How does the volume of water a plumbing system holds vary with water pressure? I know the ratio would include the modulus of elasticity of the plumbing material, the total surface area of the plumbing system, and probably the ratio of volume to area.
A quick check of a sphere and a cube shows me the volume varies as the 3/2 power of the surface area. However, I'm not sure of the relationship of surface area to pressure. Thus, I'm not sure how the volume of the pipes would change given a change in pressure.
Best Answer
The stress induced in a pipe by internal pressure is $$P_r \over t$$ where $P$ is the applied pressure, $r$ the pipe radius and $t$ the pipe wall thickness.
The strain induced in the pipe wall is $$\Delta r \over r$$ where $\Delta r$ is the change in radius.
The basic stress strain relationship is $$\sigma = E \epsilon$$ where $\sigma$ is the stress, $E$ the Young's modulus of elasticity of the pipe material and $\epsilon$ is the strain.
So from the above: $${Pr\over t} = {E {\Delta r \over r}}$$ and $${\Delta r} = {Pr^2 \over Et}\ .$$ The next step is to calculate the original and the new area of the pipe from $\pi r^2$ and $\pi(r + \Delta r)^2$.
The difference between the two areas is the change in area which when multiplied by the length of the pipe gives the change in volume under the pressure P.