- Email: [email protected]
Numerical analysis of choked refrigerant flow in adiabatic capillary tubes
Applied Thermal Engineering 24 (2004) 851–863 www.elsevier.com/locate/apthermeng
Numerical analysis of choked refrigerant flow in adiabatic capillary tubes P.K. Bansal *, G. Wang Department of Mechanical Engineering, The University of Auckland, Private Bag 92019 Auckland, New Zealand Received 18 March 2003; accepted 13 October 2003
Abstract This paper presents a homogeneous simulation model for choked flow conditions for pure refrigerants (R134a, R600a) in adiabatic capillary tubes. The model is based on the first principles of thermodynamics and fluid mechanics and some empirical relations. This study presents a fresh-look at the classical fluid flow problem, known as ÔFanno flowÕ for refrigerant flow in capillary tubes. A new diagram called the full range simulation diagram has been developed that presents a better way to understand the choked flow phenomenon graphically. The diagram is useful for design and analysis of refrigerant flow in capillary tubes. The model has been validated with published experimental data for R22, R134a and R600a and is found to agree to within ±7%. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Choked flow; Fanno flow; Metastable flow; Capillary tube; Homogeneous model; Refrigerators; R134a; R600a
1. Introduction Many investigations have been devoted to the study of refrigerant flow in capillary tubes [1,4,5,7,8]. A capillary tube is a key component of a small vapour compression refrigeration system, such as the household refrigerator and freezer, and is commonly used as the expansion and refrigerant controlling device. It is a long (between 2 and 6 m) drawn copper tube with a very small inner diameter, often less than 1 mm, which connects the outlet of the condenser to the inlet of the evaporator. It allows the pressures between the condenser and evaporator to equalize
*
Corresponding author. Tel.: +64-9-373-7599; fax: +64-9-373-7479. E-mail address: [email protected] (P.K. Bansal).
1359-4311/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2003.10.010
852
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
Nomenclature A d e f G g h K L m_ P Re s T V v x z h l q s Cp D
cross-sectional area (m2 ) diameter (m) wall roughness (m) friction factor (dimensionless) mass flux (kg s1 m2 ) gravity acceleration (m s2 ) specific enthalpy (kJ kg1 ) BoltzmannÕs constant length (m) mass flow rate (kg s1 ) pressure (kPa) Reynolds number (dimensionless) entropy (kJ kg1 K1 ) temperature (°C) velocity (m s1 ) specific volume (m3 kg1 ) vapour quality (dimensionless) vertical distance (m) inclination angle (deg) viscosity (kg m1 s1 ) density (kg m3 ) shear stress (N m2 ) specific heat (kJ kg1 K1 ) difference of
Subscripts c choked cond condenser e exit evap evaporator in inlet f liquid g gas H homogeneous m mean value meta metastable sp single-phase sc sub-cooling s isentropic s, sat saturation
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
tp v
853
two-phase vapour
during the off cycle, thus reducing the compressor starting torque requirements. The refrigerant flow through a capillary tube follows a flash process, where the state of the refrigerant changes from liquid to vapor–liquid mixture, and there exists a metastable phenomenon in the flow. This means that the inception of vaporization does not take place at the location of the thermodynamic saturation state with a pressure Ps , but at a location with a pressure Pv downstream from the thermodynamic saturation point. The pressure difference (Ps Pv ) is a characteristic quantity for the metastable flow and is designated as the under-pressure of vaporization [2]. Downstream of the flashing point, the pressure drops rapidly as a result of two-phase friction and vapor acceleration. After this point, the refrigerant is saturated liquid–vapour mixture. The continued pressure drop results in a decreasing saturation temperature and increasing refrigerant quality until the exit of the capillary tube. This results in the fluid flow velocity to increase due to the larger specific volume of the fluid until sonic velocity is reached and the Mach number becomes unity. At this point the flow becomes choked where the mass flow rate reaches an upper limit and the backpressure is called the Ôchoked pressureÕ. Any further reduction in the backpressure fails to increase the flow rate through the tube. This paper presents a simple numerical model using EES software [6] for the choked refrigerant flow in adiabatic capillary tubes. A unique Ôfull range simulation diagramÕ has been developed that presents a clear picture of the characteristics of both the choked and unchoked refrigerant flow in a capillary tube. The diagram would be useful in the design and analysis of a refrigeration system.
2. Development of a homogeneous model A ÔhomogeneousÕ model considers the flow of two phases as a single phase possessing mean fluid properties and assumes that the tube has a constant cross-sectional flow area, the flow is onedimensional and there is no work input or output. Fig. 1 shows the schematics of a typical capillary tube. 2.1. Model description The pressure distribution pattern along a tube depends on many factors such as the tube geometry (i.e. tube length, inner diameter, roughness) and refrigerant entrance conditions (i.e. condensing pressure, degree of sub-cooling and/or quality). The mass flow rate has significant influence on the fluid flow, especially in the two-phase region and affects the pressure drop and quality of the refrigerant. The exit condition, either choked or unchoked, also depends considerably on the mass flow rate. For a given condensing pressure and tube length, the mass flow rate through the capillary tube remains constant for evaporating pressure lower than the choked value.
854
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863 L meta
L sp,sc Pin
Pexit Pflash
Psat
Fluid Flow
D large
Pevap 1
Pcond inlet L sp
2 3
i
i+1
outlet
L tp
L
Schematic Diagram of a Capillary Tube
Fig. 1. Sketch of a capillary tube showing various parameters along its length.
It is well known that determining the choked mass flow rate is a complicated task due to the singularities near the choking exit plane, where the pressure gradient is large due to a very small length increment. The singularity can be avoided if pressure or temperature were used as independent variable instead of length [3]. 2.1.1. Single phase sub-cooled liquid region In this region, the fluid is assumed to be incompressible and steady. Following Bansal and Rupasinghe [1], the steady flow energy equation can be written as P1 V12 P2 V22 þ þ z1 ¼ þ þ z2 þ hloss qg 2g qg 2g
ð1Þ
where hloss ¼ k
V2 Lsp V 2 þ fsp 2g d 2g
ð2Þ
where fsp is the single phase friction factor and k is the entrance loss coefficient, defined as " 2 # d for ðd=DÞ < 0:76 ð3Þ k ¼ 0:42 1 D For a horizontal tube (i.e. z1 ¼ z2 ), the capillary tube length in the single-phase sub-cooled region can be given by Lsp;sc ¼ ½ðPcond Psat Þq=G2 ðk þ 1Þd=fsp
ð4Þ
2.1.2. Single phase metastable region Due to the metastable flow, the fluid vaporization gets delayed in the sub-cooled inlet region, which is also known as the non-thermodynamic equilibrium. The capillary length in this region is known as the metastable length. When the metastable liquid length is added to the sub-cooled length, the total single phase length increases, and hence the mass flow rate of actual capillary tube
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
855
is greater than the flow rate predicted without considering the metastable phenomenon. Eq. (5), which was first published by Chen et al. [2], provides the relationship between the flashing pressure Pflash with the saturation pressure Psat , temperature Tflash , degree of sub-cooling DTsc and the tube inner diameter d, as follows: 0:208 3:18 tg1 d 0:914 DTsc r1:5 =ðKTflash Þ0:5 =1000 Re Pflash ¼ 1000Psat 0:679 Dref tg1 tf1 Tcrit ð5Þ where Dref is the reference length given by rffiffiffiffiffiffiffiffiffiffi KTsat 104 ðK is Boltzmann’s constantÞ Dref ¼ r The metastable length can then be given by Lmeta ¼ 1000ðPsat Pflash Þ2dqsp =fsp G2
ð6Þ
The metastable length generally increases with the degree of sub-cooling and decreases with increasing condenser temperature, increasing tube inner diameter and increasing mass flow rate. 2.1.3. Two-phase region An elemental approach is adopted here to model the refrigerant flow in a capillary tube, where the total temperature drop across the two-phase region is equally divided among a number of small temperature elements. The properties of the fluid at the entrance of each element are known for its known saturation temperature and quality. However, the properties at the exit of the element (for an assumed temperature drop of DT across the element), including the tube elemental length, are evaluated by combining the conservation equations of mass, energy and momentum into a combined equation (7a) as follows: 1000hf2 þ 1000ðhg2 hf2 Þx2 þ 0:5G2 ½tf2 þ ðtg2 tf2 Þx2 2 ¼ 1000h1 þ 0:5V12
ð7aÞ
where G ¼ m_ =A and f ¼ 0:33=Re0:25
ð7bÞ
The mean velocity and the mean friction factor are used over the control volume. The only unknown in Equation (7a) is the Ôquality (x2 )Õ, which is evaluated analytically. For a known value of quality x2 ; h2 , v2 , V2 , Re and finally the elemental DL can be computed. The elemental calculation procedure is repeated downstream of the tube until the convergence condition is reached for the cumulative length of the two-phase region Ltp . The total capillary tube length is then the sum of the above three lengths in the three different regions as Ltotal ¼ Lsp;sc þ Lmeta þ Ltp
ð8Þ
If the entrance condition has two-phase flow at the inlet, the single-phase length will be zero. However, the two-phase length will never be zero unless the exit condition is unchoked at the given conditions.
