Calculate capillary back pressure, flow rate, viscosity, length, or radius with the Hagen-Poiseuille equation and unit conversions.
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Capillary Back Pressure Formula
The calculator uses the Hagen-Poiseuille equation for laminar flow through a circular capillary. The main formula for back pressure is:
The same relationship can be rearranged to solve for any one missing value:
- ΔP = capillary back pressure, usually in pascals, psi, or bar
- Q = volumetric flow rate, usually in m³/s, L/min, or gal/min
- η = dynamic viscosity of the fluid, in Pa·s or cP
- L = capillary length
- r = inside radius of the capillary
- π = pi, approximately 3.14159
To calculate back pressure, enter flow rate, viscosity, capillary length, and radius. To calculate flow rate, enter back pressure, viscosity, length, and radius. To calculate viscosity, enter back pressure, flow rate, length, and radius. To calculate length or radius, enter the other four values and leave that field blank.
The formula is very sensitive to radius because radius is raised to the fourth power. Doubling the radius lowers the back pressure by a factor of 16 if all other inputs stay the same.
Common Unit Conversions for Capillary Pressure Calculations
| Quantity | Unit | Base unit conversion |
|---|---|---|
| Flow rate | 1 L/min | 0.0000166667 m³/s |
| Flow rate | 1 gal/min | 0.0000630902 m³/s |
| Viscosity | 1 cP | 0.001 Pa·s |
| Pressure | 1 psi | 6,894.76 Pa |
| Pressure | 1 bar | 100,000 Pa |
Typical Viscosity Values for Reference
| Fluid | Approximate dynamic viscosity | Equivalent |
|---|---|---|
| Water at about 20°C | 0.001 Pa·s | 1 cP |
| Ethanol at about 20°C | 0.0012 Pa·s | 1.2 cP |
| Glycerin at about 20°C | 1.0 to 1.5 Pa·s | 1,000 to 1,500 cP |
Example Calculations
Example 1: Calculate back pressure
Suppose water flows through a capillary with these values:
- Flow rate: 1 L/min
- Viscosity: 1 cP
- Capillary length: 1 m
- Capillary radius: 0.01 m
Convert to base units: Q = 0.0000166667 m³/s, η = 0.001 Pa·s, L = 1 m, r = 0.01 m.
The result is approximately 4.244 Pa.
Example 2: Calculate radius
Suppose you want a back pressure of 100,000 Pa with these values:
- Flow rate: 1 L/min
- Viscosity: 1 cP
- Capillary length: 1 m
- Back pressure: 100,000 Pa
The result is approximately 0.000807 m, or about 0.807 mm.
FAQs
What assumptions does this capillary back pressure formula make?
It assumes steady, laminar flow of a Newtonian fluid through a straight circular capillary. It also assumes the capillary has a constant internal radius and that the fluid viscosity is known and constant. If the flow is turbulent, the tube is not circular, or the fluid is non-Newtonian, the result may not match real measured pressure drop.
Why does a small change in radius have such a large effect?
Radius appears as r4 in the denominator of the back pressure formula. That means a small decrease in capillary radius can greatly increase pressure. For example, cutting the radius in half increases back pressure by 16 times, if flow rate, viscosity, and length are unchanged.
Is back pressure the same as total system pressure?
No. This calculation gives the pressure drop across the capillary section based on viscous resistance. Total system pressure may also include losses from fittings, valves, bends, filters, elevation changes, entrance effects, and downstream restrictions.