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Rotational type of a gravitational ejector refrigerator – A system balance of the refrigerant analysis
international journal of refrigeration 33 (2010) 3–11
available at www.sciencedirect.com
w w w . i i fi i r . o r g
journal homepage: www.elsevier.com/locate/ijrefrig
Rotational type of a gravitational ejector refrigerator – A system balance of the refrigerant analysis Jacek Kasperski* Wroclaw University of Technology, Institute of Power Engineering and Fluid Mechanics, Wybrzeze Wyspianskiego 27, Wroclaw 50–370, Poland
article info
abstract
Article history:
The existing ejector systems were analyzed depending on the way in which the refrigerant
Received 20 February 2009
returns from the condenser to the generator and evaporator. The research focused on gravi-
Received in revised form
tational ejector refrigerator in which hydrostatic pressure of the refrigerant allows to equalize
9 June 2009
pressure differences between heat exchangers located on different levels. Using centrifugal
Accepted 25 August 2009
acceleration instead of gravitational allows to decrease significantly the size of a refrigerator.
Available online 3 September 2009
The name roto-gravitational refrigerator was proposed for that kind of refrigerator.
Keywords:
Surrounding temperatures when different from typical may cause drying up of the
Ejector
refrigerant in the exchangers and lead to destabilizing the refrigerator’s work. A mathe-
Rotation
matical analysis of thermal and flow processes occurring in the refrigerant has been
Gravity
conducted. A mathematical model of the refrigerant balance and its numerical solution has
Centrifugation
been proposed. The analysis of the refrigerator accelerating temperature influence on its
Refrigeration system
work parameters has been conducted for exemplary calculations.
One of the problems of small, compact refrigerators is a little amount of refrigerant.
ª 2009 Elsevier Ltd and IIR. All rights reserved.
Ejector system Improvement Operation Modelling Simulation
Un re´frige´rateur rotationnel a` e´jection gravitationnelle : analyse du frigorige`ne et l’e´quilibre du syste`me Mots cle´s : Syste`me frigorifique ; Syste`me a` e´jecteur ; Ame´lioration ; Fonctionnement ; Gravite´ ; Centrifugation ; Mode´lisation ; Simulation
1.
Introduction
The ejector refrigerators have been known in engineering since the beginning of the 20th century. There are few publications concerning gravitational ejector refrigerators. Nguyen et al.
(2004) described a refrigerator in a form of a 5-m high tower with water as a refrigerant. The refrigerator used one-way valves at the inlet of the steam generator and evaporator. Ling (2004) described a gravitational ejector refrigerator in a vertical arrangement with water as a refrigerant and a pump supply of
* Tel.: þ48 71 320 35 05; fax: þ48 71 328 38 18. E-mail address: [email protected] 0140-7007/$ – see front matter ª 2009 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2009.08.008
4
international journal of refrigeration 33 (2010) 3–11
Symbols a A F g h L _ m n p r q Q T U v V
matrix coefficient, the cross-section area of heat exchanger (m2) the area of heat transfer (m2) gravitational acceleration (m s2) enthalpy (J kg1) length (m) mass flow ratio (kg s1) rotation speed, (rev s1) pressure (Pa) radius (m) specific heat (J kg1) heat flow (W) temperature ( C) overall unit of heat transfer (W m2 K1) specific volume (m3 kg1) volume (m3)
a steam generator. In some research Srisastra and Aphornratana (2005), Srisastra et al. (2008), Huang et al. (2006) a cyclic steam generator feeding system was used. The applied systems of thermal pumping allowed to decrease the refrigerators height. Technical literature does not provide descriptions of gravitational refrigerators working without pumps, one-way valves, tanks and cyclic feeding system. The unique research concerning this area was published by Kasperski (2008). The subject of this paper is a new concept of a gravitational ejector refrigerator. An application of rotary motion allows to create significant pressures by centrifugal accelerations. Putting the whole refrigerator in rotary motion allows to construct compact refrigerators. The small size of a compact refrigerator makes it sensitive to the refrigerant drying up in heat exchangers. This phenomenon can occur during the change of surrounding temperatures, for example, the temperature in a steam generator. The following part of this paper attempts to describe and model this phenomenon.
2.
w y COP u
coefficient of free term matrix height (m) coefficient of performance rotation speed (rad s1)
Subscripts and superscripts c condenser e evaporator ej ejector en energy ext external (temperature) g steam generator liq liquid ref refrigerant sf surface vap vapour wall wall (of case)
The alternative for a pump ejector refrigerator is a gravitational one (Fig. 1b). The vertical arrangement of the heat exchangers on different levels allows to equalize the pressure differences between the exchangers with the help of the refrigerant hydrostatic pressure. The highest pressure is
Ejector refrigerator types
There are many types of ejector refrigerators. Depending on the way in which a refrigerant returns form the condenser to the generator and evaporator there are two main configurations: pump and gravitational ones. In a typical pump version of an ejector refrigerator (Fig. 1a) heat exchangers can be placed on different levels towards one another. The flow of a liquid refrigerant from the condenser to the steam generator is forced by the pump work. A throttle valve enables placing the condenser and evaporator in an optional position. The pump is the only element containing movable parts, which may cause a leak in the installation. If operating pressure of a refrigerant is higher than atmospheric, the leak usually does not cause a problem. If the refrigerant operating pressure is lower than atmospheric, even a little amount of sucked atmospheric air can destabilize processes of boiling and condensation (especially for water). In case of flammable mediums such as hydrocarbons, ethers, alcohols, sucking the air causes a significant danger of explosion.
Fig. 1 – The scheme of apparatuses system in a pumping and gravitational ejector refrigerator.
international journal of refrigeration 33 (2010) 3–11
obtained in the steam generator, which forces the lowest liquid level. The lowest pressure obtained in the evaporator causes the inflow of liquid to the highest installation level. A large diameter of liquid pipes does not allow throttling of the refrigerant flow. At the inlet and inside the evaporator space, the refrigerant unrestricted expansion occurs, depending on its local pressure. The ejector cycle allows to construct hermetic gravitational refrigerators, just as already well known in engineering, the absorptive cycle with inert gas. Hermetic refrigerators are exceptionally durable and reliable because they do not contain movable parts. Moreover, they are quiet. Installation height of a gravitational refrigerator results from the difference of internal pressures. The pressure mainly depends on the kind of applied refrigerant and the temperature in a steam generator. The refrigerants for a vapour compression cycle are useless here, however, water, alcohols, ethers, heavier hydrocarbons (pentane, hexane, heptane, octane) and some organic solvents are usable. Gravitation creates hydrostatic pressure (of a liquid), therefore, a gravitational refrigerator may have a significant height, for example, more than a 100 m. In a pump technique there is a well-known method of making significant pressures by centrifugal accelerations. It needs, however, the application of rotary motion. Putting the whole refrigerator in rotary motion allows to have its height (radius) amounting to a fraction of the gravitational refrigerator height. It demands a specific refrigerator arrangement: fixing the ejector in the axis of rotation and the concentric heat exchangers around. The conception of the gravitational refrigerator (Fig. 2a) may be developed into a rotating refrigerator (Fig. 2b) in a simple way. In the gravitational refrigerator, thanks to difference of density, the liquid flows to the lower parts. Pipe connections allow the refrigerant flow from the condenser to the evaporator and generator. According the surrounding temperature changes a temporary reverse flows are possible: from the evaporator to the condenser and/or from the generator to the condenser. Due to the lack of one-way valves, flows between the generator and evaporator are available. In the upper parts the ejector connects the three spaces filled by gas. The application of rotary motion causes the axissymmetric arrangement of liquid levels (Fig. 2b). Similarly to the gravitational refrigerator, higher steam pressure pushes the liquid level outside, and the lower steam pressure sucks the liquid inside. The total pressures of the refrigerant in the generator, condenser and evaporator balance. If the total pressure in one of the exchangers is set, the gas pressure increase causes the decrease of its hydrostatic pressure. The refrigerant density changes are of no importance. The acceleration of rotary motion may be significant, however, the pressure distribution along y parameter is a linear function, along r parameter is a square function. The similarity of both refrigerators suggests a name roto-gravitational refrigerator for the refrigerator shown in Fig. 2b. This name puts an emphasis on the similarity of liquid and vapour separation, which is created under gravitational and centrifugal accelerations. Obviously, other names may be used, for example, a rotational type of gravitational refrigerator or rotary/rotating/ centrifugal refrigerator, however, it may wrongly suggest the Ranque–Hilsh effect or scroll/screw compressor.
