Advances in Fixed-Area Expansion Devices

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ADVANCES IN FIXED-AREA EXPANSION DEVICES

Xiande Fang

ABSTRACT

Fixed-area expansion devices are widely used in refrigerators, air-conditioners, and heat

pumps of up to 3.5 kW capacity with variable operating conditions. A fixed-area expansion

device falls into one of the three categories: an orifice, a short tube, and a capillary tube. The

earliest studies on orifices began in 1900s, with the interest in metering fluid flow rate. The

research on capillary tubes began to be active in 1940s. Since then, the study of fixed-area

expansion devices has been one of main activities in refrigerating. With the increasing need to

replace the ozone-depleting CFC and HCFC refrigerants, refrigerators, air-conditioners, and

heat pump systems have to be redesigned to accommodate new replacement refrigerants so

that research activities in fixed-area expansion devices have been intensified since early

1980s. This paper gives a detailed review of the literature on fixed-area expansion devices.

The main topics include the mass flow rate through orifices, the mass flow rate through short

tubes, rating capillary tubes, metastable flow, the friction factor and friction pressure drop,

and mathematical solution of fixed-area expansion device.

1 INTRODUCTION

A fixed-area expansion device can be categorized by its length-to-diameter ratio, LID.

According to its LID, a fixed-area expansion device falls into one of the three categories: an

orifice, a short tube, and a capillary tube. An orifice has an LID < 3. A short tube is defined as

having a length-to-diameter ratio of 3 < LID < 20 (Aaron and Domanski, 990). A capillary

1

tube has an LID > 20. This is an arbitrary categorization, but acceptable. Physically,

difference is in ratio offrictionallosses and it's effects on flow.

Every refrigerating unit requires a pressure-reducing device to meter the flow of

refrigerant to the low side in accordance with demands placed on the system. In 1931, a

hermetic system was introduced in Evansville, IN, utilizing a capillary tube as an expansion

device (Schulz, 1985). With the consideration of the safe, oil-soluble, halogenated

hydrocarbon as refrigerant in the early 1940s, the applicability of fixed-area expansion

devices increased. Gradually, They achieved popularity.

Fixed-area expansion devices have advantages such as low cost, elimination of moving

parts, a pressure equalization feature to reduce starting torque, and protection for compressor

motor overloading due to a maximum flow rate. On the other hand, the fixed geometry limits

the optimum operating conditions of a refrigerating cycle to only one point.

Among fixed-area expansion devices, the capillary tube is probably most widely used in

refrigerating systems. It is preferably used in smaller unitary hermetic equipment such as

household refrigerators and freezers, dehumidifiers, and conditioners. However, the use of the

capillary tube has been extended to include larger units such as unitary air conditioners in size

up to 10 tons (35 kW) capacity (ASHRAE Handbook - Equipment, 1988). The geometry of

the capillary tube used in small refrigeration systems is D = 0.64 - 2.0 mm and L = 1 - 6 m.

The most commonly used diameter of the capillary tube is 0.79 to 1.4 mm, and the length

manufacturers recommended is between 1.5 and 4.9 m (Schulz, 1985).

The short tube is mainly used in mobile air conditioners. In recent years, besides mobile

air conditioners, it has become widely used in residential air conditioners and heat pumps

(Aaron and Domanski, 1990).

The use of the orifice as an expansion device is relatively less compared with the

capillary tube and short tube. Benjamin and Miller (1941) conducted experiments of sharp­

edged orifices of LID from 0.28 to 1 with saturated water at various upstream pressures. At an

upstream pressure of 14.1 bar, reducing downstream pressure from 14.1 to 1 bar, there was no

critical-pressure condition observed. This research suggests that a sharp-edged orifice with

2

LID <1 does not choke the flow at normal operating conditions, and therefore cannot be used

as an expansion device in vapor compression systems.

In large refrigerating systems, both the refrigerant mass flow rate and the pressure level

have to be maintained to within acceptable working limits. Numerous valve configurations

with adjustable throat areas have been developed to accomplish these controls. However, the

operation of these valves is based on sharp-edged orifices (Davies and Daniels, 1973).

Many investigations of fixed-area expansion devices have been carried out theoretically

and/or experimentally because the behavior of a fixed-area expansion device in a refrigeration

system is very complicated in spite of its simple construction. Henry (1970) summarized

several experimental observations as in Figure 1. For 0 ~ LID < 3, a superheated liquid jet is

surrounded by a vapor annulus and the pressure distribution is commensurate with the free

stream line flow of the liquid jet. For tubes between LID ~ 3 and LID ~ 12, the liquid jet

breaks up by shedding large bubbles at the surface and by forming vapor bubbles in the

middle of the jet. However, the pressure remains constant in this region. In the vicinity of LID

~ 12, the mixture takes on a thoroughly dispersed configuration.

L!C / / / {11 Vapor

Liquid _

p

~

Dispersed two-phase region

\\.---z O<UD<3 3 <UD <12 UD >12

Figure 1 Flow Patterns Characterizing the Discharge of Saturated or Subcooled Liquid through Sharp-Edged Orifices

The earliest studies on fixed-area expansion devices were related to orifices (Benjamin

and Miller, 1941; Rateau and Nostrand, 1905; and Stuart and Yarnall, 1936) with the interest

in metering fluid flow rates. The research on capillary tubes began to be active in 1940s

3

(Swart, 1946; Staebler, 1948; and Bolstad and Jordan, 1948). Since then, the study of fixed-

area expansion devices has been one of main research interests in refrigeration.

