19B.7 Time for a droplet to evaporate. A droplet of pure A of initial radius R is suspended in a large body of motionless gas B. The concentration of A in the gas phase is xAR at r = R and zero at an infinite distance from the droplet. (a) Assuming that R is constant, show that at steady state dx 1- XA AB (19B.7-1) dr where NAlER is the molar flux in the r direction at the droplet surface, c is the total molar concentration in the gas phase, and DAB temperature and pressure throughout. Show that integration of Eq. 19B.7-1 from the droplet surface to infinity gives is the diffusivity in the gas phase. Assume constant RNArlr=R = -cD AB In(1 – xAR) (19B.7-2) (b) We now let the droplet radius R be a function of time, and treat the problem as a quasi-steady one. Then the rate of decrease of moles of A within the drop can be equated to the instantaneous rate of loss of mass across the liquid-gas interface d - GTR°C) = 47R°N,rlr=R = -4rReDg In(1 – *AR) In(1 – XAR) (19B.7-3) %3D Ar r=R АВ dt where c is the molar density of pure liquid A. Show that when this equation is integrated from t = 0 to t = t, (the time for complete evaporation of the droplet), one gets to (19B.7-4) In[1/(1 – xAR)] 2cD AB
19B.7 Time for a droplet to evaporate. A droplet of pure A of initial radius R is suspended in a large body of motionless gas B. The concentration of A in the gas phase is xAR at r = R and zero at an infinite distance from the droplet. (a) Assuming that R is constant, show that at steady state dx 1- XA AB (19B.7-1) dr where NAlER is the molar flux in the r direction at the droplet surface, c is the total molar concentration in the gas phase, and DAB temperature and pressure throughout. Show that integration of Eq. 19B.7-1 from the droplet surface to infinity gives is the diffusivity in the gas phase. Assume constant RNArlr=R = -cD AB In(1 – xAR) (19B.7-2) (b) We now let the droplet radius R be a function of time, and treat the problem as a quasi-steady one. Then the rate of decrease of moles of A within the drop can be equated to the instantaneous rate of loss of mass across the liquid-gas interface d - GTR°C) = 47R°N,rlr=R = -4rReDg In(1 – *AR) In(1 – XAR) (19B.7-3) %3D Ar r=R АВ dt where c is the molar density of pure liquid A. Show that when this equation is integrated from t = 0 to t = t, (the time for complete evaporation of the droplet), one gets to (19B.7-4) In[1/(1 – xAR)] 2cD AB
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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![19B.7 Time for a droplet to evaporate. A droplet of pure A of initial radius R is suspended in a large
body of motionless gas B. The concentration of A in the
infinite distance from the droplet.
gas phase is x,
AR at r = R and zero at an
(a) Assuming that R is constant, show that at steady state
AB 2 dx
dr
R²NArlr=R
(19B.7-1)
XA
where Nl-R is the molar flux in the r direction at the droplet surface, c is the total molar
concentration in the gas phase, and DAR is the diffusivity in the gas phase. Assume constant
temperature and pressure throughout. Show that integration of Eq. 19B.7-1 from the droplet
surface to infinity gives
АВ
RNArlr=R =
= -cD
АВ
In(1 - XAR)
(19B.7-2)
(b) We now let the droplet radius R be a function of time, and treat the problem as a
quasi-steady one. Then the rate of decrease of moles of A within the drop can be equated to
the instantaneous rate of loss of mass across the liquid-gas interface
d
-TR°C) = 47R°N olar = -4rRcDg In(1 – XXAR)
(L)
AR°C'
In(1 - Х AR)
(19B.7-3)
Ar lr=R
АВ
dt
where c is the molar density of pure liquid A. Show that when this equation is integrated
from t = 0 to t = to (the time for complete evaporation of the droplet), one gets
to =
2cD
AB In[1/(1 – xAR)]
(19B.7-4)
Does this result look physically reasonable?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7bd3d356-f8d2-4558-9417-c4406f3e0630%2Fd5aa2cf4-43fa-4233-ae59-3a8d4cb8f7b1%2Flhf45rl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:19B.7 Time for a droplet to evaporate. A droplet of pure A of initial radius R is suspended in a large
body of motionless gas B. The concentration of A in the
infinite distance from the droplet.
gas phase is x,
AR at r = R and zero at an
(a) Assuming that R is constant, show that at steady state
AB 2 dx
dr
R²NArlr=R
(19B.7-1)
XA
where Nl-R is the molar flux in the r direction at the droplet surface, c is the total molar
concentration in the gas phase, and DAR is the diffusivity in the gas phase. Assume constant
temperature and pressure throughout. Show that integration of Eq. 19B.7-1 from the droplet
surface to infinity gives
АВ
RNArlr=R =
= -cD
АВ
In(1 - XAR)
(19B.7-2)
(b) We now let the droplet radius R be a function of time, and treat the problem as a
quasi-steady one. Then the rate of decrease of moles of A within the drop can be equated to
the instantaneous rate of loss of mass across the liquid-gas interface
d
-TR°C) = 47R°N olar = -4rRcDg In(1 – XXAR)
(L)
AR°C'
In(1 - Х AR)
(19B.7-3)
Ar lr=R
АВ
dt
where c is the molar density of pure liquid A. Show that when this equation is integrated
from t = 0 to t = to (the time for complete evaporation of the droplet), one gets
to =
2cD
AB In[1/(1 – xAR)]
(19B.7-4)
Does this result look physically reasonable?
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