856
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863 107.5 R134a-5.5-0.8-48.9-9.6 107.0
Fanno Line
h [kJ/kg]
106.5
106.0
105.5 Sonic velocity 105.0
104.5 0.390
0.392
0.394
0.396
0.398
0.400
0.402
0.404
0.406
0.408
0.410
s [kJ/kgK]
Fig. 2. Fanno line (h–s diagram) showing choked-flow conditions for R134a.
2.2. Calculation of choked flow conditions The choked refrigerant flow in a capillary tube can be idealized as a Fanno flow. For an adiabatic process, the Fanno line, shown in Fig. 2, is an enthalpy–entropy curve, where the enthalpy decreases and the entropy increases as the refrigerant flows through the tube. At a point along the Fanno line where the sonic velocity is reached, the refrigerant entropy tends to decrease (i.e. Ds < 0), which is in violation of the second law of thermodynamics. This point of negative entropy change can be used as a criterion to detect the choked flow condition in a capillary tube.
3. Validation of the model with experimental data The adiabatic capillary tube model has been validated with experimental data on three different refrigerants, namely R22 (from [8]), R134a (from [4]) and R600a (from [7]). While comparing the accuracy of the capillary model, the mass flow rate should be used as the comparison parameter rather than the tube length. This is due to the reason that more than 50% of the total pressure drop occurs in the last 20% of the tube length, especially near the exit plane, where the pressure decreases abruptly even with a small increase in length (e.g. 0.001 mm increment in tube length can cause up to 0.7 kPa pressure drop near the choking point for a typical flow). Fig. 3(a) shows the modelling results agree with the experimental data from Wolf et al. [8] on R22 to within +6% and )0.6%. Fig. 3(b) shows the similar comparison of the model with the experimental data of [4] on R134a, where the modelling results agree with the experimental data to within +7% and )2%. Lastly, Fig. 3(c) shows the comparison between the experimental data on R600a by Melo et al. [7] and the calculated results from the present model. The modelling results agree with the experimental data to within +9% and )2%.
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
857
0.032 D iff max= 5 99% 0.028
+6.0%
mPate [kg/s]
D iff min= -0.58%
d=1.98124mm -6.0%
l=2.0m
0.024
0.02
d=1.6764mm
0.016
l=1.5m
Model
0.012
Experiment
0.008 0.008
0.012
0.016
(a)
0.02
0.024
0.028
0.032
mmodel [kg/s] 0.0013 +5% 0.0012
mexp [kg/s]
D iff max = + 6 .6 4 % 0.0011
d=0.8mm
D iff min = + 1 .7 1 %
-5%
0.001 0.0009 0.0008
R134a
d=0.66mm
l=5.5m
0.0007
Model 0.0006
Dirik
0.0005 0.0005
0.0006
0.0007
0.0008
0.0009
0.001
0.0011
0.0012
0.0013
mmodel [kg/s]
(b) 0.0011
+10% Isobutane (R600a)
mMelo [kg/s]
0.001
d=0.77mm l=2.0m -10%
0.0009
Diff max =-8.66% 0.0008
Diff min =+1.87% Model
0.0007
Experiment 0.0007
(c)
0.0008
0.0009
0.001
0.0011
mmodel [kg/s]
Fig. 3. (a) Validation of model results with experimental data of R22 from Wolf et al. [8], (b) validation of modelling results with experimental data of R134a from Dirik et al. [4], (c) comparison of modelling results with the experimental data of R600a from Melo et al. [7].
858
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
4. Parametric analysis The variables affecting the performance of capillary tubes can be classified into two major categories, namely the geometric factors and the system operation parameters. The geometric factors include tube length and inner diameter, while the system operation parameters include entrance conditions, such as condenser pressure, temperature sub-cooling or quality, and exit conditions (choked or unchoked). In this section, the interaction and influence within those factors and parameters is presented. 4.1. Choked mass flow rate and evaporating pressure The choking phenomenon of the refrigerant in a capillary tube is quite similar to the isentropic flow of gas through a converging nozzle. Before the choked condition is reached, the mass flow rate keeps increasing as the backpressure (or evaporating pressure) is reduced. At the choking point, the Mach number equals unity, the refrigerant entropy reaches a maximum value and the mass flow rate becomes independent of the evaporating pressure. Fig. 4(a) and (b) respectively show variation of the mass flow rate and the exit pressure with the evaporating pressure. The flow condition in these figures is shown as R134a–5.5–0.66–48.9–9.6, which means that refrigerant R134a flows through a 5.5 m long tube with inner diameter of 0.66 mm, condenser temperature of 48.9 °C and sub-cooling of 9.6 K. This representation will be used consistently throughout the study. It is interesting to note from Fig. 4(a) that the absolute change in the refrigerant mass flow rate with the evaporating pressure is very small. When the evaporator pressure decreases from 200.7 kPa (unchoked region I) to 103.5 kPa (choked region II) by 48%, the mass flow rate increases by only 0.2% from 0.7104 g/s to the choked mass flow rate 0.7116 g/s. What it means is that in steady flow conditions with geometric factors and entrance conditions being fixed, the refrigerant mass flow rate is almost equal to its choked value. Fig. 4(b) shows the corresponding dimensionless diagram of Fig. 4(a), where the mass flow rate ratio ðm_ =m_ choked Þ is plotted against the pressure ratio (Pevap =Pcond ). In the unchoked region, the exit pressure equals the evaporator pressure (Pexit ¼ Pevap ) and the mass flow rate (m_ < m_ choked ) increases as the evaporator pressure decreases. At a certain value of evaporator pressure ½ðPexit ¼ Pchoked Þ > Pevap , the mass flow rate reaches a maximum value (m_ ¼ m_ choked ) through the capillary tube. 4.2. Effect of geometric and system operation parameters on mass flow rate The factors that affect the mass flow rate in a capillary tube include tube diameter and length, inlet pressure and level of inlet sub-cooling or inlet quality (d, L, Pin , DTsc =xinlet ). The objective is to investigate how each of these parameters affects the mass flow rate. The results for the base parameters in the simulation (being R134a–5.5–0.66–48.9–9.6) are shown in Tables 1 and 2 for R134a and R600a respectively. Since the trends of R600a are similar to R134a, only R134a results are discussed here. Effect of length (L): For the same operating conditions, pressure drop increases with tube length. Therefore, for a given pressure drop, the mass flow rate must be reduced if tube length is increased. It can be seen from Table 1 (rows 1 and 2) that as the length increases from 4.5 to 5.5 m,
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863 0.712
210
Region II Region I Choked Unchoked
R134a-5.5-0.66-48.9-9.6
0.7116
Pevap=200.7kPa
B
0.7112
180
Pexit
m=0.7116g/s m (g/s)
859
∆ m=+0.2% 150
∆ Pevap=-48% 0.7108
A 0.7104
Pevap=84 .4kPa 90
120
m=0.7104g/s
120
150
180
210
Pevap [kPa]
(a) 1.002
0.46 0.44
Choked region 0.998
.
