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Adiabatic capillary tube expansion devices: A comparison of the homogeneous flow and the separated flow models
Applied Thermal Engineering Vol. 16, No. 7, pp. 625-634, 1996 Copyright 0 1996 Elswicr Science Ltd Printed in Great Britain. AU rights rcscrwd 1359-4311/96 $15.00 + 0.00 13594311(!B)ooo61-5
Pergamon
ADIABATIC CAPILLARY TUBE EXPANSION DEVICES: A COMPARISON OF THE HOMOGENEOUS FLOW AND THE SEPARATED FLOW MODELS T. N. Wong and K. T. Ooi School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263, Singapore (Received in final form 20 June 1995)
Abstract-Literature shows that the homogeneous flow assumption has been commonly used in most of the adiabatic capillary tube modelling studies. The slip effect between the two phases was often not considered in this small diameter (about 1 mm) capillary tube. Due to the lack of experimental information of the slip ratio in the flow within the capillary tube, the more comprehensive separated two-phase flow model was not frequently used in theoretical studies. This paper attempts to exploit the possibility of the separated flow model. Comparisons of the predicted results between the homogeneous and the separated flow models are presented, together with experimental results from previous workers. The results show that the separated flow model using the Miropolskiy’s slip ratio combined with Lin’s frictional pressure-gradient correlations gives a better prediction compared to the homogeneous Bow model. Copyright 0 1996 Elsevier Science Ltd Keywords--Refrigeration,
expansion devices, capillary tubes, model comparison.
NOMENCLATURE A d I G h til P
Re S V unl x z
area, m* nominal pipeline diameter, m relative surface roughness friction factor mass velocity, kg/mzs specific enthalpy, Id/kg mass flow rate, kg/s pressure, N/m2 Reynolds number slip ratio velocity, m/s mixture velocity, m/s mass dryness fraction (quality) axial direction or length, m pressure gradient due to wall friction, kN/m’ pressure gradient due to flow acceleration, kN/m) total pressure gradient, kN/m3
Greek letters P
Y
P
o!
viscosity, Ns/m2 specific volume, mr/kg density, kg/m’ void fraction
Subscripts C
f g m tP
critical condition liquid saturated vapour mixture two-phase 625
626
T. N. Wong and K. T. Ooi
INTRODUCTION Small diameter (about 1 mm) capillary tubes are commonly used to meter the flow of liquid refrigerant from the high-pressure condenser to the low-pressure evaporator. As liquid refrigerant flows through a capillary tube and due to the flashing process, refrigerant changes from the liquid state to a liquid-vapour mixture. Hence the flow within a capillary tube is separated into a single-phase subcooled liquid region and a saturated two-phase flow region. In the two-phase flow region of the capillary tube, the flow was often assumed to be a homogeneous mixture by many researchers [l-lo]. This homogeneous flow phenomenon was observed by a photographic study performed by Mikol[8] and Koizumi et al. [6] in a glass capillary tube. Bubbles were observed to scatter uniformly and discretely in the liquid. In theoretical modelling, the description of an adiabatic, homogeneous equilibrium two-phase flow through the capillary tube was presented by Goldstein [7j, Melo [8], Dirik et al. [9] and Wong et al. [lo]. As reported by the authors, the predicted results compared reasonably well with experimental data. It should be noted that in all of the above-mentioned theoretical models, the slip effect between the two phases was not taken into consideration. In a two-phase liquid-vapour flow, owing to differences in the physical properties (e.g. density and viscosity) of the phases, the vapour phase tends to flow at a higher in situ velocity than the liquid phase, hence slip exists between the phases. To model the slip effect, Li et al. [l l] presented a comprehensive drift flux model, which includes the effects of the thermodynamic non-equilibrium vaporization and the relative velocity between the two phases. In their model, the local frictional pressure drop during vaporization of the refrigerant was modelled by an empirical correlation. This correlation was developed from the flow of R-12 in capillary tubes, which takes into account the tube roughness, the Reynolds number and the quality of the refrigerant. In general two-phase flow studies, the homogeneous flow and the separated flow models for the prediction of the two-phase pressure drop have been well developed [12]. In the homogeneous flow model, the two-phase mixture can be simulated as a single-phase fluid possessing mean fluid properties. Hence, the friction factor for the two-phase mixture can be defined in a similar way to the single-phase friction factor. This approach has been widely used in the modelling of the flow in capillary tubes [7-lo]. However, in the separate flow model, as slip exists between the phases and when the conservation equations are applied to the combined flow, an extra variable called the void fraction is introduced. Unlike the homogeneous flow model, which only requires experimental information in the friction effect, the separated flow model requires information of either the void fraction or slip effects and the friction effects. Though this separated flow model has been well documented [12], to the authors’s knowledge it has not been applied to the field of capillary tube study. The main reason for this may be due to the lack of experimental information on the void fraction or slip ratio for the flow within small diameter capillary tubes. This paper attempts to exploit the possibility of applying the separated flow model to the capillary tube flow. Comparison of the predictions between the homogeneous and the separated flow models in capillary tubes is presented. Attempts have also been made to compare predictions with experimental results. THEORETICAL
MODELLING
In theoretical modelling, the flow within the capillary tube is divided into a subcooled single-phase liquid and saturated two-phase flow regions. The following assumptions are made in formulating both models: (i) the capillary tube is a straight, horizontal, constant inner diameter tube, (ii) flow through the capillary tube is one-dimensional, adiabatic and thermodynamic equilibrium. Subcooled single-phase flow region
For a horizontal, subcooled single-phase flow in a tube, the conservation
= - fg
[u/3.
