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Heat and mass transfer in vertical tubular bubble absorbers for ammonia-water absorption refrigeration systems
Heat and mass transfer in vertical tubular bubble absorbers for a m m o n i a - w a t e r absorption refrigeration systems C. A. Infante Ferreira, C. Keizer and C. H. M. Machielsen Keywords: refrigeration, absorbers, ammonia, water
Transfert de chaleur et de masse dans les absorbeurs tubulaires verticaux de bulles pour les
syst mes frigorifiques 5 absorption ammoniac--eau On a mis au point un moda/e en vue du calcul des transferts simu/tan#s de chaleur et de masse dans des absorbeurs tubulaires verticaux de bulles ~ utiliser dans les r#frig#rateurs a absorption ammoniac-eau. La forme de I'absorbeur est pr#sent#e Fig. 1 et /e schema de principe
Fig. 2.
On a r#afis# des experiences pr#/iminaires en utilisant un absorbeur adiabatique, mais la plupart des experiences utilisaient une circulation annu/aire de m#thanol refroidi pour #liminer /a cha/eur de I'absorbeur. L "#coulement de gaz dans I'absorbeur passait de I~coulement mousseux a /~cou/ement diphasique interm#diaire et fina/ement a I~coulement bulles. Les tubes de I'absorbeur essay#s avaient un a/#sage de 10.0, 15,3 et 20,5mm et environ I m de long. La chute de pression mesur#e par unit# de Iongueur est compar#e par de diverses ~quadons de pr#vision au tableau 1 et la fraction de vide est compar#e tableau 2. Le coefficient global de transfert de chaleur est donn# par/~quation (12) et /e coefficient pelliculaire int#rieur par I~quation (16). Le coefficient de transfert de masse est corr#/# par I'#quation (20) I'effet de/a ha uteur du tube ~tan t d#fini, pour le domaine exp#rimental donn& par I'#quation (26).
A model is developed for calculation of simultaneous heat and mass transfer processes in vertical tubular bubble absorbers as used for ammonia-water absorption refrigeration systems. Some preliminary experiments have been performed in an absorber without heat removal, The results from these experiments are compared with the literature and give a first indication about the methods for prediction of the absorption process. Experiments have also
been performed with simultaneous heat removal. The internal diameters of the absorbers tested were 10.0, 15.3, and 20.5 mm. The mass transfer coefficients resulting from these experiments are correlated by a modified Sherwood relation. An iterative procedure is presented which allows design of vertical tubular bubble absorbers for ammonia-water absorption refrigeration systems.
The rapid inflation in energy costs in the 1 970s has resulted in the resurgence of interest in absorption machines for refrigeration and heating purposes. In thing to optimize these machines, it was recognized that the absorber has a large influence on the performance of the system. This Paper describes the modelling of combined heat and mass transfer in the absorber. The purpose of the analysis is to relate, quantitatively, the heat and mass transfer coefficients to the physical properties of the working fluids and to the geometry of the system. The preferred configuration is that of a vertical tubular absorber. This geometry wilt lead to high heat and mass transfer rates 3 and probably to absorbers which are more compact and
easier to manufacture than the conventionally applied film absorbers. Fig. 1 presents schematically the absorber configuration under study. This absorber type is characterized by a changing two-phase flow pattern: froth flow directly after the inlet nozzle followed by a slug flow region and, finally, a bubbly flow region. The absorber height is restricted to ,~ 1 m to reduce pressure drop through it. About 20% of the absorption height is froth flow and 65% slug flow. The absorption process is accompanied by large heat effects and must be calculated by an iterative procedure to allow for the wide variation of temperature experienced by the absorbent (water) as it takes in refrigerant (ammonia) and rejects heat to the cooling medium at successive levels in the absorber. When the relations for heat and mass transfer are available, one can simulate these simultaneous processes by a step-by-step procedure, implemented
CAlF and CHMM are from the Laboratoryfor Refrigerationand Indoor Climate Technology. Delft University of Technology, The Netherlands. CK is from Ingenieursbureau Het Noorden, Leeuwarden, The Netherlands.Paperreceived18 July 1983; revised 21 November 1983.
348
0140-7007/84/060348-1053.00 © 1984 Butterworth 8- Co (Publishers) Ltd and IIR
International Journal of Refrigeration
Nomenclature A As c
heat transfer area, m 2 slug surface area, m 2 molal concentration of solute in liquid phase. kmol m -3 D internal diameter of absorber tube, m D o outside diameter of absorber tube, m ID diffusivity, m 2 s -1 dO rate of heat transferred in differential element, W g gravity constant, m s-2 h specific enthalpy. J kg -1 bEG heat of solution plus vaporization, kJ kg -1 k overall heat transfer coefficient, W m -2 K-~ KL overall mass transfer coefficient, m s-1 KLa mean overall volumetric mass transfer coefficient. m s -1
rh P
Pr Re Sc Sh T AT u z
mass flow rate, kg s-1 operating pressure, kPa pressure drop in absorber, Pa Prandtl number rate of heat transferred in absorber, kW Reynolds number Schmidt number Sherwood number temperature, °C temperature driving force, °C velocity, m s-1 absorption height, m
heat transfer coefficient, W film thickness, m
m -2
K-1
on a digital computer. To check the performance calculation, a test-rig has been built. In this Paper a description will be given of the experimental set-up and measuring techniques, the model equations, the experimental results, the validation, and the resulting rules for design.
Experimental techniques
p
voidage fraction Jepsen's energy dissipation parameter, atm s-1 Jagota's energy dissipation parameter, atm s-1 viscosity, Pa s thermal conductivity. W m -1 K-1 weight concentration of solute density, kg m -3
Subscripts B C G Gs i L Lf Lp Ls L+G In vL
bubble coolant gas phase gas phase, superficial inside tube liquid phase liquid film liquid plug liquid phase, superficial based on total superficial velocity logarithm based on volumetric mass transfer coefficient
Superscripts *
equilibrium
Dimensionless numbers
Greek symbols 5
sj £JG t/
set-up
and
measuring
The experimental equipment is shown in Fig. 2. The main component of the rig is the vertical tubular absorber. In most experiments the absorber consisted of the innermost of three concentric tubes. Preliminary experiments, with adiabatic conditions, were performed using a single glass tube. The ammonia-water solution is supplied from a 185dm 3 vessel, in which equilibrium conditions are maintained by mixing the solution with its vapour. The concentration of the solution can be varied by adding water or ammonia. From the vessel the solution is pumped to the absorber. When desired, the solution can be subcooled before entering the absorber. The absorber tube is cooled by methanol, flowing in a concentric annulus surrounding the absorber tube, at temperatures down to - 4 0 ° C in counterflow to the ammonia-water solution. This low temperature of the coolant guarantees a heat flux through the glass wall
Volume 7 Number 6 November 1984
ReG= 4rhG/(~Dr/G) SCL=tlL/ (pLIDL) ShvL=KLaD2/ (pLIDL)
approximately equal to the heat flux through a steel wall, which would be used under practical conditions. A third tube surrounding both tubes is applied in which methanol is flowing too, but at a temperature being equal to the ambient temperature. The reasons are twofold: (1) to prevent frost formation on the middle tube and therefore to be able to visually observe the absorption process: and (2) to be able to m'ake heat balances around the ammonia-water flow and the low temperature methanol flow as well. The construction of the rig offers the possibility of co-current upward and downward flow and of counter-current flow, but the ranges of the mass flows of gas and liquid are too limited for practical application. For the same reason, co-current downward flow has not been used. In both cases unstable operation can quite easily occur. The experiments were therefore restricted to co-current upward flow. The different mass flows, i.e. those of the gaseous ammonia, of the solution and of the methanol flows in the middle and outer glass tubes are measured by calibrated flowrators. The temperatures of these flows and that of the solution in the 185dm 3 vessel are registered by use of copper-constantan thermocouples. The pressure of the gaseous ammonia (after expansion) and that existing in the 185 dm 3 vessel are measured by manometers. The pressure drop across the absorber is measured by a pneumatic differential
348
pressure transmitter (Honeywell NDP 22). The concentrations of ammonia in the solution before and after the absorber are determined by first loading a sample of the solution in 1.2 N hydrochloric acid. By titration or back-titration, depending on the pH-value, the concentration of ammonia is obtained.
strong solution out cooling medium in
'r
O
Preliminary investigation at adiabatic conditions
O
Preliminary experiments on hydrodynamics and mass transfer have been performed at adiabatic conditions with a glass tube of l m length and 0.01 65 m inner diameter without complete absorption of the gas phase in the liquid phase. The rate of absorption is obtained from photographic observations. These photographs show that two flow regimes should be distinguished. In the entrance region froth flow or developing slug flow exists, changing into developed slug flow. For these experiments the bubbly flow region was not attained. The following quantities were measured: rise velocity of slugs, pressure drop, and void fraction. The film thickness and the overall mass transfer coefficient were calculated from these measurements. Rise velocities of slugs, u a, have been determined at different heights, z, along the absorber tube under adiabatic conditions. From these data, relations have been derived as proposed by Nicklinl~:
,-%
q cooling ~ - - medium
For z=0.2 m;
Out
-.-
if"
-
g o s ,n Fig. 1
(1)
uB=c, (UGh+ UL~) + c2(gD ) 'j2
u8 = 1.2 (UGh+UL,) + 0.36 (gD) 1/2
weak solut ion in
(2a)
was obtained and for z=0.5 m: u8 = 1.4 (UG,+ UL,) + 0.29(gD) ~/2.
(2b)
The coefficients obtained by Nicklin do not depend on height because they are valid for stable slug flow. A comparison with Nicklin's results (c~=1.2 and
Vertical tubular bubble absorber
Fig. 1 Absorbour rubuloiro vorticol do bulles
P
S
t'pTRANSHITTER
" P ' - - - ~v
-- ---'l
~
1 /-t I---~
LOOP
LOOP
,~
i'
!
