## Analysis of Time for Water to Settle in Hose Level

For a 0.17" ID hose that is hundreds of feet long, it can take time for the water at both ends to settle to the same level. Here is the quantitative analysis.

In the well-known Poiseuille model of a fluid flowing through a cylindrical pipe, the volumetric rate of flow is

Φ=(π/8)(P/L)(R4/η)

where R is the radius of the pipe, L is the length, P is the pressure difference, and η is the viscosity. If the pressure is generated by gravity, then P=ρgΔz, where ρ is the density of the fluid, g is the acceleration due to gravity, and Δz is the difference between the fluid level at the two ends of the pipe.

As the hose level is usually used, the water level at the fixed end is maintained at a constant level and the free end is held in a vertical position and the water level in it is allowed to come to rest at the level at which the fixed end is maintained. In this situation,

Φ=πR2(dΔz/dt)

Substituting to eliminate P and Φ and rearranging leads to

dΔz/dt=(R2ρgΔz)/(8Lη)

The solution to this simple linear differential equation shows that the water level exponentially approaches equilibrium with characteristic time constant

τ=(8Lη)/(R2ρg)

The time that one must wait for the water level to equilibrate is several times this time constant.

The viscosity of water varies with temperature, but for typical outdoor temperatures it is close to 0.001 poise. For a hose with ID=0.17", τ is about 0.175 seconds per meter of hose length. That's less than a second for the 15' hose and about 2.7 seconds for the 50' hose, but it's almost 10 seconds for the 172' hose and over 30 seconds for the 575' hose! For multiple hoses connected together, you can simply add time constants. For all 4 hoses connected together, the time constant is almost 45 seconds. Although the connectors (R~=1mm, L~=4cm) have a smaller ID than the hose and restrict flow a little further, their effect on the time constant (typically 30ms) is negligible compared to the hoses themselves.

Clearly some patience is needed when using a long hose. One technique for diminishing the necessary time is to raise or lower the free end of the hose and observe whether the water level in it rises or falls, without waiting for it to come to a complete rest. The water level at the fixed end is then the level at which the water in the free end neither rises or falls.

### Fill time

This same theory can also be used to predict the time it takes to fill the hose, which can be significant. If the hose is assumed to be laid at constant slope and filled by siphoning from its upper end, simple calculus and algebra show that the time it takes to fill it completely is

τ(L/Δz)

This time can be reduced by laying the hose so that it drops as steeply as possible immediately below the siphon source at the top end, which increases the water level difference between the two ends (and hence the pressure difference and flow rate) as quickly as possible. The maximum theoretical improvement from this technique is half the fill time of the constant-slope case.