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Modeling for predicting frost behavior of a fin-tube heat exchanger with thermal contact resistance.

INTRODUCTION

Frost formation on the cold surface of a heat exchanger operated at below freezing temperature causes a decrease in the air flow rate and increase in the frost layer temperature resulting in reduced thermal performance. The complex geometry of fined tube heat exchanger coil results in uneven working fluid and air temperature distribution inside the coil and around the fins and tubes. This will lead to uneven frost formation and growth on the coil surface.

Although, much about fin-tube heat exchangers has been studied for a long time, the thermal contact resistance has not been fully investigated and often has been overlooked because the heat transfer through the interfaces is a complex phenomenon. Most of the models previously reported, did not consider dynamic variation of parameters inside the heat exchanger, such as: air temperature, humidity and fin-tube temperature.

A study on the effect of thermal contact resistance in a fin- tube heat exchanger was first attempted by Dart (1959). In his study, thermal contact resistances were tested with several samples with two passages, which were one for cold and the other for hot water. To minimize the influence of the natural convection, the tube was placed in an adiabatic chamber. The results were compared to that in soldered fins. Jones et al. (1975) studied frost formation on flat plates and found that increase of relative humidity of ambient air increases the frost formation rate, but increase of air velocity decreases the frost formation rate. Schneider (1978) investigated frost formation on cylindrical surfaces experimentally and found that frost formation is independent from Reynolds number (air velocity).

Abuebid (1984) investigated the thermal contact resistance with plate-fin tube heat exchangers placed in a vacuum. He performed an error analysis but the error band was narrower. Niederer (1986) performed experiments to investigate the frosting and defrosting effects on the heat transfer in heat exchangers. He found that frost accumulation on the coil surface reduced the air flow rate and the heat exchanger capacity. Eckels and Rabas (1987) predicted the thermal contact conductance, varying the number of fin, the fin thickness, and the diameter of tube in the wet and the dry fin-tube heat exchangers. In addition, they improved the empirical method, based on Dart's method, including error analysis.

Sami et al. (1989) developed a model to calculate the frost thickness and frost density for flat surfaces. Model results showed that increase of relative humidity or decrease of surface temperature accelerates frost formation. Kondepudi and O'Neal (1990) comprehensively discussed the effects of frost on fin efficiency, overall heat transfer coefficient, pressure drop, and surface roughness of extended surface heat exchangers. They suggested that more models are highly needed to determine effects of frost on fin performance. Rite and Crawford (1991) investigated the effects of various parameters on frost formation and performance of a domestic refrigerator-freezer fin tube evaporator. They concluded that the frosting rate increases for higher air humidity, temperature, flow rate and lower refrigerant temperature and the air side pressure drop increases as frost forms on the evaporator coil for a constant air flow rate.

Sherif et al. (1993) used a differential continuous model that includes the air side heat transfer coefficients taken from literature and the Lewis analogy to calculate frost thickness and surface temperature for flat plates operating under forced convection. Stubblefield et al. (1996) investigated the heat loss by thermal contact resistance using the insulated pipe and presented a simple method to predict the effect of contact resistance. Salgon et al. (1997) theoretically predicted the thermal contact conductance which was presented as a function of contact pressure and compared with experimental data. Lee et al. (2003) studied on flat plates experimentally and suggested correlations for frost thermal conductivity. According to their correlations frost thermal conductivity varies with its density.

