.Part 1 Tribbles have infested the cargo bay of the starship Enterprise. Assume that tribbles diffuse similarly to heat or chemicals, so they can be described by concentration T(x, t) and diffusivity D > 0, and their rate of reproduction is 1 tribble/m3 diffusivity = D impermeable x = L reproduction R = kT -x=0 proportional to their concentration, i.e. R = kT tribbles are being generated everywhere where k> 0 is a constant. Tribbles cannot penetrate the floor of the cargo bay (x = 0). Captain Kirk is frantically vacuuming up the tribbles from a catwalk at x = L, but maintains a constant concentration of 1 tribble/m³ there because he finds them too cute to completely remove. Throughout this problem it may help to substitute e.g. P = √√k/D to make the algebra less tedious. No unit conversions are needed for this problem. a. This situation continues for a very long time. Write the PDE governing the transient evolution of tribble concentration T(x, t) and then simplify it to an ODE governing the steady state concentration distribution of tribbles T(x). Also write the ODE boundary conditions mathematically. b. At steady state, show that the concentration of tribbles throughout the cargo bay height is the following, showing all steps. T(x) = cos(√k/D x) cos(√k/DL) c. Assuming k = 1 min-1, D = 40.55 m² min-1, L = 10 m, what is the steady state concentration of tribbles at the cargo bay floor?
.Part 1 Tribbles have infested the cargo bay of the starship Enterprise. Assume that tribbles diffuse similarly to heat or chemicals, so they can be described by concentration T(x, t) and diffusivity D > 0, and their rate of reproduction is 1 tribble/m3 diffusivity = D impermeable x = L reproduction R = kT -x=0 proportional to their concentration, i.e. R = kT tribbles are being generated everywhere where k> 0 is a constant. Tribbles cannot penetrate the floor of the cargo bay (x = 0). Captain Kirk is frantically vacuuming up the tribbles from a catwalk at x = L, but maintains a constant concentration of 1 tribble/m³ there because he finds them too cute to completely remove. Throughout this problem it may help to substitute e.g. P = √√k/D to make the algebra less tedious. No unit conversions are needed for this problem. a. This situation continues for a very long time. Write the PDE governing the transient evolution of tribble concentration T(x, t) and then simplify it to an ODE governing the steady state concentration distribution of tribbles T(x). Also write the ODE boundary conditions mathematically. b. At steady state, show that the concentration of tribbles throughout the cargo bay height is the following, showing all steps. T(x) = cos(√k/D x) cos(√k/DL) c. Assuming k = 1 min-1, D = 40.55 m² min-1, L = 10 m, what is the steady state concentration of tribbles at the cargo bay floor?
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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please solve step by step a b c
Transcribed Image Text:.Part 1
Tribbles have infested the cargo bay of the
starship Enterprise. Assume that tribbles diffuse
similarly to heat or chemicals, so they can be
described by concentration T(x, t) and diffusivity
D > 0, and their rate of reproduction is
1 tribble/m3
diffusivity = D
impermeable
x = L
reproduction
R = kT
-x=0
proportional to their concentration, i.e. R = kT
tribbles are being generated everywhere where
k> 0 is a constant. Tribbles cannot penetrate
the floor of the cargo bay (x = 0). Captain Kirk is
frantically vacuuming up the tribbles from a catwalk at x = L, but maintains a constant
concentration of 1 tribble/m³ there because he finds them too cute to completely
remove. Throughout this problem it may help to substitute e.g. P = √√k/D to make the
algebra less tedious. No unit conversions are needed for this problem.
a. This situation continues for a very long time. Write the PDE governing the transient
evolution of tribble concentration T(x, t) and then simplify it to an ODE governing
the steady state concentration distribution of tribbles T(x). Also write the ODE
boundary conditions mathematically.
b. At steady state, show that the concentration of tribbles throughout the cargo bay
height is the following, showing all steps.
T(x) =
cos(√k/D x)
cos(√k/DL)
c. Assuming k = 1 min-1, D = 40.55 m² min-1, L = 10 m, what is the steady state
concentration of tribbles at the cargo bay floor?
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