I i I I
t I I I
t
t
5PRO'/ NOZT.LE5 L ~,UF~'ER NH3 - H20 7 SOLUTION 4@5
COOLIN r, ME DtUPI
NH~
-
RE$1~1ANCE HEA~rlNO
Cx~
~@£
HEA'f [XCHANr-E~
¢,UPPLY
Fig. 2
HEAT E~CHANr~ER
Experimental
set-up
Fig. 2 Installation expdrirnentale
350
Revue Internationale du Froid
c2=0.35) shows that the values of c~ and c2 roughly agree. As already stated, this absorber type is characterized by changing flow patterns. Thus, it can be expected that the classical two-phase pressure drop correlation methods will only apply for each of the different patterns. Such a calculation method would
Table 1. Comparison of experimental pressure drop per unit length absorber tube with some correlating methods Tableau 1. Comparaison des mesures de/a chute de press/on exp#rimenta/e par unit# de/ongueur du tube absorbeur suivant d/verses m#thodes de corr#/ation Relative deviation Experimental AP/&z Loc kh art(Pa m -1) Mart/nell/
Hug hmarkD a v i s Pressburg Nicklin
4830 4660 5270 5040 4830 5370 5010 5010 5040 5010 5550 5550 4920 4830 5190 5270 4830 5010
-0.44 -0.43 -0.51 - 0.49 - 0.45 -0.57 -0.48 -0.47 -0.44 -0.46 -0.52 -0.52 -0.46 - 0.45 -0.47 -0.48 -0.43 -0.45
-0.82 -0.81 -0.85 - 0.84 - 0.83 -0.85 -0.84 -0.84 -0,82 -0.82 -0.86 -0.85 -0.84 -0.83 -0.83 -0.83 -0.82 -0.82
-0.45 -0.43 -0.60 - 0.51 - 0.49 -0.61 -0.58 -0.59 -0.47 -0.48 -0.59 -0.56 -0.50 - 0.49 -0.43 -0.44 -0.49 -0.51
-0.32 -0.30 -0.51 - 0.42 - 0.39 -0.51 -0.47 -0.46 -0.34 -0.34 -0.50 -0.48 -0.40 - 0.39 -0.30 -0.31 -0.41 -0.42
Mean
-0.47
-0.83
-0.51
-0.40
require the knowledge of the mass flow rate of both phases at different absorber heights. Actually, this cannot be predicted easily, which makes pressure drop calculations difficult. An attempt was made to predict the pressure drop in the system using the methods proposed by Nicklin 12. Lockhart and Mart/nell/1°, Davis ~5, and Hughmark and Pressburg 7 at the absorber inlet conditions. Table 1 presents a comparison of experimental pressure drop with the predictions from these methods, for 1 8 experiments. Also the mean value of the relative deviation of these predictions is shown. During the calculations it appeared that the wall friction term is negligible when compared with the hydrostatic pressure drop. Since the gas flow rate varies drastically with the absorber height due to absorption, it can be expected that the voidage fraction predicted for inlet conditions will be too high so that too low pressure drops are predicted. Thus, it can also be expected that the methods allowing the best prediction of the voidage fraction will also give the best results for the pressure drop. In Table 2 the experimental voidage fraction is compared with the correlating methods considered. The best results are obtained with N icklin's method, followed by Lockhart-Martinellrs method. The same trends are shown in Table 1 for the pressure drop comparison. The film thickness has been calculated with the relations of Chesters 4, Hubbard 5, Beek 1, van Paassen ~3, and Rosehart and Jagota TM. All relations give values ranging from 0,7-0.8 mm for the film thickness. Aweak increase in film thickness with decreasing gas flow along the absorber may be concluded (0.05-0.1 mm), except for that calculated by Hubbard's relation which is a rather simple one, giving a constant thickness. Overall mass transfer coefficients, K L, have also been evaluated. Evaluation of these coefficients requires rather elaborate methods: the surface area of
Table 2. Comparison of experimental voidage fraction with some correlating methods Ta/beau 2. Comparaison des mesures de/a fraction de vide suivant d/verses m~thodes de corr~/adon Relative deviation Experimental, s (1) 0.45 0.47 0.41 0.43 0.46 0.39 0.43 0.43 0.42 0.43 0.37 0.37 0.44 0.45 0.40 0.39 0.45 0.43
Experimental, s (2)
0.40 0.29
Mean
Lockhart- Mart/nell/
Davis
Hughmark-Pressburg
Nicklin
0.53 0.49 0.73 0.65 0.52 0.90 0.65 0.63 0.62 0.60 0.89 0.89 0.59 0.56 0.70 0.74 0.53 0.60
1,00 0,91 1.22 1.12 0.98 1.33 1.12 1.12 1.14 1.09 1.46 1.46 1.07 1.02 1.25 1.31 1.00 1.09
0.80 0.72 1.12 0.98 0.87 1.21 1,.00 1.00 0.93 0.91 1.30 1.30 0.93 0.89 1.00 1.05 0.89 0.98
0.40 0.34 0.80 0.58 0.48 0.87 0.70 0.67 0.50 0.47 0.89 0.86 0.55 0.51 0.45 0.49 0.53 0.60
0.66
1.15
0.99
0.59
(1) Determined from the experimental pressuredrop assuming the wall friction term is negligible when compared with the weight of gas and liquid in tube. (2) Determined from photographic data
Volume 7 Num6ro 6 Novembre 1984
351
the slugs, A,, which is the area available for mass transfer, has to be determined from photographs. The concentration driving potential has been approached by a logarithmic mean. This admittedly approximate approach, forced by the large heat effects direct after the gas inlet nozzle, gives an order of magnitude for the mass transfer coefficient, K L, which was found to be ~0.0002 m s -~ . From this preliminary analysis of the absorption process in a vertical tubular absorber of small diameter. it can be concluded that the process is rather complicated. It was decided to describe the behaviour of the system using a simplified step-by-step method which simulates the situation sufficiently well for practical purposes.
mL4
The main investigation on vertical tubular absorbers concerns those with simultaneous heat and mass transfer. The aim of this research is to obtain overall heat and mass transfer coefficients and their dependence on the different parameters, which enable the design of bubble absorbers. To process the experimental data, a model was developed which describes the processes occuring in a finite element. It is assumed that heat and mass transfer coefficients will be constant along the absorber tube. Starting from the known inlet conditions and assuming arbitrary values for the heat and mass transfer coefficients, the processes can be calculated successively along the length of the absorber tube. This will finally yield the outlet conditions. Iterative calculations with a new set of values for the heat and mass transfer coefficients will be performed until the calculated outlet conditions will be equal to the experimentally obtained outlet conditions. Thus, only overall conditions are considered.
TLJI
hL1
f
41 C
mG* d'r~G
TG÷ dT G
TL* d]" L r~L÷ dr~L
IG÷ a~ G
{L~ d~ L
hGidhG
hLid h__L
T c + dT C
- -i-?
--I____ I - tq
I NNH3 d-rnL ---"" I
Investigation involving simultaneous heat and mass transfer
~'Li
--
--[
--
_
_
L,.. d.r~qL
INH2OI _
_i
_---_
I i
_
TG
TL
r~G
r~ L
~G hG
~L hL
rhso
r~LO
! rc
GO
rh c
~LO
Fig. 3 Schematic model of differential element Fig. 3
Sch@made/'@/@mentdiff@rentiel
The overall mass balance for the ammonia yields the concentration of ammonia at the top of the absorber: ~L1 = (rhGo~GO4- rhLo~LO)/rhL1
(a) mass balance:
(3)
From the overall mass balance, the mass flow of liquid at the top may be obtained: rhL1 ----rhLo.4- rhGo
3,52
(6)
(b) mass balance for ammonia component:
For any differential element dz along the tube length, heat and mass balances can be formulated. In Fig. 3 a schematic model is presented for an element dz at an arbitrary heightz. The gaseous ammonia and the ammonia-water solution are in co-current upward flow. The cooling medium is in counter-current flow. It is assumed that the gas and liquid flows are homogeneous, i.e. the concentrations of ammonia are uniform in each phase, further the temperatures in both phases are uniform and equal to each other. When entering an element, the next variables are known from the foregoing step: thE, rh e, rh o TL, To ~L, 4]6, P. From these data, the enthalpies of liquid and gas phases hL and h G c a n be obtained. The boundary conditions at the inlet and at the outlet of the absorber are indicated by the indices 0 and 1, respectively. At the top of the absorber all gas should have been absorbed: rhG1--0
(5 )
The laws for conservation of mass and heat applied on a differential element dz give the following relations:
drh G= - drh L
Model description
t
Z
(4)
rhG~G ~- rhL~L = ([hG 4- drhG) (~G + d~G) +
(rhL+ drnL)(~L+ d~L)
(7)
(C) absorbed mass of gas: /'~ 2 drhG= --KLa (~L--4]L)~D dz
(8)
(d) enthalpy balance: rhGhG + rhLhL = (rhG + drh G) (hG + dhG) + (rhL 4- drhL)(hL4- dhL) 4- dO
(9)
When the subroutines derived from analytical relations for ammonia-water solutions, as given by Keizer and Liem 9, are used, the enthalpies at the inlet and outlet sections of the absorber can be directly calculated. (e) removed heat:
dE)=kA (T L-
Tc)
(1 O)
The calculated values determining the conditions of
International Journal of Refrigeration
the gas, the liquid and the coolant leaving the element can be taken as the values determining these flows at the entrance of the next element. This can be done successively. The calculations may be stopped when all gas has been absorbed, i.e. rhG=O. The number of elements, n, needed to absorb all gas will give the height of the absorber, z, as:
z = n dz
1
In general Experimental programme: (1)
(11 )
The calculation method presented can be applied to calculate: (1) the absorber height; and (2) the amount of heat to be transferred to the coolant, i.e. the temperature of solution and coolant, when the overall heat and mass transfer coefficients are known from experimental data. The aim of the experimental research on tubular absorbers presented here was to obtain these data. The overall resistance to heat transfer, 1/k, can be considered as the summation of partial resistance:
1
phase mass flow rate through the absorber and thus of the void fraction.
