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An homogeneous model for adiabatic capillary tubes
Pergamon
Applied Thermal Engineering Vol. 18, Nos 3~1-, pp. 207 219, 1998 ~ 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S1359-4311(97)00016-1 1359-4311/98 $19.00 + 000
A N HOMOGENEOUS MODEL FOR ADIABATIC CAPILLARY TUBES P. K. Bansal* and A. S. Rupasinghe Department of Mechanical Engineering, The University of Auckland, Private Bag, 92019, Auckland, New Zealand
(Received 25 January 1997) Abstract--This paper presents a homogeneous two-phase flow model, CAPIL, which is designed to study the performance of adiabatic capillary tubes in small vapour compression refrigeration systems, in particular household refrigerators and freezers. The model is based on the fundamental equations of conservation of mass, energy and momentum that are solved simultaneously through iterative procedure and Simpson's rule. The model uses empirical correlations for single-phase and two-phase friction factors and also accounts for the entrance effects. The model uses the REFPROP data base where the Carnahan-Starling-DeSantis equation of state is used to calculate the refrigerant properties. The model includes the effect of various design parameters, namely the tube diameter, tube relative roughness, tube length, level of subcooling and the refrigerant flow rate. The model is validated with earlier models over a range of operating conditions and is found to agree reasonably well with the available experimental data for HFC-134a. © 1997 Elsevier Science Ltd.
Keywords--Capillary tube, adiabatic, domestic refrigerator, HFC-134a, two-phase flow.
NOMENCLATURE A d 6 f G g
h h~-r AL L Lsp Ltp in [1 P Pevap Pcond
Ap
P Tcond ATsub V
cross-sectional area of the capillary tube [m2] internal diameter of capillary tube [m] relative roughness of the tube material [ ] friction factor [ ] mass flux of refrigerant flow (in~A) [kg/s.m 2] acceleration due to gravity [ms-z ] specific refrigerant enthalpy [kJ/kg] total head loss [m] elemental length [m] total length of capillary tube [m] single-phase length of capillary tube [m] two-phase length of capillary tube [m] mass flow rate of refrigerant [kg/s] dynamic viscosity of the refrigerant [kg/m.s] refrigerant pressure [kPa] evaporator pressure [kPa] condenser pressure [kPa] pressure drop across the capillary [kPa] refrigerant density [kg/m 3] Condenser saturation temperature [K] Condenser subeooling [K] Refrigerant velocity [m/s] Refrigerant specific volume [m3/kg]
INTRODUCTION Capillary tubes are c o m m o n l y used as expansion and refrigerant controlling devices in small vapour compression refrigeration systems such as household refrigerators. It is a long simple hollow tube o f drawn copper with an internal diameter ranging from 0.5 x 10 -3 to 1.5 x 10 -3 m and length from 2 to 5 m. Capillaries are o f two types, namely the adiabatic (where refrigerant *Author to whom correspondence should be addressed. Email: P.Bansal(/~!auckland.ac.nz. 207
208
P. K. Bansal and A. S. Rupasinghe
Condenser
Adiabatic Capillary Tube
[i
Compressor
Evaporator
Fig. 1. Refrigerationsystemwith an adiabatic capillary tube as an expansion valve. expands from high pressure to low pressure adiabatically, see Fig. 1) and the non-adiabatic capillary tubes (where the refrigerant expands to low pressure but the capillary tube is set up to form a heat exchange relationship with suction line). This paper specifically discusses the design and performance aspects of adiabatic capillaries. During the last 50 years, the performance of adiabatic capillary tubes has been studied extensively. Bolstad and Jordan [1] presented an analytical solution as early as 1949 for adiabatic capillary tubes based on the homogeneous flow and constant friction factor. The flow equations, based on the conservation of mass, energy and momentum were solved using simplified methods. Later Marcy [2] did a similar study except that he used liquid viscosity for the calculation of two-phase Reynolds number. In 1950, Hopkins [3] presented a graphical method to integrate flow equations used in Bolstad and Jordan's [1] and Marcy's [2] studies. Cooper et al. [4] developed rating curves based on Hopkin's [3] work for capillary tube selection. Later Rezk and Awn [5] improved these charts by using Rhomberg integration to solve flow equations. Whitesel [6] studied adiabatic capillaries assuming constant friction factor, which was determined by averaging the liquid and vapour friction factors experimentally. Whitesel's [6] analysis was later coupled with Hopkin's [3] work and Rezk and Awn's [5] work to produce the well known ASHRAE [7] charts for capillary tube selection. Erth [8] followed Whitesel's [6] model except that he used the Rhomberg integration technique for solving governing equations. All these models [1-6, 8] used a constant friction factor throughout the capillary flow. In 1981, Maczek and Krolicki [9] developed a model for adiabatic capillary tubes using variable friction factors. The model results were compared with the experimental data but there were unexplained trends in the deviations between the model predictions and the experimental data. In 1983, Maczek et al. [10] addressed the refrigerant metastable flow phenomenon with twophase flow model and showed little improvement over their simpler homogeneous two-phase flow model. Lin et al. [11] developed correlations to calculate the single and two-phase flow friction factors. These correlations were used by Li et al. [12] in their numerical model to calculate the adiabatic capillary tube length. Melo et al. [13-15] have studied temperature and pressure profiles along the capillary tubes and developed a computer modeI--CAPILAR, where the twophase flow friction factor was based on the correlation proposed by Erth [8]. Wong and Ooi [16] have developed a separated flow model using Lin et al.'s [11] frictional pressure gradient correlations and reported a better prediction than the homogeneous flow models. Recently, Bansal and Rupasinghe [17] have proposed a simple empirical model for sizing both the adiabatic and non-adiabatic capillary tubes using HFC-134a. The methodology can be extended to other refrigerants and refrigerant mixtures in domestic refrigerators.
209
Homogeneousmodel for adiabatic capillary tubes
Most of these models [1-6,8-10, 12-17] have been used for a few and specific refrigerants only, such as CFC-12, HFC-134a and HC-600a, where the refrigerant properties were supplied through empirical equations or similar methods. There is a need for formulating these models and developing a methodology where refrigerant properties can be derived from advanced property data bases such as REFPROP [18]. This is specifically the subject matter of the present study. This paper presents a two-phase homogeneous flow model--CAPIL, for studying the performance of adiabatic capillary tubes in small refrigeration systems. The model uses the REFPROP package which is based on the Carnahan-Starling-DeSantis (CSD) equation of state for calculating thermodynamic and transport properties of the refrigerants. The REFPROP data base gives the user a choice of more than 40 refrigerants which can be handled either as pure refrigerants or as a mixture of refrigerants of up to five components. The model gives users and designers the flexibility of assessing the design and comparative performance aspects of capillary tubes with a number of alternative refrigerants and their mixtures.
MODELLING This section discusses the computer model, CAPIL, that has been developed [19] to design and study the performance of adiabatic capillary tubes in household refrigerators and freezers. The model includes the effect of various design parameters, namely the tube diameter, tube relative roughness, tube length, level of subcooling and the refrigerant flow rate. The model is based on the fundamental equations of conservation of mass, energy and momentum and the solution methodology is unique in the sense that the refrigerant properties (both thermodynamic and transport) are derived from REFPROP and do not use any empirical equations. The flow through a capillary tube can generally be divided into a liquid single-phase region, where the pressure decreases linearly until the flash point; and a two-phase region where the pressure drop per unit length increases as the length increases. Figure 2 shows a simplified diagram of the capillary tube connecting the condenser and the evaporator. In Fig. 2, point 1 denotes the condenser exit and point 2 is the capillary tube inlet. The refrigerant is subcooled liquid between region 2-3. There is a slight pressure drop between points 1 and 2 (capillary tube entrance loss). Beyond point 2, pressure decreases linearly until the refrigerant becomes saturated liquid at point 3. The two-phase region lies between 3 and 4 where the pressure drop varies non-linearly with length. Point 4 is the capillary tube exit and point 5 is the evaporator inlet. The expansion process from points 1 to 5 is shown in Fig. 3 on the temperature-entropy (T-s) and the pressure-enthalpy (p h) diagrams simultaneously. The flow through the capillary tube is based on the following assumptions: 1. 2. 3. 4. 5. 6. 7.