5
Fig. 2 – The scheme of liquid refrigerant levels in gravitational and roto-gravitational ejector refrigerators.
3. Characteristic of the roto-gravitational refrigerator A simplified construction of the roto-gravitational refrigerator is shown in Fig. 3. The heat exchangers were placed in a similar way as in the refrigerators shown in above in Figs. 1 and 2. The refrigerator has geometry of a rotating cylinder. The heat transfer takes place on the side surfaces of this rotating cylinder. There are three paths of vapour streams: - The steam (marked as G in Fig. 3) coming from the generator flows into the primary nozzle; - The vapour (E) from the evaporator is sucked to the mixing chamber; - The mixed fluid (C) flowing to the condenser may condense on the liquid surface or on the casing surface, depending on the applied construction.
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international journal of refrigeration 33 (2010) 3–11
Fig. 3 – A concept of the roto-gravitational ejector refrigerator (in simplification).
In the shown example warm air heats the steam generator. The flows of the external air are made by additional fans. The refrigerator of this type demands putting it in rotary motion. That drive, however, does not have a significant influence on energetic balance. The use of centrifugal accelerations decreases significantly the size of a gravitational refrigerator. The calculating examples for different refrigerants are presented in Table 1. Comparative numerical calculations are carried out for temperatures Tg ¼ 80 C, Tc ¼ 35 C, Te ¼ 15 C. For many refrigerants the required height of the gravitational refrigerator installation is significant, for example, R410, R717, R290 it would be extremely high (w500 m) and cannot be applied. The use of rotary motion allows to decrease the refrigerator size depending on the rotation speed. The calculated sizes of the roto-gravitational refrigerator for 1000 rpm are presented in the table. The use of high-pressure refrigerants increases the
Table 1 – Comparison of gravitational and rotogravitational refrigerator dimensions. Refrigerant Gravitational
Roto-gravitational
Installation Outer diameter Rotation speed height required for required for required m rotation speed outer diameter 1000 rpm m 0.2 m rpm Water Methanol Pentane R134a R410A R717 R290 R600
4.7 22.4 53.6 187 467 594 514 147
0.19 0.41 0.63 1.18 1.87 2.07 1.94 1.05
1000 2300 3800 6300 10,000 12,000 11,500 10,400
refrigerator size to 2 m of diameter only. The use of rotary motion allows to decrease the refrigerator even by size by several hundred times. Obviously, the refrigerator size decrease depends on the applied rotation speed: the higher rotation speed, the smaller diameter of a refrigerator is. The required diameter can be reached with the appropriately high rotation speed. The calculation result for the rotation speed allowing to construct the refrigerator of diameter w0,2 m is shown in the last column of the table. The calculations show that any refrigerant may be used in a roto-gravitational refrigerator. High-pressure refrigerants will demand high rotary speeds, low-pressure refrigerants (for example, water, methanol, heavier hydrocarbons) will demand smaller. The use of low-pressure refrigerants and high rotation speed allows to decrease significantly the diameters of the refrigerator. It is possible to construct a very small refrigerator with the case diameter of even tens millimeters. The small size of a refrigerator and small refrigeration efficiency require the application of driving nozzles of small diameters. The smallest nozzle examined by the author was a nozzle of the minimum diameter of 0.8 mm and cone angle of 10 . Such small nozzles are possible to use, yet they can be exposed to flow occlusion, just as it can happen in capillary tubes in compressor refrigerators. The appropriate choice of a refrigerant requires further thermodynamic analysis. Table 1 presents the main refrigerants which have various influence on the refrigerator minimum required height and dimension. The use of water ensures the smallest size of a refrigerator. The use of water as a refrigerant has many advantages: a low price, safety, ecological advantages. The disadvantage of water is that it works in a deep vacuum, which is easily damaged by a leak in a refrigerator. A small, compact construction of the roto-gravitational refrigerator makes it easier to maintain leak-tightness.
international journal of refrigeration 33 (2010) 3–11
High rotation speed activates the Coriolis effect, which disturbs the refrigerator’s work. High rotation speed also changes the working characteristic of the ejector. Using the CFD modeling technique, the characteristic of the stationary and rotating ejectors was compared by Pietrowicz and Kasperski (2007). It was proved that the ejector’s rotating disturbs strongly the processes of steam streams mixing which take place inside it. For rotary speeds amounting to 1000 rpm the ejector may be approximately treated as immovable. For speeds over 10,000 rpm the ejector’s characteristic is strongly changed. In other researches Hong et al. (2004) concerning the influence of rotor/vane rotation up to 100,000 rpm, a similar, but slightly different, phenomenon was also was confirmed. The described effect may be advantageous, but it demands further research. There may be number of applications for the refrigerator: food storage, air-conditioning, internal cooling of rotors (generators, pump), compact cooling fans or pumps. The heat drive may be derived from the heat received from gas or liquid, and even solar radiation. The diameters of casing sections may be different. Several years of the author’s research resulted in the creation of two prototype rotating refrigerators with water as a refrigerant. The refrigerators were constructed in metal cases of diameters of 100 and 200 mm and adjusted to rotary speed of 2100 and 2400 rpm. One of the examined refrigerator construction is shown in Fig. 4. The main problem of experimental research concerning rotary refrigerators is the limited possibility to observe the occurring processes, the damage of measurement sensors caused by centrifugal acceleration, the disturbance of electric signals transmitted by sliding contacts. The application of wireless data transmission (sensor signals, video image) is hindered by a metal refrigerator case. For those reasons the processes occurring inside the refrigerator are still little known. The use of an evaporator with a transparent wall enabled the observation of the bubble generation twice, but the experiment was not repeatable. The observed changes of liquid levels in the evaporator depended, among others, on the temperature in the steam generator. One of the reasons
Fig. 4 – One of the prototypes of roto-gravitational refrigerators used in author’s experiments.
7
that causes problems with the proper refrigerator work can be unsuitably selected thickness of the liquid refrigerant layer in the exchangers. This phenomenon needs to be analyzed.