Yan et al. (2003) investigated the performance of flat plate fin tube heat exchangers under frosting conditions experimentally. They concluded that the frost formation is greater for a lower air flow rate, and the rate of pressure drop increases rapidly as the relative humidity increases. Jeong et al. (2004) evaluated the thermal contact conductance using the experimental-numerical method for various fin-tube heat exchangers with 9.52 mm tube. Their results imply that the thermal contact resistance increases as the expansion ratio and the number of fin increases. Seker et al. (2004) developed a mathematical model to calculate the unsteady thermal characteristics of fin tube heat exchangers under forced convection conditions. The model was partially based on utilizing empirical correlations for predicting frost thermal conductivity and the amount of water vapor increasing the frost density. Their results were in good agreement with reported literature. Jeong et al. (2006) also investigated the thermal contact conductance using the experimental-numerical method in the fin-tube heat exchangers with 7 mm tubes. They revealed that the factors such as fin type, manufacturing type of the tube and etc. have a large effect on the thermal contact resistance in fin-tube heat exchanger with 7 mm tube. They concluded that the thermal contact conductance in the case of wide slit fin is larger than that of normal slit fin and that in the case of plate fin is largest of all fin types. Byun et al. (2007) numerically examined the effects of the heat exchanger type, refrigerant, inner tube configuration, and fin geometry on the evaporator performance of a fin-tube heat exchanger. They found that for the given inlet air temperature of 27[degrees]C and relative humidity of 50%, the heat transfer rate of the cross counter flow type heat exchanger is about 3% higher than that of the cross-parallel flow type. Also they compared three kinds of refrigerant, R-22, R-134a, and R -410A, and their effect on performance of the heat exchanger.

The purpose of this paper is to investigate the effect of frost formation on performance of a plain-fin round-tube heat exchanger having various parameters, including temperature and relative humidity of inlet air, working fluid temperature, uneven frost thickness on along fins and thermal contact resistance between the fin collar and tube.

PROBLEM DEFINITION AND MODEL FORMULATION

Consider the problem as a multiple cross-flow, plain fin-tube heat exchanger which consists of nine circuits and four rows. The specifications of this heat exchanger are listed in Table 1. Also a figure of the heat exchanger used in model is illustrated in Figure 1.

[FIGURE 1 OMITTED]
Table 1. Coil Specifications and Operating Conditions

Description                                       Value

Coil geometry (m)                  0.4572(width) x 0.4572 (height) x
                                   0.22(depth)

Fin density (fin per meter)                     710

Tube diameter (mm)                       9.525OD, 9,195(ID)

Number of rows                                    4

Fin type                                         Flat

Inter air temperature, [degrees]C                 0.0

Inlet air relative humidity %                     80.0

Face velocity (m/s)                              0.762

Working fluid flow rate (1/min)                  24.61

Working fluid type                 50% ethylene glycol/water

Working fluid inlet temperature,                 -15.0
[degrees]C

Fin and tube material                            Brass


A quasi-steady state control volume model was developed for modeling the complex problem. To reduce the complexity of the problem to a numerically solvable model, the following assumptions are made:

1. The working fluid mass flow rate in each circuit is considered uniform, and there is no heat conduction between circuits so that the heat exchanger is simplified as one circuit.

2. The axial heat conduction within the tube wall is ignored, as transverse conduction along the fins will be dominant.

3. Frost formation process is assumed to be quasi-steady state. Steady state conditions will be assumed to exist for sufficiently small time intervals but the overall analysis will remain transient in nature.

4. The frost density at any instant is an average value of the whole layer.

5. The amount of water vapor absorbed into the layer is proportional to water vapor density in the frost.

6. The thermal contact resistance between fin collar and tube is assumed to be as a brass to brass contact.

Following the above assumptions, the governing equations for working fluid, tube wall, frost and air can be formulated. Figure 2 shows the configuration of fin tube heat exchanger and frost for numerical model.

[FIGURE 2 OMITTED]

In order to make the model closer to the practical fin-tube heat exchangers, the temperature distribution along the fin is calculated first. The fin temperature profile will result in uneven frost height profile on the fin. The governing equations are based on a single phase fluid, i.e., glycol/water as the working fluid.

We have used two distinct coordinate systems for the geometry. The first one is a Cartesian (x, y) coordinate system which is used for governing equations in the refrigerant side and the air side. x is the distance in refrigerant flow direction and y is the distance in air flow direction. The second coordinate system which is used for this geometry is a cylindrical coordinate system (r, [theta], z), which is used for governing equations for the fin and the frost layer; r is the distance from center of the tube in the fin and z is the distance perpendicular to the cold walls.

Governing Equation for the Tube Wall

The tube wall of the heat exchanger consists of the bare tube and rectangular fins. The rectangular fin is converted into an equivalent angular fin of the same thickness and the same area as shown in Figure 2. The outer radius of the circular fin is determined as:

[r.sub.fin] = [square root of ([[fin width x fin height]/[pi]])] (1)

We assume that the temperature distribution inside the tube walls is a lumped model which does not change along the spacious.