Pexit = Pevap
.
m=mchoked
0.42 0.4
m/mchoked
0.996 0.38
Unchoked region
0.994
0.36
Pexit/Pcond
1
0.992 0.34 Mass flow rate increasing
0.99 0.988 0.986 0.225
(b)
0.32 Evaporating pressure decreasing
Pexit>Pevap 0.25
0.275
0.3
0.325
0.35
0.375
0.4
0.3
0.425
0.28 0.45
Pevap/Pcond
Fig. 4. (a) Variation of simulated mass flow rate and exit pressure through a fixed length capillary tube at various evaporating pressures, (b) equivalent dimensionless diagram of Fig. 4(a).
the mass flow rate changes from 0.80 to 0.71 g/s, yielding 11.1% decrease. Both the choked pressure and choked quality remain unchanged. Inner diameter (d): The pressure drop decreases with the increase in tube internal diameter for a fixed tube length, and therefore the mass flow rate should be increased to compensate for the reduction in pressure drop. In Table 1 (rows 3 and 4), as the diameter increases by 16.7% from 0.66 to 0.77 mm, the mass flow rate increases from 0.71 and 1.05 g/s, by 47.3%. The choked qualities remain nearly unchanged. Condenser temperature (Tcond ): As the condenser temperature increases, the total pressure drop increases, thereby increasing the mass flow rate. In Table 1 (rows 5 and 6), as the condensing temperature increases from 48.9 to 58 °C, the mass flow rate changes from 0.36 to 0.41 g/s, yielding 12.9% increase. The choked pressure remains unchanged but the choked quality is a bit higher.
860
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
Table 1 Variation of mass flow rate of R134a with different geometric/operating parameters Row
Geometric factors
Entrance conditions
L (m)
d (mm)
Tcond (°C)
DTsc (K) or x
m_ (g/s)
Pcond
Pchoked Tchoked
1 2 3
4.5 5.5 5.5
0.66 0.66 0.66
48.9 48.9 48.9
9.6 9.6 9.6
0.799361 0.711619 0.711619
1283 1283 1283
116 103 103
4 5
5.5 5.5
0.77 0.5
48.9 48.9
9.6 9.6
1.048790 0.358900
6 7
5.5 5.5
0.5 0.77
58.0 48.9
9.6 5.0
8 9
5.5 5.5
0.77 1.2
48.9 58.0
10
5.5
1.2
58.0
Comparative results xchoked
Flow rate change
Variable changed
)23 )25 )25
0.32 0.33 0.33
)11.0%
L þ 22:2%
+47.3%
d þ 16:7%
1283 1283
139 )19 25.27 )52
0.32 0.34
+12.9%
Tcond þ 18:6%
0.405065 0.955871
1605 1283
25.57 )52 120 )22
0.39 0.36
+9.7%
DTsc þ 92%
9.6 0.1
1.048790 2.246765
1283 1605
139 209
)19 )9
0.32 0.51
)5.6%
x þ 100%
0.2
1.962996
1605
191
)11
0.59
xchoked
Flow rate change
Variable changed
Table 2 Variation of mass of flow rate of R600a with different geometric/operating parameters Row
Geometric factors
Entrance conditions
L (m)
d (mm)
Tcond (°C)
DTsc (K) or x
m_ (g/s)
Pcond
Pchoked Tchoked
1 2 3
2.0 4.0 4.0
0.66 0.66 0.66
48.9 48.9 48.9
5.0 5.0 5.0
0.474432 0.309303 0.309303
667.6 667.6 667.6
81 52 52
)17 )27.6 )27.6
0.28 0.34 0.34
)34.8%
L þ 100%
+17.9%
d þ 16:7%
4 5
4.0 4.0
0.77 0.77
48.9 48.9
5.0 5.0
0.448517 0.448517
667.6 667.6
74 74
)19.5 )19.5
0.33 0.33
+14.8%
Tcond þ 20:4%
6 7
4.0 2.0
0.77 0.66
58.9 48.9
5.0 1.0
0.514993 0.460890
848.2 667.6
90 46
)15 )30
0.37 0.31
+2.9%
DTsc þ 400%
8 9
2.0 3.0
0.66 0.5
48.9 58.9
5.0 0.1
0.474432 0.105354
667.6 848.2
81 76
)17 )18.8
0.28 0.56
)14.8%
x þ 100%
10
3.0
0.5
58.9
0.2
0.089798
848.2
68
)21
0.64
Comparative results
Level of sub-cooling (DTsc ): With an increase in the level of sub-cooling, the single-phase liquid length increases while the two-phase length reduces, and therefore, the quality at the outlet decreases. For a fixed choked quality, an increase in the mass flow rate should increase the acceleration pressure drop. It can be seen from Table 1 (rows 7 and 8) that as the level of sub-cooling
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
861
changes from 5 to 9.6 K, the mass flow rate changes from 0.96 to 1.05 g/s, yielding 9.7% increase. Subsequently, the choked pressure increases while the quality decreases. Quality (x): As the inlet quality increases, the vapour tends to block the liquid flow and causes the mass flow rate to decrease. But the total pressure drop level can still be maintained due to the larger quality and larger portion of the acceleration pressure drop. In Table 1 (rows 9 and 10), as the inlet quality changes from 0.1 to 0.2, the mass flow rate changes from 2.25 to 1.96 g/s, yielding 5.6% decrease. 4.3. Full range simulation diagram and application Generally two problems confront the designers of the capillary tube. One is the ÔdesignÕ problem where the Ôchoking lengthÕ is to be calculated for a given mass flow rate, and the other is the ÔsimulationÕ problem, where the choked mass flow rate is to be calculated for a given tube length. For either of these conditions and for other inputs of tube roughness, tube inner diameter, refrigerant entrance conditions and evaporator pressure; a diagram can be generated to easily understand the flow characteristics in capillary tubes. The diagram can be used to determine the mass flow rate or the tube length and also the flow exit condition (being either choked or unchoked). 4.3.1. Development of working diagrams For the given inputs of d, Tcond , DTsc or x and Tevap , the tube length and exit pressure can be plotted against the mass flow rate either as absolute values (see Fig. 5(a)) or as dimensionless ratios (see Fig. 5(b)). The dimensionless diagram shows the choking values as the Ôstar-referencevaluesÕ, defined as Lexit =L , Pexit =P , Pchoked =P , m_ =m . A dotted line passing vertically through the star point or choking point divides the diagram (Fig. 5(b)) into unchoked (left side––region I, Mach > 1) and choked region (right side––region II, Mach ¼ 1). For the tube inlet condition R134a–m–0.66–48.9–1.0, the variation of the tube length and the choking pressure against the refrigerant mass flow rate can be seen in Fig. 5(a) by the curved Line x–x0 and straight line y–y 0 respectively. The horizontal line on the figure represents corresponding variation of the evaporating pressure in the unchoked region. It may be seen from both the Fig. 5(a) and (b) that for any given flow condition, as the tube length reduces, the mass flow rate increases from the unchoked region to the choked region. Generally, for the design case, the design point should be located in the unchoked region where the optimum tube length can be found according to the specific system operating conditions as discussed further. 