A laboratory has volume V, and its atmosphere is ventilated with a volumetric extraction rate V. A supply of flammable gas in the laboratory develops a leak at time t=0, and supplies a volumetric flow rate Q of the gas into the laboratory's atmosphere. The Lower Explosive Limit (LEL) of the gas in air is 4% by volume. Assume that the atmosphere contains no flammable gas before the leak. (a) Assuming that the gas is well-mixed with the air at all times, show that the volumetric fraction F of the flammable gas in the laboratory at long times t is given by Use material balance at steady state; volumetric flow of flammable gas in Q equals flow out F.V F = V (b) Sketch a graph showing how you expect the volume-fraction F(t) of the gas in the laboratory to vary over time, indication key points on your axes. Graph for F(t) will start at origin, grow monotonically with decreasing gradient until it plateaus at steady state value Characteristic time given by VL (c) From a material balance, derive a first-order differential equation for the time dependence of FFtm. Identify volumetric flows into and out of laboratory Correctly combine to derive VL V dFFtm dt Q-V X FFtm (Note students find the material balance and correct derivation of the differential equation just a challenging as its mathematical solution)
A laboratory has volume V, and its atmosphere is ventilated with a volumetric extraction rate V. A supply of flammable gas in the laboratory develops a leak at time t=0, and supplies a volumetric flow rate Q of the gas into the laboratory's atmosphere. The Lower Explosive Limit (LEL) of the gas in air is 4% by volume. Assume that the atmosphere contains no flammable gas before the leak. (a) Assuming that the gas is well-mixed with the air at all times, show that the volumetric fraction F of the flammable gas in the laboratory at long times t is given by Use material balance at steady state; volumetric flow of flammable gas in Q equals flow out F.V F = V (b) Sketch a graph showing how you expect the volume-fraction F(t) of the gas in the laboratory to vary over time, indication key points on your axes. Graph for F(t) will start at origin, grow monotonically with decreasing gradient until it plateaus at steady state value Characteristic time given by VL (c) From a material balance, derive a first-order differential equation for the time dependence of FFtm. Identify volumetric flows into and out of laboratory Correctly combine to derive VL V dFFtm dt Q-V X FFtm (Note students find the material balance and correct derivation of the differential equation just a challenging as its mathematical solution)
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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Transcribed Image Text:A laboratory has volume V₁ and its atmosphere is ventilated with a volumetric
extraction rate V. A supply of flammable gas in the laboratory develops a leak
at time t=0, and supplies a volumetric flow rate Q of the gas into the
laboratory's atmosphere. The Lower Explosive Limit (LEL) of the gas in air is
4% by volume. Assume that the atmosphere contains no flammable gas
before the leak.
(a) Assuming that the gas is well-mixed with the air at all times, show that
the volumetric fraction Fo of the flammable gas in the laboratory at
long times t is given by
FOO
Use material balance at steady state; volumetric flow of flammable gas
in Q equals flow out F..V
Characteristic time given by
=
(b) Sketch a graph showing how you expect the volume-fraction F(t) of the
gas in the laboratory to vary over time, indication key points on your
axes.
Graph for F(t) will start at origin, grow monotonically with decreasing
gradient until it plateaus at steady state value
VL
Q
V
dFFtm
dt
VL
V
(c) From a material balance, derive a first-order differential equation for the
time dependence of FFtm.
Identify volumetric flows into and out of laboratory
Correctly combine to derive
Q - V × FFtm
(Note students find the material balance and correct derivation of the
differential equation just a challenging as its mathematical solution)
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