Surface Roughness on pressure drop

Literature Review:

The effects of surface roughness on pressure drop in circular and rectangular tubes have been studied extensively in literature.  As early as the nineteenth century, Darcy (1858) conducted careful pressure drop experiments on pipes of different materials and roughness.  Darcy made measurements on 21 pipes made of lead, glass, asphalt covered cast iron, wrought and cast iron pipes.  Except for the glass pipes, all pipes were 100 meters long with diameters ranging from 1.2 to 50 mm.  Darcy established that the flow depended on the pipe roughness, pipe diameter, and slope and introduced the concept of relative roughness.  Darcy also noted that at a certain point, friction factor is independent of the Reynolds number.  Fanning (1877) later proposed a correlation for the pressure drop as a function of surface roughness.  

The effect of surface roughness on pressure drop and heat transfer in circular and rectangular tubes have been studied extensively in literature (Put in references #a-#z).  The pioneering work of quantifying the effect of surface roughness on pressure drop was led by Nikuradse (1933).  Nikuradse established the sand grain roughness as a major parameter in defining the friction factor during laminar and turbulent flows.   To create the relative roughness on the pipes, he first sifted sand grains so that the grains were of uniform size.  He then used Japanese lacquer to stick the sand grains to the walls of the pipe.  Through exhaustive experimentation, Nikuradse studied the effect of roughness on flow characteristics in circular pipes.  Experimental data was obtained for water flowing in pipes of six different relative roughness surfaces with Re ranging from 600 to 106.  The dimensions of the test pipes were 25, 50, and 100mm.  His work helped established the effect of relative roughness (e/D), on the flow characteristics.  It should be noted that the pipe diameter used in his work was the base diameter of the pipe before applying the sand grains.

To ensure geometric similarity, Nikuradse used round pipes with the same roughness.  To do this, ordinary building sand was sifted through sieves.  The holes in the sieves varied slightly, but filtered out sand grains of the same relative size.  Once the sand was sifted, hundreds of sand grains diameters were measured using a Zeiss thickness gage accurate to 0.001mm.  From these sand grains, and arithmetical average was calculated and this was used as the surface roughness for a given pipe.

Now that the sand has been sifted and an average grain size has been computed, the sand is now ready to be adhered to the inside of a pipe.  First a pipe is stood on its end vertically with the bottom end closed.  The pipe is then filled with a think Japanese lacquer and then emptied.  The pipe is allowed to dry for 30 minutes.  This is the appropriate amount of time to allow the lacquer to become tacky, but not completely dry.  The pipe is then filled with sand of a pre-measured grain size.  Once the pipe is full, the sand is emptied out of the pipe bottom and the pipe is dried for 2-3 weeks depending on the moisture content in the air.  Uniform drying was accomplished by using an electric light bulb which circulated air through the pipe.  The pipe was then re-filled with the Japanese lacquer and allowed to dry for an additional 3-4 weeks.  This second coat of lacquer was done to better adhere the individual sand grains to the inner wall surface of the pipe.  After the pipes are completely dried, 10 centimeters of pipe was cut from the pipe end to reduce the risk of any reduction in diameter.  The pipes were then ready to be tested.

Nikuradse’s

Colebrook (1939) later published his findings on the variation in surface roughness and its effects on pressure drop.  Colebrook’s experiments included sixteen spun concrete-lined pipes and six spun bitumastic-lined pipes ranging in diameter from 4 inches to 60 inches.  Also tested was the 216-inch diameter Ontario tunnel.  The Ontario tunnel is a concrete tunnel made from steel oil forms.  When the concrete set and the forms were removed, the walls of the tunnel were rubbed down with carborundum brick to remove any deformities.   Average relative roughness values ranged from 0.04318 mm to 0.254 mm.  These pipes were tested in the transition and fully turbulent regions.  It was determined that the friction factor decreases as the velocity increases.  Once the velocity reaches the square law, the friction factor remains constant for all higher velocities.

Moody (1944) later characterized the friction factor (f) as a function of the Reynolds number and relative roughness (e/D).  Through Colebrook’s work, the Moody diagram was established which provided engineers with a convenient method of determining the pressure drop associated with the relative roughness of the pipe.  The outlined method of determining pressure drop loses its integrity when the surface roughness is on the same order of magnitude as the characteristic diameter of the pipe.  Although the current relative roughness values used in the plots go only to 5 percent, higher values are expected while dealing with flows in small diameter channels. 

More recently, work has been extended into rectangular microchannels.  Channels with hydraulic diameters ranging from 0.96 to 3200 have been investigated.  Test fluids include helium, nitrogen, argon, water, silicone oil, alcohol, air and isopropanol.  The results show that the pressure drop increases more dramatically as the relative roughness increases.  Thus fluids tend to enter the transition regime at lower Reynolds numbers in microchannels.  A literature review table of these papers has been compiled for quick reference as shown in Table 1.

Using one to one parameter mapping, the partial differential equations of motion are obtained which can fully describe a continuum fluid motion.(Eringen)  Using the right boundary conditions, these equations reduce to the Navier-Stokes which enhances the equations validity to model micro flow.  However, this simply proves that the Navier-Stokes equations fit the special case of simple micro-fluids in a continuum flow regime.

More experimental work was done by Peiyi and Little when they looked at microminiature refrigerators.  Test sections were made using lighographic processes on glass and silicon substrates.  Channels were etched into the substrate and a glass cover was bonded to the base to form a complete channel.  Using various channel dimensions and surface roughness values for each substrate, it was shown that many factors may affect the friction factor in micro channels.  The method of etching, channel dimensions, channel length, bonding methods, and test fluid characteristics may all affect the friction factor.  It is a combination of one or more of these factors that cause the data to disagree with the trends predicted by Moody.

Surface roughness has also been linked to transition from laminar to turbulent flow.  A fluid no longer reaches transition at a Reynolds number of 2300.  The increased surface roughness causes flow to transition at much smaller Reynolds numbers which may be beneficial or detrimental to the flow depending on the application.  However, it is the heat transfer characteristics that are of interest in turbulent flow.  The heat transfer co-efficient increases dramatically with the transition into turbulent flow.  Thus, a localized turbulent flow induced by just the right surface roughness may be incredibly beneficial in dissipating heat in atomic reactors, micro chip, heat exchangers and a variety of other applications.