4.3.2. Diagram applications Fig. 5(a) and (b) are helpful in understanding the capillary tube flow for choked and unchoked conditions. The application of these diagrams is illustrated by the following examples. Use of diagram 5(a): For evaporating temperature of )25.5 °C, the choked flow rate (m ) is 1.33 g/s (point a), while the choked length (L ) is 1.6 m (point b). If the evaporating temperature (Tevap ) is changed to any other value, say )18.5 °C; the corresponding values of mass flow rate (m ) and the length (L ) will be 1.83 g/s (point c) and 1.0 m (point d). For an unchoked flow condition, say the mass flow rate is reduced from its choked value of 1.33 g/s (point b) to 1.17 g/s (point a0 ), the
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863 8
160
y'
x
7
R134a-48.9-0.66-1.0 c
Lexit [m]
140
Unchoking Tevap=-18.5C
6
120
5
Tevap=-25.5C
Chocking point
4
100
a
Unchoking a'
Pexit [kPa]
862
3
L=2.0m
b'
2
m=1.17g/s
Le=1.6m
1
d x'
b
Le=1.0m y 0 0.4
0.6
80
m=1.33g/s m=1.83g/s
0.8
1
1.2
1.4
1.6
1.8
60 2
m [g/s] 4
1.4 Nonchoked Region(I)
3.5
1.3
Choked Region(II)
1.2 3
Pchoked /P*>1
1.1
Lexit /L* >1
m*,L*,P* Pexit /P* =1
Le/L*2.5
1 0.9
2 Star Point (Pexit =Pchoked =Pevap =P*)
1.5
0.8
Mach Pchoked /P* <1
lexit
0.7
Pchoke
1
Lexit/ L* <1 Lexit /L* =1
0.5 0.5
Pe/P*
0.6
0.7
0.8
0.9
1
1.1
1.2
Pexit
1.3
0.6 0.5 1.4
m/m*
Fig. 5. (a) Full range diagram showing m–L–P characteristics for R134a, (b) equivalent full range dimensionless diagram (of Fig. 5a) showing m–L–P characteristics for R134a.
tube length should increase from its choked value of 1.6 m (at point b) to unchoked value of 2.0 m (corresponding to point b0 ). Use of diagram 5(b): diagram 5(b) plots the information of diagram 5(a) as dimensionless ratios. The ratios (of the actual value to the choked value of any parameter) at the intersection points are always unity. The left side region (I) is the unchoked region, while the right side region (II) is the choked region. In region (I): Lexit =L > 1, m_ =m_ < 1, Pexit =P ¼ 1, Pchoke =P < 1, Mach < 1. As the mass flow rate increases, the length reduces while the choking pressure
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
863
increases. The exit pressure equals the evaporating pressure until the star point is reached. In region (II), the mass flow rate is at the maximum possible value and the tube length is the choked length. If the choked length is fixed, the mass flow rate can never be increased. However, for increasing mass flow rate, the length decreases and the choking pressure still increases. The exit pressure equals the choking pressure from the star point. For Example: (a) Let us consider the condition where Tevap ¼ 25:5 °C, L ¼ 1:6 m and m ¼ 1:33 g/s. Now following diagram 5(b), for a flow rate of m_ ¼ 1:0 g/s, the mass ratio rm ¼ mm_ ¼ 1:0=1:33 ¼ 0:75 and the corresponding length ratio will be rL ¼ 1:59. Then L ¼ rL L ¼ 1:59 1:6 ¼ 2:54 (m), which is in the unchoked region. (b) For L ¼ 4:8 m, the length ratio from diagram 5(b) is rL ¼ L=L ¼ 4:8=1:6 ¼ 3:0. The corresponding mass ratio rm ¼ mm_ ¼ 0:5. Then m ¼ 0:5 1:33 ¼ 0:665 g/s and Mach ¼ 0.7, means that the point of interest is in the unchoked region. (c) For L ¼ 1:0 m, the length ratio from diagram 5(b) is rL ¼ L=L ¼ 1:0=1:6 ¼ 0:63. The corresponding mass ratio is rm ¼ mm_ ¼ 1:35. Therefore, m ¼ 1:35 1:33 ¼ 1:8 g/s and Mach ¼ 1.0, meaning that the point of interest is in the choked region. Following proposed analogy, similar diagrams can be generated for other refrigerants and varying operating conditions.
5. Conclusions This study has presented a new numerical analysis to simulate adiabatic refrigerant choked flow conditions in capillary tubes. The numerical accuracy of the presented model is significantly better than previous studies and agrees with the experimental data to within ±7%. A new full range diagram has been presented that can be used for the design and analysis of choked flow characteristics of capillary tubes, where either the capillary tube length or the refrigerant mass flow rate can be determined with either of them being an input.
References [1] P.K. Bansal, A.S. Rupasinghe, A homogeneous model for adiabatic capillary tubes, J. Appl. Thermal Eng. 18 (1998) 207–219. [2] Z.H. Chen et al., A correlation for metastable flow of refrigerant 12 through capillary tube, ASHRAE Trans. 96 (1) (1990) 550–554. [3] M. Chung, A numerical procedure for simulation of Fanno flows of refrigerant mixtures in capillary tubes, ASHRAE Trans. 104 (1998) 1031–1043. [4] E. Dirik et al., Numerical and experimental studies on adiabatic and non-adiabatic capillary tubes with R-134a, in: International Refrigeration Conference at Purdue, Purdue University, USA, 1994. [5] O. Garcia-Valladares, C.D. Perez-Segarra, A. Olivia, Numerical simulation of capillary tube expansion devices behaviour with pure and mixed refrigerants considering metastable region. Part II: experimental validation and parametric studies, Appl. Thermal Eng. 22 (2002) 379–391. [6] S. Klein, F. Alvarado, Engineering Equation Solver (EES), F-Chart Software, USA, 1992. [7] C. Melo et al., An experimental analysis of adiabatic capillary tubes, J. Appl. Thermal Eng. (1999) 669–684. [8] D.A. Wolf, R.R. Bittle, M.B. Pate, Adiabatic capillary tube performance with alternative refrigerants RP-762, Final Report, ERI-95413, ASHRAE Research Project, USA, 1995.