4.
The system of internal balance
The small size of the roto-gravitational refrigerator requires small amounts of a refrigerant. The use of the flammable refrigerants enables application of little fillings, according to safety rules, but a stable work of the refrigerator demands a proper filling with the refrigerant in each of the heat exchangers. In the gravitational and roto-gravitational refrigerator the refrigerant level in the exchanger depends on the present state of balance between pressures/temperatures of a refrigerant. The refrigerant temperature increase in one of the exchangers causes the saturation pressure increase. The gas pressure increase has to be balanced by decreasing the hydrostatic pressure of the liquid. The excess of the refrigerant liquid pushed out to the other two exchangers. A small amount of the refrigerant in the heat exchangers may cause their drying up and, for example, stop the evaporation. Too large amount of the refrigerant increases the resistance of heat transfer. The balance system of the rotating refrigerator possesses a number of feedbacks. Understanding the mentioned problem allows to work out the calculating model. The model lets us predict liquid levels reactions to the changes of the surrounding’s parameters. The calculating model of the refrigerant balance inside the gravitational refrigerator was presented earlier by Kasperski (2008). This model is based on the observation of experimental installation of the gravitational ejector refrigerator, where the strong changes of liquid levels depending on the temperature in the exchangers occurred. The independent variables of temperature and the liquid levels in each of the three exchangers marked as Tg, Tc, Te, yg, yc, ye are defined for the gravitational refrigerator. The system of six balance equations, which allows to make the matrix of nonlinear coefficients for the system of linear equations is created and numerically solved. The similar procedure may be applied for the analysis of the behavior of the roto-gravitational refrigerator with the independent variables Tg, Tc, Te, r2g , r2c , r2e . The calculating model allows to predict the reactions of the refrigerator and its internal interactions. The analysis of a boiling process in the steam generator or the evaporator shows that the increase of temperature pushes the liquid down. In the gravitational refrigerator this situation causes the decrease of the heat transfer (side) area F( y). In the roto-gravitational refrigerator shown in Fig. 3 the (side/ bottom) area F(r) does not change (Fig. 5). The area of heat transfer may be described as a mathematical dependence F( y) for the gravitational refrigerator and F(r) for the roto-gravitational refrigerator. In a similar way the liquid surface may be described as functions A( y) and A(r). These functions allow to model the processes of the heat transfer for any level of the liquid in the exchanger. The liquid pushed out of the exchanger flows in the limited volume of the refrigerator; therefore, it increases the level (decreases the radius) in the other two exchangers. It makes the system of feedbacks of all
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international journal of refrigeration 33 (2010) 3–11
After putting it to the formula (1), it may be presented in a general form of a linear equation depending on the six unknown variables: Fg rg Ug Tg þ Fc ðrc ÞUc Tc þ Fe ðre ÞUe Te ¼ wen
(5)
where the word wen describes the balance dependence on outside temperatures tg_ext, tc_ext, te_ext stimulating the refrigerator wen ¼ Fg rg Ug Tg
4.2.
ext
þ Fc ðrc ÞUc Tc
ext
þ Fe ðre ÞUe Te
ext
(6)
Ejector equation
The basic parameter describing working properties of the ejector is the entrainment ratio defining the relation between the mass flow ratio of the driving steam and vapour sucked in: ER ¼
_e m _g m
(7)
Coefficient of performance (COP) for the ejector refrigerator is the relation between the heat flow of the driving refrigerant and the heat flow occurred in the evaporator. In case of the ejector cycle both operating characteristics of the ejector and thermodynamic properties of the refrigerant have an influence on this parameter. It may be described by the following dependence: COP ¼
_ e Dhe Qe m Dhe ¼ ¼ ER _ g Dhg Qg m Dhg
(8)
and then transformed to the following form: Dhe Qe ER Qg ¼ 0 Dhg Fig. 5 – The relations between the refrigerant level in the steam generator or the evaporator and temperature change.
Generally, the characteristic of an ejector is a mathematical function: ER Tg ; Tc ; Te ; ejector geometry; u ;
three heat exchangers and it is difficult to predict whether the liquid pushed out from the steam generator flows in the most part to the evaporator or to the condenser. The cross-section area of the exchangers A(r) is necessary to define the balance of the liquid refrigerant volume. The calculating model for the roto-gravitational refrigerator demands making the system of balance equations.
4.1.
Energy balance equation
According to energy balance: Qc Qg Qe ¼ 0
(1)
Each of the heat fluxes may be described by the dependence resulting from heat transfer processes: Qg ¼ Fg rg Ug Tg
ext
Tg
Qc ¼ Fc ðrc ÞUc ðTc Tc
ext Þ
(2)
ext
Te Þ
Dhe ðTc ; Te Þ;
(11)
Dhg Tg ; Tc
(12)
In that case, the general dependence (9), after taking into consideration the conditions of heat exchange in the generator (2) and the evaporator (4), may be transformed to the form dependent on the unknown variables:
(3) (4)
(10)
where each T represents the saturation temperature corresponding to the refrigerant pressure. Function (10) could be determined through experimental data, numerical calculations or one of the models proposed by other authors. Thermodynamic properties of the refrigerant and the realized cycle are calculated by specific heat of vaporization in the evaporator and producing the driving steam in the generator:
ER Qe ¼ Fe ðre ÞUe ðTe
(9)
Dhe Fg rg Ug Tg Fe ðre ÞUe Te ¼ wej Dhg
(13)
9
international journal of refrigeration 33 (2010) 3–11
where the expression wej describes the system dependence on external conditions Dhe wej ¼ ER Fg rg Ug Tg Dhg
ext
þ Fe ðre ÞUe Te
(14)
ext
It was assumed that the stream from the generator and evaporator are saturated steam.
4.3.
Equation of the refrigerant volume
a) Generator-evaporator balance 62 2 62 2 rg þ r ¼0 p Tg pðTe Þ 2vliq g 2vliq e e
(20)
b) Generator-condenser balance
Closed working space of the refrigerant generates the strongest correlation of the system balance: the liquid pushed out of one heat exchanger is going to be pushed into the two remaining exchangers. (15) Vg rg þ Vc ðrc Þ þ Ve ðre Þ ¼ Vliq ref It was assumed that for the cylindrical shape of the heat exchangers shown in Fig. 5, where rwall is the radius of wall with the length L, the equation (15) may be transformed to the form PLg r2g þ PLc r2c þ PLe r2e ¼ Vvap
Three conditions of balance occur between the three vessels. These conditions may be described by the following pairs:
ref
62 2 62 2 p Tg pðTc Þ r þ r ¼0 2vliq g g 2vliq c c
(21)
c) Condenser-evaporator balance pðTc Þ pðTe Þ
62 2 62 2 r þ r ¼0 2vliq c c 2vliq e e
(22)
(16)
In each of the equations there is the function p(T ) which may be replaced by the following function:
(17)
pðTÞ ¼
where: Vvap
ref
¼ PLg r2g
wall
þ PLc r2c
wall
þ PLe r2e
wall
Vliq
ref
The expression Vvap_ref determines the associative volume of the refrigerant vapour, whereas Vliq_ref means the liquid volume. If the geometry of the exchangers is not ring-shaped, the equations (16,17) may be transformed using a substitute length: L ðrÞ ¼
AðrÞ 2Pr
(18)
pðTÞ T T
(23)
allowing to depend the equations on the unknown variables Tg, Tc, Te. The variability of the calculated expressions in the analyzed range of temperature amounts, strongly decomposes numerical calculations; therefore, the expression of the left side of the equations (20–22) should be replaced by the linear formula:
It was assumed, that the steam density is significantly smaller than the liquid density.
