# How to design a capillary tube: the simplest and most reliable throttling device

Every refrigerating system requires a pressure-reducing device to meter refrigerant flow from the highpressure side to the low-pressure side according to load demand. The use of capillary tube is especially popular for smaller single-compressor/single-evaporator systems such as household refrigerators and freezers, dehumidifiers, and room air conditioners. Capillary tube use may extend to larger singlecompressor/single-evaporator systems, such as unitary air conditioners up to 35 kW capacity.

The capillary tube is a piece of drawn copper tube with a small inner diameter. When used for controlling refrigerant flow, it connects the outlet of the condenser to the inlet of the evaporator. The term “capillary tube” is actually a misnomer because the inner bore, though narrow, is much too large to allow capillary action.

A capillary tube passes liquid much more readily than vapour because of the latter’s increased volume; as a result, it is a practical metering device. When a capillary tube is sized to permit the desired flow of refrigerant, the liquid seals its inlet. If the system becomes unbalanced, some vapour (uncondensed refrigerant) enters the capillary tube. This vapour reduces the mass flow of refrigerant considerably, which increases condenser pressure and causes subcooling at the condenser exit and capillary tube inlet. The result is increased mass flow of refrigerant through the capillary tube. If properly sized for the application, the capillary tube compensates automatically for load and system variations and gives acceptable performance over a limited range of operating conditions.

A common flow condition is to have subcooled liquid at the entrance to the capillary tube.

With subcooled liquid entering the capillary tube, the pressure distribution along the tube is similar to that shown in Figure 1. At the entrance to the tube, section 0-1, since the fluid is in liquid phase, a slight pressure drop occurs.

From point 1 to point 2, the pressure drop is linear. In the portion of the tube 0-1-2, the refrigerant is entirely in the liquid state, and at point 2, the first bubble of vapour forms.

From point 2 to the end of the tube, the pressure drop is not linear, and the pressure drop per unit length increases as the end of the tube is approached. For this portion of the tube, both the saturated liquid and saturated vapour phases are present, with the percent and volume of vapour increasing in the direction of flow. In most of the runs, a significant pressure drop occurred from the end of the tube into the evaporator space.

The temperature is constant for the first portion of the tube 0-1-2. At point 2, the pressure has dropped to the saturation pressure corresponding to this temperature. Further pressure drop beyond point 2 is accompanied by a corresponding drop in temperature, the temperature being the saturation temperature corresponding to the pressure. As a consequence, the pressure and temperature lines coincide from point 2 to the end of the tube. The point 2 at which the first gas bubble appears is called the bubble point. The preceding portion of capillary tube is called the liquid length, and that following is called the two-phase length.

The rate of refrigerant flow through a capillary tube always increases with an increase in inlet pressure. Flow rate also increases with a decrease in external outlet pressure down to a certain critical value, below which flow does not change (choked flow). Figure 1 illustrates a case in which outlet pressure inside the capillary tube has reached the critical value (point 3), which is higher than the external pressure (point 4), which can be considered the typical for normal operation.

*Figure 1: Pressure and temperature distribution along a typical capillary tube.*

According to the ASHRAE Handbook (2006), there are two ways to select or design a capillary tubes: one involves the use of diagrams while the latter permits an analytical calculation. The aim of these pages is to give a brief overview of both methods showing some selection/design examples.

*Figure 2: Mass flow rate for the reference capillary tube. L=3.3 m and d=0.86. R134a fluid.*

The two diagrams are used as follows: for given inlet pressure and subcooling level/vapour quality, the mass flow rate for the reference capillary tube can be estimated from Figure 2; then, for a given geometry, the flow rate correction is obtained from Figure 3. The mass flow rate elaborated is given by the product of the reference flow rate and the correction factor. For instance, considering a inlet pressure of 16 bar and a subcooling of 15 K, the reference mass flow rate is 8 kg/h; thus, for a capillary tube 3 m long with a d=1.1 mm, the correction factor is 2 (Figure 3); therefore, actual mass flow rate for this system is equal to 16 kg/h. Differently, if 24 kg/h were needed of flow rate, we should select a capillary tube with a correction factor equal to 3, which according to Figure 3 has a d=1.25 mm and it is 2.625 m long. The ASHRAE Handbook also reports the diagrams relative to R410A and R22.

The second approach involves the definition of different dimensionless parameters calculated as a function of the thermophysical properties, operating test conditions and geometrical characteristics. The next table lists the definition of the dimensionless parameters that can be used to design a capillary tube.

In the case of subcooled liquid at the inlet (1 K< ∆ts <17 K) , the design equation is:

In the case of two-phase mixture at the inlet (0.03<x<0.25), the design equation becomes:

The previous equations (valid also for R410A and R22) are the ones that lets you design a capillary tube; on the other side the following examples can be useful to describe the procedure above.

For instance, considering a capillary tube 3.5 m long with d=1.0 mm. The inlet pressure is 15 bar and the subcooling is 15 K. Calculate the elaborated mass flow rate of R134a.

The corresponding saturation inlet temperature at 15 bar is 55.23 °C but we must estimate the thermophysical properties at: Tin= Tsat (pin) – Fts= 55.23 – 15= 40.23 °C . The thermophysical properties are:

The same capillary tube can be fed with two-phase mixture with X=0.10; in this case, the thermophysical properties must be estimated at 15 bar or at 55.23 °C of saturation temperature and they are:

These two examples highlight the effect of the inlet conditions on the refrigerant mass flow rate elaborated by the same capillary tube. In the design process, for given inlet conditions, through an iterative procedure, it is possible to calculate different solutions as a function of the ratio between L and d.

Considering an inlet pressure of 14 bar and 10 K of subcooling, the capillary tube, which guarantees 15 kg/h of R134a at the inlet of the evaporator, is that has L/d=1450. Therefore, the following solutions are valid: L=1.45 m and d= 1mm, L=0.945 m and d=0.65 mm; L=2.9 m and d=2 mm, etc.

*References*

- ASHRAE Handbook, 2006, Refrigerant – Control Devices, Ch. 44