An extension of a steady-state model for fin-and-tube heat exchangers to include those using capillary tubes for flow control.

INTRODUCTION

Multipath fin-and-tube heat exchangers (MPHX) are widely used in refrigeration or air-conditioning applications, and the refrigerant distribution in the paths is very important to the performance of the heat exchangers. Unsuitable refrigerant distribution in evaporators may result in dry-out of circuits and finally result in poor heat transfer and waste of the heat transfer area, while unsuitable refrigerant distribution in condensers may create zones of reduced heat transfer due to high liquid loading. Moreover, unsuitable refrigerant distribution may lead to a high temperature difference between adjacent tubes, which causes the reversed heat conduction through the fins and degrades the performance of the heat exchangers (Romero-Mendez et al. 1997; Wang et al. 1999). In order to get suitable refrigerant distribution, the methods of changing the refrigerant circuit pattern (Romero-Mendez et al. 1997; Wang et al. 1999; Liang et al. 2000, 2001) and changing the refrigerant distributor geometries (Jiao et al. 2003) have been used. Besides these two methods, adding capillary tubes to the paths is an effective way to adjust the refrigerant distribution in different paths for MPHX, which is low in cost and especially effective for subtle adjustment. Therefore, research on simulation of MPHX should include those using capillary tubes for flow control.

Many models and algorithms for MPHX were presented, but they either neglected the refrigerant distributions among the paths (Martins and Parise 1993; Jia et al. 1999; Judge and Radermacher 1997; Lee et al. 2003) or only considered the refrigerant distributions among the paths without capillary tubes (Domanski 1991; Lee and Domanski 1997; Bensafi et al. 1997; Liang et al. 2001). We presented a general steady-state model (GSSM) for MPHX with an arbitrary refrigerant circuit (Liu et al. 2004) to simultaneously evaluate the effects of the refrigerant distribution among the paths in any refrigerant circuit, the heat conduction between the adjacent tubes with continuous fins, air-side maldistribution, etc., but it still did not consider the refrigerant distribution among the paths of MPHX using capillary tubes for flow control. As to capillary tubes, the mathematical models for a single capillary tube acting as throttling equipment in refrigeration or air-conditioning systems were well developed, such as Zhang and Ding (2004), Bansal and Rupasinghe (1996), Bittle and Pate (1996), Chung (1998), Garcia-Valladares et al. (2002), etc., but models on the multicapillary-tube flow or MPHX using capillary tubes for flow control are not available. However, if we directly extend the GSSM to MPHX using capillary tubes for flow control by simply adding the model of a single capillary tube, incorrect results or abnormal termination of the simulation process may occur due to the choked characteristics of the capillary tube. Therefore, measures must be taken to extend the GSSM to MPHX using capillary tubes for flow control.

The main difficulty in extending the GSSM for MPHX to those using capillary tubes for flow control is to solve the refrigerant distribution among the paths containing capillary tubes. The GSSM solves the refrigerant distribution among the paths by using iteration calculations. As capillary tubes have the choked characteristics, an extra constraint on the maximum refrigerant mass flow rate is introduced into the refrigerant distribution equations of the MPHX using capillary tubes for flow control. The simulation process may abnormally terminate or output obvious incorrect results if the assumed value of the mass flow rate of refrigerant in one capillary tube is larger than the maximum value during the iteration process. Therefore, the choked characteristics of the capillary tubes should be simultaneously considered during the process of solving equations for the refrigerant distribution among the paths of MPHX using capillary tubes for flow control.

The purpose of this paper is to extend the GSSM to MPHX using capillary tubes for flow control. In this paper, the description of the simulation object and analysis on the problem of directly applying the GSSM to MPHX using capillary tubes for flow control are given at first, then the extended GSSM is presented. Finally, experimental validation and conclusions are given.

DESCRIPTION OF THE SIMULATION OBJECT

The simulation object considered in this paper is MPHX using capillary tubes for flow control. The capillary tubes can be added on several or all inlet paths of the MPHX, and the MPHX can have arbitrary divergent or confluent refrigerant paths. All of the geometrical parameters of the simulation object as well as the inlet states of both refrigerant and air are known. The target of the simulation is to obtain the general performance of the simulation object and the states of the refrigerant and air on each part of the simulation object.