of momentum gives:
(1)
Adiabatic capillary tube expansion devices
621
The roughness of the capillary tube wall, as reported by Mikol [5], cannot be assumed smooth. Its value depends very much on the manufacturing processes, such as plug-drawn, wire-drawn and sunk tube. It is well known that the friction coefficient, for both laminar and turbulent pipe flow conditions, can be described by Colebrook’s correlation [13]: --$=
[
1.14-210g
(e/d)+
Rt>.
1
(2)
As the subcooled liquid refrigerant flows along the tube, the fluid pressure drops and, hence, the saturated temperature of the liquid decreases. The commencement of the two-phase flow region occurs when the saturated temperature of the liquid approaches the inlet subcooled liquid temperature. Two-phase flow region
Since the flow is assumed adiabatic in the two-phase flow region, liquid flashes into vapour arising purely from the reduction in pressure. Mass, momentum and energy equations are formulated for one-dimensional flow. The energy equation takes account of the proportion of liquid which has to evaporate to maintain thermal equilibrium. The momentum equation includes the marked fluid acceleration effect caused by evaporation. In both the homogeneous and the separated flow models the two-phase pressure drop along the tube can be expressed as the sum of the pressure drop due to tube wall friction, as well as the fluid acceleration effect:
Homogeneous flow model. This model ignores slip effects and considers the two phases as being intimately mixed. The governing equations for a homogeneous capillary tube model [lo] are briefly expressed as:
dx -= dZ
(4)
(A)(B)(W - (A)(C)
and
(5) j&(D) where
B = hti + G2v,v6 C = G*v,~
x$$ +(1-x)$! vgf = vg -
Vf
hti = h, - hr,
1
’
T. N.
628
Wongand K. T. Ooi
where the local frictional pressure gradient is defined as
(6) The homogeneous two-phase friction factor can be determined from the Colebrook equation with the Reynolds number being defined as Re=
s,
(7)
CLtpVm
mixture viscosity is defined as Dukler et al. [14]: kP
=
Al
-
PI
+
(8)
.&PI
where
xvg
p = xv, +
(9)
(1 - x)v, .
The homogeneous mixture velocity (Ua) of the vapour-liquid refrigerant can be obtained from the consideration of mass conservation. It can be expressed as UH = ; vm= ; [xv, + (1 - x)v&
(10)
Separatedflow model. The separated flow model considers the two phases to be segregated with different properties and different mean velocities. For equilibrium two-phase flow, the change in the quality of the fluid along the tube can be obtained from the energy equation as follows:
ril
i
& [xh, + (1 -
X)hf + ;xE
+ ; (1 - x)V]
(11)
= 0. 1
For adiabatic flow this becomes
x$!
1
+&FZ
+(1-x)$
+x+
+(l
-x)Vfg
+ ;(I+
l+)FZ
=o.
(12)
Since h,, hr, ps, pr are functions of pressure, then, (13) and dVr = dZ
-
&[,,-x,q~)T-vf~ + y$Jg].
