# Differential Equations

Topics: Fluid dynamics, Reynolds number, Viscosity Pages: 4 (645 words) Published: May 23, 2014
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1 LIST OF SYMBOLS

Symbol
Description
Unit
T
Temperature
K
ΔP
Pressure Drop
Pa

ρ
Density
kg/m3
µ
Kinematic Viscosity
N*s/m2
V
Bulk Velocity
m/s
D
Diameter
m
A
Area
m2

Flow Rate
m3/s
Re
Reynolds Number
-
f
Friction Factor
-
L
Length
m

2 CALCULATIONS
For the sample calculations, we looked at the first sample point of the flow in Pipe 1, the smallest diameter smooth copper tube:

The first step in determining the properties of the flow is finding the density and kinematic viscosity of the water. At 296.51 K, water has the following properties1:

From this we can determine the bulk velocity of the stream using Equation 1.
(Eqn. 1)
Where is the flowrate in m3/s and A is the cross-sectional area of the pipe. To find the flowrate, we multiply the flowmeter reading by the constant and convert from gallons to cubic meters as follows:

The cross sectional area of the 7.75mm pipe is
Plugging these values into Equation 1, we obtain a bulk velocity .

With the bulk velocity value, we can find the Reynolds number of the flow using Equation 2.
(Eqn. 2)

Plugging in known values to Equation 2, we find:

The experimental friction factor of the pipe can be calculated as:
(Eqn. 3)
Using the pressure drop for the chosen sample from smallest smooth copper pipe across the known distance L, we obtain an experimental friction factor

The theoretical friction factor for smooth pipes can be calculated with the Petukhov formula:
(Petukhov Formula)
Using this formula with our calculated Reynolds number yields a theoretical friction factor of

Because Pipe 4 is a rough pipe, this Petukhov Formula does not apply and we must perform additional sample calculations. From the first data point for the fourth pipe we obtain the following flow properties:

Using Equations 2 and 3 we can find the...