p Tg pðTe Þ p Tg pðTe Þ Tg Te p Tg pðTe Þ ¼ Tg Te Tg Te
(24)
4.4.
p Tg pðTc Þ p Tg pðTc Þ Tg Tc p Tg pðTc Þ ¼ Tg Tc Tg Tc
(25)
Pressure balance equations
In each of the three heat exchangers of the roto-gravitational refrigerator the pressure balance of the connected vessels takes place. Total pressures in the outer (bottom) part of the vessels are equal. Between the two heat exchangers the system of balance, which results from the addition of the steam pressure and hydrostatic pressure, occurs. Hydrostatic pressure of the rotating ring of the liquid may be calculated as follows: 62 2 r r2liq Dp ¼ 2vliq
(19)
sf
pðTc Þ pðTe Þ ¼
4.5.
pðTc Þ pðTe Þ pðTc Þ pðTe Þ Tc Te Tc Te Tc Te
(26)
Linear equation system
Mathematical formulas (5–26) allow to determine a system of six linear equations, describing the dependence of the unknown variables Tg, Tc, Te, r2g , r2c , r2e on the surrounding parameters Tg_ext, Tc_ext, Te_ext. The calculated parameters
Table 2 – Main working conditions of the refrigerator for calculation example. Heat exchanger (part)
Generator Evaporator Condenser (tank) Condenser (film)
Radius
Length
m
m
Liq.–gas surface
Wall
0.0963 0.0296 0.0400 0.0350
0.1010 0.0350 0.0450 0.0350
0.10 0.20 0.03 0.40
Refrigerant sat. temp.
COP
Vref
n
u
C
–
cm3
rpm
rad s1
90 22
0.16
554
1,000
105
Sat. pressure
Heat flux
C
kPa
W
80 15 35 35
47.4 1.70 5.63 5.63
500 79
–
Ext. air temp.
– 579
30
10
international journal of refrigeration 33 (2010) 3–11
Fig. 6 – Changes of temperatures and liquid levels inside the refrigerator as a reaction to the change of the external temperature of the steam generator – the results of the numerical calculation.
with the expressions wen, wej, Vvap_ref and the working properties F(r), A(r) of the heat exchangers allow to determine the unknown parameters of the balance occurring in the refrigerator. In the matrix record the system of six balance equations may be presented as follows: energy balance eq: ejector balance eq: liquid refrig: balance eq: gen:-evap: press: bal: eq: gen:-cond: press: bal: eq: con:-evap: press: bal: eq:
2
a0 a1 a2 0 6 a6 0 a8 0 6 6 0 0 0 a15 6 6 a18 0 a20 a21 6 4 a24 a25 0 a27 0 a31 a32 0 2 3 2 3 tg wen 6 tc 7 6 wej 7 6 7 6 7 6 te 7 6 Vvap ref 7 6 7 7 ¼ 6 6 r2g 7 6 0 7 6 7 6 7 4 r2 5 4 0 5 c 0 r2e
0 0 a16 0 a28 a34
3 0 0 7 7 a17 7 7 a23 7 7 0 5 a35
(27)
where the matrix of free terms determines the surrounding influence on balance conditions. The system of six linear equations may be solved numerically and the proper computer software has been built for this purpose. The equation (27) solved once does not produce a correct result, which results from the dependence of the coefficients a0–a35 and unknown variables. Numerous equation solution allows to reach good accuracy of the unknown variables, which was described by Kasperski (2008).
5.
Calculating example
The calculating example has been made for the gravitational refrigerator with water as a refrigerant. The heat exchanger temperatures were adjusted to the air-conditioning use of the refrigerator (Table 2) just as in one of the refrigerators experimentally examined by the author. The equation (10) was determined by numerical calculations according to the designed experiment. The computer software was used for
testing the influence of the external air inlet temperature on the balance parameters inside the refrigerator. The compact results of calculations are presented in Fig. 6 on the example of the air temperature heating the steam generator. It can be shown that along with the air temperature increase, the saturation temperature in the steam generator increases as well. It slightly increases in the condenser. In the evaporator the temperature decreases, which results from the ejector’s properties. Along with the increase of steam generator temperature the radius of the liquid surface increases. The liquid pushed out of the generator flows into the evaporator and condenser. The cooling capacity increases along with the temperature of the steam generator. Together with the change of the generator’s surrounding temperature, the refrigerant’s temperature in the generator and evaporator changes – it influences the process of the heat exchange, and, therefore, the change of COP and its local maximum. The model showed the ability to calculate the states of balance inside the roto-gravitational refrigerator in response to the surrounding temperature disturbance. For simulations it is recommended to keep a few millimeters of thickness of the refrigerant layers in each of the exchangers. Such thickness allows to self-regulate the refrigerator’s levels for the changes of a few C in the steam generator surrounding temperature. The accuracy of the calculations conduced on the described model requires further experimental validation. The refrigerator’s construction should enable an observation and measurement of the refrigerant liquid in all heat exchangers. The working stations constructed by the author did not offer such a possibility.
6.
Conclusion
The conception of the roto-gravitational refrigerator is a direct development of the gravitational refrigerator. The accelerations of rotary motion are bigger than the gravitational
international journal of refrigeration 33 (2010) 3–11
acceleration and allow to decrease significantly the indispensable size of the gravitational refrigerator. Therefore, in the roto-gravitational refrigerator it is possible to use a highpressure refrigerant with a significant rotation speed. A very small size of the refrigerator with a small amount of the refrigerant is possible. The small size may cause the refrigerant drying up in the exchanger under the influence of the change of surrounding parameters, for example, the temperature of the steam generator surrounding. The calculating model of the balance allows to test the phenomena of drying up of the exchangers.
references
Hong, W.J., Alhussan, K., Zhang Jr., H., Garris, C.A., 2004. A novel thermally driven rotor-vane/pressure-exchange ejector refrigeration system with environmental benefits and energy efficiency. Energy 29, 2331–2345.
