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Numerical simulation of a variable speed refrigeration system
International Journal of Refrigeration 24 (2001) 192±200
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Numerical simulation of a variable speed refrigeration system R.N.N. Koury a, L. Machado a, K.A.R. Ismail b,* a
Mechanical Engineering Department, UFMG, Av. AntoÃnio Carlos, 6627, CEP 31270-910, Belo Horizonte, Minas Gerais, Brazil Thermal and Fluid Engineering Department, FEM, UNICAMP, BaraÄo Geraldo, CEP 13081-970, Campinas, SaÄo Paulo, Brazil
b
Abstract This work presents two numerical models to simulate the transient and steady state behavior of a vapor compression refrigeration system. The condenser and the evaporator were divided into a number of control volumes. Time dependent partial dierential equations system was obtained from the mass, energy and momentum balances for each control volume. As the expansion valve and the compressor both have very small thermal inertia, the steady state models were applied for these components. Transient and steady state models numerical predictions were compared and good agreement was found. Further simulations were performed with the objective of verifying the possibility of controlling the refrigeration system and the superheating of the refrigerant in the evaporator outlet by varying the compressor speed and the throttling valve sectional area. The results indicate that the proposed models can be used to formulate an algorithm for controlling a refrigeration system. # 2001 Elsevier Science Ltd and IIR. All rights reserved. Keywords: Refrigerating system; Compression system; Operating; Steady state; Transient behavior; Modelling
Simulation numeÂrique d'un systeÁme frigori®que aÁ vitesse variable ReÂsume Le principal objectif de ce travail est de preÂsenter deux modeÁles eÂlaboreÂs a®n de simuler le fonctionnement en reÂgime transitoire et permanent d'un systeÁme frigori®que aÁ compression de vapeur. Le condenseur et l'eÂvaporateur ont eÂte diviseÂs en plusieurs sous-volumes. Un systeÁme d'eÂquations dieÂrentielles a eÂte obtenu aÁ partir de l'application, pour chaque sousvolume, des eÂquations de bilans eÂnergeÂtique, massique et de quantite de mouvement. Puisque le deÂtendeur et le compresseur ont une faible inertie thermique, ces composants ont eÂte modeleÂs en reÂgime permanent. Les preÂvisions des modeÁles en reÂgime transitoire et permanent ont eÂte compareÂs et on a pu montrer une bonne concordance entre ces reÂsultats. Des simulations suppleÂmentaires ont eÂte reÂaliseÂes a®n de veÂri®er la possibilite d'eectuer la reÂgulation de la puissance frigori®que et de la surchaue du frigorigeÁne aÁ la sortie de l'eÂvaporateur aÁ partir de la vitesse du compresseur et de la section de l'obturateur. Les reÂsultats obtenus indiquent que les modeÁles peuvent eÃtre utiles dans le deÂveloppement d'un algorithme de reÂgulation d'un systeÁme frigori®que. # 2001 Elsevier Science Ltd and IIR. All rights reserved. Mots cleÂs : SysteÁme frigori®que ; SysteÁme aÁ compression ; Fonctionnement ; ReÂgime permanent ; ReÂgime transitoire ; ModeÂlisation
* Corresponding author. Fax: +55-19-239-3722. E-mail address: [email protected] (K.A.R. Ismail). 0140-7007/01/$20.00 # 2001 Elsevier Science Ltd and IIR. All rights reserved. PII: S0140-7007(00)00014-1
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Nomenclature Latin symbols A section (m2) c speci®c heat (J/kg K) Cr compressor clearance ratio G mass ¯ux (kg/m2 s) h enthalpy (J/kg) K expansion device characteristic constant : m mass ¯ow rate (kg/s) N compressor speed (r.p.s.) P pressure (Pa) p perimeter (m) T temperature ( C or K) t time (s) V piston displacement volume (m3) v speci®c volume (m3/kg) x vapor quality y variable z abscissa axis (m) Greek symbols heat-transfer coecient (W/m2 K)
1. Introduction The experimental investigation of any refrigeration system is usually very complicated, mainly due to the ®nancial costs and the large number of variables involved. The use of numerical models can reduce the costs and also facilitate understanding the phenomena related to the problem. Refrigeration systems models are divided in two broad classes: steady-state models and transient models. Refrigeration systems steady-state behavior and models seem to be very well understood and widely used as a convenient tool to investigate different ®eld problems such as the replacement of the conventional refrigerants by ecological ones as well as optimization of air-conditioning systems [1±4]. Full understanding of models to simulate the dynamic operation of refrigeration systems is still lacking and many eorts are devoted to achieve more progress. These models, as the steady-state ones, are based on modular formulation in which the components of the system (evaporator, condenser, compressor and expansion valve) are separately modeled. During the transient operation the heat exchangers (evaporator and condenser) are important in the system behavior because they retain initially almost all the system refrigerant charge. Generally, to investigate the dynamic behavior of refrigeration systems, usually the heat exchangers are treated by transient models while the expansion valve as well as the compressor are considered in steady state.
l
193
void fraction property variation eciency inclination angle of heat exchangers pipes (rad) density (kg/m3)
Subscripts a secondary ¯uid c condenser cp constant pressure cv constant volume cpa speci®c heat of air cpp pipe material speci®c heat e evaporator f refrigerant ¯uid F friction l liquid p pipe wall v vapor v volumetric Superscripts o precedent instant of time
The separate models of each component constitute the whole system model, coupled by the convergence of the inlet and outlet variables in the refrigeration circuit. The main objective of this work is to present two numerical models to simulate the dynamic and steady state behavior of a vapor compression refrigeration system. The simulations were performed with the refrigerants R12 and R134a. 2. Mathematical model The refrigeration system modeled in this work was based on a prototype, which is under construction in the Mechanical Engineering Department of the Federal University of Minas Gerais-Brazil, supported ®nancially by the Foundation for the Research Support of State of Minas Gerais-FAPEMIG. The plant was designed to operate with the R134a and is composed of an open speed-controlled compressor, a manual expansion valve, an evaporator, a condenser and a subcooller. The condenser is made up of two concentric copper tubes. The refrigerant ¯ows inside the internal tube and water ¯ows in the annular space in the opposite direction. The evaporator is composed of an external shell and three internal copper tubes. The refrigerant ¯ows inside the three internal tubes and the secondary ¯uid, a blend of water and ethylene glycol, ¯ows into the external shell in the opposite direction.