With the increasing need to replace ozone-depleting CFC and HCFC refrigerants,

refrigerators, air-conditioners, and heat pump systems have to be redesigned to accommodate

new replacement refrigerants. Consequently, research activities in fixed-area expansion

devices have been intensified recent two decades (Aaron and Domanski, 1990; Dirik et aI.,

1994; Kim and O'Neal, 1993, 1994a, and 1994b; Mei, 1982; Melo et aI., 1994 and 1995; and

Meyer and Dunn, 1996). It is the purpose of this paper to summarize activities published in

the literature regarding fixed-area expansion devices with emphases on the following topics:

• Mass flow rate through orifices;

• Mass flow rate through short tubes;

• Rating capillary tubes;

• Metastable flow;

• Friction factor and friction pressure drop;

• mathematical solution of fixed-area expansion device.

2 MASS FLOW RATE THROUGH ORIFICES

When liquid flows through an orifice without phase change, the mass flow rate equation

could be derived straight from the Berlouli equation. One form is given in ASHRAE

Handbook - Fundamentals (1997):

(1)

When the ratio of orifice diameter to conduit diameter p ::s; 0.2, the discharge coefficient

of sharp-edged orifices, Cd' is fitted based on the curve provided by ASHRAE Handbook -

Fundamentals (1997) as follows:

4

{0.9199 - 0.14256IogRe+ 0.016185(logRe)2

Cd = 0.6

(2)

The published papers about the mass flow rate of a fluid having partially evaporated at

upstream of, or during passage through an orifice are limited. Furthermore, the majority of

them are focused to the flow of steam-water mixtures through standard measuring orifices

(Benjamin and Miller, 1941; Burnell, 1947; Murdock, 1962; Romig et aI., 1966; Chisholm,

1967a, 1967b; and Collins, 1978). Work done in that area at ASHRAE will be presented soon.

Benjamin and Miller (1941) reported that the mass flow rate of saturated water through a

sharp-edged orifice could be calculated with sufficient accuracy by the formula used to

determine the flow of cold water through an orifice, and the discharge coefficient found for

saturated water was approximately the same as that generally used for cold water.

Some earlier researchers (Chisholm and Wastson, 1965; Romig et al., 1966; and Davies

and Daniels, 1973) thought that it was convenient to retain the form of Equation 1 when

dealing with two-phase situations. For a two-phase flow situation, partial vaporization takes

place at upstream of, during passage through, and/or at downstream of an orifice. The effect

of vapor-liquid composition on the flow rate was consequently reflected in the value of the

expansion factor. This approach yields

where y is defined as an expansion factor.

2p(Pup - Pc/own)

1- f34 (3)

The value of the expansion factor is unity if the flowing fluid has a constant density and

therefore no vaporization occurs during the whole flowing process. From the experimental

5

data with steam-water mixtures, Romig et al. (1966) correlated the following equation for the

expansion factor:

{ 0.968 + 0.112/3

y-0.94 + 0.20v'-0.0Iv'2

entirely liquid at upstream (4)

partially vaporized at upstream

Chisholm (1967a) developed a model for determining the expansion factor under the

condition where the density change of the gas or liquid through an orifice is negligible. The

theoretical development allows for the interfacial shear force between the phases. The model

can be used where the pressure drop over an orifice is small relative to the pressure at the

orifice. The difficulty of determining the expansion factor makes the Chisholm model less

usable. Davies and Daniels (1973) tried to use the Chisholm model for R-12 passing through

sharp-edged orifices, and no direct results were obtained from the Chisholm model. In the

case of the entirely liquid at upstream, they obtained the following equation based on

correlating experimental data:

y = 1-I.4xdown (5)

Equation 5 correlated the majority of the experimental data to within ±10%. Because the

data points are limited, the agreement of the predictions with the experimental data seems not

satisfied.

Studying boiling water flowing through nozzles, orifices and pipes, Burnell (1947)

proposed

(6)

6

where C is a surface-tension-dependent constant, which is

(7)

where K is an experimentally determined coefficient. In Burnell's study, K = 0.264.

Using more accurate values of water surface tension, Kinderman and Wales (1957)

proposed the following modification to Equation 7:

C=K~ (8) 0"200

where K = 0.284

Krakow and Lin (1988) observed that the mass flow rate of a refrigerant through an

orifice or a short tube, used as a throttling device in a heat pump, was primarily dependent on

the upstream not on the downstream conditions, that indicates choked flow conditions in their

data. The characteristics of the flow through an orifice were thus similar to those of the flow

through a capillary tube. Furthermore, observed mass flow rates were greater than those

calculated, assuming that the flow is sonic at a plane of the orifice. They developed the same

theoretical model for refrigerant flow through an orifice and a short tube based on the

assumption that when the upstream refrigerant is subcooled, the exit plane of the orifice is

saturated liquid, and when the upstream refrigerant is two-phase, the exit plane is sonic. Their

model is consistent with capillary tube models.

