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Chapter 1: Problem 62

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 \(\Delta x,\) derive the heat equation, $$ \frac{\partial T}{\partial t}=K \frac{\partial^{2} T}{\partial x^{2}} $$ where \(K=k_{t} / c \rho, c\) is the specific heat of the material, and \(\rho\) 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 $$ T(x, t)=T_{0}+\frac{A}{\sqrt{t}} e^{-x^{2} / 4 K t} $$ where \(T_{0}\) 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.

Short Answer

Expert verified
The heat equation is derived using conservation of energy principles. The given function is a solution representing diffusive heat spread over time.

Step by step solution

01

Understand Heat Conduction

Consider a segment of the rod of length \(\Delta x\) positioned between \(x\) and \(x + \Delta x\). The heat flowing into this segment from the left at position \(x\) is \(-k_t \frac{\partial T}{\partial x} \bigg|_x\), and the heat flowing out from the right at position \(x + \Delta x\) is \(-k_t \frac{\partial T}{\partial x} \bigg|_{x+\Delta x}\).
02

Apply Conservation of Energy

The rate at which heat accumulates in the segment is equal to the difference between the heat flowing in and flowing out: \[ k_t \left(\frac{\partial T}{\partial x} \bigg|_{x+\Delta x} - \frac{\partial T}{\partial x} \bigg|_x\right) \approx k_t \frac{\partial^2 T}{\partial x^2} \Delta x. \]
03

Relate Heat Flow to Temperature Change

By conservation of energy, the change in thermal energy \(\Delta Q\) in the segment over time \(\Delta t\) is \[ \Delta Q = c \rho \Delta x \Delta T \], where \(\Delta T = \frac{\partial T}{\partial t} \Delta t\). Match this to the heat flow change: \[ k_t \frac{\partial^2 T}{\partial x^2} \Delta x \Delta t = c \rho \Delta x \frac{\partial T}{\partial t} \Delta t. \]
04

Derive Heat Equation

Cancel \(\Delta x\) and \(\Delta t\), leaving \[ \frac{\partial T}{\partial t} = \frac{k_t}{c \rho} \frac{\partial^2 T}{\partial x^2} = K \frac{\partial^2 T}{\partial x^2}. \] This confirms the heat equation: \[ \frac{\partial T}{\partial t} = K \frac{\partial^2 T}{\partial x^2}. \]
05

Verify Special Solution

Substitute \(T(x, t) = T_0 + \frac{A}{\sqrt{t}} e^{-x^2 / 4Kt} \) into the heat equation to verify it is a solution. Compute the time derivative: \[ \frac{\partial T}{\partial t} = -\frac{A}{2t^{3/2}} e^{-x^2 / 4Kt} - \frac{A x^2}{4Kt^{5/2}} e^{-x^2 / 4Kt}. \] Compute the second spatial derivative and show they balance according to the heat equation.
06

Interpret the Solution Physically

The solution describes a heat pulse spreading out over time. At initial times, \(t\) small, the temperature is high and concentrated around \(x = 0\). As \(t\) increases, the pulse spreads out more widely, leading to a broader, lower temperature distribution over space. This illustrates diffusive spreading of heat within the rod.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Conduction
Heat conduction is the process through which thermal energy moves from a region of higher temperature to one of lower temperature. It happens in solid materials, like the rod in our example, through collisions between particles and their immediate neighbors, which transfer kinetic energy. At the microscopic level, think about how fast-moving atoms in a hotter region collide with slower-moving atoms in a cooler region and speed them up. This transfer of energy is what we call heat conduction.
For the scenario involving our rod, we are interested in a small segment of it, \( \Delta x \). The heat flowing into this segment from the left can be calculated as \(-k_t \frac{\partial T}{\partial x} \bigg|_x\). This represents how much heat is entering the segment based on the temperature gradient at position \( x \). Meanwhile, the heat exiting on the other side (at \( x + \Delta x \)) is similarly calculated as \(-k_t \frac{\partial T}{\partial x} \bigg|_{x+\Delta x}\).
It's important to remember that because the movement of energy relies on the temperature differential, the greater the temperature change, the more significant the flow of thermal energy, or heat conduction, in the material.
Specific Heat
Specific heat refers to the amount of heat energy required to change the temperature of a unit mass of a substance by one degree Celsius (or equivalently, one Kelvin). It is a property that differs between materials because some substances require more energy to change their temperature than others.
In our scenario with the rod, we denote the specific heat of the material by \( c \). The specific heat affects how much thermal energy (\

