometer & Its Surroundings = Instantaneous Temperature of the fluid in bulb

12. Bulb of A Thermometer Tg,o Tg,i Ttf

13. Design Criteria : How Long it takes to achieve Zeroth Law? Conservation of Energy during a time dt Heat in = Change in energy of thermometric substance in the bulb = Instantaneous Temperature of the fluid in bulb

14. Governing ODE for Thermometer Define Time constant

15. Design for Constant System Temperature Define relative temperature of the bulb as Initial condition Equilibrium condition

16. Instantaneous Relative Temperature of Bulb Initial condition

17. Response of A Thermometer bulb in A Constant Temperature System

18. Need for Inhomogeneous First Order ODE Ts(t)=Ct

19. Response of Thermometers: Periodic Loading If the input is a sine-wave, the output response is quite different; but again, it will be found that there is a general solution for all situations of this kind.

20. Important Applications of Particle Movement in A Field • What is the effect of gravity on the movement of a monocyte in blood? • How does sedimentation vary with the size of the sediment particles? • How rapidly do enzyme-coated beads move in a bioreactor? • What electric field is required to move a charged particle in electrophoresis? • What g force is required to centrifuge cells in a given amount of time?

22. Stokes Flow • In a remarkable 1851 scientific paper, Sir G. G. Stokes first derived the basic formula for the drag of a sphere. Derived a simple formula for drag using above velocity field. CD Re • The formula is strictly valid only for Re << 1 but agrees with experiment up to about Re = 1.

23. Important Applications of Stokes Solution in Thermofluids • What electric field is required to move a charged particle in electrophoresis? • What g force is required to centrifuge cells in a given amount of time? • What is the effect of gravity on the movement of a monocyte in blood? • How does sedimentation vary with the size of the sediment particles? • How rapidly do enzyme-coated beads move in a bioreactor?

24. Instantaneous Velocity of A Settling Particle Initial condition: ball stats from rest: Separate the variables:

25. Velocity of SettlingParticle

26. A Travel into Space….. Generation of Cheapest Solutions thru Simple ODEs….

27. Basic Forces Acting on A Rocket • T = Rocket thrust • D = Rocket Dynamic Drag • Vr = Velocity of rocket • mejects = Mass flow rate of ejects • mr= Mass of the rocket

28. MOMENTUM BALANCE FOR A ROCKET Rocket mass X Acceleration = Thrust – Drag -gravity effect Drag

29. Force Balance on A Rocket Conservation of mass:

30. Finite Duration of Flying A rocket is designed for a finite duration of flying, known as time of burnout, tb.

31. Requirements to REACH An ORBIT • For a typical launch vehicle headed to an orbit, aerodynamic drag losses are in the order of 100 to 500 m/sec. • Gravitational losses are larger, generally ranging from 700 to 1200 m/sec depending on the shape of the trajectory to orbit. • By far the largest term is the equation for the space velocity increment. • The lowest altitude where a stable orbit can be maintained, is at an altitude of 185 km.

32. Geostationary orbit • A circular geosynchronous orbit in the plane of the Earth's equator has a radius of approximately 42,164 km from the center of the Earth. • A satellite in such an orbit is at an altitude of approximately 35,786 km above mean sea level. • It maintains the same position relative to the Earth's surface. • If one could see a satellite in geostationary orbit, it would appear to hover at the same point in the sky. • Orbital velocity is 11,066 km/hr= 3.07 km/sec.

33. Series Stage Rocket 3rd Stage Thrusting

34. Clustered Rocket in First Stage

35. Travel Cycle of Modern Spacecrafts