(14)
The void fraction is defined as xvg c(= xv,+S(l
(15)
_X)Vf’
since the void fraction depends on the quality and pressure, then
0 aa
z p= xv,
+
8- x)v, S”(l
[xv~;s~f-T)vf]2 (16) B
-
Adiabatic capillary tube expansiondevices
629
xgfXVI +S(1 -X)Vf - xv* 1 aa x= [xv, +S( 0ap dvr + S(l -x)-djj
(17)
1 - x)vr]’
Putting equations (13) and (14) into equation (12) gives dx dZ=-
-x)@
x3GZv, dv (1 - xYG*vr dvr a* 5 + (1 - a)2 dP
+
br+ ;(C-E)+a2 dx,’ >I
x3G2v2 s
-
dhr
x*G*v; _ (1 - x)*G*v: (1 - a)*
(I - x)'G*v?da (1 - a)3
a3
(1 - x)‘G’v: da (1 - a)’ >(
(18)
The total pressure gradient in the separated flow [12] is expressed as
(19)
Substituting for
0
dP z T in equation (18) using equation (19), gives the two governing equations
for the flow: dx -= dZ
(20)
and
(21) $
(Dl)
where ~1
=x$
B1 =h,r+
+(I
-x)!$
;(I,+
Dl = 1 + G*
ATE W-F
?‘?)+
+ ?%$!h !$ + (1;x)‘F -a
x*G*vi (1 _
a2
x)*G*v: _
(1 - a)’
$g _ x3G2v: _ (1 -x)~G*v: (1 - aY
a3
I(>
_ (1 - :)‘G3*v: (1
$
P
I(ap >x aa
630
T. N. Wong and K. T. Ooi
The local frictional pressure gradient (dP/dZ)F in the two-phase flow region can be obtained from Lin et al. [15]:
(22) where (dP/dZ),, is the single-phase pressure gradient and r#&is the multiplier for the two-phase flow region, which can be expressed by
(23) where
>3 16
(7/Re1J0.’1 0.27(e/d)
(24)
(25)
Re lo = g k
(26) 16
(7/Retp)0.p: 0.27(e/6)
(27)
(28) the mixture viscosity and Reynolds number are defined as
(29 Re, = g
.
(30)
kP
The single-phase pressure gradient is expressed as
(31) The mixture velocity and density are defined, respectively, as (32) and pm= ap, + (1 -
a)pf.
(33)
Adiabatic capillary tube expansion devices
631
Since the void fraction correlation for capillary tube flow is not available, in this study the slip ratio developed by Miropolskiy [ 161 was employed: 13.5(1 - P/PC)
S=l+
FrARei
(34) ’
where Fr=
G2v:
(35)
Gd
(36)
gd
and Re =
Pf
Sonic velocity in the two-phase region. The fluid velocity increases in the flow direction due to pressure drop. A condition is reached when the fluid velocity reaches the local sonic velocity, and the flow is said to have achieved the critical flow condition or choked. The sonic velocity may be obtained from the momentum equation (5) and equation (19). Choking occurs as the denominator of equations (5) and (19) approaches zero, i.e (dP/dZ), + co. Solution method. In the above analysis, equation (1) is applied to the single-phase subcooled liquid region with the known pressure and mass flow rate at the inlet of the capillary tube. Once the saturated condition is reached, the governing equations (4) and (5) for the homogeneous flow model and equations (19) and (20) for the separated flow model are solved using the standard fourth order Rung+Kutta technique with a 1 mm length increment. As the flow reaches the critical flow condition the computation is terminated.
RESULTS
AND
DISCUSSION
To validate the simulation models comparisons results published by Li et al. [l l] and Mikol [8].
Comparison
with experimental
have been made with available experimental
data
Experimental results for adiabatic two-phase flow of R-12 in capillary tube was presented by Li et al. [l 11. Pressure distributions along the capillary were measured by 10 pressure transducers. The flow conditions are given in Table 1. Figure l(a) and (b) shows the pressure distribution along the tube for both predicted and results published by Li et al. The results indicate that both the equilibrium homogeneous and the two separated two-phase flow models give reasonably good predictions. However, the separated flow model gives better predictions. Figure 2 shows a comparison of the pressure distribution along the capillary tube for the two models with the experimental results reported by Mikol [5]. Again better agreement was obtained by using the separated flow model. The above results show that better agreement with experimental results was achieved with the separated flow model, even though the non-thermodynamic equilibrium effect was ignored in the model. It was also noticed that when using the homogeneous flow model, greater discrepancies between the prediction and measured results were observed, particularly at the region near the choked condition.
Table I. Flow conditions for R-12 in capillary tube Inlet pressure (bar)
Inlet temperature (“C)
Mass flow rate (kg/s)
Diameter (mm)
Relative roughness
8.85 8.40
30.0 33.8
4.35 x IO-3 3.4 x 10-1
1.17 1.17
3.0 x lo-’ 3.0 x IO-’
T. N. Wong and K. T. Ooi
632
Fluid: R12 Pi=8.85 bar Tic30.0 Oc m=4.35xlO-J kg/s e/d=3.0xl O-3 A-, ,_)--
(4
-0.0
0.2
0.4
0.6
--.--’
Homogeneous
0.8
1.0
Distance
1.2
flow model
1.4
1.6
(m)
Fluid: R12 Pis8.40 bar
@I
Tis33.8
‘X
m=3.4xlO-3 kg/s e/d=3.Oxl O-5
9L---hd=1 8-
--.-.-.
Homogeneous
flow model
*l 7 mm ----L_ A --._
32 0.0
P 0.2
0.4
0.6
0.8 Distance
1.0
1.2
1.4
1 1.6
(m)
Fig. 1. Experimental and theoretical pressure distributions.