Obermeier (1990) developed theoretical equations for critical flow rates of refrigerants

flowing through tubes and orifices. They concluded that for a given thermodynamic state, not

one, but a range of critical flow rates existed.

7

We use the Aaron and Domanski (1990) model (see Equation 12) for R-134a flowing

through small orifices with diameters from 0.03 to 0.103 mm, and achieve success. The work

will be reported on another paper.

3 MASS FLOW RATE THROUGH SHORT TUBES

Bailey (1951) studied the flow of saturated and nearly saturated water through short

tubes with 5 < LID < 20. When keeping the upstream pressure to be constant and varying the

downstream pressure, he noted three distinct flow regimes as shown in Figure 2. Curve AB

represents the conventional single-phase flow relationship described by Equation 1, where the

fluid is subcooled at all points in the tube. Near point B, the pressure near the vena contracta

(the smallest flow area caused by sudden contraction) was at a value very close to Ps• When the

downstream pressure was varied near points B and C, Bailey observed an instability in the

flow and the operating points shifting back and forth from curve AB to CD. Further reduction

of the downstream pressure from point C to D, the flow relationship described by Equation 1

was reestablished. D is the point where the critical flow occurs. Continuing to decrease

downstream pressure from point D has no effect on the mass flow rate.

Zaloudek (1963) investigated water flowing through short tubes with LID < 6, and

reported the same findings as in Figure 2 except the stable operating points between curves

AB and CD, as represented by the dotted line. Observing no difference in the mass flow rate

from the transitional region between curves AB and CD, Zaloudek termed the phenomenon as

first-step critical flow or simply first-stage choking. The region from D to E, which Bailey

termed as critical flow, was termed as second-step critical flow or simply second-stage

choking by Zaloudek. Zaloudek suggested that the mass flow rate in the first-step critical

region could be calculated by

(9)

8

where Cd is approximately 0.61 to 0.64 for water.

D E

o~---------------------------Prssure differential across short tube

Figure 2 Operating Curves for a Short Tube at Constant Upstream Pressure, Bailey (1951)

Mei (1982) investigated the flow of initially subcooled R-22 flowing through short tubes

with 7.5 < LID <11.9. With experiments based on a three-ton split-system heat pump, he

observed that the first-stage choking occurred at the liquid subcooling temperature of 22.2 °C

and no second-stage choking took place. Utilizing Equation I as the model for AT < 22.2 °C,

and Equation 9 as the model for AT > 22.2 °C, Mei obtained the following equations based on

his own experimental data:

Cd = 0.4 - 0.007364(~Pup - Pt/own -.J1034.2) + 0.0108AT AT 5:. 22.2 (10)

and

Cd = 0.9175 -0.00585AT 22.2 < AT < 27.8 (11)

Aaron and Domanski (1990) adopted Equation I to short tubes with neglecting ~,

correcting Cd' and substituting Pf , the pressure before flashing occurs, for P down' Equation I is

9

used for incompressible flow. Using Pf instead of P down avoids violating the incompressible

flow assumption. They proposed the following equation as the mass flow rate model of short

tubes:

(12a)

which is in essence

(12b)

where Cd and Cs (or Pf) are correlated with experimental data.

Aaron and Domanski correlated their experimental data with R-22 in the range of 5 <

LID < 20, 1.09 < D < 1.7 mm, 5.6 < AT < 13.9 °C, 14.48 < Pup < 20.06 bar, 2.07 bar < Pdown

< Ps, and 0 ~ Depth < 0.508 mm, and obtained:

2 Cs = 1+ 12. 599Subl.293 - 0.122ge-o.0169(L I D) - 0.04753EvapO.6192 (13a)

and for sharp-edged and 45° chamfered geometries

Cd = 1+ 0.0551(LI D) 0.5844 (Depth 1 D)0.2967 (13b)

where

Sub = (Ts-Tup) 1 Ts (T in absolute temperatures, K)

Evap = (P s -P down) 1 P s (T in absolute pressures, K)

For sharp-edged tubes, Depth = 0, therefore Cd =1.

10

Aaron and Domanski claimed that their correlation described ninety-five percent of 929

experimental data to be within ±5%.

Using the same form of mass flow rate as Aaron and Domanki's, Kim and O'Neal

obtained the correlations of C for sharp-edged and 45° chamfered short tubes with R-22 s

(l994a)

(p J-O.485( )-0179 (P J2.7I6

1.005 + 5.7367 ;c'P ~. SUbcO.9948 + 0.268 ;c'P -Cs = (l4a)

2 0.226e -0.021(D I Dref)(L/ D) _ 0.092Evapc

Cd =1+0.02655(LI D) 0.70775 (Depth 1 D) 0.22684 (l4b)

and for sharp-edged short tubes with R-134a (l994b)

Cs = (

p J-O.3296( )-0.1758

1.0156 + 10.0612 ;c'P ~ SUbcl.0831 -(l5a)

2.9596 0.1802e -0.00214(DI Dref)(L/ D) _ 0.0754Evapc

I ])~

-04474 0.38210( .£ )

Cd = ~-2.6519X..{1+5.7705(~)· (1~:" ~;:J D (l5b)

where

Subc = (Ts-Tup) 1 Tc (T in absolute temperatures, K)

Evapc = (Pc-P down) 1 Pc (T in absolute pressures, K)

11

Some experimental conditions of Equations 14 and 15 are 5 <LID < 20, 1.092 ::::; D ::::;

1.717 mm, and ~T::::; 13.9 °C to "up::::; 0.1. For Equation 14, 14.48::::; Pup::::; 20.06 bar, and for

Equation 15, 8.96 bar ::::; Pup::::; 14.48 bar. The equations described approximately ninety-five

percent of their experimental data of the sharp-edged short tubes and ninety percent of those

of the 45° chamfered short tubes to be within ±5%.