DID o
D1
k - ~ 4-~-~n~+~O~c
(12)
where cq represents the mean heat transfer coefficient on the solution side. As stated above, at a difference element both the gas and liquid phases are assumed to be at the same mean temperature. The coefficient =, applies then to the heat transfer from the wall temperature to the mean solution temperature. Actually, as can be seen from Fig. 1, in the film flow region there will be a transfer of heat from the interface to the bulk of the film, conduction through the film and, finally, convection heat transfer from the liquid film to the absorber wall. The coefficient cq has been calculated as proposed below. The validity of this assumption has been evaluated by calculating the overall heat transfer coefficient, k, from the experimental results:
~2=kA&T,n
(1 3)
with O -----rhLoh LO+ rhGohG0 -- rhLlh L1
( 1 4)
The logarithmic mean temperature difference can only poorly represent the actual temperature difference profile along the absorber but no better alternative is suggested since the temperature profile cannot easily be measured. The overall volumetric mass transfer coefficient, Kta, has been defined by Equation (8). The parameters determining the conditions of the solution vary with the height z of the tube: concentrations ~L and ~*, temperature TL and pressure vary with z. For a differential element of height dz these parameters are assumed constant. By a trial and error method, a value for KLa can be found for which the calculated absorption height equals the experimentally obtained one. Actually, the assumption of a constant KLa is not in agreement with the experiments. The volumetric mass transfer coefficient will vary from very large values directly after the inlet nozzle to very small values in the bubbly flow region. However, as a first approach this assumption will be maintained. The assumption of a constant K,a will lead to poor prediction of the gas
Volume 7 Number 6 November 1984
(2) (3)
(4) (5) (6) (7)
variation of the tube geometry, i.e. the diameterD of the tube and the absorption height z; variation of the mass flows of gas and liquid, i.e. of the flow regimes which can be encountered; variation of the heat flux through the tube wall, i.e. of the amount of heat which is removed from the solution; variation of the concentration of ammonia in the solution; variation of the temperature of the solution, i.e. of the driving force for heat and mass transfer; variation of the gas temperature; variation of the absorber pressure.
Tubes with diameters of 1 0.0, 1 5.3, and 20.5 mm have been investigated. The absorption height is limited to 1 m. Table 3 presents the minimum and maximum values of the most relevant parameters. Neither the nozzle geometry nor the working fluids have been varied in this work. A change of the working fluids, i.e. mainly in the viscosity of the solution, would not only be of interest with respect to the application in absorption refrigeration machines but also for the
Table 3. M i n i m u m and m a x i m u m values of parameters Tableau 3. Va/eurs minima/es et maxima/es des param#tres Parameter
Minimum
Maximum
rh G (g s -1) rh L (g S-1) ~Lin (kg kg -1)
0.018 2.1 0.341 0.342 0.001 0.005 8.2 9.5 1.26 0.35 0.02 1.05 0.0007 230 92 447 1890 7 1 410 0 0.03 0.162 0.5 2.8 0.2 4.5 500 1 360 6.8
0.480 26.2 0.451 0.510 0.103 0.170 32.0 33.5 3.33 4.31 1.04 5.39 0.0017 4600 2535 5113 17910 14 1 1090 1.1 0.58 0.706 12.6 1 5.6 1.1 263.7 15280 19400 90.0
~Loot (kg kg -1) (~Lout--~Lin) (kg kg -1) (¢~n--(~Lin) (kg kg -1) TG (°C) TL ('C) P (bar) UGs (m s-1) ULs (m s-1) u U (m s-1) (m) ReGs ReLs Reu ReL+G PrL Sc G Sc L ~), (kW) k (kW m -2 K-t) ¢c (kW m -2 K -1) ~i (kW m -2 K-1) ~L, (kW m -2 K-1) ~j~_(kW m -2 K-1) KLa (kg m -3 s-1) ShvL Re (Equation (19)) z/D
353
verification of the universal value of the results. Only one nozzle geometry (annular-type inlet) has been studied. The gas phase was introduced in the centre of the absorber using a tube with 4 mm internal diameter. The inlet mixer design will affect the resulting liquid-gas flow pattern but, for the type of absorbers studied in which the flow pattern rapidly varies from churn to slug and, finally, to bubbly flow, it can be expected that this effect will be reduced. The minimum and maximum values of KLa and of the mean driving potential for mass transfer are given in Table 4. The mean driving potential for mass transfer is defined as: z
(15)
( ¢ t - eL) = Z l ( ~ - - ~L)dz
k~,, = (kca,c- ke,p) /ke×o
In total 1 74 experimental runs were performed: 63 with the 10mm tube, 53 with the 15 mm tube, and 59 with the 20mm tube.
Evaluation of t h e experimental results on heat transfer The partial heat transfer coefficient, a, is assumed to be determined by the following equation: ai=~aLf+ (1 -~)aLp
(1 6)
i.e. two regions are distinguished for which the heat transfer shows different behaviour. In the part of the absorber in which slug and froth flow prevails, a, is
Table 4. M i n i m u m and maximum values of the volumetric mass transfer coefficient and of the mean driving potential for mass transfer Tableau 4. Valeurs rain/males et max/males du coefficient de transfert de masse vo/um#trique et du potent/el d'entrainement moyen pour le transfert de masse Parameter
D (mm)
Minimum
Maximum
KLa (kg m-3 s-1 )
10 15 20 10 15 20
11.8 7.0 4.5 0.019 0.019 0.021
91.3 67.0 263.7 0.155 0.171 0.131
( 17 )
The relative deviation kdiff is given by kdiff + and kdiff_, respectively, for the cases in which cq is included or omitted in the calculation of k. From Table 5 can be concluded that a, should be involved in the calculation of the k value. This conclusion is supported by the increase of the standard error e, if a, is omitted. The resistance to heat transfer is essentially affected by the heat transfer coefficient at the solution side if: (1) the partial heat transfer coefficient, ac, assumes rather high values; or (2) the void fraction is relatively low. Then a, is to an increased extent determined by the heat transfer coefficient alp, corresponding with single phase flow. It can be concluded that the proposed model on heat transfer, involving the partial heat transfer coefficient, cq, enables a quite accurate prediction of the overall heat transfer coefficient, k.
0
(~--~L) (kg kg -~)
supposed to be identical to the heat transfer coefficient in a thin liquid film. The size of this part of the absorber corresponds with the averaged void fraction ~. In the other part of the absorber, in which liquid slugs prevail, the heat transfer is supposed to correspond with that for single phase flow in a tube. For the calculation of au the correlation proposed by Horn 8 for a cooled falling film has been used. In Horn's correlation, the characteristic length is the film thickness and has been calculated from Brauer 2. In Table 5 the minimum and maximum values for the calculated partial heat transfer coefficients alp, a u, and a~ are depicted. Also shown are the mean values and the standard errors of the relative deviations kdiff with:
Evaluation of the experimental results on mass transfer Dimensional analysis For a general comparison with other data, especially on mass transfer, the data should preferably be available in the same, commonly used, form. A first attempt was made to correlate the data using an energy dissipation method as proposed by Jepsen 8. The efforts have been unfruitful. Mass transfer, without simultaneous heat transfer, is often described according to the following relation, which can be derived from dimensional analysis: ShvL = aRe~Sc~
( 1 8)
Table 5. Calculated heat transfer coefficients and relative deviations between calculated and experimental values of the overall heat transfer coefficient Tableau 5. Coefficients de transfert de chaleur calculus et 6carts relatifs entre /es va/eurs exp~rimentales du coefficient global de transfert de chaleur C~Lp(W m -2 K-1)
=Lf (W m -2 K-~)
cq (W m -2 K-1)
D (mm)
Minimum
Maximum
Minimum
Maximum
Minimum
Maximum kdif~+ (%)
e+ (%) kdi,_ (%)
~
10 15 20
310 220 1 50
1120 590 410
2810 4310 6470
13880 12930 15630
530 470 920
10190 10020 12550
18.8 8.8 14.9
35.4 1 5.8 22.9
354
0.5 - 24.5 - 10.7
22.6 - 17.1 2.9
(%)
Revue Internationale du Froid
/ ShvLII.~
• -- O= 40.Omm
40000
• ~
• -- B - 4~.&mm
(1)
/
(2)
• 0
(3)
5DOQ
• • • V~• • •
•
k
~,&& 4000
•
• •
•
,,~
(4)
•
The following Equation (20):
• . . . . . . .
-
SO0~ V 500
assumez has a determined value (a first approach can be obtained from Equation (26)): use this value to calculate KLa from Equations (19) or (20): use the model described in the model description section to estimate the absorber height; in general, this absorber height will differ from that assumed. The correct solution occurs when a value of the absorber height, z, is reached which is equal to the value calculated from the model. This may be achieved by an iterative procedure in which successively better guesses for the absorber height are made.
