Dynamic characteristics of thermostatic expansion valves

K. H i g u c h i and M . H a y a n o

Key words: valve, dynamic characterist ics, thermosta t ic expansion valve

Caract ristiques dynamiques des d6tendeurs thermo- statiques Les AA. ont effectu# des recherches analytiques et exp#rimenta/es sur les facteurs intervenant clans/es caract#ristiques de r#ponse dynamique au d#bit des d#tendeurs thermostatiques.

D'apr#s les dquafions fondamentales les r#ponses du pointeau au d#bit peuvent #tre d#cr/tes par des fonctions de transfert et le diagramme de Bode. On peut r#aliser un dispositif de mesure permettant d'obtenir les r#ponses fr#quent/elles de d#bit d'un pointeau. La comparaison entre les valeurs exp#rimentales obtenues d'apr#s les mesures de r#ponse fr#quentie/le et /es va/eurs calcu/#es

' obtenues d'apr#s/'analyse a fair ressortir une bonne concordance.

Analytical and experimental investigations of the factors which affect the dynamic flow response characteristics of thermostatic expansion valves have been made.

From the basic equations, the flow responses of a valve can be described by transfer functions and the Bode diagram. A measuring

device can be developed to obtain the frequency responses of the f low rate of a valve. The comparison between the experimental values obtained from the frequency response measurements and the calculated values obtained from the analysis showed good agreement.

Nomencla ture

A AD AE As

GR

g H

K

k = k 1 + k 2 + k D I

PD Pa

thermal equivalent to work diaphragm effective force area rod cross-sectional area in Fig. 1 valve effective force area specific heat of sensing bulb wall

inside diameters in Fig. 1

f low rates in Fig. 1

gravitational constant specific enthalpy of refrigerant in sensing bulb determined gain by the specification of superheat adjustable mechanism composition spring constant capillary tube length

pressures in Fig. 1

The authors are from the Department of Mechanical Engineering, Tokyo University of Agriculture and Technology. Tokyo, Japan. Paper received 28 October 1981.

R E

R

RA

Rw

s

v

W WB X ®A

®B ®s

pressures in Fig. 1

rates of heat f low in Fig. 1

gas constant of refrigerant in sensing bulb thermal resistance between the sensing bulb and its ambient air thermal resistance between the sensing bulb and the outlet tube wall of evaporator Laplace operator

chamber volumes in Fig. 1

specific volume of refrigerant in sensing bulb weight of refrigerant in sensing bulb sensing bulb wall weight valve opening ambient air temperature of sensing bulb sensing bulb temperature degrees of superheat setting of expansion valve

01 40-7007/82/04021 6-0583.00 © 1982 Butterworth & Co (Publishers) Ltd and IIR 211} International Journal of Refrigeration

Nomenclature (continued)

~)w

#

outlet tube wall temperature of evaporator viscosity of saturated vapour in capillary tube

Superscripts

' saturated liquid " saturated vapour

Subscript

O value for equilibrium state

F(" ,-IH'l ( al l" av" l

-AVB o+CBWa time constant relating the heat

capacity of sensing bulb

128~/A~ TD-- ~d4g~ time constant relating the flow

resistance of capillary tube

, VDo (~qPB "} TE=Rw{H"-H }o#7-t~-~o... o~- -o> time constant

interacting the pressure and the temperature of charged refrigerant in sensing bulb

The thermostatic expansion valve is widely used in refrigerating and air-conditioning systems. The testing method of rated capacity of this valve is established by ASHRAE Standard 1 7-75 for static flow characteristics. However, this testing method is as yet unestablished for dynamic flow characteristics.

Therefore, we developed an experimental device and technique to obtain the flow characteristics of the thermostatic expansion valve under working conditions. The static flow and the frequency flow characteristics were measured with the same testing device. The experimental results were then compared with the analysis. We have quantitatively established the parameters relating the frequency characteristics of the valve.

Basic equations on dynamic characteristics

In this report, the dynamic analysis was accomplished on the ordinary liquid charged type expansion valve with internal or external equalizer. 1 A schematic cross-section of the thermostatic expansion valve is shown in Fig. 1.