Mei (1982) found first-stage choking phenomena in his experiments but pointed out that,

second-stage choking did not occur. Aaron and Domanski (1990) claimed second-stage

choking was repeated in their experiments but the flow was found to be slightly sensitive to

the downstream pressure once the critical pressure was reached. The mass flow rate typically

increased 3% to 8% from when the downstream pressure was near Ps to when the downstream

pressure was at the minimum value tested (about 200 kPa), and 100% choked flow was not

obtained. They used the term "choked flow" in order to maintain the same terminology as

customarily used in the previous literature on refrigerant flow through short tubes and

capillary tubes, where small flow dependency on downstream pressure has typically been

assumed negligible. For example, experimental data from Bolstad and Jordan (1948) and Pate

and Tree (1988) show that the flow of refrigerant through capillary tubes is not ideally

choked. Aaron and Domanski did not find the first-stage choking in their experiments. They

thought it was because their study concentrated on conditions where the downstream pressure

was below the saturation pressure. Kim and O'Neal (1994) also pointed out that the

refrigerant flow through short tubes did not satisfy ideal choked-flow conditions. They found

the mass flow rate increased 1% for LID=19 and 7% for LID=5.5, with the downstream

pressure reductions beyond Ps to the minimum downstream pressure (490 kPa).

4 RATING CAPILLARY TUBES

For mass flow through capillary tubes, no simple empirical correlations like those of

orifices and short tubes are found. Assuming adiabatic flow with saturated liquid or two-

12

phase fluid at the upstream and critical pressure at downstream, Whitesel (1957) developed a

formula for R -12 and R -22 flowing through capillary tubes

(16)

where

a=

[ J0.333

190 47.9 {1_XO.167)exp( ~p XO. 333 X 10-6] ~p up 47.9 up

[ JO.333

190 47.9 {1_xo.333 )2 exp(1.2Pup XO. 333 XI0-6] ~p up 47.9 up

for R-12

(17)

for R-22

Hopkins (1950) presented comprehensive ratings covering capillary tube refrigerant flow

capacities for R-12 and R-22. The rating curves presented cover the variables of tube

diameter, length, saturated condensing temperature, saturated evaporator temperature, amount

of liquid subcooling, and flow rate. The curves can be used to calculate flow rate directly

from a given capillary selection and flow condition or used to determine a capillary selection

from a flow condition and flow rate.

Based on the Hopkins' work (1950) for capillary tube selection, Whitesel (1957) and

Cooper et al. (1957) developed rating curves. Later, Rezk and Awn (1979) improved these

charts. Whitesel's work (1957) was later coupled with Hopkins' work, yielding the well

known ASHRAE rating curves for R-12 and R-22 (ASHRAE Handbook - Equipment, 1988).

13

5 METASTABLE FLOW

When an initially subcooled refrigerant flows through an expansion device in a

refrigerating system, the thermodynamic equilibrium flash point occurs not at but after the

point where the refrigerant pressure equals the saturation pressure corresponding to the

refrigerant temperature. In other words, the pressure at the point where flashing begins is

lower than the saturated pressure. This state of the fluid is classified as the metastable state, as

differentiated from a stable equilibrium state. The metastable phenomenon has remarkable

effect on the mass flow rate, and the downstream pressure and quality. Based on the

nucleation theory and experimental data of refrigerant-12, Chen et al. (1990) correlated the

following equation for calculation the underpressure of vaporization, (Ps-Pv):

(p p) fkT (v J ( J-O.208( )-3.18 8 - v '1/1\,.1 8 = 0.679 g ReO.914 AT8C ~

(]"3/2 V - v T D g 1 C

(18)

where D' is the reference length which is defmed as

(19)

Equation 18 is valid in the region of0.464xl04 < Re < 3.76x104, 0 < ATsc < 30.6 °C, and

0.66 < D <1.17 mm.

The Chen model has been used as a metastable model by several studies. Li et al. (1990)

used it to their mathematical model for capillary tubes with R-12. Dirik et al. (1994) and Yin

(1998) used it to their mathematical models for capillary tubes and short tubes with R-134a,

respectively. All of them reported the agreement of the model predictions with experimental

data.

14

Bittle and Pate (1996) incorporated Equation 18 into the flow model for predicting R-

134a, R-22, R-152a, and R-41Oa flowing through capillary tubes. They found it improved

mass flow rate predictions between 1 % and 5.5% for mass flux less than 0.42 kgl(s.m2) and

roughly within 1 % for mass flux greater than 0.42 kgl(s.m2). At the higher mass flux levels

measured with R-410a and R-22, the flow model consistently underpredicted the measured

data by 4% to 6%. They thought this was because the flow range went beyond that of the

Chen model.