Straight capillary tube with constant inner diameter and roughness. Homogeneous and one-dimensional flow through the capillary tube. Capillary tube is fully insulated. The metastable flow phenomena is neglected. Refrigerant is free of oil. Flow through the capillary tube is fully developed turbulent flow. Refrigerant is either saturated or subcooled liquid at the capillary entrance. Capillary Tube
q
GI I I I I I
L sp
lll:Jb,!2 IIIii
I I I I I I I
Jill Condenser
Ltp
(~
_-
AL~ .
3
r
x
4]5 [, .
y
.
.
.
Evaporator
Fig. 2. Simplifieddiagram of the adiabatic capillary tube.
210
P. K. Bansal and A. S. Rupasinghe d
i
T
P 11
2,1 3'
/
5
41
\ s
h
Fig. 3. Location of state points 1-5 in T - s and p-h diagrams. The refrigerant flow in the capillary can be divided into single-phase and two-phase regions, as described below.
Single-phase region The steady flow energy equation between points 2 and 3 may be written as:
P2g
V2
p3 P3g
2g
+hl .
(1)
For single-phase flow, assuming that V2~ 113= V and z3 = z2, the term hlx (total head loss) can be written as [20]:
hiT :fsp ( - ~ )
(~-~)
(2)
where Lsp and f~p are the single-phase length and the single-phase friction factor, respectively. Substituting Equation 2 in Equation 1 and rearranging the terms, t h e following equation may be obtained: P2--
\ P--33]
(3)
The pressure drop due to sharp entrance into the capillary can be determined from the following equation; -
-
-
.
(4)
Saturation pressure at 3, P3, can be determined by knowing the level of subcooling at the capillary entrance i.e. T3 = Tl-ATsub. Therefore, adding Equations 3 and 4, and assuming P2 P3, the single-phase length, Lsp, o f the capillary tube can be determined from: =
Lsp= (Pl - P3) ~--~-
~
(5)
where G (tn/A) is the refrigerant mass flow rate per unit area, and the single-phase friction factor, f~p, can be calculated from the Churchill [21] correlation, as shown below:
8 [ ( 8 ,~12 fsp =
L\~ee ]
1 + ( A +})3/2
]1/12 (6)
Homogeneous model for adiabatic capillary tubes
where
7o.I
A = { 2.457 In
(Ree]
+0.27e
37530] 6 Re )
B=\
]}
211
16
(6a)
(6b)
and
tnd ,uA
Re =
-
(6c)
-
where c is relative roughness of the tube, d is the tube internal diameter and I~ is the dynamic viscosity of the liquid refrigerant.
Two-phase region The fundamental equations applicable to the control volume bounded by points x and y (see Fig. 2) in the two-phase region are the conservation of mass, the conservation of energy, and the conservation of momentum. The equation for conservation of mass states that tn
AV3 -
-
1;3
AV4 -
-
-
-
(7)
1;4
Neglecting the elevation difference and the heat transfer in and out of the tube, the equation for energy conservation may be written as: V~.
V2
h,-+7: : h + 7 -
(s)
where h and V are, respectively, the enthalpy and the velocity at any point in the two-phase region. From the conservation of momentum equation, the difference in forces applied to the element due to drag and pressure difference on opposite ends of the element should be equal to that needed to accelerate the fluid, i.e.
As the refrigerant flows through the capillary tube, its pressure and saturation temperature progressively drops and the vapour fraction, x, continuously increases. At any point, h = h/(1 - x) + hxx v = vi(l - x) + vex.
(10)
From Equation 9, it can be assumed that: p.,. - p , . = dp; p~.~p,.; V~ - V,. = dV. Therefore, the momentum Equation 9 can be rewritten as:
By rearranging the terms and substituting G -- in~A, the following equation can be obtained:
212
P.K. Bansaland A. S. Rupasinghe
In the above equation, the term (p/G) dV is equal to dV/V, which is equal to dp/p. By substituting these values and integrating between sections 3 and 4 of the two-phase flow of the capillary (Fig. 1), the two-phase length can be obtained as follows:
f4 p dp
2dlf4dp ttp
:f~p/J3 p
- J3 -G-T }"
(13)
In Equation 13, the integral J'34pdp is calculated using Simpson's rule. The two-phase flow friction factor (ftp) was calculated by the Lin et al. [11] correlation as given below: J~p = q~2fsp( Vsp"~ \ Vtp]
(14)
where fsp is expressed by Equation 6 and the multiplier, q~2, is given by:
-
8
12
_l
]
where the terms with suffixes 'sp' and 'tp' correspond to the single-phase and two-phase regions, respectively. A and B may be calculated using Equations 6a-b, respectively. To evaluate ftp, it is necessary to know the two-phase viscosity--law (to evaluate Re) and the specific volume in the two-phase region (Vtp). By knowing the refrigerant saturation pressure at condenser (Pcond), the vapour quality can be calculated from: /
(-hfg-G2vfvfg+~(hfg
+
G 2V2
-
x-
(16)
The vapour quality, x, in conjunction with Pcond, enables the calculation of two-phase viscosity (latp) from the REFPROP [17] and therefore, the variable friction factor in two-phase flow.
Solution methodology The governing equations presented so far were solved by writing a computer program in FORTRAN 77. The required refrigerant properties were taken from the REFPROP database. In the single-phase region, after calculating the Reynolds number, single-phase friction factor and single-phase length were calculated. Calculation of the two-phase region length (Ltp) required a numerical solution due to the term ~ p dp in Equation 13. The two-phase region 3-4 was divided into a number of small elements as shown in Fig. 4. For each element, the pressure (Pi), the temperature (Ti), the dryness fraction (xi), the entropy (s) and the two-phase friction factor (ftp) were calculated. At each element, the calculated entropy (si) was compared with the entropy of the previous element (si_ 0, to make sure that the entropy increased as the refrigerant flowed through the capillary tube. The entropy should increase through the capillary expansion process due to the fact that it is an adiabatic irrevers1
2
3
4
5
Pl P2 P3 P4 P5 refrigerant in
6
7
8
9
10
Capillary Tube
i-2 i-1
i
i+l
Pi
Fig. 4, Grid systemfor the numericalsolution.
4~ refrigerant out
Homogeneous model for adiabatic capillary tubes [ 1. Pcond or Read Inputs: 2. Pevap or I 3. Con6m~r
Teond Teyap ~ubc~li~
4. Capillary Tube Diameter 5. Roughness of Tube 6. Mass Flow Rate of R©f~crant
Using k (enlmnee loss factor) Calculate P2 Using Teond and ATsub
ic o ,o
I I
Calculate Lsp(equalion 5)
I I
Divide two phase region into small number of elements /
Calculate x, p, s and f tp at each element
L
Is entropy (s) rnazdmum?
]
F
L
i
+
I
Calculate Ixessure
[(Pi)s
]
where the entropy is maximum
Is [(Pi)smax] > Pevap?
P4 = ( Pi )Smax I Calculate average friction f ~ t ~ (fav )
Celedate L tp based on fay
~ l l s ~ y I~gfla L = L 8p+ L tp
Fig. 5. Flow diagram for adiabatic capillary tube model (CAP1L).
