Question

Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector \(\mathbf{F}\) at a point is proportional to the negative gradient of the temperature; that is, \(\mathbf{F}=-k \nabla T,\) which means that heat energy flows from hot regions to cold regions. The constant \(k>0\) is called the conductivity, which has metric units of \(\mathrm{J} / \mathrm{m}-\mathrm{s}-\mathrm{K}\) A temperature function for a region \(D\) is given. Find the net outward heat flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=-k \iint_{S} \nabla T \cdot \mathbf{n} d S\) across the boundary \(S\) of \(D\) In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that \(k=1\) $$\begin{aligned} &T(x, y, z)=100+x^{2}+y^{2}+z^{2}\\\ &D=\\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\\} \end{aligned}$$

Short answer

Answer: The net outward heat flux across the boundary is -6.

Step-by-step solution

Compute \(\nabla T\)

The gradient of the temperature function \(T(x, y, z) = 100 + x^{2} + y^{2} + z^{2}\) is: $$\nabla T = \left( \frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}, \frac{\partial T}{\partial z} \right) = (2x, 2y, 2z)$$

Compute \(\mathbf{F}\)

Now, we will compute the heat flux vector \(\mathbf{F}=-k \nabla T\). Since \(k=1\), we have: $$\mathbf{F}=-\nabla T = (-2x, -2y, -2z)$$

Apply the Divergence Theorem

We will use the Divergence Theorem to compute the surface integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} dS\), where \(\mathbf{n}\) is the unit normal vector on the boundary \(S\) of \(D\). The Divergence Theorem states: $$\iint_{S} \mathbf{F} \cdot \mathbf{n} dS= \iiint_{D} div(\mathbf{F})dV$$ We first need to compute the divergence of \(\mathbf{F}\). $$div(\mathbf{F})=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}=-2-2-2=-6$$ Now, we evaluate the triple integral over the region D: $$\iiint_{D} div(\mathbf{F})dV = -6 \iiint_{D} dV$$ Since D is the unit cube, this triple integral represents the volume of D, which is \(1^3 = 1\). Hence, $$\iiint_{D} div(\mathbf{F})dV = -6$$

Conclusion

Applying the Divergence Theorem, we find that the net outward heat flux across the boundary \(S\) of the region \(D\) is: $$\iint_{S} \mathbf{F} \cdot \mathbf{n} dS = -6$$

Key concepts

Heat Conduction

Heat conduction is a fundamental concept in the study of thermal processes. It describes how heat energy moves through materials. This movement happens due to the difference in temperature between different parts of a substance. Warm regions transfer heat to cooler ones until an equilibrium is established. Fourier's Law of heat conduction formalizes this idea mathematically. It states that the heat flow vector \(\mathbf{F}\) at any point in space is proportional to the negative gradient of the temperature, which is expressed as \(\mathbf{F} = -k abla T\). Here, \(k\) represents the thermal conductivity of the material and is always a positive value. The negative sign indicates that heat flows from hot to cold. Understanding heat conduction allows for better control and management of thermal systems in engineering applications.

In our exercise, this principle is applied using a temperature function \(T(x, y, z)\) over a defined region \(D\). Calculating the gradient of this function is a key step in determining how temperature changes throughout the specified region.

Gradient

The gradient is a vector operation that shows how a scalar field varies in space. For temperature, it tells us the direction and rate at which temperature changes. If we have a temperature function \(T(x, y, z) = 100 + x^2 + y^2 + z^2\), the gradient is calculated as \(abla T = \left( \frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}, \frac{\partial T}{\partial z} \right)\). This represents a vector with partial derivatives showing changes along the \(x\), \(y\), and \(z\) axes.

In our example, the gradient \(abla T\) becomes \((2x, 2y, 2z)\). This implies that at any point within the region \(D\), the temperature changes most rapidly along the vector indicated by these values. The gradient is thus crucial in defining the direction of heat flow, and allows us to move on to calculate the heat flux \(\mathbf{F}\).

Divergence Theorem

The Divergence Theorem is a powerful tool in vector calculus, connecting the flow of a vector field through a surface to the behavior of the vector field inside the surface. In simpler terms, it relates a surface integral of a vector field over a closed surface, to a volume integral of the divergence of the field within the enclosed volume. Mathematically, it's given by:\[\iint_{S} \mathbf{F} \cdot \mathbf{n} \ dS = \iiint_{D} \text{div}(\mathbf{F}) \ dV\] where \(\mathbf{F}\) is the vector field, \(S\) is the closed surface, \(\mathbf{n}\) is the normal vector, and \(D\) is the volume enclosed by \(S\).

This theorem is particularly useful in simplifying calculations. For our problem, it translates the task of evaluating a surface integral into a volume integral over the region \(D\). Since \(\text{div}(\mathbf{F}) = -6\) as calculated, the Divergence Theorem helps us ascertain the net outward heat flux, making complex surface integrals more manageable.

Heat Flux

Heat flux represents the rate of heat energy passing through a surface. It's the tangible quantity demonstrating how heat moves around or out of a domain. In practice, this is crucial for maintaining energy efficiency and safety in systems. The heat flux \(\mathbf{F}\) in Fourier's context is deduced from the gradient: \(\mathbf{F} = -k abla T\). Since \(k = 1\) in our example, \(\mathbf{F} = (-2x, -2y, -2z)\). This vector tells us the intensity and direction of heat transfer for any point in region \(D\).

Our ultimate goal was to find the net outward heat flux across the boundary \(S\) of \(D\). Using calculus tools like the Divergence Theorem, we see this as \(\iint_{S} \mathbf{F} \cdot \mathbf{n} \ dS = -6\). This calculation exemplifies how heat flux is analyzed over volume, demonstrating its importance in larger systems and processes.