Melo et a1. (1995a) compared predictions of Equation 18 with their own experimental

data with R-600a, R-134a, and R-12 flowing through capillary tubes of 0.606 :::;; D:::;; 1.05 mm

• in the range of ~ T = 2 -16°C, m = 2 - 6.5 kglh, and Re = 5300 - 9600. They found that the

predicted underpressures of vaporization varied only slightly with Re. However, the

experimental data appeared to vary randomly. The Chen model was unable to predict the

experimental data properly even allowing a relative error of ± 26%. Melo et a1. thought this

was due to the random variation of the reference length and was not related to the refrigerant

type.

Meyer and Dunn (1996 and 1998) examined experimentally the mass flow rate ofR-22

in an adiabatic capillary tube. They concluded that the metastable region of an operating

capillary tube was much more predictable than previously reported in the literature. Taking

the history of system operation into account, their experimental curves revealed a hysteresis

effect in the mass flow rate as the level of inlet subcooling is increased and decreased. As the

inlet level of subcooling is decreasing, it is possible to create and lengthen a metastable region

in which liquid flow exists where it might otherwise be two-phase. The longer liquid length

resulted in the higher mass flow rate. For their particular refrigerant-tube combination, the

mass flow rate for a given state point while the subcooling is decreasing can be as much as

about 10% higher than the same conditions while the subcooling is increasing. Meyer and

Dunn's contributions are helpful to understand the random variation of the mass flow rate at a

given state point.

15

Liu and Bullard (1997) observed that the mass flow hysteresis effect on a non-diabatic

capillary tube was as large as 7%, and the effect of capillary tube hysteresis on refrigerator

system thermodynamic cycle efficiency was as large as 3%.

6 FRICTION FACTOR AND FRICTION PRESSURE DROP

Two-phase friction pressure drop calculation is needed for studying fixed-area expansion

devices in many aspects (Hopkins, N. E., 1990; Mikol, E. P., 1963; and Meyer and Dunn,

1996). The common approach for calculating two-phase friction pressure drop is to utilize

single-phase friction factor models with some modifications. The single-phase friction factor

can be expressed as either the Darcy-Weisbach friction factor or the Fanning friction factor.

The former is four times of the latter.

6.1 Single-Phase Friction Factor

Many equations for the Darcy-Weisbach friction factor have been developed. The

Blasius' equation (20) and Filonenko's equation (21) are widely used for the turbulent flow in

smooth tubes (Zukauskas and Karni, 1989).

(20)

f = {1.821og Re-l.64y2 (21)

The "smooth" here means that the wall roughness elements are so small that their

influence does not extend beyond the laminar sublayer.

There are other opinions about the applicable Reynolds number range of Blasius'

equation (20) and Filonenko's equation (21). For example, Incropera and DeWitt (1996)

16

introduced Re :s; 2x104 for Blasius' equation (20) and 3000 :s; Re :s; 5x106 for Filonenko's

equation (21).

Moody and Princeton (1944) introduced Colebrook's equation. Colebrook, in

collaboration with C. M. White, developed an equation which agrees with two extremes of

roughness in transitional zone.

1 21 (8 2.51 J 10.5 =- og 3.7D + 10.5Re (22)

Since Colebrook's equation (22) cannot be solved explicitly for f, Althul has developed

an explicit formula for turbulent flow which was modified by Tsal (ASHRAE Handbook­

Fundamentals, 1993)

/ =0.11('!"-+ 68)°·25 D Re

{/ 1= I = 0.0028 + 0.85/

(23)

if / ~ 0.018

if / < 0.018

Friction factors obtained from Althul's modified equation are within 1.6% of those

obtained by Colebrook's equation.

Churchill (1977) proposed an equation that combines laminar and turbulent regime and is

applicable to all relative roughness, which agrees with the Moody diagram (Moody and

Princeton, 1944)

17

{ }

11I2 12 16 16 -3/2

-8 8 + 2457ln 1 + 37,530 f - (RJ [(. (7!Re)" + 0.278/ D) (Re)] (24)

Figure 3 compares Blasius' equation (20), Filonenko's equation (21), Althul's modified

equation (23), and Churchill's equation (24). It is seen that Blasius' equation (20) can be valid

for Re <1.5x105, and Filonenko's equation (21) can be used within Re > 8xl03.

Althul's modified equation (23) has apparently lower prediction than Churchill's

equation (24) for large relative roughness conditions. When relative roughness & ID ~ 0.001,

the predictions by Althul's modified equation (23) and Churchill's equation (24) differ only

slightly. It is better to limit the application of Equation (23) to & ID ~ 0.001.

-...... 0.07

0.06

o t5 ~ 0.05 c o U 0.04 ·c u.

0.03

0.02

Relative roughness = 0.05

/ 0.01

-e-Blasius --*- Filonenko -tr-Churchill -Altshul's modified

O. 0 ~ 0'-::3~--'----L--'-"""""-'-'-.L...L...:---L---'---"-""""'~"",--="-....&.-.--'-....L-.JL.....L...JU:::W1 0 6

Reynolds number, Re

Figure 3 The Comparison of Friction Factor Equations

6.2 Two-Phase Friction Pressure Drop

18

Martinelli et a1. (1944) proposed the following equation for the pressure drop of

isothermal two-phase flow:

(25)

pressure drop per unit length which would exist if the gaseous phase is assumed to flow alone,

and ¢ g is a parameter which is a function of a dimensionless variable, I(, • The variable I(, is a

function of the ratio of mass flow rates of the liquid and gas, the ratio of densities of the

liquid and gas, the ratio of viscosities of the liquid and gas, and tube diameter.

