Logic unconstrained multi-zone evaporator and condenser models

International Journal of Refrigeration 30 (2007) 926e931 www.elsevier.com/locate/ijrefrig

Logic unconstrained multi-zone evaporator and condenser models Liang-Liang Shaoa, Chun-Lu Zhangb,* a

Institute of Refrigeration and Cryogenics, Shanghai Jiaotong University, Shanghai 200240, China b China R&D Center, Carrier Corporation, 3239 Shen Jiang Road, Shanghai 201206, China Received 26 May 2006; received in revised form 21 August 2006; accepted 2 November 2006 Available online 22 December 2006

Abstract The multi-zone model is widely used in evaporator and condenser modeling. The basic two-zone evaporator model and three-zone condenser model might reduce to one-zone or two-zone model depending on the operating conditions. In the traditional multi-zone models, logic constraints are invoked to separate the possible running modes and trigger mode changes from one to another, which will lead to the complicated logic in implementation. A logic unconstrained multi-zone model has been developed in this paper. In the new model, all the logic constraints are transformed into continuously differentiable equations. As a result, all the equations can be solved simultaneously without any logic constraints. Numerical examples show that the new logic unconstrained multi-zone models are in good agreement with the traditional ones. Ó 2006 Elsevier Ltd and IIR. All rights reserved. Keywords: Refrigeration; Air conditioning; Process; Modelling; Evaporator; Condenser

Mode´lisation multi-zone de l’e´vaporation et de la condensation sans contraintes logiques Mots cle´s : Froid ; Conditionnement d’air ; Proce´de´ ; Mode´lisation ; E´vaporateur ; Condenseur

1. Introduction All kinds of evaporators and condensers are widely used in the refrigeration and air-conditioning systems. Normally, based on the refrigerant state, an evaporator can be divided into two zones, namely the two-phase region and the superheated region. Likewise, a condenser can be sectioned into * Corresponding author. Tel.: þ86 21 5899 7971; fax: þ86 21 5899 7973. E-mail address: [email protected] (C.-L. Zhang). 0140-7007/$35.00 Ó 2006 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2006.11.007

three zones, namely the desuperheated region, the two-phase region, and the subcooled region. Accordingly, lots of researchers developed the steady-state and transient multizone models of evaporators or condensers [1e10]. Under certain operating conditions or disturbances, the zones might appear or disappear. For instance, with the increase of mass flow rate, the superheated region in the evaporator will be shortened and eventually vanish. Therefore, the multi-zone model should be able to handle such kind of model reduction. In the traditional multi-zone models, logic constraints were used to separate the possible running

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927

Nomenclature a C D f G h k L m_ _ t DT x d D p r

Subscripts 0,1,2,3 interface position a air in inlet l saturated liquid m mean value max maximum out outlet p pressure r refrigerant SC subcooled section SH superheated/desuperheated section TP two-phase section v saturated vapor

shape factor defined in Eq. (2) constraint function diameter (mm) friction factor mass flux (kg m2 s1) enthalpy (kJ/kg) heat transfer coefficient (W/(m2 K)) length (m) mass flow rate (kg/h) temperature ( C) temperature difference ( C) unknown variable relative error absolute error circumference ratio density (kg m3)

modes and force mode changes from one to another. For instance, when the superheated region in the evaporator disappears, the equations that describe the superheated region will be removed. However, sometimes it might be not as simple as expected [10]. Complicated logic constraints have to be added in the mode switch. The objective of this work is to transform the logic constraints into continuously differentiable equations and develop the generalized logic unconstrained multizone modeling methodology for evaporators and condensers. For brevity, only the steady-state counter flow evaporator and condenser models are demonstrated in this paper. 2. General constraint equations Generally, logic constraints can be expressed by some single or multiple ‘‘IF-ELSE’’ structures in implementation. Let’s consider the single ‘‘IF-ELSE’’ structure first. As shown in Fig. 1, if x > b, the model will step to equations F1(x). Else, equations F2(x) will be invoked. Therefore, it can be expressed as FðxÞ ¼ CðxÞF1 ðxÞ þ ½1  CðxÞF2 ðxÞ

ð1Þ

x>b Y

N F1

F2

Fig. 1. Single ‘‘IF-ELSE’’ structure.

where the logic constraint function  1 x>b CðxÞ ¼ 0 xb Therefore, the logic constraint function of the single ‘‘IF-ELSE’’ structure is actually a step function. The step function is discontinuous at x ¼ b. Unfortunately, most simultaneously solving methods, e.g. the commonly used NewtoneRaphson method, require the solved equation(s) to be continuous, even continuously differentiable. Hence, the step function cannot be directly applied to equation solving. To overcome the shortage of the step function, a continuously differentiable function is used to approximate the step function. That’s the sigmoid function below. CðxÞ ¼

