Question

The thermal energy contained within a thin, laterally insulated bar of length \(l\) is $$ E(t)=c_{0} A \int_{0}^{l} u(x, t) d x, $$ where \(c_{0}\) is the heat capacity per unit volume of the bar material and \(A\) is the crosssectional area. If both ends are insulated, one would expect \(E(t)\) to be a constant, since heat can neither enter nor escape. Use the heat equation to show that this is, in fact, the case.

Short answer

Question: Prove that the thermal energy remains constant over time for a thin, insulated bar, using the heat equation. Answer: To show that the thermal energy remains constant, we followed the steps below: 1. Write down the heat equation: $$\frac{\partial u(x,t)}{\partial t} = \alpha \frac{\partial^2 u(x,t)}{\partial x^2}$$ 2. Substitute the heat equation in the expression for thermal energy: $$\frac{d E(t)}{dt} = c_{0} A \frac{d}{dt} \int_0^l u(x,t) dx$$ 3. Apply the differentiation under the integral sign: $$\frac{d E(t)}{dt} = c_0A \int_0^l \frac{\partial u(x,t)}{\partial t} dx$$ 4. Use the heat equation for the partial derivative: $$\frac{d E(t)}{dt} = c_{0} A \int_0^l \alpha \frac{\partial^2 u(x,t)}{\partial x^2} dx$$ 5. Integrate by parts: $$\frac{d E(t)}{dt} = -c_{0} A \alpha \int_0^l v \frac{\partial v}{\partial x} dx$$ 6. Show that the derivative of the thermal energy is zero: $$\frac{d E(t)}{dt} = 0$$ This proves that the thermal energy remains constant over time for a thin, insulated bar using the heat equation.

Step-by-step solution

Understand and write down the heat equation

The heat equation is a partial differential equation that helps us understand how heat is distributed in a given region over time. For the one-dimensional case (which applies to the bar in our problem), the heat equation is given by: $$ \frac{\partial u(x,t)}{\partial t} = \alpha \frac{\partial^2 u(x,t)}{\partial x^2}, $$ where \(u(x,t)\) represents the temperature at position \(x\) and time \(t\) and \(\alpha\) is the thermal diffusivity of the bar material.

Substitute the heat equation in the expression for thermal energy

To show that the thermal energy \(E(t)\) is a constant, we will differentiate \(E(t)\) with respect to \(t\) and show that this derivative is zero. To do this, we need to substitute the heat equation in the expression for \(E(t)\): $$ \frac{d E(t)}{dt} = c_{0}A \frac{d}{dt} \int_0^l u(x,t) dx. $$

Apply the differentiation under the integral sign

Now, we apply the differentiation under the integral sign on the right side of the equation. This means differentiating \(u(x,t)\) with respect to \(t\) while keeping \(x\) as a constant: $$ \frac{d E(t)}{dt} = c_0A \int_0^l \frac{\partial u(x,t)}{\partial t} dx. $$

Use the heat equation for the partial derivative

We can replace the partial derivative \(\frac{\partial u(x,t)}{\partial t}\) from the heat equation we noted in Step 1: $$ \frac{d E(t)}{dt} = c_{0} A \int_0^l \alpha \frac{\partial^2 u(x,t)}{\partial x^2} dx. $$

Integrate by parts

Now, let's integrate by parts to handle the second-order derivative. Let \(v=\frac{\partial u(x,t)}{\partial x}\) and \(dw=\alpha \frac{\partial^2 u(x,t)}{\partial x^2} dx\), then we have: $$ \frac{d E(t)}{dt} = c_{0} A [\alpha uv\big|_0^l - \alpha \int_0^l v \frac{\partial v}{\partial x} dx]. $$ Since both ends of the bar are insulated, there's no heat flow at \(x = 0\) and \(x = l\), which means that \(\frac{\partial u}{\partial x}=0\) at these points, and so \(uv|_0^l=0\). Therefore, the equation simplifies to: $$ \frac{d E(t)}{dt} = -c_{0} A \alpha \int_0^l v \frac{\partial v}{\partial x} dx. $$

Show that the derivative of the thermal energy is zero

Since at both ends of the bar, the derivative of the temperature is zero (the bar is insulated), then the expression \(\int_0^l v \frac{\partial v}{\partial x} dx\) is always a positive value. Therefore, the equation becomes: $$ \frac{d E(t)}{dt} = 0. $$ This means that the thermal energy \(E(t)\) is invariant over time, as expected for an insulated bar.

Key concepts

Thermal Energy

Thermal energy within a body represents the total energy of its particles due to their motion and position. This form of energy is tightly connected with temperature as it is a measure of the internal movements of particles. In our case, when considering a thin insulated bar, thermal energy is described by integrating the temperature distribution over its length. This integration provides a measure for the total heat contained in the bar at any given time.

• It is dependent on the material's heat capacity.
• It relies on the cross-sectional area where heat is distributed.

By understanding thermal energy in this manner, we see why the energy content in an insulated bar could remain constant, as changes in temperature distribution would not alter the net energy with no heat entering or leaving the system.

Insulated Bar

An insulated bar is a simple physical model used to study heat flow. Think of it as a one-dimensional object that has no heat exchange with its environment. Because the ends of the bar are insulated, no thermal energy can escape or enter, making this model an excellent illustration of constant energy.

• Insulation ensures no energy loss over time.
• It sets the boundary conditions, where the temperature gradient is zero at both ends.

This lack of external heat exchange is critical in the context of calculating and showing that thermal energy remains constant over time for different internal conditions.

Partial Differential Equation

The heat equation, a type of partial differential equation (PDE), describes how temperature changes over time within a body. For the one-dimensional model of our insulated bar, this PDE relates the rate of change of temperature with respect to time to its spatial distribution.It is given by:\[\frac{\partial u(x,t)}{\partial t} = \alpha \frac{\partial^2 u(x,t)}{\partial x^2},\]where:

• \( u(x,t) \) is the temperature at any position \( x \) and time \( t \).
• \( \alpha \) is the thermal diffusivity, a material-specific constant.

PDEs like the heat equation are vital in modeling how different variables interact across both space and time. In essence, they illustrate the interconnectedness of thermal distributions within a static system.

Constant Energy

The constancy of thermal energy in a system is significant for understanding heat flow within insulated systems. In simple terms, for our insulated bar, ensuring that the thermal energy remains invariant means the system perfectly balances its energy internally without losses or gains.

• Energy constancy due to no heat flow through the boundaries.
• Demonstrated by showing the derivative of thermal energy with respect to time equals zero.

This concept not only confirms our theoretical understanding but also serves as a practical checking point for ensuring calculations in thermal systems are correct. With insulation, once set into a particular thermal state, the bar maintains this state unless an external force acts upon it.