Improving Equation (25), Lockhart and Martinelli (1949) proposed the following

equation for the pressure drop of isothermal two-phase flow:

(26)

where (~) I is the pressure drop per unit length which would exist if the liquid phase is

assumed to flow alone, and ¢l is a two-phase frictional multiplier which is a function of a

dimensionless variable, X, defined as

(27)

The variable X, similar to the variable 1(" is also a function of the ratio of mass flow rates

of the liquid and gas, the ratio of densities of the liquid and gas, the ratio of viscosities of the

19

liquid and gas, and tube diameter. The correlations of the variables X and X were given by the

authors (Martinelli et al. 1944; and Lockhart and Martinelli, 1949).

The Lockhart and Martinelli approach is successful for may flow conditions (Baroczy,

1966; Chen and Spedding, 1981; Chisholm, 1968; Davis, 1990; Dukler et aI., 1964; Marie,

1987; Lin et al., 1991; Souza et aI., 1993 and 1995; and Chang and Ro, 1996). Chisholm

(1967 -68) evaluated "Despite the large volume of work published in the field of two-phase

flow during the last 20 years, a recent investigation has shown that one of the earliest

procedures for predicting friction pressure gradient in two-phase flow, that of Lockhart and

Martinelli, remains the most successful."

Based on experimental data of air and liquids including benzene, kerosene, water, and

various oils in pipes varying in diameter from 1.5-25.8 mm, Lockhart and Martinelli provided

coordinates of ~ and f vs. Parameter X. De Souza et al. (1993, 1995) made some

improvements based on experimental data with refrigerants R-12, R-22, R-134a, MP-39, and

R-32/125. Li et al. (1990), Lin et al. (1991),and Chang and Ro (1996) presented different

models for the two-phase frictional multiplier, based on which we obtain a generalized

equation for the two-phase frictional multiplier as follows:

(28)

where two-phase friction factor ~ is calculated by a single-phase friction factor model with

the Reynolds number calculated by

Re= 4m PI D

Two-phase viscosity models reported in the published literature include

(i) the McAdams model (McAdams et al. 1942), given by

20

(29)

(ii)

1 1-x x -=--+-PI PI Pg

(iii) the Cicchitti model (Cicchitti et aI., 1960), given by

(iv)

(v) the Dukler model (Dukler et aI., 1964), given by

(vi)

xv g Pg + (1- x)v I PI PI = ---"'--~-----

vm

(30)

(31)

(32)

Another main approach for determining two-phase pressure drop due to the friction is

using the following equation (Niaz and de Vahl Davis; Bittle and Pate, 1996; and Wijaya,

1991):

(33)

where two-phase friction factor ~ is determined in the same way as just described.

Mikol (1963) analyzed that the inside diameters of refrigeration restrictor tubes were

large enough to be considered as "usual pipe", not "capillary" in the sense used by physicists

because even the smallest of the tube bores is approximately 100 or more times larger than the

mean free path of molecules. In other words, the friction factor correlations developed for

"usual pipes" are valid for refrigerant expansion devices. Mikol also pointed out that the

largest contributors to error in friction factors were tube inside diameter measurement and

21

mass flow rate measurement. A I % error in either measurements could result in ±5% and

±2% error in the friction factors, respectively.

6.3 Tube Roughness and Diameter Uncertainties

Tube roughness may cause a large uncertainty in the determination of friction factors. A

number of researchers (Mikol, 1963; Scott, 1976; and Sweedyk, 1981) suggested that short

tubes and capillary tubes made of drawn copper tubing of small bore, in general, could not be

considered smooth for purposes of friction factor selection.

Moody and Princeton (1944) estimated that the roughness of drawn brass and copper

tubing as of the order of 6xl0-5 in. Mikol (1963) experimentally determined the roughness of

0.055 inch drawn copper tubing to be 2.1xl0-5 in., which combined with the mean inside

diameter gives a relative roughness 3.8xl0-4. Sweedyk (1981) measured the roughnesses of

tubes from different manufacturers and different processes, and reported the roughnesses from

3.5x 10-6 to 7.3xl0-5 in. The variability is surprisingly large.

Meyer and Dunn (1996) implied that the Moody diagram (1944) would be successful in

giving an accurate surface roughness of a tube. In other words, the tube roughness can be

determined by measuring its single-phase friction factor and then reading the Moody diagram

according to the friction factor.

Liu and Bullard (1997) pointed out that it was necessary to determine actual diameters of

capillary tubes before analyzing the data from them because manufacturer's ± 0.001 inches of

tolerance in tube diameters could introduce up to ±10% uncertainty in cross sectional area and

mass flow rates.

7 MATHEMATICAL SOLUTION OF FIXED-AREA EXPANSION DEVICES

In the mathematical solution of engineering problems, one can identify three major levels

of approaches:

(1) Exact or rigorous solution

22

(2) Modeling

(3) Experimental correlation

7.1 Exact or Rigorous Solution

The term exact or rigorous solution in fluid mechanics is used for the exact solution of

the Navier-Stokes equation. Solutions of the Euler equation, that is, the Navier-Stokes

equation for zero viscosity are also considered exact solutions. Exact solutions of the Navier­

Stokes equation are very rare and only simple problem can be solved exactly. The flow of

refrigerants through fixed-area expansion devices is generally turbulent and two-phase, where

the exact solution is not realistic therefore no related literature is searched.