213
214
P.K. Bansal and A. S. Rupasinghe
ible process. As the calculation of properties moves towards the capillary exit point 4 (see Fig. 4), there may be a point where the entropy is maximum. Beyond this point, the entropy decreases as the length increases. Therefore, it is not necessary to do any further element property calculations beyond this point. At this stage the element at which the entropy is maximum, its pressure [(Pi)smax] is compared with the evaporator pressure (Pevap). If the evaporator pressure (Pevap) is less than the element pressure, the pressure at point 4 (P4) is taken as (Pi)sm~x" If evaporator pressure (Pevap) is greater than the element pressure, P4 is taken as the pressure of the evaporator (Pevap). This may be due to the reason that for a fixed condenser pressure (Pcond), the mass flow rate through the capillary tube increases as the evaporator pressure (Pevap) decreases. There is a critical evaporator pressure [(Pevap)crit] at which the mass flow rate through the capillary tube is maximum. If the evaporator pressure decreases beyond this value [(Pevap)crit], the mass flow rate does not increase and it remains unchanged. This condition is known as the critical flow condition of the capillary tube. In this analysis, the evaporator pressure was given as an input. If this pressure (Pevap) was less than (Pevap)crit, the flow condition through the capillary tube was maximum and the pressure (Pevap)crit should be used for the calculations. If the input evaporator pressure (Pevap) was greater than (Pevap)crit, then the flow through the capillary tube had not reached the critical (maximum) condition and Pevap could be used for the calculations. After determining the pressure at point 4, Equation 13 could be used to calculate the two-phase length (Ltp) where the two-phase friction factor (ftp) varies along the capillary tube. In this study, an averaged value for the two-phase friction factor (fav) was used, based on the following procedure:
fav =J~p' -'}-ftp2 -'}-'''-'}-ftpi-, -']-ftpi "l-ftp,, n
(17)
Now, Equation 13 can be re-written as:
Ltp = fa-~ 1 Jp3
P
Jp3
G2
•
The flow diagram for the adiabatic capillary tube model is shown in Fig. 5.
R E S U L T S AND D I S C U S S I O N This section discusses the validity of the computer m o d e l - - C A P I L , with earlier models [1315, 22] and experimental data [14,23]. Although the model is able to handle any alternative refrigerants (pure or mixture), the present calculations are only limited to HFC-134a due to the availability of test data on this refrigerant in the literature. The effect of different variables on the adiabatic capillary tube length is also discussed.
Validation of the computer model ( CAPIL) To validate the computer model, comparisons were made with limited available experimental and modelling data from Wijaya [23], Melo et al. [13, 14] and Wong et al. [22]. Wijaya [23] measured mass flow rates through a range of capillary tubes with different internal diameters. A range of condensing temperatures from 37.8 to 54.4°C with subcoolings from 5.5 to 16.7 K were chosen for the operating conditions. A number of capillaries were tested of lengths between 1.52 and 3.05 m. For the internal diameter of 0.66 mm, the tube lengths were limited to 2.74 m and the degrees of subcooling were limited from 11.1 to 16.7 K because the capillary tube exit pressure was below the atmospheric pressure beyond these ranges. To obtain relevant e which was not given in Wijaya, one set of his experimental data (e.g. d = 0.84mm, A T s u b = l l . l K, To=310 K) together with an assumed value for the relative roughness (c) was used to calculate the output from the current model (CAPIL), by varying the mass flow rate. For this value of c, the modelling results for capillary lengths were compared with the experimental data of Wijaya in Fig. 6. The figure shows the variation of mass flow rate versus tube length for d = 0.84 mm and ATsub = 16.7 K but for a number of condensing temperatures. It can be seen that the mass
Homogeneous model for adiabatic capillary tubes
215
13
d -- 0 . 8 4 m m 12
ATmb
16.7K
o Tcond = 310.8K • Tcond = 316.3K • Tcond = 321.9K
11
[] T c o n d = 3 2 7 . 4 K
i9 8
7
6 1.3
I
I
I
I
1.8
2.3
2.8
3.3
Adiabatic
Capillary
Tube
Length
(m)
Fig. 6. Comparison of the computer modelling results from CAP|L (solid lines), with the experimental data of Wijaya [23] (shown by points), for mass flow rate versus adiabatic capillary tube length.
flow rate decreases with condenser temperature and also with the capillary tube length. It is obvious that any increase in the tube length will evidently increase the pressure drop unless the pressure drop (AP) is not controlled externally. Therefore, to maintain the same pressure drop (AP), the mass flow rate has to be reduced significantly. The lengths (L) and the corresponding mass flow rates (in) are: 3.05 m, 1.52 m, 6.45 kg/h and 10.89 kg/h, respectively. The predicted results compare reasonably well within +7% with the experimental data. Melo et al. [13] carried out a mathematical study on adiabatic capillary tubes to predict the mass flow rate (in) through a 0.6 mm internal diameter capillary tube at a condensing temperature (Tcond) of 54.4 °C, evaporating temperature (Tevap) o f - 2 3 . 3 °C and condenser subcooling (ATsub) of 5.5 K. Their results are compared in Fig. 7 with CAPIL for HFC-134a. It can be seen that the mass flow rate decreases rapidly as the length increases from 0.5 to 2 m. The modelling results from CAPIL always underpredict Melo et al.'s [13] model by about 8%. The percentage error is small for shorter capillaries. Later in 1993, the same group (Melo et al. [14]) carried out an experimental study on six adiabatic capillary tubes. These capillaries had two different lengths and three different diameters. Tests were performed to measure the mass flow rate through these capillary tubes. By varying the condenser pressure (Pcona) and the amount of subcooling (ATsub), a number of mass flow rates were measured. For the comparison of modelling results (from CAPIL) with the experimental data, the length required for the measured mass flow rate (in) was calculated using
P. K. Bansal and A. S. Rupasinghe
216
fl = 0.60 m m
7
T,.o,d = 327.4 K
,.6
Tt.vap = 250 K AT.__ub~ 5.5 K
1 0
I 0.5
0
I 1
I 1.5
I 2
I 2.5
I 3
I 3.5
I 4
Adiabatic Capillary Tube Length (m) Fig. 7. Comparison of the modelling results from CAPIL (dotted line) with Melo et al. [13] (solid line), for mass flow rate versus adiabatic capillary tube length.
CAPIL and was compared with the capillary tube length from Melo et al. [14]. Fig. 8 shows the results of such a comparison for the capillary tube having d = 0.77mm, L = 2.926m, c = 9.74 x 10-4. It can be noted from Fig. 8 that the maximum percentage error is 9.3% when the subcooling (ATsub) is 6 K. For condenser pressure (Pcond) of 9 bar, the calculated results were overpredicted for subcooling (ATsub) between 2 and 7 K and 11 and 14 K. For the condenser pressure (Pcond) of 11 bar, calculated values for length (L) were overpredicted for subcooling (ATsub) between 2 and 3 K and 7 and 14 K. Figure 9 shows the comparison of the modelling results of Wong et al. [22] and CAPIL for the mass flow rate (in) versus capillary tube length (L). Wong et al. did not specify subcooling
15 ...... w#...°
r i i
~
5
Pcond = 9 bar [ Pcond = 11 bar[ .'''"
....
o.....
'6 i
~-5
-10
o.°
Condenser Subcooling (K) Fig. 8. Comparison of modelling results from CAPIL with the experimental data of Melo et al. [14].