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2.1. Compressor In recent years, there have been a great deal of results and investigations concerning models elaborated to simulate the behavior of open or hermetic refrigeration compressors. Some of these models are very complex and take into account the refrigerant mass ¯ow conditions through the inlet and outlet valves and the heat exchange in the various internal components of the compressor [5,6]. Other models do not take into account the mass variation in the compressor, thus considering the same mass ¯ow at the inlet and outlet [7]. This last kind of model was adopted in this work, in order to compute the mass ¯ow and enthalpy at the compressor outlet. To calculate the refrigerant mass ¯ow imposed : by the compressor, mf , the following equation was used: : mf NVf v
1
where N, V and v are the rotational speed (rps), the piston displacement volume and the volumetric eciency of the compressor, respectively, and f is the refrigerant density at the compressor inlet. The volumetric eciency was calculated by the following equation: v 1 cr ÿ cr
ccv =ccp Pc Pe
2
where cr is the compressor clearance ratio, Pc and Pe are the refrigerant condensing and evaporation pressures and ccv and ccp are the constant volume and constant pressure speci®c heats at the compressor inlet. The enthalpy at the compressor outlet was calculated at the ®rst moment considering the compression process as being isentropic. The calculated dierence between the enthalpies at inlet and outlet of the compressor (compression work) was divided by the compression eciency, supposed equal to 70% for all operation points of the system. The temperatures and enthalpies values were recalculated to represent now the actual compression work. The adopted value for the compression eciency was based on the experimental value due to Mongey et al. [8]. 2.2. Expansion valve The expansion device is one of the most important components of the refrigeration system. Its function is to reduce the pressure and to regulate the refrigerant mass ¯ow rate. The widely utilized expansion devices are the capillary tubes and the thermostatic expansion valves. The capillary tube is a small-bore tube, which acts as a restriction and reduces the pressure. The thermostatic expansion is a valve for controlling the refrigerant ¯ow by a sensor bulb placed in the evaporator
discharge line and hence control the mass rate by the degree of superheat. Various experimental studies on expansion devices, are available, as in Tamainot-Telto [9]. The majority of expansion devices models have the advantage of simplicity [10]. In fact, because of their very small volume, the expansion process can be considered as being adiabatic without any refrigerant mass variation inside it. The present study considers the constant enthalpy process and the refrigerant mass ¯ow is calculated by the following equation: p : mf K Pf
3
where K is the expansion device characteristic constant (dependent mainly on the throttling section), P is the dierence between condensing and evaporation pressures and f is the refrigerant density at the expansion device inlet. 2.3. Heat exchangers The condenser and the evaporator are the components of the refrigeration system that most in¯uence the transient thermal performance of the system. The models to simulate the behavior of evaporators and condensers can be classi®ed into four categories: black box models, one-zone models, two zone models and distributed models. Black box modeling is based on the theory of system automation and control, being utilized mainly for the evaporator [11,12]. In this kind of model the system component can be represented by a set of transfer functions with several constants which are determined through experiments (gain, time constant and retardation). So, the black box models have the advantage of simplicity but rely on experimentation to obtain the necessary data for simulation. One-zone, two-zone and distributed models are established from the physical laws of energy, mass conservation and momentum. In the one-zone models, the heat exchanger is analyzed from the variables which represent the component in the general form [13,14]. In these models, only one value of refrigerant enthalpy is utilized to express the energy contained in the heat exchanger. In the same way, one value is utilized for the refrigerant density, secondary ¯uid temperatures, etc. The heat-transfer coecients between the refrigerant, the secondary ¯uid and the tube wall are supposed constant in the entire heat exchanger. Two-zone models are the most utilized [15,16]. The evaporator is divided into two zones (evaporation and superheating zones) and the condenser is subdivided into three zones (superheating, condensation and subcooling zones). In this case the equations of continuity, energy and momentum can be established with higher precision and considering the use of one heat-transfer
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coecient for each zone, the physical problem can be easily represented. The distributed models were used extensively during the last years [17±21]. In these models, the heat exchanger is divided into various small control volumes. The application of the mass, energy and momentum balances in each one of these control volumes, together with the use of local heat-transfer coecients allows to obtain better agreement with the physical situation, when compared with the two-zone models. The choice of the model to be utilized, must be made mainly in terms of the heat exchangers geometry. Therefore, for shell and tube evaporators or condensers, with the evaporation process occurring on the outside surface of the tube, the one-zone or two-zone models can be used to represent it adequately. However, for concentric tubes evaporators and condensers, because of the complexity in the twophase ¯ow zone, a distributed model is required. Thus, the distributed model was adopted in this work. The evaporator and condenser models are presented together because they have common characteristics. The following simpli®cations are considered: (i) the physical properties related to refrigerant, secondary ¯uid and pipe wall are considered uniform in the heat exchanger transversal section, (ii) the refrigerant liquid and vapor phases are in thermodynamic equilibrium, (iii) the heat exchangers have a perfect thermal insulation, (iv) the axial heat conduction in the pipes is not taken into account (v) the refrigerant and secondary ¯uid potential energy variations are also not considered. The model is established by applying for each heat exchanger control volume, the energy conservation equation (refrigerant, secondary ¯uid and pipe wall), as well the mass and momentum conservation equations (refrigerant). This procedure generated the following set of dierential equations. . Refrigerant ¯uid (energy, mass and momentum conservation) Af
ÿ @ @ f hf ÿ Pf vf ÿAf Gf hf f pf Tp ÿ Tf @t @z 4
@f @Gf 0 @t @z 2 @ x vv 1 ÿ x2 vl Pf G2f @z l 1ÿl @Gf dP ÿ gf sin ÿ dz F @t
5
6
. Secondary ¯uid and pipe wall (energy conservation) p Ap cpp
ÿ ÿ @Tp e pe Ta ÿ Tp ÿ f npf Tp ÿ Tf @t
7
a Aa cpa
ÿ @Ta @Ta ÿGa Aa cpa ÿ a pa Ta ÿ Tp @t @z
195
8
The subscripts f, p and a are related, respectively, to the refrigerant, pipe wall and secondary ¯uid. A, G, h, T, P, x, v, l, g, , e, are, respectively, the transversal section, the mass ¯ux, the entalphy, the temperature, the pressure, the vapor quality in the two-phase ¯ow, the speci®c volume, the void fraction, the gravity acceleration, the density and the pipe inclination related to the horizontal line. The character when indexed ( f and a ) represents the heat-transfer coecients between the refrigerant and secondary ¯uids with the pipe walls. The refrigerant density was computed by the following relationship, where l and v are the liquid and vapor densities, respectively. l l v ÿ l
9
The correlations utilized in this work to calculate the convective heat-transfer coecients, void fraction and pressure drop due to friction were obtained experimentally from the technical literature available. A detailed investigation of the principles of mass, momentum and heat-transfer in a refrigeration system was performed by Machado [22]. In the present work, the correlation proposed by Dengler and Addoms [23] was adopted to compute the evaporation heat-transfer coecient and the correlation proposed by Shah [24] was used to compute the condensation heat-transfer coecient. To calculate the heat transfer coecient in the one-phase zone (refrigerant and secondary ¯uid) the Dittus±BoÈelter correlation was adopted. The void fraction was computed by the Hughmark correlation. The pressure drop along the tubes for the case of two-phase ¯ow was calculated by the Lockhart±Martinelli correlation while the Fanning correlation was adopted for the single phase ¯ow. Because of the fact that the refrigerant and water ¯ow in opposite directions, the simultaneous solution of the ®ve equations (4)±(8) is very complicated. Hence, Eqs. (4)±(6) were solved separately by the fourth order Runge±Kutta method, along the z axis, while the temperature pro®le was determined from Eqs. (7) and (8) by the ®nite dierence implicit method. This sequence was repeated until achieving convergence. The time-dependent derivatives in the refrigerant equations were determined from the following expression: @y y ÿ yo @t t
10
where y represents the variables related to the refrigerant in the present time (pressure, enthalpy, density and mass ¯ux) and t is the time step. The superscript ``o'' refers to the values of these variables at the instant t ÿ t.
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In the evaporator and condenser models, the enthalpies and mass ¯ux values at inlet and outlet, were determined from the models of the expansion device and compressor. Starting from arbitrary values of evaporation and condensation pressures and utilizing the initial conditions of the problem, the spatial pro®les of the refrigerant enthalpy, mass ¯ux and density can be calculated at each point of the heat exchanger. The transition between the single-phase zone to the two-phase-zone and vice versa are determined by computing the refrigerant vapor quality along the entire length of the heat exchanger. At the outlet of the evaporator and condenser, the refrigerant mass ¯ux must be equal to the assumed value obtained from the compressor and expansion device models. If convergence is not achieved, a new arbitrary value for the pressure is determined from Newton±Raphson modi®ed method. This sequence is repeated, until the mass ¯ux convergence is achieved. 2.4. Refrigeration system 2.4.1. Transient model The condenser model is initiated after establishing the geometrical parameters of the refrigeration system, the initial conditions, arbitrary values for the evaporation and condensing pressures as well as the degree of superheat at the condenser inlet. The outlet values of the condenser model are utilized to initiate the evaporator model. Then, the evaporation pressure and degree of superheat calculated by the evaporator model are compared with the corresponding values utilized as input data in the condenser model. This calculation procedure is repeated until achieving convergence. The ®nal values obtained are then utilized for the calculations during the subsequent time instant. 2.4.2. Steady-state model Using an in®nite time step (107s) in Eqs. (4)±(8) and modifying the criterion of convergence of the transient model, the steady-state model can be obtained. The values adopted for the evaporation pressure and for the degree of superheat at the evaporator outlet, are used to initiate the condenser model as in the case of the transient model. However, the evaporation pressure computed by the evaporator model is now determined from converging the degree of superheat at the evaporator outlet. The value utilized as input in the condenser model is then compared with the calculated value from the evaporator model. The refrigerant mass utilized as input is now compared with the value calculated by the model. If convergence is not achieved, a new value of the degree of superheat is determined from the Newton±Rhapson method. As in the transient model, an iterative procedure is needed to ensure that the values of the evaporation pressure and the degree of superheat calculated by the evaporator model, used as input to the condenser model, are in agreement.