Figure 3 shows the quality variation along the tube length. As expected, the quality increases along the tube as more liquid flashes into vapour due to the rapid pressure drop. The results show that the two models give very similar predictions at low quality values, discrepancies become significant as the flow approaches the choked condition. CONCLUSIONS
In the single-phase flow region of the capillary tube, the comparison between theoretical prediction and the experimental results shows that Moody’s friction factor is applicable in predicting the pressure drop in small diameter capillary tube flow systems.
Adiabatic capillary tube expansion devices
633
I
Fluid: R12
9-
‘....‘..’
Separated
flow model
--.-.-.
Homogeneous
flow model
t 0-
:
32..‘.‘~‘.‘.‘.‘~‘.‘.‘.’ 0.0 0.2 0.4
0.6
0.8
1 .O
1.2
1.4
1.6
1.8
2.0
Distance(m)
Fig. 2. Experimental and theoretical pressure distributions.
Fluid: R12 pi=8.40 bar Ti=33.8
Oc
m=3.4xlO-J e/d=3.0xlO-3 0.25
-
0.20
-
d=l
-_-.-.
Homogeneous
flow model
kg/s
1
.17 mm
:. ;
: ;
x
.e 4 u
: i
0.15
,:‘,.:
0.10
.:‘ i I .’
_’I
.:;.’ . )’
-
0.0
:
0.2
0.4
0.6
0.8 Distance
1.0
1.2
1.4
1.6
(m)
Fig. 3. Variation of quality along the tube.
In the two-phase flow region of the capillary tube, the results show that both the equilibrium homogeneous and the separated two-phase flow models may be used adequately to predict the flow of refrigerants in capillary tubes. The separated flow model using Miropolskiy’s slip ratio combined with Lin’s frictional pressure gradient correlations gives better predictions compared to the homogeneous flow model. It was also noticed that greater discrepancies exist at the region near the choked condition when using the homogeneous flow model. The results also show a non-linear quality variation along the tube length for an adiabatic capillary tube flow.
634
T. N. Wong and K. T. Ooi
REFERENCES 1. M. M. Bolstad and R. C. Jordan, Theory and use of the capillary tube expansion device. Refrigerufing Engng, 519-523 (1948). 2. H. A. Whitesel, Capillary two phase flow. Refrigeruting Engng, 4243, 98 (1957). 3. L. Cooper, C. K. Chu and W. R. Brisken, Simple selection method for capillaries derived from physical flow conditions. Refrigerating Engng, 37-41 (1957). 4. W. F. Stoecker, Refrigeration and Air Conditioning. McGraw-Hill, New York (1958). 5. E. P. Mikol, Adiabatic I. 5, 7>88 (1963). single and two phase flow in small bore tubes. ASHRAE 6. H. Koizumi and K. Yokoyama, Characteristics of refrigerant flow in a capillary tube. ASHRAE Trans. 86, 19-27 (1980). I. S. D. Goldstein, A computer simulation method for describing two-phase flashing flow in small diameter tubes. ASHRAE Trans. 87, 5160 (1981). 8. C. Melo, R. T. S. R. Ferreira and H. Pereira, Modelling adiabatic capillary tubes: a critical analysis. Int. Refrigerafion Conf., Vol. 1, pp. 113-122 (1992). 9. E. Dirik, C. Inan and M. Y. Tanes, Numerical and experimental studies on adiabatic and non-adiabatic capillary tubes with HFCl34a. Int. Refrigeration Conf., pp. 365370 (1994). 10. T. N. Wong and K. T. Ooi, A study on capillary tube flow. Int. Refrigeration Conf., pp. 371-376 (1994). non-equilibrium flow of refrigerant through 11. R. Y. Li, S. Lin and Z. H. Chen, Numerical modelling of thermodynamic capillary tubes. ASHRAE Trans. 96, 542-549 (1990). 12. G. B. Wallis, One-dimension Two-phase Flow. McGraw-Hill, New York (1969). Introducrion to Fluid Mechanics, 3rd Edn. Wiley, New York (1985). 13. R. W. Fox and A. T. McDonald, 14. A. E. Dukler, M. Wicks and R. G. Cleveland, Frictional pressure drop in two phase flow: a comparison of existing correlations from pressure loss and holdup. AICHE .I. 10(l), 3843 (1964). 15. S. Lin, C. C. K. Kwork, R. Y. Li, Z. H. Chen and Z. Y. Chen, Local frictional pressure drop during vaporization for R-12 through capillary tubes. Int. .I. Multiphase Flow 17(l), 995-102 (1991). 16. Z. L. Miropolskiy, R. I. Shneyerova and A. I. Karamysheva, Vapour void fraction in steam fluid mixture flowing in heated and unheated channels paper B4.7. Inf. Hear Trans. Conf., Vol. 5, Paris (1970).
