How this instrument works
Mass flow rate here is how much refrigerant, in kilograms, crosses the capillary tube's throat every second. The formula comes straight from Bernoulli's equation applied to a fluid accelerating through a restriction: pressure energy converts to kinetic energy, so the exit velocity scales with the square root of the pressure drop, and multiplying that velocity by density and cross-sectional area gives mass flow. The discharge coefficient Cd is the correction factor engineers add because a real jet contracts slightly past the entrance and loses a little energy to friction that the idealized derivation ignores.
A refrigeration technician sizing a replacement capillary tube for a domestic fridge or window air conditioner uses exactly this relationship, often run in reverse from a target mass flow back to a bore diameter, because the tube has no moving parts to adjust after installation — get the length and bore wrong and the compressor either starves for refrigerant or floods the evaporator with liquid. It is the deliberately simple counterpart to a thermostatically controlled expansion valve: cheaper and more reliable, but fixed the moment it's soldered in.
The formula's real limit is that it assumes single-phase liquid the whole way through, which only holds near the tube's entrance. Further down the bore, pressure falls below the refrigerant's local saturation pressure and the liquid starts flashing into vapor — a two-phase, choked-flow condition this orifice equation does not capture. Treat the result as the entrance flow rate under subcooled liquid conditions, the same starting assumption ASHRAE's own capillary tube charts use before switching to empirical two-phase correlations for the rest of the bore.
- Enter the Discharge coefficient — typically 0.6 to 0.8 for a capillary tube's rounded entrance; use 0.7 without a manufacturer figure.
- Enter the Capillary tube cross-sectional area in mm² — a 1.0 mm bore is about 0.785 mm², a 0.8 mm bore about 0.503 mm².
- Enter the Refrigerant liquid density in kg/m³ at the tube's inlet — the subcooled liquid leaving the condenser, not a vapor value.
- Enter the Pressure drop across the tube in kPa — condenser pressure minus evaporator pressure.
- Read the Mass flow rate, kg/s result and compare it against the compressor's rated flow to check the tube is sized correctly.
Worked example — a 0.5 mm² bore in a domestic refrigerator
Take a discharge coefficient of 0.7, a bore area of 0.5 mm² (5×10⁻⁷ m²), liquid refrigerant at 1,200 kg/m³, and an 800 kPa pressure drop between condenser and evaporator — realistic numbers for a household refrigerator's capillary tube. Convert the pressure drop to pascals first, since the formula wants SI units throughout: 800 kPa becomes 800,000 Pa.
Inside the square root: 2 × 1,200 × 800,000 = 1,920,000,000, and the square root of that is about 43,817.8. Multiply by the area and the coefficient: 0.7 × 5×10⁻⁷ × 43,817.8 ≈ 0.015336 kg/s — call it 15.3 grams of refrigerant metered through the tube every second, matched to what the compressor draws in on the low side of the circuit.
Questions
Why does a capillary tube have no moving parts, unlike a thermostatic expansion valve?
Because it relies on fixed geometry rather than a mechanically adjusted orifice. Bore diameter and length are chosen once, at design time, to match a specific compressor and refrigerant charge; there is no sensing bulb, spring, or diaphragm to fail. That simplicity is why capillary tubes dominate mass-produced refrigerators and window air conditioners, where low cost and reliability outweigh the finer flow control a TXV provides.
Why does the formula use the square root of pressure drop rather than the pressure drop itself?
Because it descends from Bernoulli's principle: the fluid's kinetic energy at the throat equals the pressure energy released, and kinetic energy scales with velocity squared. Solving for velocity puts a square root on the pressure term, so quadrupling the pressure drop only doubles the flow — a relationship this instrument's own test cases confirm directly.
What does the discharge coefficient actually correct for?
It corrects for the gap between the idealized, frictionless flow the raw formula assumes and the real jet, which contracts slightly past the entrance (a vena contracta) and loses some energy to viscous friction along the bore. A value near 0.7 is typical for a capillary tube's rounded entrance; a sharper edge or a longer, rougher bore pushes it lower.
Does this formula account for the refrigerant flashing into vapor inside the tube?
No — it models single-phase liquid flow, which holds only near the entrance while the refrigerant is still subcooled. Further along, pressure drops below the local saturation pressure and the liquid begins flashing to vapor, a two-phase condition real capillary tube design handles with empirical correlations rather than this orifice equation. Treat this result as the entrance-region flow rate, not a full-length simulation.
How sensitive is mass flow rate to the tube's bore area?
Linearly sensitive: double the cross-sectional area and the mass flow rate doubles exactly, everything else held constant, because area enters the formula to the first power. That makes bore diameter the most direct lever for re-sizing a tube — a roughly 40% increase in diameter nearly doubles the area, since area scales with diameter squared, and so nearly doubles the flow.
What refrigerant density value should I use?
Use the liquid density at the tube's inlet condition — the subcooled liquid leaving the condenser, not a vapor or saturated-mixture figure. For common refrigerants this typically falls between roughly 1,000 and 1,250 kg/m³ depending on condensing temperature; a pressure-enthalpy chart or refrigerant property table gives the exact figure for the operating conditions at hand.