Problem 32 - Thomas Calculus

Question

The heat diffusion equation Let \(T(t, x, y, z)\) be a function with continuous second derivatives giving the temperature at time \(t\) at the point \((x, y, z)\) of a solid occupying a region \(D\) in space. If the solid's heat capacity and mass density are denoted by the constants \(c\) and \(\rho,\) respectively, the quantity \(c \rho T\) is called the solid's heat energy per unit volume. a. Explain why - \(\nabla T\) points in the direction of heat flow. b. Let \(-k \nabla T\) denote the energy flux vector. (Here the constant \(k\) is called the conductivity.) Assuming the Law of Conservation of Mass with \(-k \nabla T=\mathbf{v}\) and \(c \rho T=p\) in Exercise \(31,\) derive the diffusion (heat) equation $$\frac{\partial T}{\partial t}=K \nabla^{2} T$$ where \(K=k /(c \rho)>0\) is the diffusivity constant. (Notice that if \(T(t, x)\) represents the temperature at time \(t\) at position \(x\) in a uniform conducting rod with perfectly insulated sides, then \(\nabla^{2} T=\partial^{2} T / \partial x^{2}\) and the diffusion equation reduces to the one-dimensional heat equation in Chapter 14 's Additional Exercises.)

Step-by-Step Solution

Answer

Heat flows along \(-\nabla T\), controlled by \(-k \nabla T\), resulting in \(\frac{\partial T}{\partial t} = K \nabla^2 T\).

Step 1: Understanding Heat Flow Direction

The negative gradient of temperature, \(-abla T\), indicates the direction of maximum decrease in temperature. Heat naturally flows from regions of higher temperature to regions of lower temperature, so the direction of \(-abla T\) corresponds to the direction of heat flow.

Step 2: Identifying Energy Flux

Given the energy flux vector is \(-k abla T\), where \(k\) is the thermal conductivity, this vector direction determines how heat energy is transmitted through a material. The magnitude of this vector represents the rate at which energy per unit area per unit time is transferred.

Step 3: Applying Conservation of Mass and Energy

Using the Law of Conservation of Mass, and noting that any change in energy density \(c \rho T\) over time must equal the negative divergence of the energy flux \(-k abla T\), we have \(\frac{\partial p}{\partial t} = -abla \cdot \mathbf{v}\) with \(p = c \rho T\) and \(\mathbf{v} = -k abla T\).

Step 4: Deriving the Heat Equation from Energy Balance

Substituting \(p = c \rho T\) and \(\mathbf{v} = -k abla T\) into \(\frac{\partial p}{\partial t} = -abla \cdot \mathbf{v}\), we have \(c \rho \frac{\partial T}{\partial t} = abla \cdot (k abla T)\). Simplifying, \(\frac{\partial T}{\partial t} = \frac{k}{c \rho} abla^2 T\), which simplifies to \(\frac{\partial T}{\partial t} = K abla^2 T\), with \(K = \frac{k}{c \rho}\).

Step 5: Special Case - One Dimensional Heat Equation

For a one-dimensional rod, the Laplacian \(abla^2 T\) simplifies to \(\frac{\partial^2 T}{\partial x^2}\), reducing the diffusion equation to \(\frac{\partial T}{\partial t} = K \frac{\partial^2 T}{\partial x^2}\). This is the one-dimensional heat equation.

Key Concepts

Temperature Gradient

The temperature gradient is a fundamental concept in understanding how heat moves through a material. Imagine a scenario where different points in a solid have varying temperatures. The temperature gradient is essentially the vector that points in the direction where the temperature decreases the most rapidly. This is represented by abla T. In the problem at hand, we are interested in abla T's opposite, that is, - abla T, because heat naturally flows from hot to cold regions. Therefore, the vector - abla T shows us the direction in which heat is flowing. This concept is important as it underpins much of thermal physics, making it a valuable tool for analyzing complex systems.

Energy Flux

Energy flux refers to the flow of energy through a surface per unit area per unit time. In this context, the energy flux vector abla T, demonstrates how thermal energy is being conducted through a material. The energy flux vector is defined by the expression, -k abla T, where k represents the thermal conductivity, which we'll discuss in the next section. This vector direction shows us the way energy is flowing, while its magnitude tells us the rate at which this energy transfer occurs. - The direction of the vector signifies movement of energy - Magnitude helps in determining the flow rate of energy Understanding energy flux allows us to manage and manipulate the thermal flow in practical applications, like designing more effective heat sinks or insulating materials.

Thermal Conductivity

Thermal conductivity is a material property that quantifies the ability of a substance to conduct heat. It is denoted by the variable k in the energy flux vector -k abla T. A high thermal conductivity signifies that a material can efficiently transfer heat, such as metals like copper and aluminum. On the other hand, materials like wood or fiberglass have low thermal conductivity, making them good insulators. Key points to remember include:

• Higher values of k imply better heat transfer
• Metals are typically very good at conducting heat
• Insulators have low thermal conductivity, preventing energy flow

In engineering, selecting materials with the appropriate thermal conductivity is crucial for effective temperature control in devices.

Law of Conservation of Mass

The law of conservation of mass is a core principle in physics stating that mass cannot be created or destroyed in a closed system. When applied to heat transfer, it helps ensure that thermal energy changes within a system are fully accounted for. In the example provided, any variation in energy density (cpT where c is specific heat capacity and ρ density) over time is balanced by changes in the energy flux. This is represented in the equation:\[\frac{\partial p}{\partial t} = -abla \cdot \mathbf{v}\]where p is the energy density, and v is the energy flux vector. This equation reinforces the idea that as energy shifts within a system, it should balance the input or release of energy elsewhere, adhering to the law of conservation. This principle is utilized when calculating how heat moves through various forms of media.

One-Dimensional Heat Equation

The one-dimensional heat equation simplifies the general form of the heat diffusion equation to model heat flow in a single direction. This simplification is valuable in scenarios such as examining a rod with insulated sides, where heat can only travel along the length of the rod. Here, the temperature changes just with one spatial variable, usually x.The equation is expressed as:\[\frac{\partial T}{\partial t} = K \frac{\partial^2 T}{\partial x^2}\]Where K is the diffusivity constant. This equation helps in understanding how temperature evolves over time in one-dimensional systems. It's crucial in fields like engineering and physics, where it helps predict how temperature distributions change. This makes it very useful for understanding the thermal behavior of machinery, buildings, and other materials.