When we design pumping systems, we are out to pick a pump
and pipe combination that helps us to be efficient. In terms of the flow rate,
we are trying to achieve the amount of pressure we need to get there and to get
the pump to operate in an efficient range of it is operating rate.
There are several approaches to doing this, ranging from the
Hazen-Williams equation (an empirical relationship which relates the flow of water
in a pipe with the physical properties of the pipe and the pressure drop caused
by friction) to the Darcy-Weisbach method (which is does essentially the same
thing, but takes into account fluid density as well as viscosity, so it can be
used with fluids other than water). So, this brings us to the question, which
one should we use when we are trying to estimate manure flow rates?
The truth is even though Hazen-Williams is only for water, in
most cases it will be accurate enough for manure and give us an idea of what is
happening, but at least in theory the Darcy-Weisbach method is more accurate if
viscosity becomes high. This leads us to a discussion of viscosity, how manures
are different than water as a fluid, and what this means.
Viscosity is a measure of a fluid’s resistance to motion
under an applied force. Our classic example of this is to compare water and
honey; honey has a thicker consistency so when we coat our spoon with it and
tip the spoon it flows off slowly, while the water slides right off, so we say
the honey is more viscous than water. From this it may become clear that if we
were trying to pump honey with the same setup we were pumping water with, the
flow rate would be much lower, because the viscosity makes the honey harder to
pump.
But what about manure – this is where things get even more
complicated. Water and honey are what we call Newtonian fluids. Their viscosity is dependent only on the
temperature of the fluid, but manure is more complex than that and is a
shear-thinning fluid, meaning its viscosity if dependent on temperature, but
also on the shear rate (basically the flow rate). The faster we shear it the
easier it becomes to shear it.
Here we are going to look at a slightly different question,
how does the solids content of the manure, impact viscosity. You can see this
relationship in figure 1, basically higher solids content leads to more
viscosity. For reference, the viscosity of water is essentially 1 centipoise so
if we take all the solids out of the liquid manure, it is really close to
water, but a solids content of about 8% which is pretty typical of deep pit
finishing swine manures we have a viscosity of 3.65 centipoise (or higher than
the viscosity of water). This may sound like a lot, but it is essentially the
same as kerosene (honey on the other hand is 1,500).
Figure 1. Relationship between solids content in swine
finishing manure and the measured viscosity.
To see what this might mean, let’s assume we have an 8-inch
diameter pipe and the fluid inside is flowing at 10 feet/second, we’ll just
assume smooth pipe that is 1-mile long. Our question will be how much extra
pressure is needed if the viscosity of the manure is 3.65 centipoise instead of
the 1 centipoise it would be for water. This can be calculated using equation
1, where hf is the head loss in feet, f is the Darcy friction
factor, L is the length of the pipe, D is the diameter, V is the velocity of
the fluid, and g is the gravitational constant.
hf = fLV2/(2Dg) (1)
You may note that viscosity doesn’t show up anywhere in this
equation and wonder what all the fuss was about, but it is hidden in the Darcy
friction factor, f, which can be approximated using equation 2.
1/sqrt(f) = 2log(Re*sqrt(f)) -0.8 (2)
For
water under these conditions, I’d estimate we need to supply about 136 feet of
pressure head. If we repeat the calculation for manure assuming a viscosity
that is 3.65 times higher like we measured on in our viscosity analysis it
would only reduce the pressure loss to 172 feet, or by about 25%. Just for fun,
if you compare this to what we’d get using the Hazen-Williams equation, you’d
estimate something like 160 feet of heat loss in those flow conditions. All this to say, while in theory it seems like
it would be nice to incorporate viscosity into our flow estimation, generally
the uncertainties in all the variables and the variability of manure make using
the Hazen-Williams approach good enough for flow and pumping problems.