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Most popular questions from this chapter

Imagine some helium in a cylinder with an initial volume of 1 liter and an initial pressure of 1 atm. Somehow the helium is made to expand to a final volume of 3 liters, in such a way that its pressure rises in direct proportion to its volume.

Problem 1.61. Geologists measure conductive heat flow out of the earth by drilling holes (a few hundred meters deep) and measuring the temperature as a function of depth. Suppose that in a certain location the temperature increases by \(20^{\circ} \mathrm{C}\) per kilometer of depth and the thermal conductivity of the rock is \(2.5 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) What is the rate of heat conduction per square meter in this location? Assuming that this value is typical of other locations over all of earth's surface, at approximately what rate is the earth losing heat via conduction? (The radius of the earth is \(6400 \mathrm{km} .\) )

Even at low density, real gases don't quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion, $$P V=n R T\left(1+\frac{B(T)}{(V / n)}+\frac{C(T)}{(V / n)^{2}}+\cdots\right)$$ where the functions \(B(T), C(T),\) and so on are called the virial coefficients. When the density of the gas is fairly low, so that the volume per mole is large, each term in the series is much smaller than the one before. In many situations it's sufficient to omit the third term and concentrate on the second, whose coefficient \(B(T)\) is called the second virial coefficient (the first coefficient being 1 ). Here are some measured values of the second virial coefficient for nitrogen \(\left(\mathrm{N}_{2}\right):\) $$\begin{array}{cc} T(\mathrm{K}) & B\left(\mathrm{cm}^{3} / \mathrm{mol}\right) \\ \hline 100 & -160 \\ 200 & -35 \\ 300 & -4.2 \\ 400 & 9.0 \\ 500 & 16.9 \\ 600 & 21.3 \end{array}$$ (a) For each temperature in the table, compute the second term in the virial equation, \(B(T) /(V / n),\) for nitrogen at atmospheric pressure. Discuss the validity of the ideal gas law under these conditions. (b) Think about the forces between molecules, and explain why we might expect \(B(T)\) to be negative at low temperatures but positive at high temperatures. (c) Any proposed relation between \(P, V,\) and \(T,\) like the ideal gas law or the virial equation, is called an equation of state. Another famous equation of state, which is qualitatively accurate even for dense fluids, is the van der Waals equation, $$\left(P+\frac{a n^{2}}{V^{2}}\right)(V-n b)=n R T$$ where \(a\) and \(b\) are constants that depend on the type of gas. Calculate the second and third virial coefficients \((B \text { and } C)\) for a gas obeying the van der Waals equation, in terms of \(a\) and \(b\). (Hint: The binomial expansion says that \((1+x)^{p} \approx 1+p x+\frac{1}{2} p(p-1) x^{2},\) provided that \(|p x| \ll 1 .\) Apply this approximation to the quantity \([1-(n b / V)]^{-1}\).) (d) Plot a graph of the van der Waals prediction for \(B(T),\) choosing \(a\) and \(b\) so as to approximately match the data given above for nitrogen. Discuss the accuracy of the van der Waals equation over this range of conditions. (The van der Waals equation is discussed much further in Section 5.3.)

Calculate the mass of a mole of dry air, which is a mixture of \(\mathrm{N}_{2}\) \((78 \% \text { by volume }), \mathrm{O}_{2}(21 \%),\) and argon \((1 \%)\).

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