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Simulation of transient behaviour in refrigeration plant pressure vessels: mathematical models and experimental validation
International Journal of Refrigeration 26 (2003) 170–179 www.elsevier.com/locate/ijrefrig
Simulation of transient behaviour in refrigeration plant pressure vessels: mathematical models and experimental validation S. Estrada-Floresa,*, D.J. Clelandb, A.C. Clelandc, R.W. Jamesd a
Supply Chain Innovation, Food Science Australia, PO Box 52, North Ryde, NSW 1670, Australia b Institute of Technology and Engineering, Massey University, Palmerston North, New Zealand c The Institution of Professional Engineers New Zealand. PO Box 12241, Wellington, New Zealand d School of Engineering Systems & Design, South Bank University, 103 Borough Rd, London SE1 0AA, UK Received 11 April 2002; accepted 19 August 2002
Abstract Four dynamic models of different degrees of complexity were derived to represent a typical industrial refrigeration intercooler (pressure vessel). The models were validated against temperature and pressure data from two pilot plant calorimeters containing R-134a under a variety of transient operating conditions. The measured response rate was strongly influenced by sensible heat storage in the calorimeter shell and liquid refrigerant. Little difference in predictions by the four models was obtained in spite of major simplifying assumptions made to develop the less complex models. A model considering only the thermal capacity in the shell and liquid refrigerant predicted rates of temperature change within 10% of predictions by the other models, and also close to experimental data. # 2003 Elsevier Science Ltd and IIR. All rights reserved. Keywords: Refrigerating circuit; Pressure vessel; Transient state; Operating; Modelling; Measurement
Simulation du comportement transitoire des re´cipients sous pression des installations frigorifiques : mode´les mathe´matiques et validation expe´rimentale Mots cle´s : Circuit frigorifique ; Re´servoir sous pression ; Regime transitoire ; Fonctionnement ; Mode´lisation ; Mesure
1. Introduction Pressure vessels such as intercoolers, receivers and surge drums are often used in refrigeration plants. The
* Corresponding author. E-mail address: silvia.estrada-fl[email protected] (S. EstradaFlores).
transients of parameters such as pressures, temperatures and refrigerant mass distribution in such vessels can be used as indicators of the overall performance of a plant. Complex dynamic phenomena, such as de-superheating of vapour from low stage compressor, flashing of high pressure liquid and the heat flow from external (ambient) conditions occur in such vessels (Fig. 1). Two approaches have been traditionally used for developing vessel models: a) detailed models with time
0140-7007/03/$20.00 # 2003 Elsevier Science Ltd and IIR. All rights reserved. PII: S0140-7007(02)00081-6
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171
Nomenclature A cp E h M m q p r1 r2 r3
sc T t U u v w x Z
area (m2) specific heat capacity (J kg1 K1) energy (J) enthalpy (J kg 1) mass (kg) mass flow rate (kg s 1) heat flow (W) pressure (Pa) internal radius, pressure vessel (m) external radius (internal space plus metallic shell), pressure vessel (m) total external radius (internal space plus metallic shell plus insulation), pressure vessel (m) subcooling ( C) temperature ( C) time (s) internal energy (J) specific internal energy (J kg 1) specific volume (m3 kg 1) specific work (W) thickness (m) length (m)
Greek letters heat transfer coefficient (W m2 K 1) l thermal conductivity (W m 1 K1) density (kg m-3)
Subscripts 1 State at the beginning of the transient period 2 State at the end of the transient period amb!vessel ambient environment to the external surface of a vessel con condensation coil cooling coil ev evaporation g vapour insul insulation l liquid s shaft sat saturated condition p constant pressure T total Vessel!zone internal surface of a vessel to a liquid or vapour refrigerant zone
Fig. 1. Physical representation of a closed intercooler.
and spatial dependence using partial differential equations (PDEs); b) a lumped thermal modelling approach, using ordinary differential equations (ODEs). Most existing PDE and ODE based models are ‘‘thermal models’’ that ignore refrigerant hydrodynamics and mostly concentrating on properties such as thermal capacity and thermal resistance [1]. Published modelling approaches applied for vessels range between lumping the thermal capacity with a flooded evaporator to multiple zoning of the vessel, normally using zones selected according to the refrigerant physical states coexisting in the vessel. Marshall and James [2] modelled an intercooler, assuming four zones: vapour, liquid, vapour bubbles temporarily in the liquid and the subcooling coil. The ODEs for the refrigerant zones were based on energy and mass balances. James and James [3] used a similar approach, modelling a liquid receiver by separating zones at the vapour–metal, liquid–metal and vapour– liquid interfaces. Some disagreement between predicted and experimental values was found during plant startup and shutdown [4]. Cleland [5] assumed that the liquid mass present in a vessel was constant and that the liquid phase thermal capacity was significantly larger than that of the vapour phase. The vessel dynamics could then be determined from the liquid refrigerant properties and the net inflow of energy to the vessel. Lovatt [6] used this model for the REFSIM simulation package. While the overall simulation package performed well, the specific performance of the vessels model was not investigated. The apparent lack of development of pressure vessels models may be attributable to their assumed low thermal mass and relatively ‘‘static’’ behaviour. Published models generally ignored the linking elements or their thermal capacities were lumped with main equipment. Hence, research directed at modelling unsteady-state energy transfer in pressure vessels might provide a better understanding of these phenomena in the dynamic behaviour of refrigeration plants.
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The objectives of this paper were to develop new models for pressure vessels of different levels of complexity using thermal modelling approaches, and to validate the models against experimental data. The four models developed were: A) A rigorous thermodynamic approach, considering transitions between states for the liquid and vapour zones in detail, B) a less complex model following the assumptions made by Marshall and James [2] but still considering the transition between states, C) the same approach as B, but neglecting the transition between states, and D) a simple model following the assumptions of Cleland [5].
2. Physical and conceptual model Fig. 1 shows a generalised closed intercooler vessel for a two-stage refrigeration plant. The intercooler receives superheated gas from the low stage compressors. The vapour is cooled and a flow of vapour is drawn away from the head space of the intercooler by the high stage compressors. The vessel is fed via a high-pressure refrigerant liquid line. Prior to its entrance to the vessel, the liquid pressure is reduced to the intercooler pressure,
thus forming a mixture of vapour and liquid. A second liquid line may pass through a coil in the vessel. The liquid in the coil is subcooled by boiling of the refrigerant surrounding the coil, creating vapour. Refrigerant from evaporators or secondary vessels might also return to the intercooler. The vessel may be insulated and covered with a metallic vapour barrier layer, to lower the rate of radiant and convective heat transfer from the external environment. Fig. 2 presents a conceptual model of the major phenomena occurring in such a pressure vessel. The basic principles and assumptions were: 1) Vessel contents were divided into perfectly mixed liquid and vapour zones. 2) Although no external work was done by the system as a whole, a net exchange of work between zones due to liquid level change was possible. 3) Instantaneous condensation of subcooled vapour and evaporation of superheated liquid upon entering vessel, to re-establish a saturation state was assumed (although liquid and vapour zones could be subcooled and superheated respectively).