Homogeneous model for adiabatic capillary tubes
~
3.5 3
~
2.5
~
1.5
•
217
- - .x. - - d = 0 . 6
mm
-- ~--d=0.7
mm
--v--o=u.B
i m
~q
u 0.5
0
I
I
I
I
I
I
I
0.5
1
1.5
2
2.5
3
3.5
Adiabatic
Capillary
Tube
Length
4
(m)
Fig. 9. Comparison of the modelling results of CAP1L (dotted line), with Wong et al. [22] (solid line) between mass flow rate (in) and adiabatic capillary tube length for three different diameters•
x • Subcooling = 0 K
3.5 o
•
i
- - ~ - - Subcooling
= 5 K
.... D .... S u b c o o l i n g
ffi 10 K
- - o - -Subcooling
= 15 K
- - ~ - - Subcooling
= 20 K
~2.5
~
1.5
I
I
I
I
I
0.2
0.4
0.6
0.8
1
Adiabatic
Capillary
Tube
Length
(m)
Fig. 10. Comparison of modelling results, from CAPIL (dotted line) and Wong et al. [22] (solid line), between mass flow rate (in) and adiabatic capillary tube length for a number of subcoolings. (ATsub) a n d the relative r o u g h n e s s (c). These values were d e t e r m i n e d by curve-fitting one set o f W o n g e t a l ' s d a t a (i.e. d = 0.6 m m , Pcond = 9.7 bar, Pevap = 1 bar) with C A P I L m o d e l l i n g results a n d were used for the rest o f the c o m p a r i s o n . It can be seen f r o m Fig. 9 t h a t the results o f C A P I L o v e r p r e d i c t the W o n g e t a l . ' s m o d e l for d = 0.6 mm. This was the d a t a used to d e t e r m i n e subc o o l i n g (AT~ub) a n d the relative r o u g h n e s s (e). T h e r e f o r e the p e r c e n t a g e e r r o r was between 2 a n d 7 % for this case. H o w e v e r , C A P I L u n d e r p r e d i c t s the W o n g e t al. m o d e l l i n g results for the o t h e r two cases o f d = 0.7 m m a n d d = 0.8 ram. F o r d = 0.7 m m , the p e r c e n t a g e e r r o r was between - 2 a n d - 8 . 5 % b u t for d = 0.8 m m , the e r r o r was between - 4 a n d - 1 1 % . It can also be n o t e d that the m a s s flow rate decreases with a decrease in the diameter. This is due to the
218
P.K. Bansal and A. S. Rupasinghe
10
•~
7 l
[ 1
[
0 0
Wong et al •
I
Experimental (Li et al) ] I 0.5
I 1
1.5
Distance from the Capillary Inlet (m) Fig. 11. Comparison of the computer modelling results (shown by a dotted line), with experimental data [12] (shown by points) and with the earlier model of Wong et al. [22] (shown by a solid line) for a set of given data. fact that pressure drop (AP) increases as the diameter decreases. The increase in pressure drop requires comparatively less mass flow rate for the desired operating conditions. Figure 10 shows the comparison of modelling results for R134a from CAPIL with Wong et al.'s results for mass flow rate (in) versus capillary tube length (L) with subcooling (ATsub)varying from 0 to 20 K and for 0.6 mm diameter capillary tube. The relative roughness (~) was back calculated by curve fitting Wong et al.'s data with the calculated values from CAPIL. The data for zero subcooling (A Tsub = 0 K) was used to determine the relative roughness; which was then used for the rest of the calculations. For data where subcoolings (ATsub) were 0, 5, 15 and 20 K; the modelling results overpredict the Wong et al. results with the average error being around 11%. The calculated values for subcooling (ATsub) 10 K underpredicted the Wong et al. results with an average error within 8.5%. Figure l 1 shows the comparison of modelling results of CAPIL with the experimental data of Li et al. [12] and with the modelling results of Wong et al. [22], for pressure along the capillary tube. As the capillary length increases, the pressure decreases linearly for length between 0 and 0.8 m (i.e. during the single-phase region). The rate of change of decrease in pressure increases beyond this point (i.e. two-phase region). The error between CAPIL results and the experimental data of Li et al. [12] was +_8%. It can be noted here that the percentage difference between the experimental data and CAPIL data increases as the length increases. The error between Wong et al. [21] and CAPIL results was between - 2 to + 11%.
CONCLUSIONS A homogeneous two-phase flow model, CAPIL, is developed to study the performance and design aspects of adiabatic capillary tubes for household refrigerators and freezers. A methodology is presented where these models can be implemented as subroutines in advanced refrigeration simulation programs, such as BICYCLE [24] for selecting proper capillaries. These programs can be based on large refrigerant property data bases such as R E F P R O P [17] (rather than empirical equations) that give accurate properties at a faster speed. The model is validated with earlier experimental data and simulation models and is found to agree within +_10%. Such
Homogeneous model for adiabatic capillary tubes a m o d e l will h e l p d e s i g n e r s o f c a p i l l a r y t u b e s t o u n d e r s t a n d
the thermo-physical
219 behaviour of
refrigerant flow and design better capillaries. REFERENCES 1. M. M. Bolstad and R. C. Jordan, Theory and use of the capillary tube expansion device. Part II, Nonadiabatic flow. Refrigerating Engineering 57, 572 583 (1949). 2. G. P. Marcy, Pressure drop with change of phase in a capillary tube. R~J'r~eeratingEngineering 57(1), 53--57 (1949). 3. N. E. Hopkins, Rating the restrictor tube. Refrigerating Engineering 58(11), 1087 1095 (1950). 4. L. Cooper, C. K. Chu and W. R. Brisken, Simple selection method for capillaries derived from physical flow conditions. Refrigerating Engineering 65(7), 37-46 (1957). 5. A. Rezk and A. Awn, Investigation on flow of RI2 through capillary tubes. In Proceedings o1"the 15th International Congress of Re/?~geration, Commission, B2, Vol. 2, pp. 443452 (1979). 6. H. A. Whitesel, Capillary two-phase flow, part II. Refrigerating Engineering 65(9), 35 47 (1957). 7. ASHRAE, 1988 ASHRAE Handbook Equipment, chap. 19. American Society of Heating, Refrigerating, and AirConditioning Engineers, Inc., Atlanta (1988). 8. R. A. Erth, Two-phase flow in refrigeration capillary tubes: analysis and prediction. Ph.D. thesis, Purdue University, West Lafayette, USA (1969). 9. K. Maczek and Z. Krolicki, Non-adiabatic process in throttling capillary tubes used in packaged units. In Proceedings of the International Institute t~f Refrigeration, Commission B2, Paris, pp. 134-142 (1981 ). 10. K. Maczek, Z. Krolicki and E. Sochanecka, Model of throttling capillary tube with metastable process. In Proceedings of the XVth International Congress of Refrigeration, Commission B2, Paris, 154-161 (1983). 11. S. Lin, C. C. K. Kwok, R. Y. Li, Z. H. Chen and Z. Y. Chen, Local frictional pressure drop during vaporisation of R 12 through capillary tubes. International Journal o["Multiphase Flaw 17(1), 95 102 (1991). 12. R. Y. Li, S. Lin, Z. Y. Chen and Z. H. Chen, Metastable flow of R12 through capillary tubes. International Journal ~?/Re/'rigeration 13, 181 186 (1990). 13. C. Melo, R. T. S. Ferreira and R. H. Pereira, Modelling adiabatic capillary tubes: a critical analysis. In Proceedings t?['the HR-Purdue Con/~,rence, West Lafayette, USA, pp. 113 123 (1992). 14. C. Melo, R. T. S. Ferreira, C. Boabaid Neto, J. M. Goncalves, R. H. Pereira and M. R. Thiessen, Evaluation of HC-600a, HFC-134a and CFC-12 mass flow rates through capillary tubes. In Proceedings of the HR Coq~,rence, Hannover, Germany, pp. 621 630 (1994). 15. C. Melo, R. T. S. Ferreira, C. B. Neto and J. M. Goncalves, Experimentation and analysis of refrigerant flow through adiabatic capillary tubes. In Proceedings ASME Int. Mech. Engrs Congress and Exposition, AES, Vol. 34, pp. 19-30 (1995). 16. T. N. Wong and K. T. Ooi, Adiabatic capillary tube expansion devices: a comparison of the homogeneous flow and the separated flow models. Applied Thermal Engineering 16(7), 625-634 (1996). 17. P. K. Bansal and A. S. Rupasinghe, An empirical correlation for sizing capillary tubes. International Journal of Re#~geration 19(8), 497 505 (1996). 18. M. O. McLinden, J. G. Gallagher, M. L. Huber and G. Morrison, REFPROP refrigerant properties database: capabilities, limitations, and future directions. ASHRAE NIST Refi'igerants Conference, Gaithersburg, Maryland, pp. 59 71 (1993). 19. A. S. Rupasinghe, Modelling of capillary tubes for household refrigerators and freezers. M.E. Thesis, University of Auckland, Auckland, New Zealand (1996). 20. F. M. White, Viscous Fluid Flow. McGraw-Hill, New York (1974). 21. S. W. Churchill, Frictional equation spans all fluid flow regimes. Chemical Engineering 84, 91-92 (1977). 22. T. N. Wong, K. T. Ooi and C. T. Khoo, A study on capillary tube flow. In Proceedings ~?f the llR-Purdue Conl~,rence, West Lafayette, USA, pp. 371-376 (1992). 23. H. Wijaya, Adiabatic capillary tube test data for HFC-134a. In International Refrigeration Conference on Energy E[h'ciemT and New Re/?igerants, Purdue University, West Lafayette, USA, Vol. 1, pp. 63-71 (1992). 24. P. K. Bansal and C. K. Rice, Thermodynamic model for R-12 alternatives in domestic refrigerators--initial analysis. Proceedings of the HR Conference on Cold Chain Refrigeration Equipment by Design held at Massey University, New Zealand, pp. 95 104 (1993).
