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Consider a uniform rod of material whose temperature
Chapter 1, Problem 62P(choose chapter or problem)
Problem 62P
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 62P
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:
Solution 62P
To solve this question, we shall have to consider the Fourier heat conduction law. The mathematical equation for Fourier heat conduction law is written as …..(1)
Here, amount of heat
time taken for the heat to flow
thermal conductivity
area of cross-section
change in temperature
change in length
Step 1
We are given in the question that heat flows from both directions of the rod.
Using equation (1),
For heat coming from one direction,
…..(2)
For heat conduction from the other direction.
…..(3)
Therefore, the net heat flow can be subtracting equation (3) from equation (2),
…..(2), where
…..(4)