Question

many hot-water heating systems have a reservoir tank connected directly to the pipeline, so as to allow for expansion when the water becomes hot. the heating system of a house has 52.1 m of copper pipe whose inside radius is 9.68 × 10⁻³ m. when the water and pipe are heated from 23.3 to 70.2 °c, what must be the minimum volume of the reservoir tank to hold the overflow of water?
number
units
m³

Explanation:

Step1: Calculate the initial volume of water

The volume of a cylinder is \(V = \pi r^{2}L\). Given \(r = 9.68\times10^{- 3}\text{ m}\) and \(L = 52.1\text{ m}\).

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$$V_{0}\approx1.54\times 10^{-2}\text{ m}^{3}$$

Step2: Use the formula for volume expansion

The formula for volume expansion is \(\Delta V = V_{0}(\beta_{water}-\beta_{copper})\Delta T\). The coefficient of volume expansion for water \(\beta_{water}=207\times 10^{-6}\text{ C}^{\circ - 1}\) and for copper \(\beta_{copper}=51\times 10^{-6}\text{ C}^{\circ - 1}\), and \(\Delta T=T_{2}-T_{1}=70.2 - 23.3=46.9^{\circ}\text{C}\)

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Answer:

\(1.13\times 10^{-4}\)