Applied Thermal Engineering Vol. 18, Nos 3~1-, pp. 207 219, 1998 ~ 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S1359-4311(97)00016-1 1359-4311/98 $19.00 + 000
A N HOMOGENEOUS MODEL FOR ADIABATIC CAPILLARY TUBES P. K. Bansal* and A. S. Rupasinghe Department of Mechanical Engineering, The University of Auckland, Private Bag, 92019, Auckland, New Zealand
(Received 25 January 1997) Abstract--This paper presents a homogeneous two-phase flow model, CAPIL, which is designed to study the performance of adiabatic capillary tubes in small vapour compression refrigeration systems, in particular household refrigerators and freezers. The model is based on the fundamental equations of conservation of mass, energy and momentum that are solved simultaneously through iterative procedure and Simpson's rule. The model uses empirical correlations for single-phase and two-phase friction factors and also accounts for the entrance effects. The model uses the REFPROP data base where the Carnahan-Starling-DeSantis equation of state is used to calculate the refrigerant properties. The model includes the effect of various design parameters, namely the tube diameter, tube relative roughness, tube length, level of subcooling and the refrigerant flow rate. The model is validated with earlier models over a range of operating conditions and is found to agree reasonably well with the available experimental data for HFC-134a. © 1997 Elsevier Science Ltd.
Keywords--Capillary tube, adiabatic, domestic refrigerator, HFC-134a, two-phase flow.
NOMENCLATURE A d 6 f G g
h h~-r AL L Lsp Ltp in [1 P Pevap Pcond
Ap
P Tcond ATsub V
cross-sectional area of the capillary tube [m2] internal diameter of capillary tube [m] relative roughness of the tube material [ ] friction factor [ ] mass flux of refrigerant flow (in~A) [kg/s.m 2] acceleration due to gravity [ms-z ] specific refrigerant enthalpy [kJ/kg] total head loss [m] elemental length [m] total length of capillary tube [m] single-phase length of capillary tube [m] two-phase length of capillary tube [m] mass flow rate of refrigerant [kg/s] dynamic viscosity of the refrigerant [kg/m.s] refrigerant pressure [kPa] evaporator pressure [kPa] condenser pressure [kPa] pressure drop across the capillary [kPa] refrigerant density [kg/m 3] Condenser saturation temperature [K] Condenser subeooling [K] Refrigerant velocity [m/s] Refrigerant specific volume [m3/kg]
INTRODUCTION Capillary tubes are c o m m o n l y used as expansion and refrigerant controlling devices in small vapour compression refrigeration systems such as household refrigerators. It is a long simple hollow tube o f drawn copper with an internal diameter ranging from 0.5 x 10 -3 to 1.5 x 10 -3 m and length from 2 to 5 m. Capillaries are o f two types, namely the adiabatic (where refrigerant *Author to whom correspondence should be addressed. Email: P.Bansal(/~!auckland.ac.nz. 207
208
P. K. Bansal and A. S. Rupasinghe
Condenser
Adiabatic Capillary Tube
[i
Compressor
Evaporator
Fig. 1. Refrigerationsystemwith an adiabatic capillary tube as an expansion valve. expands from high pressure to low pressure adiabatically, see Fig. 1) and the non-adiabatic capillary tubes (where the refrigerant expands to low pressure but the capillary tube is set up to form a heat exchange relationship with suction line). This paper specifically discusses the design and performance aspects of adiabatic capillaries. During the last 50 years, the performance of adiabatic capillary tubes has been studied extensively. Bolstad and Jordan [1] presented an analytical solution as early as 1949 for adiabatic capillary tubes based on the homogeneous flow and constant friction factor. The flow equations, based on the conservation of mass, energy and momentum were solved using simplified methods. Later Marcy [2] did a similar study except that he used liquid viscosity for the calculation of two-phase Reynolds number. In 1950, Hopkins [3] presented a graphical method to integrate flow equations used in Bolstad and Jordan's [1] and Marcy's [2] studies. Cooper et al. [4] developed rating curves based on Hopkin's [3] work for capillary tube selection. Later Rezk and Awn [5] improved these charts by using Rhomberg integration to solve flow equations. Whitesel [6] studied adiabatic capillaries assuming constant friction factor, which was determined by averaging the liquid and vapour friction factors experimentally. Whitesel's [6] analysis was later coupled with Hopkin's [3] work and Rezk and Awn's [5] work to produce the well known ASHRAE [7] charts for capillary tube selection. Erth [8] followed Whitesel's [6] model except that he used the Rhomberg integration technique for solving governing equations. All these models [1-6, 8] used a constant friction factor throughout the capillary flow. In 1981, Maczek and Krolicki [9] developed a model for adiabatic capillary tubes using variable friction factors. The model results were compared with the experimental data but there were unexplained trends in the deviations between the model predictions and the experimental data. In 1983, Maczek et al. [10] addressed the refrigerant metastable flow phenomenon with twophase flow model and showed little improvement over their simpler homogeneous two-phase flow model. Lin et al. [11] developed correlations to calculate the single and two-phase flow friction factors. These correlations were used by Li et al. [12] in their numerical model to calculate the adiabatic capillary tube length. Melo et al. [13-15] have studied temperature and pressure profiles along the capillary tubes and developed a computer modeI--CAPILAR, where the twophase flow friction factor was based on the correlation proposed by Erth [8]. Wong and Ooi [16] have developed a separated flow model using Lin et al.'s [11] frictional pressure gradient correlations and reported a better prediction than the homogeneous flow models. Recently, Bansal and Rupasinghe [17] have proposed a simple empirical model for sizing both the adiabatic and non-adiabatic capillary tubes using HFC-134a. The methodology can be extended to other refrigerants and refrigerant mixtures in domestic refrigerators.
209
Homogeneousmodel for adiabatic capillary tubes
Most of these models [1-6,8-10, 12-17] have been used for a few and specific refrigerants only, such as CFC-12, HFC-134a and HC-600a, where the refrigerant properties were supplied through empirical equations or similar methods. There is a need for formulating these models and developing a methodology where refrigerant properties can be derived from advanced property data bases such as REFPROP [18]. This is specifically the subject matter of the present study. This paper presents a two-phase homogeneous flow model--CAPIL, for studying the performance of adiabatic capillary tubes in small refrigeration systems. The model uses the REFPROP package which is based on the Carnahan-Starling-DeSantis (CSD) equation of state for calculating thermodynamic and transport properties of the refrigerants. The REFPROP data base gives the user a choice of more than 40 refrigerants which can be handled either as pure refrigerants or as a mixture of refrigerants of up to five components. The model gives users and designers the flexibility of assessing the design and comparative performance aspects of capillary tubes with a number of alternative refrigerants and their mixtures.
MODELLING This section discusses the computer model, CAPIL, that has been developed [19] to design and study the performance of adiabatic capillary tubes in household refrigerators and freezers. The model includes the effect of various design parameters, namely the tube diameter, tube relative roughness, tube length, level of subcooling and the refrigerant flow rate. The model is based on the fundamental equations of conservation of mass, energy and momentum and the solution methodology is unique in the sense that the refrigerant properties (both thermodynamic and transport) are derived from REFPROP and do not use any empirical equations. The flow through a capillary tube can generally be divided into a liquid single-phase region, where the pressure decreases linearly until the flash point; and a two-phase region where the pressure drop per unit length increases as the length increases. Figure 2 shows a simplified diagram of the capillary tube connecting the condenser and the evaporator. In Fig. 2, point 1 denotes the condenser exit and point 2 is the capillary tube inlet. The refrigerant is subcooled liquid between region 2-3. There is a slight pressure drop between points 1 and 2 (capillary tube entrance loss). Beyond point 2, pressure decreases linearly until the refrigerant becomes saturated liquid at point 3. The two-phase region lies between 3 and 4 where the pressure drop varies non-linearly with length. Point 4 is the capillary tube exit and point 5 is the evaporator inlet. The expansion process from points 1 to 5 is shown in Fig. 3 on the temperature-entropy (T-s) and the pressure-enthalpy (p h) diagrams simultaneously. The flow through the capillary tube is based on the following assumptions: 1. 2. 3. 4. 5. 6. 7.