Pergamon
ADIABATIC CAPILLARY TUBE EXPANSION DEVICES: A COMPARISON OF THE HOMOGENEOUS FLOW AND THE SEPARATED FLOW MODELS T. N. Wong and K. T. Ooi School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263, Singapore (Received in final form 20 June 1995)
Abstract-Literature shows that the homogeneous flow assumption has been commonly used in most of the adiabatic capillary tube modelling studies. The slip effect between the two phases was often not considered in this small diameter (about 1 mm) capillary tube. Due to the lack of experimental information of the slip ratio in the flow within the capillary tube, the more comprehensive separated two-phase flow model was not frequently used in theoretical studies. This paper attempts to exploit the possibility of the separated flow model. Comparisons of the predicted results between the homogeneous and the separated flow models are presented, together with experimental results from previous workers. The results show that the separated flow model using the Miropolskiy’s slip ratio combined with Lin’s frictional pressure-gradient correlations gives a better prediction compared to the homogeneous Bow model. Copyright 0 1996 Elsevier Science Ltd Keywords--Refrigeration,
expansion devices, capillary tubes, model comparison.
NOMENCLATURE A d I G h til P
Re S V unl x z
area, m* nominal pipeline diameter, m relative surface roughness friction factor mass velocity, kg/mzs specific enthalpy, Id/kg mass flow rate, kg/s pressure, N/m2 Reynolds number slip ratio velocity, m/s mixture velocity, m/s mass dryness fraction (quality) axial direction or length, m pressure gradient due to wall friction, kN/m’ pressure gradient due to flow acceleration, kN/m) total pressure gradient, kN/m3
Greek letters P
Y
P
o!
viscosity, Ns/m2 specific volume, mr/kg density, kg/m’ void fraction
Subscripts C
f g m tP
critical condition liquid saturated vapour mixture two-phase 625
626
T. N. Wong and K. T. Ooi
INTRODUCTION Small diameter (about 1 mm) capillary tubes are commonly used to meter the flow of liquid refrigerant from the high-pressure condenser to the low-pressure evaporator. As liquid refrigerant flows through a capillary tube and due to the flashing process, refrigerant changes from the liquid state to a liquid-vapour mixture. Hence the flow within a capillary tube is separated into a single-phase subcooled liquid region and a saturated two-phase flow region. In the two-phase flow region of the capillary tube, the flow was often assumed to be a homogeneous mixture by many researchers [l-lo]. This homogeneous flow phenomenon was observed by a photographic study performed by Mikol[8] and Koizumi et al. [6] in a glass capillary tube. Bubbles were observed to scatter uniformly and discretely in the liquid. In theoretical modelling, the description of an adiabatic, homogeneous equilibrium two-phase flow through the capillary tube was presented by Goldstein [7j, Melo [8], Dirik et al. [9] and Wong et al. [lo]. As reported by the authors, the predicted results compared reasonably well with experimental data. It should be noted that in all of the above-mentioned theoretical models, the slip effect between the two phases was not taken into consideration. In a two-phase liquid-vapour flow, owing to differences in the physical properties (e.g. density and viscosity) of the phases, the vapour phase tends to flow at a higher in situ velocity than the liquid phase, hence slip exists between the phases. To model the slip effect, Li et al. [l l] presented a comprehensive drift flux model, which includes the effects of the thermodynamic non-equilibrium vaporization and the relative velocity between the two phases. In their model, the local frictional pressure drop during vaporization of the refrigerant was modelled by an empirical correlation. This correlation was developed from the flow of R-12 in capillary tubes, which takes into account the tube roughness, the Reynolds number and the quality of the refrigerant. In general two-phase flow studies, the homogeneous flow and the separated flow models for the prediction of the two-phase pressure drop have been well developed [12]. In the homogeneous flow model, the two-phase mixture can be simulated as a single-phase fluid possessing mean fluid properties. Hence, the friction factor for the two-phase mixture can be defined in a similar way to the single-phase friction factor. This approach has been widely used in the modelling of the flow in capillary tubes [7-lo]. However, in the separate flow model, as slip exists between the phases and when the conservation equations are applied to the combined flow, an extra variable called the void fraction is introduced. Unlike the homogeneous flow model, which only requires experimental information in the friction effect, the separated flow model requires information of either the void fraction or slip effects and the friction effects. Though this separated flow model has been well documented [12], to the authors’s knowledge it has not been applied to the field of capillary tube study. The main reason for this may be due to the lack of experimental information on the void fraction or slip ratio for the flow within small diameter capillary tubes. This paper attempts to exploit the possibility of applying the separated flow model to the capillary tube flow. Comparison of the predictions between the homogeneous and the separated flow models in capillary tubes is presented. Attempts have also been made to compare predictions with experimental results. THEORETICAL
MODELLING
In theoretical modelling, the flow within the capillary tube is divided into a subcooled single-phase liquid and saturated two-phase flow regions. The following assumptions are made in formulating both models: (i) the capillary tube is a straight, horizontal, constant inner diameter tube, (ii) flow through the capillary tube is one-dimensional, adiabatic and thermodynamic equilibrium. Subcooled single-phase flow region
For a horizontal, subcooled single-phase flow in a tube, the conservation
= - fg
[u/3.