The energy equation for the tube wall side is:

[c.sub.p,w][M.sub.w][[partial derivative][T.sub.w]/[partial derivative]t] = [q.sub.t] + [q.sub.fin] - [U.sub.t,r][A.sub.i,t] ([T.sub.w] - [T.sub.r]) (2)

Where

[q.sub.t] for condition without frost is

[q.sub.t] = [[U.sub.o][A.sub.o,t]/[c.sub.p,a]]([h.sub.a] - [h.sub.w]) (3)

And for condition with frost is

[q.sub.t] = [k.sub.fst][[partial derivative][T.sub.fst]/[partial derivative]z][A.sub.o,t] (4)

The heat transfer rate within the fin can be determined from governing equation for the fin derived in the following section.

Governing Equation for the Fin

As thickness of the fins is too small compared with the length of the fins, we consider the temperature distribution along the fin as a function of length (r). And neglect heat transfer through the thickness of the fin. So the heat transfer rate within the fin can be determined from:

[[rho].sub.fin][c.sub.p,fin][[partial derivative][T.sub.fin]/[partial derivative]t] = [k.sub.fin][[1/[r.sub.fin]][[partial derivative]/[partial derivative][r.sub.fin]]([r.sub.fin][[partial derivative][T.sub.fin]/[partial derivative][r.sub.fin]])] + S (5)

Where

The heat transfer term for condition without frost is

S= 2A[[U.sub.o]/[c.sub.p,a]]([h.sub.a] - [h.sub.fin]) [right arrow] (heat source per lenght) S = [2/[[delta].sub.fin]][[U.sub.o/[c.sub.p,a]]([h.sub.a] - [h.subfin]) (6)

And for condition with frost is

S = [2/[[delta].sub.fin]][K.sub.fst][[partial derivative][T.sub.fst]/[partial derivative]z] (7)

Governing Equation for the Working Fluid

For the working fluid side we consider the temperature distribution as a function of x direction and neglect the temperature variation along the y direction. So energy equation for the working fluid side is

[[rho].sub.r][c.sub.p,r][[partial derivative][T.sub.r]/[partial derivative]t] + [[rho].sub.r][u.sub.r][c.sub.p,r][[partial derivative][T.sub.r]/[partial derivative]x] = [2[r.sub.t,o]/[r.sub.t,i.sup.2]][U.sub.t,r]([T.sub.w,o] - [T.sub.r]) (8)

It is not a far assumption to neglect the thermal conductive resistance of tube walls and consider, [k.sub.w] = [infinity]. So [h.sub.r] is represented for the [U.sub.t,r].

U = [1/[[SIGMA].sup.RA]], [R.sub.1] = [L/k] = 0([k.sub.tube] = [infinity]), [R.sub.2] = [1/hA] [right arrow] [U.sub.t,r] = [h.sub.r] (9)

The correlations used to estimate the convective heat transfer coefficient of the working fluid in a single-phase region is the Dittus-Boelter heat transfer correlation. This correlation is applicable when forced convection is the only mode of heat transfer; i.e., there is no boiling, condensation, significant radiation, etc. The accuracy of this correlation is anticipated to be +/-15%.

For a liquid flowing in a straight circular pipe with a Reynolds number between 10000 and 120000 (in the turbulent pipe flow range), when the liquid's Prandtl number is between 0.7 and 120, for a location far from the pipe entrance (more than 10 pipe diameters; more than 50 diameters according to many authors) or other flow disturbances, and when the pipe surface is hydraulically smooth, the heat transfer coefficient can be expressed as:

[[h.sub.r][d.sub.i]/k] = 0.023[Re.sup.0.8][Pr.sup.0.4] (10)

The Reynolds number of the working fluid flowing inside the tubes is obtained from the following equation:

Re = [[u.sub.r][d.sub.i,t]/[upsilon]] (11)

Governing Equation for Air

The energy balance for the air side is; For condition without frost:

[[m.sub.a]d[h.sub.a]/dy] = [[U.sub.o][A.sub.o]/[c.sub.p,a]]([h.sub.a] - [h.sub.w]) (12)

And for condition with frost:

[[m.sub.a]d[h.sub.a]/dy] = [[U.sub.o][A.sub.o]/[c.sub.p,a]]([h.sub.a] - [h.subfst,s]) (13)

The mass balance energy for the air side is; For condition without frost:

[[m.sub.a]d[[omega].sub.a]/dy] = [U.sub.m][A.sub.o]([[omega].sub.a] - [[omega].sub.w]) (14)

And for condition with frost:

[[m.sub.a]d[[omega].sub.a]/dy] = [U.sub.m]([A.sub.o]([[omega].sub.a] - [[omega].sub.fst,s]) (15)

[U.sub.m] is the mass transfer coefficient which can be determined using the Lewis correlation.

[U.sub.m] = [[U.sub.o]/[c.sub.p,a]Le (16)

Where the Lewis number, Le, is taken to be 1 and 0.905 for condition without frost and frosting condition, respectively. Many researchers assumed that the Lewis number is equal to unity for the calculation of the mass transfer coefficient for convenience of analysis. However, the correlation obtained from the Lee et al. (2003) numerical results is given as Le = 0.905 [+ or -] 0.005. Therefore, it is believed that the previous works obtained from the model using the assumption of "Le = 1" may lead to yield more error, compared with those of the Lee et al. model.

In above equations we need to obtain [U.sub.0] that is equal to [h.sub.out], which will be obtained through following equations.

Air Side Heat Transfer Correlation for the Coil

The air side heat transfer correlation for the coil is calculated by Wang and Chi (2000)

[j.sub.a] = [Nu/Re[Pr.sup.[1/3]] (17)

Where, Colburn transfer factor is

For [N.sub.r] = 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Where

[b.sub.1] = 1.9 - 0.23 ln (Re)

[b.sub.2] = 0.236 + 0.126 ln (Re)

for [N.sub.r] [greater than or equal to] 2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When frost is formed on the heat exchanger coil, mass velocity, fin spacing, tube diameter and thickness of fin change with frost growth and are all incorporated into the model.

[U.sub.o] = [h.sub.o] [j.sub.a] = [Nu/Re[Pr.sup.[1/3]]] = [[[h.sub.o]L/[k.sub.air]]/[[V.sub.a]L/[[upsilon].sub.air]][Pr.sup.[1/3]]] = [[h.sub.o][[upsilon].sub.air]/[V.sub.a][k.sub.air][Pr.sup.[1/3]]] (20)

The values of [[upsilon].sub.air] and [k.sub.air] can be obtained using thermodynamic tables at an average temperature. [V.sub.a] can be determined using the inlet air mass flux, [m.sub.a], and the inlet face area.

Pressure Drop Across the Coil

The pressure drop across the coil can be computed by Wang and Chi (2000)

[DELTA][P.sub.a] = [[G.sub.max.sup.2]/2[[rho].sub.a,i]][[f.sub.a]([[A.sub.T]/[A.sub.min]]*[[[rho].sub.a,i]/[[rho].sub.m]]) + (1 + [[sigma].sup.2])([[[rho].sub.a,i]/[[rho].sub.a,o]] -1)] (21)

[[rho].sub.m] calculated at average temperature of air inlet and outlet and [f.sub.a] can be calculated by following correlation.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Pressure drop across the coil is a valuable parameter that can easily be compared with experiments to validate the model.

Governing Equations for the Frost Layer

The total heat transfer rate at frost air interface is given by:

[q".sub.a] = [[U.sub.a]/[c.sub.p,a]]([h.sub.a] - [h.sub.fst,s]) + [U.sub.m][h.sub.sg]([[omega].sub.a] - [[omega].sub.fst,s]) (23)

The transferred mass flux of water vapor from air to frost can be expressed as:

[m".sub.v] = [U.sub.m]([[omega].sub.a] - [[omega].sub.fst,s]) (24)

The mass transfer from the air contributes partly to frost densification and the rest increases the frost thickness. Each layer of frost forms in different temperature condition at each time step; therefore, the layers of frost piled up on each other have different density. So we can assume that the mass of frost layers to be changed in two ways; first, increasing the height of the frost ([m".sub.[delta]]), second, increasing the density of layers due to the temperature change, ([m".sub.[rho]]).

