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Control with a thermostatic expansion valve
Control with a thermostatic expansion valve P. M . T. B roersen
Key w o r d s : t h e r m o s t a t i c valves, r e f r i g e r a t i o n , c o n t r o l s y s t e m s
R6gulation par d6tendeur thermostatique Les recherches sur /a stab#it# des syst#mes frigorifiques s'appuient gdn#ra/ement sur une description du ddtendeur en rant que r#gu/ateur proportionnel de la surchauffe. Ceci est raisonnab/e parce que /e rdg/age du ddbit de/iquide est
proportionne/ ~ la diff#rence de press/on sur /e diaphragme du ddtendeur. Cependant cette diffdrence de pression est lide ~ un signal obtenu en consid#rant s#par#ment deux temp#ratures dont /a diffdrence est /a surchauffe et /a fonction de transfert du bulbe, sensible ~ /a tempdrature, n'agit que sur /'un des deux composants. Cet article traite de/a dynamique et prdsente une mdthode permettant de simplifier le probl#me de rdgulation ~ variables multiples.
Investigations of the stability of refrigeration systems are usually based on a description of a thermostatic expansion valve as a proportional superheat controller. This is reasonable because the controlled liquid flow is proportional to the pressure difference on the diaphragm in the valve. However, this pressure difference is related to a signal
obtained by considering separately t w o temperatures whose difference is the superheat, and the transfer function of the temperature sensing bulb acts only on one of the t w o components. This paper treats the dynamics and presents a method to simplify the multivariable control problem.
A thermostatic expansion valve is often used for the control of dry evaporators. This valve is represented as a proportional controller of the superheat in most theoretical studies, 1 but this simple model cannot take into account the different treatment of the two temperatures that together define the superheat. The behaviour of the bulb transforming the temperature of the superheated vapour into pressure variations has been determined experimentally. 2,3 This dynamic valve description can be introduced into a mathematical model for the control behaviour of a simple refrigeration cycle. 4 The experimentally verified dynamics of the valve is combined with the mathematical model to investigate the stability of the complete refrigeration system.
temperature has a static relation with the evaporation pressure, according to Clapeyron's law; this pressure creates the force on one side of the diaphragm in the valve. The bulb pressure acts on the other side and it has a similar static relation with the bulb temperature. But the relation between the bulb temperature Tb and the temperature of the superheated vapour Ts is given by the transfer function of the bulb according to:
Control system
Volume 5 Number 4 July 1982
(1)
We assume that small variations of the controlled liquid flow q~ through the valve are linear with the fluctuations in the pressure difference on the diaphragm: e~=7(P~-Pe)
The control of a refrigeration process is a multivariable control problem. The superheat used in the control is the difference between two components: the temperature of the superheated vapour at the end of the evaporator mi.nus the evaporation temperature. The evaporati£n The author is from the Department of Applied Physics, Delft University of Technology, PO Box 5046. 2600 GA Delft. The Netherlands. Paper received 26 November 1981.
Tb=HbT~
(2)
So the control of the valve is proportional with respect to the pressure difference. Furthermore:
Pe=C'l me
(3)
Pb = c~ Tb
(4)
I
#
where c 1 and c 2 are constants in a linearization for small fluctuations around a given operating point. 0140-7007/82/040209-0483.00 © 1982 Butterworth & Co (Publishers) Ltd and IIR
209
Moreover, c'1 and c'2 are equal if the bulb is filled with the same refrigerant as the evaporator.
I
By taking
(5)
c, = 7c[
(6)
~ l = C 2 H b T s - C 1 Te
q
This last equation shows that a thermostatic expansion valve is not a proportional superheat controller, because e)is not proportional to the superheat Ts-Te; the variations in both Ts and Te should be specified to find el. In other words, the same superheat fluctuations can cause different fluctuations in the liquid flow, depending on whether Ts or Te was varying.