All of the paths of the simulation object are simply classified into two types--the main path (MP) and the subpath (SP)--in order to conveniently describe the refrigerant distribution. The MP refers to a group of paths that have same inlet tube and same outlet tube, and the SP refers to one of the subpaths in one MP. All of the MPs in the MPHX are coded as [MP.sub.i] (i = 1, 2, ..., [N.sub.m]) in line based on depth-first search rule (Liu et al. 2004), and all the SPs in the MPHX are coded as [SP.sub.ij] (i = 1, 2, ..., [N.sub.m], and j = 1, 2, ..., m) in line based on breadth-first search rule (Liu et al. 2004) in each [MP.sub.i] (i = 1, 2, ..., [N.sub.m]). In order to identify each tube in the heat exchanger, the tubes are numbered from 1 to N in the order from front row to back row and from bottom to top of the heat exchanger, and the inlet collecting tube and outlet collecting tube are numbered as 0 and N + 1, respectively.

Figure 1a shows the schematic of an i-column and j-row MPHX using capillary tubes for flow control, and Figure 1b shows the associated MPs and SPs.

[FIGURE 1 OMITTED]

The refrigerant distribution in the No. 1 MP is especially focused on since the capillary tubes are only added on [MP.sub.1] for helping to adjust the refrigerant distribution among the paths. It is better to further simplify the diagram of [MP.sub.1] to Figure 2 for the convenience of the analysis.

[FIGURE 2 OMITTED]

In Figure 2, the refrigerant inlet parameters [M.sub.r,in], [p.sub.r,in], [h.sub.r,in] are given; [M.sub.r,out], [p.sub.r,out], [h.sub.r,out], and [M.sub.r,1j] (j = 1, 2, ..., m) are unknown parameters that need to be calculated.

PROBLEM IN DIRECTLY APPLYING THE GSSM TO MPHX USING CAPILLARY TUBES FOR FLOW CONTROL

In GSSM, the mass flow rates of refrigerant in paths are iteratively adjusted until the pressure drops of refrigerant in all paths from inlet to outlet are equal. The following equation was used to reflect the relationship between the mass flow rate and the pressure drop of refrigerant in [MP.sub.i]:

[DELTA][p.sub.r,ij] = [S.sub.r,ij][M.sub.[r,ij].sup.2](j = 1,2, ..., m) (1)

where

[DELTA][p.sub.r,ij] = the pressure drop of refrigerant in [SP.sub.ij]

[S.sub.r,ij] = equivalent flow resistance of refrigerant in [SP.sub.ij]

[M.sub.r,ij] = mass flow rate of refrigerant in [SP.sub.ij].

The distribution of refrigerant in each SP of the same MP was adjusted to ensure the same pressure drop of refrigerant in all SPs, and the ratio of the mass flow rate of refrigerant in each SP was expressed as

[M.sub.r,i1]:[M.sub.r,i2]: ...:[M.sub.r,im] = [S.sub.[r,i1].sup.[ - 0.5]]:[S.sub.[r,i2].sup.[ - 0.5]]: ...:[S.sub.[r,im].sup.[ - 0.5]]. (2)

Since the sum of the mass flow rate of refrigerant is equal to the known value of the total mass flow rate of refrigerant in [MP.sub.i], the ratio of the mass flow rate of refrigerant in [SP.sub.ij] was calculated as

[[epsilon].sub.r,ij] = [[S.sub.[r,ij].sup.[ - 0.5]]/[[summation over (term)](j = 1) m][S.sub.[r,ij].sup.[ - 0.5]]]] (3)

where

[[epsilon].sub.r, ij] = the ratio of the mass flow rate of refrigerant in [SP.sub.ij] to the total mass flow rate of refrigerant in [MP.sub.i].

And then, the mass flow rate of refrigerant in each SP of [MP.sub.i] can be determined as:

[M.sub.r,ij] = [[epsilon].sub.r,ij][M.sub.r,i](j = 1,2, ..., m), (4)

Where

[M.sub.r,i] = the total mass flow rate of refrigrant in [MP.sub.i].

The calculation process from Equations 1 to 4 is repeated until the pressure drops of refrigerant in all subpaths are equal.

Incorrect results may occur if we directly apply the above GSSM to the MPHX using capillary tubes for flow control, especially for the cases that the mass flow rate of refrigerant calculated from Equation 4 for one capillary tube is larger than the choked value. For [MP.sub.1] containing capillary tubes, the total mass flow rate of refrigerant may decrease after the xth iteration calculation

[M.sub.r,1.sup.(x)] = [M.sub.r,1.sup.(x - 1)] - [DELTA][M.sub.r,1.sup.(x)] (5a)

[DELTA][M.sub.r,1.sup.(x)] = [summation]([M.sub.r,1k.sup.(0)] - [M.sub.r,1k,c]),([M.sub.r,1k.sup.(0)]>[M.sub.r,1k,c]andk[menber of][SP.sub.c]) (5b)

where

[M.sub.r, 1.sup.(x-1)] and [M.sub.r, 1.sup.(x)] = the total mass flow rate of refrigerant in [MP.sub.1] after the (x-1)th and the xth iteration calculation

