A water droplet evaporates before they reach the ground. Figure 1: Water droplets (source] In this situation, a few assumptions are made: a) At initial point, a typical water droplet is in sphere shape with radius r and remain spherical while evaporating. b) The rate of evaporation (when it loses mass (m) is proportional to the surface area, S. c) There is no air-resistance and downward direction is the positive direction. To describe this problem, given that p is the mass density of water, rais the radius of water before it drops, m is the water mass, V is the water volume and k is the constant of proportionality. QUESTION: (1) From assumption (b), show that the radius of the water droplet at time t is (Hint: m = pV,V = r², S = 4ar*). (2) From assumption (c), the velocity (v) at time (t) of the water droplet is +r. If the water drops from stationary, solve for v (t). (3) Determine the time when the water droplet has evaporated entirely, given that To = 3mm. Then, 10 seconds after the water drops, the radius r = 2mm.
A water droplet evaporates before they reach the ground. Figure 1: Water droplets (source] In this situation, a few assumptions are made: a) At initial point, a typical water droplet is in sphere shape with radius r and remain spherical while evaporating. b) The rate of evaporation (when it loses mass (m) is proportional to the surface area, S. c) There is no air-resistance and downward direction is the positive direction. To describe this problem, given that p is the mass density of water, rais the radius of water before it drops, m is the water mass, V is the water volume and k is the constant of proportionality. QUESTION: (1) From assumption (b), show that the radius of the water droplet at time t is (Hint: m = pV,V = r², S = 4ar*). (2) From assumption (c), the velocity (v) at time (t) of the water droplet is +r. If the water drops from stationary, solve for v (t). (3) Determine the time when the water droplet has evaporated entirely, given that To = 3mm. Then, 10 seconds after the water drops, the radius r = 2mm.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![A water droplet evaporates before they reach the ground.
Figure 1: Water droplets (source]
In this situation, a few assumptions are made:
a) At initial point, a typical water droplet is in sphere shape with radius r and remain spherical
while evaporating.
b) The rate of evaporation (when it loses mass (m)) is proportional to the surface area, S.
c) There is no air-resistance and downward direction is the positive direction.
To describe this problem, given that p is the mass density of water, rois the radius of water before it
drops, m is the water mass, V is the water volume and k is the constant of proportionality.
QUESTION:
(1) From assumption (b), show that the radius of the water droplet at time t is
r(t) = () t+ ro-
(Hint: m = pV,V =r³, S = 4zr²).
(2) From assumption (c), the velocity (v) at time (t) of the water droplet is
3 ()
v= g.
t +ro
If the water drops from stationary, solve for v (t).
(3) Determine the time when the water droplet has evaporated entirely, given that
ro = 3mm. Then, 10 seconds after the water drops, the radius r = 2mm.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec84e97f-6283-4e56-a301-8d740ee431d2%2F53e50541-a210-4678-b04c-c168d2109c95%2F6mvjod6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A water droplet evaporates before they reach the ground.
Figure 1: Water droplets (source]
In this situation, a few assumptions are made:
a) At initial point, a typical water droplet is in sphere shape with radius r and remain spherical
while evaporating.
b) The rate of evaporation (when it loses mass (m)) is proportional to the surface area, S.
c) There is no air-resistance and downward direction is the positive direction.
To describe this problem, given that p is the mass density of water, rois the radius of water before it
drops, m is the water mass, V is the water volume and k is the constant of proportionality.
QUESTION:
(1) From assumption (b), show that the radius of the water droplet at time t is
r(t) = () t+ ro-
(Hint: m = pV,V =r³, S = 4zr²).
(2) From assumption (c), the velocity (v) at time (t) of the water droplet is
3 ()
v= g.
t +ro
If the water drops from stationary, solve for v (t).
(3) Determine the time when the water droplet has evaporated entirely, given that
ro = 3mm. Then, 10 seconds after the water drops, the radius r = 2mm.
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