3. Discussion of results The models presented in the preceding section were used to simulate the behavior of the system for two modes of operation. In the ®rst mode, some numerical simulations were realized to determine the performance of the system at the start-up of the compressor while all the system is in equilibrium. In the second mode, we investigated the response of the system due to a step variation in the rotational speed of the compressor and a step variation of the sectional area expansion valve section while the system is under steady-state conditions. 3.1. Compressor start-up The start-up operation of a refrigeration system implies heavy demands from all the components and controls due to the large temperatures and pressures variations and hence the large time and space derivatives of the governing parameters. This requires a correct choice of the time step and space grid. Initial numerical test realized to optimize the computational grid showed that 15 s was a suitable time increment and that 400 control volumes divided between the evaporator and the condenser allowed for 2% relative error in the pressure and mass ¯ow of the refrigerant and 0.1 C in the refrigerant temperature. Figs. 1 and 2 present simulations of the start-up mode of the compressor with R12 as a refrigerant ¯uid. Fig. 1 presents the timewise evolution of the mass ¯ow of the refrigerant imposed by the compressor and the expansion valve as well as the mass of the refrigerant contained in the condenser and evaporator. Fig. 2 presents the time-wise evolutions of the condensation and boiling temperatures,
Fig. 1. Dynamic behavior of mass ¯ow rate and refrigerant (R12) charge system after start-up. Fig. 1. Comportement dynamique du ¯ux massique et de la charge en frigorigeÁne (R12) suite au deÂmarrage.
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Fig. 2. Dynamic behavior of refrigerant (R12) temperatures after start-up. Fig. 2. Comportement dynamique des tempeÂratures du frigorigeÁne (R12) suite au deÂmarrage.
the degree of superheat at the evaporator exit and the degree of subcooling at the condenser exit. The steadystate values are indicated in both Figs. 1 and 2 by the symbol (*). In Fig. 1 we can observe that after the start-up the compressor draws out the refrigerant from the evaporator and delivers it to the condenser and hence the refrigerant mass decreases in the evaporator and increases in the condenser as shown in Fig. 1. As a result the pressure and temperature values of the refrigerant in the evaporator decrease while they increase in the condenser. The mass ¯ow rate through the expansion valve depends on the pressure dierence between the condensation and boiling processes as in Eq. (3) whose value varies from zero to a maximum value corresponding to the steady-state operation and hence its variation follows what is depicted in Fig. 1. The analysis of the variation of the mass rate in the compressor is more complex because it depends directly on the compressor speed, the ratio of the boiling to the condensing pressures as well as the density of the refrigerant at the evaporator exit as in Eqs. (1) and (2). Soon after the compressor start-up, the refrigerant mass ¯ow rate at the compressor inlet increases due to the increase of the compressor speed. The subsequent decrease of the mass ¯ow is due to the increase of the pressure rate, and also to the decrease of the two phase ¯ow density at the compressor inlet.
197
The control by varying the rotational speed of the compressor improves the general performance and adequates the rotational speed to the required refrigeration capacity. Studies show that controlling the power by varying the rotational speed leads to a strong variation of the refrigerant superheat at the evaporator exit. As the thermostatic expansion valves have slow response, electronic valves are more adequate in this case, [12]. With the objective of developing a control algorithm for the power and the degree of superheat of a variable speed refrigeration machine, operation simulations were performed varying the rotational speed and the refrigerant ¯ow area of the expansion valve. These simulations represent a practical situation where the temperature of the secondary ¯uid in the evaporator circuit is high. The increase of the mass ¯ow rate and the refrigeration capacity by varying the rotational speed of the compressor causes a reduction of the temperature of the secondary ¯ow. This action causes an increase in the degree of superheat of the refrigeration ¯uid at the evaporator exit and reduces the COP of the system. The increase of the ¯ow sectional area of the expansion valve leads to decrease the degree of superheat and consequently improves the performance of the system. The refrigeration ¯uid used in these simulations is R134a. Figs. 3 and 4 present the behavior of the system due to step variation of the rotational speed from 1500 to 1800 rpm. Fig. 3 shows the time-wise variation of the mass ¯ow rate of the refrigerant ¯uid imposed by the compressor and the expansion valve as well as the degree of superheat at the evaporator exit. Fig. 4 presents the time-wise variation of the power consumed by the compressor which is essentially dependent on the mass ¯ow rate of the refrigerant. Consequently, the observed
3.2. Step variation of compressor speed The power control of small and medium refrigeration machines is usually done by the classic on±o system. This control system usually impairs the performance of the machine, reduces the useful life of the compressor and components and increases the power consumption.
Fig. 3. Dynamic behavior of the mass ¯ow rate and superheat in response to a step increase of the compressor rpm from 1500 to 1800. Fig. 3. Comportement dynamique du ¯ux massique et de la surchaue suite aÁ une augmentation de la vitesse du compresseur (de 1500 aÁ 1800 tours par minute).
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3.3. Step variation of the throttling sectional area of the expansion valve
Fig. 4. Dynamic behavior of the power system in response to a step increase of the compressor rpm from 1500 to 1800. Fig. 4. Comportement dynamique du systeÁme d'alimentation en eÂnergie suite aÁ une augmentation de la vitesse du compresseur (de 1500 aÁ 1800 tours par minute).
increase in the refrigeration capacity is proportional to the increase in the mass ¯ow rate of the refrigerant (both the refrigeration capacity and the mass ¯ow rate increased by about 7%). On the other hand, the compressor power consumption is directly dependent on the mass ¯ow rate of the refrigerant and inversely proportional to its density at the evaporator exit. Knowing that the density increases with the increase of the degree of superheat and with the reduction of the boiling pressure, the compressor power consumption increases much more than the refrigeration capacity (about 27%), and consequently the COP of the system is decreased. Fig. 5 shows a comparison between the present model and the experimental results due to Wang [25]. When the compressor work with R12, its rotational speed was varied from 712 rpm to 1463 rpm. We can observe a fair agreement between the present predictions and Wang's experimental results.
As we observed before, the increase of the refrigeration capacity due to the increase of the compressor rotational speed leads to an increase of the degree of superheat and consequently the reduction of the COP of the machine. Hence, it is necessary that the expansion valve corrects this action by increasing the throttling area. Figs. 6±10 present the behavior of the system due to 10% increase in the throttling area of the expansion valve. Fig. 6 shows, immediately after the changing of the expansion valve sectional area, the mass ¯ow rate of the refrigerant increased at both inlet and exit of the evaporator and a consequent reduction of the degree of superheat due to the increase of the mass ¯ow rate in the evaporator. Also as a consequence of changing the mass ¯ow rate, one can observe a reduction of the condensation pressure, an increase of the boiling pressure and ®nally the reduction of the dierence between the two
Fig. 6. Dynamic behavior of the mass ¯ow rate and the superheat in response to a step variation of the compressor rpm and of the throttling section area. Fig. 6. Comportement dynamique du ¯ux massique et de la surchaue suite aÁ une variation de vitesse du compresseur et de la section du deÂtendeur.
