EXERCISES 1.4
1.4.1. Determine the equilibrium temperature distribution for a one-dimensional
rod with constant thermal properties with the following sources and boundary
conditions:

(a) Q = 0, u(0) = 0, u(L)=T
(b) Q = 0, u(0) = T, u(L)=0
(c) Q = 0, u'(0) = 0, u(L)=T
(d) Q = 0, u(0) = T, u'(L)=alpha
(e) Q/K0 = 1, u(0) = T1, u(L) = T2
(f) Q/K0 = x^2, u(0) = T, u'(L) = 0
(g) Q = 0, u(0) = T, u'(L) + u(L) = 0
(h) Q=0, u'(0)-[u(0)-T]=0, u'(L)=alpha

In these you may assume that u(x,0) = f(x).

1.4.2. Consider the equilibrium temperature distribution for a uniform one-dimensional
rod with sources Q/Ko = x of thermal energy, subject to the boundary
conditions u(0) = 0 and u(L) = 0.
*
(a) Determine the heat energy generated per unit time inside the entire
rod.

(b) Determine the heat energy flowing out of the rod per unit time at x = 0
and at x = L.

(c) What relationships should exist between the answers in parts (a) and
(b)?

1.4.3. Determine the equilibrium temperature distribution for a one-dimensional
rod composed of two different materials in perfect thermal contact at x = 1.
For x=0 to x=1, there is one material (c rho = 1, K0 = 1) with a constant
source (Q = 1), whereas for the other x=1 to x=2 there are no sources
(Q = 0, c rho = 2, K0 = 2) with u(O) = 0 and u(2) = 0. See Exercise 1.3.2.

1.4.4. If both ends of a rod are insulated, derive from the partial differential equation
that the total thermal energy in the rod is constant.

1.4.5. Consider a one-dimensional rod x=0 to x=L of known
constant thermal properties without sources. Suppose that the temperature
is an unknown constant T at x = L. Determine T if we know the steady
state temperature and the heat flow at x = 0.

1.4.6. The two ends of a uniform rod of length L are insulated. There is a constant
nonzero source of thermal energy Q0, and the temperature is initially u(x,0)=f(x).

(a) Show mathematically that there does not exist any equilibrium temperature
distribution. Briefly explain physically.

(b) Calculate the total thermal energy in the entire rod.

1.4.7. For the following problems, determine an equilibrium temperature distribution
(if one exists). For what values of beta are there solutions? Explain
physically.

(a) du/dt = (d/dx)^2 u + 1, u(x,0)=f(x), (du/dx)(0,t)=1, (du/dx)(L,t)=beta

(b) du/dt = (d/dx)^2 u, u(x,0)=f(x), (du/dx)(0,t)=1, (du/dx)(L,t)=beta

(b) du/dt = (d/dx)^2 u + x - beta, u(x,0)=f(x), (du/dx)(0,t)=0, (du/dx)(L,t)=0


1.4.8. Express the integral conservation law for the entire rod with constant thermal
properties. Assume the heat flow is known to be different constants at
both ends. By integrating with respect to time and using the initial condition, determine the total thermal
energy in the rod. Consider separately these two cases.

(a) Assume there are no sources.

(b) Assume the sources of thermal energy are constant.

1.4.9. Derive the integral conservation law for the entire rod with constant thermal
properties by integrating the heat equation (1.2.10), assuming no sources, which is the equation
du/dt = k (d/dx)^2 u.
Show the result is equivalent to (1.2.4), which is the equation

(d/dt) int(e,x=a..b) = phi(a,t) - phi(b,t) + int(Q,x=a..b).

1.4.10. Suppose du/dt = (d/dx)^2 u + 4, u(x,0) = f(x), (du/dx)(0, t) = 5, (du/dx)(L, t) = 6. Calculate the total thermal energy at time t in the one-dimensional rod.

1.4.11. Suppose du/dt = (d/dx)^2 u + x, u(x,0) = f(x), (du/dx)(0,t) = beta, (du/dx)(L,t) = 7.

(a) Calculate the total thermal energy at time t in the one-dimensional rod.

(b) From part (a), determine a value of beta for which an equilibrium exists.
For this value of beta, determine the limit of u(x,t) at t=infinity.

1.4.12. Suppose the concentration u(x,t) of a chemical satisfies Fick's law (1.2.13),
which is the equation phi = -k (du/dx),
and suppose the initial concentration is given by u(x,0) = f(x). Consider a region
x=0 to x=L in which the flow is specified at both ends:  -k (du/dx)(0, t) = alpha and
-k (du/dx)(L, t)=beta. Assume alpha and beta are constants.

(a) Display the conservation law for the entire region.

(b) Determine, using the initial condition, the total amount of chemical in the region as a function of time.

(c) Under what conditions is there an equilibrium chemical concentration and what is it?

1.4.13. Do Exercise 1.4.12 if alpha and beta are functions of time.