A thermal engineer must design a heat-delivery system that will raise the temperature…
PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics
Problem
A thermal engineer must design a heat-delivery system that will raise the temperature of a 0.30kg glycerin sample from 22∘C to 60∘C in no more than 5.0min. The proposed design uses two solid cylindrical rods, placed in parallel between a 180∘C thermal reservoir and the glycerin container.
• Rod A is copper, length 0.40m, cross-sectional area 1.5×10−4m2.
• Rod B will be made of aluminum and will have the same length as Rod A, but its cross-sectional area AB can be chosen freely.
Assume the lateral surfaces of both rods are perfectly insulated and that no energy is lost from the glycerin except through the rods.
Useful data:
• Copper: thermal conductivity kCu=400W⋅m−1⋅K−1
• Aluminum: thermal conductivity kAl=240W⋅m−1⋅K−1
• Glycerin: specific heat cgly=2400J⋅kg−1⋅K−1
(a) Determine the minimum total thermal power that must reach the glycerin so that the stated temperature rise can be achieved within the 5.0min time limit.
(b) Calculate the thermal power conducted through Rod A alone when the glycerin is still near its initial temperature.
(c) Derive an expression for the required cross-sectional area AB of Rod B that will ensure the total power found in part (a) is met. Then compute the numerical value of AB.
(d) Suppose both rods were instead manufactured to be 0.80m long while their cross-sectional areas remained the same as in parts (a)–(c). Explain qualitatively and quantitatively whether the area AB you found would still meet the 5.0min criterion.