Straight capillary tube with constant inner diameter and roughness. Homogeneous and one-dimensional flow through the capillary tube. Capillary tube is fully insulated. The metastable flow phenomena is neglected. Refrigerant is free of oil. Flow through the capillary tube is fully developed turbulent flow. Refrigerant is either saturated or subcooled liquid at the capillary entrance. Capillary Tube
q
GI I I I I I
L sp
lll:Jb,!2 IIIii
I I I I I I I
Jill Condenser
Ltp
(~
_-
AL~ .
3
r
x
4]5 [, .
y
.
.
.
Evaporator
Fig. 2. Simplifieddiagram of the adiabatic capillary tube.
210
P. K. Bansal and A. S. Rupasinghe d
i
T
P 11
2,1 3'
/
5
41
\ s
h
Fig. 3. Location of state points 1-5 in T - s and p-h diagrams. The refrigerant flow in the capillary can be divided into single-phase and two-phase regions, as described below.
Single-phase region The steady flow energy equation between points 2 and 3 may be written as:
P2g
V2
p3 P3g
2g
+hl .
(1)
For single-phase flow, assuming that V2~ 113= V and z3 = z2, the term hlx (total head loss) can be written as [20]:
hiT :fsp ( - ~ )
(~-~)
(2)
where Lsp and f~p are the single-phase length and the single-phase friction factor, respectively. Substituting Equation 2 in Equation 1 and rearranging the terms, t h e following equation may be obtained: P2--
\ P--33]
(3)
The pressure drop due to sharp entrance into the capillary can be determined from the following equation; -
-
-
.
(4)
Saturation pressure at 3, P3, can be determined by knowing the level of subcooling at the capillary entrance i.e. T3 = Tl-ATsub. Therefore, adding Equations 3 and 4, and assuming P2 P3, the single-phase length, Lsp, o f the capillary tube can be determined from: =
Lsp= (Pl - P3) ~--~-
~
(5)
where G (tn/A) is the refrigerant mass flow rate per unit area, and the single-phase friction factor, f~p, can be calculated from the Churchill [21] correlation, as shown below:
8 [ ( 8 ,~12 fsp =
L\~ee ]
1 + ( A +})3/2
]1/12 (6)
Homogeneous model for adiabatic capillary tubes
where
7o.I
A = { 2.457 In
(Ree]
+0.27e
37530] 6 Re )
B=\
]}
211
16
(6a)
(6b)
and
tnd ,uA
Re =
-
(6c)
-
where c is relative roughness of the tube, d is the tube internal diameter and I~ is the dynamic viscosity of the liquid refrigerant.
Two-phase region The fundamental equations applicable to the control volume bounded by points x and y (see Fig. 2) in the two-phase region are the conservation of mass, the conservation of energy, and the conservation of momentum. The equation for conservation of mass states that tn
AV3 -
-
1;3
AV4 -
-
-
-
(7)
1;4
Neglecting the elevation difference and the heat transfer in and out of the tube, the equation for energy conservation may be written as: V~.
V2
h,-+7: : h + 7 -
(s)
where h and V are, respectively, the enthalpy and the velocity at any point in the two-phase region. From the conservation of momentum equation, the difference in forces applied to the element due to drag and pressure difference on opposite ends of the element should be equal to that needed to accelerate the fluid, i.e.
As the refrigerant flows through the capillary tube, its pressure and saturation temperature progressively drops and the vapour fraction, x, continuously increases. At any point, h = h/(1 - x) + hxx v = vi(l - x) + vex.
(10)
From Equation 9, it can be assumed that: p.,. - p , . = dp; p~.~p,.; V~ - V,. = dV. Therefore, the momentum Equation 9 can be rewritten as:
By rearranging the terms and substituting G -- in~A, the following equation can be obtained:
212
P.K. Bansaland A. S. Rupasinghe
In the above equation, the term (p/G) dV is equal to dV/V, which is equal to dp/p. By substituting these values and integrating between sections 3 and 4 of the two-phase flow of the capillary (Fig. 1), the two-phase length can be obtained as follows:
f4 p dp
2dlf4dp ttp
:f~p/J3 p
- J3 -G-T }"
(13)
In Equation 13, the integral J'34pdp is calculated using Simpson's rule. The two-phase flow friction factor (ftp) was calculated by the Lin et al. [11] correlation as given below: J~p = q~2fsp( Vsp"~ \ Vtp]
(14)
where fsp is expressed by Equation 6 and the multiplier, q~2, is given by:
-
8
12
_l
]
where the terms with suffixes 'sp' and 'tp' correspond to the single-phase and two-phase regions, respectively. A and B may be calculated using Equations 6a-b, respectively. To evaluate ftp, it is necessary to know the two-phase viscosity--law (to evaluate Re) and the specific volume in the two-phase region (Vtp). By knowing the refrigerant saturation pressure at condenser (Pcond), the vapour quality can be calculated from: /
(-hfg-G2vfvfg+~(hfg
+
G 2V2
-
x-
(16)
The vapour quality, x, in conjunction with Pcond, enables the calculation of two-phase viscosity (latp) from the REFPROP [17] and therefore, the variable friction factor in two-phase flow.
Solution methodology The governing equations presented so far were solved by writing a computer program in FORTRAN 77. The required refrigerant properties were taken from the REFPROP database. In the single-phase region, after calculating the Reynolds number, single-phase friction factor and single-phase length were calculated. Calculation of the two-phase region length (Ltp) required a numerical solution due to the term ~ p dp in Equation 13. The two-phase region 3-4 was divided into a number of small elements as shown in Fig. 4. For each element, the pressure (Pi), the temperature (Ti), the dryness fraction (xi), the entropy (s) and the two-phase friction factor (ftp) were calculated. At each element, the calculated entropy (si) was compared with the entropy of the previous element (si_ 0, to make sure that the entropy increased as the refrigerant flowed through the capillary tube. The entropy should increase through the capillary expansion process due to the fact that it is an adiabatic irrevers1
2
3
4
5
Pl P2 P3 P4 P5 refrigerant in
6
7
8
9
10
Capillary Tube
i-2 i-1
i
i+l
Pi
Fig. 4, Grid systemfor the numericalsolution.
4~ refrigerant out
Homogeneous model for adiabatic capillary tubes [ 1. Pcond or Read Inputs: 2. Pevap or I 3. Con6m~r
Teond Teyap ~ubc~li~
4. Capillary Tube Diameter 5. Roughness of Tube 6. Mass Flow Rate of R©f~crant
Using k (enlmnee loss factor) Calculate P2 Using Teond and ATsub
ic o ,o
I I
Calculate Lsp(equalion 5)
I I
Divide two phase region into small number of elements /
Calculate x, p, s and f tp at each element
L
Is entropy (s) rnazdmum?
]
F
L
i
+
I
Calculate Ixessure
[(Pi)s
]
where the entropy is maximum
Is [(Pi)smax] > Pevap?
P4 = ( Pi )Smax I Calculate average friction f ~ t ~ (fav )
Celedate L tp based on fay
~ l l s ~ y I~gfla L = L 8p+ L tp
Fig. 5. Flow diagram for adiabatic capillary tube model (CAP1L).
213
214
P.K. Bansal and A. S. Rupasinghe
ible process. As the calculation of properties moves towards the capillary exit point 4 (see Fig. 4), there may be a point where the entropy is maximum. Beyond this point, the entropy decreases as the length increases. Therefore, it is not necessary to do any further element property calculations beyond this point. At this stage the element at which the entropy is maximum, its pressure [(Pi)smax] is compared with the evaporator pressure (Pevap). If the evaporator pressure (Pevap) is less than the element pressure, the pressure at point 4 (P4) is taken as (Pi)sm~x" If evaporator pressure (Pevap) is greater than the element pressure, P4 is taken as the pressure of the evaporator (Pevap). This may be due to the reason that for a fixed condenser pressure (Pcond), the mass flow rate through the capillary tube increases as the evaporator pressure (Pevap) decreases. There is a critical evaporator pressure [(Pevap)crit] at which the mass flow rate through the capillary tube is maximum. If the evaporator pressure decreases beyond this value [(Pevap)crit], the mass flow rate does not increase and it remains unchanged. This condition is known as the critical flow condition of the capillary tube. In this analysis, the evaporator pressure was given as an input. If this pressure (Pevap) was less than (Pevap)crit, the flow condition through the capillary tube was maximum and the pressure (Pevap)crit should be used for the calculations. If the input evaporator pressure (Pevap) was greater than (Pevap)crit, then the flow through the capillary tube had not reached the critical (maximum) condition and Pevap could be used for the calculations. After determining the pressure at point 4, Equation 13 could be used to calculate the two-phase length (Ltp) where the two-phase friction factor (ftp) varies along the capillary tube. In this study, an averaged value for the two-phase friction factor (fav) was used, based on the following procedure:
fav =J~p' -'}-ftp2 -'}-'''-'}-ftpi-, -']-ftpi "l-ftp,, n
(17)
Now, Equation 13 can be re-written as:
Ltp = fa-~ 1 Jp3
P
Jp3
G2
•
The flow diagram for the adiabatic capillary tube model is shown in Fig. 5.
