Diffusional growth of droplets



EAS 6140 - Thermodynamics of Atmospheres and Oceans

Worksheet 12 (5.2)

Diffusional growth of droplets

Diffusional growth of water drops

1. Once a drop is activated (has reached its critical radius), it will grow spontaneously by diffusion. This growth is associated with (circle all that are correct)

a) diffusion of heat towards the drop

b) diffusion of heat away from the drop

c) diffusion of water vapor towards the drop

d) diffusion of water vapor away from the drop

2. The equation for the diffusion of water vapor is

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Doing dimensional analysis, determine the units of D.

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3. The volume of a sphere is _____[pic]________________

4. The surface area of a sphere is ___[pic]_______________

5. The droplet growth rate as influenced by both diffusion of water vapor and heat is

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Are curvature and solute effects included in this equation? Why or why not?

No, their impact is negligible once the drop size has increased past a few microns

6. Assume that S, K, D in #5 are constant. Integrate #5 from an initial drop size ro to find r(t).

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7. Large cloud droplets will increase their radius (faster, slower, at the same rate) by diffusion than small cloud drops

8. Consider Table 5.5. How long does it typically take to grow a drop to 50 microns radius by diffusion? Is it realistic to expect rain drops to form by diffusional growth?

Between 41,500 and 44,500 seconds, this is not realistic for rain drops

9. Consider a cloud with a concentration of 500 drops per cm3 volume of air that are growing by diffusion, in an air parcel initially with S=1.01. As a result of the diffusional growth, the saturation ratio will (increase, decrease, remain the same).

10. What processes determine how supersaturation varies with time.

the production of supersaturation (via cooling) and condensation

11. What are some physical mechanisms that would cause relative humidity (and supersaturation) to increase? Cooling in adiabatic ascent, rising motion

12. Consider the following equation (5.28).

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In the derivation of the expression of A given in (5.29)- (5.37), circle all of the thermodynamical “laws” or equations that are used directly in this derivation:

a) ideal gas law

b) Dalton’s law of partial pressures

c) hydrostatic equation’

d) first law of thermodynamics

e) adiabatic lapse rate

f) second law of thermodynamics

g) Clausius-Clapeyron equation

13. Consider Fig. 5.6, which was derived from a simple model that included both diffusional growth and the saturation ratio equation, in the presence of a steady updraft and a prescribed spectra of CCN. Above a height of about 10 above cloud base, why does supersaturation start to decrease?

Diffusional growth leads to decreasing supersaturation

14. Discuss why saturation ratios in clouds are rarely observed to exceed 1.01.

Supersaturation comes into balance with diffusion growth

15. Assuming that droplets are activated at different values of r* associated with different sizes of aerosol particles, a spectrum of drop of drop sizes will result. As a result of diffusional growth of this spectrum of drops, the spread in drop sizes will (increase, decrease, remain the same).

17. The spread of observed drop size spectra are observed to (increase, decrease, remain the same).

18. List two processes that may contribute to the broadening of drop size spectra.

Giant particles acting as embyos for large drops, small-scale turbulence, entrainment of dry air

Diffusional growth of ice crystals

19. What ice crystal habit would you expect to have at -20oC and a relative humidity (with respect to water) of 95%? Solid to Skeleton Thin Plates

20. Ice crystals of non-spherical shape grow by diffusion (faster than, slower than, at the same rate as) spherical ice crystals

21. Refer to Table 4.4. At a temperature of –20oC and RH=100%, what is the saturation ratio

a) with respect to water 1

b) with respect to ice 1.22

22. At relative humidity of 101% with respect to liquid water, which particle will grow faster, a water drop or an ice crystal? Ice crystal

23. If water drops and ice crystals exist in the same air parcel and the temperature is slightly below freezing, what will be the final equilibrium state? The water drops will change into ice crystals

24 Assume that you have a rising parcel of air at T = -5°C and p = 800 mb. Assume that a slight supersaturation exists with H = 100.5% (with respect to liquid).

a) Compute how long it would take to grow a cloud droplet from an initial radius of 1 μm to a droplet radius of 10 μm and 100 μm, (use eq 5.27)

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b) Write a version of (5.27) that is suitable for the growth of ice crystals. Which factors in the equation differ for the growth of ice vs. liquid particles?

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Factors that differ for the growth of the ice vs liquid particles are related to the calculation of Ki and Di. Llv becomes Liv ew becomes ei, and ρl becomes ρi.

c) Compute how long it would take to grow a spherical ice ball from an initial radius of 1 μm to a radius of 10 μm and 100 μm. (adapt eq. 5.27 for ice crystals)

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