Fig. 5. Comparison between theoretical results and experimental data of Wang [25] for the COP behavior in response to a step increase of the compressor rpm 712 to 1463. Fig. 5. Comparaison des reÂsultats theÂoriques et expeÂrimentaux obtenus par Wang [25] pour le COP en fonction d' une augmentation de la vitesse du compresseur (de 712 aÁ 1463 tours par minute).
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199
Fig. 7. Dynamic behavior of the system power in response to a step variation of the compressor rpm and the throttling area. Fig. 7. Comportement dynamique du systeÁme d'alimentation en eÂnergie suite aÁ une variation de vitesse du compresseur et de la section du deÂtendeur. Fig. 9. Response of the refrigerant temperature in the evaporator to a step variation of the compressor rpm and of the throttling section area. Fig. 9. TempeÂrature du frigorigeÁne dans l'eÂvaporateur suite aÁ une variation de vitesse du compresseur et de la section du deÂtendeur.
Fig. 8. Dynamic behavior of the system performance in response to a step variation of the compressor rpm and the throttling area. Fig. 8. Comportement dynamique de la performance du systeÁme suite aÁ une variation de vitesse du compresseur et de la section du deÂtendeur.
pressures and their ratio. The mass ¯ow rate peak and the subsequent stabilization can be attributed to the reduction of the pressure dierence between boiling and condensation. As a consequence of the increase of the mass ¯ow rate, the compressor power and refrigeration capacity increased as shown in Fig. 7. Irrespective of the increase of the mass ¯ow rate, the change of the enthalpy of the refrigeration ¯uid is reduced and this leads to nearly constant power consumption as indicated by Fig. 7. The reduction in the enthalpy is due to a reduction in the condensing pressure and an increase in the evaporation pressure. Fig. 8 shows the eect of the increasing of the throttling area on the COP of the refrigeration system. The increase in the COP can be attributed to the increase of the refrigeration capacity while the compressor power consumption is constant. Naturally, the variation of the
Fig. 10. Response of the refrigerant temperature in the condenser to a step variation of the compressor rpm and of the throttling section area. Fig. 10. TempeÂrature du frigorigeÁne dans le condenseur suite aÁ une variation de vitesse du compresseur et de la section du deÂtendeur.
compressor rotational speed and the expansion valve throttling area must be repeated as frequently as necessary until the refrigeration system delivers the required refrigeration capacity associated with the adequate degree of superheat, ensuring good performance of the system. Figs. 9 and 10 show the temperatures distribution in the evaporator and the condenser indicating the same behavior as mentioned before.
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4. Conclusions In this study, two models were presented to simulate the steady-state and transient behavior of a refrigeration system of the water±water type. The ®rst simulation realized corresponds to the start-up of the machine initially at thermal equilibrium. Comparison between the terminal values of the transient model seems to agree with the predictions from the steady-state model. Control of the refrigeration capacity by varying the compressor rotational speed was also investigated and it was found that this control leads to increasing the degree of superheat and hence impair the COP of the system. An alternative method is to increase the throttling area to alleviate the increase of the degree of superheat and improve the COP of the system. This model can be used to elaborate an algorithm to control the refrigeration capacity and the degree of superheat of the system. Acknowledgements
[9]
[10] [11] [12]
[13]
[14]
The authors wish to thank the CNPq and Fapesp for the ®nancial support and research scholarships to the authors.
[15]
References
[17]
[1] Domanski PA, Mclinden MO. A simpli®ed cycle simulation model for the performance rating of refrigerants and refrigerant mixtures. Rev Int Froid 1992;15:81±8. [2] Pelletier O, BjoÈrn P. Performance of plate heat exchangers and compressor in a domestic heat pump using propane. In: Proceedings of Applications for Natural Refrigerants, Denmark, 1996. p. 11.16±20. [3] Rogstam HJ. Theoretical and experimental investigation of cyclopropane as refrigerant. Proceedings of Applications for Natural Refrigerants, Denmark, 1996. p. 10.15± 19. [4] Lemos MJS, Zaparoli EL. Steady-state numerical solution of vapor compression refrigeration units. In: Int. Refrigeration Conference at Purdue, Purdue University, 1996. 235±40. [5] Yasuda H, Touber S, Machielsen CHM. Simulation model of a vapor compression refrigeration system. ASHRAE Trans 1983;98(2a):408±25. [6] Parise JAR. Theoretical and experimental analysis of a diesel driven heat pump. PhD thesis, University of Manchester, UK, 1983. p. 325. [7] Conde MR. A steady state mathematical simulation model for air-to-water heat pumps (Report NEFF). Zurich: Swiss Federal Institute of Technology, 1985, p. 236. [8] Mongey B, Mcmullan JT, Mcnerlin MG. An examination of hydrocarbon mixtures for use in high temperature heat
[16]
[18] [19] [20] [21] [22] [23] [24] [25]
pump applications. In: Proceedings of Applications for Natural Refrigerants, Denmark, 1996. p. 11.11±15. Tamainot-Telto Z. DeÂtendeurs thermostatiques de machines frigori®ques. Comportement dynamique et modeÂlisation. TheÁse de doctorat, INSA, Lyon, 1993. p. 150. Votsis PP, Tassou SA, Wilson DR, Marquand CJ. Dynamic characteristics of an air-to-water heat-pump system. Int J Refrig 1992;15(2):89±94. Wenxue H, Kraft H. A mathematical model of an evaporator based on the step exciting method. In: XVIIIth Int. Cong. Refrigeration, Montreal, 1991. p. 1217±31. Outtagarts A, Haberschill P, Lallemand M. Comportement dynamique d'un eÂvaporateur de machine frigori®que soumis aÁ des variations de deÂbit. In: 19th Int. Cong. Refrigeration, vol. IIIa, La Haye, Netherlands, August 1995, p. 413±20. Darrow JB, Lovatt SJ, Cleland AC. Assessment of a simple mathematical model for predicting the transient behavior of a refrigeration system. In: XVIIIth Int. Cong. Refrigeration, Montreal, 1991, p. 1189±92. Murphy WE, Goldschmidt VW. Cycling characteristics of a residential air conditioner. Modeling of shutdown transients. ASHRAE Trans 1986;92(1a):186±201. Welsby P, Devotta S, Diggory PJ. Steady- and dynamicstate simulations of heat-pumps. Part I: literature review. Applied Energy 1988;31:189±203. Salim M, Sadasivam M, Balakrishnan AR. Transient analysis of heat pump assisted ditillation systems I. The heat pump. Int J Energy Research 1991;15:123±35. Macarthur JW, Grald EW. Unsteady compressible twophase ¯ow model for predicting cyclic heat pump performance and a comparison with experimental data. Int J Refrig 1989;12:29±41. Wang H, Touber S. Distributed and non-steady-state modelling of air cooler. International Journal of Refrigeration 1991;1:98±110. Nyers J, Stoyan G. A dynamical model adequate for controlling the evaporator of a heat pump. Int J Refrig 1994;17(2):101±8. Jia X, Tso CP, Chia PK, Jolly P. A distributed model for prediction of the transient response of an evaporator. Int J Refrig 1995;18(5):336±42. Ferreira CI, Kalkman H. Transient behavior of economizers for multi-stage refrigeration plant. In: Int. Refrigeration Conference at Purdue, Purdue University, 1996. p. 241±46. Machado L. Modele de simulation et etude experimentale d'un evaporateur de machine frigori®que en regime transitoire. These de Docteur, INSA/LYON, 1996. p. 160. Dengler C, Addoms JN. Heat transfer mechanism for vaporization of water in a vertical tube. Chem Eng Prog Symp Series 1956;52(18):95±103. Shah MM. Chart correlation for saturated boiling heat tranfer: equations and further study. ASHRAE Trans 1982;88:185±96. Wang H. Modelling of a refrigerating system coupled with a refrigerated room. TheÁse de doctorat, Delf University of Technology, Netherlands, 1991. p. 230.
