A solid cylindrical rod made from a center cylinder of lead and an outer concentric jacket of copper.Except for its ends,the rod is insulated,so that the loss of heat from the curved surface is negligible.When a temperature difference is maintained between its ends,this rod conducts 1/2 the amount of heat that a copper would conducts.Determine the ration of the radii r1/r2.
Thermal conductivity of copper=390,lead=35
To determine the ratio of the radii r1/r2, we need to use the concept of thermal conductivity.
The heat conducted through a material is given by the formula:
Q = ((k * A * ΔT))/d
Where:
Q is the amount of heat conducted
k is the thermal conductivity of the material
A is the cross-sectional area perpendicular to the direction of heat flow
ΔT is the temperature difference
d is the thickness of the material in the direction of heat flow
Given that the rod conducts 1/2 the amount of heat that a copper rod would conduct, we can set up the following equation:
(Q_lead)/(Q_copper) = 1/2
Since the rod is made up of a center cylinder of lead and an outer concentric jacket of copper, the area of lead (A_lead) is given by:
A_lead = π * (r1^2 - (r1 - d)^2)
where r1 is the radius of the outer copper jacket and d is the thickness of the copper jacket.
The area of copper (A_copper) is given by:
A_copper = π * (r2^2 - r1^2)
where r2 is the radius of the inner lead cylinder.
Using the formula for heat conducted, we can rewrite the equation as:
((k_lead * A_lead * ΔT))/d_lead) / ((k_copper * A_copper * ΔT))/d_copper)) = 1/2
Substituting the given thermal conductivities:
((35 * A_lead * ΔT)/d_lead) / ((390 * A_copper * ΔT)/d_copper)) = 1/2
Now, let's solve for the ratio r1/r2:
((35 * (π * (r1^2 - (r1 - d)^2)) * ΔT)/d_lead) / ((390 * (π * (r2^2 - r1^2)) * ΔT)/d_copper)) = 1/2
Simplifying the equation further:
((35 * (r1^2 - (r1 - d)^2)) / (390 * (r2^2 - r1^2)) = (d_copper * 2) / (d_lead)
Now, we can substitute the value of d_copper / d_lead with the thickness ratio, r1 / r2:
((35 * (r1^2 - (r1 - d)^2)) / (390 * (r2^2 - r1^2)) = (r1 / r2) * (2 / 1)
Simplifying the equation:
35 * (r1^2 - (r1 - d)^2) = 780 * (r1^2 - r2^2)
Now, we can solve this equation to find the value of r1/r2.
To determine the ratio of the radii (r1/r2) of the cylindrical rod, we need to use the formula for the thermal conductivity of a composite material.
The rate of heat conduction through a cylindrical rod is given by:
Q = (kAΔT)/L
Where:
Q = Rate of heat conduction
k = Thermal conductivity
A = Area of cross-section
ΔT = Temperature difference
L = Length of the rod
Let's assume the outer radius of the rod is r1 and the inner radius is r2.
The area of the outer surface (copper) is given by:
A1 = 2πr1L
The area of the inner surface (lead) is given by:
A2 = 2πr2L
Since the rod is insulated on the curved surface, we neglect the heat loss from the curved surface, so the heat conduction only occurs from the ends of the rod, which have the same temperature difference.
Given that the rod conducts half the amount of heat that a pure copper rod would conduct, we can conclude that the thermal conductivity of the composite rod is half that of pure copper. In other words:
(k1A1ΔT)/L = (1/2) * (k2A2ΔT)/L
Where:
k1 = Thermal conductivity of copper
k2 = Thermal conductivity of lead
Substituting the values, the equation becomes:
(390 * 2πr1L * ΔT)/L = (1/2) * (35 * 2πr2L * ΔT)/L
Simplifying the equation:
390 * 2πr1 = 1/2 * 35 * 2πr2
390r1 = 17.5r2
Finally, we can determine the ratio of the radii (r1/r2) by dividing both sides of the equation by r2:
(r1/r2) = 17.5/390
After calculating, the ratio of the radii (r1/r2) is approximately 0.0449.