1 1 þ eaðxbÞ

ð2Þ

where a is the shape factor. The differential of C(x) is dC ¼ aCð1  CÞ dx

ð3Þ

According to Eq. (3), we know the shape of the sigmoid function depends on the shape factor a. As shown in Fig. 2, the larger the shape factor is, the sharper the C(x) will be at x ¼ b. Accordingly, the sigmoid function will be closer to the step function. If we use the sigmoid function as the general constraint equation, there will be some approximation error. In the case x  b, the approximate value is CF1 þ (1  C)F2 in comparison with the true value F2. The difference is D ¼ CF1 þ ð1  CÞF2  F2 ¼ CðF1  F2 Þ ðx  bÞ Consequently, the relative error becomes

ð4Þ

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928

C 1 x1 > b1 N

Y

a = 100

x2 > b2 a = 10

Y

F1

a=1

N

F2

F3

Fig. 3. Multiple ‘‘IF-ELSE’’ structure.

b

x

Fig. 2. The sigmoid constraint equation.

d¼

    D F1 1 F1 ¼C 1 ¼  1 ðx  bÞ F2 1 þ eaðxbÞ F2 F2 ð5Þ

Similarly if x > b, the difference between the approximate CF1 þ (1  C)F2 and the true one F1 becomes D ¼ CF1 þ ð1  CÞF2  F1 ¼ ð1  CÞðF1  F2 Þ ðx > bÞ ð6Þ And the relative error is   D F2 d ¼ ¼ ð1  CÞ 1  F1 F1   aðxbÞ e F2 1 ðx > bÞ ¼ 1 þ eaðxbÞ F1

3. Evaporator model Fig. 4 shows the typical temperature profile in a counter flow evaporator. Generally, refrigerant enters the evaporator at low quality two-phase state and is heated to high quality two-phase state or superheated gas when leaves. For the superheated leaving state, two-zone evaporator model will be applied. For the two-phase leaving state, the model will reduce to one-zone mode. To decide which mode should be invoked in calculation, the constraint equation will be included. Equations of the superheated region: m_ r ðhr2  hr1 Þ ¼ m_ a ðha0  ha1 Þ

ð9Þ

kSH pDLSH DTm;SH ¼ m_ r ðhr2  hr1 Þ   fSH LSH G2 1 1 þ G2  pr1  pr2 ¼ rr2 rr1 2Drm;SH

ð11Þ

Equations of the two-phase region: ð7Þ

m_ r ðhr1  hr0 Þ ¼ m_ a ðha1  ha2 Þ

ð12Þ

kTP pDLTP DTm;TP ¼ m_ r ðhr1  hr0 Þ

ð13Þ

Therefore, any required accuracy can be achieved by adjusting a. The multiple ‘‘IF-ELSE’’ structures can be regarded as the combination of some single ‘‘IF-ELSE’’ structures. For instance, an ‘‘IF-ELSEIF-ELSE’’ structure shown in Fig. 3 can be expressed as below.

Air

ta0

ta1

F ¼ C1 F1 þ ð1  C1 Þ½C2 F2 þ ð1  C2 ÞF3  ¼ C1 F1 þ ð1  C1 ÞC2 F2 þ ð1  C1 Þð1  C2 ÞF3

ð10Þ

ð8Þ

tr2

ta2

where the constraint equations are 1 1 þ ea1 ðxb1 Þ 1 C2 ¼ 1 þ ea2 ðxb2 Þ

C1 ¼

Likewise, the accuracy can be controlled by the shape factors.

tr0

tr1 Refrigerant Two-phase

Superheated

Fig. 4. Temperature profile of a counter flow evaporator.

L.-L. Shao, C.-L. Zhang / International Journal of Refrigeration 30 (2007) 926e931