Te=HI~ I
(7)
f s = H2~ I
(8)
where H 1 and H 2 are the frequency dependent transfer functions describing the dynamics of the evaporator. H 1 and H 2 a r e completely determined by known or measurable quantities like the length of the evaporator and the enthalpy of the refrigerant. 4 The equations (6), (7) and (8) together yield
(9)
By denoting (10)
we can treat this system as a control problem with the liquid flow as the only variable in a feedback loop. In this way we can use the simple criteria for
210
/
---
Fig. 1 Block diagram with the two branches of the thermostatic expansion valve
Equation (6) gives a relation for the liquid flow, but it is clear that both evaporation temperature and the temperature of the superheated vapour depend dynamically on the liquid flow. Broersen 4 determined a mathematical model for the response of an evaporator to variations of the liquid flow by linearization of the governing differential equations for small fluctuations around an operating point. The compressor and the condenser could be neglected in this model, because the condenser was controlled at a constant pressure and temperature, whereas the valve in the suction line of the compressor caused a supercritical vapour flow eliminating the influence of the compressor on the evaporator dynamics. The large static non-linearities defining the operating point as well as smaller ones due to hysteresis are omitted in the linearized model or replaced by linear dynamics, the philosophy being that the linearized system has to be stable for small fluctuations and the non-linear system will have an acceptable behaviour for larger variations. The dynamics of a system consisting of an evaporator controlled by a thermostatic expansion valve can be represented in the block diagram of Fig. 1.4 The equations are:
Hto t = c 1H 1 - C2HbH 2
H2
-1
it follows from (1) to (4):
qh(1 4-c1H 1 -c2HbH2)=0
J
Fig. 1 Diagramrne de principe avec les deux branches du d#tendeur
stability from univariate control theory for this multivariable problem. We will demand that the phase margin must be greater than 45 ° for stability in practice.
Transfer functions Broersen 4 described the transfer function of the bulb, relating the pressure on the diaphragm in the valve to the temperature of the superheated vapour, as a third order system with three separate heat capacities: the wall of the evaporator, the bulb wall and finally the content of the bulb. However, identification experiments 23 showed that the bulb together with its content behaves as one single time constant, denoted T2, which is the product of the heat resistance between the evaporator wall and the bulb with the heat capacity of the complete bulb. A second time constant'[ 1 is given by the product of the heat resistance from superheated vapour to evaporator wall with the heat capacity of the evaporator wall. The transfer function of the bulb becomes: ~'2 H b = 2 1 C2 '[ 1`[2S + (T1 _FT2 4_ 0~t.1 ) S + 1
(11)
with ~r~ as a coupling term rising in two coupled passive first order RC members. '[2 is 1 6.2 s for a Danfoss type TKE 3 valve 3 for the refrigerant R 12. Wobst 2 found about 1 8 s for comparable valves. However, t2 depends heavily on the connection between the bulb and the evaporator because of the variation of the heat resistance; it can be as high as 60 s if the bulb is loosely taped, and about 20 s if the bulb is fixed with the normal bulb clamp. The 16.2 s has been found with a very tight metal clamp. The time constant'[~ of the evaporator wall depends on the velocity of the refrigerant, the thickness of the wall etc.; it will be in the order of 25 s. The constant ~ in (5) is the ratio of the heat capacity of the bulb and the heat capacity of that part of the evaporator (or suction line) wall that is
Revue Internatlonale du Froid
in heat contact with the bulb. This ratio ~ depends on the relative magnitudes of heat transports inside the pipe wall and into the pipe wall. Consequently, cc depends on the heat resistance between the bulb and the pipe and on the velocity of the superheated vapour. Values have been estimated in different situations, giving = between 0.2 and 1. It is advised to take ==0.5. The transfer functions have been calculated for % = 2 2 s, ~2=16.2 s and ~=0.5. Htot as well as its two components c~H~ and -c2HbH 2 are shown in the Figs 2 and 3; c~H~ itself is always stable because it remains in the 4th quadrant, but -c2HbH 2 alone would be unstable as it encircles - 1 . Together they give Hto t with a phase margin of 45 °. The transfer function of the bulb, shown in Fig. 4, is important because it is the only element that can be changed in the unstable branch w i t h o u t influencing the stabilizing effect of the other. The table gives the phase margin as a function of the Re
