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Molecular Transport Equations

Molecular Transport Equations. Outline. Molecular Transport Equations Viscosity of Fluids Fluid Flow. Molecular Transport. “Each molecule of a system has a certain quantity of mass, thermal energy, and momentum associated with it.” – Foust

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Molecular Transport Equations

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  1. Molecular Transport Equations

  2. Outline Molecular Transport Equations Viscosity of Fluids Fluid Flow

  3. Molecular Transport “Each molecule of a system has a certain quantity of mass, thermal energy, and momentum associated with it.” – Foust • What happens when a difference in the concentration of these properties occur from one region to another? • How is transport different in a solid, gas, and a liquid?

  4. Molecular Transport We need a simple physical model to describe molecular transport - one that does not take into account the structural differences of the three states.

  5. Molecular Transport A driving force is needed to overcome resistancein order to transport a property. Recall: Ohm’s Law from Physics 72

  6. Molecular Transport

  7. Flux Define: FLUX : amount of property  being transferred per unit time through a cross-sectional area Mathematically, Is the equation dimensionally consistent? What are the units of: ψz?δ? Γ?

  8. Flux Flux in the z-direction: amount of property transferred per unit time per cross-sectional area perpendicular to the z-direction of flow δ: diffusivity, proportionality constant

  9. Flux If the transport process is at steady state, what happens to the flux?

  10. General Property Balance If the transport process is at steady state, what happens to the flux? 0 0

  11. Flux at Steady State At steady-state:

  12. Flux What happens when you have an unsteady-state transport process?

  13. General Property Balance Assume: • Transport occurs in the z-direction only. • Volume element has a unit cross-sectional area. • R = rate of generation of property (concentration per unit time)

  14. General Property Balance Assume: • Transport occurs in the z-direction only. • Volume element has a unit cross-sectional area. • R = rate of generation of property (amount per unit time per unit volume) WHY?

  15. General Property Balance Assume: • Transport occurs in the z-direction only. • Volume element has a unit cross-sectional area. • R = rate of generation of property (amount per unit time per unit volume) WHY?

  16. General Property Balance Assume: • Transport occurs in the z-direction only. • Volume element has a unit cross-sectional area. • R = rate of generation of property (amount per unit time per unit volume) WHY?

  17. General Property Balance

  18. General Property Balance General equation for momentum, energy, and mass conservation (molecular transport mechanism only)

  19. Momentum Transport • Imagine two parallel plates, with area A, separated by a distance Y, with a fluid in between. • Imagine the fluid made up of many layers – like a stack of cards.

  20. Momentum Transport Driving Force – change in velocity

  21. Momentum Transport Flux of x-directed momentum in the y-direction

  22. Momentum Transport

  23. Heat Transport • Imagine two parallel plates, with area A, separated by a distance Y, with a slab of solid in between. • What will happen if it was a fluid instead of a solid slab?

  24. Heat Transport Driving Force – change in temperature

  25. Heat Transport Heat flux in the y-direction

  26. Heat Transport

  27. Mass Transport • Imagine a slab of fused silica, with thickness Y and area A. • Imagine the slab is covered with pure air on both surfaces.

  28. Mass Transport Driving Force – change in concentration

  29. Mass Transport Mass flux in the y-direction

  30. Analogy HEAT MOMENTUM MASS

  31. Assignment • Compute the steady-state momentum flux τyx in lbf/ft2 when the lower plate velocity V is 1 ft/s in the positive x- direction, the plate separation Y is 0.001 ft, and the fluid viscosity µ is 0.7 cp.

  32. Assignment • Compute the steady-state momentum flux τyx in lbf/ft2 when the lower plate velocity V is 1 ft/s in the positive x- direction, the plate separation Y is 0.001 ft, and the fluid viscosity µ is 0.7 cp. ANS: 1.46 x 10-2lbf/ft2

  33. Assignment • A plastic panel of area A = 1 ft2 and thickness Y = 0.252 in. was found to conduct heat at a rate of 3.0 W at steady state with temperatures To = 24.00°C and T1 = 26.00°C imposed on the two main surfaces. What is the thermal conductivity of the plastic in cal/cm-s-K at 25°C?

  34. Assignment • A plastic panel of area A = 1 ft2 and thickness Y = 0.252 in. was found to conduct heat at a rate of 3.0 W at steady state with temperatures To = 24.00°C and T1 = 26.00°C imposed on the two main surfaces. What is the thermal conductivity of the plastic in cal/cm-s-K at 25°C? ANS: 2.47 x 10-4cal/cm-s-K

  35. Assignment • Calculate the steady-state mass flux jAy of helium for the system at 500°C. The partial pressure of helium is 1 atm at y = 0 and zero at the upper surface of the plate. The thickness Y of the Pyrex plate is 10-2 mm, and its density ρ(B)is 2.6 g/cm3. The solubility and diffusivity of helium in pyrex are reported as 0.0084 volumes of gaseous helium per volume of glass, and DAB = 0.2 10-7 cm2/s, respectively.

  36. Assignment ANS: 1.05 x 10-11 g/cm2-s

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