4000
5000
~no00
5hvLclL~ Fig. 4 Experimentaland predicted Sherwood numbers, according to Equation (I 9)
Fig. 4 Nombres de Sherwood exp~rimenrauxet calculus, d'aprbs I'#quation 19
In this particular case of fast mass transfer, and consequently a changing flow pattern along the absorber tube, the mass transfer may be expected to depend on the height of absorption, z. As two-phase flow is expected to influence mass transfer, one may expect both the liquid phase and gas phase velocities to play a role. A multiple linear regression has been applied to determine the coefficients and the exponents appearing in Equation (1 8). A statistical analysis of the correlation results for Equation (1 8) and modified equations of its type led to the following conclusions: (1) gas phase velocity and absorption height are important in the description of mass transfer; and (2) the mass flow rate of liquid can be neglected as a parameter. The following modified Sherwood relation was obtained which correlates the 1 74 experiments with the absorber tubes of 1 0.0, 1 5.3, and 20.5 mm:
conclusions
/rh
-,0.853 1 353~t-~0.50
PL
/UL qO,53
1 zD,8~,
(20)
Equation (1 9) does not contain direct information about the rate of mass transfer since the absorber height, z, is unknown. The various stages to arrive at the absorber height are as follows:
Volume 7 Num~ro 6 Novembre 1984
from
_0.83/t-~0.50
KLa,,,4.6rhOi33 PL /uL
1
(21
zOSOO 133
(19)
The correlation coefficient was 0.86. The mean relative deviation of the calculated Sherwood number was + 0.050 for the 1 74 observations. Since the liquid mass flow rate appeared not to affect the mass transfer coefficient, the two-phase Reynolds number has been defined asRe= (pLUGsD)/tlL with Uos the superficial gas velocity at the absorber inlet. In Fig. 4 the experimental values of the Sherwood number and those predicted by the regression line, Equation (19), are shown as a function of the right hand side of Equation (19). Equation (1 9) can also be written as: KLa = 2.91~--~G~ "\PG)
be drawn
(1) Effect of gas phase volume f l o w rate. As can be seen from Fig. 1, the gas phase volume flow rate varies from inlet conditions, rh G, to zero at the top of the absorber. The local volume flow rate will be a function of the absorber height. From Equation (20), a short absorber will have a larger mean gas phase volume flow rate than a long absorber. The greatest part of the absorption process occurs in the slug flow regime. In this region, heat and mass transfer occur in the falling film which surrounds the side part of the gas bubbles, From a study by Griffith and Wallis 17, if the liquid film is laminar and the liquid phase volume flow rate is negligible when compared with the gas volume flow rate, the liquid mass flow rate in this film is approximately proportional to ( r h G / P G ) pLDIS/tlL. It is known from the penetration theory (Higbie ~8) that a large interfacial velocity leads to large mass transfer coefficients. If the gas flow rate is increased, the liquid flow rate in the film increases, the mean and interfacial velocities in the film increase and the mass transfer coefficient also increases, in agreement with Equation (20). (2) Effect of the physical properties. If Higbie's theory is applied to a laminar falling film the following relation is obtained:
17017
ShvL = exp (0.8 6 307 ) Re °853Sc °5°
can
where rhLf is the liquid mass flow rate in the film andzf is the length of the falling film. If the approximate equation for rhLf, as given by Griffith and Wallis 17, is substituted in Equation (21), we obtain: /"h
GI
Kta=0.283 7 6
/ 0.33
1 17.D0.50 ,OL / L
~
1
zO~ODOB 3
(22)
where rhG. is the local gas phase mass flow rate. Equation (22) allows a comparison with the experimental correlation (20). The effect of the liquid phase properties on the mass transfer coefficient agrees with the effect predicted from the penetration theory. (3) Effect of absorber height and tube diameter. Equation (20) shows that long absorber tubes present low volumetric mass transfer coefficients and also that KLa decreases with the tube diameter. In the type of absorbers in study three different flow regimes are encountered and the absorber height, z, and the
3515
interfacial area, a, do not correspond with the length, zf, and interfacial area of a falling film. Nevertheless, the same trends are found for the effect o f z and D on the mass transfer coefficient (compare Equations (20) and (22)). S t e p w i s e m u l t i p l e linear regression
The method proposed in the last section, for the calculation of the absorber height, requires an iterative procedure. The question arises if a description of the experimental data could be obtained that should lead to a direct calculation of the absorber height. Stepwise multiple linear regression can be applied to assess the relative influences of the different parameters on the dependent variable, i.e. on, for instance, the absorber height z. The latter has been taken as the dependent variable in the hypothesis which is arbitrarily assumed as: (23)
z=aXb'Xb2 . . . X b~
This is actually a more general approach to correlate the experimental data on z. However, if only indepent variables are taken as X., the results will not be applicable for systems other than ammonia-water. The independent parameters X., which eventually might affect z, are identified as follows: D,FhL,rhG,P, TG,~ L TL,TLoo, (or O).
A remark can be made with respect to the last parameter TLo,t (orO). This variable should be known at the absorber height z; actually, the liquid phase temperature is only measured at the outlet of the test section (1 m from the inlet). The heat transfer rate until the absorption height z, as obtained from the model presented above, O was used. A correlation coefficient r=0.93 and F-value F=156 were obtained. The dependence of z on the parameters which have a significant effect can be expressed as: 21
Z=133.6~L O
06
z")- 1 0 r ~ ~ 4.-~01 ,,,G F u
0 5.¢., - 0 04
'"u
(24)
The following trends can be differed from Equation (24): (1) an increase of the system pressure, of the absorber diameter and of the liquid phase mass flow rate lead to smaller absorbers; and (2) an increase of the liquid phase concentration, of the liquid phase temperature, of the gas phase mass flow rate, and of the transferred heat lead to larger absorbers. All these trends are in agreement with the conclusions that can be drawn from an enthalpy-concentration diagram for the system studied. The influence of the diameter can be drawn directly from Equation (8). It must be noted that a diameter increase does not necessarily imply a shorter absorber since (~T is also a function of the absorber diameter. As the liquid solution temperature, TL, decreases the absorber height, z, decreases with a limit of z = 0 for T~=0~C. Actually Equation (24) only applies for the conditions tested and, as shown in Table 3, only the range 9.5
358
the absorber height, <0.15m. This equation constitutes thus an alternative for Equation (1 9). It must also be remarked that Equation (24) requires an iterative calculation: the heat transfer through the absorber tube wall (3 must be calculated as presented in the model description section. The advantage of Equation (24) is that the heat transfer through the absorber tube C5 can be approached by C5=rhGhLG
(25)
where hLG is the heat of solution plus vaporization, so that, although approximately, z can be estimated directly.
Conclusions The proposed model on heat transfer enables a quite accurate prediction of the overall heat transfer coefficient. From Equation (20) or, in dimensionless form, Equation (1 9), a mean volumetric mass transfer coefficient can be estimated. The absorber height, the parameter in which we are especially interested for design of vertical tubular bubble absorbers, can be calculated by the iterative procedure presented in the dimensional analysis section. A first approach of the absorber height, for the system ammonia-water, as a function of the inlet conditions, can be obtained from: z= 1 ~ ~-~21r~O7F~O6"FO4D-10/-)-O5~-°04 (26) ~'V~L
"~G
"LG ~ L ~
~"
H~L
Acknowledgements The authors wish to express their thanks to Prof A. L. Stolk for guidance during the course of the investigation and Prof J . M . Smith for helpful discussions and suggestions during the preparation of the manuscript.
References 1 2 3 4 5 6 7 8 9
10
Beek,W. J. Stofoverdracht door beweeglijke grensvlakken, Thesis, Delft University of Technology (1962) Brauer, H. Grundlagen der Einphasen- und Mehrphasenstr6mungen, Verlag Sauerl~nder, Aarau (1971) Charl~ntier, J. C. What's new in absorption with chemical reaction? Transactions of the Institution of Chemical Engineers 60 (1982) 131-156 Cheltit$, A. K. Stromingen warmteoverdracht. Department of Mechanical Engineering, Delft University of Technology (1976) Gregory, G. A., Scott, D. S. Physical and chemical mass transfer in horizontalcocurrent gas-liquid slug flow, Chemical Engineering Journal 2 (1971) 287-295 Horn, R. K. Messungenzum W~rme- und StoffQbergang am Rieselfilm, Thesis. Univ. Karlsruhe (1970) Hughmark, e. A., Pressburg, B. S. Holdup and pressure drop with gas-liquid flow in a vertical pipe.AIChEJ 7 (1961 ) 677-682 Jel~en, J. C. Masstransfer in two-phase flow in horizontal pipelines. AIChE J 16 (1970) 705-710 Keizer, C., Liem, S. H. Absorption refrigeration machine driven by solar heat. Internal report K82. Laboratory for Refrigeration and Indoor Climate Technology, Delft University of Technoloy. (1979) Lockhart, R. W., Martinelli, R. C. Proposedcorrelation for
International Journal of Refrigeration
11 12 13
14
isothermal two-phase, two-component flow in pipes, Chem Eng Progr 46 (1949) 39-48 Nicklin, D. J. Two phase bubble flow, ChemicalEngineering Science 17 (1962) 693-702 Nicklin, D. J. The air-lift pump: theory and optimisation. Trans Inst Chem Engrs 41 (1963) 29-35 van Paa~sen, J. A. De absorptie van ammoniakgas vanuit opstijgende bellen in een ammoniakwater oplossing. Internal report S701. Laboratory for Refrigeration and Indoor Climate Technology, Delft University of Technology, (1979) Rosehsrt, R. G. D., Jagota, A. K. A method for the determination of individual film and droplet volumetric mass
Volume 7 Number 6 November 1984
15
16 17
transfer coefficients and interfacial areasof gas-liquid annular cocurrent flow. Cocurrent gas-liquid flow. (Eds Rhodes, E., Scott, D. S.). Plenum Press, New York (1969) 667 Scott, D. S. Properties of cocurrent gas-liquid flow, Advances in Chemical Engineering 4 (1963) 199-277. (Eds Drew, T. B, Hoopes. J. W., Vermeuler, T.) Academic Press, New York Higbie, R. The rate of absorption of a pure gas into a still liquid during short period of exposure, TransAm Inst Chem Engng 31 (1935) 365-389 Griffith, P., Wallis, C. B. Two-phase slug flow, Journal of Heat Transfer Trans. of the ASME (1961 ) 307-320
357
Transfert de chaleur et de masse dans les absorbeurs tubulaires verticaux de bulles pour les
syst mes frigorifiques 5 absorption ammoniac--eau On a mis au point un moda/e en vue du calcul des transferts simu/tan#s de chaleur et de masse dans des absorbeurs tubulaires verticaux de bulles ~ utiliser dans les r#frig#rateurs a absorption ammoniac-eau. La forme de I'absorbeur est pr#sent#e Fig. 1 et /e schema de principe
Fig. 2.