J 2o
.o
:o
8o
Fig. 2 The real and the imaginary parts of the transfer functions describing the behaviour of the controlled evaporator as a function of the frequency •Fig. 2 Partie rOe/le et partie imaginaire des foncdons de transfert d#crivant le comportement de I'#vaporateur en foncdon de/a fr#quence
I
-C~Hb H2 Re
I
2
/
/ Ht°t
-4
Fig. 3 Detail of the transfer functions of Fig. 2 near the origin; o - f = 0 . 0 0 6 Hz, x - f = 0 . 0 0 9 Hz Fig. 3 D#taff des fonctions de transfert de/a Fig. 2 au volsinage de/'origine. e-f=O.O06 Hz. x-f=O.O09 Hz
Volume 5 Num6ro 4 Juillet 1982
T a b l e 1. Phase m a r g i n as a f u n c t i o n o f t h e t i m e c o n s t a n t o f t h e pipe w a l l
Tab/eau 1. Dfffdrence de phase en fonction de/a constante du temps de/a paroi du tube "[1' S
phase margin, °
0
57
2
48
5
40
10
37
15
39
20
44
22
45
25
48
30
53
50
75
evaporator wall time constant~ 1, forT2=16.2 s and ~=0.5. Table 1 shows that both smaller and greater time constants give a larger phase margin. This conclusion agrees with previous results about variation of time constants in the bulb transfer function. It is clear that in practice stability with a phase margin greater than 45 ° can only be achieved with the slow dynamics; time constants smaller than 5 s are not feasible. The tendency to find a greater phase margin for slower and for quicker bulb dynamics (Table 1 ) can be understood by reference to Figs 2 and 3. In the case of slower bulb dynamics, the contribution of -C2HbH2to Htot is diminished for a certain frequency, so t]qe stable clH ~ has more influence o n Hto t. On the other hand, a smaller time constant gives less phase shift in the unstable branch for a given frequency, so -c2HbH 2 hardly comes into the second quadrant and H,ot remains stable. This explanation makes explicit use of the fact that Hb appears in one branch only, thus indicating the necessity of the presented model in understanding the stability of systems with a thermostatic expansion valve. The table is calculated for one specific set of values for all parameters appearing in H~ and H2; detailed results may be different for other parameter values. But the general conclusion will be that slower dynamics in the bulb transfer function will improve the behaviour of the refrigeration system as far as stability is concerned. The stabilizing effect of c~H~ is explicitly used in a thermostatic expansion valve with external equalizing. This valve is used when the pressure drop over the evaporator is too great to justify the assumption of a constant pressure over the total length of the evaporator. The external equalizing line is connected to that part of the evaporator (or the suction line) where the bulb is also clamped. In this way, a maximum part of the superheat fluctuations can be controlled by the stable feedback branch of the evaporation pressure. Feedback of the evaporation pressure with ciH~ is essential for stable
211
systems, because the feedback branch of the bulb w th - c 2 H b H ~ is always unstable for every Hb. This can easdy be venfied: putting a constant pressure on the external equalizing line instead of the evaporation pressure will always result in an unstable system in practice. Conclusions
An evaporator controlled by a thermostatic expansion valve should be modelled with two separate feedback branches, in one of which appears the transfer function of the bulb. The stability of the complete system depends strongly on this transfer function. Explanations for the stability dealing with a proportional superheat controller as model are insufficient as the two components of the superheat must be treated differently in their respective feedback loops. The contact between the bulb and the suction line is important. The lucky factor is that a bad contact increases the phase margin and consequently has a favourable effect on the stability. The same improvement can be reached by using a thick wall of the suction line under the bulb.
Corrigenda:
Computer-based refrigerant thermodynamic properties• Part 2: program listings, by C. Y. Chan and G. G. Haselden, volume 4, No 2, March 1981, 52-60, For refrigerant R 113 only the constants given for the liquid density equation are incorrect. The corrected values are given below.