Fig. 2. Conceptual model for pressure vessels.
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4) All mixed refrigerant flows entering the vessel separated instantly, with liquid flows entering the liquid zone and vapour flows entering the vapour zone. 5) In the liquid zone, because pv is negligibly small, the internal energy u equals the enthalpy h. 6) Heat flow through vessel walls was modelled considering the insulation layer as a pure resistive material and the metallic shell as two pure capacitive layers (one in contact with liquid and one in contact with vapour) 7) No thermal capacity was attributed to coil pipes or to heating elements. 8) Liquid refrigerant density was assumed constant. 9) Kinetic and potential energy effects were neglected. Fig. 2 shows the different sources of energy and mass exchange considered: a) Flow in items entering the vessel b) Flow out items leaving the vessel, with the thermodynamic characteristics of either the liquid or vapour phase. c) Object items, interacting with the vessel contents via convective heat exchange only. d) Latent items, such as evaporation or condensation flow between phases.
0 vl X X ml;flows in ml;flows out B vg B X B B þ ðmev mcon Þ 1 vl þ mg;flows B vg B X B 1 B mg;flows out d B vg B ¼B B dt B AðZT Zl Þ B B B B B B @
1 C C C C inC C C C C C C C C C C C C C C A ð2Þ
The energy equation for the liquid zone was obtained from a full energy balance according to the 1st Law of Thermodynamics, including condensation and evaporation phenomena. The left hand side of the energy balance for a zone containing Ml kg of homogeneous liquid is: dUl dðMl ul Þ dul dMl ¼ Ml ¼ þ ul dt dt dt dt
ð3Þ
Using Eq. (1), Eq. (3) becomes: dUl dul ¼ Ml dt dt X X ml;flows in þul ml;flows
out þmcon mev
ð4Þ
3. Mathematical models Full details of model derivations are given by Estrada-Flores [7]. 3.1. Model A): two-refrigerant-zone model based on a full energy balance The main characteristics of this model were: a) the work exchange between liquid and vapour phases was included; b) the full derivative of internal h i energy respect to time was used, i.e.,
dug dt
¼
dhg dt
dðdtpvÞ ; c) the model
simulated transition between saturated and non-saturated states of liquid and vapour; and d) the full thermal capacities of the liquid and vapour phases and the metallic shell were considered. The changes in the liquid level respect to time was derived by mass balance, noting that Ml=A Zl/vl: P P ml;flows in ml;flows out mev þ mcon dZl ¼ l dt A ð1Þ A similar mass balance for the vapour zone, noting that Mg=A Zg/vg and Zg=ZTZl, yields:
dul dhl , so the full For the liquid zone, ul hl and dt dt energy balance equation is: X X dhl Ml ml;flows in þ hl ml;flows out þ mcon mev dt X X ml;flows out hl ¼ ml;flows in hl;flows in X þ qobject ws mev hg;sat þ mcon hl;sat ð5Þ The external work ws is that done on the vapour phase: X X ml;flows in ml;flows out mev þ mcon ws ¼ pg vl ð6Þ Rearranging Eq. (6) yields: X dhl X ml;flows in hl;flows in Ml ml;flows dt þ qobject ws mev hg;sat hl;sat þ mcon hl;sat hl X X ml;flows in hl ml;flows
out
out
hl
ð7Þ
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The energy balance equation for the vapour zone was derived in a similar fashion, but the dðdtpvÞ term was dðpvÞ dp dv retained. Noting that dt g ¼ vg dtg þ pg dtg , the resulting energy balance equation was: P B1 þ qobject ws B2 hg pg vg þ B3 dhg ¼ dt Mg dpg dvg þ pg þ vg ð8Þ dt dt where: X B1 ¼ mg;flows X mg;flows B2 ¼
X hg;flows in mg;flows out hg X m in g;flows out þ mev mcon in
B3 ¼ mev hg;sat mcon hl;sat
ð8aÞ ð8bÞ ð8cÞ
3.1.1. Modelling of temperatures and pressures for the liquid and vapour zones The thermodynamic property routines of Cleland [8,9] were adopted. For the liquid zone, two states were recognised:
In a similar fashion, the shell model for the vapour zone was: dTshell;vapour zone Mshell cp;shell dt qamb ! vessel;vapour zone qvessel ! vapour zone ¼ Zl 1 ZT ð12Þ The heat flow terms qamb!vessel and qvessel!zone for the two zones were calculated using the generic equation for cylindrical and slab conduction/convection combined: qamb ! vessel 2
13
0
B 6 B 6 B ¼6 D þ 1 B 6 @ 4
1 1
2r3 amb ! vessel Zzone
ln þ
r3 r2
C7 C7 C7 C7 A5
2linsul Zzone
ðTamb Tshell Þ ð13Þ
Saturated liquid, where hl=hl sat and Tl=Tl sat at pg. Subcooled liquid, where hl,sc is found from Eq. (10) and sc was calculated using: hl;sat hl sc ¼ ð9Þ cpl
r23
D1 ¼
1 amb ! vessel
þ
xinsul linsul
ð13aÞ
and:
Hence, the temperature of the liquid was: Tl ¼ Tl;sat sc
where:
ð10Þ
For the vapour zone, a methodology to calculate pg whilst knowing only hg and vg of the vapour phase was required. For saturated vapour this was straightforward, but for superheated vapours an iterative routine to find the extend of superheat, pg, vg, Tg and hg was developed [7]. 3.1.2. Modelling of energy exchange with the external environment through the vessel shell Eqs. (11) and (12) were used to calculate the change in temperature of the metal shell in contact with the liquid and vapour zones, respectively. These equations assume that changes in Mshell, shell mass in contact with each phase with respect to time as a consequence of the variation of Zl are sufficiently small that they can be ignored in the heat balance. dTshell;liquid zone Mshell cp;shell dt qamb ! vessel;liquid zone qvessel ! liquid zone ¼ Zl ZT ð11Þ
qvessel ! zone ¼ vessel ! zone 2 r1 Zzone þ r21
ðTshell Tzone Þ
ð13bÞ
3.1.3. Modelling of latent items The criteria used to define the state of the liquid and vapour zones were: a)Vapour: if ug > ug,sat at pg then the vapour was superheated and mcon=0. Otherwise, vapour was saturated and mcon50. b)Liquid: ul < ul,sat at pg then the liquid was subcooled and mev=0. Otherwise, liquid was saturated and mev50. For the liquid zone, when the transition criteria signalled a change to saturated liquid, the rate of evaporation between zones to maintain saturation was calculated using: P mev ¼
dhl ws qobject Ml dt hg;sat hl;sat
ð14Þ
S. Estrada-Flores et al. / International Journal of Refrigeration 26 (2003) 170–179
Similarly, Eq. (15) was used to calculate the rate of condensation for the vapour zone: X dhg dpg dvg qobject Mg vg pg dt dt dt dMg þpg vg þws dt ð15Þ mcon ¼ hl;sat hg;sat The use of Eqs. (14) and (15) implies the estimation of five derivative terms:
dhl dt
;
dhg dt
;
dpg dt
;
dMg dt
and
dvg dt
by means
other than the original ODEs [Eqs. (1), (2), (7) and (8)]. First estimates of these terms were made using backward difference approximations of the form: dy ytime¼t ytime¼t#t ¼ dt #t