[[DELTA]M.sub.r, 1.sup.(x)] = the discrepant total mass flow rate of refrigerant after the xth iteration calculation

[M.sub.r, 1k.sup.(0)] = the mass flow rate of refrigerant calculated from Equation 4 for [SP.sub.1k]

[M.sub.r, 1k, c] = the choked mass flow rate of refrigerant in capillary tube in [SP.sub.1k]

[SP.sub.c] = the collection of the no. of the subpaths containing choked capillary tubes

According to Equations 5a and 5b, the total mass flow rate of refrigerant may be "lost" during the simulation process. Figure 3 schematically shows this refrigerant-lost problem. The simulation result with the GSSM is incorrect due to the unavoidable refrigerant-lost problem in the algorithm. Therefore, new algorithm for solving the refrigerant distributions among the paths of MPHX using capillary tubes for flow control must be developed to overcome this refrigerant-lost problem.

[FIGURE 3 OMITTED]

BASIC IDEA IN EXTENDING THE GSSM TO MPHX USING CAPILLARY TUBES FOR FLOW CONTROL

The refrigerant-lost problem that occurred in GSSM is mainly caused by the extra constraint on the maximum mass flow rate of refrigerant for MPHX using capillary tubes for flow control, and the cause of the refrigerant-lost problem is different for different cases of the choked refrigerant flows. Therefore, auxiliary equations should be introduced in order to calculate the equations for refrigerant distribution among the paths, and different auxiliary equations should be used for different cases.

For the case that not all refrigerant flows in capillary tubes are choked during the iteration process for solving the refrigerant distribution among the paths, auxiliary equations should be introduced to redistribute the discrepant refrigerant. The reason for the occurring refrigerant-lost problem in GSSM for this case is that the assumed value is larger than the choked value for [m.sub.c] number of refrigerant flows, as shown in Equation 5b. Therefore, the [M.sub.r,1.sup.(x)] discrepant refrigerant should be redistributed to other paths in which the refrigerant flows are not choked, and the value of the total mass flow rate of refrigerant may not change after the redistribution of the [M.sub.r, 1.sup.(x)] discrepant refrigerant.

For the case that all refrigerant flows in capillary tubes are choked during the iteration process for solving the refrigerant distribution among the paths, auxiliary equations should be introduced to determine the right value of the mass flow rate of refrigerant in each path. The reason for producing discrepant mass flow rate of refrigerant in GSSM for this case is that the given total mass flow rate of refrigerant in the heat exchanger conflicts with the inlet state of refrigerant and the geometry parameters of the capillary tubes. The inlet state of refrigerant and the geometry parameters of the capillary tubes determine the choked mass flow rate of refrigerant in an adiabatic capillary tube. If the given total mass flow rate of refrigerant in the heat exchanger is larger than the sum of the all-choked mass flow rate of refrigerant, all of the capillary tubes may be choked. Equations 6-9 show the relationship between the given total mass flow rate of refrigerant and the sum of the all-choked mass flow rates of refrigerant, and Equation 9 shows the discrepant total mass flow rate of refrigerant after the xth iteration calculation for this case:

[M.sub.r,in] = [summation][M.sub.r, 1j.sup.(0)](j = 1,2, ...,m) (6)

[M.sub.r,1,max] = [summation][M.sub.r,1j,c](j = 1,2, ...,m) (7)

[M.sub.r,in]>[M.sub.r,1,max]([M.sub.r,1j].sup.(0)]>[M.sub.r,1j,c]andj = 1,2, ...,m) (8)

[DELTA][M.sub.r,1.sup.(x)] = [summation]([M.sub.r,1j.sup.(0)] - [M.sub.r,1j,c])([M.sub.r,1j.sup.(0)]>[M.sub.r,1j,c]andj = 1,2, ...,m) (9)

where

[M.sub.r,in] = the given total mass flow rate of refrigerant in the heat exchanger

[M.sub.r,1,max] = the sum of the all-choked mass flow rates of refrigerant, respectively

The refrigerant-lost problem may repeat even after redistributing the [DELTA][M.sub.r,1.sup.(x)] refrigerant to other paths because all other capillary tubes are choked and the new discrepant refrigerant may be lost again. Finally, a very small value of the total mass flow rate of refrigerant may be obtained after the simulation with the GSSM. Therefore, auxiliary equations should be introduced to determine the right value of the mass flow rate of refrigerant in each path to avoid such repeat of the refrigerant-lost problem occurring in GSSM.