of momentum gives:
(1)
Adiabatic capillary tube expansion devices
621
The roughness of the capillary tube wall, as reported by Mikol [5], cannot be assumed smooth. Its value depends very much on the manufacturing processes, such as plug-drawn, wire-drawn and sunk tube. It is well known that the friction coefficient, for both laminar and turbulent pipe flow conditions, can be described by Colebrook’s correlation [13]: --$=
[
1.14-210g
(e/d)+
Rt>.
1
(2)
As the subcooled liquid refrigerant flows along the tube, the fluid pressure drops and, hence, the saturated temperature of the liquid decreases. The commencement of the two-phase flow region occurs when the saturated temperature of the liquid approaches the inlet subcooled liquid temperature. Two-phase flow region
Since the flow is assumed adiabatic in the two-phase flow region, liquid flashes into vapour arising purely from the reduction in pressure. Mass, momentum and energy equations are formulated for one-dimensional flow. The energy equation takes account of the proportion of liquid which has to evaporate to maintain thermal equilibrium. The momentum equation includes the marked fluid acceleration effect caused by evaporation. In both the homogeneous and the separated flow models the two-phase pressure drop along the tube can be expressed as the sum of the pressure drop due to tube wall friction, as well as the fluid acceleration effect:
Homogeneous flow model. This model ignores slip effects and considers the two phases as being intimately mixed. The governing equations for a homogeneous capillary tube model [lo] are briefly expressed as:
dx -= dZ
(4)
(A)(B)(W - (A)(C)
and
(5) j&(D) where
B = hti + G2v,v6 C = G*v,~
x$$ +(1-x)$! vgf = vg -
Vf
hti = h, - hr,
1
’
T. N.
628
Wongand K. T. Ooi
where the local frictional pressure gradient is defined as
(6) The homogeneous two-phase friction factor can be determined from the Colebrook equation with the Reynolds number being defined as Re=
s,
(7)
CLtpVm
mixture viscosity is defined as Dukler et al. [14]: kP
=
Al
-
PI
+
(8)
.&PI
where
xvg
p = xv, +
(9)
(1 - x)v, .
The homogeneous mixture velocity (Ua) of the vapour-liquid refrigerant can be obtained from the consideration of mass conservation. It can be expressed as UH = ; vm= ; [xv, + (1 - x)v&
(10)
Separatedflow model. The separated flow model considers the two phases to be segregated with different properties and different mean velocities. For equilibrium two-phase flow, the change in the quality of the fluid along the tube can be obtained from the energy equation as follows:
ril
i
& [xh, + (1 -
X)hf + ;xE
+ ; (1 - x)V]
(11)
= 0. 1
For adiabatic flow this becomes
x$!
1
+&FZ
+(1-x)$
+x+
+(l
-x)Vfg
+ ;(I+
l+)FZ
=o.
(12)
Since h,, hr, ps, pr are functions of pressure, then, (13) and dVr = dZ
-
&[,,-x,q~)T-vf~ + y$Jg].