On a r#afis# des experiences pr#/iminaires en utilisant un absorbeur adiabatique, mais la plupart des experiences utilisaient une circulation annu/aire de m#thanol refroidi pour #liminer /a cha/eur de I'absorbeur. L "#coulement de gaz dans I'absorbeur passait de I~coulement mousseux a /~cou/ement diphasique interm#diaire et fina/ement a I~coulement bulles. Les tubes de I'absorbeur essay#s avaient un a/#sage de 10.0, 15,3 et 20,5mm et environ I m de long. La chute de pression mesur#e par unit# de Iongueur est compar#e par de diverses ~quadons de pr#vision au tableau 1 et la fraction de vide est compar#e tableau 2. Le coefficient global de transfert de chaleur est donn# par/~quation (12) et /e coefficient pelliculaire int#rieur par I~quation (16). Le coefficient de transfert de masse est corr#/# par I'#quation (20) I'effet de/a ha uteur du tube ~tan t d#fini, pour le domaine exp#rimental donn& par I'#quation (26).
A model is developed for calculation of simultaneous heat and mass transfer processes in vertical tubular bubble absorbers as used for ammonia-water absorption refrigeration systems. Some preliminary experiments have been performed in an absorber without heat removal, The results from these experiments are compared with the literature and give a first indication about the methods for prediction of the absorption process. Experiments have also
been performed with simultaneous heat removal. The internal diameters of the absorbers tested were 10.0, 15.3, and 20.5 mm. The mass transfer coefficients resulting from these experiments are correlated by a modified Sherwood relation. An iterative procedure is presented which allows design of vertical tubular bubble absorbers for ammonia-water absorption refrigeration systems.
The rapid inflation in energy costs in the 1 970s has resulted in the resurgence of interest in absorption machines for refrigeration and heating purposes. In thing to optimize these machines, it was recognized that the absorber has a large influence on the performance of the system. This Paper describes the modelling of combined heat and mass transfer in the absorber. The purpose of the analysis is to relate, quantitatively, the heat and mass transfer coefficients to the physical properties of the working fluids and to the geometry of the system. The preferred configuration is that of a vertical tubular absorber. This geometry wilt lead to high heat and mass transfer rates 3 and probably to absorbers which are more compact and
easier to manufacture than the conventionally applied film absorbers. Fig. 1 presents schematically the absorber configuration under study. This absorber type is characterized by a changing two-phase flow pattern: froth flow directly after the inlet nozzle followed by a slug flow region and, finally, a bubbly flow region. The absorber height is restricted to ,~ 1 m to reduce pressure drop through it. About 20% of the absorption height is froth flow and 65% slug flow. The absorption process is accompanied by large heat effects and must be calculated by an iterative procedure to allow for the wide variation of temperature experienced by the absorbent (water) as it takes in refrigerant (ammonia) and rejects heat to the cooling medium at successive levels in the absorber. When the relations for heat and mass transfer are available, one can simulate these simultaneous processes by a step-by-step procedure, implemented
CAlF and CHMM are from the Laboratoryfor Refrigerationand Indoor Climate Technology. Delft University of Technology, The Netherlands. CK is from Ingenieursbureau Het Noorden, Leeuwarden, The Netherlands.Paperreceived18 July 1983; revised 21 November 1983.
348
0140-7007/84/060348-1053.00 © 1984 Butterworth 8- Co (Publishers) Ltd and IIR
International Journal of Refrigeration
Nomenclature A As c
heat transfer area, m 2 slug surface area, m 2 molal concentration of solute in liquid phase. kmol m -3 D internal diameter of absorber tube, m D o outside diameter of absorber tube, m ID diffusivity, m 2 s -1 dO rate of heat transferred in differential element, W g gravity constant, m s-2 h specific enthalpy. J kg -1 bEG heat of solution plus vaporization, kJ kg -1 k overall heat transfer coefficient, W m -2 K-~ KL overall mass transfer coefficient, m s-1 KLa mean overall volumetric mass transfer coefficient. m s -1
rh P
Pr Re Sc Sh T AT u z
mass flow rate, kg s-1 operating pressure, kPa pressure drop in absorber, Pa Prandtl number rate of heat transferred in absorber, kW Reynolds number Schmidt number Sherwood number temperature, °C temperature driving force, °C velocity, m s-1 absorption height, m
heat transfer coefficient, W film thickness, m
m -2
K-1
on a digital computer. To check the performance calculation, a test-rig has been built. In this Paper a description will be given of the experimental set-up and measuring techniques, the model equations, the experimental results, the validation, and the resulting rules for design.
Experimental techniques
p
voidage fraction Jepsen's energy dissipation parameter, atm s-1 Jagota's energy dissipation parameter, atm s-1 viscosity, Pa s thermal conductivity. W m -1 K-1 weight concentration of solute density, kg m -3
Subscripts B C G Gs i L Lf Lp Ls L+G In vL
bubble coolant gas phase gas phase, superficial inside tube liquid phase liquid film liquid plug liquid phase, superficial based on total superficial velocity logarithm based on volumetric mass transfer coefficient
Superscripts *
equilibrium
Dimensionless numbers
Greek symbols 5
sj £JG t/
set-up
and
measuring
The experimental equipment is shown in Fig. 2. The main component of the rig is the vertical tubular absorber. In most experiments the absorber consisted of the innermost of three concentric tubes. Preliminary experiments, with adiabatic conditions, were performed using a single glass tube. The ammonia-water solution is supplied from a 185dm 3 vessel, in which equilibrium conditions are maintained by mixing the solution with its vapour. The concentration of the solution can be varied by adding water or ammonia. From the vessel the solution is pumped to the absorber. When desired, the solution can be subcooled before entering the absorber. The absorber tube is cooled by methanol, flowing in a concentric annulus surrounding the absorber tube, at temperatures down to - 4 0 ° C in counterflow to the ammonia-water solution. This low temperature of the coolant guarantees a heat flux through the glass wall
Volume 7 Number 6 November 1984
ReG= 4rhG/(~Dr/G) SCL=tlL/ (pLIDL) ShvL=KLaD2/ (pLIDL)
approximately equal to the heat flux through a steel wall, which would be used under practical conditions. A third tube surrounding both tubes is applied in which methanol is flowing too, but at a temperature being equal to the ambient temperature. The reasons are twofold: (1) to prevent frost formation on the middle tube and therefore to be able to visually observe the absorption process: and (2) to be able to m'ake heat balances around the ammonia-water flow and the low temperature methanol flow as well. The construction of the rig offers the possibility of co-current upward and downward flow and of counter-current flow, but the ranges of the mass flows of gas and liquid are too limited for practical application. For the same reason, co-current downward flow has not been used. In both cases unstable operation can quite easily occur. The experiments were therefore restricted to co-current upward flow. The different mass flows, i.e. those of the gaseous ammonia, of the solution and of the methanol flows in the middle and outer glass tubes are measured by calibrated flowrators. The temperatures of these flows and that of the solution in the 185dm 3 vessel are registered by use of copper-constantan thermocouples. The pressure of the gaseous ammonia (after expansion) and that existing in the 185 dm 3 vessel are measured by manometers. The pressure drop across the absorber is measured by a pneumatic differential
348
pressure transmitter (Honeywell NDP 22). The concentrations of ammonia in the solution before and after the absorber are determined by first loading a sample of the solution in 1.2 N hydrochloric acid. By titration or back-titration, depending on the pH-value, the concentration of ammonia is obtained.
strong solution out cooling medium in
'r
O
Preliminary investigation at adiabatic conditions
O
Preliminary experiments on hydrodynamics and mass transfer have been performed at adiabatic conditions with a glass tube of l m length and 0.01 65 m inner diameter without complete absorption of the gas phase in the liquid phase. The rate of absorption is obtained from photographic observations. These photographs show that two flow regimes should be distinguished. In the entrance region froth flow or developing slug flow exists, changing into developed slug flow. For these experiments the bubbly flow region was not attained. The following quantities were measured: rise velocity of slugs, pressure drop, and void fraction. The film thickness and the overall mass transfer coefficient were calculated from these measurements. Rise velocities of slugs, u a, have been determined at different heights, z, along the absorber tube under adiabatic conditions. From these data, relations have been derived as proposed by Nicklinl~:
,-%
q cooling ~ - - medium
For z=0.2 m;
Out
-.-
if"
-
g o s ,n Fig. 1
(1)
uB=c, (UGh+ UL~) + c2(gD ) 'j2
u8 = 1.2 (UGh+UL,) + 0.36 (gD) 1/2
weak solut ion in
(2a)
was obtained and for z=0.5 m: u8 = 1.4 (UG,+ UL,) + 0.29(gD) ~/2.
(2b)
The coefficients obtained by Nicklin do not depend on height because they are valid for stable slug flow. A comparison with Nicklin's results (c~=1.2 and
Vertical tubular bubble absorber
Fig. 1 Absorbour rubuloiro vorticol do bulles
P
S
t'pTRANSHITTER
" P ' - - - ~v
-- ---'l
~
1 /-t I---~
LOOP
LOOP
,~
i'
!