This may be expressed as:

[m".sub.[upsilon]] = [m".sub.[delta]] + [m".sub.[rho]] (25)

[m".sub.[delta]] = [[rho].sub.fst](d[[delta].sub.fst]/dt) (26)

[m".sub.[rho]] = [[delta].sub.fst](d[[rho].sub.fst]/dt) (27)

The water-vapor diffusion in the frost layer is expressed as:

D[[d.sup.2][[[rho].sub.[upsilon]]/d[z.sup.2]] = [[epsilon].sub.fst][[rho].sub.[upsilon]] (28)

Considering energy conservation in the control volume, the energy transfer mechanism is the heat transferred from the air conducted through frost layer and the latent heat transfer due to phase change from water vapor to frost crystal. This can be written as follows:

[k.sub.fst][[d.sup.2][T.sub.fst]/d[z.sup.2]] = -[[epsilon].sub.fst][h.sub.sg][[rho].sub.[upsilon]] (29)

[k.sub.fst] is obtained from Yonko and Sepsy's (1967). Their correlation was based on data obtained with frost surface temperature ranging from -30[degrees] C to -5.7[degrees]C.

[k.sub.fst] = [[0.02422 + 7.214 x [10.sup.-4] x [[rho].sub.fst] + 1.1797 x [10.sup.-6] x [[rho].sub.fst.sup.2]/1000] (30)

[[rho].sub.fst] will update at each time step.

The amount of water-vapor absorbed into a frost layer is given by

[m".sub.[rho]] = D[d[rho]/dz] = [[z = [delta]].[integral](z = 0)] [[epsilon].sub.fst][[rho].sub.[upsilon]]dz (31)

For each time step, the changes of frost density and thickness are the increment value added to previous time step value as follows:

[[rho].sub.fst,t] + [DELTA]t = [[rho].sub.fst, t] + [m".sub.[rho]]/[[delta].sub.fst][DELTA]t (32)

[[delta].sub.fst,t + [DELTA]t] = [[delta].sub.fst,t] + [[m".sub.[rho]]/[[rho].sub.fst]][DELTA]t (33)

Initial and Boundary Conditions

Working Fluid Side and Air Side. The initial conditions for air and working fluid are assumed to be in the steady state. In the steady state, the time derivatives in Equations 2, 5, and 8 are set to zero, with the solutions of the basic governing equations in the steady state being used as the initial conditions of the time-dependent system. So we should solve the steady state forms of equations in condition without frost to get the initial conditions of time dependent equations under frosting condition.

The boundary conditions for working fluid and air conditions onto the coil are; for inlet working fluid conditions,

[m.sub.r](x = 0) = [m.sub.r, in] [T.sub.r](x = 0) = [T.sub.r, in] (34)

And for inlet air conditions,

[m.sub.a](y = 0) = [m.sub.a, in]

[T.sub.a](y = 0) = [T.sub.a, in]

[[omega].sub.a](y = 0) = [[omega].sub.a,in] (35)

Fin Side. The initial conditions for the fin is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

In this study we assume the contact between the fin and tube as a brass to brass contact. So

[R".sub.t,c] = 0.025 [approximately equal to] 0.14 (37)

Frost Layer Side. The boundary conditions for Equation 28 are given by:

at z = 0, [d[[rho].sub.[upsilon]]/dz] = 0 [[rho].sub.[upsilon]] = [[rho].sub.[upsilon],sat]([T.sub.w]) at z = [[delta].sub.fst], [[rho].sub.[upsilon]] = [[rho].sub.[upsilon],sat] ([T.sub.fst,s]) (38)

Solving Equation 28 with conditions given in Equation 36, the water vapor density and absorption coefficient in the frost layer can be shown as:

[[rho].sub.[upsilon]](z) = [[rho].sub.[upsilon],sat]([T.sub.w]) cosh[phi]z (39)