ð16Þ
where y can be hl, hg, pg, vg or Mg. To improve these estimates, an iterative procedure fully described in Estrada-Flores [7] that gave feasible computation times and stable results was adopted. 3.2. Models B and C: Two-refrigerant-zone models based on simplified energy balances These models were derived by applying two simplifications to model A, which were: a) the work between liquid and vapour phases was neglected and b) the du dh approximation dtg ¼ dtg was used [2]. Hence, the full expression for the liquid enthalpy, Eq. (7), was simplified to: dhl X ¼ qobject mev hg;sat hl;sat Ml dt ð17Þ mcon hl;sat hl The vapour zone energy balance of Eq. (8) was simplified to: dhg dt P ¼
dMg qobject þ mev hg;sat hg mcon hl;sat hg þ pg vg dt Mg ð18Þ
In both models B and C, the calculation of the liquid level and density of vapour were by Eqs. (1) and (2). As in model A, model B recognised transition between phases and latent items were modelled. Model C neglected these, always calculating hl through Eq. (17). For both models B and C, the modelling of energy exchange with the external environment through the vessel applied as described for model A.
175
3.3. Model D: one-refrigerant-zone model based on an enthalpy balance for the liquid zone This model had the following characteristics: a) All refrigerant mass was treated as liquid; b) thermodynamic equilibrium was assumed between zones (vapour and liquid were at the same temperature); c) mass flow rates entering and leaving the vessel all interacted with the liquid zone; d) the level of liquid was constant; all the simplifying assumptions of model C also applied. Thus, only one energy balance equation for the liquid phase was needed: dul dTl Ml ¼ Ml cp;l dt dt X X ¼ mflowsin hflow sin mflowsout hflows out ð19Þ þ
X
qobject
The modelling of energy exchange with the external environment through the vessel shell comprised Eqs. (11)–(13), but only one zone containing all the metal shell was considered. 3.4. Models implementation The mathematical models were implemented in a TurboPascal programme. A numerical solution using the 4th order Runge–Kutta procedure was selected.
4. Experimental validation The models were tested by comparison of predictions to the measured time-temperature responses of refrigerant contained in two pilot plant calorimeters. These calorimeters are shown in Figs. 3 and 4, along with physical data of the systems. A detailed description of both systems is given in Estrada-Flores [7]. During the experimental trials, the calorimeter were switched on, selecting the condenser water flow rate and temperature, the evaporation temperature, heater input and suction and discharge pressures. The temperatures of the vessel contents were monitored until steady-state conditions were achieved. Once the conditions were steady, a first step change in operating conditions was carried out by changing the energy input of the heater. The system then underwent a period of transient behaviour before reaching a new steady-state condition. Several step changes were carried out in a sequence. Several experimental trials using different conditions and different step change strategies were performed. A global energy balance was adopted as an indicator of the quality of the experimental data, described by:
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ð t2 ET;2 ET;1 ¼
ðqheater þ qamb ! vessel qcoil Þdt X ¼ qnet #t t1
ð20Þ
The term ET,1ET,2 was estimated as: ET;2 ET;1 ¼ Mshell cp;shell Tshell;2 Tshell;1 þ Mcoil cp;coil Tcoil;2 Tcoil;1 þ Ml;1 ul;2 ul;1 þ Mg;1 ug;2 ug;1 þ Mg;2 Mg;1
ug;2 ul;1
Fig. 3. Schematic diagram of calorimeter 1.
ð21Þ
The term qamb!vessel was calculated through Eq. (13). The term qheater was directly controlled and known. The term qcoil was calculated using a linear change of the coil heat removal respect to time, since experimental coil temperature data showed a linear change in temperature difference across the coil. Experimental trials with energy balance errors of more than 15% were regarded as unacceptable for comparison purposes.
Fig. 4. Schematic diagram of calorimeter 2.
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5. Results and discussion Table 1 presents the conditions of the six trials that met the relative error criterion. Similar energy balances to those described for the experimental trials were performed on the predicted behaviour of all four models, as illustrated in Table 2 (trial 1). Figs. 5 and 6 illustrate the experimental and simulated liquid and vapour temperature profiles for trials 1 and 3. The energy balances of model D were apparently better than those of models A, B and C, due to the additive effect of incomplete convergence of the iterative procedures required to calculate several of the liquid and vapour thermodynamic parameters related to saturated and non-saturated states in the 3 more complex models. Model D also computed faster than models A, B and C: for the former, a typical simulation using a time step of 1 s was adequate; for the latter, a time step of 0.1 s and more than 12 iterations for vapour superheat calculations were required to ensure a good accuracy.
In Calorimeters 1 and 2, the contribution of the energy storage in the metal shell to the global energy balance was approximately 77 and 66%, respectively. The thermal mass of the liquid was next most important. The thermal mass of both coil and vapour components contributed less than 5%. The contribution of the latent heat component did not exceed 7% in trials 1 and 2, and it was less than 3% in the rest of the trials. Model D overpredicted the rate of the change of temperatures of the shell and liquid, due to the lack of latent items to account for latent heat at evaporation. However, lumping the thermal mass of the vapour with the liquid did not lead to significant disagreements when compared with models A, B and C. In all trials, the predicted temperatures showed deviations within 0.3 to 2.2 C of the experimental values for all models. The analysis of qnet showed that the heat transfer through the insulation and metal shell accounted for less than 30% of qnet in all trials except for trial 2 in which temperature profile was totally dependent on the heat
Table 1 Summary of selected experimental trials for model validation Trial
Heater input (W)a Duration (s) Temperatures in refrigeration plant
Vessel pressure (R-134a,bar) Troom ( C)
Evaporation R-22 ( C) Condensing R-22 ( C) 1 1. Calorimeter 2 I. 2. II. 3.