I i I I
t I I I
t
t
5PRO'/ NOZT.LE5 L ~,UF~'ER NH3 - H20 7 SOLUTION 4@5
COOLIN r, ME DtUPI
NH~
-
RE$1~1ANCE HEA~rlNO
Cx~
~@£
HEA'f [XCHANr-E~
¢,UPPLY
Fig. 2
HEAT E~CHANr~ER
Experimental
set-up
Fig. 2 Installation expdrirnentale
350
Revue Internationale du Froid
c2=0.35) shows that the values of c~ and c2 roughly agree. As already stated, this absorber type is characterized by changing flow patterns. Thus, it can be expected that the classical two-phase pressure drop correlation methods will only apply for each of the different patterns. Such a calculation method would
Table 1. Comparison of experimental pressure drop per unit length absorber tube with some correlating methods Tableau 1. Comparaison des mesures de/a chute de press/on exp#rimenta/e par unit# de/ongueur du tube absorbeur suivant d/verses m#thodes de corr#/ation Relative deviation Experimental AP/&z Loc kh art(Pa m -1) Mart/nell/
Hug hmarkD a v i s Pressburg Nicklin
4830 4660 5270 5040 4830 5370 5010 5010 5040 5010 5550 5550 4920 4830 5190 5270 4830 5010
-0.44 -0.43 -0.51 - 0.49 - 0.45 -0.57 -0.48 -0.47 -0.44 -0.46 -0.52 -0.52 -0.46 - 0.45 -0.47 -0.48 -0.43 -0.45
-0.82 -0.81 -0.85 - 0.84 - 0.83 -0.85 -0.84 -0.84 -0,82 -0.82 -0.86 -0.85 -0.84 -0.83 -0.83 -0.83 -0.82 -0.82
-0.45 -0.43 -0.60 - 0.51 - 0.49 -0.61 -0.58 -0.59 -0.47 -0.48 -0.59 -0.56 -0.50 - 0.49 -0.43 -0.44 -0.49 -0.51
-0.32 -0.30 -0.51 - 0.42 - 0.39 -0.51 -0.47 -0.46 -0.34 -0.34 -0.50 -0.48 -0.40 - 0.39 -0.30 -0.31 -0.41 -0.42
Mean
-0.47
-0.83
-0.51
-0.40
require the knowledge of the mass flow rate of both phases at different absorber heights. Actually, this cannot be predicted easily, which makes pressure drop calculations difficult. An attempt was made to predict the pressure drop in the system using the methods proposed by Nicklin 12. Lockhart and Mart/nell/1°, Davis ~5, and Hughmark and Pressburg 7 at the absorber inlet conditions. Table 1 presents a comparison of experimental pressure drop with the predictions from these methods, for 1 8 experiments. Also the mean value of the relative deviation of these predictions is shown. During the calculations it appeared that the wall friction term is negligible when compared with the hydrostatic pressure drop. Since the gas flow rate varies drastically with the absorber height due to absorption, it can be expected that the voidage fraction predicted for inlet conditions will be too high so that too low pressure drops are predicted. Thus, it can also be expected that the methods allowing the best prediction of the voidage fraction will also give the best results for the pressure drop. In Table 2 the experimental voidage fraction is compared with the correlating methods considered. The best results are obtained with N icklin's method, followed by Lockhart-Martinellrs method. The same trends are shown in Table 1 for the pressure drop comparison. The film thickness has been calculated with the relations of Chesters 4, Hubbard 5, Beek 1, van Paassen ~3, and Rosehart and Jagota TM. All relations give values ranging from 0,7-0.8 mm for the film thickness. Aweak increase in film thickness with decreasing gas flow along the absorber may be concluded (0.05-0.1 mm), except for that calculated by Hubbard's relation which is a rather simple one, giving a constant thickness. Overall mass transfer coefficients, K L, have also been evaluated. Evaluation of these coefficients requires rather elaborate methods: the surface area of
Table 2. Comparison of experimental voidage fraction with some correlating methods Ta/beau 2. Comparaison des mesures de/a fraction de vide suivant d/verses m~thodes de corr~/adon Relative deviation Experimental, s (1) 0.45 0.47 0.41 0.43 0.46 0.39 0.43 0.43 0.42 0.43 0.37 0.37 0.44 0.45 0.40 0.39 0.45 0.43
Experimental, s (2)
0.40 0.29
Mean
Lockhart- Mart/nell/
Davis
Hughmark-Pressburg
Nicklin
0.53 0.49 0.73 0.65 0.52 0.90 0.65 0.63 0.62 0.60 0.89 0.89 0.59 0.56 0.70 0.74 0.53 0.60
1,00 0,91 1.22 1.12 0.98 1.33 1.12 1.12 1.14 1.09 1.46 1.46 1.07 1.02 1.25 1.31 1.00 1.09
0.80 0.72 1.12 0.98 0.87 1.21 1,.00 1.00 0.93 0.91 1.30 1.30 0.93 0.89 1.00 1.05 0.89 0.98
0.40 0.34 0.80 0.58 0.48 0.87 0.70 0.67 0.50 0.47 0.89 0.86 0.55 0.51 0.45 0.49 0.53 0.60
0.66
1.15
0.99
0.59
(1) Determined from the experimental pressuredrop assuming the wall friction term is negligible when compared with the weight of gas and liquid in tube. (2) Determined from photographic data
Volume 7 Num6ro 6 Novembre 1984
351
the slugs, A,, which is the area available for mass transfer, has to be determined from photographs. The concentration driving potential has been approached by a logarithmic mean. This admittedly approximate approach, forced by the large heat effects direct after the gas inlet nozzle, gives an order of magnitude for the mass transfer coefficient, K L, which was found to be ~0.0002 m s -~ . From this preliminary analysis of the absorption process in a vertical tubular absorber of small diameter. it can be concluded that the process is rather complicated. It was decided to describe the behaviour of the system using a simplified step-by-step method which simulates the situation sufficiently well for practical purposes.
mL4
The main investigation on vertical tubular absorbers concerns those with simultaneous heat and mass transfer. The aim of this research is to obtain overall heat and mass transfer coefficients and their dependence on the different parameters, which enable the design of bubble absorbers. To process the experimental data, a model was developed which describes the processes occuring in a finite element. It is assumed that heat and mass transfer coefficients will be constant along the absorber tube. Starting from the known inlet conditions and assuming arbitrary values for the heat and mass transfer coefficients, the processes can be calculated successively along the length of the absorber tube. This will finally yield the outlet conditions. Iterative calculations with a new set of values for the heat and mass transfer coefficients will be performed until the calculated outlet conditions will be equal to the experimentally obtained outlet conditions. Thus, only overall conditions are considered.
TLJI
hL1
f
41 C
mG* d'r~G
TG÷ dT G
TL* d]" L r~L÷ dr~L
IG÷ a~ G
{L~ d~ L
hGidhG
hLid h__L
T c + dT C
- -i-?
--I____ I - tq
I NNH3 d-rnL ---"" I
Investigation involving simultaneous heat and mass transfer
~'Li
--
--[
--
_
_
L,.. d.r~qL
INH2OI _
_i
_---_
I i
_
TG
TL
r~G
r~ L
~G hG
~L hL
rhso
r~LO
! rc
GO
rh c
~LO
Fig. 3 Schematic model of differential element Fig. 3
Sch@made/'@/@mentdiff@rentiel
The overall mass balance for the ammonia yields the concentration of ammonia at the top of the absorber: ~L1 = (rhGo~GO4- rhLo~LO)/rhL1
(a) mass balance:
(3)
From the overall mass balance, the mass flow of liquid at the top may be obtained: rhL1 ----rhLo.4- rhGo
3,52
(6)
(b) mass balance for ammonia component:
For any differential element dz along the tube length, heat and mass balances can be formulated. In Fig. 3 a schematic model is presented for an element dz at an arbitrary heightz. The gaseous ammonia and the ammonia-water solution are in co-current upward flow. The cooling medium is in counter-current flow. It is assumed that the gas and liquid flows are homogeneous, i.e. the concentrations of ammonia are uniform in each phase, further the temperatures in both phases are uniform and equal to each other. When entering an element, the next variables are known from the foregoing step: thE, rh e, rh o TL, To ~L, 4]6, P. From these data, the enthalpies of liquid and gas phases hL and h G c a n be obtained. The boundary conditions at the inlet and at the outlet of the absorber are indicated by the indices 0 and 1, respectively. At the top of the absorber all gas should have been absorbed: rhG1--0
(5 )
The laws for conservation of mass and heat applied on a differential element dz give the following relations:
drh G= - drh L
Model description
t
Z
(4)
rhG~G ~- rhL~L = ([hG 4- drhG) (~G + d~G) +
(rhL+ drnL)(~L+ d~L)
(7)
(C) absorbed mass of gas: /'~ 2 drhG= --KLa (~L--4]L)~D dz
(8)
(d) enthalpy balance: rhGhG + rhLhL = (rhG + drh G) (hG + dhG) + (rhL 4- drhL)(hL4- dhL) 4- dO
(9)
When the subroutines derived from analytical relations for ammonia-water solutions, as given by Keizer and Liem 9, are used, the enthalpies at the inlet and outlet sections of the absorber can be directly calculated. (e) removed heat:
dE)=kA (T L-
Tc)
(1 O)
The calculated values determining the conditions of
International Journal of Refrigeration
the gas, the liquid and the coolant leaving the element can be taken as the values determining these flows at the entrance of the next element. This can be done successively. The calculations may be stopped when all gas has been absorbed, i.e. rhG=O. The number of elements, n, needed to absorb all gas will give the height of the absorber, z, as:
z = n dz
1
In general Experimental programme: (1)
(11 )
The calculation method presented can be applied to calculate: (1) the absorber height; and (2) the amount of heat to be transferred to the coolant, i.e. the temperature of solution and coolant, when the overall heat and mass transfer coefficients are known from experimental data. The aim of the experimental research on tubular absorbers presented here was to obtain these data. The overall resistance to heat transfer, 1/k, can be considered as the summation of partial resistance:
1
phase mass flow rate through the absorber and thus of the void fraction.
