### Velocity Profile in a Circular Pipe

This Excel spreadsheet calculates and plots the velocity profile of a power-law fluid in a circular pipe. The spreadsheet is valid for Newtonian, pseudoplastic and dilatant fluids.

The velocity profile of a Power-law fluid is given by this equation.

where
• r is the radial position i.e. distance from the center line (m)
• R is the pipe radius (m)
• u(r) is the axial velocity (m s-1)
• dp/dz is the axial pressure gradient (Pa m-1)
• K is the flow consistency index (Pa sn)
• n is the flow behavior index
The equation above is only valid for developed steady-state laminar flow in a circular pipe. The entrance length (that is, the pipe length needed to develop a steady-state velocity profile) for laminar flow is given by this correlation.

The value of n determines the relationship between the axial velocity and radial distance. If
• n = 1, the fluid is Newtonian
• n < 1, the fluid is pseudoplastic
• n > 1, the fluid is dilatant
The chart below gives the velocity profile for different values of n (with all other parameters constant).

The spreadsheet is simple to use. Simply enter the parameters

Excel will then plot the velocity profile from the centerline to the surface of the pipe. As expected, the velocity profile is symmetrical.

The transient velocity profile in laminar flow is more difficult to calculate; this requires the numerical solution of a partial differential equation via a finite difference scheme. This will be the subject of another spreadsheet.

### Calculate Gas Viscosity with Sutherland's Law

This Excel spreadsheet calculates the viscosity of gasses with Sutherland's Law. The spreadsheet contains constants suitable for air, but can be used for other gases.

William Sutherland was an Australian scientist who studied the temperature-dependence of ideal gases. In 1893, he developed an empirical-theoretical relationship between the temperature and viscosity of an ideal gas.

Essentially, the method uses a known reference viscosity and temperature to find another viscosity at a specified temperature.

Sutherland's formula is

where
• μ is the dynamic viscosity (Pa s or kg m-1 s-1)
• T is the temperature (K)
• μref is a reference viscosity (Pa s or kg m-1 s-1)
• Tref is a reference temperature (K)
• S is the Sutherland Constant for the gas (K)

The Sutherland Constant is characteristic for the gas. For example,
• air at 323 K has a viscosity of 1.716 x 10-5 Pa s, with a Sutherland Constant of 110 K
• helium at 273K has a viscosity of 1.9 x 10-5 Pa s, with a Sutherland Constant of 79.4 K

• finds the density of the gas with the Ideal Gas Law
• calculates the kinematic viscosity by dividing the dynamic viscosity by the gas density

The Ideal Gas Law is given by this formula

where
• R is the Universal Gas Constant (8314.4 J kmol-1 K-1)
• M is the molecular weight of the gas (kg kmol-1)
• P is the pressure (Pa)
• ρ is the density (kg m-3)
The molecular weight of air, for example, is 28.96 kg kmol-1.

Sutherland's Law is accurate for moderate temperatures and pressures, and also describes the viscosity-temperature relationship of gasses at hypersonic speeds.

### Historical Crude Oil Prices

These two Excel spreadsheets automatically download historical crude oil prices straight into Excel. Both tools are fully automated; the most you do is supply dates and click a button.

Some VBA then downloads the data from a 3rd party website, straight into Excel. After the data is in Excel, you can do whatever you want - visualize price historical, perform economic analysis, correlate oil prices against the cost of other raw materials, and more.