EXTENDED GSSM

Governing Equations for the Refrigerant Distribution in the Paths Containing Capillary Tubes

As shown in Figure 2, only the subpaths of [MP.sub.1] contain capillary tubes. The following equations are used to describe the refrigerant flow among the [SP.sub.1j](j = 1,2, ..., m):

[M.sub.r,in] = [summation][M.sub.r,1j](j = 1,2, ...,m) (10)

[M.sub.r,1j][less than or equal to][M.sub.r,1j,c](j = 1,2, ...,m) (11)

[h.sub.r,in] = [h.sub.r,1j](j = 1,2, ...,m) (12)

[p.sub.r,in] = [p.sub.r,1j](j = 1,2, ...,m) (13)

where

[M.sub.r,in] = the mass flow rate

[h.sub.r,in] = specific enthalpy 

[p.sub.r,in] = pressure of refrigerant at the inlet of the [MP.sub.1]

[M.sub.r,1j] = the mass flow rate

[M.sub.r,1j,c] = the choked mass flow rate

[h.sub.r,1j] = inlet specific enthalpy

[p.sub.r,1j] = inlet pressure of refrigerant in [SP.sub.1j] (j = 1, 2, ..., m)

Equation 14 is used to calculate the pressure drop of refrigerant in [SP.subj.1j] in order to reflect the effect of the capillary tubes on refrigerant side pressure drop:

[[DELTA][p.sub.r,1j]] = [[DELTA][p.sub.r,1j,cap]] + [[DELTA][p.sub.r,1j,tube]] + [[DELTA][p.sub.r,1j,cs]](j = 1,2, ..., m) (14)

where

[[DELTA]p.sub.r, 1j] = the total pressure drop of refrigerant in [SP.sub.1j]

[[DELTA]p.sub.r, 1j, cap], [[DELTA]p.sub.r, 1j, tube] and [[DELTA]p.sub.r, 1j, cs] = the pressure drop of refrigerant in capillary tubes, in finned tubes and at the connecting section between the capillary and finned tube due to the enlarge cross-area from capillary to finned tube

If capillary tube is not used, [DELTA][p.sub.r, 1j, cap] and [DELTA][p.sub.r, 1j, dc] in Equation 14 are equal to 0, Equation 14 becomes the same as that in the previously developed GSSM (Liu et al. 2004), and the mass flow rate of refrigerant in [SP.sub.1j] can still be calculated with the previously developed GSSM.

If refrigerant flows in capillary tubes are not choked during the iteration process of solving the refrigerant distribution among the paths, Equation 11 is always satisfied, and the mass flow rate of refrigerant in [SP.sub.1j] can be calculated with Equations 1-4, 10, and 12-14.

Auxiliary Equations for the Case that One or Several Refrigerant Flows in Capillary Tubes Are Choked When Solving the Refrigerant Distribution Among the Paths

If the refrigerant flows in one or several capillary tubes are choked during the iteration process of solving the refrigerant distribution among the paths, the[[DELTA]M.sub.r,1.sup.(x)] refrigerant, which is determined by Equation 5a, should be redistributed among other SPs in order to reflect the real refrigerant distribution characteristics. Since the ratio of the mass flow rate of refrigerant in each path has already been determined by Equation 2, distributing the [[DELTA]M.sub.r,1.sup.(x)] refrigerant equally to the SPs with an unchoked capillary tube may not violently disturb the distribution ratio of the mass flow rate of refrigerant among the SPs. Therefore, a share-distribution strategy, which distributes the [[DELTA]M.sub.r,1.sup.(x)] refrigerant equally to the SPs containing an unchoked capillary tube, is applied to redistribute the above redundant refrigerant. By using the share-distribution strategy,

[M.sub.r,1j] = [M.sub.r,1j.sup.(0)] + ([summation][M.sub.r,1j.sup.(0)] - [summation][M.sub.r,1k])/(m - [m.sub.c])(j = 1,2, ..., m and j[not member of][SP.sub.c],k[member of]S[P.sub.c]) (15)

[M.sub.r,1k] = [M.sub.r,1k,c]/B(k[member of][m.sub.c]) (16)

where

[M.sub.r,1j.sup.(0)] = the value obtained by Equations 1-4, 10, and 12-14

B = an experiential correction factor and should be in the range of 1.0-2.0

The refrigerant-lost problem can be avoided after redistributing the redundant refrigerant with Equations 15-16. The correction factor B in Equation 16 is necessary and should be larger than 1.0 in order to avoid the abnormal termination of the simulation process. The pressure drop of refrigerant in the choked capillary tube is very large, and sometimes is near the value of the pressure of inlet refrigerant. If B equals 1.0 in Equation 16, the calculated pressure drop of refrigerant on the subpath containing the choked capillary tube may be larger than the pressure of inlet refrigerant, which is wrong and may abnormally terminate the simulation program. If B is very large, the assigned mass flow rate of refrigerant in the subpath containing the choked capillary tube may become very small, which will lead to a smaller pressure drop of refrigerant in the subpath and will prolong the iteration process for solving the refrigerant distribution among the paths.