Constants symbols RL(1 )
Numerical values as printed in paper 160.563004
RL(2)
0.00
RL(3) RL(4)
0.00 - 86.6076088
Corrected values 62.7292956 0,00 0.00 109.058809
RL(5)
0.00
0.00
RL(6)
0.00
0.00
RL(7)
48.4166044
-48.9166044
The mathematical model of this paper can give a theoretical explanation for the stability phenomena. But it can also be used to investigate which improvements can be reached with new valve constructions. Re
<
0.2
0.6
I.O
Reprints Reprints of all articles in this journal are available in quantities of 100 or more.
0.6
Reprints are essential -Fig 4 The real and the imaginary part of the transfer function of the bulb describing the relation betweenthe superheat temperature and the bulb temperature as a function of the frequency Fig. 4 La pattie r#elle et la partie imagina/re de la fonction de transfert du bulbe d#crivant la relation entre la temp#rature de surchauffe et la temperature du bulbe en fonction de la fr#quence
R eferences
1 James, R. W. Refrigeration and air conditioning systems. Ch. 3 of Modelling of Dynamical Systems. ed H Nicholson (1980) Peter Peregrinus Ltd, Stevenage, UK 2 Wobst, E. RegelungstechnischeKennwertedes thermostatischen Expansionsventils. Luft und Kaeltetechnik 15 (1979) 83-85 3 Broersen, P. M. T,, ten Napel, J. Identification of a thermostatic expansion valve. Preprints IFAC Symposium on Identification and System Parameter Estimation, Washington (1982) 4 Broersen, P. M. T,, van der Jagt, M. F. G. Hunting of evaporators controlled by a thermostatic expansion valve
• for the company that wants to distribute impartial comment on its activities to potential customers and clients • for the company that wants to up-date its technical staff on new techniques and new technologies
For full details of prices and availability of reprints. please write to The Reprint Department Butterworth Scientific Limited -- Journals Division PO Box63 Westbury House Bury Street Guildford Surrey GU2 5BH England
ASME Journal of Dynamic Systems, Measurement and Control 102 (1980) 130-135
212
International Journal of Refrigeration
Key w o r d s : t h e r m o s t a t i c valves, r e f r i g e r a t i o n , c o n t r o l s y s t e m s
R6gulation par d6tendeur thermostatique Les recherches sur /a stab#it# des syst#mes frigorifiques s'appuient gdn#ra/ement sur une description du ddtendeur en rant que r#gu/ateur proportionnel de la surchauffe. Ceci est raisonnab/e parce que /e rdg/age du ddbit de/iquide est
proportionne/ ~ la diff#rence de press/on sur /e diaphragme du ddtendeur. Cependant cette diffdrence de pression est lide ~ un signal obtenu en consid#rant s#par#ment deux temp#ratures dont /a diffdrence est /a surchauffe et /a fonction de transfert du bulbe, sensible ~ /a tempdrature, n'agit que sur /'un des deux composants. Cet article traite de/a dynamique et prdsente une mdthode permettant de simplifier le probl#me de rdgulation ~ variables multiples.
Investigations of the stability of refrigeration systems are usually based on a description of a thermostatic expansion valve as a proportional superheat controller. This is reasonable because the controlled liquid flow is proportional to the pressure difference on the diaphragm in the valve. However, this pressure difference is related to a signal
obtained by considering separately t w o temperatures whose difference is the superheat, and the transfer function of the temperature sensing bulb acts only on one of the t w o components. This paper treats the dynamics and presents a method to simplify the multivariable control problem.
