Question

A thin copper rod, 4 meters in length, is heated at its midpoint, and the ends are held at a constant temperature of \(0^{\circ} .\) When the temperature reaches equilibrium, the temperature profile is given by \(T(x)=40 x(4-x),\) where \(0 \leq x \leq 4\) is the position along the rod. The heat flux at a point on the rod equals \(-k T^{\prime}(x),\) where \(k>0\) is a constant. If the heat flux is positive at a point, heat moves in the positive \(x\) -direction at that point, and if the heat flux is negative, heat moves in the negative \(x\) -direction. a. With \(k=1,\) what is the heat flux at \(x=1 ?\) At \(x=3 ?\) b. For what values of \(x\) is the heat flux negative? Positive? c. Explain the statement that heat flows out of the rod at its ends.

Short answer

Explain the statement "heat flows out of the rod at its ends." Answer: For \(0 \le x < 2\), the heat flux is negative, while for \(2 < x \le 4\), the heat flux is positive. The statement "heat flows out of the rod at its ends" means that heat is being transferred from the midpoint of the rod (at x=2) to both ends, where the left end of the rod has a negative heat flux, indicating the heat is flowing in the negative x-direction, and the right end of the rod has a positive heat flux, indicating the heat is flowing in the positive x-direction.

Step-by-step solution

Let's use the product rule of differentiation: \(T^{\prime}(x) = 40(4-x) - 40x\) This gives us the derived function for the temperature's rate of change. #Step 2: Determine the heat flux function# Now that we have the derivative of the temperature function, we need to multiply by a negative constant \(-k\) to obtain the heat flux function. In this case, \(k=1\).

Multiply the derived temperature function by the negative constant \(-k\) to obtain the heat flux function: \(H(x) = -k T^{\prime}(x) = -1 \times (40(4-x) - 40x) = -40(4-x) + 40x\) #Step 3: Determine the heat flux at x=1 and x=3# Now that we have the heat flux function, \(H(x) = -40(4-x) + 40x\), we can find the heat flux at x=1 and x=3.

Plug x=1 into the heat flux function: \(H(1) = -40(4-1) + 40(1) = -120 + 40 = -80\)

Plug x=3 into the heat flux function: \(H(3) = -40(4-3) + 40(3) = -40 + 120 = 80\) The heat flux at x=1 is -80 and at x=3 is 80. #Step 4: Determine the values of x for positive and negative heat flux# We have the heat flux function \(H(x) = -40(4-x) + 40x\). To find the values of \(x\) where the heat flux is positive or negative, we can solve for \(x\) when \(H(x) = 0\).

Let H(x) = 0 and solve for x: \(0 = -40(4-x) + 40x\) \(x = 2\) The heat flux changes sign at x=2. For \(0 \le x < 2\), the heat flux is negative, and for \(2 < x \le 4\), the heat flux is positive. #Step 5: Explain the statement# Now we need to explain the statement: "heat flows out of the rod at its ends."

Since heat flux is negative at \(x=0\) and positive at \(x=4\), it means that the heat is flowing in the negative x-direction at the left end of the rod and in the positive x-direction at the right end of the rod. Consequently, the heat is being transferred from the midpoint (at x=2, where heat flux changes sign) to both ends of the rod, confirming the statement that heat flows out of the rod at its ends.

Key concepts

Temperature Profile

Temperature profile refers to the variation of temperature along a body or medium. In this particular exercise, we are looking at a thin copper rod with a specific temperature profile that indicates how temperature changes from one end of the rod to the other. The temperature profile is mathematically represented as a function: \(T(x) = 40x(4-x)\), where \(0 \leq x \leq 4\).

This function clearly shows dependence on the position \(x\) along the rod. At both ends of the rod, where \(x = 0\) and \(x = 4\), the temperature is zero, highlighting that heat is not added or removed at the ends. Meanwhile, the peak temperature occurs halfway down the rod, defining a parabolic shape typical of quadratic functions. Understanding this profile helps in predicting how heat is distributed within the rod at steady state.

Differential Equations

Differential equations play a crucial role in understanding how variables change with respect to one another. In the case of the copper rod, the derivative of the temperature profile \(T(x)\), denoted as \(T'(x)\), helps us comprehend how temperature varies along the length of the rod.

To find \(T'(x)\), we differentiate the temperature function: \(T(x) = 40x(4-x)\). This differentiation involves applying the product rule, which is a method for finding the derivative of two multiplied functions. Using the product rule, we ascertain:

• The instantaneous rate of change of temperature at any point \(x\),
• Which is given by the derivative \(T'(x) = 40(4-x) - 40x = 160 - 80x\).

Ultimately, solving these equations enables us to derive other vital metrics like heat flux, which are critical in studying the dynamics of thermal systems.

Heat Transfer

Heat transfer explains how energy moves from one part of an object to another or between different objects. The concept is essential in determining how the temperature is balanced in the copper rod. In this example, we focus on heat flux, which is the rate of heat flow through a surface. It is determined by the formula: \(H(x) = -kT'(x)\), where \(k\) is the heat conductivity constant, given as 1 in this scenario.

The heat flux conveys the direction and rate of thermal energy spreading through the rod.

• At \(x=1\), the heat flux is calculated as \(H(1) = -80\) which implies heat moves towards the left end of the rod,
• While at \(x=3\), a value of \(H(3) = 80\) shows heat flows right.

This directional flow, influenced by the sign of \(H(x)\), is key in investigating thermal behaviors in materials.

Thermodynamics

Thermodynamics concepts, particularly those pertaining to the flow and conservation of energy, are fundamental in this exercise. The copper rod problem illustrates the redistribution of thermal energy through its demonstration of heat flowing from the central point outwards. This is explained by the thermodynamic principle that energy tends to spread or 'flow' from regions of higher concentration to lower concentration.

In simpler terms, heat energy introduced at the rod's center moves towards the less energetic ('colder') ends.

• For \(0 \le x < 2\), the heat flux is negative, signifying an outward flow toward the rod's starting point,
• While for \(2 < x \le 4\), positive heat flux indicates energy moving towards the other end.

This demonstrates thermodynamics' role in explaining how and why heat redistributes itself in such systems, striving for equilibrium where temperature differences are minimized.