A heat-conducting rod consists of an aluminum section, 0.20 m long, and a copper section, long. Both sections have a cross-sectional area of The aluminum end of the rod is maintained at a temperature of , while the copper end is maintained at What is the temperature at the point where the aluminum and copper sections meet? The thermal conductivity of aluminum is 205 W/m ∙ K, and of copper is 385 W/m ∙ K.
80 celcious
To find the temperature at the point where the aluminum and copper sections meet in the heat-conducting rod, we can use the concept of thermal conductance.
Thermal conductance (C) is a measure of how well a material conducts heat. It is defined as the rate of heat conduction through a unit area of a material per unit temperature difference across it.
Mathematically, thermal conductance (C) can be calculated using the formula:
C = (k x A) / L,
where C is the thermal conductance, k is the thermal conductivity, A is the cross-sectional area, and L is the length of the material.
In the given problem, we are given the thermal conductivity (k), cross-sectional area (A), and length (L) for both the aluminum and copper sections of the rod. However, we do not have the temperature difference (ΔT) across each section.
To solve for the temperature at the point where the aluminum and copper sections meet, we need to assume steady-state conditions, where the heat flow is constant throughout the rod.
The heat flow (Q) through each section of the rod can be calculated using Fourier's law of heat conduction:
Q = (k x A x ΔT) / L,
where Q is the heat flow, k is the thermal conductivity, A is the cross-sectional area, ΔT is the temperature difference across the section, and L is the length of the section.
Since the heat flow through each section of the rod is the same and assuming there is no heat loss or gain along the length of the rod, we can equate the heat flow equations for the aluminum and copper sections:
(k_aluminum x A_aluminum x ΔT_aluminum) / L_aluminum = (k_copper x A_copper x ΔT_copper) / L_copper,
where k_aluminum and k_copper are the thermal conductivities of aluminum and copper, A_aluminum and A_copper are the cross-sectional areas of aluminum and copper, ΔT_aluminum and ΔT_copper are the temperature differences across the aluminum and copper sections, and L_aluminum and L_copper are the lengths of the aluminum and copper sections.
We can rearrange this equation to solve for ΔT_copper, the temperature difference across the copper section:
ΔT_copper = (k_aluminum x A_aluminum x ΔT_aluminum x L_copper) / (k_copper x A_copper x L_aluminum).
Finally, we can use the formula for temperature difference to find the temperature at the point where the aluminum and copper sections meet:
T_meet = T_aluminum + ΔT_copper,
where T_meet is the temperature at the point where the aluminum and copper sections meet, and T_aluminum is the temperature at the aluminum end of the rod.
Plug in the given values for the thermal conductivity, lengths, cross-sectional areas, and temperature at the aluminum end, and you can find the temperature at the point where the aluminum and copper sections meet using the above equation.
To find the temperature at the point where the aluminum and copper sections meet, we can use the principle of thermal conduction. The heat flow through each section of the rod can be calculated using Fourier's Law:
Heat flow, Q = (Thermal conductivity) x (Cross-sectional area) x (Temperature difference) / (Length)
Let's assume the temperature at the point of junction is T.
The heat flow through the aluminum section can be expressed as:
Q_aluminum = (205 W/m ∙ K) x (cross-sectional area) x (T - )
And the heat flow through the copper section can be expressed as:
Q_copper = (385 W/m ∙ K) x (cross-sectional area) x ( - T)
Since the heat flow through the entire rod is the same, we can equate the two expressions:
(205 W/m ∙ K) x (cross-sectional area) x (T - ) = (385 W/m ∙ K) x (cross-sectional area) x ( - T)
The cross-sectional area cancels out, so we have:
205 (T - ) = 385 ( - T)
Now, let's simplify the equation:
205T - 205T = - 385 + 385T
+ 385T - 205T = 385
180T = 385
T = 385 / 180
T ≈ 2.139 °C
Therefore, the temperature at the point where the aluminum and copper sections meet is approximately 2.139 °C.