A thermostatic expansion valve is often used for the control of dry evaporators. This valve is represented as a proportional controller of the superheat in most theoretical studies, 1 but this simple model cannot take into account the different treatment of the two temperatures that together define the superheat. The behaviour of the bulb transforming the temperature of the superheated vapour into pressure variations has been determined experimentally. 2,3 This dynamic valve description can be introduced into a mathematical model for the control behaviour of a simple refrigeration cycle. 4 The experimentally verified dynamics of the valve is combined with the mathematical model to investigate the stability of the complete refrigeration system.
temperature has a static relation with the evaporation pressure, according to Clapeyron's law; this pressure creates the force on one side of the diaphragm in the valve. The bulb pressure acts on the other side and it has a similar static relation with the bulb temperature. But the relation between the bulb temperature Tb and the temperature of the superheated vapour Ts is given by the transfer function of the bulb according to:
Control system
Volume 5 Number 4 July 1982
(1)
We assume that small variations of the controlled liquid flow q~ through the valve are linear with the fluctuations in the pressure difference on the diaphragm: e~=7(P~-Pe)
The control of a refrigeration process is a multivariable control problem. The superheat used in the control is the difference between two components: the temperature of the superheated vapour at the end of the evaporator mi.nus the evaporation temperature. The evaporati£n The author is from the Department of Applied Physics, Delft University of Technology, PO Box 5046. 2600 GA Delft. The Netherlands. Paper received 26 November 1981.
Tb=HbT~
(2)
So the control of the valve is proportional with respect to the pressure difference. Furthermore:
Pe=C'l me
(3)
Pb = c~ Tb
(4)
I
#
where c 1 and c 2 are constants in a linearization for small fluctuations around a given operating point. 0140-7007/82/040209-0483.00 © 1982 Butterworth & Co (Publishers) Ltd and IIR
209
Moreover, c'1 and c'2 are equal if the bulb is filled with the same refrigerant as the evaporator.
I
By taking
(5)
c, = 7c[
(6)
~ l = C 2 H b T s - C 1 Te
q
This last equation shows that a thermostatic expansion valve is not a proportional superheat controller, because e)is not proportional to the superheat Ts-Te; the variations in both Ts and Te should be specified to find el. In other words, the same superheat fluctuations can cause different fluctuations in the liquid flow, depending on whether Ts or Te was varying.
Te=HI~ I
(7)
f s = H2~ I
(8)
where H 1 and H 2 are the frequency dependent transfer functions describing the dynamics of the evaporator. H 1 and H 2 a r e completely determined by known or measurable quantities like the length of the evaporator and the enthalpy of the refrigerant. 4 The equations (6), (7) and (8) together yield
(9)
By denoting (10)
we can treat this system as a control problem with the liquid flow as the only variable in a feedback loop. In this way we can use the simple criteria for
210
/
---
Fig. 1 Block diagram with the two branches of the thermostatic expansion valve
Equation (6) gives a relation for the liquid flow, but it is clear that both evaporation temperature and the temperature of the superheated vapour depend dynamically on the liquid flow. Broersen 4 determined a mathematical model for the response of an evaporator to variations of the liquid flow by linearization of the governing differential equations for small fluctuations around an operating point. The compressor and the condenser could be neglected in this model, because the condenser was controlled at a constant pressure and temperature, whereas the valve in the suction line of the compressor caused a supercritical vapour flow eliminating the influence of the compressor on the evaporator dynamics. The large static non-linearities defining the operating point as well as smaller ones due to hysteresis are omitted in the linearized model or replaced by linear dynamics, the philosophy being that the linearized system has to be stable for small fluctuations and the non-linear system will have an acceptable behaviour for larger variations. The dynamics of a system consisting of an evaporator controlled by a thermostatic expansion valve can be represented in the block diagram of Fig. 1.4 The equations are:
Hto t = c 1H 1 - C2HbH 2
H2
-1
it follows from (1) to (4):
qh(1 4-c1H 1 -c2HbH2)=0
J
Fig. 1 Diagramrne de principe avec les deux branches du d#tendeur
stability from univariate control theory for this multivariable problem. We will demand that the phase margin must be greater than 45 ° for stability in practice.