www.elsevier.com/locate/ijrefrig
Numerical simulation of a variable speed refrigeration system R.N.N. Koury a, L. Machado a, K.A.R. Ismail b,* a
Mechanical Engineering Department, UFMG, Av. AntoÃnio Carlos, 6627, CEP 31270-910, Belo Horizonte, Minas Gerais, Brazil Thermal and Fluid Engineering Department, FEM, UNICAMP, BaraÄo Geraldo, CEP 13081-970, Campinas, SaÄo Paulo, Brazil
b
Abstract This work presents two numerical models to simulate the transient and steady state behavior of a vapor compression refrigeration system. The condenser and the evaporator were divided into a number of control volumes. Time dependent partial dierential equations system was obtained from the mass, energy and momentum balances for each control volume. As the expansion valve and the compressor both have very small thermal inertia, the steady state models were applied for these components. Transient and steady state models numerical predictions were compared and good agreement was found. Further simulations were performed with the objective of verifying the possibility of controlling the refrigeration system and the superheating of the refrigerant in the evaporator outlet by varying the compressor speed and the throttling valve sectional area. The results indicate that the proposed models can be used to formulate an algorithm for controlling a refrigeration system. # 2001 Elsevier Science Ltd and IIR. All rights reserved. Keywords: Refrigerating system; Compression system; Operating; Steady state; Transient behavior; Modelling
Simulation numeÂrique d'un systeÁme frigori®que aÁ vitesse variable ReÂsume Le principal objectif de ce travail est de preÂsenter deux modeÁles eÂlaboreÂs a®n de simuler le fonctionnement en reÂgime transitoire et permanent d'un systeÁme frigori®que aÁ compression de vapeur. Le condenseur et l'eÂvaporateur ont eÂte diviseÂs en plusieurs sous-volumes. Un systeÁme d'eÂquations dieÂrentielles a eÂte obtenu aÁ partir de l'application, pour chaque sousvolume, des eÂquations de bilans eÂnergeÂtique, massique et de quantite de mouvement. Puisque le deÂtendeur et le compresseur ont une faible inertie thermique, ces composants ont eÂte modeleÂs en reÂgime permanent. Les preÂvisions des modeÁles en reÂgime transitoire et permanent ont eÂte compareÂs et on a pu montrer une bonne concordance entre ces reÂsultats. Des simulations suppleÂmentaires ont eÂte reÂaliseÂes a®n de veÂri®er la possibilite d'eectuer la reÂgulation de la puissance frigori®que et de la surchaue du frigorigeÁne aÁ la sortie de l'eÂvaporateur aÁ partir de la vitesse du compresseur et de la section de l'obturateur. Les reÂsultats obtenus indiquent que les modeÁles peuvent eÃtre utiles dans le deÂveloppement d'un algorithme de reÂgulation d'un systeÁme frigori®que. # 2001 Elsevier Science Ltd and IIR. All rights reserved. Mots cleÂs : SysteÁme frigori®que ; SysteÁme aÁ compression ; Fonctionnement ; ReÂgime permanent ; ReÂgime transitoire ; ModeÂlisation
* Corresponding author. Fax: +55-19-239-3722. E-mail address: [email protected] (K.A.R. Ismail). 0140-7007/01/$20.00 # 2001 Elsevier Science Ltd and IIR. All rights reserved. PII: S0140-7007(00)00014-1
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Nomenclature Latin symbols A section (m2) c speci®c heat (J/kg K) Cr compressor clearance ratio G mass ¯ux (kg/m2 s) h enthalpy (J/kg) K expansion device characteristic constant : m mass ¯ow rate (kg/s) N compressor speed (r.p.s.) P pressure (Pa) p perimeter (m) T temperature ( C or K) t time (s) V piston displacement volume (m3) v speci®c volume (m3/kg) x vapor quality y variable z abscissa axis (m) Greek symbols heat-transfer coecient (W/m2 K)
1. Introduction The experimental investigation of any refrigeration system is usually very complicated, mainly due to the ®nancial costs and the large number of variables involved. The use of numerical models can reduce the costs and also facilitate understanding the phenomena related to the problem. Refrigeration systems models are divided in two broad classes: steady-state models and transient models. Refrigeration systems steady-state behavior and models seem to be very well understood and widely used as a convenient tool to investigate different ®eld problems such as the replacement of the conventional refrigerants by ecological ones as well as optimization of air-conditioning systems [1±4]. Full understanding of models to simulate the dynamic operation of refrigeration systems is still lacking and many eorts are devoted to achieve more progress. These models, as the steady-state ones, are based on modular formulation in which the components of the system (evaporator, condenser, compressor and expansion valve) are separately modeled. During the transient operation the heat exchangers (evaporator and condenser) are important in the system behavior because they retain initially almost all the system refrigerant charge. Generally, to investigate the dynamic behavior of refrigeration systems, usually the heat exchangers are treated by transient models while the expansion valve as well as the compressor are considered in steady state.
l
193
void fraction property variation eciency inclination angle of heat exchangers pipes (rad) density (kg/m3)
Subscripts a secondary ¯uid c condenser cp constant pressure cv constant volume cpa speci®c heat of air cpp pipe material speci®c heat e evaporator f refrigerant ¯uid F friction l liquid p pipe wall v vapor v volumetric Superscripts o precedent instant of time
The separate models of each component constitute the whole system model, coupled by the convergence of the inlet and outlet variables in the refrigeration circuit. The main objective of this work is to present two numerical models to simulate the dynamic and steady state behavior of a vapor compression refrigeration system. The simulations were performed with the refrigerants R12 and R134a. 2. Mathematical model The refrigeration system modeled in this work was based on a prototype, which is under construction in the Mechanical Engineering Department of the Federal University of Minas Gerais-Brazil, supported ®nancially by the Foundation for the Research Support of State of Minas Gerais-FAPEMIG. The plant was designed to operate with the R134a and is composed of an open speed-controlled compressor, a manual expansion valve, an evaporator, a condenser and a subcooller. The condenser is made up of two concentric copper tubes. The refrigerant ¯ows inside the internal tube and water ¯ows in the annular space in the opposite direction. The evaporator is composed of an external shell and three internal copper tubes. The refrigerant ¯ows inside the three internal tubes and the secondary ¯uid, a blend of water and ethylene glycol, ¯ows into the external shell in the opposite direction.
