Differential Equations

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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 following Reynolds number and experimental friction factor:

The theoretical friction factor for a rough pipe can be found by calculating the parallel rib roughness of the pipe (Equation 4) and using Nikuradse’s formula. (Eqn. 4) (Nikuradse’s Formula)
The rib height and pitch were listed as h = 0.000305m and p = 0.00308m.
We can then use Equation 4 to find the pipe’s surface roughness,

Plugging this roughness into Nikuradse’s formula yields

Linear fit and Power Law
Extra data points were taken for the smallest pipe, Pipe 1. and are found using the above methods. The log base 10 of both of these taken, and a least squares fit for the data is found.

This is then converted into a power equation.

From these results, standard error is fit. For this calculation, . . In the following formula i represents each of the N sample points, in this case, 13.

From here the precision uncertainty of each measurement as well as the precision uncertainty of the curve fit at each point can be found. Below this is demonstrated for the first point.

Bias Error
The analysis conducted is subject to the above precision uncertainties, but below are calculations that take into account the error in the measuring equipment used. In this experiment, the errors in the measuring equipment are propagated through into the Reynold’s number, and the experimental friction factor.
The max bias error in the flow meter was given as ±0.2% of its nominal reading. The max bias error in the pressure transducer is given as ±0.5% of full scale (200kPa). This equals a ±1kPa max bias error for any measurement, and therefore a constant max absolute error.
Since the Reynolds number is a constant expression multiplied by the flow meter reading, its relative error is the same; ±0.2% of the Reynolds number.

The first data point thus has the value: . Propagating the bias error in measured values to the experimental friction factor occurs as follows:

An example of the first data point calculation is as follows:

The first data point is thus: .