Numerical analysis of choked refrigerant flow in adiabatic capillary tubes P.K. Bansal *, G. Wang Department of Mechanical Engineering, The University of Auckland, Private Bag 92019 Auckland, New Zealand Received 18 March 2003; accepted 13 October 2003
Abstract This paper presents a homogeneous simulation model for choked flow conditions for pure refrigerants (R134a, R600a) in adiabatic capillary tubes. The model is based on the first principles of thermodynamics and fluid mechanics and some empirical relations. This study presents a fresh-look at the classical fluid flow problem, known as ÔFanno flowÕ for refrigerant flow in capillary tubes. A new diagram called the full range simulation diagram has been developed that presents a better way to understand the choked flow phenomenon graphically. The diagram is useful for design and analysis of refrigerant flow in capillary tubes. The model has been validated with published experimental data for R22, R134a and R600a and is found to agree to within ±7%. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Choked flow; Fanno flow; Metastable flow; Capillary tube; Homogeneous model; Refrigerators; R134a; R600a
1. Introduction Many investigations have been devoted to the study of refrigerant flow in capillary tubes [1,4,5,7,8]. A capillary tube is a key component of a small vapour compression refrigeration system, such as the household refrigerator and freezer, and is commonly used as the expansion and refrigerant controlling device. It is a long (between 2 and 6 m) drawn copper tube with a very small inner diameter, often less than 1 mm, which connects the outlet of the condenser to the inlet of the evaporator. It allows the pressures between the condenser and evaporator to equalize
*
Corresponding author. Tel.: +64-9-373-7599; fax: +64-9-373-7479. E-mail address: [email protected] (P.K. Bansal).
1359-4311/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2003.10.010
852
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
Nomenclature A d e f G g h K L m_ P Re s T V v x z h l q s Cp D
cross-sectional area (m2 ) diameter (m) wall roughness (m) friction factor (dimensionless) mass flux (kg s1 m2 ) gravity acceleration (m s2 ) specific enthalpy (kJ kg1 ) BoltzmannÕs constant length (m) mass flow rate (kg s1 ) pressure (kPa) Reynolds number (dimensionless) entropy (kJ kg1 K1 ) temperature (°C) velocity (m s1 ) specific volume (m3 kg1 ) vapour quality (dimensionless) vertical distance (m) inclination angle (deg) viscosity (kg m1 s1 ) density (kg m3 ) shear stress (N m2 ) specific heat (kJ kg1 K1 ) difference of
Subscripts c choked cond condenser e exit evap evaporator in inlet f liquid g gas H homogeneous m mean value meta metastable sp single-phase sc sub-cooling s isentropic s, sat saturation
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
tp v
853
two-phase vapour
during the off cycle, thus reducing the compressor starting torque requirements. The refrigerant flow through a capillary tube follows a flash process, where the state of the refrigerant changes from liquid to vapor–liquid mixture, and there exists a metastable phenomenon in the flow. This means that the inception of vaporization does not take place at the location of the thermodynamic saturation state with a pressure Ps , but at a location with a pressure Pv downstream from the thermodynamic saturation point. The pressure difference (Ps Pv ) is a characteristic quantity for the metastable flow and is designated as the under-pressure of vaporization [2]. Downstream of the flashing point, the pressure drops rapidly as a result of two-phase friction and vapor acceleration. After this point, the refrigerant is saturated liquid–vapour mixture. The continued pressure drop results in a decreasing saturation temperature and increasing refrigerant quality until the exit of the capillary tube. This results in the fluid flow velocity to increase due to the larger specific volume of the fluid until sonic velocity is reached and the Mach number becomes unity. At this point the flow becomes choked where the mass flow rate reaches an upper limit and the backpressure is called the Ôchoked pressureÕ. Any further reduction in the backpressure fails to increase the flow rate through the tube. This paper presents a simple numerical model using EES software [6] for the choked refrigerant flow in adiabatic capillary tubes. A unique Ôfull range simulation diagramÕ has been developed that presents a clear picture of the characteristics of both the choked and unchoked refrigerant flow in a capillary tube. The diagram would be useful in the design and analysis of a refrigeration system.
2. Development of a homogeneous model A ÔhomogeneousÕ model considers the flow of two phases as a single phase possessing mean fluid properties and assumes that the tube has a constant cross-sectional flow area, the flow is onedimensional and there is no work input or output. Fig. 1 shows the schematics of a typical capillary tube. 2.1. Model description The pressure distribution pattern along a tube depends on many factors such as the tube geometry (i.e. tube length, inner diameter, roughness) and refrigerant entrance conditions (i.e. condensing pressure, degree of sub-cooling and/or quality). The mass flow rate has significant influence on the fluid flow, especially in the two-phase region and affects the pressure drop and quality of the refrigerant. The exit condition, either choked or unchoked, also depends considerably on the mass flow rate. For a given condensing pressure and tube length, the mass flow rate through the capillary tube remains constant for evaporating pressure lower than the choked value.
854
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863 L meta
L sp,sc Pin
Pexit Pflash
Psat
Fluid Flow
D large
Pevap 1
Pcond inlet L sp
2 3
i
i+1
outlet
L tp
L
Schematic Diagram of a Capillary Tube
Fig. 1. Sketch of a capillary tube showing various parameters along its length.
It is well known that determining the choked mass flow rate is a complicated task due to the singularities near the choking exit plane, where the pressure gradient is large due to a very small length increment. The singularity can be avoided if pressure or temperature were used as independent variable instead of length [3]. 2.1.1. Single phase sub-cooled liquid region In this region, the fluid is assumed to be incompressible and steady. Following Bansal and Rupasinghe [1], the steady flow energy equation can be written as P1 V12 P2 V22 þ þ z1 ¼ þ þ z2 þ hloss qg 2g qg 2g
ð1Þ
where hloss ¼ k
V2 Lsp V 2 þ fsp 2g d 2g
ð2Þ
where fsp is the single phase friction factor and k is the entrance loss coefficient, defined as " 2 # d for ðd=DÞ < 0:76 ð3Þ k ¼ 0:42 1 D For a horizontal tube (i.e. z1 ¼ z2 ), the capillary tube length in the single-phase sub-cooled region can be given by Lsp;sc ¼ ½ðPcond Psat Þq=G2 ðk þ 1Þd=fsp
ð4Þ
2.1.2. Single phase metastable region Due to the metastable flow, the fluid vaporization gets delayed in the sub-cooled inlet region, which is also known as the non-thermodynamic equilibrium. The capillary length in this region is known as the metastable length. When the metastable liquid length is added to the sub-cooled length, the total single phase length increases, and hence the mass flow rate of actual capillary tube
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
855
is greater than the flow rate predicted without considering the metastable phenomenon. Eq. (5), which was first published by Chen et al. [2], provides the relationship between the flashing pressure Pflash with the saturation pressure Psat , temperature Tflash , degree of sub-cooling DTsc and the tube inner diameter d, as follows: 0:208 3:18 tg1 d 0:914 DTsc r1:5 =ðKTflash Þ0:5 =1000 Re Pflash ¼ 1000Psat 0:679 Dref tg1 tf1 Tcrit ð5Þ where Dref is the reference length given by rffiffiffiffiffiffiffiffiffiffi KTsat 104 ðK is Boltzmann’s constantÞ Dref ¼ r The metastable length can then be given by Lmeta ¼ 1000ðPsat Pflash Þ2dqsp =fsp G2
ð6Þ
The metastable length generally increases with the degree of sub-cooling and decreases with increasing condenser temperature, increasing tube inner diameter and increasing mass flow rate. 2.1.3. Two-phase region An elemental approach is adopted here to model the refrigerant flow in a capillary tube, where the total temperature drop across the two-phase region is equally divided among a number of small temperature elements. The properties of the fluid at the entrance of each element are known for its known saturation temperature and quality. However, the properties at the exit of the element (for an assumed temperature drop of DT across the element), including the tube elemental length, are evaluated by combining the conservation equations of mass, energy and momentum into a combined equation (7a) as follows: 1000hf2 þ 1000ðhg2 hf2 Þx2 þ 0:5G2 ½tf2 þ ðtg2 tf2 Þx2 2 ¼ 1000h1 þ 0:5V12
ð7aÞ
where G ¼ m_ =A and f ¼ 0:33=Re0:25
ð7bÞ
The mean velocity and the mean friction factor are used over the control volume. The only unknown in Equation (7a) is the Ôquality (x2 )Õ, which is evaluated analytically. For a known value of quality x2 ; h2 , v2 , V2 , Re and finally the elemental DL can be computed. The elemental calculation procedure is repeated downstream of the tube until the convergence condition is reached for the cumulative length of the two-phase region Ltp . The total capillary tube length is then the sum of the above three lengths in the three different regions as Ltotal ¼ Lsp;sc þ Lmeta þ Ltp
ð8Þ
If the entrance condition has two-phase flow at the inlet, the single-phase length will be zero. However, the two-phase length will never be zero unless the exit condition is unchoked at the given conditions.