DID o
D1
k - ~ 4-~-~n~+~O~c
(12)
where cq represents the mean heat transfer coefficient on the solution side. As stated above, at a difference element both the gas and liquid phases are assumed to be at the same mean temperature. The coefficient =, applies then to the heat transfer from the wall temperature to the mean solution temperature. Actually, as can be seen from Fig. 1, in the film flow region there will be a transfer of heat from the interface to the bulk of the film, conduction through the film and, finally, convection heat transfer from the liquid film to the absorber wall. The coefficient cq has been calculated as proposed below. The validity of this assumption has been evaluated by calculating the overall heat transfer coefficient, k, from the experimental results:
~2=kA&T,n
(1 3)
with O -----rhLoh LO+ rhGohG0 -- rhLlh L1
( 1 4)
The logarithmic mean temperature difference can only poorly represent the actual temperature difference profile along the absorber but no better alternative is suggested since the temperature profile cannot easily be measured. The overall volumetric mass transfer coefficient, Kta, has been defined by Equation (8). The parameters determining the conditions of the solution vary with the height z of the tube: concentrations ~L and ~*, temperature TL and pressure vary with z. For a differential element of height dz these parameters are assumed constant. By a trial and error method, a value for KLa can be found for which the calculated absorption height equals the experimentally obtained one. Actually, the assumption of a constant KLa is not in agreement with the experiments. The volumetric mass transfer coefficient will vary from very large values directly after the inlet nozzle to very small values in the bubbly flow region. However, as a first approach this assumption will be maintained. The assumption of a constant K,a will lead to poor prediction of the gas
Volume 7 Number 6 November 1984
(2) (3)
(4) (5) (6) (7)
variation of the tube geometry, i.e. the diameterD of the tube and the absorption height z; variation of the mass flows of gas and liquid, i.e. of the flow regimes which can be encountered; variation of the heat flux through the tube wall, i.e. of the amount of heat which is removed from the solution; variation of the concentration of ammonia in the solution; variation of the temperature of the solution, i.e. of the driving force for heat and mass transfer; variation of the gas temperature; variation of the absorber pressure.
Tubes with diameters of 1 0.0, 1 5.3, and 20.5 mm have been investigated. The absorption height is limited to 1 m. Table 3 presents the minimum and maximum values of the most relevant parameters. Neither the nozzle geometry nor the working fluids have been varied in this work. A change of the working fluids, i.e. mainly in the viscosity of the solution, would not only be of interest with respect to the application in absorption refrigeration machines but also for the
Table 3. M i n i m u m and m a x i m u m values of parameters Tableau 3. Va/eurs minima/es et maxima/es des param#tres Parameter
Minimum
Maximum
rh G (g s -1) rh L (g S-1) ~Lin (kg kg -1)
0.018 2.1 0.341 0.342 0.001 0.005 8.2 9.5 1.26 0.35 0.02 1.05 0.0007 230 92 447 1890 7 1 410 0 0.03 0.162 0.5 2.8 0.2 4.5 500 1 360 6.8
0.480 26.2 0.451 0.510 0.103 0.170 32.0 33.5 3.33 4.31 1.04 5.39 0.0017 4600 2535 5113 17910 14 1 1090 1.1 0.58 0.706 12.6 1 5.6 1.1 263.7 15280 19400 90.0
~Loot (kg kg -1) (~Lout--~Lin) (kg kg -1) (¢~n--(~Lin) (kg kg -1) TG (°C) TL ('C) P (bar) UGs (m s-1) ULs (m s-1) u U (m s-1) (m) ReGs ReLs Reu ReL+G PrL Sc G Sc L ~), (kW) k (kW m -2 K-t) ¢c (kW m -2 K -1) ~i (kW m -2 K-1) ~L, (kW m -2 K-1) ~j~_(kW m -2 K-1) KLa (kg m -3 s-1) ShvL Re (Equation (19)) z/D
353
verification of the universal value of the results. Only one nozzle geometry (annular-type inlet) has been studied. The gas phase was introduced in the centre of the absorber using a tube with 4 mm internal diameter. The inlet mixer design will affect the resulting liquid-gas flow pattern but, for the type of absorbers studied in which the flow pattern rapidly varies from churn to slug and, finally, to bubbly flow, it can be expected that this effect will be reduced. The minimum and maximum values of KLa and of the mean driving potential for mass transfer are given in Table 4. The mean driving potential for mass transfer is defined as: z
(15)
( ¢ t - eL) = Z l ( ~ - - ~L)dz
k~,, = (kca,c- ke,p) /ke×o
In total 1 74 experimental runs were performed: 63 with the 10mm tube, 53 with the 15 mm tube, and 59 with the 20mm tube.
Evaluation of t h e experimental results on heat transfer The partial heat transfer coefficient, a, is assumed to be determined by the following equation: ai=~aLf+ (1 -~)aLp
(1 6)
i.e. two regions are distinguished for which the heat transfer shows different behaviour. In the part of the absorber in which slug and froth flow prevails, a, is
Table 4. M i n i m u m and maximum values of the volumetric mass transfer coefficient and of the mean driving potential for mass transfer Tableau 4. Valeurs rain/males et max/males du coefficient de transfert de masse vo/um#trique et du potent/el d'entrainement moyen pour le transfert de masse Parameter
D (mm)
Minimum
Maximum
KLa (kg m-3 s-1 )
10 15 20 10 15 20
11.8 7.0 4.5 0.019 0.019 0.021
91.3 67.0 263.7 0.155 0.171 0.131
( 17 )
The relative deviation kdiff is given by kdiff + and kdiff_, respectively, for the cases in which cq is included or omitted in the calculation of k. From Table 5 can be concluded that a, should be involved in the calculation of the k value. This conclusion is supported by the increase of the standard error e, if a, is omitted. The resistance to heat transfer is essentially affected by the heat transfer coefficient at the solution side if: (1) the partial heat transfer coefficient, ac, assumes rather high values; or (2) the void fraction is relatively low. Then a, is to an increased extent determined by the heat transfer coefficient alp, corresponding with single phase flow. It can be concluded that the proposed model on heat transfer, involving the partial heat transfer coefficient, cq, enables a quite accurate prediction of the overall heat transfer coefficient, k.
0
(~--~L) (kg kg -~)
supposed to be identical to the heat transfer coefficient in a thin liquid film. The size of this part of the absorber corresponds with the averaged void fraction ~. In the other part of the absorber, in which liquid slugs prevail, the heat transfer is supposed to correspond with that for single phase flow in a tube. For the calculation of au the correlation proposed by Horn 8 for a cooled falling film has been used. In Horn's correlation, the characteristic length is the film thickness and has been calculated from Brauer 2. In Table 5 the minimum and maximum values for the calculated partial heat transfer coefficients alp, a u, and a~ are depicted. Also shown are the mean values and the standard errors of the relative deviations kdiff with:
Evaluation of the experimental results on mass transfer Dimensional analysis For a general comparison with other data, especially on mass transfer, the data should preferably be available in the same, commonly used, form. A first attempt was made to correlate the data using an energy dissipation method as proposed by Jepsen 8. The efforts have been unfruitful. Mass transfer, without simultaneous heat transfer, is often described according to the following relation, which can be derived from dimensional analysis: ShvL = aRe~Sc~
( 1 8)
Table 5. Calculated heat transfer coefficients and relative deviations between calculated and experimental values of the overall heat transfer coefficient Tableau 5. Coefficients de transfert de chaleur calculus et 6carts relatifs entre /es va/eurs exp~rimentales du coefficient global de transfert de chaleur C~Lp(W m -2 K-1)
=Lf (W m -2 K-~)
cq (W m -2 K-1)
D (mm)
Minimum
Maximum
Minimum
Maximum
Minimum
Maximum kdif~+ (%)
e+ (%) kdi,_ (%)
~
10 15 20
310 220 1 50
1120 590 410
2810 4310 6470
13880 12930 15630
530 470 920
10190 10020 12550
18.8 8.8 14.9
35.4 1 5.8 22.9
354
0.5 - 24.5 - 10.7
22.6 - 17.1 2.9
(%)
Revue Internationale du Froid
/ ShvLII.~
• -- O= 40.Omm
40000
• ~
• -- B - 4~.&mm
(1)
/
(2)
• 0
(3)
5DOQ
• • • V~• • •
•
k
~,&& 4000
•
• •
•
,,~
(4)
•
The following Equation (20):
• . . . . . . .
-
SO0~ V 500
assumez has a determined value (a first approach can be obtained from Equation (26)): use this value to calculate KLa from Equations (19) or (20): use the model described in the model description section to estimate the absorber height; in general, this absorber height will differ from that assumed. The correct solution occurs when a value of the absorber height, z, is reached which is equal to the value calculated from the model. This may be achieved by an iterative procedure in which successively better guesses for the absorber height are made.