(14)
The void fraction is defined as xvg c(= xv,+S(l
(15)
_X)Vf’
since the void fraction depends on the quality and pressure, then
0 aa
z p= xv,
+
8- x)v, S”(l
[xv~;s~f-T)vf]2 (16) B
-
Adiabatic capillary tube expansiondevices
629
xgfXVI +S(1 -X)Vf - xv* 1 aa x= [xv, +S( 0ap dvr + S(l -x)-djj
(17)
1 - x)vr]’
Putting equations (13) and (14) into equation (12) gives dx dZ=-
-x)@
x3GZv, dv (1 - xYG*vr dvr a* 5 + (1 - a)2 dP
+
br+ ;(C-E)+a2 dx,’ >I
x3G2v2 s
-
dhr
x*G*v; _ (1 - x)*G*v: (1 - a)*
(I - x)'G*v?da (1 - a)3
a3
(1 - x)‘G’v: da (1 - a)’ >(
(18)
The total pressure gradient in the separated flow [12] is expressed as
(19)
Substituting for
0
dP z T in equation (18) using equation (19), gives the two governing equations
for the flow: dx -= dZ
(20)
and
(21) $
(Dl)
where ~1
=x$
B1 =h,r+
+(I
-x)!$
;(I,+
Dl = 1 + G*
ATE W-F
?‘?)+
+ ?%$!h !$ + (1;x)‘F -a
x*G*vi (1 _
a2
x)*G*v: _
(1 - a)’
$g _ x3G2v: _ (1 -x)~G*v: (1 - aY
a3
I(>
_ (1 - :)‘G3*v: (1
$
P
I(ap >x aa
630
T. N. Wong and K. T. Ooi
The local frictional pressure gradient (dP/dZ)F in the two-phase flow region can be obtained from Lin et al. [15]:
(22) where (dP/dZ),, is the single-phase pressure gradient and r#&is the multiplier for the two-phase flow region, which can be expressed by
(23) where
>3 16
(7/Re1J0.’1 0.27(e/d)
(24)
(25)
Re lo = g k
(26) 16
(7/Retp)0.p: 0.27(e/6)
(27)
(28) the mixture viscosity and Reynolds number are defined as
(29 Re, = g
.
(30)
kP
The single-phase pressure gradient is expressed as
(31) The mixture velocity and density are defined, respectively, as (32) and pm= ap, + (1 -
a)pf.
(33)
Adiabatic capillary tube expansion devices
631
Since the void fraction correlation for capillary tube flow is not available, in this study the slip ratio developed by Miropolskiy [ 161 was employed: 13.5(1 - P/PC)
S=l+
FrARei
(34) ’
where Fr=
G2v:
(35)
Gd
(36)
gd
and Re =
Pf
Sonic velocity in the two-phase region. The fluid velocity increases in the flow direction due to pressure drop. A condition is reached when the fluid velocity reaches the local sonic velocity, and the flow is said to have achieved the critical flow condition or choked. The sonic velocity may be obtained from the momentum equation (5) and equation (19). Choking occurs as the denominator of equations (5) and (19) approaches zero, i.e (dP/dZ), + co. Solution method. In the above analysis, equation (1) is applied to the single-phase subcooled liquid region with the known pressure and mass flow rate at the inlet of the capillary tube. Once the saturated condition is reached, the governing equations (4) and (5) for the homogeneous flow model and equations (19) and (20) for the separated flow model are solved using the standard fourth order Rung+Kutta technique with a 1 mm length increment. As the flow reaches the critical flow condition the computation is terminated.
RESULTS
AND
DISCUSSION
To validate the simulation models comparisons results published by Li et al. [l l] and Mikol [8].
Comparison
with experimental
have been made with available experimental
data
Experimental results for adiabatic two-phase flow of R-12 in capillary tube was presented by Li et al. [l 11. Pressure distributions along the capillary were measured by 10 pressure transducers. The flow conditions are given in Table 1. Figure l(a) and (b) shows the pressure distribution along the tube for both predicted and results published by Li et al. The results indicate that both the equilibrium homogeneous and the two separated two-phase flow models give reasonably good predictions. However, the separated flow model gives better predictions. Figure 2 shows a comparison of the pressure distribution along the capillary tube for the two models with the experimental results reported by Mikol [5]. Again better agreement was obtained by using the separated flow model. The above results show that better agreement with experimental results was achieved with the separated flow model, even though the non-thermodynamic equilibrium effect was ignored in the model. It was also noticed that when using the homogeneous flow model, greater discrepancies between the prediction and measured results were observed, particularly at the region near the choked condition.
Table I. Flow conditions for R-12 in capillary tube Inlet pressure (bar)
Inlet temperature (“C)
Mass flow rate (kg/s)
Diameter (mm)
Relative roughness
8.85 8.40
30.0 33.8
4.35 x IO-3 3.4 x 10-1
1.17 1.17
3.0 x lo-’ 3.0 x IO-’
T. N. Wong and K. T. Ooi
632
Fluid: R12 Pi=8.85 bar Tic30.0 Oc m=4.35xlO-J kg/s e/d=3.0xl O-3 A-, ,_)--
(4
-0.0
0.2
0.4
0.6
--.--’
Homogeneous
0.8
1.0
Distance
1.2
flow model
1.4
1.6
(m)
Fluid: R12 Pis8.40 bar
@I
Tis33.8
‘X
m=3.4xlO-3 kg/s e/d=3.Oxl O-5
9L---hd=1 8-
--.-.-.
Homogeneous
flow model
*l 7 mm ----L_ A --._
32 0.0
P 0.2
0.4
0.6
0.8 Distance
1.0
1.2
1.4
1 1.6
(m)
Fig. 1. Experimental and theoretical pressure distributions.