7.2 Modeling

A two-phase flow is governed by equations representing conservation of mass,

conservation of momentum, and conservation of energy, the equation of state, the equation of

entropy (the second law of thermodynamics), and equations representing boundary conditions

such as the equation of critical mass flux (or the equation of sound velocity). Some

simplifications to the equations are needed in order to make a two-phase flow problem

tractable by analytical and numerical means. These simplifications will result in some kind of

approximations. Solving engineering problems by this approach is termed modeling. In the

quest for more accurate, but yet, sufficiently simple approaches to the solution of two-phase

flow problems, modeling is of utmost importance.

Approximated models for the modeling of fixed-area expansion devices range all the way

from homogeneous eqUilibrium model (HEM) to methods that attempt to represent all of the

non-equilibrium phenomena. In between are many hybrid models that treat some of the non­

equilibrium aspects by using assumptions or empiricism. Among them, the HEM is the

earliest and most widely used one (Bolstad and Jordan, 1948; Marcy, 1949; Henry, 1970;

Leung et al., 1986, 1988, and 1990; Krakow and Lin, 1988; Lahey and Drew, 1990; Dirik et

al., 1994; Bansal and Rupasinghe, 1998; and Sami and Tribes, 1998).

23

There is no absolute criterion for a good model. It depends on the purpose of the model,

the accuracy needed, and the constraints such as time and cost to obtain a solution (Taite I and

Tel-Aviv, 1994). Hundreds of figures appear in the literature showing comparisons among

model predictions and/or experimental data. However there is no systematic evaluation of all

data to determined if quantitative criteria can be developed to guide the user who wishes to

know how accurate a certain estimate is likely to be under various circumstances (Wallis,

1980).

The HEM treats a two-phase mixture as a pseudo-fluid that can be described by the same

equations as an equivalent single-phase flow. The two phases are everywhere in equilibrium

with equal velocities and temperatures. Refrigerant properties such as quality, specific

volume, and enthalpy can be obtained from data tables or some general equations of state.

Nowadays, some standard software such as EES and REFPROP make the calculation of

refrigerant properties fairly easy.

Assuming that the flow is one-dimensional steady flow, gravity effect is negligible, no

shaft work exists, and D = constant, one can obtain the following equations of the HEM:

where

~G=O dz

du dP GA-=-A--r 7rD

dz dz W

24

conservation of mass (34)

conservation of momentum (35)

conservation of energy (36 )

(37)

The relation between the shearing force 'tw and the Darcy-Weisbach friction factor f is as

follows (Pai, 1981):

f=8~ pu2 (38)

Substituting the above expression into Equation 35, the momentum equation reduces to

G 2 d v m = _ dP _ G 2!, v m

dz dz 2D (39)

where

(40)

If ~q in energy equation (36) equals to zero, there is no heat conduction through the wall

of the expansion device. That is, the flow is adiabatic. Otherwise, it is non-adiabatic, and the

heat transfer equations representing the boundary condition must be given to solve the

problem.

The HEM works well for predicting the critical mass flux, Gc, in capillary tubes where

there is sufficient time for equilibrium to be achieved. However, little larger errors will occur

for short pipes and orifices, where there is insufficient time for the vapor formation to proceed

to equilibrium (Wallis, 1980). All of the two-phase critical flow theories are based on some

form of the single-phase critical flow rate equation:

(41)

25

The critical flow of a single-phase gas usually occurs when the Mach Number is equal to

one at the smallest cross-section. Even though speeds are high, molecular relaxation

phenomena are sufficiently rapid for the gas to be regarded as in thermodynamic equilibrium.

Two-phase critical flow is more complicated. The relaxation time for the formation of

new interfaces, heat, mass, and momentum transfer, and the evolution of flow patterns is

comparable with the time spent by the fluid in the "critical" region of rapid property change.

Although it may be possible to defme a mathematical condition of criticality at one location,

an entire region (that may include parts of the upstream system) plays a role in determining

how this condition is approached (Walls, 1980).

Just as the HEM is based on the ideal case of complete interphase equilibrium, it is

possible to derive a set of other models by making other limiting assumption, such as the

homogeneous frozen model (HFM) and the homogeneous isentropic model (HIM). This

avoids the necessity for taking any account of the details of the non-equilibrium phenomena.

7.3 Experimental Correlation

The approach of the experimental correlation, which uses equations correlated from

experimental data to solve engineering problems, is very attractive and useful. It is preferred

to correlate experimental data as a relation among dimensionless parameters so that the

relation is universally correct in the experimental range of the dimensionless parameters,

which may drastically reduce experimental load and difficulties. For example, the pressure

drop for a single-phase flow in a tube is of the form:

oP - = F(u,D,p,f-l,s) oz

In dimensionless form this relation can be replaced by

26

(42)

f = F(Re,&/ D) (43)

However, for some two-phase flow problems it is very difficult to obtain a dimensionless

relation because of the large number of parameters so that a dimensional relation has to be

chosen.