R E S U L T S AND D I S C U S S I O N This section discusses the validity of the computer m o d e l - - C A P I L , with earlier models [1315, 22] and experimental data [14,23]. Although the model is able to handle any alternative refrigerants (pure or mixture), the present calculations are only limited to HFC-134a due to the availability of test data on this refrigerant in the literature. The effect of different variables on the adiabatic capillary tube length is also discussed.
Validation of the computer model ( CAPIL) To validate the computer model, comparisons were made with limited available experimental and modelling data from Wijaya [23], Melo et al. [13, 14] and Wong et al. [22]. Wijaya [23] measured mass flow rates through a range of capillary tubes with different internal diameters. A range of condensing temperatures from 37.8 to 54.4°C with subcoolings from 5.5 to 16.7 K were chosen for the operating conditions. A number of capillaries were tested of lengths between 1.52 and 3.05 m. For the internal diameter of 0.66 mm, the tube lengths were limited to 2.74 m and the degrees of subcooling were limited from 11.1 to 16.7 K because the capillary tube exit pressure was below the atmospheric pressure beyond these ranges. To obtain relevant e which was not given in Wijaya, one set of his experimental data (e.g. d = 0.84mm, A T s u b = l l . l K, To=310 K) together with an assumed value for the relative roughness (c) was used to calculate the output from the current model (CAPIL), by varying the mass flow rate. For this value of c, the modelling results for capillary lengths were compared with the experimental data of Wijaya in Fig. 6. The figure shows the variation of mass flow rate versus tube length for d = 0.84 mm and ATsub = 16.7 K but for a number of condensing temperatures. It can be seen that the mass
Homogeneous model for adiabatic capillary tubes
215
13
d -- 0 . 8 4 m m 12
ATmb
16.7K
o Tcond = 310.8K • Tcond = 316.3K • Tcond = 321.9K
11
[] T c o n d = 3 2 7 . 4 K
i9 8
7
6 1.3
I
I
I
I
1.8
2.3
2.8
3.3
Adiabatic
Capillary
Tube
Length
(m)
Fig. 6. Comparison of the computer modelling results from CAP|L (solid lines), with the experimental data of Wijaya [23] (shown by points), for mass flow rate versus adiabatic capillary tube length.
flow rate decreases with condenser temperature and also with the capillary tube length. It is obvious that any increase in the tube length will evidently increase the pressure drop unless the pressure drop (AP) is not controlled externally. Therefore, to maintain the same pressure drop (AP), the mass flow rate has to be reduced significantly. The lengths (L) and the corresponding mass flow rates (in) are: 3.05 m, 1.52 m, 6.45 kg/h and 10.89 kg/h, respectively. The predicted results compare reasonably well within +7% with the experimental data. Melo et al. [13] carried out a mathematical study on adiabatic capillary tubes to predict the mass flow rate (in) through a 0.6 mm internal diameter capillary tube at a condensing temperature (Tcond) of 54.4 °C, evaporating temperature (Tevap) o f - 2 3 . 3 °C and condenser subcooling (ATsub) of 5.5 K. Their results are compared in Fig. 7 with CAPIL for HFC-134a. It can be seen that the mass flow rate decreases rapidly as the length increases from 0.5 to 2 m. The modelling results from CAPIL always underpredict Melo et al.'s [13] model by about 8%. The percentage error is small for shorter capillaries. Later in 1993, the same group (Melo et al. [14]) carried out an experimental study on six adiabatic capillary tubes. These capillaries had two different lengths and three different diameters. Tests were performed to measure the mass flow rate through these capillary tubes. By varying the condenser pressure (Pcona) and the amount of subcooling (ATsub), a number of mass flow rates were measured. For the comparison of modelling results (from CAPIL) with the experimental data, the length required for the measured mass flow rate (in) was calculated using
P. K. Bansal and A. S. Rupasinghe
216
fl = 0.60 m m
7
T,.o,d = 327.4 K
,.6
Tt.vap = 250 K AT.__ub~ 5.5 K
1 0
I 0.5
0
I 1
I 1.5
I 2
I 2.5
I 3
I 3.5
I 4
Adiabatic Capillary Tube Length (m) Fig. 7. Comparison of the modelling results from CAPIL (dotted line) with Melo et al. [13] (solid line), for mass flow rate versus adiabatic capillary tube length.
CAPIL and was compared with the capillary tube length from Melo et al. [14]. Fig. 8 shows the results of such a comparison for the capillary tube having d = 0.77mm, L = 2.926m, c = 9.74 x 10-4. It can be noted from Fig. 8 that the maximum percentage error is 9.3% when the subcooling (ATsub) is 6 K. For condenser pressure (Pcond) of 9 bar, the calculated results were overpredicted for subcooling (ATsub) between 2 and 7 K and 11 and 14 K. For the condenser pressure (Pcond) of 11 bar, calculated values for length (L) were overpredicted for subcooling (ATsub) between 2 and 3 K and 7 and 14 K. Figure 9 shows the comparison of the modelling results of Wong et al. [22] and CAPIL for the mass flow rate (in) versus capillary tube length (L). Wong et al. did not specify subcooling
15 ...... w#...°
r i i
~
5
Pcond = 9 bar [ Pcond = 11 bar[ .'''"
....
o.....
'6 i
~-5
-10
o.°
Condenser Subcooling (K) Fig. 8. Comparison of modelling results from CAPIL with the experimental data of Melo et al. [14].