Auxiliary Equations for the Case that All Refrigerant Flows in Capillary Tubes are Choked When Solving the Refrigerant Distribution Among the Paths

If all refrigerant in capillary tubes on all paths are choked during the iteration process of solving the refrigerant distribution among the paths, an incorrect and much smaller value of the mass flow rate of refrigerant may be obtained after simulation with the GSSM because the [M.sub.r,1] refrigerant is always lost during the iteration process for determining the refrigerant distribution among the subpaths. In fact, when refrigerant flows in all capillary tubes are choked, the pressure drop of refrigerant in the MP is dominated by the choked refrigerant flow in the SP with minimum pressure drop of refrigerant. So the refrigerant distribution method should be based on the SP that has minimum pressure drop of refrigerant. Let's denote the SP that has minimum pressure drop of refrigerant as the standard path and the corresponding minimum pressure drop of refrigerant as [p.sub.r,s]. Then the following equation can be used to determine the mass flow rates of refrigerant in other SPs based on the pressure drop of the standard path:

[M.sub.r, 1j] = [M.sub.r, 1j,c](j = s) (17)

[M.sub.r,1j] = f([[DELTA][p.sub.r,s]]) = f([[DELTA][p.sub.r,1j,cap]] + [[DELTA][p.sub.r,1j,tube]] + [[DELTA][p.sub.r,1j,cs]])(j = 1,2, ..., m and j[not equal to]s) (18)

where function f is derived from Equation 14.

The refrigerant-lost problem can be avoided after redistributing the redundant refrigerant with Equations 17 and 18, but [M.sub.r, 1.sup.(x)]<[M.sub.r, in.sup.(0)] for this case.

Other Governing Equations

The refrigerant flow in the capillary tube is considered as adiabatic one-dimensional homogenous flow. The details of the governing equations for single capillary tube and fin-and-tube are provided by Liu et al. (2004) and Zhang and Ding (2004), and are summarized in the appendix.

Algorithm Description 

The whole MPHX using capillary tubes for flow control is divided into many control volumes. The simulation process begins with the first control volume of the inlet tube and ends with the last control volume of the outlet tube with a section-by-section method. A heat transfer and pressure drop alternative iteration algorithm is applied for the simulation of the heat exchanger using capillary tubes for flow control.

Since the refrigerant flow in the capillary tube is considered as adiabatic one-dimensional homogenous flow, the heat transfer via capillary tubes can be neglected. So the heat-transfer calculation algorithm in the previously developed GSSM (Liu et al. 2004) is still applied.

The process of pressure drop calculation starts from the inlet tubes, and then comes to the capillary tube and heat exchanger tube along the SPs of each MP. The pressure drop of refrigerant in each SP is calculated one by one based on the known refrigerant states obtained from the heat-transfer calculation process.

In the pressure drop calculation module, different methods of refrigerant distribution are used to determine the refrigerant distribution among the paths according to the states of the capillary tubes on different paths. Figure 4 shows the flow chart of the pressure drop calculation module developed in this paper. The parts with the gray background are the added parts, as compared to the flow chart of the previously developed GSSM (Liu et al. 2004).

[FIGURE 4 OMITTED]

EXPERIMENTAL VALIDATION

The right way to validate the extended GSSM is to validate the submodels of the GSSM for heat exchanger without capillary tubes and the model for single capillary tube at first, and then to validate the effect of the integrated model of the two submodels in the extended GSSM. The validation of the GSSM for heat exchanger without capillary tubes and the model for single capillary tube were already reported in the published papers (Liu et al. 2004; Zhang and Ding 2004), which showed that the cooling capacity calculation deviations and the refrigerant-side pressure drop calculation deviations of GSSM for a MPHX using R-22 were less than [+ or -]10% and [+ or -]20% individually, while the deviation of the model for single capillary tube was within [+ or -]15%. Therefore, these two models are not validated in this paper, and only the effect of the integrated model and algorithm in the extended GSSM is verified.

Experimental System

In order to verify the extended GSSM for the MPHX using capillary tubes for flow control, a test rig has been set up and series of experiments have been designed and performed. The schematic of the experimental rig and the refrigerant circuit of the test coil are shown in Figure 5. The experimental conditions are shown in Table 1.