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2.1. Compressor In recent years, there have been a great deal of results and investigations concerning models elaborated to simulate the behavior of open or hermetic refrigeration compressors. Some of these models are very complex and take into account the refrigerant mass ¯ow conditions through the inlet and outlet valves and the heat exchange in the various internal components of the compressor [5,6]. Other models do not take into account the mass variation in the compressor, thus considering the same mass ¯ow at the inlet and outlet [7]. This last kind of model was adopted in this work, in order to compute the mass ¯ow and enthalpy at the compressor outlet. To calculate the refrigerant mass ¯ow imposed : by the compressor, mf , the following equation was used: : mf NVf v
1
where N, V and v are the rotational speed (rps), the piston displacement volume and the volumetric eciency of the compressor, respectively, and f is the refrigerant density at the compressor inlet. The volumetric eciency was calculated by the following equation: v 1 cr ÿ cr
ccv =ccp Pc Pe
2
where cr is the compressor clearance ratio, Pc and Pe are the refrigerant condensing and evaporation pressures and ccv and ccp are the constant volume and constant pressure speci®c heats at the compressor inlet. The enthalpy at the compressor outlet was calculated at the ®rst moment considering the compression process as being isentropic. The calculated dierence between the enthalpies at inlet and outlet of the compressor (compression work) was divided by the compression eciency, supposed equal to 70% for all operation points of the system. The temperatures and enthalpies values were recalculated to represent now the actual compression work. The adopted value for the compression eciency was based on the experimental value due to Mongey et al. [8]. 2.2. Expansion valve The expansion device is one of the most important components of the refrigeration system. Its function is to reduce the pressure and to regulate the refrigerant mass ¯ow rate. The widely utilized expansion devices are the capillary tubes and the thermostatic expansion valves. The capillary tube is a small-bore tube, which acts as a restriction and reduces the pressure. The thermostatic expansion is a valve for controlling the refrigerant ¯ow by a sensor bulb placed in the evaporator
discharge line and hence control the mass rate by the degree of superheat. Various experimental studies on expansion devices, are available, as in Tamainot-Telto [9]. The majority of expansion devices models have the advantage of simplicity [10]. In fact, because of their very small volume, the expansion process can be considered as being adiabatic without any refrigerant mass variation inside it. The present study considers the constant enthalpy process and the refrigerant mass ¯ow is calculated by the following equation: p : mf K Pf
3
where K is the expansion device characteristic constant (dependent mainly on the throttling section), P is the dierence between condensing and evaporation pressures and f is the refrigerant density at the expansion device inlet. 2.3. Heat exchangers The condenser and the evaporator are the components of the refrigeration system that most in¯uence the transient thermal performance of the system. The models to simulate the behavior of evaporators and condensers can be classi®ed into four categories: black box models, one-zone models, two zone models and distributed models. Black box modeling is based on the theory of system automation and control, being utilized mainly for the evaporator [11,12]. In this kind of model the system component can be represented by a set of transfer functions with several constants which are determined through experiments (gain, time constant and retardation). So, the black box models have the advantage of simplicity but rely on experimentation to obtain the necessary data for simulation. One-zone, two-zone and distributed models are established from the physical laws of energy, mass conservation and momentum. In the one-zone models, the heat exchanger is analyzed from the variables which represent the component in the general form [13,14]. In these models, only one value of refrigerant enthalpy is utilized to express the energy contained in the heat exchanger. In the same way, one value is utilized for the refrigerant density, secondary ¯uid temperatures, etc. The heat-transfer coecients between the refrigerant, the secondary ¯uid and the tube wall are supposed constant in the entire heat exchanger. Two-zone models are the most utilized [15,16]. The evaporator is divided into two zones (evaporation and superheating zones) and the condenser is subdivided into three zones (superheating, condensation and subcooling zones). In this case the equations of continuity, energy and momentum can be established with higher precision and considering the use of one heat-transfer
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coecient for each zone, the physical problem can be easily represented. The distributed models were used extensively during the last years [17±21]. In these models, the heat exchanger is divided into various small control volumes. The application of the mass, energy and momentum balances in each one of these control volumes, together with the use of local heat-transfer coecients allows to obtain better agreement with the physical situation, when compared with the two-zone models. The choice of the model to be utilized, must be made mainly in terms of the heat exchangers geometry. Therefore, for shell and tube evaporators or condensers, with the evaporation process occurring on the outside surface of the tube, the one-zone or two-zone models can be used to represent it adequately. However, for concentric tubes evaporators and condensers, because of the complexity in the twophase ¯ow zone, a distributed model is required. Thus, the distributed model was adopted in this work. The evaporator and condenser models are presented together because they have common characteristics. The following simpli®cations are considered: (i) the physical properties related to refrigerant, secondary ¯uid and pipe wall are considered uniform in the heat exchanger transversal section, (ii) the refrigerant liquid and vapor phases are in thermodynamic equilibrium, (iii) the heat exchangers have a perfect thermal insulation, (iv) the axial heat conduction in the pipes is not taken into account (v) the refrigerant and secondary ¯uid potential energy variations are also not considered. The model is established by applying for each heat exchanger control volume, the energy conservation equation (refrigerant, secondary ¯uid and pipe wall), as well the mass and momentum conservation equations (refrigerant). This procedure generated the following set of dierential equations. . Refrigerant ¯uid (energy, mass and momentum conservation) Af
ÿ @ @ f hf ÿ Pf vf ÿAf Gf hf f pf Tp ÿ Tf @t @z 4
@f @Gf 0 @t @z 2 @ x vv 1 ÿ x2 vl Pf G2f @z l 1ÿl @Gf dP ÿ gf sin ÿ dz F @t
5
6
. Secondary ¯uid and pipe wall (energy conservation) p Ap cpp
ÿ ÿ @Tp e pe Ta ÿ Tp ÿ f npf Tp ÿ Tf @t
7
a Aa cpa
ÿ @Ta @Ta ÿGa Aa cpa ÿ a pa Ta ÿ Tp @t @z
195
8
The subscripts f, p and a are related, respectively, to the refrigerant, pipe wall and secondary ¯uid. A, G, h, T, P, x, v, l, g, , e, are, respectively, the transversal section, the mass ¯ux, the entalphy, the temperature, the pressure, the vapor quality in the two-phase ¯ow, the speci®c volume, the void fraction, the gravity acceleration, the density and the pipe inclination related to the horizontal line. The character when indexed ( f and a ) represents the heat-transfer coecients between the refrigerant and secondary ¯uids with the pipe walls. The refrigerant density was computed by the following relationship, where l and v are the liquid and vapor densities, respectively. l l v ÿ l
9
The correlations utilized in this work to calculate the convective heat-transfer coecients, void fraction and pressure drop due to friction were obtained experimentally from the technical literature available. A detailed investigation of the principles of mass, momentum and heat-transfer in a refrigeration system was performed by Machado [22]. In the present work, the correlation proposed by Dengler and Addoms [23] was adopted to compute the evaporation heat-transfer coecient and the correlation proposed by Shah [24] was used to compute the condensation heat-transfer coecient. To calculate the heat transfer coecient in the one-phase zone (refrigerant and secondary ¯uid) the Dittus±BoÈelter correlation was adopted. The void fraction was computed by the Hughmark correlation. The pressure drop along the tubes for the case of two-phase ¯ow was calculated by the Lockhart±Martinelli correlation while the Fanning correlation was adopted for the single phase ¯ow. Because of the fact that the refrigerant and water ¯ow in opposite directions, the simultaneous solution of the ®ve equations (4)±(8) is very complicated. Hence, Eqs. (4)±(6) were solved separately by the fourth order Runge±Kutta method, along the z axis, while the temperature pro®le was determined from Eqs. (7) and (8) by the ®nite dierence implicit method. This sequence was repeated until achieving convergence. The time-dependent derivatives in the refrigerant equations were determined from the following expression: @y y ÿ yo @t t
10
where y represents the variables related to the refrigerant in the present time (pressure, enthalpy, density and mass ¯ux) and t is the time step. The superscript ``o'' refers to the values of these variables at the instant t ÿ t.
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In the evaporator and condenser models, the enthalpies and mass ¯ux values at inlet and outlet, were determined from the models of the expansion device and compressor. Starting from arbitrary values of evaporation and condensation pressures and utilizing the initial conditions of the problem, the spatial pro®les of the refrigerant enthalpy, mass ¯ux and density can be calculated at each point of the heat exchanger. The transition between the single-phase zone to the two-phase-zone and vice versa are determined by computing the refrigerant vapor quality along the entire length of the heat exchanger. At the outlet of the evaporator and condenser, the refrigerant mass ¯ux must be equal to the assumed value obtained from the compressor and expansion device models. If convergence is not achieved, a new arbitrary value for the pressure is determined from Newton±Raphson modi®ed method. This sequence is repeated, until the mass ¯ux convergence is achieved. 2.4. Refrigeration system 2.4.1. Transient model The condenser model is initiated after establishing the geometrical parameters of the refrigeration system, the initial conditions, arbitrary values for the evaporation and condensing pressures as well as the degree of superheat at the condenser inlet. The outlet values of the condenser model are utilized to initiate the evaporator model. Then, the evaporation pressure and degree of superheat calculated by the evaporator model are compared with the corresponding values utilized as input data in the condenser model. This calculation procedure is repeated until achieving convergence. The ®nal values obtained are then utilized for the calculations during the subsequent time instant. 2.4.2. Steady-state model Using an in®nite time step (107s) in Eqs. (4)±(8) and modifying the criterion of convergence of the transient model, the steady-state model can be obtained. The values adopted for the evaporation pressure and for the degree of superheat at the evaporator outlet, are used to initiate the condenser model as in the case of the transient model. However, the evaporation pressure computed by the evaporator model is now determined from converging the degree of superheat at the evaporator outlet. The value utilized as input in the condenser model is then compared with the calculated value from the evaporator model. The refrigerant mass utilized as input is now compared with the value calculated by the model. If convergence is not achieved, a new value of the degree of superheat is determined from the Newton±Rhapson method. As in the transient model, an iterative procedure is needed to ensure that the values of the evaporation pressure and the degree of superheat calculated by the evaporator model, used as input to the condenser model, are in agreement.
