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 57.4 m of copper pipe whose inside radius is 7.52 x 10- 3 m. When the water and pipe are heated from 22.7 to 72.8 °C, what must be the minimum volume of the reservoir tank to hold the overflow of water?
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 pipe when heated.
Step 1: Calculate the change in volume of the pipe:
Use the formula for the volume of a cylinder: V = πr^2h, where r is the radius and h is the height.
Given:
Inside radius of the pipe (r) = 7.52 x 10^-3 m
Initial temperature of the water and pipe (T1) = 22.7 °C
Final temperature of the water and pipe (T2) = 72.8 °C
Calculate the initial volume (V1) of the pipe at T1:
V1 = πr^2h1
Calculate the final volume (V2) of the pipe at T2:
V2 = πr^2h2
Step 2: Calculate the change in volume of the water:
The change in volume of the water can be calculated using the formula:
ΔV = βV0ΔT
where ΔV is the change in volume, β is the volumetric thermal expansion coefficient of water, V0 is the initial volume of water, and ΔT is the change in temperature.
Given:
Initial temperature of the water (T1) = 22.7 °C
Final temperature of the water (T2) = 72.8 °C
Calculate the volumetric thermal expansion coefficient of water (β):
β = 3.0 x 10^-4 °C^-1 (approximately)
Calculate the initial volume of the water (V0) at T1:
V0 = volume of the pipe at T1 (V1)
Calculate the change in volume of the water (ΔV):
ΔV = βV0ΔT
Step 3: Calculate the minimum volume of the reservoir tank:
The minimum volume of the reservoir tank must be able to accommodate the change in volume of both the pipe and the water. Therefore, the minimum volume of the reservoir tank (V_min) is calculated as:
V_min = ΔV_pipe + ΔV_water
where ΔV_pipe is the change in volume of the pipe and ΔV_water is the change in volume of the water.
Plug in the calculated values to find V_min.
To determine the minimum volume of the reservoir tank required to hold the overflow of water, we need to calculate the expansion of the water and the copper pipe separately.
First, let's calculate the expansion of the water:
1. Calculate the initial volume of the water:
- The initial temperature of the water is 22.7 °C.
- The density of water is approximately 1000 kg/m^3.
- The initial mass of water can be determined using the equation:
Mass = Density * Volume
- Since we're only interested in the volume, we rearrange the equation to:
Volume = Mass / Density
2. Calculate the final volume of the water:
- The final temperature of the water is 72.8 °C.
- To calculate the final volume, we can use the equation for thermal expansion:
ΔV = V * β * ΔT
- Where ΔV is the change in volume, V is the initial volume, β is the volume expansion coefficient of water (which is approximately 0.000210), and ΔT is the change in temperature.
3. Calculate the expansion of the water:
- The expansion of the water can be determined by subtracting the initial volume from the final volume:
Expansion of water = Final Volume - Initial Volume
Next, let's calculate the expansion of the copper pipe:
1. Calculate the initial volume of the copper pipe:
- The inside radius of the copper pipe is given as 7.52 x 10^(-3) m.
- The length of the copper pipe is 57.4 m.
- The initial volume of the pipe can be calculated using the equation:
Volume = π * r^2 * length
2. Calculate the final volume of the copper pipe:
- The copper pipe expands due to thermal expansion as well.
- The thermal expansion coefficient of copper is approximately 0.000016.
- The change in volume of the copper pipe can be calculated using the formula:
ΔV = V * β * ΔT
3. Calculate the expansion of the copper pipe:
- The expansion of the copper pipe can be determined by subtracting the initial volume from the final volume:
Expansion of pipe = Final Volume - Initial Volume
Finally, to determine the minimum volume of the reservoir tank required, we add the expansion of the water and the expansion of the pipe.
Minimum Volume of the Reservoir Tank = Expansion of water + Expansion of pipe