564 1000 577 1000 630
1. I. 2. II. 3.
1320 960 4380 2580 1760
2. 1. Calorimeter 2 I. 2.
2400 0 0b
1. I. 2.
3. 1. Calorimeter 1 I. 2. II. 3.
1700 2500 2500 3000
4. 1. Calorimeter 1 I. 2. II.
1.
4.30
27.0
2.
5.69
3.
27.0
3.
6.58
0.01 1500 50
1.
0.0
1. 55.0
1.
7.64
2.
b
2.b
2.
5.58
1. I. 2. II. 3.
1620 630 1320 460 980
1.
15.0
10.5
1.
2.43
2.
8.0
2.
3.10
3.
4.0
3.
3.98
1500 1500 3000 3000
1. I. 2. II.
2600 580 2600 3030
1.
4.0
1.
3.95
2.
17.5
2.
2.18
5. 1. Calorimeter 1 I. 2. II.
2300 2300 3000 3000
1. I. 2. II.
3090 490 3090 490
1.
4.5
1.
3.93
2.
10.0
2.
3.11
6. 1. Calorimeter 1 I. 2. II.
2300 2300 3000 3000
1. I. 2. II.
8350 1020 2580 1020
1.
37.4
2.
14.0
a b
1.
27.0
2.
55.0
10.5
11.6
6.8
Arabic numbers represent steady-state periods. The calorimeter was switched off and only ambient heat flow entered the system.
7.60 2.27
25
18
18
16
15
12
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S. Estrada-Flores et al. / International Journal of Refrigeration 26 (2003) 170–179
Table 2 Energy balance in experimental calorimeter 2 and models A, B, C and D: trial 1 Energy components
System Experiment
Initial temperature (liq, vap), C Final temperature (liq, vap), C Sensible heat taken up by shell, kJ Sensible heat taken up by coil, kJ Sensible heat taken up by liquid phase, kJ Sensible heat taken up by vapour phase, kJ Latent heat from evap/condens, kJ Total 1 (ET,2ET,1), kJ Total energy input by heater, kJ Total energy removal by coil, kJ Energy infiltrated through shell, kJ Total 2 ($qnet #t), kJ % Error (internal energy balance)
11.1, 13.3 24.6, 24.9 1003.3 47.4 236.2 9.8 82.7 1379.5 7770.1 6357.3 71.6 1484.4 7.6
transfer through the insulation. The heater and coil contributions were the most important heat transfer pathways and, since the measured values were fed into the programme that implemented the models, these values were the same for experimental and simulated trials. Models A, B and C had similar internal energy balances. Differences between these models, related to modelling of phase transitions and work between phases, did not significantly affect the predictions obtained. Fig. 5 shows a temperature difference of up to 2 C between liquid and vapour phases. In this trial, 3 quasisteady-state periods occurred, during which condensation on the coil in the vapour and evaporation induced by the heater in the liquid were occurring at the same time and rate. These conditions would have led to a vapour phase close to saturation, yet the measured data suggests superheat. The two most likely explanations are incomplete mixing in the vapour zone, with the
Simulations Model A
Model B
Model C
Model D
11.1, 11.1 25.9, 25.9 1124.1 0 259.6 8.0 106.7 1498.4 7770.1 6357.3 73.6 1486.4 0.8
11.1, 11.1 25.9, 25.9 1123.9 0 259.6 8.0 106.7 1498.2 7770.1 6357.3 73.6 1486.4 0.8
11.1, 11.1 25.9, 25.9 1123.9 0 259.5 8.0 106.6 1498.2 7770.1 6357.3 73.6 1486.4 0.8
11.1, 11.1 26.6 1186.7 0 296.2 0 0 1483.0 7770.1 6357.3 70.2 1483.0 0.002
thermocouple being located in an area away from the coil, or measurement error due to thermal bridging between the thermocouple and the metallic shell. These explanations could not be checked without dissembling the plant, which was not possible. The differences between measured and experimental data must be interpreted in relation to the experimental energy balances. For example, as summarized in Table 2, in trial 1 the measured temperature data suggested that ET,1ET,2 was 7.6% less than qnet. All models overpredicted the measured temperature change by about 10% when using essentially the same qnet as measured. The six experimental energy balances were all in error by 5 to 12%. Considering this, the agreement between experimental and simulated trials was satisfactory. Overall, because all the models accounted for the effect of the dominant thermal masses (liquid and metallic shell) in the systems investigated, the good performance of all models regarding overall temperature
Fig. 5. Experimental and simulated liquid and vapour temperatures for Trial 1.
S. Estrada-Flores et al. / International Journal of Refrigeration 26 (2003) 170–179
179
Fig. 6. Experimental and simulated liquid and vapour temperatures for Trial 3.
profiles was expected. The most complex models A, B and C did not display a noticeably higher level of accuracy than the simpler model D. The operating conditions covered by the experimental work described in this paper do not encompass all situations that may be encountered industrially, particularly in respect to liquid level, deviation from saturation and extent of pressure variations. However, an industrial case study described by Estrada-Flores [7] suggests that the results described in this paper may be valid for wider ranges than those investigated experimentally.
6. Conclusions Dynamic models for pressure vessels in small systems should consider the energy storage of the metallic shell and liquid phase as the most important components in a global energy balance. For vessels operating with good liquid level control and close to saturation, the assumptions of Marshall and James [2] and Cleland [5] related to work and enthalpy terms introduced almost no errors. Hence, models B, C and D are expected to perform satisfactorily in many situations commonly found in industrial pressure vessels. The simpler one refrigerant-zone (liquid only) model D, which did not consider latent phenomena, predicted rates of temperature change that were less than 10% different from those predicted by other models. Model D also offered considerable advantages in terms of ease of computer implementation and computational cost.
Acknowledgements The experimental work was carried out at South Bank University and Star Refrigeration (UK). The authors wish to express their gratitude to Dr. John Missenden
and Dr. Forbes Pearson. The theoretical work was conducted at Massey University, whilst Silvia EstradaFlores was supported by a UNAM scholarship. This financial support is also gratefully acknowledged.