[[epsilon].sub.fst] = D[{[1/[[delta].sub.fst]][cosh.sup.-1] [[[[rho].sub.[upsilon],sat]([T.sub.fst,s])/[[rho].sub.[upsilon],sat]([T.sub.w])]]}.sup.2] (40)

Where

[phi] = [square root of ([[epsilon].sub.fst]/D)]

The boundary conditions for Equation 29 are given by:

at z = 0, T = [T.sub.w] at z = [[delta].sub.fst], [k.sub.fst][dT/dz] = [q".sub.a] (41)

Solving Equation 29 using conditions in Equation 41 will result in temperature distribution inside the frost layer as:

T(z) = [[[epsilon].sub.fst]/[k.sub.fst][[phi].sup.2]][h.sub.sg][[rho].sub.[upsilon],sat]([T.sub.w])(z[phi]sinh[phi][[delta].sub.fst] - cosh[phi]z + 1) + [[q".sub.a]z/[k.sub.fst]] + [T.sub.w] (42)

The initial conditions for frost height and density on cold surface are important as results are sensitive to their selections. Jones and Parker (1975) tested the initial conditions by changing the values of the initial frost thickness and density. They found that the prediction results of the frost growth rate would not be affected significantly if the initial frost thickness value approaches a low value ([approximately equal to]2x [10.sup.-2]mm). They also found that as long as the initial value of the frost density is significantly smaller than the frost density during growth, it will not affect the solution for the frost growth rate of densification. Hence, in this work, the initial conditions for the frost temperature, thickness and density are fixed as:

Temperature [T.sub.fst] = [T.sub.w]

Thickness [[delta].sub.fst] = 2 x [10.sup.-2] mm (43)

Density [[rho].sub.fst] = 30 kg/[m.sup.3]

NUMERICAL SOLUTION

The governing partial differential equations are solved using the finite difference formulation. The coil is divided into a number of continuous non-overlapping control volumes as shown in Figure 3a. Equations 2, 5, 8, 13, and 15 form the basic dynamic system of the heat exchanger. To solve this set of equations for five unknowns [T.sub.r],[T.sub.w],[T.sub.a],[T.sub.fin] and [[omega].sub.a] other equations are required to calculate the thermodynamic properties of both working fluid and air side.

[FIGURE 3 OMITTED]

The variables of the refrigerant and the tube are placed at the cell center, while the air temperature and humidity are placed at the cell walls, as shown in Figure 3b. The equations are divided into two levels of iteration.

Frost mass and the temperature variation of the frost layer are obtained analytically in previous sections (Equations 39, 40, and 42, respectively). The pressure drop across the coil is obtained using Equation 21 in each time step. The flow chart of the numerical algorithm is shown in Figure 4.

FIGURE 4 OMITTED

Level 1: Solving for Variables at Working Fluid and Tube Wall Side

At this level, the air conditions ([T.sub.a], [[omega].sub.a]) onto each cell are assumed known, and a cell-by-cell method beginning from the working fluid inlet is used to solve for [T.sub.r] and [T.sub.w], using Equations 2 and 5. Newton-Raphson iteration algorithm (Williams et al. 1986) is employed to solve the set. The calculation starts from the working fluid inlet and proceeds cell-by-cell until all the cells at a particular row are covered.

Level 2: Solving for Variables at Air Side

At level 2, air temperature and humidity are calculated using Equations 13 and 15. And the variables obtained for tube wall and working fluid from the level 1 are used in this level. Newton-Raphson iteration algorithm is used to solve the equations.

RESULTS AND DISCUSSION

The simulation is conduced based on test conditions of Kondepudi and O'Neal (1993). The test coil was a single circuit, flat, fin tube heat exchanger. The working fluid was 50% ethylene glycol/water mixture. The coil specification and operating condition are shown in Table 1. The simulation was performed for the fin tube heat exchanger which was divided into 40 cells per coil numerically using a time-step of 10 s.

A comparison between the results of our model and Kondepudi and O'Neal (1993) experiments and Tso et al (2006) model has been performed. Parameters selected for comparison were frost mass accumulation, pressure drop and energy transfer coefficient.