Basic equations on the sensing bulb are derived by using assumptions as follows: one, the charged liquid refrigerant stays in the bottom of sensing bulb; two, the charged liquid and vapour refrigerants are equal to the sensing bulb wall temperature, and stay saturated; three, the sensing bulb has a uniform temperature.

Under these assumptions, we can derive the basic equations for describing the dynamic flow characteristics of the expansion valve. That is, by using equations such as the heat energy balance equation and the principle of mass conservation on the sensing bulb, the equilibrium equation applied to the fluid force and the spring force on the diaphragm, the equation of state of the charged refrigerant in the sensing bulb, and so on.

vo,p ~ , d,l .m. ,~.p,,w,':~ 1

. . . . . . . i . . . . \ - - C-,;o i..,,, o.s.

p,

Fig. 1 Thermostatic expansion valve

Fig. I D#tendeur thermostatique

,. II ,alol -

I , , ' P . J o l -

+ ÷

7 Ao 1 72( ,o- o '

Fig. 2 Block diagram for thermostatic expansion valve

Fig. 2 Organigramme du d#tendeur thermostatique

The variables in the basic equations can be written as sums of the equilibrium state and the perturbation quantity. Thus the expansion valve behaviour is considered in terms of small perturbations from the

Volume 5 Number 4 July 1982 217

equilibrium state, using linearizations and Laplace transformations of the equations.

By combining these equations, the transfer function relating the flow response of the expansion valve with external equalizer can be obtained as

] 1 8A(S) --aB(S)}+~w(S) (1 + TBS ) (1 + T~S) + TES

-- { TETDS2 (1 + TBS ) (1 + T~s) + T~s

} 1 ,J'a® B'} + 1 (1 + TDS)~.~P--BB]O

+~-ps(S)

k Xo (@OB] mD 2(mdo-mso) J ~]~Bj'o{Pd(S)

--AD GR ° ~ R(s) (1)

From this equation, the block diagram describing the dynamic characteristics of the expansion valve with external equalizer are shown in Fig. 2. And, when Ps----PE, the transfer function and the block diagram agree with those of the expansion valve with internal equalizer.

C a l c u l a t e d f l o w r e s p o n s e s a n d i ts c o n s i d e r a t i o n s

Using the analytical equation, the nature of the various flow responses are now considered.

Effect of out/et tube wall temperature I~ w of evaporator. When refrigerating system pressure p,, pe and the equalizer line pressure PE and also the ambient air temperature OA of sensing bulb does not change, the frequency response of f low rate gR due to perturbation of the outlet tube wall temperature ®w of the evaporator is shown in Fig. 3. The curves in this figure are calculated by (2) which is derived from (1).

g.(s) R, Ao aP4, G.o

RAT;% a%Jo 0 (2) Ow(s)

1+ TBS ( I+TDS)+RA+R ~

Where the calculation used the determined values by the static f low measurements, the results were as follows:

RA - - = 0 . 8 9 6 5 RA+Rw

AD{aPB] GRo ~-~ ~j~o-~-o = 0.001 60 kgs 1°C -'

TB=32.0 S, To=2.0 s, TE=0.9 s (time constants obtained analytically).

Goi~°-°-°'O-o-o.o.. -60 0,%

--'~ -"---"~'~4L o, 0 ~° ~'i'-e \ -3o

_~" -6o

o,.w o \ 1-, o I I

O.001 0.OI 0,1 I Angulor frequency ~, rod s -I

Fig. 3 Analytical frequency response and experimental result for gR/eW

Fig. 3 R#ponse fr#quentiel/e analytique et r#sultats exp#nmentaux pour gR/OW

-6C ~ ~,

0

E -60

A ,60.o \ ~ . % ~ . , K \ -9o =~

B 64.0 \ \-'q~",,%. -12o o. -BC C 32.0 \ \ \ ~

9 ,6.o_ %:2.o, \ \ \ \" -,5o

O.OOI Ol.OI 011 Angulor frequency ~,rod s -~

Fig. 4 Effect of time constant T B on frequency response gR/EW

Fig. 4 /nfluence de/a constante de temps T B sur /a r#ponse fr#quentie/le gR/gW

-60 " ~

o

-- - 7 0 - 3 0 .~.

