高等輸送二 — 質傳Lecture 8Forced convection 郭修伯 助理教授
Forced convection • Forced convection • The flow is determined by factors other than diffusion, factors like pressure gradients and wetted area • exist whether or not diffusion occurs • Analyzing tools • simple physical models • elaborate analytical mathematics
Liquid film p1 c1i Gas Bulk liquid c1 z = 0 z = l The film theory (for interfacial mass transfer) • Assuming a stagnant film exists near interface, a solute present at high dilution is slowly diffusing across this film. • At steady-state: or
Example Carbon dioxide is being scrubbed out of a gas using water flowing through a packed bed of 1 cm Berl saddles. The carbon dioxide is absorbed at a rate of 2.3 x 10-6 mol/cm2 sec. The carbon dioxide is present at a partial pressure of 10 atm, the Henry’s law coefficient H is 600 atm, and the diffusion coefficient of carbon dioxide in water is 1.9 x 10-5 cm2/sec. Find the film thickness. 9.3 x 10-4 mol/cm3 The interfacial concentration: 1/18 mol/cm3 10 atm 600 atm Mass transfer k = 2.5 x 10-3 cm/sec typical order: 10-2 cm
p1 c1i Gas Bulk liquid c1 z = z = 0 Penetration theory (Higbie, 1935) • The falling film is very thick. In the z direction, diffusion is much more important than convection, and in the x direction, diffusion is much less important then convection. • Flux at the interface: The flux averaged over x is:
Sherwood number Peclet number Reynolds number Schmidt number
Surface-renewal theory (Dankwerts, 1951) • It consists of two regions: • Interfacial region: mass transfer occurs by penetration theory. • Renewal region: constantly exchanged with new elements from a second bulk region. p1 Bulk liquid The length of time that small fluid elements spend in the interfacial region is the key. c1i Gas c1 Residence time distribution z = 0
The probability that a given surface element will be at the surface for time t E(t)dt = Residence time distribution The fraction of surface elements remaining at time t : The residence time distribution of surface element : In the interfacial region, the flux is that for diffusion into a infinite slab: The average flux is:
Comparison of the three theories • The film theory • The penetration theory • The surface-renewal theory Film thickness Contact time Surface residence time
Boundary layer theory • A more complete description of mass transfer • Based on parallel with earlier studies of fluid mechanics and heat transfer • The sharp-edged plate made of a sparingly soluble solute is immersed in a rapid flowing solvent. • A boundary layer is formed. • The boundary layer is usually defined as the locus of distance over which 99% of the disruptive effect occurs. • When the flow pattern develops, the solute dissolves off the plate. turbulent region laminar region turbulent region
Layers caused by flow and by mass transfer • The distance that the solute penetrates produces a new concentration boundary layer c , but this layer is not the same as that observed for flow . The two layers influence each other. • When the dissolving solute is only sparingly soluble, the boundary layer caused by the flow is unaffected. Assuming flow varies as a power series in the boundary layer thickness find c ?
Find the boundary layer for flow first: Assuming the fluid flowing parallel to the flat plate follows: The boundary conditions are: The fluid sticks to the plate. The plate is solid and the stress on it is constant. Far from the plate, the plate has no effect.
turbulent region Mass balance on the control volume of the width W, the thickness x and the height l: Dividing Wx x 0 x-momentum balance gives: Dividing Wx x 0
We have now and use to find vx. How about c ?
Find the boundary layer for concentration: Assuming the concentration profile parallel to the flat plate follows: The boundary conditions are: The concentration and flux are constant at the plate. Far from the plate, the plate has no effect.
turbulent region Mass balance on the control volume of the width W, the thickness x and the height l: Dividing Wx x 0
Mass transfer coefficient? Similar to film theory Schmidt number:
Averaged over length L • Valid for a flat plate when the boundary layer is laminar (I.e., Re < 300,000) • ,between the prediction of the film theory and the penetration/surface-renewal theories.
Water flows at 10 cm/sec over a sharp-edged plate of benzoic acid. The dissolution of benzoic acid is diffusion-controlled, with a diffusion coefficient of 1.0 x 10-5 cm2/sec. Find (a) the distance at which the laminar boundary layer ends, (b) the thickness of the flow and concentration boundary layers at that point, and (c) the local mass transfer coefficients at the leading edge and at the position of transition, as well as the average mass transfer coefficient over this length. (a) the length before the turbulent region begins: x = 300 cm x = 300 cm
r z Graetz-Nusselt problem • Mass transfer across the walls of a pipe containing fluid in laminar flow. • Find the dissolution rate as a function of quantities like Reynolds and Schmidt numbers Flow conditions Diffusion conditions Sparing soluble solute
Solute accumulation by convection Solute accumulation by diffusion Solute accumulation = r z Fixed solute concentration at the wall of a short tube Mass balance for the solute in a constant-density fluid on a washer-shaped region:
Averaged over length L: incomplete gamma function How to find the mass transfer coefficient? Schmidt 1.62 Sherwood Reynolds diameter length
Water is flowing at 6.1 cm/sec through a pipe of 2.3 cm in diameter. The walls of a 14-cm section of this pipe are made of benzoic acid, whose diffusion coefficient in water is 1.0 x 10-5 cm2/sec. Find the average mass transfer coefficient over this section. Check before ending the question! Laminar flow? Short pipe? OK!
Concentrated solutions • In most application, correlations for dilute solutions can also be applied to concentrated solutions. • In a few cases, k is a function of the driving force: ? mass transfer coefficient for rapid mass transfer dilute system
Bulk liquid c1i p1 Gas c1 z = l z = 0 A mass balance on a thin film shell z thick shows that the total flux is a constant: or The relation between the dilute mass transfer coefficient k0 and the concentrated mass transfer coefficient k.
Benzene is evaporating from a flat porous plate into pure flowing air. Using the film theory, find N1/k0c1i and k/k0 as a function of the concentration of benzene at the surface of the plate. The benzene evaporates off the plate into air flowing parallel to the plate : In dilute solution:
Summary of forced convection • Table 13.6-1 • all predictions cluster around experimentally observed values • In most cases. Sh Re1/2 and Sc1/3 • Recommend: film and penetration theories