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Imagine a narrow pipe, filled with fluid, in which the
Chapter 1, Problem 69P(choose chapter or problem)
Problem 69P
Imagine a narrow pipe, filled with fluid, in which the concentration of a certain type of molecule varies only along the length of the pipe (in the x direction). By considering the flux of these particles from both directions into a short segment Δx, derive Fick’s second law,
Noting the similarity to the heat equation derived in Problem, discuss the implications of this equation in some detail.
Problem:
Consider a uniform rod of material whose temperature varies only along its length, in the x direction. By considering the heat flowing from both directions into a small segment of length Δx, derive the heat equation,
where K= κt/cρ, c is the specific heat of the material, and ρ is its density. (Assume that the only motion of energy is heat conduction within the rod; no energy enters or leaves along the sides.) Assuming that K is independent of temperature, show that a solution of the heat equation is
where T0 is a constant background temperature and A is any constant. Sketch (or use a computer to plot) this solution as a function of x, for several values of t. Interpret this solution physically, and discuss in some detail how energy spreads through the rod as time passes.
Questions & Answers
QUESTION:
Problem 69P
Imagine a narrow pipe, filled with fluid, in which the concentration of a certain type of molecule varies only along the length of the pipe (in the x direction). By considering the flux of these particles from both directions into a short segment Δx, derive Fick’s second law,
Noting the similarity to the heat equation derived in Problem, discuss the implications of this equation in some detail.
Problem:
Consider a uniform rod of material whose temperature varies only along its length, in the x direction. By considering the heat flowing from both directions into a small segment of length Δx, derive the heat equation,
where K= κt/cρ, c is the specific heat of the material, and ρ is its density. (Assume that the only motion of energy is heat conduction within the rod; no energy enters or leaves along the sides.) Assuming that K is independent of temperature, show that a solution of the heat equation is
where T0 is a constant background temperature and A is any constant. Sketch (or use a computer to plot) this solution as a function of x, for several values of t. Interpret this solution physically, and discuss in some detail how energy spreads through the rod as time passes.
ANSWER:
Step 1 of 5
Fick’s first law states that the flux of the particles across any surface in any particular direction is directly proportional to the gradient of particle concentration in that particular direction.
Mathematically it can be expressed as;
Here 
Here the minus sign indicates that if the gradient of the particle concentration is positive in a particular direction then the flux will be negative in that direction.