References [1] Cleland AC. Food refrigeration processes. Analysis, design and simulation. London: Elsevier Applied Science; 1990. [2] Marshall SA, James RW. Dynamic analysis of an industrial refrigeration system to investigate capacity control. Proc Inst Mech Eng 1979;189:437–44. [3] James, K. A. and James, R. W. Dynamic analysis of a heat pump using established modelling techniques. Research memorandum 98, Inst Env Eng. South Bank University, London, 1986. [4] James, K. A. Dynamic mathematical modelling of refrigeration systems and heat pumps. PhD thesis, South Bank University, London, 1988. [5] Cleland AC. Simulation of industrial refrigeration plants under time-variable conditions. Int J Refrig 1983;6:11–19. [6] Lovatt, S. T. Development of a dynamic modelling methodology for the simulation of industrial refrigeration systems. PhD thesis, Massey University, New Zealand, 1992. [7] Estrada-Flores, S. Evaluation of dynamic models for refrigeration system components. PhD thesis, Massey University, New Zealand, 1996. [8] Cleland AC. Computer subroutines for rapid evaluation of refrigerant thermodynamic properties. Int J Refrig 1986;9:346–51. [9] Cleland AC. Polynomial curve-fits for refrigerant thermodynamic properties: extension to include R 134a. Int J Refrig 1994;17:245–9. [10] Bejan A. Heat Transfer. New York: John Wiley & Sons; 1993. [11] Armaflex II. Standard ASTM C 534. Amstrong Industry Products Division: 1991. [12] ASHRAE. (1995). ASHRAE handbook of fundamentals. Atlanta (GA): Amer. Soc. of Heat,. Refrig., and Air Cond. Eng.
Simulation of transient behaviour in refrigeration plant pressure vessels: mathematical models and experimental validation S. Estrada-Floresa,*, D.J. Clelandb, A.C. Clelandc, R.W. Jamesd a
Supply Chain Innovation, Food Science Australia, PO Box 52, North Ryde, NSW 1670, Australia b Institute of Technology and Engineering, Massey University, Palmerston North, New Zealand c The Institution of Professional Engineers New Zealand. PO Box 12241, Wellington, New Zealand d School of Engineering Systems & Design, South Bank University, 103 Borough Rd, London SE1 0AA, UK Received 11 April 2002; accepted 19 August 2002
Abstract Four dynamic models of different degrees of complexity were derived to represent a typical industrial refrigeration intercooler (pressure vessel). The models were validated against temperature and pressure data from two pilot plant calorimeters containing R-134a under a variety of transient operating conditions. The measured response rate was strongly influenced by sensible heat storage in the calorimeter shell and liquid refrigerant. Little difference in predictions by the four models was obtained in spite of major simplifying assumptions made to develop the less complex models. A model considering only the thermal capacity in the shell and liquid refrigerant predicted rates of temperature change within 10% of predictions by the other models, and also close to experimental data. # 2003 Elsevier Science Ltd and IIR. All rights reserved. Keywords: Refrigerating circuit; Pressure vessel; Transient state; Operating; Modelling; Measurement
Simulation du comportement transitoire des re´cipients sous pression des installations frigorifiques : mode´les mathe´matiques et validation expe´rimentale Mots cle´s : Circuit frigorifique ; Re´servoir sous pression ; Regime transitoire ; Fonctionnement ; Mode´lisation ; Mesure
1. Introduction Pressure vessels such as intercoolers, receivers and surge drums are often used in refrigeration plants. The
* Corresponding author. E-mail address: silvia.estrada-fl[email protected] (S. EstradaFlores).
transients of parameters such as pressures, temperatures and refrigerant mass distribution in such vessels can be used as indicators of the overall performance of a plant. Complex dynamic phenomena, such as de-superheating of vapour from low stage compressor, flashing of high pressure liquid and the heat flow from external (ambient) conditions occur in such vessels (Fig. 1). Two approaches have been traditionally used for developing vessel models: a) detailed models with time
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Nomenclature A cp E h M m q p r1 r2 r3
sc T t U u v w x Z
area (m2) specific heat capacity (J kg1 K1) energy (J) enthalpy (J kg 1) mass (kg) mass flow rate (kg s 1) heat flow (W) pressure (Pa) internal radius, pressure vessel (m) external radius (internal space plus metallic shell), pressure vessel (m) total external radius (internal space plus metallic shell plus insulation), pressure vessel (m) subcooling ( C) temperature ( C) time (s) internal energy (J) specific internal energy (J kg 1) specific volume (m3 kg 1) specific work (W) thickness (m) length (m)
Greek letters heat transfer coefficient (W m2 K 1) l thermal conductivity (W m 1 K1) density (kg m-3)
Subscripts 1 State at the beginning of the transient period 2 State at the end of the transient period amb!vessel ambient environment to the external surface of a vessel con condensation coil cooling coil ev evaporation g vapour insul insulation l liquid s shaft sat saturated condition p constant pressure T total Vessel!zone internal surface of a vessel to a liquid or vapour refrigerant zone
Fig. 1. Physical representation of a closed intercooler.
and spatial dependence using partial differential equations (PDEs); b) a lumped thermal modelling approach, using ordinary differential equations (ODEs). Most existing PDE and ODE based models are ‘‘thermal models’’ that ignore refrigerant hydrodynamics and mostly concentrating on properties such as thermal capacity and thermal resistance [1]. Published modelling approaches applied for vessels range between lumping the thermal capacity with a flooded evaporator to multiple zoning of the vessel, normally using zones selected according to the refrigerant physical states coexisting in the vessel. Marshall and James [2] modelled an intercooler, assuming four zones: vapour, liquid, vapour bubbles temporarily in the liquid and the subcooling coil. The ODEs for the refrigerant zones were based on energy and mass balances. James and James [3] used a similar approach, modelling a liquid receiver by separating zones at the vapour–metal, liquid–metal and vapour– liquid interfaces. Some disagreement between predicted and experimental values was found during plant startup and shutdown [4]. Cleland [5] assumed that the liquid mass present in a vessel was constant and that the liquid phase thermal capacity was significantly larger than that of the vapour phase. The vessel dynamics could then be determined from the liquid refrigerant properties and the net inflow of energy to the vessel. Lovatt [6] used this model for the REFSIM simulation package. While the overall simulation package performed well, the specific performance of the vessels model was not investigated. The apparent lack of development of pressure vessels models may be attributable to their assumed low thermal mass and relatively ‘‘static’’ behaviour. Published models generally ignored the linking elements or their thermal capacities were lumped with main equipment. Hence, research directed at modelling unsteady-state energy transfer in pressure vessels might provide a better understanding of these phenomena in the dynamic behaviour of refrigeration plants.
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The objectives of this paper were to develop new models for pressure vessels of different levels of complexity using thermal modelling approaches, and to validate the models against experimental data. The four models developed were: A) A rigorous thermodynamic approach, considering transitions between states for the liquid and vapour zones in detail, B) a less complex model following the assumptions made by Marshall and James [2] but still considering the transition between states, C) the same approach as B, but neglecting the transition between states, and D) a simple model following the assumptions of Cleland [5].