856
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863 107.5 R134a-5.5-0.8-48.9-9.6 107.0
Fanno Line
h [kJ/kg]
106.5
106.0
105.5 Sonic velocity 105.0
104.5 0.390
0.392
0.394
0.396
0.398
0.400
0.402
0.404
0.406
0.408
0.410
s [kJ/kgK]
Fig. 2. Fanno line (h–s diagram) showing choked-flow conditions for R134a.
2.2. Calculation of choked flow conditions The choked refrigerant flow in a capillary tube can be idealized as a Fanno flow. For an adiabatic process, the Fanno line, shown in Fig. 2, is an enthalpy–entropy curve, where the enthalpy decreases and the entropy increases as the refrigerant flows through the tube. At a point along the Fanno line where the sonic velocity is reached, the refrigerant entropy tends to decrease (i.e. Ds < 0), which is in violation of the second law of thermodynamics. This point of negative entropy change can be used as a criterion to detect the choked flow condition in a capillary tube.
3. Validation of the model with experimental data The adiabatic capillary tube model has been validated with experimental data on three different refrigerants, namely R22 (from [8]), R134a (from [4]) and R600a (from [7]). While comparing the accuracy of the capillary model, the mass flow rate should be used as the comparison parameter rather than the tube length. This is due to the reason that more than 50% of the total pressure drop occurs in the last 20% of the tube length, especially near the exit plane, where the pressure decreases abruptly even with a small increase in length (e.g. 0.001 mm increment in tube length can cause up to 0.7 kPa pressure drop near the choking point for a typical flow). Fig. 3(a) shows the modelling results agree with the experimental data from Wolf et al. [8] on R22 to within +6% and )0.6%. Fig. 3(b) shows the similar comparison of the model with the experimental data of [4] on R134a, where the modelling results agree with the experimental data to within +7% and )2%. Lastly, Fig. 3(c) shows the comparison between the experimental data on R600a by Melo et al. [7] and the calculated results from the present model. The modelling results agree with the experimental data to within +9% and )2%.
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
857
0.032 D iff max= 5 99% 0.028
+6.0%
mPate [kg/s]
D iff min= -0.58%
d=1.98124mm -6.0%
l=2.0m
0.024
0.02
d=1.6764mm
0.016
l=1.5m
Model
0.012
Experiment
0.008 0.008
0.012
0.016
(a)
0.02
0.024
0.028
0.032
mmodel [kg/s] 0.0013 +5% 0.0012
mexp [kg/s]
D iff max = + 6 .6 4 % 0.0011
d=0.8mm
D iff min = + 1 .7 1 %
-5%
0.001 0.0009 0.0008
R134a
d=0.66mm
l=5.5m
0.0007
Model 0.0006
Dirik
0.0005 0.0005
0.0006
0.0007
0.0008
0.0009
0.001
0.0011
0.0012
0.0013
mmodel [kg/s]
(b) 0.0011
+10% Isobutane (R600a)
mMelo [kg/s]
0.001
d=0.77mm l=2.0m -10%
0.0009
Diff max =-8.66% 0.0008
Diff min =+1.87% Model
0.0007
Experiment 0.0007
(c)
0.0008
0.0009
0.001
0.0011
mmodel [kg/s]
Fig. 3. (a) Validation of model results with experimental data of R22 from Wolf et al. [8], (b) validation of modelling results with experimental data of R134a from Dirik et al. [4], (c) comparison of modelling results with the experimental data of R600a from Melo et al. [7].
858
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
4. Parametric analysis The variables affecting the performance of capillary tubes can be classified into two major categories, namely the geometric factors and the system operation parameters. The geometric factors include tube length and inner diameter, while the system operation parameters include entrance conditions, such as condenser pressure, temperature sub-cooling or quality, and exit conditions (choked or unchoked). In this section, the interaction and influence within those factors and parameters is presented. 4.1. Choked mass flow rate and evaporating pressure The choking phenomenon of the refrigerant in a capillary tube is quite similar to the isentropic flow of gas through a converging nozzle. Before the choked condition is reached, the mass flow rate keeps increasing as the backpressure (or evaporating pressure) is reduced. At the choking point, the Mach number equals unity, the refrigerant entropy reaches a maximum value and the mass flow rate becomes independent of the evaporating pressure. Fig. 4(a) and (b) respectively show variation of the mass flow rate and the exit pressure with the evaporating pressure. The flow condition in these figures is shown as R134a–5.5–0.66–48.9–9.6, which means that refrigerant R134a flows through a 5.5 m long tube with inner diameter of 0.66 mm, condenser temperature of 48.9 °C and sub-cooling of 9.6 K. This representation will be used consistently throughout the study. It is interesting to note from Fig. 4(a) that the absolute change in the refrigerant mass flow rate with the evaporating pressure is very small. When the evaporator pressure decreases from 200.7 kPa (unchoked region I) to 103.5 kPa (choked region II) by 48%, the mass flow rate increases by only 0.2% from 0.7104 g/s to the choked mass flow rate 0.7116 g/s. What it means is that in steady flow conditions with geometric factors and entrance conditions being fixed, the refrigerant mass flow rate is almost equal to its choked value. Fig. 4(b) shows the corresponding dimensionless diagram of Fig. 4(a), where the mass flow rate ratio ðm_ =m_ choked Þ is plotted against the pressure ratio (Pevap =Pcond ). In the unchoked region, the exit pressure equals the evaporator pressure (Pexit ¼ Pevap ) and the mass flow rate (m_ < m_ choked ) increases as the evaporator pressure decreases. At a certain value of evaporator pressure ½ðPexit ¼ Pchoked Þ > Pevap , the mass flow rate reaches a maximum value (m_ ¼ m_ choked ) through the capillary tube. 4.2. Effect of geometric and system operation parameters on mass flow rate The factors that affect the mass flow rate in a capillary tube include tube diameter and length, inlet pressure and level of inlet sub-cooling or inlet quality (d, L, Pin , DTsc =xinlet ). The objective is to investigate how each of these parameters affects the mass flow rate. The results for the base parameters in the simulation (being R134a–5.5–0.66–48.9–9.6) are shown in Tables 1 and 2 for R134a and R600a respectively. Since the trends of R600a are similar to R134a, only R134a results are discussed here. Effect of length (L): For the same operating conditions, pressure drop increases with tube length. Therefore, for a given pressure drop, the mass flow rate must be reduced if tube length is increased. It can be seen from Table 1 (rows 1 and 2) that as the length increases from 4.5 to 5.5 m,
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863 0.712
210
Region II Region I Choked Unchoked
R134a-5.5-0.66-48.9-9.6
0.7116
Pevap=200.7kPa
B
0.7112
180
Pexit
m=0.7116g/s m (g/s)
859
∆ m=+0.2% 150
∆ Pevap=-48% 0.7108
A 0.7104
Pevap=84 .4kPa 90
120
m=0.7104g/s
120
150
180
210
Pevap [kPa]
(a) 1.002
0.46 0.44
Choked region 0.998
.