pr0  pr1 ¼

  fTP LTP G2 1 1 þ G2  rr1 rr0 2Drm;TP

Subcooled

ð14Þ

Two-phase

929

Desuperheated tr0

Refrigerant

Geometry equation: 0 ¼ L  LTP  LSH

tr2

ð15Þ

tr1

Constraint equation: hr1 ¼ Cout hv;pr1 þ ð1  Cout Þhr2

ta3

ð16Þ

tr3

ta2

where Cout ¼

1 1 þ eaðhr2 =hv;pr2 1Þ

ta1

Eq. (16) is based on the following fact. When the refrigerant leaving enthalpy hr2 is greater than hv;pr2 , the saturated vapor enthalpy related to the leaving pressure, the refrigerant leaving state is superheated. Hence, the enthalpy at the interface of two-phase and superheated regions, hr1, is equal to the saturated vapor enthalpy at the local pressure, namely hv;pr1 . When hr2 is less than hv;pr2 , the leaving refrigerant is two-phase and hr1 ¼ hr2. In Eq. (16), the factor a is determined by the required accuracy. For the superheated leaving state, the relative error of using Eq. (16), according to Eq. (7), is   eaðhr2 =hv;pr2 1Þ hr2 d¼ 1  hv;pr1 1 þ eaðhr2 =hv;pr2 1Þ   eaðhr2 =hv;pr2 1Þ hr2 hv;pr2 1  ð17Þ ¼ hv;pr2 hv;pr1 1 þ eaðhr2 =hv;pr2 1Þ Apparently in Fig. 2, the maximum error occurs in the neighborhood of hr2 ¼ hv;pr2 . In this neighborhood, pr2 and pr1 are pretty close to each other. So do the corresponding saturated enthalpies, hv;pr2 and hv;pr1 . Therefore, Eq. (17) could be simplified as   eaðhr2 =hv;pr2 1Þ hr2  1 ð18Þ d¼ 1 þ eaðhr2 =hv;pr2 1Þ hv;pr2 The relative error will reach the maximum when hr2 =hv;pr2 ¼ 1 þ 1:2785=a. 0:2785 dmax ¼ a

Air

ta0 Fig. 5. Temperature profile of a counter flow condenser.

superheated entering and two-phase leaving, superheated entering and leaving, two-phase entering and subcooled leaving, two-phase entering and leaving, and subcooled entering and leaving. Therefore, there is a multiple ‘‘IF-ELSE’’ structure in the condenser modeling. Fig. 6 shows the flowchart of the condenser model with multiple ‘‘IF-ELSE’’ structures. First, the refrigerant entering state is identified, and finally the leaving state will be figured out.

Start

Inlet SH? Y

N

SH-TP-SC Inlet TP? N

Y LSC>0 Y N

TP-SC

SH-TP SC

ð19Þ

Similarly, we can get the same maximum relative error for the two-phase leaving state. Generally, d < 0.1% could be accurate enough and accordingly a ¼ 3000 is recommended for the evaporator model.

LTP>0 LTP>0

N N

Y

Y SC

SH

4. Condenser model As shown in Fig. 5, the condenser model is more complicated than the evaporator model. In the condenser, there might be six combinations of refrigerant entering and leaving states: superheated entering and subcooled leaving,

End Fig. 6. Flow chart of logic bounded condenser model.

L.-L. Shao, C.-L. Zhang / International Journal of Refrigeration 30 (2007) 926e931

930

Equations of the desuperheated region: m_ r ðhr0  hr1 Þ ¼ m_ a ðha3  ha2 Þ

ð20Þ

kSH pDLSH DTm;SH ¼ m_ r ðhr0  hr1 Þ   fSH LSH G2 1 1 2 þG  pr0  pr1 ¼ 2Drm;SH rr1 rr0

ð21Þ ð22Þ

Equations of the two-phase region: m_ r ðhr1  hr2 Þ ¼ m_ a ðha2  ha1 Þ

ð23Þ

kTP pDLTP DTm;TP ¼ m_ r ðhr1  hr2 Þ   fTP LTP G2 1 1 þ G2  pr1  pr2 ¼ rr2 rr1 2Drm;TP

ð24Þ ð25Þ

Equations of the subcooled region: m_ _r ðhr2

 hr3 Þ ¼ m_ a ðha1  ha0 Þ

kSC pDLSC DTm;SC ¼ m_ r ðhr2  hr3 Þ   fSC LSC G2 1 1 þ G2  pr2  pr3 ¼ rr3 rr2 2Drm;SC

ð26Þ ð27Þ

5. Case study

ð28Þ

Case study here will focus on the comparison between the new logic unconstrained model and the traditional model. The prediction accuracy of the models will mainly base on the selection of empirical heat transfer and pressure drop correlations, which won’t be discussed in this paper. Therefore, constant empirical coefficients are used in the following simulation. In addition, REFPROP 6 [11] is employed to calculate the fluid thermal properties. The equation solver is the NewtoneRaphson method from Numerical Recipes [12]. With the increase of refrigerant mass flow rate, the leaving state of the evaporator will change from two-phase to superheat. The simulation conditions are set below.

Geometry equation: 0 ¼ L  LSH  LTP  LSC

means not. Cinl represents whether the entering state is subcooled, Cinl ¼ 0 means subcooled and Cinl ¼ 1 means not. Coutv represents whether the leaving state is superheated, Coutv ¼ 1 means superheated and Coutv ¼ 0 means not. Coutl represents whether the leaving state is subcooled, Coutl ¼ 0 means subcooled and Coutl ¼ 1 means not. Accordingly, in Eqs. (30) and (31), if CinvCinl(1  Coutv)(1  Coutl) ¼ 1, it represents the combination of superheated entering and subcooled leaving is available. Similarly, CinvCinl(1  Coutv)Coutl, CinvCinlCoutvCoutl, (1  Cinv)Cinl(1  Coutv)(1  Coutl), (1  Cinv)Cinl(1  Coutv)Coutl, and (1  Cinv)(1  Cinl) (1  Coutv)(1  Coutl) represent the rest five combinations of refrigerant entering and leaving states, respectively. Similar to the error analysis in the evaporator model, a ¼ 3000 could make the approximate errors below 0.1%.