2. Physical and conceptual model Fig. 1 shows a generalised closed intercooler vessel for a two-stage refrigeration plant. The intercooler receives superheated gas from the low stage compressors. The vapour is cooled and a flow of vapour is drawn away from the head space of the intercooler by the high stage compressors. The vessel is fed via a high-pressure refrigerant liquid line. Prior to its entrance to the vessel, the liquid pressure is reduced to the intercooler pressure,
thus forming a mixture of vapour and liquid. A second liquid line may pass through a coil in the vessel. The liquid in the coil is subcooled by boiling of the refrigerant surrounding the coil, creating vapour. Refrigerant from evaporators or secondary vessels might also return to the intercooler. The vessel may be insulated and covered with a metallic vapour barrier layer, to lower the rate of radiant and convective heat transfer from the external environment. Fig. 2 presents a conceptual model of the major phenomena occurring in such a pressure vessel. The basic principles and assumptions were: 1) Vessel contents were divided into perfectly mixed liquid and vapour zones. 2) Although no external work was done by the system as a whole, a net exchange of work between zones due to liquid level change was possible. 3) Instantaneous condensation of subcooled vapour and evaporation of superheated liquid upon entering vessel, to re-establish a saturation state was assumed (although liquid and vapour zones could be subcooled and superheated respectively).
Fig. 2. Conceptual model for pressure vessels.
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4) All mixed refrigerant flows entering the vessel separated instantly, with liquid flows entering the liquid zone and vapour flows entering the vapour zone. 5) In the liquid zone, because pv is negligibly small, the internal energy u equals the enthalpy h. 6) Heat flow through vessel walls was modelled considering the insulation layer as a pure resistive material and the metallic shell as two pure capacitive layers (one in contact with liquid and one in contact with vapour) 7) No thermal capacity was attributed to coil pipes or to heating elements. 8) Liquid refrigerant density was assumed constant. 9) Kinetic and potential energy effects were neglected. Fig. 2 shows the different sources of energy and mass exchange considered: a) Flow in items entering the vessel b) Flow out items leaving the vessel, with the thermodynamic characteristics of either the liquid or vapour phase. c) Object items, interacting with the vessel contents via convective heat exchange only. d) Latent items, such as evaporation or condensation flow between phases.
0 vl X X ml;flows in ml;flows out B vg B X B B þ ðmev mcon Þ 1 vl þ mg;flows B vg B X B 1 B mg;flows out d B vg B ¼B B dt B AðZT Zl Þ B B B B B B @
1 C C C C inC C C C C C C C C C C C C C C A ð2Þ
The energy equation for the liquid zone was obtained from a full energy balance according to the 1st Law of Thermodynamics, including condensation and evaporation phenomena. The left hand side of the energy balance for a zone containing Ml kg of homogeneous liquid is: dUl dðMl ul Þ dul dMl ¼ Ml ¼ þ ul dt dt dt dt
ð3Þ
Using Eq. (1), Eq. (3) becomes: dUl dul ¼ Ml dt dt X X ml;flows in þul ml;flows
out þmcon mev
ð4Þ
3. Mathematical models Full details of model derivations are given by Estrada-Flores [7]. 3.1. Model A): two-refrigerant-zone model based on a full energy balance The main characteristics of this model were: a) the work exchange between liquid and vapour phases was included; b) the full derivative of internal h i energy respect to time was used, i.e.,
dug dt
¼
dhg dt
dðdtpvÞ ; c) the model
simulated transition between saturated and non-saturated states of liquid and vapour; and d) the full thermal capacities of the liquid and vapour phases and the metallic shell were considered. The changes in the liquid level respect to time was derived by mass balance, noting that Ml=A Zl/vl: P P ml;flows in ml;flows out mev þ mcon dZl ¼ l dt A ð1Þ A similar mass balance for the vapour zone, noting that Mg=A Zg/vg and Zg=ZTZl, yields:
dul dhl , so the full For the liquid zone, ul hl and dt dt energy balance equation is: X X dhl Ml ml;flows in þ hl ml;flows out þ mcon mev dt X X ml;flows out hl ¼ ml;flows in hl;flows in X þ qobject ws mev hg;sat þ mcon hl;sat ð5Þ The external work ws is that done on the vapour phase: X X ml;flows in ml;flows out mev þ mcon ws ¼ pg vl ð6Þ Rearranging Eq. (6) yields: X dhl X ml;flows in hl;flows in Ml ml;flows dt þ qobject ws mev hg;sat hl;sat þ mcon hl;sat hl X X ml;flows in hl ml;flows
out
out
hl
ð7Þ
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The energy balance equation for the vapour zone was derived in a similar fashion, but the dðdtpvÞ term was dðpvÞ dp dv retained. Noting that dt g ¼ vg dtg þ pg dtg , the resulting energy balance equation was: P B1 þ qobject ws B2 hg pg vg þ B3 dhg ¼ dt Mg dpg dvg þ pg þ vg ð8Þ dt dt where: X B1 ¼ mg;flows X mg;flows B2 ¼
X hg;flows in mg;flows out hg X m in g;flows out þ mev mcon in
B3 ¼ mev hg;sat mcon hl;sat
ð8aÞ ð8bÞ ð8cÞ
3.1.1. Modelling of temperatures and pressures for the liquid and vapour zones The thermodynamic property routines of Cleland [8,9] were adopted. For the liquid zone, two states were recognised:
In a similar fashion, the shell model for the vapour zone was: dTshell;vapour zone Mshell cp;shell dt qamb ! vessel;vapour zone qvessel ! vapour zone ¼ Zl 1 ZT ð12Þ The heat flow terms qamb!vessel and qvessel!zone for the two zones were calculated using the generic equation for cylindrical and slab conduction/convection combined: qamb ! vessel 2
13
0
B 6 B 6 B ¼6 D þ 1 B 6 @ 4
1 1
2r3 amb ! vessel Zzone
ln þ
r3 r2
C7 C7 C7 C7 A5
2linsul Zzone
ðTamb Tshell Þ ð13Þ
Saturated liquid, where hl=hl sat and Tl=Tl sat at pg. Subcooled liquid, where hl,sc is found from Eq. (10) and sc was calculated using: hl;sat hl sc ¼ ð9Þ cpl
r23
D1 ¼
1 amb ! vessel
þ
xinsul linsul
ð13aÞ
and:
Hence, the temperature of the liquid was: Tl ¼ Tl;sat sc
where:
ð10Þ
For the vapour zone, a methodology to calculate pg whilst knowing only hg and vg of the vapour phase was required. For saturated vapour this was straightforward, but for superheated vapours an iterative routine to find the extend of superheat, pg, vg, Tg and hg was developed [7]. 3.1.2. Modelling of energy exchange with the external environment through the vessel shell Eqs. (11) and (12) were used to calculate the change in temperature of the metal shell in contact with the liquid and vapour zones, respectively. These equations assume that changes in Mshell, shell mass in contact with each phase with respect to time as a consequence of the variation of Zl are sufficiently small that they can be ignored in the heat balance. dTshell;liquid zone Mshell cp;shell dt qamb ! vessel;liquid zone qvessel ! liquid zone ¼ Zl ZT ð11Þ