Figure 3 shows the quality variation along the tube length. As expected, the quality increases along the tube as more liquid flashes into vapour due to the rapid pressure drop. The results show that the two models give very similar predictions at low quality values, discrepancies become significant as the flow approaches the choked condition. CONCLUSIONS
In the single-phase flow region of the capillary tube, the comparison between theoretical prediction and the experimental results shows that Moody’s friction factor is applicable in predicting the pressure drop in small diameter capillary tube flow systems.
Adiabatic capillary tube expansion devices
633
I
Fluid: R12
9-
‘....‘..’
Separated
flow model
--.-.-.
Homogeneous
flow model
t 0-
:
32..‘.‘~‘.‘.‘.‘~‘.‘.‘.’ 0.0 0.2 0.4
0.6
0.8
1 .O
1.2
1.4
1.6
1.8
2.0
Distance(m)
Fig. 2. Experimental and theoretical pressure distributions.
Fluid: R12 pi=8.40 bar Ti=33.8
Oc
m=3.4xlO-J e/d=3.0xlO-3 0.25
-
0.20
-
d=l
-_-.-.
Homogeneous
flow model
kg/s
1
.17 mm
:. ;
: ;
x
.e 4 u
: i
0.15
,:‘,.:
0.10
.:‘ i I .’
_’I
.:;.’ . )’
-
0.0
:
0.2
0.4
0.6
0.8 Distance
1.0
1.2
1.4
1.6
(m)
Fig. 3. Variation of quality along the tube.
In the two-phase flow region of the capillary tube, the results show that both the equilibrium homogeneous and the separated two-phase flow models may be used adequately to predict the flow of refrigerants in capillary tubes. The separated flow model using Miropolskiy’s slip ratio combined with Lin’s frictional pressure gradient correlations gives better predictions compared to the homogeneous flow model. It was also noticed that greater discrepancies exist at the region near the choked condition when using the homogeneous flow model. The results also show a non-linear quality variation along the tube length for an adiabatic capillary tube flow.
634
T. N. Wong and K. T. Ooi
REFERENCES 1. M. M. Bolstad and R. C. Jordan, Theory and use of the capillary tube expansion device. Refrigerufing Engng, 519-523 (1948). 2. H. A. Whitesel, Capillary two phase flow. Refrigeruting Engng, 4243, 98 (1957). 3. L. Cooper, C. K. Chu and W. R. Brisken, Simple selection method for capillaries derived from physical flow conditions. Refrigerating Engng, 37-41 (1957). 4. W. F. Stoecker, Refrigeration and Air Conditioning. McGraw-Hill, New York (1958). 5. E. P. Mikol, Adiabatic I. 5, 7>88 (1963). single and two phase flow in small bore tubes. ASHRAE 6. H. Koizumi and K. Yokoyama, Characteristics of refrigerant flow in a capillary tube. ASHRAE Trans. 86, 19-27 (1980). I. S. D. Goldstein, A computer simulation method for describing two-phase flashing flow in small diameter tubes. ASHRAE Trans. 87, 5160 (1981). 8. C. Melo, R. T. S. R. Ferreira and H. Pereira, Modelling adiabatic capillary tubes: a critical analysis. Int. Refrigerafion Conf., Vol. 1, pp. 113-122 (1992). 9. E. Dirik, C. Inan and M. Y. Tanes, Numerical and experimental studies on adiabatic and non-adiabatic capillary tubes with HFCl34a. Int. Refrigeration Conf., pp. 365370 (1994). 10. T. N. Wong and K. T. Ooi, A study on capillary tube flow. Int. Refrigeration Conf., pp. 371-376 (1994). non-equilibrium flow of refrigerant through 11. R. Y. Li, S. Lin and Z. H. Chen, Numerical modelling of thermodynamic capillary tubes. ASHRAE Trans. 96, 542-549 (1990). 12. G. B. Wallis, One-dimension Two-phase Flow. McGraw-Hill, New York (1969). Introducrion to Fluid Mechanics, 3rd Edn. Wiley, New York (1985). 13. R. W. Fox and A. T. McDonald, 14. A. E. Dukler, M. Wicks and R. G. Cleveland, Frictional pressure drop in two phase flow: a comparison of existing correlations from pressure loss and holdup. AICHE .I. 10(l), 3843 (1964). 15. S. Lin, C. C. K. Kwork, R. Y. Li, Z. H. Chen and Z. Y. Chen, Local frictional pressure drop during vaporization for R-12 through capillary tubes. Int. .I. Multiphase Flow 17(l), 995-102 (1991). 16. Z. L. Miropolskiy, R. I. Shneyerova and A. I. Karamysheva, Vapour void fraction in steam fluid mixture flowing in heated and unheated channels paper B4.7. Inf. Hear Trans. Conf., Vol. 5, Paris (1970).