Problem 424 - Calculus Volume 3

Question

For the following exercises, Fourier's law of heat transfer 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 hot regions to cold regions. The constant \(k>0\) is called the conductivity, which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region \(D\) is given. Use the divergence theorem to find net outward heat \(\quad\) 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 \(D,\) where \(k=1\). \(T(x, y, z)=100+x+2 y+z ;\) \(D=\mid(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\\}\)

Step-by-Step Solution

Answer

The net outward heat flux across the boundary is 0.

Step 1: Determine the Gradient of the Temperature

The gradient of the temperature function \( T(x, y, z) = 100 + x + 2y + z \) is calculated by taking the partial derivatives with respect to \( x \), \( y \), and \( z \). \( abla T = \left( \frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}, \frac{\partial T}{\partial z} \right) = (1, 2, 1) \).

Step 2: Apply Fourier's Law of Heat Transfer

Fourier's Law states that the heat flow vector \( \mathbf{F} = -k abla T \). Substituting \( k=1 \), we find \( \mathbf{F} = -(1, 2, 1) = (-1, -2, -1) \).

Step 3: Compute the Divergence of the Heat Flow Vector

The divergence of \( \mathbf{F}\) is \( abla \cdot \mathbf{F} = \frac{\partial (-1)}{\partial x} + \frac{\partial (-2)}{\partial y} + \frac{\partial (-1)}{\partial z} = 0 \). The divergence of \( \mathbf{F} \) within the region \( D \) is zero.

Step 4: Use the Divergence Theorem

The divergence theorem relates the surface integral over \( S \) to a volume integral over \( D \). Since \( abla \cdot \mathbf{F} = 0 \), the volume integral \( \iiint_{D} abla \cdot \mathbf{F} \ dV = 0 \).

Step 5: Conclude the Heat Flux Calculation

According to the divergence theorem, \( \iint_{S} \mathbf{F} \cdot \mathbf{N} \ dS = \iiint_{D} abla \cdot \mathbf{F} \ dV \). Given that \( abla \cdot \mathbf{F} = 0 \), the net outward heat flux across \( S \) is \( 0 \).

Key Concepts

Gradient of Temperature

The gradient of temperature is an essential concept that helps us understand how temperature changes in a particular region. In calculus, the gradient (\( abla T \)) of a temperature function represents the direction and rate of the most significant increase in temperature. It is expressed as a vector that contains partial derivatives with respect to each variable in the temperature function.

For the example function \( T(x, y, z) = 100 + x + 2y + z \), this involves finding the derivatives with respect to \( x \), \( y \), and \( z \). The resulting gradient is \( abla T = (1, 2, 1) \), indicating how the temperature changes in the region along these axes.

• The gradient tells us that the temperature increases at a rate of 1 unit per meter when moving along the x-axis,
• 2 units per meter along the y-axis,
• and 1 unit per meter along the z-axis.

Such information is vital in physics and engineering to understand heat distribution and dynamics.

Divergence Theorem

The divergence theorem is a critical tool in vector calculus, connecting the flux of a vector field across a surface to the behavior of the vector field inside the surface's boundary. It states that the net flow (flux) of a vector field through a closed surface is equal to the integral of the divergence over the volume bounded by the surface.

In this context, the theorem is used to calculate heat flux across the boundary \( S \) of any given region \( D \). By considering the divergence of the heat flow vector, \( abla \cdot \mathbf{F} \), we learn about the circulation of heat within the region. For the given exercise, since the divergence is \( 0 \), the heat is evenly balanced inside the region, with no net production or loss internal to \( D \).

The practical use of the divergence theorem allows us to avoid directly computing complex surface integrals, simplifying the problem to a volume integral, which can often be more straightforward to solve.

Heat Flux Calculation

Heat flux calculation is about determining how much heat energy passes through a surface in a given time. It relies heavily on Fourier's Law of Heat Transfer, which tells us that the heat flow vector \( \mathbf{F} \) is related to the temperature gradient. Understanding this helps determine how rapidly and in what direction heat will flow from hot to cold areas.

In the problem, Fourier's Law gives us \( \mathbf{F} = -k abla T \). Here, since \( k = 1 \), we found \( \mathbf{F} = (-1, -2, -1) \), indicating the direction and rate at which heat travels.

The net heat flux problem then asks us to understand this vector field's behavior across a closed surface \( S \). With the divergence theorem applied, showing \( abla \cdot \mathbf{F} = 0 \), the net outward heat flux through \( S \) turns out to be zero. Thus, within region \( D \), there is no net gain or loss in heat, leading to temperature stability across the boundary.