~ -6o o ~ =" %,s \ ' ~ - E -90 ~

A I0.0 B 4.0 -120 a.

-80 C 2.0 D 1.0 T8 =32.0s -150 E 0.4 T E = 0.9 s -180

o.o0, o'.o, o'., Angulor frequency ~, rod S - I

Fig. 5 Effect of time constant T o on frequency response gR/OW

Fig. 5 /nfluence de/a constante de temps T D sur /a r#ponse fr~quentie//e gR/OW

Figs 4, 5 and 6 show the frequency response of the flow through the expansion valve for various values of the time constants. From these analyses, the instability (ie hunting) of the combined system of the thermostatic expansion valve and evaporator are

218 Revue lnternationale du Froid

greatly affected by the values of the time constants Te and TD, and it was shown that the selection of these values is critical.

Effect of outlet and inlet pressure p, and Pd on expansion valve flow. In this study, only the outlet or inlet refrigerant pressure p,, Pd of the expansion valve is perturbed, using the internal equalizer (ps=PE). From (1), the transfer functions are respectively

gR(s) p,(s) - RA AD-AsoGRo

RA+Rw k Xo TDTES2

(3)

AD-AsoGRo 1 GRo k X o (1 +TDS) 2(Pdo-Pso)

g.(s) Pd(S ) -- RA Aso ~ o

RA+Rw k -,oTDTES2

R - - A 7-ES}( {(1 + R A ~ T B S ) ( 1 +TDS)4 RA+R w 1 + T ~ )

Aso GRo 1 GRo 4

k X o (1 +TDs) 2(Pdo-P~o )

(4)

The frequency responses in these cases are shown in Figs 7 and 8. The gain constants in the calculations had the following values:

AD-Aso GRo k Xo =0.01406 cm 2 5 -1

Aso G Ro =0.00009356 cm 2 S -1 k Xo

GRo 2 (Pdo-- Pso)

=0.00601 cm 2 s -1

Gain

-6C

~ ~ B

%1= -7c A B _._~=

c

-8C B O. 18 T = 2.0 s ~ B"

\ \ 180

o.o01 0Ol O. I Angular frequency oJ, rod s - I

Fig. 6 Effect of time constant T E on frequency response gR/~W

Fig. 6 Influence de la constante de temps T E sur la r#ponse fr#quentie/le gR/Ow

o == -30 1=,

19-

-60 E

-90 g.

-120

-150

-~-4C

rE= o.gs

"o

£

o O.

Gain

rD.s A I0.0

4.0 2 .0 I.o 0.4

-45 ~ 0

-20

0 0 0 1 0.01 O. I I I0 4 0 Angular frequency ~ , rod s -t

Fig. 7 Effect of time constant T D on frequency response gRIPs

Fig. 7 Influence de/a constante de temps T D sur /a r~ponse fr@quentielle gRIPs

-44.4

-44.!

-44.E

c

@

Gain

TD, s

A I0.0 B 2.0 T 8 = 32 .0 s C 0.4 TE= 0.9s

A B C

i I I

o.6oJ o.ol oJ i io

Angular frequency ~ , rod s -I

Fig. 8 Effect of time constant T D on frequency response gR/Pd

Fig. 8 Influence de la constante de temps T D sur /a r#ponse fr@quentielle gR/Pd

#

3.5

0

From these transfer functions and values, the following facts became evident. The frequency of flow response is decided approximately by the values of the time constant T O . On the other hand, when the values T B and T e differ very much the effect of the magnitudes of these time constants is very small, and the difference of responses has not been discernable.

E x p e r i m e n t s on f l o w c h a r a c t e r i s t i c s

The liquid charged type thermostatic expansion valve with internal equalizer (for R 1 2, rated capacity of 900 kcal h -1) was used in this study. This valve is the same as was used in the analysis. Using water as the working fluid, experiments were performed on flow characteristics under static and fluctuating conditions. These experimental flow characteristics were also used to check the validity of the analytical study. 2

The experimental equipment is shown in Fig. 9 and the control device for varying the sensing bulb temperature is shown in Fig. 10.