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Huang, B.J., Hu, S.S., Lee, S.H., 2006. Development of an ejector cooling system with thermal pumping effect. International Journal of Refrigeration 29, 476–484. Kasperski, J., 2008. Mathematical model of thermal and substance equilibrium in installation of gravitational ejector refrigerator. Chemical and Process Engineering 29, 997–1011. Ling, Z., 2004. A study on the new separate heat pipe refrigerator and heat pump. Applied Thermal Engineering 24, 2737–2745. Nguyen, V.M., Riffat, S.B., Doherty, P.S., 2004. Development of a solar-powered passive ejector cooling system. Applied Thermal Engineering 21, 157–168. Pietrowicz, S., Kasperski, J., 2007. The numerical modeling of thermo – flow processes in high – speed rotation ejector used in refrigerating system, The 22nd International Congress of Refrigeration, China, ICR07-B1-1076. Srisastra, P., Aphornratana, S., 2005. A circulating system for a steam jet refrigeration system. Applied Thermal Engineering 25, 2247–2257. Srisastra, P., Aphornratana, S., Sriveerakul, T., 2008. Development of a circulating system for a jet refrigeration cycle. International Journal of Refrigeration 31, 921–929.
available at www.sciencedirect.com
w w w . i i fi i r . o r g
journal homepage: www.elsevier.com/locate/ijrefrig
Rotational type of a gravitational ejector refrigerator – A system balance of the refrigerant analysis Jacek Kasperski* Wroclaw University of Technology, Institute of Power Engineering and Fluid Mechanics, Wybrzeze Wyspianskiego 27, Wroclaw 50–370, Poland
article info
abstract
Article history:
The existing ejector systems were analyzed depending on the way in which the refrigerant
Received 20 February 2009
returns from the condenser to the generator and evaporator. The research focused on gravi-
Received in revised form
tational ejector refrigerator in which hydrostatic pressure of the refrigerant allows to equalize
9 June 2009
pressure differences between heat exchangers located on different levels. Using centrifugal
Accepted 25 August 2009
acceleration instead of gravitational allows to decrease significantly the size of a refrigerator.
Available online 3 September 2009
The name roto-gravitational refrigerator was proposed for that kind of refrigerator.
Keywords:
Surrounding temperatures when different from typical may cause drying up of the
Ejector
refrigerant in the exchangers and lead to destabilizing the refrigerator’s work. A mathe-
Rotation
matical analysis of thermal and flow processes occurring in the refrigerant has been
Gravity
conducted. A mathematical model of the refrigerant balance and its numerical solution has
Centrifugation
been proposed. The analysis of the refrigerator accelerating temperature influence on its
Refrigeration system
work parameters has been conducted for exemplary calculations.
One of the problems of small, compact refrigerators is a little amount of refrigerant.
ª 2009 Elsevier Ltd and IIR. All rights reserved.
Ejector system Improvement Operation Modelling Simulation
Un re´frige´rateur rotationnel a` e´jection gravitationnelle : analyse du frigorige`ne et l’e´quilibre du syste`me Mots cle´s : Syste`me frigorifique ; Syste`me a` e´jecteur ; Ame´lioration ; Fonctionnement ; Gravite´ ; Centrifugation ; Mode´lisation ; Simulation
1.
Introduction
The ejector refrigerators have been known in engineering since the beginning of the 20th century. There are few publications concerning gravitational ejector refrigerators. Nguyen et al.
(2004) described a refrigerator in a form of a 5-m high tower with water as a refrigerant. The refrigerator used one-way valves at the inlet of the steam generator and evaporator. Ling (2004) described a gravitational ejector refrigerator in a vertical arrangement with water as a refrigerant and a pump supply of
* Tel.: þ48 71 320 35 05; fax: þ48 71 328 38 18. E-mail address: [email protected] 0140-7007/$ – see front matter ª 2009 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2009.08.008
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international journal of refrigeration 33 (2010) 3–11
Symbols a A F g h L _ m n p r q Q T U v V
matrix coefficient, the cross-section area of heat exchanger (m2) the area of heat transfer (m2) gravitational acceleration (m s2) enthalpy (J kg1) length (m) mass flow ratio (kg s1) rotation speed, (rev s1) pressure (Pa) radius (m) specific heat (J kg1) heat flow (W) temperature ( C) overall unit of heat transfer (W m2 K1) specific volume (m3 kg1) volume (m3)
a steam generator. In some research Srisastra and Aphornratana (2005), Srisastra et al. (2008), Huang et al. (2006) a cyclic steam generator feeding system was used. The applied systems of thermal pumping allowed to decrease the refrigerators height. Technical literature does not provide descriptions of gravitational refrigerators working without pumps, one-way valves, tanks and cyclic feeding system. The unique research concerning this area was published by Kasperski (2008). The subject of this paper is a new concept of a gravitational ejector refrigerator. An application of rotary motion allows to create significant pressures by centrifugal accelerations. Putting the whole refrigerator in rotary motion allows to construct compact refrigerators. The small size of a compact refrigerator makes it sensitive to the refrigerant drying up in heat exchangers. This phenomenon can occur during the change of surrounding temperatures, for example, the temperature in a steam generator. The following part of this paper attempts to describe and model this phenomenon.
2.
w y COP u
coefficient of free term matrix height (m) coefficient of performance rotation speed (rad s1)
Subscripts and superscripts c condenser e evaporator ej ejector en energy ext external (temperature) g steam generator liq liquid ref refrigerant sf surface vap vapour wall wall (of case)
The alternative for a pump ejector refrigerator is a gravitational one (Fig. 1b). The vertical arrangement of the heat exchangers on different levels allows to equalize the pressure differences between the exchangers with the help of the refrigerant hydrostatic pressure. The highest pressure is
Ejector refrigerator types
There are many types of ejector refrigerators. Depending on the way in which a refrigerant returns form the condenser to the generator and evaporator there are two main configurations: pump and gravitational ones. In a typical pump version of an ejector refrigerator (Fig. 1a) heat exchangers can be placed on different levels towards one another. The flow of a liquid refrigerant from the condenser to the steam generator is forced by the pump work. A throttle valve enables placing the condenser and evaporator in an optional position. The pump is the only element containing movable parts, which may cause a leak in the installation. If operating pressure of a refrigerant is higher than atmospheric, the leak usually does not cause a problem. If the refrigerant operating pressure is lower than atmospheric, even a little amount of sucked atmospheric air can destabilize processes of boiling and condensation (especially for water). In case of flammable mediums such as hydrocarbons, ethers, alcohols, sucking the air causes a significant danger of explosion.
Fig. 1 – The scheme of apparatuses system in a pumping and gravitational ejector refrigerator.
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obtained in the steam generator, which forces the lowest liquid level. The lowest pressure obtained in the evaporator causes the inflow of liquid to the highest installation level. A large diameter of liquid pipes does not allow throttling of the refrigerant flow. At the inlet and inside the evaporator space, the refrigerant unrestricted expansion occurs, depending on its local pressure. The ejector cycle allows to construct hermetic gravitational refrigerators, just as already well known in engineering, the absorptive cycle with inert gas. Hermetic refrigerators are exceptionally durable and reliable because they do not contain movable parts. Moreover, they are quiet. Installation height of a gravitational refrigerator results from the difference of internal pressures. The pressure mainly depends on the kind of applied refrigerant and the temperature in a steam generator. The refrigerants for a vapour compression cycle are useless here, however, water, alcohols, ethers, heavier hydrocarbons (pentane, hexane, heptane, octane) and some organic solvents are usable. Gravitation creates hydrostatic pressure (of a liquid), therefore, a gravitational refrigerator may have a significant height, for example, more than a 100 m. In a pump technique there is a well-known method of making significant pressures by centrifugal accelerations. It needs, however, the application of rotary motion. Putting the whole refrigerator in rotary motion allows to have its height (radius) amounting to a fraction of the gravitational refrigerator height. It demands a specific refrigerator arrangement: fixing the ejector in the axis of rotation and the concentric heat exchangers around. The conception of the gravitational refrigerator (Fig. 2a) may be developed into a rotating refrigerator (Fig. 2b) in a simple way. In the gravitational refrigerator, thanks to difference of density, the liquid flows to the lower parts. Pipe connections allow the refrigerant flow from the condenser to the evaporator and generator. According the surrounding temperature changes a temporary reverse flows are possible: from the evaporator to the condenser and/or from the generator to the condenser. Due to the lack of one-way valves, flows between the generator and evaporator are available. In the upper parts the ejector connects the three spaces filled by gas. The application of rotary motion causes the axissymmetric arrangement of liquid levels (Fig. 2b). Similarly to the gravitational refrigerator, higher steam pressure pushes the liquid level outside, and the lower steam pressure sucks the liquid inside. The total pressures of the refrigerant in the generator, condenser and evaporator balance. If the total pressure in one of the exchangers is set, the gas pressure increase causes the decrease of its hydrostatic pressure. The refrigerant density changes are of no importance. The acceleration of rotary motion may be significant, however, the pressure distribution along y parameter is a linear function, along r parameter is a square function. The similarity of both refrigerators suggests a name roto-gravitational refrigerator for the refrigerator shown in Fig. 2b. This name puts an emphasis on the similarity of liquid and vapour separation, which is created under gravitational and centrifugal accelerations. Obviously, other names may be used, for example, a rotational type of gravitational refrigerator or rotary/rotating/ centrifugal refrigerator, however, it may wrongly suggest the Ranque–Hilsh effect or scroll/screw compressor.