Pexit = Pevap
.
m=mchoked
0.42 0.4
m/mchoked
0.996 0.38
Unchoked region
0.994
0.36
Pexit/Pcond
1
0.992 0.34 Mass flow rate increasing
0.99 0.988 0.986 0.225
(b)
0.32 Evaporating pressure decreasing
Pexit>Pevap 0.25
0.275
0.3
0.325
0.35
0.375
0.4
0.3
0.425
0.28 0.45
Pevap/Pcond
Fig. 4. (a) Variation of simulated mass flow rate and exit pressure through a fixed length capillary tube at various evaporating pressures, (b) equivalent dimensionless diagram of Fig. 4(a).
the mass flow rate changes from 0.80 to 0.71 g/s, yielding 11.1% decrease. Both the choked pressure and choked quality remain unchanged. Inner diameter (d): The pressure drop decreases with the increase in tube internal diameter for a fixed tube length, and therefore the mass flow rate should be increased to compensate for the reduction in pressure drop. In Table 1 (rows 3 and 4), as the diameter increases by 16.7% from 0.66 to 0.77 mm, the mass flow rate increases from 0.71 and 1.05 g/s, by 47.3%. The choked qualities remain nearly unchanged. Condenser temperature (Tcond ): As the condenser temperature increases, the total pressure drop increases, thereby increasing the mass flow rate. In Table 1 (rows 5 and 6), as the condensing temperature increases from 48.9 to 58 °C, the mass flow rate changes from 0.36 to 0.41 g/s, yielding 12.9% increase. The choked pressure remains unchanged but the choked quality is a bit higher.
860
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
Table 1 Variation of mass flow rate of R134a with different geometric/operating parameters Row
Geometric factors
Entrance conditions
L (m)
d (mm)
Tcond (°C)
DTsc (K) or x
m_ (g/s)
Pcond
Pchoked Tchoked
1 2 3
4.5 5.5 5.5
0.66 0.66 0.66
48.9 48.9 48.9
9.6 9.6 9.6
0.799361 0.711619 0.711619
1283 1283 1283
116 103 103
4 5
5.5 5.5
0.77 0.5
48.9 48.9
9.6 9.6
1.048790 0.358900
6 7
5.5 5.5
0.5 0.77
58.0 48.9
9.6 5.0
8 9
5.5 5.5
0.77 1.2
48.9 58.0
10
5.5
1.2
58.0
Comparative results xchoked
Flow rate change
Variable changed
)23 )25 )25
0.32 0.33 0.33
)11.0%
L þ 22:2%
+47.3%
d þ 16:7%
1283 1283
139 )19 25.27 )52
0.32 0.34
+12.9%
Tcond þ 18:6%
0.405065 0.955871
1605 1283
25.57 )52 120 )22
0.39 0.36
+9.7%
DTsc þ 92%
9.6 0.1
1.048790 2.246765
1283 1605
139 209
)19 )9
0.32 0.51
)5.6%
x þ 100%
0.2
1.962996
1605
191
)11
0.59
xchoked
Flow rate change
Variable changed
Table 2 Variation of mass of flow rate of R600a with different geometric/operating parameters Row
Geometric factors
Entrance conditions
L (m)
d (mm)
Tcond (°C)
DTsc (K) or x
m_ (g/s)
Pcond
Pchoked Tchoked
1 2 3
2.0 4.0 4.0
0.66 0.66 0.66
48.9 48.9 48.9
5.0 5.0 5.0
0.474432 0.309303 0.309303
667.6 667.6 667.6
81 52 52
)17 )27.6 )27.6
0.28 0.34 0.34
)34.8%
L þ 100%
+17.9%
d þ 16:7%
4 5
4.0 4.0
0.77 0.77
48.9 48.9
5.0 5.0
0.448517 0.448517
667.6 667.6
74 74
)19.5 )19.5
0.33 0.33
+14.8%
Tcond þ 20:4%
6 7
4.0 2.0
0.77 0.66
58.9 48.9
5.0 1.0
0.514993 0.460890
848.2 667.6
90 46
)15 )30
0.37 0.31
+2.9%
DTsc þ 400%
8 9
2.0 3.0
0.66 0.5
48.9 58.9
5.0 0.1
0.474432 0.105354
667.6 848.2
81 76
)17 )18.8
0.28 0.56
)14.8%
x þ 100%
10
3.0
0.5
58.9
0.2
0.089798
848.2
68
)21
0.64
Comparative results
Level of sub-cooling (DTsc ): With an increase in the level of sub-cooling, the single-phase liquid length increases while the two-phase length reduces, and therefore, the quality at the outlet decreases. For a fixed choked quality, an increase in the mass flow rate should increase the acceleration pressure drop. It can be seen from Table 1 (rows 7 and 8) that as the level of sub-cooling
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
861
changes from 5 to 9.6 K, the mass flow rate changes from 0.96 to 1.05 g/s, yielding 9.7% increase. Subsequently, the choked pressure increases while the quality decreases. Quality (x): As the inlet quality increases, the vapour tends to block the liquid flow and causes the mass flow rate to decrease. But the total pressure drop level can still be maintained due to the larger quality and larger portion of the acceleration pressure drop. In Table 1 (rows 9 and 10), as the inlet quality changes from 0.1 to 0.2, the mass flow rate changes from 2.25 to 1.96 g/s, yielding 5.6% decrease. 4.3. Full range simulation diagram and application Generally two problems confront the designers of the capillary tube. One is the ÔdesignÕ problem where the Ôchoking lengthÕ is to be calculated for a given mass flow rate, and the other is the ÔsimulationÕ problem, where the choked mass flow rate is to be calculated for a given tube length. For either of these conditions and for other inputs of tube roughness, tube inner diameter, refrigerant entrance conditions and evaporator pressure; a diagram can be generated to easily understand the flow characteristics in capillary tubes. The diagram can be used to determine the mass flow rate or the tube length and also the flow exit condition (being either choked or unchoked). 4.3.1. Development of working diagrams For the given inputs of d, Tcond , DTsc or x and Tevap , the tube length and exit pressure can be plotted against the mass flow rate either as absolute values (see Fig. 5(a)) or as dimensionless ratios (see Fig. 5(b)). The dimensionless diagram shows the choking values as the Ôstar-referencevaluesÕ, defined as Lexit =L , Pexit =P , Pchoked =P , m_ =m . A dotted line passing vertically through the star point or choking point divides the diagram (Fig. 5(b)) into unchoked (left side––region I, Mach > 1) and choked region (right side––region II, Mach ¼ 1). For the tube inlet condition R134a–m–0.66–48.9–1.0, the variation of the tube length and the choking pressure against the refrigerant mass flow rate can be seen in Fig. 5(a) by the curved Line x–x0 and straight line y–y 0 respectively. The horizontal line on the figure represents corresponding variation of the evaporating pressure in the unchoked region. It may be seen from both the Fig. 5(a) and (b) that for any given flow condition, as the tube length reduces, the mass flow rate increases from the unchoked region to the choked region. Generally, for the design case, the design point should be located in the unchoked region where the optimum tube length can be found according to the specific system operating conditions as discussed further. 