8 SUMMARY

Fixed-area expansion devices are widely used in refrigerators, air-conditioners, and heat

pumps of up to 3.5 kW capacity with variable operating conditions. A fixed-area expansion

device can be categorized by its length-to-diameter ratio, LID. An orifice has an LID < 3. A

short tube is defined as having a length-to-diameter ratio of3 < LID < 20. A capillary tube has

an LID > 20. This paper gives a detailed review of the literature on fixed-area expansion

devices with emphases on topics of the mass flow rate through orifices, the mass flow rate

through short tubes, metastable flow correction, the friction factor of single-phase flow, and

the pressure drop of two-phase flow.

Two-phase mass flow rate through an orifice can be calculated by Equation 3 and

Burnell's equation (6). Aaron and Domanski's equation (12) is a successful model for two­

phase mass flow rate through a short tube.

For mass flow through capillary tubes, no simple empirical correlations like those of

orifices and short tubes are found. The rating charts are one of useful approach for capillary

tube selection.

The metastable phenomenon has remarkable effect on the mass flow rate, and the

downstream pressure and quality. The mass flow rate in the metastable region varies

randomly. The experimental curves reveal a hysteresis effect on the mass flow rate as the

level of inlet subcooling is increased and decreased. As the inlet level of subcooling is

decreasing, it is possible to create and lengthen a metastable region in which liquid flow

exists where it might otherwise be two-phase. The longer liquid length resulted in the higher

27

mass flow rate. The mass flow rate for a given state point while the subcooling is decreasing

can be as much as about 10% higher than the same conditions while the subcooling is

increasing. The Chen equation (18), developed from data for R-12, is the only model for

correcting metastable flow. It can improve mass flow rate predictions up to 5.5%. In some

flow range, the Chen equation (18) is not satisfied. Further research on metastable flow

correction is needed.

There are a number of equations for single-phase friction factor. Figure 3 compares

several single-phase friction factor equations. Churchill's equation 24 is complicated but has

the widest applicable range. There are two common approaches for determining two-phase

friction pressure drop. One is the Lockhart and Martinelli approach, which utilizes the two­

phase frictional multiplier to calculate two-phase friction pressure drop. The other is similar to

single-phase pressure drop calculation method, where the two-phase friction factor is the

same form as the single-phase friction factor with Reynolds number calculated by two-phase

viscosity. The Lockhart and Martinelli approach is more successful. However, more studies

should be done to extend the Lockhart and Martinelli approach to wider refrigerant

application range.

In the mathematical solution of engineering problems, one can identify three major levels

of approaches: exact or rigorous solution, modeling, and experimental correlation. For

refrigerants flowing through expansion devices, the approach of the exact or rigorous solution

is not realistic, the approach of the modeling is of utmost importance, and the approach of the

experimental correlation is very useful in some aspects. Of the approximated models for the

modeling, the HEM is the earliest and most widely used one

The presentations provided here are by no means complete. The purpose here is just to

provide readers useful links to previous relative work. For more details, one has to consult the

original papers.

ACKNOWLEDGMENTS

28

The work is in the guidance of Professors Predrag S. Hrnjak and Clark W. Bullard, who

made many contributions to this paper. Their contributions are gratefully acknowledged.

NOMENCLATURE

Cd = discharge coefficient

C =P IP s f s

D = throttle diameter, m, except pointing out

D' = reference length

Dref = 1.35xlO-3 m

Depth = inlet chamfer depth, m, except pointing out

Evap = (Ps-Pout) / Ps

Evapc = (Pc-PouJ / Pc

(T in absolute pressures)

(T in absolute pressures)

f = Darcy-Weisbach fraction factor

ff = Fanning fraction factor

G = mass velocity (= density x velocity)

k = Boltzman's constant (= 1.380662xlO-23 JIK)

L = length, m, except pointing out

m = mass flow rate

P = pressure, Pa, except pointing out

Re = Reynolds number

Sub = (Ts-Tin) / Ts

Subc = (Ts-Tin) / Tc

(T in absolute temperatures, K)

(T in absolute temperatures, K)

T = Temperature, K, except pointing out

u = velocity

v = specific volume

v' = ratio of the actual specific volume at the upstream tap of the test orifice to the

29

specific volume of pure liquid at the upstream temperature.

x = refrigerant quality

X = two-phase flow modules defined by Equation 14

dP = pressure drop

( ~ ) g ~ pressure drop per unit length due to gas flowing at a rate ;" g with a density P,

(~} = pressure drop per unit length due to liquid flowing at a rate :nl with a density PI

( ~ ) t = pressure drop per unit length during two-phase flow

d T = subcooling

Greek Symbols

p = ratio of orifice diameter to conduit diameter

P = density

8 = roughness oftubes

'tw = shearing force

<J = surface tension

<J175 = liquid surface tension at 12.07xl05 Pa (175 psia)

<J200 = liquid surface tension at 13.79xl05 Pa (200 psia)

~ = two-phase frictional multiplier

X = two-phase flow modules

Subscripts

c = critical state, critical flow

down = downstream

f = at the point before flashing occurs

g = gas or vapor

I = liquid

30

m = average

s = saturated state

t = two-phase

up = upstream

v = vaporization

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37