3. Discussion of results The models presented in the preceding section were used to simulate the behavior of the system for two modes of operation. In the ®rst mode, some numerical simulations were realized to determine the performance of the system at the start-up of the compressor while all the system is in equilibrium. In the second mode, we investigated the response of the system due to a step variation in the rotational speed of the compressor and a step variation of the sectional area expansion valve section while the system is under steady-state conditions. 3.1. Compressor start-up The start-up operation of a refrigeration system implies heavy demands from all the components and controls due to the large temperatures and pressures variations and hence the large time and space derivatives of the governing parameters. This requires a correct choice of the time step and space grid. Initial numerical test realized to optimize the computational grid showed that 15 s was a suitable time increment and that 400 control volumes divided between the evaporator and the condenser allowed for 2% relative error in the pressure and mass ¯ow of the refrigerant and 0.1 C in the refrigerant temperature. Figs. 1 and 2 present simulations of the start-up mode of the compressor with R12 as a refrigerant ¯uid. Fig. 1 presents the timewise evolution of the mass ¯ow of the refrigerant imposed by the compressor and the expansion valve as well as the mass of the refrigerant contained in the condenser and evaporator. Fig. 2 presents the time-wise evolutions of the condensation and boiling temperatures,
Fig. 1. Dynamic behavior of mass ¯ow rate and refrigerant (R12) charge system after start-up. Fig. 1. Comportement dynamique du ¯ux massique et de la charge en frigorigeÁne (R12) suite au deÂmarrage.
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Fig. 2. Dynamic behavior of refrigerant (R12) temperatures after start-up. Fig. 2. Comportement dynamique des tempeÂratures du frigorigeÁne (R12) suite au deÂmarrage.
the degree of superheat at the evaporator exit and the degree of subcooling at the condenser exit. The steadystate values are indicated in both Figs. 1 and 2 by the symbol (*). In Fig. 1 we can observe that after the start-up the compressor draws out the refrigerant from the evaporator and delivers it to the condenser and hence the refrigerant mass decreases in the evaporator and increases in the condenser as shown in Fig. 1. As a result the pressure and temperature values of the refrigerant in the evaporator decrease while they increase in the condenser. The mass ¯ow rate through the expansion valve depends on the pressure dierence between the condensation and boiling processes as in Eq. (3) whose value varies from zero to a maximum value corresponding to the steady-state operation and hence its variation follows what is depicted in Fig. 1. The analysis of the variation of the mass rate in the compressor is more complex because it depends directly on the compressor speed, the ratio of the boiling to the condensing pressures as well as the density of the refrigerant at the evaporator exit as in Eqs. (1) and (2). Soon after the compressor start-up, the refrigerant mass ¯ow rate at the compressor inlet increases due to the increase of the compressor speed. The subsequent decrease of the mass ¯ow is due to the increase of the pressure rate, and also to the decrease of the two phase ¯ow density at the compressor inlet.
197
The control by varying the rotational speed of the compressor improves the general performance and adequates the rotational speed to the required refrigeration capacity. Studies show that controlling the power by varying the rotational speed leads to a strong variation of the refrigerant superheat at the evaporator exit. As the thermostatic expansion valves have slow response, electronic valves are more adequate in this case, [12]. With the objective of developing a control algorithm for the power and the degree of superheat of a variable speed refrigeration machine, operation simulations were performed varying the rotational speed and the refrigerant ¯ow area of the expansion valve. These simulations represent a practical situation where the temperature of the secondary ¯uid in the evaporator circuit is high. The increase of the mass ¯ow rate and the refrigeration capacity by varying the rotational speed of the compressor causes a reduction of the temperature of the secondary ¯ow. This action causes an increase in the degree of superheat of the refrigeration ¯uid at the evaporator exit and reduces the COP of the system. The increase of the ¯ow sectional area of the expansion valve leads to decrease the degree of superheat and consequently improves the performance of the system. The refrigeration ¯uid used in these simulations is R134a. Figs. 3 and 4 present the behavior of the system due to step variation of the rotational speed from 1500 to 1800 rpm. Fig. 3 shows the time-wise variation of the mass ¯ow rate of the refrigerant ¯uid imposed by the compressor and the expansion valve as well as the degree of superheat at the evaporator exit. Fig. 4 presents the time-wise variation of the power consumed by the compressor which is essentially dependent on the mass ¯ow rate of the refrigerant. Consequently, the observed
3.2. Step variation of compressor speed The power control of small and medium refrigeration machines is usually done by the classic on±o system. This control system usually impairs the performance of the machine, reduces the useful life of the compressor and components and increases the power consumption.
Fig. 3. Dynamic behavior of the mass ¯ow rate and superheat in response to a step increase of the compressor rpm from 1500 to 1800. Fig. 3. Comportement dynamique du ¯ux massique et de la surchaue suite aÁ une augmentation de la vitesse du compresseur (de 1500 aÁ 1800 tours par minute).
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3.3. Step variation of the throttling sectional area of the expansion valve
Fig. 4. Dynamic behavior of the power system in response to a step increase of the compressor rpm from 1500 to 1800. Fig. 4. Comportement dynamique du systeÁme d'alimentation en eÂnergie suite aÁ une augmentation de la vitesse du compresseur (de 1500 aÁ 1800 tours par minute).
increase in the refrigeration capacity is proportional to the increase in the mass ¯ow rate of the refrigerant (both the refrigeration capacity and the mass ¯ow rate increased by about 7%). On the other hand, the compressor power consumption is directly dependent on the mass ¯ow rate of the refrigerant and inversely proportional to its density at the evaporator exit. Knowing that the density increases with the increase of the degree of superheat and with the reduction of the boiling pressure, the compressor power consumption increases much more than the refrigeration capacity (about 27%), and consequently the COP of the system is decreased. Fig. 5 shows a comparison between the present model and the experimental results due to Wang [25]. When the compressor work with R12, its rotational speed was varied from 712 rpm to 1463 rpm. We can observe a fair agreement between the present predictions and Wang's experimental results.
As we observed before, the increase of the refrigeration capacity due to the increase of the compressor rotational speed leads to an increase of the degree of superheat and consequently the reduction of the COP of the machine. Hence, it is necessary that the expansion valve corrects this action by increasing the throttling area. Figs. 6±10 present the behavior of the system due to 10% increase in the throttling area of the expansion valve. Fig. 6 shows, immediately after the changing of the expansion valve sectional area, the mass ¯ow rate of the refrigerant increased at both inlet and exit of the evaporator and a consequent reduction of the degree of superheat due to the increase of the mass ¯ow rate in the evaporator. Also as a consequence of changing the mass ¯ow rate, one can observe a reduction of the condensation pressure, an increase of the boiling pressure and ®nally the reduction of the dierence between the two
Fig. 6. Dynamic behavior of the mass ¯ow rate and the superheat in response to a step variation of the compressor rpm and of the throttling section area. Fig. 6. Comportement dynamique du ¯ux massique et de la surchaue suite aÁ une variation de vitesse du compresseur et de la section du deÂtendeur.
