GNG1106 – Fundamentals of Engineering Computation
Determining fluid flow through pipes and tubes has great relevance in many areas of engineering
and science. In mechanical and aerospace engineering, typical applications include the flow of
liquids and gases through cooling systems.
The resistance to flow in such conduits is parameterized by a dimensionless number called the
friction factor. For turbulent flow, the Colebrook equation provides a means to calculate the
friction factor,
1 2 . 5 10 2 . 0 l o g 3 . 7 ReDff
Equation 1
where
f = friction factor
ε = the roughness (m),
D = diameter (m),
and Re = the Reynolds number, defined by the following equation,
Re VD Equation 2
where
ρ = the fluid’s density [kg/m3],
V = its velocity [m/s], and
μ = dynamic viscosity [N s/m2].
In addition to appearing in Equation 2, the Reynolds number also serves as the criterion for
whether flow is turbulent (Re > 4000); the Colebrook equation apply under this condition.
The various pipe parameters have values that range as follows:
Roughness ε (in millimeters): 0.0001 to 3
(e.g. cement lined cast iron pipe: 1.5, Commercial steel
pipe: 0.046, drawn tubing: 0.0015)
Fluid density ρ (in kg/m3): 0.5 (air is 1.2) to 2000 (95% sulfuric acid is 1839)
Dynamic viscosity (in N s/m2): 10-6 (air is 18.6x10-6) to 300 (peanut butter is 250)
Finding the value of friction factor f from Equation (1) using analytical means is far from
trivial. Numerical methods can provide very good results by using a root finding method, that is,
finding the root of the function g(f) defined as
1 2 .5 1( ) 2 .0 l o g 3 .7 Regf Dff
Equation 3
will give the friction factor f according to the Colebrook equation of Equation 1.
The bisection method for finding roots, which is alternatively called binary chopping,
interval halving, or Bolzano’s method, is one type of incremental search method in which the
interval is always divided in half. If a function changes sign over an interval, the function value
at the midpoint is evaluated. The location of the root is then determined as lying at the midpoint
of the subinterval within which the sign change occurs. The process is repeated to obtain refined
estimates. A simple algorithm for the bisection calculation to finding the root of the function f(x)
consists of the following steps.
Step 1: Choose lower xL and upper xU guesses for the root such that the function changes
sign over the interval. This can be checked by ensuring that f(xL)f(xU) 0, the root lies in the upper subinterval. Therefore, set xL = xR and
return to step 2.
(c) If |f(xL)f(xR)| 4000, which meets the condition of turbulence to apply the Colebrook equation). Using
the Swamee-Jain equation, the friction factor can be estimated as: Note that friction factors range from about 0.008 to 0.08. Use this information to develop your
test cases. Do not forget to add this information in section 2 of your project document.