qvessel ! zone ¼ vessel ! zone 2 r1 Zzone þ r21
ðTshell Tzone Þ
ð13bÞ
3.1.3. Modelling of latent items The criteria used to define the state of the liquid and vapour zones were: a)Vapour: if ug > ug,sat at pg then the vapour was superheated and mcon=0. Otherwise, vapour was saturated and mcon50. b)Liquid: ul < ul,sat at pg then the liquid was subcooled and mev=0. Otherwise, liquid was saturated and mev50. For the liquid zone, when the transition criteria signalled a change to saturated liquid, the rate of evaporation between zones to maintain saturation was calculated using: P mev ¼
dhl ws qobject Ml dt hg;sat hl;sat
ð14Þ
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Similarly, Eq. (15) was used to calculate the rate of condensation for the vapour zone: X dhg dpg dvg qobject Mg vg pg dt dt dt dMg þpg vg þws dt ð15Þ mcon ¼ hl;sat hg;sat The use of Eqs. (14) and (15) implies the estimation of five derivative terms:
dhl dt
;
dhg dt
;
dpg dt
;
dMg dt
and
dvg dt
by means
other than the original ODEs [Eqs. (1), (2), (7) and (8)]. First estimates of these terms were made using backward difference approximations of the form: dy ytime¼t ytime¼t#t ¼ dt #t
ð16Þ
where y can be hl, hg, pg, vg or Mg. To improve these estimates, an iterative procedure fully described in Estrada-Flores [7] that gave feasible computation times and stable results was adopted. 3.2. Models B and C: Two-refrigerant-zone models based on simplified energy balances These models were derived by applying two simplifications to model A, which were: a) the work between liquid and vapour phases was neglected and b) the du dh approximation dtg ¼ dtg was used [2]. Hence, the full expression for the liquid enthalpy, Eq. (7), was simplified to: dhl X ¼ qobject mev hg;sat hl;sat Ml dt ð17Þ mcon hl;sat hl The vapour zone energy balance of Eq. (8) was simplified to: dhg dt P ¼
dMg qobject þ mev hg;sat hg mcon hl;sat hg þ pg vg dt Mg ð18Þ
In both models B and C, the calculation of the liquid level and density of vapour were by Eqs. (1) and (2). As in model A, model B recognised transition between phases and latent items were modelled. Model C neglected these, always calculating hl through Eq. (17). For both models B and C, the modelling of energy exchange with the external environment through the vessel applied as described for model A.
175
3.3. Model D: one-refrigerant-zone model based on an enthalpy balance for the liquid zone This model had the following characteristics: a) All refrigerant mass was treated as liquid; b) thermodynamic equilibrium was assumed between zones (vapour and liquid were at the same temperature); c) mass flow rates entering and leaving the vessel all interacted with the liquid zone; d) the level of liquid was constant; all the simplifying assumptions of model C also applied. Thus, only one energy balance equation for the liquid phase was needed: dul dTl Ml ¼ Ml cp;l dt dt X X ¼ mflowsin hflow sin mflowsout hflows out ð19Þ þ
X
qobject
The modelling of energy exchange with the external environment through the vessel shell comprised Eqs. (11)–(13), but only one zone containing all the metal shell was considered. 3.4. Models implementation The mathematical models were implemented in a TurboPascal programme. A numerical solution using the 4th order Runge–Kutta procedure was selected.
4. Experimental validation The models were tested by comparison of predictions to the measured time-temperature responses of refrigerant contained in two pilot plant calorimeters. These calorimeters are shown in Figs. 3 and 4, along with physical data of the systems. A detailed description of both systems is given in Estrada-Flores [7]. During the experimental trials, the calorimeter were switched on, selecting the condenser water flow rate and temperature, the evaporation temperature, heater input and suction and discharge pressures. The temperatures of the vessel contents were monitored until steady-state conditions were achieved. Once the conditions were steady, a first step change in operating conditions was carried out by changing the energy input of the heater. The system then underwent a period of transient behaviour before reaching a new steady-state condition. Several step changes were carried out in a sequence. Several experimental trials using different conditions and different step change strategies were performed. A global energy balance was adopted as an indicator of the quality of the experimental data, described by:
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ð t2 ET;2 ET;1 ¼
ðqheater þ qamb ! vessel qcoil Þdt X ¼ qnet #t t1
ð20Þ
The term ET,1ET,2 was estimated as: ET;2 ET;1 ¼ Mshell cp;shell Tshell;2 Tshell;1 þ Mcoil cp;coil Tcoil;2 Tcoil;1 þ Ml;1 ul;2 ul;1 þ Mg;1 ug;2 ug;1 þ Mg;2 Mg;1
ug;2 ul;1
Fig. 3. Schematic diagram of calorimeter 1.
ð21Þ
The term qamb!vessel was calculated through Eq. (13). The term qheater was directly controlled and known. The term qcoil was calculated using a linear change of the coil heat removal respect to time, since experimental coil temperature data showed a linear change in temperature difference across the coil. Experimental trials with energy balance errors of more than 15% were regarded as unacceptable for comparison purposes.
Fig. 4. Schematic diagram of calorimeter 2.
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5. Results and discussion Table 1 presents the conditions of the six trials that met the relative error criterion. Similar energy balances to those described for the experimental trials were performed on the predicted behaviour of all four models, as illustrated in Table 2 (trial 1). Figs. 5 and 6 illustrate the experimental and simulated liquid and vapour temperature profiles for trials 1 and 3. The energy balances of model D were apparently better than those of models A, B and C, due to the additive effect of incomplete convergence of the iterative procedures required to calculate several of the liquid and vapour thermodynamic parameters related to saturated and non-saturated states in the 3 more complex models. Model D also computed faster than models A, B and C: for the former, a typical simulation using a time step of 1 s was adequate; for the latter, a time step of 0.1 s and more than 12 iterations for vapour superheat calculations were required to ensure a good accuracy.
