This calculator allows laminar flow and turbulent flow of a particular solvent composition through
a certain volume of piping to be plotted,
providing insight into the cause of irregular pressure readouts
in HPLC tubing
Often times in HPLC, very narrow tubing is used in the instrument. Although HPLC systems are generally assumed to behave according to
Laminar Flow Theory under optimal conditions, occasionally instruments will exhibit pressures much higher than those predicted by this theory.
However, the unexpectedly high pressure does correspond to pressures predicted assuming that the mobile phase in the column behaves
according to the turbulent flow model. This calculator allows the user to compare the pressures of their tubing under both Turbulent and
Laminar flow conditions in order to better understand the behavior of their tubing under the specific conditions of their experiment as
the flow rate is increased.
To determine the pressure of your tubing, fill in the following four fields and click submit:
The "Length" field should contain the length of the HPLC tubing in meters.
The "Diameter" field should contain the diameter in microns of the inside of the tubing.
The "Composition" field should contain the composition your mobile phase on a 0-1 scale indicating the the ratio
of acetonitrile to water in the solution.
The "Temperature" field contains the temperature of the liquid phase in degrees Celcius.
Before submitting, you also have the option to check a box that will include the laminar flow of the tubing at an ambient pressure
on the graph. Without checking this box, the ambient flow option will be hidden on the
graph, but still acessable by clicking on the "Laminar Flow (ambient pressure)" icon in the legend of the graph. The text of the
legend object should darken and another line should appear in blue on the graph. It should be noted that any of the plots can be
hidden or shown in the same way.
At any point, any one of the fields can be changed and the criteria for your tubing resubmitted.
The page must be
refreshed to clear all of the fields and reset to default. However,
successfully running the calculator does not require that this
happen at any point.
The pressure prediction graph plots points with Flow (mL/min) on the x-axis and Pressure (bar) on the y-axis
There are three lines plotted on the graph:
Turbulent flow, which plots the pressure predicted for a length of tubing under input conditions that is experiencing turbulent flow.
Laminar flow (Viscosity at actual pressure), which plots the pressure predicted for tubing that is acting under laminar flow.
Laminar flow (Viscosity at ambient pressure) is hidden unless requested, and plots the pressure prediction for tubing acting under laminar flow if the viscosity of the liquid phase is not dependent on pressure.
Once graphed, you can hover over the points on the graph to determine pressure at the corresponding
flow rate, as the data will
appear as a flag next to your cursor.
Using this graph, the pressure of tubing at a particular flow rate can be compared under turbulent and laminar conditions to help
determine the cause
of unexpected pressure readings, particularly in narrow tubing where unexpected pressures may be the result of
turbulent flow in narrow tubing.
The calculations below were used to calculate the pressure for each point on the plot
SetupA number of calculations and conversions had to take place before the pressure of each point could be calculated.
The input temperature was converted into Kelvin and the input tubing diameter was converted from microns into centimeters,
as both were needed to determine the final pressure. The density of the mobile phase was determined using the composition of the mobile phase
and the calculation is shown below:
ρ = A1P2 + A2P + A3 + (B1P2 + B2P + B3)Φ + (C1P2 + C2P + C3)T + (D1P2 + D2P + D3)Φ2 +(E1P2 + E2P + E3)ΦT + (F1P2 + F2P + F3)T2
In this equation,
P = pressure in bar
T = temperature in K
Φ = volume fraction of ACN on 0-1 scale
| A | B | C | D | E | F | |
| 1 | -8.84*10-8 | 5.06*10-8 | 4.98*10-10 | -1.57*10-8 | -1.63*10-10 | -7.29*10-13 |
| 2 | 1.15*10-4 | -1.75*10-4 | -3.18*10-7 | 2.75*10-5 | 6.07*10-7 | 2.66*10-10 |
| 3 | 9.63*10-1 | 1.96*10-2 | 7.17*10-4 | -5.57*10-2 | -6.37*10-4 | -1.95*10-6 |
Because the viscosity of a fluid is dependant on the pressure of the system, and because an increase in viscosity in the tubing
causes an increase in the pressure, the viscosity was calculated several times to ensure that the viscosity and pressure determined were
both relevant to a real world scenario
First, the velocity of the mobile phase was calculated using from the diameter of the tubing in centimeters (cm) and the flow rate for
that particular point using the equation below:
velocity =
| flow |
| (diameter/2)2*Π*60 |
Second, the viscosity was calculated using the input temperature in K, the mobile phase composition and the pressure according to the following equation:
ln(η) = A1P2+A2P + A3 + (B1P2+B2P + B3)Φ + (C1P2+C2P + C3)T + (D1P2+D2P + D3)Φ2 + (E1P2+E2P + E3)ΦT + (F1P2+F2P + F3)T2 + (G1P2+G2P + G3)Φ3 +(H1P2+H2P + H3)Φ2T + (I1P2+I2P + I3)ΦT2
| A | B | C | D | E | F | G | H | I | |
| 1 | 4.82*10-7 | -7.41*10-7 | -2.77*10-9 | 1.04*10-6 | 2.54*10-9 | 4.00*10-12 | -4.57*10-7 | -1.10*10-9 | -4.05*10-12 |
| 2 | -2.17*10-3 | 4.44*10-3 | 1.19*10-5 | -2.01*10-3 | -1.91*10-5 | -1.59*10-8 | 6.43*10-4 | 3.37*10-6 | 2.38*10-8 |
| 3 | 1.64*10-1 | -6.20*10 | -8.81*10-2 | -5.88*10 | 4.57*10-2 | 1.10*10-4 | 3.92*10-1 | 1.24*10-2 | -7.43*10-5 |
The Reynolds number could then be calculated as follows to determine the friction factor for the flow:
reynolds number = density *
| flow*diameter |
| 15 000*(viscosity)3 |
Once the Reynolds number has been determined, it was rounded to the nearest whole number and the friction factor for the tubing was
was found in a table according to its corresponding Reynolds number. The turbulent pressure of the piping could be determined using the friction factor
turbulent pressure = friction factor*tubing diameter(μm)*
| density*(velocity)2 |
| 2 000*diameter |
The calculations outlined in the "Determining Turbulent Flow" section were run several times, substituting the pressure from the previous iteration with the
newly calculated pressure for the next set of calculations. The calculations were repeated until the difference between the viscosity used to determine the Reynolds number
in the previous calculation and the viscosity determined using that Reynolds number and its corresponding pressure value had a difference of less than 0.0001. Once the
viscosity was determined to be unchanging, the most recent calculated value of pressure was stored to be plotted as a turbulent pressure point on the graph
Having determined the viscosity value and the Turbulent flow pressure value for this particular flow rate value, the only thing left to do is calculate the
pressure of the tubing under Laminar flow conditions.
laminar flow pressure = 128*length*
| viscosity*flow |
| 188 495 559.2*[diameter (cm)]2 |
It should be noted that the determination of laminar flow pressure requires the same calculation as described above regardless of how viscosity was calculated. This means
that the calculation for laminar flow assuming viscosity at ambient pressure will be the equation used above. The only difference is that the calculation takes place before
the calculations used to determine the friction factor (previous section), when the pressure has not yet been changed by increasing viscosity.
The graph was made using plotly (https://plot.ly), an open-source graphing library for javascript If you have any questions, comments or concerns, please contact Dr. Dwight Stoll at multidlc(at)multidlc.com or dstoll@gustavus.edu
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