[FIGURE 5 OMITTED]

Table 1. Experimental Conditions

(a) Six Capillary Tubes                (b) Heat Exchanger

                                         Length/width      635/26.6/336
                                         /height, mm

                            1.5/150,     Row number/         2/16
                                         column
                            3.0/300,     number

Inner Diameter/Length,      4.7/300,     Path number          2
mm                          3.0/150,

                            1.5/300,     Tube diameter,   7.0/enhanced
                            4.7/300      mm/type

                                         Fin type/fin       Slit/1.2
                                         pitch, mm

(c) Measuring Condition                  (d)Refrigerant Condition

Air Inlet Temperature       300/292      Refrigerant type   R410A
[T.sub.dry]/[T.sub.wet], K

Air Velocity,               1.0, 1.4,    Enthalpy          249, 260
m[s.sup.-1]                 2.0          of EEV
                                         outlet,
                                         kJ k[g.sup.-1]

Medium Temperature of
Condensation, K             308, 317     Mass flow rate,     62-94
                                         kg [h.sup.-1]

The test system consists of four subsystems of wind tunnel, refrigerant circuit, air and refrigerant property control system, and data acquisition system. A variable-speed blower is used to control the airflow rate. The temperature and humidity of air are controlled in the temperature and humidity control room. The mass flow rate of air is measured with a standard nozzle. The temperatures of air and refrigerant are measured with Type T thermocouples. The temperature measuring system has been calibrated with a standard platinum resistance thermometer, and the uncertainty is [+ or -]0.05 K for the entire temperature range of 173.15-473.15 K. The mass flow rate of refrigerant is measured by a mass flowmeter with a maximum error less than [+ or -]0.12% in the full range of 0-200 kg.[h.sup.-1]. The refrigerant pressure is measured using absolute pressure transducer with an error less than [+ or -]0.12% in the full range of 0-5 MPa. The mass flow rate of refrigerant is regulated by a pulse-controller for the electric expansion valve (EEV). The enthalpy of the coil inlet refrigerant is determined by the measured pressure and temperature at the inlet of the EEV. The air duct and connecting pipes are well insulated to prevent the heat exchange with the surroundings. The cooling capacities are measured with air enthalpy method with a maximum error less than [+ or -]4% in the full range of 0-5 kW. The experimental data are collected after the air-side dry-bulb temperature and wet-bulb temperature of the enthalpy potential system reach their steady states. The experimental data are collected every five minutes by using the data acquisitions system, a total of seven times for each set. Then the mean value of the seven-time-collected data is used as the actual experimental data.

Validation Results of the Extended GSSM

The experimental data are compared with the simulation results of the extended GSSM using the heat transfer and pressure drop correlations listed in Table 2. The simulation results are compared with the experimental results. The comparison results of the cooling capacity and refrigerant-side pressure drop between the simulation and experimental data are shown in Figures 6 and 7, respectively, which show that the difference between the calculated cooling capacity and the experimented one is less than [+ or -]5%, while the difference between the calculated refrigerant-side pressure drop and the experimented one is less than [+ or -]15%.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Table 2. List of Applied Heat Transfer and Pressure
Drop Correlations

Application
Region        Items          Correlations

                            Yun et al. (2002) for
                            two-phase flow, enhanced tube
             Heat-transfer
             coefficient
                            Dittus-Boelter equation
                            (Shah 1979) for single-phase
                            flow

                            Goto et al. (2001) for
                            two-phase flow,
                            enhanced tube

Refrigerant                 Chisholm and Sutherland
Side                        (1969) for two-phase
                            flow at the connecting
                            section between the
                            capillary and finned tube
             Pressure drop
                            Colebrook-White
                            equation (Smith et al. 2001)
                            for single-phase flow

                            Churchill (1977) for the
                            friction factor of capillary

             Heat-transfer  Nakayama and Xu (1983)
             coefficient    for slit fin
Air Side
             Pressure drop  Wang et al. (2001)
                            for slit fin