4.3.2. Diagram applications Fig. 5(a) and (b) are helpful in understanding the capillary tube flow for choked and unchoked conditions. The application of these diagrams is illustrated by the following examples. Use of diagram 5(a): For evaporating temperature of )25.5 °C, the choked flow rate (m ) is 1.33 g/s (point a), while the choked length (L ) is 1.6 m (point b). If the evaporating temperature (Tevap ) is changed to any other value, say )18.5 °C; the corresponding values of mass flow rate (m ) and the length (L ) will be 1.83 g/s (point c) and 1.0 m (point d). For an unchoked flow condition, say the mass flow rate is reduced from its choked value of 1.33 g/s (point b) to 1.17 g/s (point a0 ), the
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863 8
160
y'
x
7
R134a-48.9-0.66-1.0 c
Lexit [m]
140
Unchoking Tevap=-18.5C
6
120
5
Tevap=-25.5C
Chocking point
4
100
a
Unchoking a'
Pexit [kPa]
862
3
L=2.0m
b'
2
m=1.17g/s
Le=1.6m
1
d x'
b
Le=1.0m y 0 0.4
0.6
80
m=1.33g/s m=1.83g/s
0.8
1
1.2
1.4
1.6
1.8
60 2
m [g/s] 4
1.4 Nonchoked Region(I)
3.5
1.3
Choked Region(II)
1.2 3
Pchoked /P*>1
1.1
Lexit /L* >1
m*,L*,P* Pexit /P* =1
Le/L*2.5
1 0.9
2 Star Point (Pexit =Pchoked =Pevap =P*)
1.5
0.8
Mach Pchoked /P* <1
lexit
0.7
Pchoke
1
Lexit/ L* <1 Lexit /L* =1
0.5 0.5
Pe/P*
0.6
0.7
0.8
0.9
1
1.1
1.2
Pexit
1.3
0.6 0.5 1.4
m/m*
Fig. 5. (a) Full range diagram showing m–L–P characteristics for R134a, (b) equivalent full range dimensionless diagram (of Fig. 5a) showing m–L–P characteristics for R134a.
tube length should increase from its choked value of 1.6 m (at point b) to unchoked value of 2.0 m (corresponding to point b0 ). Use of diagram 5(b): diagram 5(b) plots the information of diagram 5(a) as dimensionless ratios. The ratios (of the actual value to the choked value of any parameter) at the intersection points are always unity. The left side region (I) is the unchoked region, while the right side region (II) is the choked region. In region (I): Lexit =L > 1, m_ =m_ < 1, Pexit =P ¼ 1, Pchoke =P < 1, Mach < 1. As the mass flow rate increases, the length reduces while the choking pressure
P.K. Bansal, G. Wang / Applied Thermal Engineering 24 (2004) 851–863
863
increases. The exit pressure equals the evaporating pressure until the star point is reached. In region (II), the mass flow rate is at the maximum possible value and the tube length is the choked length. If the choked length is fixed, the mass flow rate can never be increased. However, for increasing mass flow rate, the length decreases and the choking pressure still increases. The exit pressure equals the choking pressure from the star point. For Example: (a) Let us consider the condition where Tevap ¼ 25:5 °C, L ¼ 1:6 m and m ¼ 1:33 g/s. Now following diagram 5(b), for a flow rate of m_ ¼ 1:0 g/s, the mass ratio rm ¼ mm_ ¼ 1:0=1:33 ¼ 0:75 and the corresponding length ratio will be rL ¼ 1:59. Then L ¼ rL L ¼ 1:59 1:6 ¼ 2:54 (m), which is in the unchoked region. (b) For L ¼ 4:8 m, the length ratio from diagram 5(b) is rL ¼ L=L ¼ 4:8=1:6 ¼ 3:0. The corresponding mass ratio rm ¼ mm_ ¼ 0:5. Then m ¼ 0:5 1:33 ¼ 0:665 g/s and Mach ¼ 0.7, means that the point of interest is in the unchoked region. (c) For L ¼ 1:0 m, the length ratio from diagram 5(b) is rL ¼ L=L ¼ 1:0=1:6 ¼ 0:63. The corresponding mass ratio is rm ¼ mm_ ¼ 1:35. Therefore, m ¼ 1:35 1:33 ¼ 1:8 g/s and Mach ¼ 1.0, meaning that the point of interest is in the choked region. Following proposed analogy, similar diagrams can be generated for other refrigerants and varying operating conditions.
5. Conclusions This study has presented a new numerical analysis to simulate adiabatic refrigerant choked flow conditions in capillary tubes. The numerical accuracy of the presented model is significantly better than previous studies and agrees with the experimental data to within ±7%. A new full range diagram has been presented that can be used for the design and analysis of choked flow characteristics of capillary tubes, where either the capillary tube length or the refrigerant mass flow rate can be determined with either of them being an input.
References [1] P.K. Bansal, A.S. Rupasinghe, A homogeneous model for adiabatic capillary tubes, J. Appl. Thermal Eng. 18 (1998) 207–219. [2] Z.H. Chen et al., A correlation for metastable flow of refrigerant 12 through capillary tube, ASHRAE Trans. 96 (1) (1990) 550–554. [3] M. Chung, A numerical procedure for simulation of Fanno flows of refrigerant mixtures in capillary tubes, ASHRAE Trans. 104 (1998) 1031–1043. [4] E. Dirik et al., Numerical and experimental studies on adiabatic and non-adiabatic capillary tubes with R-134a, in: International Refrigeration Conference at Purdue, Purdue University, USA, 1994. [5] O. Garcia-Valladares, C.D. Perez-Segarra, A. Olivia, Numerical simulation of capillary tube expansion devices behaviour with pure and mixed refrigerants considering metastable region. Part II: experimental validation and parametric studies, Appl. Thermal Eng. 22 (2002) 379–391. [6] S. Klein, F. Alvarado, Engineering Equation Solver (EES), F-Chart Software, USA, 1992. [7] C. Melo et al., An experimental analysis of adiabatic capillary tubes, J. Appl. Thermal Eng. (1999) 669–684. [8] D.A. Wolf, R.R. Bittle, M.B. Pate, Adiabatic capillary tube performance with alternative refrigerants RP-762, Final Report, ERI-95413, ASHRAE Research Project, USA, 1995.