4000
5000
~no00
5hvLclL~ Fig. 4 Experimentaland predicted Sherwood numbers, according to Equation (I 9)
Fig. 4 Nombres de Sherwood exp~rimenrauxet calculus, d'aprbs I'#quation 19
In this particular case of fast mass transfer, and consequently a changing flow pattern along the absorber tube, the mass transfer may be expected to depend on the height of absorption, z. As two-phase flow is expected to influence mass transfer, one may expect both the liquid phase and gas phase velocities to play a role. A multiple linear regression has been applied to determine the coefficients and the exponents appearing in Equation (1 8). A statistical analysis of the correlation results for Equation (1 8) and modified equations of its type led to the following conclusions: (1) gas phase velocity and absorption height are important in the description of mass transfer; and (2) the mass flow rate of liquid can be neglected as a parameter. The following modified Sherwood relation was obtained which correlates the 1 74 experiments with the absorber tubes of 1 0.0, 1 5.3, and 20.5 mm:
conclusions
/rh
-,0.853 1 353~t-~0.50
PL
/UL qO,53
1 zD,8~,
(20)
Equation (1 9) does not contain direct information about the rate of mass transfer since the absorber height, z, is unknown. The various stages to arrive at the absorber height are as follows:
Volume 7 Num~ro 6 Novembre 1984
from
_0.83/t-~0.50
KLa,,,4.6rhOi33 PL /uL
1
(21
zOSOO 133
(19)
The correlation coefficient was 0.86. The mean relative deviation of the calculated Sherwood number was + 0.050 for the 1 74 observations. Since the liquid mass flow rate appeared not to affect the mass transfer coefficient, the two-phase Reynolds number has been defined asRe= (pLUGsD)/tlL with Uos the superficial gas velocity at the absorber inlet. In Fig. 4 the experimental values of the Sherwood number and those predicted by the regression line, Equation (19), are shown as a function of the right hand side of Equation (19). Equation (1 9) can also be written as: KLa = 2.91~--~G~ "\PG)
be drawn
(1) Effect of gas phase volume f l o w rate. As can be seen from Fig. 1, the gas phase volume flow rate varies from inlet conditions, rh G, to zero at the top of the absorber. The local volume flow rate will be a function of the absorber height. From Equation (20), a short absorber will have a larger mean gas phase volume flow rate than a long absorber. The greatest part of the absorption process occurs in the slug flow regime. In this region, heat and mass transfer occur in the falling film which surrounds the side part of the gas bubbles, From a study by Griffith and Wallis 17, if the liquid film is laminar and the liquid phase volume flow rate is negligible when compared with the gas volume flow rate, the liquid mass flow rate in this film is approximately proportional to ( r h G / P G ) pLDIS/tlL. It is known from the penetration theory (Higbie ~8) that a large interfacial velocity leads to large mass transfer coefficients. If the gas flow rate is increased, the liquid flow rate in the film increases, the mean and interfacial velocities in the film increase and the mass transfer coefficient also increases, in agreement with Equation (20). (2) Effect of the physical properties. If Higbie's theory is applied to a laminar falling film the following relation is obtained:
17017
ShvL = exp (0.8 6 307 ) Re °853Sc °5°
can
where rhLf is the liquid mass flow rate in the film andzf is the length of the falling film. If the approximate equation for rhLf, as given by Griffith and Wallis 17, is substituted in Equation (21), we obtain: /"h
GI
Kta=0.283 7 6
/ 0.33
1 17.D0.50 ,OL / L
~
1
zO~ODOB 3
(22)
where rhG. is the local gas phase mass flow rate. Equation (22) allows a comparison with the experimental correlation (20). The effect of the liquid phase properties on the mass transfer coefficient agrees with the effect predicted from the penetration theory. (3) Effect of absorber height and tube diameter. Equation (20) shows that long absorber tubes present low volumetric mass transfer coefficients and also that KLa decreases with the tube diameter. In the type of absorbers in study three different flow regimes are encountered and the absorber height, z, and the
3515
interfacial area, a, do not correspond with the length, zf, and interfacial area of a falling film. Nevertheless, the same trends are found for the effect o f z and D on the mass transfer coefficient (compare Equations (20) and (22)). S t e p w i s e m u l t i p l e linear regression
The method proposed in the last section, for the calculation of the absorber height, requires an iterative procedure. The question arises if a description of the experimental data could be obtained that should lead to a direct calculation of the absorber height. Stepwise multiple linear regression can be applied to assess the relative influences of the different parameters on the dependent variable, i.e. on, for instance, the absorber height z. The latter has been taken as the dependent variable in the hypothesis which is arbitrarily assumed as: (23)
z=aXb'Xb2 . . . X b~
This is actually a more general approach to correlate the experimental data on z. However, if only indepent variables are taken as X., the results will not be applicable for systems other than ammonia-water. The independent parameters X., which eventually might affect z, are identified as follows: D,FhL,rhG,P, TG,~ L TL,TLoo, (or O).
A remark can be made with respect to the last parameter TLo,t (orO). This variable should be known at the absorber height z; actually, the liquid phase temperature is only measured at the outlet of the test section (1 m from the inlet). The heat transfer rate until the absorption height z, as obtained from the model presented above, O was used. A correlation coefficient r=0.93 and F-value F=156 were obtained. The dependence of z on the parameters which have a significant effect can be expressed as: 21
Z=133.6~L O
06
z")- 1 0 r ~ ~ 4.-~01 ,,,G F u
0 5.¢., - 0 04
'"u
(24)
The following trends can be differed from Equation (24): (1) an increase of the system pressure, of the absorber diameter and of the liquid phase mass flow rate lead to smaller absorbers; and (2) an increase of the liquid phase concentration, of the liquid phase temperature, of the gas phase mass flow rate, and of the transferred heat lead to larger absorbers. All these trends are in agreement with the conclusions that can be drawn from an enthalpy-concentration diagram for the system studied. The influence of the diameter can be drawn directly from Equation (8). It must be noted that a diameter increase does not necessarily imply a shorter absorber since (~T is also a function of the absorber diameter. As the liquid solution temperature, TL, decreases the absorber height, z, decreases with a limit of z = 0 for T~=0~C. Actually Equation (24) only applies for the conditions tested and, as shown in Table 3, only the range 9.5
358
the absorber height, <0.15m. This equation constitutes thus an alternative for Equation (1 9). It must also be remarked that Equation (24) requires an iterative calculation: the heat transfer through the absorber tube wall (3 must be calculated as presented in the model description section. The advantage of Equation (24) is that the heat transfer through the absorber tube C5 can be approached by C5=rhGhLG
(25)
where hLG is the heat of solution plus vaporization, so that, although approximately, z can be estimated directly.
Conclusions The proposed model on heat transfer enables a quite accurate prediction of the overall heat transfer coefficient. From Equation (20) or, in dimensionless form, Equation (1 9), a mean volumetric mass transfer coefficient can be estimated. The absorber height, the parameter in which we are especially interested for design of vertical tubular bubble absorbers, can be calculated by the iterative procedure presented in the dimensional analysis section. A first approach of the absorber height, for the system ammonia-water, as a function of the inlet conditions, can be obtained from: z= 1 ~ ~-~21r~O7F~O6"FO4D-10/-)-O5~-°04 (26) ~'V~L
"~G
"LG ~ L ~
~"
H~L
Acknowledgements The authors wish to express their thanks to Prof A. L. Stolk for guidance during the course of the investigation and Prof J . M . Smith for helpful discussions and suggestions during the preparation of the manuscript.
References 1 2 3 4 5 6 7 8 9
10
Beek,W. J. Stofoverdracht door beweeglijke grensvlakken, Thesis, Delft University of Technology (1962) Brauer, H. Grundlagen der Einphasen- und Mehrphasenstr6mungen, Verlag Sauerl~nder, Aarau (1971) Charl~ntier, J. C. What's new in absorption with chemical reaction? Transactions of the Institution of Chemical Engineers 60 (1982) 131-156 Cheltit$, A. K. Stromingen warmteoverdracht. Department of Mechanical Engineering, Delft University of Technology (1976) Gregory, G. A., Scott, D. S. Physical and chemical mass transfer in horizontalcocurrent gas-liquid slug flow, Chemical Engineering Journal 2 (1971) 287-295 Horn, R. K. Messungenzum W~rme- und StoffQbergang am Rieselfilm, Thesis. Univ. Karlsruhe (1970) Hughmark, e. A., Pressburg, B. S. Holdup and pressure drop with gas-liquid flow in a vertical pipe.AIChEJ 7 (1961 ) 677-682 Jel~en, J. C. Masstransfer in two-phase flow in horizontal pipelines. AIChE J 16 (1970) 705-710 Keizer, C., Liem, S. H. Absorption refrigeration machine driven by solar heat. Internal report K82. Laboratory for Refrigeration and Indoor Climate Technology, Delft University of Technoloy. (1979) Lockhart, R. W., Martinelli, R. C. Proposedcorrelation for
International Journal of Refrigeration
11 12 13
14
isothermal two-phase, two-component flow in pipes, Chem Eng Progr 46 (1949) 39-48 Nicklin, D. J. Two phase bubble flow, ChemicalEngineering Science 17 (1962) 693-702 Nicklin, D. J. The air-lift pump: theory and optimisation. Trans Inst Chem Engrs 41 (1963) 29-35 van Paa~sen, J. A. De absorptie van ammoniakgas vanuit opstijgende bellen in een ammoniakwater oplossing. Internal report S701. Laboratory for Refrigeration and Indoor Climate Technology, Delft University of Technology, (1979) Rosehsrt, R. G. D., Jagota, A. K. A method for the determination of individual film and droplet volumetric mass
Volume 7 Number 6 November 1984
15
16 17
transfer coefficients and interfacial areasof gas-liquid annular cocurrent flow. Cocurrent gas-liquid flow. (Eds Rhodes, E., Scott, D. S.). Plenum Press, New York (1969) 667 Scott, D. S. Properties of cocurrent gas-liquid flow, Advances in Chemical Engineering 4 (1963) 199-277. (Eds Drew, T. B, Hoopes. J. W., Vermeuler, T.) Academic Press, New York Higbie, R. The rate of absorption of a pure gas into a still liquid during short period of exposure, TransAm Inst Chem Engng 31 (1935) 365-389 Griffith, P., Wallis, C. B. Two-phase slug flow, Journal of Heat Transfer Trans. of the ASME (1961 ) 307-320
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