Molecular Transport Equations

Outline 1.Molecular Transport Equations 2.Viscosity of Fluids 3.Fluid Flow

Molecular Transport “Each molecule of a system has a certain quantity of mass, thermal energy, and momentum associated with it.” – Foust 1.What happens when a difference in the concentration of these properties occur from one region to another? 2.How is transport different in a solid, gas, and a liquid?

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.

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

Molecular Transport

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: ψ z ? δ? Γ?

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

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

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

Flux at Steady State At steady-state:

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

General Property Balance Assume: 1.Transport occurs in the z- direction only. 2.Volume element has a unit cross-sectional area. 3.R = rate of generation of property (concentration per unit time)

General Property Balance Assume: 1.Transport occurs in the z- direction only. 2.Volume element has a unit cross-sectional area. 3.R = rate of generation of property (amount per unit time per unit volume) WHY?

General Property Balance Assume: 1.Transport occurs in the z- direction only. 2.Volume element has a unit cross-sectional area. 3.R = rate of generation of property (amount per unit time per unit volume) WHY?

General Property Balance Assume: 1.Transport occurs in the z- direction only. 2.Volume element has a unit cross-sectional area. 3.R = rate of generation of property (amount per unit time per unit volume) WHY?

General Property Balance

General equation for momentum, energy, and mass conservation (molecular transport mechanism only)

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.

Momentum Transport Driving Force – change in velocity

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

Momentum Transport

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?

Heat Transport Driving Force – change in temperature

Heat Transport Heat flux in the y-direction

Heat Transport

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.

Mass Transport Driving Force – change in concentration

Mass Transport Mass flux in the y-direction

Analogy MOMENTUM HEAT MASS

Assignment Compute the steady-state momentum flux τ yx in lb f /ft 2 when the lower plate velocity V is 1 ft/s in the positive x- direction, the plate separation Y is ft, and the fluid viscosity µ is 0.7 cp.

Assignment Compute the steady-state momentum flux τ yx in lb f /ft 2 when the lower plate velocity V is 1 ft/s in the positive x- direction, the plate separation Y is ft, and the fluid viscosity µ is 0.7 cp. ANS: 1.46 x lb f /ft 2

Assignment A plastic panel of area A = 1 ft 2 and thickness Y = in. was found to conduct heat at a rate of 3.0 W at steady state with temperatures T o = 24.00°C and T 1 = 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?

Assignment A plastic panel of area A = 1 ft 2 and thickness Y = in. was found to conduct heat at a rate of 3.0 W at steady state with temperatures T o = 24.00°C and T 1 = 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 cal/cm-s-K

Assignment Calculate the steady-state mass flux j Ay 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 mm, and its density ρ (B) is 2.6 g/cm 3. The solubility and diffusivity of helium in pyrex are reported as volumes of gaseous helium per volume of glass, and D AB = 0.2  cm 2 /s, respectively.

Assignment ANS: 1.05 x g/cm 2 -s