Transfer functions Broersen 4 described the transfer function of the bulb, relating the pressure on the diaphragm in the valve to the temperature of the superheated vapour, as a third order system with three separate heat capacities: the wall of the evaporator, the bulb wall and finally the content of the bulb. However, identification experiments 23 showed that the bulb together with its content behaves as one single time constant, denoted T2, which is the product of the heat resistance between the evaporator wall and the bulb with the heat capacity of the complete bulb. A second time constant'[ 1 is given by the product of the heat resistance from superheated vapour to evaporator wall with the heat capacity of the evaporator wall. The transfer function of the bulb becomes: ~'2 H b = 2 1 C2 '[ 1`[2S + (T1 _FT2 4_ 0~t.1 ) S + 1
(11)
with ~r~ as a coupling term rising in two coupled passive first order RC members. '[2 is 1 6.2 s for a Danfoss type TKE 3 valve 3 for the refrigerant R 12. Wobst 2 found about 1 8 s for comparable valves. However, t2 depends heavily on the connection between the bulb and the evaporator because of the variation of the heat resistance; it can be as high as 60 s if the bulb is loosely taped, and about 20 s if the bulb is fixed with the normal bulb clamp. The 16.2 s has been found with a very tight metal clamp. The time constant'[~ of the evaporator wall depends on the velocity of the refrigerant, the thickness of the wall etc.; it will be in the order of 25 s. The constant ~ in (5) is the ratio of the heat capacity of the bulb and the heat capacity of that part of the evaporator (or suction line) wall that is
Revue Internatlonale du Froid
in heat contact with the bulb. This ratio ~ depends on the relative magnitudes of heat transports inside the pipe wall and into the pipe wall. Consequently, cc depends on the heat resistance between the bulb and the pipe and on the velocity of the superheated vapour. Values have been estimated in different situations, giving = between 0.2 and 1. It is advised to take ==0.5. The transfer functions have been calculated for % = 2 2 s, ~2=16.2 s and ~=0.5. Htot as well as its two components c~H~ and -c2HbH 2 are shown in the Figs 2 and 3; c~H~ itself is always stable because it remains in the 4th quadrant, but -c2HbH 2 alone would be unstable as it encircles - 1 . Together they give Hto t with a phase margin of 45 °. The transfer function of the bulb, shown in Fig. 4, is important because it is the only element that can be changed in the unstable branch w i t h o u t influencing the stabilizing effect of the other. The table gives the phase margin as a function of the Re
J 2o
.o
:o
8o
Fig. 2 The real and the imaginary parts of the transfer functions describing the behaviour of the controlled evaporator as a function of the frequency •Fig. 2 Partie rOe/le et partie imaginaire des foncdons de transfert d#crivant le comportement de I'#vaporateur en foncdon de/a fr#quence
I
-C~Hb H2 Re
I
2
/
/ Ht°t
-4
Fig. 3 Detail of the transfer functions of Fig. 2 near the origin; o - f = 0 . 0 0 6 Hz, x - f = 0 . 0 0 9 Hz Fig. 3 D#taff des fonctions de transfert de/a Fig. 2 au volsinage de/'origine. e-f=O.O06 Hz. x-f=O.O09 Hz
Volume 5 Num6ro 4 Juillet 1982
T a b l e 1. Phase m a r g i n as a f u n c t i o n o f t h e t i m e c o n s t a n t o f t h e pipe w a l l
Tab/eau 1. Dfffdrence de phase en fonction de/a constante du temps de/a paroi du tube "[1' S
phase margin, °
0
57
2
48
5
40
10
37
15
39
20
44
22
45
25
48
30
53
50
75
evaporator wall time constant~ 1, forT2=16.2 s and ~=0.5. Table 1 shows that both smaller and greater time constants give a larger phase margin. This conclusion agrees with previous results about variation of time constants in the bulb transfer function. It is clear that in practice stability with a phase margin greater than 45 ° can only be achieved with the slow dynamics; time constants smaller than 5 s are not feasible. The tendency to find a greater phase margin for slower and for quicker bulb dynamics (Table 1 ) can be understood by reference to Figs 2 and 3. In the case of slower bulb dynamics, the contribution of -C2HbH2to Htot is diminished for a certain frequency, so t]qe stable clH ~ has more influence o n Hto t. On the other hand, a smaller time constant gives less phase shift in the unstable branch for a given frequency, so -c2HbH 2 hardly comes into the second quadrant and H,ot remains stable. This explanation makes explicit use of the fact that Hb appears