Earth Tube Equations
Posted on March 16, 2014 by
The Equations!
Pressure Drop
Professional engineers and HVAC designers calculate the pressure drop (head loss) for a given duct system. You can use similar equations for earth tube design. If the pressure losses in the system exceed the pressure driving the flow (passive or active), the flow will stop and the duct will become useless. When air flows through a duct, there is pressure drop due to friction losses as well as dynamic losses which are caused by change in direction or velocity (usually at the fittings). For a commercial HVAC duct system, 1 Pa/m loss is typical.
Pressure drop can kill a passive system or require an active system to use a lot more electricity (and make more noise) than would other wise be necessary.
Frictional Pressure Losses
I will move the actual calculation to this page (link coming).
However, even without entering numbers, the equations tell most of what you need to know about the relationship between the parameters. Understanding these relationships leads to design insight.
The friction losses are due to viscous interactions between the air and the pipe walls and can be expressed with the D’ArcyWeisbach Equation. In this equation;
Some things are immediately apparent from the equation; for instance, the pressure drop is proportional to the friction factor. In other words, the rougher the pipe wall, the higher the frictional pressure loss (which seems pretty obvious). This should affect your material choice; PVC is 200 times smoother than concrete, HDPE is even smoother. Increased length is also a factor; while we want length to provide more contact for heat exchange, too much and the flow could stop. Velocity is very important because this is squared. In other words, if you double the velocity, the pressure drop is affected by a power of 4. If we reduce our velocity from 700 to 175 ft/min, we reduce the velocity by a factor of 4, and our pressure losses by a factor of 16. Passive systems tend to move the air relatively slowly, but adding a fan to increase the velocity may actually be counter productive (choose a fan with high pressure rather than high velocity ratings).
On the bottom of the expression, we see the hydraulic diameter (proportional to pipe radius), which means increasing the pipe radius reduces the frictional pressure drop.
Dynamic Pressure Losses
The dynamic pressure losses can usually be found in tables. For instance, for a given velocity, the pressure drop around a 90 degree mitered turn may be 50%. The same flow around a smooth bend may only lose 15%. The problem is that most of the tables are for higher velocities used in home or industrial HVAC systems. However, looking at these charts, you quickly get the idea that fewer fittings is a good idea. You can also see that lower velocities have lower losses for a given fitting.
Gentle curves are much better than tight turns. A gentle turn has a radius at least 6 times the pipe diameter. However, even gentle curves or serpentine layouts cause the flow to change direction and, therefore, induce dynamic losses that will reduce your flow rate or increase your energy costs.
The charts also show that diverging or converging sections should be as gradual as possible. Most HVAC texts suggest that divergence should not exceed 12° and convergence should not exceed 30°.
Reynolds Number and Turbulence
A second equation, important for understanding earth tubes, is “Reynolds number”. When scientists were studying flow, they knew that it was sometimes laminar and then as the velocity would increase, it would transition to turbulent. They could graph it but it took until 1883 before Osborn Reynolds showed that the change depended on ρVd/μ, which was named “Reynolds Number” in his honor (and usually shown as “Re”).
This is important because the turbulence of the flow affects both the pressure drop and heat transfer rate of the system. By calculating the Reynolds number for a given design, you can predict if the flow will be turbulent or laminar. In a viscous flow, friction is able to stop the air molecules adjacent to the wall. If the flow is laminar, layers form with each layer away from the wall moving a little faster than the layer below it. This forms a “boundary layer profile”. The flow is called “laminar” because these layers are stable. Streamlines stay nice and straight and never cross. While the flow at the walls is stopped, the majority of the flow moves easily and smoothly thru the duct/pipe. The problem is that heat transfer between the walls and the majority of the flow is greatly reduced. In turbulent flow, the fluid is always mixing and the system is better able to transfer heat from the walls to the majority of the flow or vice versa.
Many Earth tube designers incorrectly assume that their flow will be laminar. They tell you that you need to “add turbulence” to increase heat transfer by choosing tubes with rough or corrugated surfaces, laying the tubes in serpentine patterns, etc. These “enhancements” increase pressure drop dramatically. A few quick Re calculations show that they are not necessary for most earth tubes.
Reynolds number is proportional to the density, velocity and diameter and inversely proportional to the dynamic viscosity. The density and viscosity are properties of the fluid (such as air) and are both inversely proportional to the temperature. Thicker fluids (like syrup) have higher viscosity and tend to form laminar flows (Low Re), air is not very “viscous” and goes turbulent easily. The velocity and diameter are aspects of the duct design, increasing either parameter will increase your Reynolds number and turbulence.

ρ = fluid density
 V = mean flow velocity
 d = hydrolic diameter (inside tube diameter) (keep in mind that this may be different from the “nominal diameter”)
 μ = dynamic viscosity of the fluid
Lower Reynolds number flows are laminar. Higher Reynolds number flows are turbulent. For a round duct/pipe, this transition happens around Re~2300. We can easily calculate a table of Reynolds numbers for various nominal duct sizes (actual diameters would vary based on duct material). In this chart (above), I have colored Reynolds numbers >2300 red. These are turbulent flows.
Using the velocity and the nominal diameters (again, the actual internal diameters would vary based on duct material), we would get this table showing cubic feet per minute. Again, the “turbulent” flows are colored red. If you can stay below these flow rates, you may have laminar flow (flow could still be made turbulent by seams/joints, dirt, upstream turbulence from the fan, etc.) which would flow with less resistance, but much less heat transfer (much less slope for pressure drop over velocity).
A related question is how quickly the turbulence will form. Assuming the flow enters the duct as laminar flow (unlikely), how far will it go before it becomes fully turbulent?
Friction between the flow and the walls (friction exists even in a relatively smooth pipe) will bring the molecules immediately adjacent to the wall to a stop. This slows the flow next to it, and the flow next to that, etc. The result is a growing boundary layer profile that shows the gradient between the stopped flow at the wall and the free stream velocity. This boundary layer grows as the flow moves down the pipe until it meets in the middle and a stable flow profile develops. If the viscosity is high enough that the Re<2300, the flow can remain laminar. However, if the flow is not viscous enough, the friction at the wall can actually cause some flow reversal (wall roughness can cause this to happen even sooner). This flow reversal starts transition to a fully turbulent flow. A boundary profile can still develop in a turbulent flow, but it is really the “mean” turbulent velocity profile; the average of many small fluctuations in velocity and direction. Since this average is relatively constant, the resulting wall shear is constant and the pressure drop becomes linear with X.
The distance before this stable profile develops is a function of the Reynolds number and Diameter and can also be calculated. Often this distance is expressed over the diameter. For all the Reynolds numbers on the above chart, this works out to between 18 and 20 times the diameter, which for these pipes is between 6 and 20 ft. Any ridges, fans, screens or other upstream obstacles will only cause this to happen sooner.
Just incase my point got lost in the engineer speak… here it is plainly. Turbulence is good for heat conduction, but trying to intentionally induce additional turbulence is unnecessary and bad for pressure loss. In designing your system, you can assume the flow in your earth tubes will be turbulent no matter how smooth the walls are. There is no need to add features to increase turbulence, they will only increase your back pressure and reduce your flow.