ð29Þ

Constraint equations: hr1 ¼ Cinv Cinl ð1  Coutv Þð1  Coutl Þhv;pr1 þ Cinv Cinl ð1  Coutv ÞCoutl hv;pr1 þ Cinv Cinl Coutv Coutl hr3 þ ð1  Cinv ÞCinl ð1  Coutv Þð1  Coutl Þhr0 þ ð1  Cinv ÞCinl ð1  Coutv ÞCoutl hr0 þ ð1  Cinv Þð1  Cinl Þð1  Coutv Þð1  Coutl Þhr0 hr2 ¼ Cinv Cinl ð1  Coutv Þð1  Coutl Þhl;pr2

Given: mr ¼ 71e73 kg/h, hr0 ¼ 250 kJ/kg, pr0 ¼ 570 kPa, ma ¼ 503.2 kg/h, ta0 ¼ 20  C, L ¼ 13.728 m, D ¼ 8.732 mm,

þ Cinv Cinl ð1  Coutv ÞCoutl hr3 þ Cinv Cinl Coutv Coutl hr3

þ ð1  Cinv Þð1  Cinl Þð1  Coutv Þð1  Coutl Þhr0 ð31Þ

Cinv ¼ Cinl ¼

1 1þe

aðhr0 =hv;pr0 1Þ

1 1 þ eaðhr0 =hl;pr0 1Þ

1 1 þ eaðhr3 =hv;pr3 1Þ 1 Coutl ¼ 1 þ eaðhr3 =hl;pr3 1Þ Cinv represents whether the entering state of condenser is superheated, Cinv ¼ 1 means superheated and Cinv ¼ 0

Refrigerant outlet enthalpy (kJ/kg)

þ ð1  Cinv ÞCinl ð1  Coutv ÞCoutl hr3

where

5.8

410

þ ð1  Cinv ÞCinl ð1  Coutv Þð1  Coutl Þhl;pr2

superheated outlet

two-phase outlet

408 5.6 406 5.4 404

Coutv ¼

400

5.2

h_traditional model h_new model T_traditional model T_new model

402

71

72

73

Air outlet temperature (ºC)

ð30Þ

5

Refrigerant mass flow rate (kg/h) Fig. 7. Refrigerant outlet enthalpy and air outlet temperature of evaporator.

L.-L. Shao, C.-L. Zhang / International Journal of Refrigeration 30 (2007) 926e931 33.8

250 two-phase outlet

248 33.6 246 33.4 244 33.2

h_traditional model h_new model T_traditional model T_new model

242

240 67

67.5

68

68.5

69

Air outlet temperature (ºC)

Refrigerant outlet enthalpy (kJ/kg)

subcooled outlet

33

Refrigerant mass flow rate (kg/h) Fig. 8. Refrigerant outlet enthalpy and air outlet temperature of condenser.

kSH ¼ 263 W/(m2 K), kTP ¼ 612 W/(m2 K), fSC ¼ 0.02. Unknown: hr2, pr2, ta2, LTP, and LSH.

and

fTP ¼

Fig. 7 shows the changes of the leaving refrigerant enthalpy and the leaving air temperature against the refrigerant mass flow rate. As expected, the new model gives the almost same results as the traditional one. Similarly, if we change the refrigerant mass flow rate, the leaving state of the condenser will change from subcooled to two-phase. The calculation conditions are as follows. Given: mr ¼ 67e69 kg/h, hr0 ¼ 445.5 kJ/kg, pr0 ¼ 1390 kPa, ma ¼ 986.7, ta0 ¼ 20  C, L ¼ 13.728 m, D ¼ 8.732 mm, kSH ¼ 263 W/(m2 K), kTP ¼ 612 W/(m2 K), kSC ¼ 263 W/ (m2 K), and fSH ¼ fTP ¼ fSC ¼ 0.02. Unknown: hr3, pr3, ta3, LSH, LTP, and LSC. Fig. 8 shows the new model is in good agreement with the traditional one as well. In addition, the CPU time of the two methods is almost same as expected because the number of equations involved is same in common use. 6. Conclusion New logic unconstrained evaporator and condenser models have been developed. The logic constraint functions

931

are transformed into the continuously differentiable sigmoid functions. Accordingly, all the equations in the model can be simultaneously solved. Therefore, the new models are more flexible in implementation than the traditional logic bounded ones.

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