Homogeneous model for adiabatic capillary tubes
~
3.5 3
~
2.5
~
1.5
•
217
- - .x. - - d = 0 . 6
mm
-- ~--d=0.7
mm
--v--o=u.B
i m
~q
u 0.5
0
I
I
I
I
I
I
I
0.5
1
1.5
2
2.5
3
3.5
Adiabatic
Capillary
Tube
Length
4
(m)
Fig. 9. Comparison of the modelling results of CAP1L (dotted line), with Wong et al. [22] (solid line) between mass flow rate (in) and adiabatic capillary tube length for three different diameters•
x • Subcooling = 0 K
3.5 o
•
i
- - ~ - - Subcooling
= 5 K
.... D .... S u b c o o l i n g
ffi 10 K
- - o - -Subcooling
= 15 K
- - ~ - - Subcooling
= 20 K
~2.5
~
1.5
I
I
I
I
I
0.2
0.4
0.6
0.8
1
Adiabatic
Capillary
Tube
Length
(m)
Fig. 10. Comparison of modelling results, from CAPIL (dotted line) and Wong et al. [22] (solid line), between mass flow rate (in) and adiabatic capillary tube length for a number of subcoolings. (ATsub) a n d the relative r o u g h n e s s (c). These values were d e t e r m i n e d by curve-fitting one set o f W o n g e t a l ' s d a t a (i.e. d = 0.6 m m , Pcond = 9.7 bar, Pevap = 1 bar) with C A P I L m o d e l l i n g results a n d were used for the rest o f the c o m p a r i s o n . It can be seen f r o m Fig. 9 t h a t the results o f C A P I L o v e r p r e d i c t the W o n g e t a l . ' s m o d e l for d = 0.6 mm. This was the d a t a used to d e t e r m i n e subc o o l i n g (AT~ub) a n d the relative r o u g h n e s s (e). T h e r e f o r e the p e r c e n t a g e e r r o r was between 2 a n d 7 % for this case. H o w e v e r , C A P I L u n d e r p r e d i c t s the W o n g e t al. m o d e l l i n g results for the o t h e r two cases o f d = 0.7 m m a n d d = 0.8 ram. F o r d = 0.7 m m , the p e r c e n t a g e e r r o r was between - 2 a n d - 8 . 5 % b u t for d = 0.8 m m , the e r r o r was between - 4 a n d - 1 1 % . It can also be n o t e d that the m a s s flow rate decreases with a decrease in the diameter. This is due to the
218
P.K. Bansal and A. S. Rupasinghe
10
•~
7 l
[ 1
[
0 0
Wong et al •
I
Experimental (Li et al) ] I 0.5
I 1
1.5
Distance from the Capillary Inlet (m) Fig. 11. Comparison of the computer modelling results (shown by a dotted line), with experimental data [12] (shown by points) and with the earlier model of Wong et al. [22] (shown by a solid line) for a set of given data. fact that pressure drop (AP) increases as the diameter decreases. The increase in pressure drop requires comparatively less mass flow rate for the desired operating conditions. Figure 10 shows the comparison of modelling results for R134a from CAPIL with Wong et al.'s results for mass flow rate (in) versus capillary tube length (L) with subcooling (ATsub)varying from 0 to 20 K and for 0.6 mm diameter capillary tube. The relative roughness (~) was back calculated by curve fitting Wong et al.'s data with the calculated values from CAPIL. The data for zero subcooling (A Tsub = 0 K) was used to determine the relative roughness; which was then used for the rest of the calculations. For data where subcoolings (ATsub) were 0, 5, 15 and 20 K; the modelling results overpredict the Wong et al. results with the average error being around 11%. The calculated values for subcooling (ATsub) 10 K underpredicted the Wong et al. results with an average error within 8.5%. Figure l 1 shows the comparison of modelling results of CAPIL with the experimental data of Li et al. [12] and with the modelling results of Wong et al. [22], for pressure along the capillary tube. As the capillary length increases, the pressure decreases linearly for length between 0 and 0.8 m (i.e. during the single-phase region). The rate of change of decrease in pressure increases beyond this point (i.e. two-phase region). The error between CAPIL results and the experimental data of Li et al. [12] was +_8%. It can be noted here that the percentage difference between the experimental data and CAPIL data increases as the length increases. The error between Wong et al. [21] and CAPIL results was between - 2 to + 11%.
CONCLUSIONS A homogeneous two-phase flow model, CAPIL, is developed to study the performance and design aspects of adiabatic capillary tubes for household refrigerators and freezers. A methodology is presented where these models can be implemented as subroutines in advanced refrigeration simulation programs, such as BICYCLE [24] for selecting proper capillaries. These programs can be based on large refrigerant property data bases such as R E F P R O P [17] (rather than empirical equations) that give accurate properties at a faster speed. The model is validated with earlier experimental data and simulation models and is found to agree within +_10%. Such
Homogeneous model for adiabatic capillary tubes a m o d e l will h e l p d e s i g n e r s o f c a p i l l a r y t u b e s t o u n d e r s t a n d
the thermo-physical
219 behaviour of
refrigerant flow and design better capillaries. REFERENCES 1. M. M. Bolstad and R. C. Jordan, Theory and use of the capillary tube expansion device. Part II, Nonadiabatic flow. Refrigerating Engineering 57, 572 583 (1949). 2. G. P. Marcy, Pressure drop with change of phase in a capillary tube. R~J'r~eeratingEngineering 57(1), 53--57 (1949). 3. N. E. Hopkins, Rating the restrictor tube. Refrigerating Engineering 58(11), 1087 1095 (1950). 4. L. Cooper, C. K. Chu and W. R. Brisken, Simple selection method for capillaries derived from physical flow conditions. Refrigerating Engineering 65(7), 37-46 (1957). 5. A. Rezk and A. Awn, Investigation on flow of RI2 through capillary tubes. In Proceedings o1"the 15th International Congress of Re/?~geration, Commission, B2, Vol. 2, pp. 443452 (1979). 6. H. A. Whitesel, Capillary two-phase flow, part II. Refrigerating Engineering 65(9), 35 47 (1957). 7. ASHRAE, 1988 ASHRAE Handbook Equipment, chap. 19. American Society of Heating, Refrigerating, and AirConditioning Engineers, Inc., Atlanta (1988). 8. R. A. Erth, Two-phase flow in refrigeration capillary tubes: analysis and prediction. Ph.D. thesis, Purdue University, West Lafayette, USA (1969). 9. K. Maczek and Z. Krolicki, Non-adiabatic process in throttling capillary tubes used in packaged units. In Proceedings of the International Institute t~f Refrigeration, Commission B2, Paris, pp. 134-142 (1981 ). 10. K. Maczek, Z. Krolicki and E. Sochanecka, Model of throttling capillary tube with metastable process. In Proceedings of the XVth International Congress of Refrigeration, Commission B2, Paris, 154-161 (1983). 11. S. Lin, C. C. K. Kwok, R. Y. Li, Z. H. Chen and Z. Y. Chen, Local frictional pressure drop during vaporisation of R 12 through capillary tubes. International Journal o["Multiphase Flaw 17(1), 95 102 (1991). 12. R. Y. Li, S. Lin, Z. Y. Chen and Z. H. Chen, Metastable flow of R12 through capillary tubes. International Journal ~?/Re/'rigeration 13, 181 186 (1990). 13. C. Melo, R. T. S. Ferreira and R. H. Pereira, Modelling adiabatic capillary tubes: a critical analysis. In Proceedings t?['the HR-Purdue Con/~,rence, West Lafayette, USA, pp. 113 123 (1992). 14. C. Melo, R. T. S. Ferreira, C. Boabaid Neto, J. M. Goncalves, R. H. Pereira and M. R. Thiessen, Evaluation of HC-600a, HFC-134a and CFC-12 mass flow rates through capillary tubes. In Proceedings of the HR Coq~,rence, Hannover, Germany, pp. 621 630 (1994). 15. C. Melo, R. T. S. Ferreira, C. B. Neto and J. M. Goncalves, Experimentation and analysis of refrigerant flow through adiabatic capillary tubes. In Proceedings ASME Int. Mech. Engrs Congress and Exposition, AES, Vol. 34, pp. 19-30 (1995). 16. T. N. Wong and K. T. Ooi, Adiabatic capillary tube expansion devices: a comparison of the homogeneous flow and the separated flow models. Applied Thermal Engineering 16(7), 625-634 (1996). 17. P. K. Bansal and A. S. Rupasinghe, An empirical correlation for sizing capillary tubes. International Journal of Re#~geration 19(8), 497 505 (1996). 18. M. O. McLinden, J. G. Gallagher, M. L. Huber and G. Morrison, REFPROP refrigerant properties database: capabilities, limitations, and future directions. ASHRAE NIST Refi'igerants Conference, Gaithersburg, Maryland, pp. 59 71 (1993). 19. A. S. Rupasinghe, Modelling of capillary tubes for household refrigerators and freezers. M.E. Thesis, University of Auckland, Auckland, New Zealand (1996). 20. F. M. White, Viscous Fluid Flow. McGraw-Hill, New York (1974). 21. S. W. Churchill, Frictional equation spans all fluid flow regimes. Chemical Engineering 84, 91-92 (1977). 22. T. N. Wong, K. T. Ooi and C. T. Khoo, A study on capillary tube flow. In Proceedings ~?f the llR-Purdue Conl~,rence, West Lafayette, USA, pp. 371-376 (1992). 23. H. Wijaya, Adiabatic capillary tube test data for HFC-134a. In International Refrigeration Conference on Energy E[h'ciemT and New Re/?igerants, Purdue University, West Lafayette, USA, Vol. 1, pp. 63-71 (1992). 24. P. K. Bansal and C. K. Rice, Thermodynamic model for R-12 alternatives in domestic refrigerators--initial analysis. Proceedings of the HR Conference on Cold Chain Refrigeration Equipment by Design held at Massey University, New Zealand, pp. 95 104 (1993).