In Calorimeters 1 and 2, the contribution of the energy storage in the metal shell to the global energy balance was approximately 77 and 66%, respectively. The thermal mass of the liquid was next most important. The thermal mass of both coil and vapour components contributed less than 5%. The contribution of the latent heat component did not exceed 7% in trials 1 and 2, and it was less than 3% in the rest of the trials. Model D overpredicted the rate of the change of temperatures of the shell and liquid, due to the lack of latent items to account for latent heat at evaporation. However, lumping the thermal mass of the vapour with the liquid did not lead to significant disagreements when compared with models A, B and C. In all trials, the predicted temperatures showed deviations within 0.3 to 2.2 C of the experimental values for all models. The analysis of qnet showed that the heat transfer through the insulation and metal shell accounted for less than 30% of qnet in all trials except for trial 2 in which temperature profile was totally dependent on the heat
Table 1 Summary of selected experimental trials for model validation Trial
Heater input (W)a Duration (s) Temperatures in refrigeration plant
Vessel pressure (R-134a,bar) Troom ( C)
Evaporation R-22 ( C) Condensing R-22 ( C) 1 1. Calorimeter 2 I. 2. II. 3.
564 1000 577 1000 630
1. I. 2. II. 3.
1320 960 4380 2580 1760
2. 1. Calorimeter 2 I. 2.
2400 0 0b
1. I. 2.
3. 1. Calorimeter 1 I. 2. II. 3.
1700 2500 2500 3000
4. 1. Calorimeter 1 I. 2. II.
1.
4.30
27.0
2.
5.69
3.
27.0
3.
6.58
0.01 1500 50
1.
0.0
1. 55.0
1.
7.64
2.
b
2.b
2.
5.58
1. I. 2. II. 3.
1620 630 1320 460 980
1.
15.0
10.5
1.
2.43
2.
8.0
2.
3.10
3.
4.0
3.
3.98
1500 1500 3000 3000
1. I. 2. II.
2600 580 2600 3030
1.
4.0
1.
3.95
2.
17.5
2.
2.18
5. 1. Calorimeter 1 I. 2. II.
2300 2300 3000 3000
1. I. 2. II.
3090 490 3090 490
1.
4.5
1.
3.93
2.
10.0
2.
3.11
6. 1. Calorimeter 1 I. 2. II.
2300 2300 3000 3000
1. I. 2. II.
8350 1020 2580 1020
1.
37.4
2.
14.0
a b
1.
27.0
2.
55.0
10.5
11.6
6.8
Arabic numbers represent steady-state periods. The calorimeter was switched off and only ambient heat flow entered the system.
7.60 2.27
25
18
18
16
15
12
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Table 2 Energy balance in experimental calorimeter 2 and models A, B, C and D: trial 1 Energy components
System Experiment
Initial temperature (liq, vap), C Final temperature (liq, vap), C Sensible heat taken up by shell, kJ Sensible heat taken up by coil, kJ Sensible heat taken up by liquid phase, kJ Sensible heat taken up by vapour phase, kJ Latent heat from evap/condens, kJ Total 1 (ET,2ET,1), kJ Total energy input by heater, kJ Total energy removal by coil, kJ Energy infiltrated through shell, kJ Total 2 ($qnet #t), kJ % Error (internal energy balance)
11.1, 13.3 24.6, 24.9 1003.3 47.4 236.2 9.8 82.7 1379.5 7770.1 6357.3 71.6 1484.4 7.6
transfer through the insulation. The heater and coil contributions were the most important heat transfer pathways and, since the measured values were fed into the programme that implemented the models, these values were the same for experimental and simulated trials. Models A, B and C had similar internal energy balances. Differences between these models, related to modelling of phase transitions and work between phases, did not significantly affect the predictions obtained. Fig. 5 shows a temperature difference of up to 2 C between liquid and vapour phases. In this trial, 3 quasisteady-state periods occurred, during which condensation on the coil in the vapour and evaporation induced by the heater in the liquid were occurring at the same time and rate. These conditions would have led to a vapour phase close to saturation, yet the measured data suggests superheat. The two most likely explanations are incomplete mixing in the vapour zone, with the
Simulations Model A
Model B
Model C
Model D
11.1, 11.1 25.9, 25.9 1124.1 0 259.6 8.0 106.7 1498.4 7770.1 6357.3 73.6 1486.4 0.8
11.1, 11.1 25.9, 25.9 1123.9 0 259.6 8.0 106.7 1498.2 7770.1 6357.3 73.6 1486.4 0.8
11.1, 11.1 25.9, 25.9 1123.9 0 259.5 8.0 106.6 1498.2 7770.1 6357.3 73.6 1486.4 0.8
11.1, 11.1 26.6 1186.7 0 296.2 0 0 1483.0 7770.1 6357.3 70.2 1483.0 0.002
thermocouple being located in an area away from the coil, or measurement error due to thermal bridging between the thermocouple and the metallic shell. These explanations could not be checked without dissembling the plant, which was not possible. The differences between measured and experimental data must be interpreted in relation to the experimental energy balances. For example, as summarized in Table 2, in trial 1 the measured temperature data suggested that ET,1ET,2 was 7.6% less than qnet. All models overpredicted the measured temperature change by about 10% when using essentially the same qnet as measured. The six experimental energy balances were all in error by 5 to 12%. Considering this, the agreement between experimental and simulated trials was satisfactory. Overall, because all the models accounted for the effect of the dominant thermal masses (liquid and metallic shell) in the systems investigated, the good performance of all models regarding overall temperature
Fig. 5. Experimental and simulated liquid and vapour temperatures for Trial 1.
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179
Fig. 6. Experimental and simulated liquid and vapour temperatures for Trial 3.
profiles was expected. The most complex models A, B and C did not display a noticeably higher level of accuracy than the simpler model D. The operating conditions covered by the experimental work described in this paper do not encompass all situations that may be encountered industrially, particularly in respect to liquid level, deviation from saturation and extent of pressure variations. However, an industrial case study described by Estrada-Flores [7] suggests that the results described in this paper may be valid for wider ranges than those investigated experimentally.
6. Conclusions Dynamic models for pressure vessels in small systems should consider the energy storage of the metallic shell and liquid phase as the most important components in a global energy balance. For vessels operating with good liquid level control and close to saturation, the assumptions of Marshall and James [2] and Cleland [5] related to work and enthalpy terms introduced almost no errors. Hence, models B, C and D are expected to perform satisfactorily in many situations commonly found in industrial pressure vessels. The simpler one refrigerant-zone (liquid only) model D, which did not consider latent phenomena, predicted rates of temperature change that were less than 10% different from those predicted by other models. Model D also offered considerable advantages in terms of ease of computer implementation and computational cost.
Acknowledgements The experimental work was carried out at South Bank University and Star Refrigeration (UK). The authors wish to express their gratitude to Dr. John Missenden
and Dr. Forbes Pearson. The theoretical work was conducted at Massey University, whilst Silvia EstradaFlores was supported by a UNAM scholarship. This financial support is also gratefully acknowledged.
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