The calculated mass flow rates of refrigerant in each path of four test cases are listed in Table 3 in order to demonstrate the difference between the equal distribution case modeled for heat exchangers only, and the maldistribution case modeled for heat exchangers using capillary tubes for flow control. Among the four test cases, Case 1 does not use capillary tubes, Case 2 and Case 3 use different capillary tubes on different paths and have the same total mass flow rate of refrigerant as that of Case 1, and Case 4 uses the same capillary tubes as those used in Case 3 and has larger total mass flow rate of refrigerant than that of Case 3. In Table 3, the calculation results of Case 1 show that there is an equal refrigerant distribution in the heat exchanger without using capillary tubes. The calculation results of Case 2 and Case 3 show that varying the length or inner diameter of capillary tubes on different paths can control the refrigerant distribution among the paths, and the degree of the refrigerant maldistribution decreases with the increase of the inner diameter of the capillary tubes. The calculation results of Case 3 and Case 4 show that the degree of the refrigerant maldistribution increases with the increase of the total mass flow rate of the refrigerant. Therefore, by considering the effect of the capillary tubes on refrigerant distribution among the paths, the engineering designers can flexibly use the extended GSSM to control the refrigerant distribution in the MPHX by using different kinds of capillary tubes.

Table 3. The Effect of Using Capillary Tube to Control the Refrigerant
Distribution in the MPHX


                     Parameters of
                    Capillary Tubes

Case  Subpath    Inner         Length, mm  Mass Flow Rate    Ratio of
No.              Diameter, mm                   of          Refrigerant

                                           Refrigerant,    Distribution,
                                           kg[h.sup.-1]         %

1       SP12         --           --           31.2           50.0

        SP11          --          --           31.2           50.0

2       SP12         1.5         300           29.3           47.0

        SP11         1.5         150           33.1           53.0

3       SP12         3.0         300           31.0           49.7

        SP11         3.0         150           31.4           50.3

4       SP12         3.0         300           45.8           49.2

        SP11         3.0         150           47.2           50.8

CONCLUSIONS

By introducing two auxiliary equations to solve the refrigerant-lost problem in the previously developed GSSM (Liu et al. 2004) for the MPHX without capillary tubes, an improved model and algorithm for MPHX using capillary tubes for flow control is developed. The developed model and algorithm is suitable for not only the MPHX using capillary tubes for flow control, but also the MPHX without capillary tubes. An experimental rig is set up, and series of experiments on heat transfer and pressure drop characteristics of the heat exchanger using capillary tubes for flow control are performed to evaluate the developed model and algorithm. The evaluation results show that the difference between the calculated cooling capacity and the experimented one is less than [+ or -]5%, while the difference between the calculated refrigerant-side pressure drop and the experimented one is less than [+ or -]15%. Using different capillary tubes on different paths can help to control the refrigerant distribution among the paths in MPHX.

This study can theoretically contribute to modeling the MPHX using capillary tubes for flow control and other multicapillary systems, such as the multicapillary domestic refrigeration system, etc. In engineering practice, the associated software can give the users and designers the flexibility of analyzing the performance of the MPHX using capillary tubes for flow control, as well as analyzing the performance of the traditional MPHX without capillary tubes.

NOMENCLATURE 

[A.sub.o] = total surface area on air side, [m.sup.2]

[A.sub.i] = inside tube surface area, [m.sup.2]

d = inside diameter, m

f = friction factor or a function

G = mass flux, kg [m.sup.-2] [s.sup.-1])

h = specific enthalpy, kJ [kg.sup.-1]

L = length, m

M = mass flow rate, kg [s.sup.-1]

m = subpath number

[N.sub.m] = main path number

N = total tube number

p = pressure, kPa

[SP.sub.c] = collection of the no. for the sub paths containing choked capillary tubes

T = temperature, K

x = quality

Greek

[alpha] = heat transfer coefficient, kW[m.sup.-2][K.sup.-1]

[DELTA]p = pressure drop, Pa

[[eta].sub.0] = fin surface efficiency

[rho] = density, kg [m.sup.-3]

Subscript 

a = air

acc = acceleration

c = choke 

cap = capillary tube

cs = connecting section between the capillary tube and finned tube

fri = friction

m = mean value

max = maximum

in = inlet

out = outlet

r = refrigerant

s = standard path

tube = tube or fin-and-tube

wall = tube wall

Acronyms

EEV = electric expansion valve

GSSM = general steady-state model

MPHX = multipath fin-and-tube heat exchanger

MP = main path

SP = subpath

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APPENDIX

The refrigerant flow in the capillary tube is considered as adiabatic one-dimensional homogenous flow. Thermal equilibrium in the flashing process is assumed, and the flow resistance for the contraction is neglected. In this paper, the capillary tube is set after the EEV; thus, the refrigerant flow inside capillary tube is assumed to have no subcooling. The choked refrigerant mass flow rate is calculated according to the following criteria:

[dL.sub.cap]/[dp.sub.r,cap] = 0 (A1)

-d[p.sub.r,cap] = [G.sub.r,out,cap.sup.2](1/[[rho].sub.r,out,cap] - 1/[[rho].sub.r,in,cap]) + [[[f.sub.m][G.sub.r,out,cap.sup.2]/[2d.sub.cap][[rho].sub.r,m,cap]]]d[L.sub.cap] (A2)

The refrigerant flow inside the tube is considered a one-dimensional axial flow, and the axial conduction along the tubes is neglected.