in one branch only, thus indicating the necessity of the presented model in understanding the stability of systems with a thermostatic expansion valve. The table is calculated for one specific set of values for all parameters appearing in H~ and H2; detailed results may be different for other parameter values. But the general conclusion will be that slower dynamics in the bulb transfer function will improve the behaviour of the refrigeration system as far as stability is concerned. The stabilizing effect of c~H~ is explicitly used in a thermostatic expansion valve with external equalizing. This valve is used when the pressure drop over the evaporator is too great to justify the assumption of a constant pressure over the total length of the evaporator. The external equalizing line is connected to that part of the evaporator (or the suction line) where the bulb is also clamped. In this way, a maximum part of the superheat fluctuations can be controlled by the stable feedback branch of the evaporation pressure. Feedback of the evaporation pressure with ciH~ is essential for stable
211
systems, because the feedback branch of the bulb w th - c 2 H b H ~ is always unstable for every Hb. This can easdy be venfied: putting a constant pressure on the external equalizing line instead of the evaporation pressure will always result in an unstable system in practice. Conclusions
An evaporator controlled by a thermostatic expansion valve should be modelled with two separate feedback branches, in one of which appears the transfer function of the bulb. The stability of the complete system depends strongly on this transfer function. Explanations for the stability dealing with a proportional superheat controller as model are insufficient as the two components of the superheat must be treated differently in their respective feedback loops. The contact between the bulb and the suction line is important. The lucky factor is that a bad contact increases the phase margin and consequently has a favourable effect on the stability. The same improvement can be reached by using a thick wall of the suction line under the bulb.
Corrigenda:
Computer-based refrigerant thermodynamic properties• Part 2: program listings, by C. Y. Chan and G. G. Haselden, volume 4, No 2, March 1981, 52-60, For refrigerant R 113 only the constants given for the liquid density equation are incorrect. The corrected values are given below.
Constants symbols RL(1 )
Numerical values as printed in paper 160.563004
RL(2)
0.00
RL(3) RL(4)
0.00 - 86.6076088
Corrected values 62.7292956 0,00 0.00 109.058809
RL(5)
0.00
0.00
RL(6)
0.00
0.00
RL(7)
48.4166044
-48.9166044
The mathematical model of this paper can give a theoretical explanation for the stability phenomena. But it can also be used to investigate which improvements can be reached with new valve constructions. Re
<
0.2
0.6
I.O
Reprints Reprints of all articles in this journal are available in quantities of 100 or more.
0.6
Reprints are essential -Fig 4 The real and the imaginary part of the transfer function of the bulb describing the relation betweenthe superheat temperature and the bulb temperature as a function of the frequency Fig. 4 La pattie r#elle et la partie imagina/re de la fonction de transfert du bulbe d#crivant la relation entre la temp#rature de surchauffe et la temperature du bulbe en fonction de la fr#quence
R eferences
1 James, R. W. Refrigeration and air conditioning systems. Ch. 3 of Modelling of Dynamical Systems. ed H Nicholson (1980) Peter Peregrinus Ltd, Stevenage, UK 2 Wobst, E. RegelungstechnischeKennwertedes thermostatischen Expansionsventils. Luft und Kaeltetechnik 15 (1979) 83-85 3 Broersen, P. M. T,, ten Napel, J. Identification of a thermostatic expansion valve. Preprints IFAC Symposium on Identification and System Parameter Estimation, Washington (1982) 4 Broersen, P. M. T,, van der Jagt, M. F. G. Hunting of evaporators controlled by a thermostatic expansion valve
• for the company that wants to distribute impartial comment on its activities to potential customers and clients • for the company that wants to up-date its technical staff on new techniques and new technologies
For full details of prices and availability of reprints. please write to The Reprint Department Butterworth Scientific Limited -- Journals Division PO Box63 Westbury House Bury Street Guildford Surrey GU2 5BH England
ASME Journal of Dynamic Systems, Measurement and Control 102 (1980) 130-135
212
International Journal of Refrigeration