Control of the sensing bulb temperature. In the control of sensing bulb temperature, the heat flow between the sensing bulb and the environment must be as close as possible to the condition when the

Volume 5 Num~ro 4 Juillet 1982 219

sensing bulb is attached to the evaporator outlet tube wall. Thus the sensing bulb was attached by a connecting band, modelled on the tube wall, to the thermo-electric module (maximum heat absorption rate of 32 W) as shown in Fig. 10. By this means the sensing bulb temperature could be controlled by feedback means constantly and sinusoidally.

Frequency flow characteristics of the thermostatic expansion va/ve. By using the apparatus shown in Fig. 9, frequency response experiments on the thermostatic expansion valve, without hysteresis, were accomplished with either the sensing bulb or the base plate temperature changing sinusoidally. No difference could be detected, so the results in this study have the base plate temperature changing sinusoidally. The frequency used was from 0.1 to 0.0005 Hz, and the responses of the base plate temperature, the sensing bulb temperature and the flow rate were recorded.

From the experimental results the Bode diagram was derived as shown in Fig. 3. In this figure, the calculated results are represented by the solid lines, and there is good agreement with the measured results.

In addition, the various static characteristics of the expansion valve, required in this analysis of the dynamic f low characteristics of this valve, were measured.

Conclusions

A measuring device can be developed to obtain the frequency responses of the f low rate of a thermostatic expansion valve. It was shown that this device was extremely useful in showing the static and dynamic characteristics of the valve.

The dynamic flow response characteristics of a thermostatic expansion valve relating to

L Float type [ ] displacement

transducer

I Temperature control device

Laminar flow type flow meter

Pressure regulotind valve

Measuring Sam )led cylinder expansion

valve

Pressure regulating valve

Pressurized pump

Fig. 9 Experimental rig for f low characteristic studies on the expansion valve

Fig 9 Dispositif exp#rimenta/ pour /'#tude des caract#ristiques d'#cou/ement sur le d#tendeur

Connecting b a n d ~ ~ c c thermocouple ~ S e n s i n g bulb

B o s e p l o t e ~ J. ! _~_ - ~ I- I

Thermo-electric J I module , --'=P-- J Woterjacket J "-,=---- Cooling woter

Fig. 10 Temperature control device for sensing bulb

FI~7. 10 Dispositif de r#gu/ation de la tempdrature pour le bu/be

perturbations of the outlet tube wall temperature, the inlet and outlet refrigerant pressure, and the ambient temperature of the sensing bulb, can be obtained analytically. These characteristics can be described by transfer functions.

In previous investigations on the expansion valve it has been suggested that the flow characteristics may be expressed by the first order lag element 1/(1 +TBs) relating the heat capacity of the sensing bulb. From the results of our investigations, it became evident that we must consider other elements. These include the first order lag element 1/(1 + TDS) relating the charged refrigerant resistance in the capillary tube, the interactive element TES for both the sensing bulb temperature and the charged refrigerant pressure and the rate of the heat transfer from ambient air to the sensing bulb qA(s) = {eA(s) - ~ B(S)}iRA, etc.

The comparison between the experimental values obtained from the frequency response measurements and the calculated values obtained from analysis showed good agreement.

The effects of the magnitudes of the gain constants and the time constants relating the flow characteristics of the thermostatic expansion valve were clearly demonstrated.

If the dynamic characteristics of the evaporator can be obtained, the hunting phenomenon sometimes present in the combined system of evaporator and expansion valve can be elucidated lending to the possibility of improved dynamic characteristics.

References

1 Higuch i , K., Hayano, M. Dynamic characteristics of thermostatic expansion valve, 1 st report: Analytical study, Refrigeration, Japanese Association of Refrigeration 55-636 (1 980) 859

2 Higuchi , K., Hayano, a . Dynamic characteristics of thermostatic expansion valve, 2nd report: Experimental study, Refrigeration, Japanese Association of Refrigeration 55-636 (1980) 869

220 International Journal of Refrigeration