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Fig. 2 – The scheme of liquid refrigerant levels in gravitational and roto-gravitational ejector refrigerators.
3. Characteristic of the roto-gravitational refrigerator A simplified construction of the roto-gravitational refrigerator is shown in Fig. 3. The heat exchangers were placed in a similar way as in the refrigerators shown in above in Figs. 1 and 2. The refrigerator has geometry of a rotating cylinder. The heat transfer takes place on the side surfaces of this rotating cylinder. There are three paths of vapour streams: - The steam (marked as G in Fig. 3) coming from the generator flows into the primary nozzle; - The vapour (E) from the evaporator is sucked to the mixing chamber; - The mixed fluid (C) flowing to the condenser may condense on the liquid surface or on the casing surface, depending on the applied construction.
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Fig. 3 – A concept of the roto-gravitational ejector refrigerator (in simplification).
In the shown example warm air heats the steam generator. The flows of the external air are made by additional fans. The refrigerator of this type demands putting it in rotary motion. That drive, however, does not have a significant influence on energetic balance. The use of centrifugal accelerations decreases significantly the size of a gravitational refrigerator. The calculating examples for different refrigerants are presented in Table 1. Comparative numerical calculations are carried out for temperatures Tg ¼ 80 C, Tc ¼ 35 C, Te ¼ 15 C. For many refrigerants the required height of the gravitational refrigerator installation is significant, for example, R410, R717, R290 it would be extremely high (w500 m) and cannot be applied. The use of rotary motion allows to decrease the refrigerator size depending on the rotation speed. The calculated sizes of the roto-gravitational refrigerator for 1000 rpm are presented in the table. The use of high-pressure refrigerants increases the
Table 1 – Comparison of gravitational and rotogravitational refrigerator dimensions. Refrigerant Gravitational
Roto-gravitational
Installation Outer diameter Rotation speed height required for required for required m rotation speed outer diameter 1000 rpm m 0.2 m rpm Water Methanol Pentane R134a R410A R717 R290 R600
4.7 22.4 53.6 187 467 594 514 147
0.19 0.41 0.63 1.18 1.87 2.07 1.94 1.05
1000 2300 3800 6300 10,000 12,000 11,500 10,400
refrigerator size to 2 m of diameter only. The use of rotary motion allows to decrease the refrigerator even by size by several hundred times. Obviously, the refrigerator size decrease depends on the applied rotation speed: the higher rotation speed, the smaller diameter of a refrigerator is. The required diameter can be reached with the appropriately high rotation speed. The calculation result for the rotation speed allowing to construct the refrigerator of diameter w0,2 m is shown in the last column of the table. The calculations show that any refrigerant may be used in a roto-gravitational refrigerator. High-pressure refrigerants will demand high rotary speeds, low-pressure refrigerants (for example, water, methanol, heavier hydrocarbons) will demand smaller. The use of low-pressure refrigerants and high rotation speed allows to decrease significantly the diameters of the refrigerator. It is possible to construct a very small refrigerator with the case diameter of even tens millimeters. The small size of a refrigerator and small refrigeration efficiency require the application of driving nozzles of small diameters. The smallest nozzle examined by the author was a nozzle of the minimum diameter of 0.8 mm and cone angle of 10 . Such small nozzles are possible to use, yet they can be exposed to flow occlusion, just as it can happen in capillary tubes in compressor refrigerators. The appropriate choice of a refrigerant requires further thermodynamic analysis. Table 1 presents the main refrigerants which have various influence on the refrigerator minimum required height and dimension. The use of water ensures the smallest size of a refrigerator. The use of water as a refrigerant has many advantages: a low price, safety, ecological advantages. The disadvantage of water is that it works in a deep vacuum, which is easily damaged by a leak in a refrigerator. A small, compact construction of the roto-gravitational refrigerator makes it easier to maintain leak-tightness.
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High rotation speed activates the Coriolis effect, which disturbs the refrigerator’s work. High rotation speed also changes the working characteristic of the ejector. Using the CFD modeling technique, the characteristic of the stationary and rotating ejectors was compared by Pietrowicz and Kasperski (2007). It was proved that the ejector’s rotating disturbs strongly the processes of steam streams mixing which take place inside it. For rotary speeds amounting to 1000 rpm the ejector may be approximately treated as immovable. For speeds over 10,000 rpm the ejector’s characteristic is strongly changed. In other researches Hong et al. (2004) concerning the influence of rotor/vane rotation up to 100,000 rpm, a similar, but slightly different, phenomenon was also was confirmed. The described effect may be advantageous, but it demands further research. There may be number of applications for the refrigerator: food storage, air-conditioning, internal cooling of rotors (generators, pump), compact cooling fans or pumps. The heat drive may be derived from the heat received from gas or liquid, and even solar radiation. The diameters of casing sections may be different. Several years of the author’s research resulted in the creation of two prototype rotating refrigerators with water as a refrigerant. The refrigerators were constructed in metal cases of diameters of 100 and 200 mm and adjusted to rotary speed of 2100 and 2400 rpm. One of the examined refrigerator construction is shown in Fig. 4. The main problem of experimental research concerning rotary refrigerators is the limited possibility to observe the occurring processes, the damage of measurement sensors caused by centrifugal acceleration, the disturbance of electric signals transmitted by sliding contacts. The application of wireless data transmission (sensor signals, video image) is hindered by a metal refrigerator case. For those reasons the processes occurring inside the refrigerator are still little known. The use of an evaporator with a transparent wall enabled the observation of the bubble generation twice, but the experiment was not repeatable. The observed changes of liquid levels in the evaporator depended, among others, on the temperature in the steam generator. One of the reasons
Fig. 4 – One of the prototypes of roto-gravitational refrigerators used in author’s experiments.
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that causes problems with the proper refrigerator work can be unsuitably selected thickness of the liquid refrigerant layer in the exchangers. This phenomenon needs to be analyzed.
4.