Fig. 5. Comparison between theoretical results and experimental data of Wang [25] for the COP behavior in response to a step increase of the compressor rpm 712 to 1463. Fig. 5. Comparaison des reÂsultats theÂoriques et expeÂrimentaux obtenus par Wang [25] pour le COP en fonction d' une augmentation de la vitesse du compresseur (de 712 aÁ 1463 tours par minute).
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Fig. 7. Dynamic behavior of the system power in response to a step variation of the compressor rpm and the throttling area. Fig. 7. Comportement dynamique du systeÁme d'alimentation en eÂnergie suite aÁ une variation de vitesse du compresseur et de la section du deÂtendeur. Fig. 9. Response of the refrigerant temperature in the evaporator to a step variation of the compressor rpm and of the throttling section area. Fig. 9. TempeÂrature du frigorigeÁne dans l'eÂvaporateur suite aÁ une variation de vitesse du compresseur et de la section du deÂtendeur.
Fig. 8. Dynamic behavior of the system performance in response to a step variation of the compressor rpm and the throttling area. Fig. 8. Comportement dynamique de la performance du systeÁme suite aÁ une variation de vitesse du compresseur et de la section du deÂtendeur.
pressures and their ratio. The mass ¯ow rate peak and the subsequent stabilization can be attributed to the reduction of the pressure dierence between boiling and condensation. As a consequence of the increase of the mass ¯ow rate, the compressor power and refrigeration capacity increased as shown in Fig. 7. Irrespective of the increase of the mass ¯ow rate, the change of the enthalpy of the refrigeration ¯uid is reduced and this leads to nearly constant power consumption as indicated by Fig. 7. The reduction in the enthalpy is due to a reduction in the condensing pressure and an increase in the evaporation pressure. Fig. 8 shows the eect of the increasing of the throttling area on the COP of the refrigeration system. The increase in the COP can be attributed to the increase of the refrigeration capacity while the compressor power consumption is constant. Naturally, the variation of the
Fig. 10. Response of the refrigerant temperature in the condenser to a step variation of the compressor rpm and of the throttling section area. Fig. 10. TempeÂrature du frigorigeÁne dans le condenseur suite aÁ une variation de vitesse du compresseur et de la section du deÂtendeur.
compressor rotational speed and the expansion valve throttling area must be repeated as frequently as necessary until the refrigeration system delivers the required refrigeration capacity associated with the adequate degree of superheat, ensuring good performance of the system. Figs. 9 and 10 show the temperatures distribution in the evaporator and the condenser indicating the same behavior as mentioned before.
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4. Conclusions In this study, two models were presented to simulate the steady-state and transient behavior of a refrigeration system of the water±water type. The ®rst simulation realized corresponds to the start-up of the machine initially at thermal equilibrium. Comparison between the terminal values of the transient model seems to agree with the predictions from the steady-state model. Control of the refrigeration capacity by varying the compressor rotational speed was also investigated and it was found that this control leads to increasing the degree of superheat and hence impair the COP of the system. An alternative method is to increase the throttling area to alleviate the increase of the degree of superheat and improve the COP of the system. This model can be used to elaborate an algorithm to control the refrigeration capacity and the degree of superheat of the system. Acknowledgements
[9]
[10] [11] [12]
[13]
[14]
The authors wish to thank the CNPq and Fapesp for the ®nancial support and research scholarships to the authors.
[15]
References
[17]
[1] Domanski PA, Mclinden MO. A simpli®ed cycle simulation model for the performance rating of refrigerants and refrigerant mixtures. Rev Int Froid 1992;15:81±8. [2] Pelletier O, BjoÈrn P. Performance of plate heat exchangers and compressor in a domestic heat pump using propane. In: Proceedings of Applications for Natural Refrigerants, Denmark, 1996. p. 11.16±20. [3] Rogstam HJ. Theoretical and experimental investigation of cyclopropane as refrigerant. Proceedings of Applications for Natural Refrigerants, Denmark, 1996. p. 10.15± 19. [4] Lemos MJS, Zaparoli EL. Steady-state numerical solution of vapor compression refrigeration units. In: Int. Refrigeration Conference at Purdue, Purdue University, 1996. 235±40. [5] Yasuda H, Touber S, Machielsen CHM. Simulation model of a vapor compression refrigeration system. ASHRAE Trans 1983;98(2a):408±25. [6] Parise JAR. Theoretical and experimental analysis of a diesel driven heat pump. PhD thesis, University of Manchester, UK, 1983. p. 325. [7] Conde MR. A steady state mathematical simulation model for air-to-water heat pumps (Report NEFF). Zurich: Swiss Federal Institute of Technology, 1985, p. 236. [8] Mongey B, Mcmullan JT, Mcnerlin MG. An examination of hydrocarbon mixtures for use in high temperature heat
[16]
[18] [19] [20] [21] [22] [23] [24] [25]
pump applications. In: Proceedings of Applications for Natural Refrigerants, Denmark, 1996. p. 11.11±15. Tamainot-Telto Z. DeÂtendeurs thermostatiques de machines frigori®ques. Comportement dynamique et modeÂlisation. TheÁse de doctorat, INSA, Lyon, 1993. p. 150. Votsis PP, Tassou SA, Wilson DR, Marquand CJ. Dynamic characteristics of an air-to-water heat-pump system. Int J Refrig 1992;15(2):89±94. Wenxue H, Kraft H. A mathematical model of an evaporator based on the step exciting method. In: XVIIIth Int. Cong. Refrigeration, Montreal, 1991. p. 1217±31. Outtagarts A, Haberschill P, Lallemand M. Comportement dynamique d'un eÂvaporateur de machine frigori®que soumis aÁ des variations de deÂbit. In: 19th Int. Cong. Refrigeration, vol. IIIa, La Haye, Netherlands, August 1995, p. 413±20. Darrow JB, Lovatt SJ, Cleland AC. Assessment of a simple mathematical model for predicting the transient behavior of a refrigeration system. In: XVIIIth Int. Cong. Refrigeration, Montreal, 1991, p. 1189±92. Murphy WE, Goldschmidt VW. Cycling characteristics of a residential air conditioner. Modeling of shutdown transients. ASHRAE Trans 1986;92(1a):186±201. Welsby P, Devotta S, Diggory PJ. Steady- and dynamicstate simulations of heat-pumps. Part I: literature review. Applied Energy 1988;31:189±203. Salim M, Sadasivam M, Balakrishnan AR. Transient analysis of heat pump assisted ditillation systems I. The heat pump. Int J Energy Research 1991;15:123±35. Macarthur JW, Grald EW. Unsteady compressible twophase ¯ow model for predicting cyclic heat pump performance and a comparison with experimental data. Int J Refrig 1989;12:29±41. Wang H, Touber S. Distributed and non-steady-state modelling of air cooler. International Journal of Refrigeration 1991;1:98±110. Nyers J, Stoyan G. A dynamical model adequate for controlling the evaporator of a heat pump. Int J Refrig 1994;17(2):101±8. Jia X, Tso CP, Chia PK, Jolly P. A distributed model for prediction of the transient response of an evaporator. Int J Refrig 1995;18(5):336±42. Ferreira CI, Kalkman H. Transient behavior of economizers for multi-stage refrigeration plant. In: Int. Refrigeration Conference at Purdue, Purdue University, 1996. p. 241±46. Machado L. Modele de simulation et etude experimentale d'un evaporateur de machine frigori®que en regime transitoire. These de Docteur, INSA/LYON, 1996. p. 160. Dengler C, Addoms JN. Heat transfer mechanism for vaporization of water in a vertical tube. Chem Eng Prog Symp Series 1956;52(18):95±103. Shah MM. Chart correlation for saturated boiling heat tranfer: equations and further study. ASHRAE Trans 1982;88:185±96. Wang H. Modelling of a refrigerating system coupled with a refrigerated room. TheÁse de doctorat, Delf University of Technology, Netherlands, 1991. p. 230.