Energy conservation equation for refrigerant flow in tubes is as follows:

[Q.sub.1r] = [Q.sub.2r] (A3)

Where

[Q.sub.1r] = [M.sub.r]([h.sub.r,in] - [h.sub.r,out]) (A4)

[Q.sub.2r] = [[alpha].sub.r][A.sub.i]([[T.sub.r,in] + [T.sub.r,out]]/2 - [T.sub.wall]) (A5)

where [[alpha].sub.r] is calculated from selected empirical correlations.

The continuity equation for refrigerant flow in tubes is as follows:

[G.sub.r,out] = [G.sub.r,in] (A6)

The momentum conservation equation for refrigerant flow in tubes is as follows:

[[DELTA][p.sub.r,tube]] = [[DELTA][p.sub.r,f]] + [[DELTA][p.sub.r,acc]] (A7)

where [p.sub.r,f] and [p.sub.r,acc] are calculated from selected empirical correlations.

The energy conservation equation for air is as follows:

[Q.sub.1a] = [Q.sub.2a] (A8)

Where

[Q.sub.1a] = [M.sub.a].([h.sub.a,in] - [h.sub.a,out]) (A9)

[Q.sub.2a] = [[alpha].sub.a][A.sub.o][[eta].sub.o]([[[T.sub.a,in] + [T.sub.a,out]]/2] - [T.sub.wall]) (A10)

where air mass flow rate [M.sub.a] is calculated based on upstream control volumes in front row; [[alpha].sub.a] is calculated from selected empirical correlations.

The continuity equation for air is as follows:

[G.sub.a,out] = [G.sub.a,in] (A11)

The momentum conservation equation for air is as follows:

[[DELTA]p.sub.a]] = [[DELTA]p.sub.a, fin]] + [[DELTA]p.sub.a, tube]] (A12)

where [p.sub.a,fin] is the air-side pressure drop due to the fin surface; [p.sub.a,tube] is the air-side pressure drop due to the tube surface.

The energy conservation equation for fin-and-tube is as follows:

[Q.sub.1r] + [Q.sub.1a] + [Q.sub.cond] = 0 (A13)

[Q.sub.cond] = [Q.sub.front] + [Q.sub.back] + [Q.sub.top] + [Q.sub.bottom] (A14)

where [Q.sub.cond] is the total heat conduction by fins; [Q.sub.front], [Q.sub.back], [Q.sub.top], and [Q.sub.bottom] are heat conductions by fins from nearest front row, back row, upper column, and bottom column, respectively.

For coil with divergence or confluence, the governing equations at the divergence or confluence points are needed in order to determine the inlet state parameters of refrigerant in downstream branches. Equations 10, 12, 13 are used for No. i divergence flow after replacing "1" with "i," and the following equations are used for No. i confluence flows:

[M.sub.r,i] = [m.[summation over (j = 1)][M.sub.r,ij](j = 1,2, ...,m) (A15)

[h.sub.r,i] = [[m.[summation over (j = 1)]][h.sub.r,ij][M.sub.r,ij]]/[[[m.[summation over (j = 1)]])][M.sub.r,ij]](j = 1,2, ...,m) (A16)

[p.sub.r,i] = [p.sub.r,ij](j = 1,2, ...,m) (A17)

where

[M.sub.r,i] = the mass flow rate at the point after the of No. i confluence

[h.sub.r,i] = specific enthalpy at the point after the of No. i confluence

[p.sub.r,i] = pressure of refrigerant at the point after the of No. i confluence

[M.sub.r,ij] = the mass flow rate in the No. j sub path of No. i confluence

[h.sub.r,ij] = inlet specific enthalpy in the No. j sub path of No. i confluence

[p.sub.r,ij] = inlet pressure of refrigerant in the No. j sub path of No. i confluence

Zhigang Wu

Guoliang Ding, PhD

Kaijian Wang

Masaharu Fukaya

Guoliang Ding is a professor in the Institute of Refrigeration and Cryogenics Engineering and Zhigang Wu is a PhD candidate at Shanghai Jiao Tong University, Shanghai, China. Kaijian Wang is senior engineer and Masaharu Fukaya is an engineer at the Fujitsu General Institute of Air-Conditioning Technology Limited, Kawasaki, Japan.

Received November 16, 2006; accepted August 9, 2007