We all have experienced the result of someone spraying perfume in the corner of a large room. Within a fairly short amount of time, most people in the room can smell the perfume. In this case, we say that the perfume has diffused from a region of the room in which it was highly concentrated to a region of the room in which its concentration was initially low. Diffusion is the process by which the molecules naturally move from regions where they are highly concentrated to regions where they are not as concentrated. This process can be investigated using a diffusion model (also known as a heat equation): ди at อใน əx², u(0, t) = 20, u(30, t) = 50, u(x, 0) = 60 - 2x, t> 0, 0 < x < 30, (1) with u = u(x, t) corresponds to the concentration of perfume particles moving over time, t, along a one-dimensional spatial domain, x. a) Calculate the steady-state concentration of perfume particles. b) Formulate the initial-boundary value problem that determines the transient distribution of perfume particles. c) Hence, using the Fourier series techniques, derive the solution u(x, t) of the PDE (1).
We all have experienced the result of someone spraying perfume in the corner of a large room. Within a fairly short amount of time, most people in the room can smell the perfume. In this case, we say that the perfume has diffused from a region of the room in which it was highly concentrated to a region of the room in which its concentration was initially low. Diffusion is the process by which the molecules naturally move from regions where they are highly concentrated to regions where they are not as concentrated. This process can be investigated using a diffusion model (also known as a heat equation): ди at อใน əx², u(0, t) = 20, u(30, t) = 50, u(x, 0) = 60 - 2x, t> 0, 0 < x < 30, (1) with u = u(x, t) corresponds to the concentration of perfume particles moving over time, t, along a one-dimensional spatial domain, x. a) Calculate the steady-state concentration of perfume particles. b) Formulate the initial-boundary value problem that determines the transient distribution of perfume particles. c) Hence, using the Fourier series techniques, derive the solution u(x, t) of the PDE (1).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:We all have experienced the result of someone spraying perfume in the corner of a large
room. Within a fairly short amount of time, most people in the room can smell the perfume.
In this case, we say that the perfume has diffused from a region of the room in which it was
highly concentrated to a region of the room in which its concentration was initially low.
Diffusion is the process by which the molecules naturally move from regions where they are
highly concentrated to regions where they are not as concentrated. This process can be
investigated using a diffusion model (also known as a heat equation):
ди
at
=
อน
əx²¹
u(0,t) = 20, u(30, t) = 50, t> 0,
u(x,0) = 60 2x,
0 < x < 30,
(1)
with u = u(x, t) corresponds to the concentration of perfume particles moving over time, t,
along a one-dimensional spatial domain, x.
a) Calculate the steady-state concentration of perfume particles.
b) Formulate the initial-boundary value problem that determines the transient distribution of
perfume particles.
c) Hence, using the Fourier series techniques, derive the solution u(x, t) of the PDE (1).
d) By plotting the solution obtained in part (c) using any computer package, display the initial
particle concentration, the final (or steady state) particle concentration, and the particle
concentration at t = 3 and t = 25 i.e., the intermediate particle concentration.
(For part (d), enclose together the coding of computer package that your group employs in
your report).
e) From the plot in (d), produce some plausible explanations on the intermediate particle
concentration with respect to its behaviour at the boundary conditions and also the
observations as t increases.
Expert Solution
Trending now
This is a popular solution!
Step by step
Solved in 9 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
