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 74 m of copper pipe whose inside radius is 9.10 10-3 m. When the water and pipe are heated from 20 to 75°C, what must be the minimum volume of the reservoir tank to hold the overflow of water?

m3

The volume of the pipe is

pi R^2 L = 1.93*10^-2 m^2 = 19.3 liters

Calculate how much that volume of water increases when being heated from 20 to 75 C. You will need the bulk thermal expansion coefficient of water. It varies a lot from 20 to 75 C, but a suitable mean value is 4*10^-4 K^-1. See
http://hypertextbook.com/physics/thermal/expansion/

delta V = V*4*10^-4*(75 - 20)
= 0.43 liters

That volume expansion will be the minimum reservoir volume needed.

To find the minimum volume of the reservoir tank required to hold the overflow of water, we need to calculate the change in volume of the water and copper pipe when heated from 20 to 75°C.

First, we calculate the initial volume of water in the copper pipe using the formula for the volume of a cylinder:

V_initial = π * r^2 * h

Where:
V_initial is the initial volume of water
π is a mathematical constant approximately equal to 3.14159
r is the inside radius of the copper pipe
h is the length of the copper pipe

Given that the inside radius of the copper pipe is 9.10 x 10^-3 m, and the length of the pipe is 74 m, we can plug these values into the formula:

V_initial = π * (9.10 x 10^-3)^2 * 74

Next, we calculate the final volume of water in the copper pipe after it has been heated to 75°C. We use the same formula, but with the final radius of the pipe (which will be larger due to expansion) and the same length:

V_final = π * r_final^2 * h

To find the final radius of the copper pipe, we can use the formula for linear thermal expansion:

ΔL = α * L_initial * ΔT

Where:
ΔL is the change in length of the copper pipe
α is the coefficient of linear expansion for copper (approximately 0.000016)
L_initial is the initial length of the copper pipe
ΔT is the change in temperature

Given that the change in temperature ΔT is from 20 to 75°C, we can calculate ΔL:

ΔL = α * 74 * (75 - 20)

Now, we can calculate the final radius of the copper pipe:

r_final = r_initial + (ΔL / h)

Finally, we can plug the values for r_final and h into the formula to calculate V_final.

The change in volume, ΔV, will be the difference between V_final and V_initial:

ΔV = V_final - V_initial

This change in volume represents the overflow of water that needs to be accommodated by the reservoir tank. Therefore, the minimum volume of the reservoir tank required will be equal to ΔV.