The system of internal balance
The small size of the roto-gravitational refrigerator requires small amounts of a refrigerant. The use of the flammable refrigerants enables application of little fillings, according to safety rules, but a stable work of the refrigerator demands a proper filling with the refrigerant in each of the heat exchangers. In the gravitational and roto-gravitational refrigerator the refrigerant level in the exchanger depends on the present state of balance between pressures/temperatures of a refrigerant. The refrigerant temperature increase in one of the exchangers causes the saturation pressure increase. The gas pressure increase has to be balanced by decreasing the hydrostatic pressure of the liquid. The excess of the refrigerant liquid pushed out to the other two exchangers. A small amount of the refrigerant in the heat exchangers may cause their drying up and, for example, stop the evaporation. Too large amount of the refrigerant increases the resistance of heat transfer. The balance system of the rotating refrigerator possesses a number of feedbacks. Understanding the mentioned problem allows to work out the calculating model. The model lets us predict liquid levels reactions to the changes of the surrounding’s parameters. The calculating model of the refrigerant balance inside the gravitational refrigerator was presented earlier by Kasperski (2008). This model is based on the observation of experimental installation of the gravitational ejector refrigerator, where the strong changes of liquid levels depending on the temperature in the exchangers occurred. The independent variables of temperature and the liquid levels in each of the three exchangers marked as Tg, Tc, Te, yg, yc, ye are defined for the gravitational refrigerator. The system of six balance equations, which allows to make the matrix of nonlinear coefficients for the system of linear equations is created and numerically solved. The similar procedure may be applied for the analysis of the behavior of the roto-gravitational refrigerator with the independent variables Tg, Tc, Te, r2g , r2c , r2e . The calculating model allows to predict the reactions of the refrigerator and its internal interactions. The analysis of a boiling process in the steam generator or the evaporator shows that the increase of temperature pushes the liquid down. In the gravitational refrigerator this situation causes the decrease of the heat transfer (side) area F( y). In the roto-gravitational refrigerator shown in Fig. 3 the (side/ bottom) area F(r) does not change (Fig. 5). The area of heat transfer may be described as a mathematical dependence F( y) for the gravitational refrigerator and F(r) for the roto-gravitational refrigerator. In a similar way the liquid surface may be described as functions A( y) and A(r). These functions allow to model the processes of the heat transfer for any level of the liquid in the exchanger. The liquid pushed out of the exchanger flows in the limited volume of the refrigerator; therefore, it increases the level (decreases the radius) in the other two exchangers. It makes the system of feedbacks of all
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After putting it to the formula (1), it may be presented in a general form of a linear equation depending on the six unknown variables: Fg rg Ug Tg þ Fc ðrc ÞUc Tc þ Fe ðre ÞUe Te ¼ wen
(5)
where the word wen describes the balance dependence on outside temperatures tg_ext, tc_ext, te_ext stimulating the refrigerator wen ¼ Fg rg Ug Tg
4.2.
ext
þ Fc ðrc ÞUc Tc
ext
þ Fe ðre ÞUe Te
ext
(6)
Ejector equation
The basic parameter describing working properties of the ejector is the entrainment ratio defining the relation between the mass flow ratio of the driving steam and vapour sucked in: ER ¼
_e m _g m
(7)
Coefficient of performance (COP) for the ejector refrigerator is the relation between the heat flow of the driving refrigerant and the heat flow occurred in the evaporator. In case of the ejector cycle both operating characteristics of the ejector and thermodynamic properties of the refrigerant have an influence on this parameter. It may be described by the following dependence: COP ¼
_ e Dhe Qe m Dhe ¼ ¼ ER _ g Dhg Qg m Dhg
(8)
and then transformed to the following form: Dhe Qe ER Qg ¼ 0 Dhg Fig. 5 – The relations between the refrigerant level in the steam generator or the evaporator and temperature change.
Generally, the characteristic of an ejector is a mathematical function: ER Tg ; Tc ; Te ; ejector geometry; u ;
three heat exchangers and it is difficult to predict whether the liquid pushed out from the steam generator flows in the most part to the evaporator or to the condenser. The cross-section area of the exchangers A(r) is necessary to define the balance of the liquid refrigerant volume. The calculating model for the roto-gravitational refrigerator demands making the system of balance equations.
4.1.
Energy balance equation
According to energy balance: Qc Qg Qe ¼ 0
(1)
Each of the heat fluxes may be described by the dependence resulting from heat transfer processes: Qg ¼ Fg rg Ug Tg
ext
Tg
Qc ¼ Fc ðrc ÞUc ðTc Tc
ext Þ
(2)
ext
Te Þ
Dhe ðTc ; Te Þ;
(11)
Dhg Tg ; Tc
(12)
In that case, the general dependence (9), after taking into consideration the conditions of heat exchange in the generator (2) and the evaporator (4), may be transformed to the form dependent on the unknown variables:
(3) (4)
(10)
where each T represents the saturation temperature corresponding to the refrigerant pressure. Function (10) could be determined through experimental data, numerical calculations or one of the models proposed by other authors. Thermodynamic properties of the refrigerant and the realized cycle are calculated by specific heat of vaporization in the evaporator and producing the driving steam in the generator:
ER Qe ¼ Fe ðre ÞUe ðTe
(9)
Dhe Fg rg Ug Tg Fe ðre ÞUe Te ¼ wej Dhg
(13)
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where the expression wej describes the system dependence on external conditions Dhe wej ¼ ER Fg rg Ug Tg Dhg
ext
þ Fe ðre ÞUe Te
(14)
ext
It was assumed that the stream from the generator and evaporator are saturated steam.
4.3.
Equation of the refrigerant volume
a) Generator-evaporator balance 62 2 62 2 rg þ r ¼0 p Tg pðTe Þ 2vliq g 2vliq e e
(20)
b) Generator-condenser balance
Closed working space of the refrigerant generates the strongest correlation of the system balance: the liquid pushed out of one heat exchanger is going to be pushed into the two remaining exchangers. (15) Vg rg þ Vc ðrc Þ þ Ve ðre Þ ¼ Vliq ref It was assumed that for the cylindrical shape of the heat exchangers shown in Fig. 5, where rwall is the radius of wall with the length L, the equation (15) may be transformed to the form PLg r2g þ PLc r2c þ PLe r2e ¼ Vvap
Three conditions of balance occur between the three vessels. These conditions may be described by the following pairs:
ref
62 2 62 2 p Tg pðTc Þ r þ r ¼0 2vliq g g 2vliq c c
(21)
c) Condenser-evaporator balance pðTc Þ pðTe Þ
62 2 62 2 r þ r ¼0 2vliq c c 2vliq e e
(22)
(16)
In each of the equations there is the function p(T ) which may be replaced by the following function:
(17)
pðTÞ ¼
where: Vvap
ref
¼ PLg r2g
wall
þ PLc r2c
wall
þ PLe r2e
wall
Vliq
ref
The expression Vvap_ref determines the associative volume of the refrigerant vapour, whereas Vliq_ref means the liquid volume. If the geometry of the exchangers is not ring-shaped, the equations (16,17) may be transformed using a substitute length: L ðrÞ ¼
AðrÞ 2Pr
(18)
pðTÞ T T
(23)
allowing to depend the equations on the unknown variables Tg, Tc, Te. The variability of the calculated expressions in the analyzed range of temperature amounts, strongly decomposes numerical calculations; therefore, the expression of the left side of the equations (20–22) should be replaced by the linear formula:
It was assumed, that the steam density is significantly smaller than the liquid density.
p Tg pðTe Þ p Tg pðTe Þ Tg Te p Tg pðTe Þ ¼ Tg Te Tg Te
(24)
4.4.
p Tg pðTc Þ p Tg pðTc Þ Tg Tc p Tg pðTc Þ ¼ Tg Tc Tg Tc
(25)
Pressure balance equations
In each of the three heat exchangers of the roto-gravitational refrigerator the pressure balance of the connected vessels takes place. Total pressures in the outer (bottom) part of the vessels are equal. Between the two heat exchangers the system of balance, which results from the addition of the steam pressure and hydrostatic pressure, occurs. Hydrostatic pressure of the rotating ring of the liquid may be calculated as follows: 62 2 r r2liq Dp ¼ 2vliq
(19)
sf
pðTc Þ pðTe Þ ¼
4.5.
pðTc Þ pðTe Þ pðTc Þ pðTe Þ Tc Te Tc Te Tc Te
(26)
Linear equation system
Mathematical formulas (5–26) allow to determine a system of six linear equations, describing the dependence of the unknown variables Tg, Tc, Te, r2g , r2c , r2e on the surrounding parameters Tg_ext, Tc_ext, Te_ext. The calculated parameters
Table 2 – Main working conditions of the refrigerator for calculation example. Heat exchanger (part)
Generator Evaporator Condenser (tank) Condenser (film)
Radius
Length
m
m
Liq.–gas surface
Wall
0.0963 0.0296 0.0400 0.0350
0.1010 0.0350 0.0450 0.0350
0.10 0.20 0.03 0.40
Refrigerant sat. temp.
COP
Vref
n
u
C
–
cm3
rpm
rad s1
90 22
0.16
554
1,000
105
Sat. pressure
Heat flux
C
kPa
W
80 15 35 35
47.4 1.70 5.63 5.63
500 79
–
Ext. air temp.
– 579
30
10
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Fig. 6 – Changes of temperatures and liquid levels inside the refrigerator as a reaction to the change of the external temperature of the steam generator – the results of the numerical calculation.
with the expressions wen, wej, Vvap_ref and the working properties F(r), A(r) of the heat exchangers allow to determine the unknown parameters of the balance occurring in the refrigerator. In the matrix record the system of six balance equations may be presented as follows: energy balance eq: ejector balance eq: liquid refrig: balance eq: gen:-evap: press: bal: eq: gen:-cond: press: bal: eq: con:-evap: press: bal: eq:
2
a0 a1 a2 0 6 a6 0 a8 0 6 6 0 0 0 a15 6 6 a18 0 a20 a21 6 4 a24 a25 0 a27 0 a31 a32 0 2 3 2 3 tg wen 6 tc 7 6 wej 7 6 7 6 7 6 te 7 6 Vvap ref 7 6 7 7 ¼ 6 6 r2g 7 6 0 7 6 7 6 7 4 r2 5 4 0 5 c 0 r2e
0 0 a16 0 a28 a34
3 0 0 7 7 a17 7 7 a23 7 7 0 5 a35
(27)
where the matrix of free terms determines the surrounding influence on balance conditions. The system of six linear equations may be solved numerically and the proper computer software has been built for this purpose. The equation (27) solved once does not produce a correct result, which results from the dependence of the coefficients a0–a35 and unknown variables. Numerous equation solution allows to reach good accuracy of the unknown variables, which was described by Kasperski (2008).
5.
Calculating example
The calculating example has been made for the gravitational refrigerator with water as a refrigerant. The heat exchanger temperatures were adjusted to the air-conditioning use of the refrigerator (Table 2) just as in one of the refrigerators experimentally examined by the author. The equation (10) was determined by numerical calculations according to the designed experiment. The computer software was used for
testing the influence of the external air inlet temperature on the balance parameters inside the refrigerator. The compact results of calculations are presented in Fig. 6 on the example of the air temperature heating the steam generator. It can be shown that along with the air temperature increase, the saturation temperature in the steam generator increases as well. It slightly increases in the condenser. In the evaporator the temperature decreases, which results from the ejector’s properties. Along with the increase of steam generator temperature the radius of the liquid surface increases. The liquid pushed out of the generator flows into the evaporator and condenser. The cooling capacity increases along with the temperature of the steam generator. Together with the change of the generator’s surrounding temperature, the refrigerant’s temperature in the generator and evaporator changes – it influences the process of the heat exchange, and, therefore, the change of COP and its local maximum. The model showed the ability to calculate the states of balance inside the roto-gravitational refrigerator in response to the surrounding temperature disturbance. For simulations it is recommended to keep a few millimeters of thickness of the refrigerant layers in each of the exchangers. Such thickness allows to self-regulate the refrigerator’s levels for the changes of a few C in the steam generator surrounding temperature. The accuracy of the calculations conduced on the described model requires further experimental validation. The refrigerator’s construction should enable an observation and measurement of the refrigerant liquid in all heat exchangers. The working stations constructed by the author did not offer such a possibility.
6.
Conclusion
The conception of the roto-gravitational refrigerator is a direct development of the gravitational refrigerator. The accelerations of rotary motion are bigger than the gravitational
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acceleration and allow to decrease significantly the indispensable size of the gravitational refrigerator. Therefore, in the roto-gravitational refrigerator it is possible to use a highpressure refrigerant with a significant rotation speed. A very small size of the refrigerator with a small amount of the refrigerant is possible. The small size may cause the refrigerant drying up in the exchanger under the influence of the change of surrounding parameters, for example, the temperature of the steam generator surrounding. The calculating model of the balance allows to test the phenomena of drying up of the exchangers.
references
Hong, W.J., Alhussan, K., Zhang Jr., H., Garris, C.A., 2004. A novel thermally driven rotor-vane/pressure-exchange ejector refrigeration system with environmental benefits and energy efficiency. Energy 29, 2331–2345.
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Huang, B.J., Hu, S.S., Lee, S.H., 2006. Development of an ejector cooling system with thermal pumping effect. International Journal of Refrigeration 29, 476–484. Kasperski, J., 2008. Mathematical model of thermal and substance equilibrium in installation of gravitational ejector refrigerator. Chemical and Process Engineering 29, 997–1011. Ling, Z., 2004. A study on the new separate heat pipe refrigerator and heat pump. Applied Thermal Engineering 24, 2737–2745. Nguyen, V.M., Riffat, S.B., Doherty, P.S., 2004. Development of a solar-powered passive ejector cooling system. Applied Thermal Engineering 21, 157–168. Pietrowicz, S., Kasperski, J., 2007. The numerical modeling of thermo – flow processes in high – speed rotation ejector used in refrigerating system, The 22nd International Congress of Refrigeration, China, ICR07-B1-1076. Srisastra, P., Aphornratana, S., 2005. A circulating system for a steam jet refrigeration system. Applied Thermal Engineering 25, 2247–2257. Srisastra, P., Aphornratana, S., Sriveerakul, T., 2008. Development of a circulating system for a jet refrigeration cycle. International Journal of Refrigeration 31, 921–929.