After validating the presented model, the effects of face velocity (air flow rate), air relative humidity, and inlet working fluid temperature on overall heat transfer coefficient and air side pressure drop of the heat exchanger operating under frosting condition were investigated.

Frost Growth

The experimental results of Kondepudi and O'Neal (1993) shown in Figure 5 indicate that the mass of frost increases almost linearly with time. Higher values of air humidity, air temperature and lower working fluid temperature boost the frost growth.

[FIGURE 5 OMITTED]

For the experiment, the total mass accumulated after 50 min was 0.43 kg. The model predicted similar trend with the predicted mass of 0.44 kg accumulated at the end of the experiment. A total of approximately 2.5% discrepancy is found. The present model predicts results closer to experimental results than the simulation results of Tso et al (2006). The model predicts that after two hours the amount of accumulated frost on the heat exchanger is about 1 kg, which will cause decreasing in the performance of heat exchanger and increasing in the air side pressure drop. Therefore a way of defrosting is needed for the heat exchangers operating under frosting condition.

The simulated result reveals that the frost is thicker as it gets closer to the working fluid inlet point. Figure 6 shows the variation of frost height on tube in the direction of working fluid flow at t=50 min. The calculated frost height at cell 1 is 0.314 mm and 0.288 mm at cell 40. It is found that the difference is about 7.8%. The wall temperature is colder at the working fluid inlet and increases as it moves toward the outlet point. The calculated wall temperature at cell 1 is -8.36[degrees] C and exit at cell 40 at -7.98[degrees] C. This phenomenon agrees with other researchers in that frost growth is more rapid with a decrease in the temperature of the heat transfer surface.

[FIGURE 6 OMITTED]

The frost height varies along the fin due to the difference in fin surface temperature. Figure 7 shows the variation of fin surface temperature and frost height along the fin. The fin surface temperature is colder at the base with temperature of -8.01[degrees]C, whereas at fin tip it is -6.92[degrees]C; with a difference of 1[degrees]C. It is worth to indicate that the base temperature of the fin is different from the temperature of the tube wall at the same time step. This shows the effect of thermal contact resistance between the fin and the tube in the model which causes difference between the results of our model and Tso et al (2006) results. Owing to the difference along fin temperature, the frost height decreases toward the fin tip. The frost height at fin base is 0.3 mm and frost height is 0.2 mm at fin tip. Figure 7.

[FIGURE 7 OMITTED]

Air Side Pressure Drop Across Heat Exchanger Coil

The deposition and growth of frost on the surface of heat exchanger lead to a narrower fin pitch and a smaller air flow passage area. Lower air flow area directly affects the air side condition by increasing the pressure drop across the coil. Variables that affect the frost growth, such as working fluid temperature, humidity and air temperature have influence on the air side pressure drop across the coil. The higher the frost growth rate, the higher the air side pressure drop. Other factors that affect the pressure drop are the fin pitch and face velocity. In the experimental results shown in Figure 8, the air side pressure drop increases with time. The increasing trend is due to the frost height continuing to grow, thus reducing the air flow passage cross-sectional area. The experimental value for pressure drop increases from 0.49 to 10.77 Pa at the end of the experiment. The model predicts the same trend. The values are within 5% between the experimental and simulated results. The present model predicts closer results to experiment, as compared with the model presented by Tso et al in (2006).

[FIGURE 8 OMITTED]

Energy Transfer Coefficient

Kondepudi and O'Neal (1989) proposed that the energy transfer coefficient can be expressed as follow:

[E.sub.o] = [[Q x [c.sub.p,a]]/[LMED x [A.sub.o]]] (44)

The proposed energy transfer coefficient is a primary measure of the heat exchanger thermal performance which is similar in concept to the heat transfer coefficient, but includes the latent component of the energy transfer which occurs between air and heat exchanger. The energy transfer coefficient can be influenced by variables that affect frost growth such as humidity, air velocity, working fluid temperature and air temperature. Other factors such as fin spacing can also affect the energy transfer coefficient. High air humidity and air temperature, and low working fluid temperature will increase the energy transfer coefficient.

The variation of energy transfer coefficient with time for the experimental results is shown in Figure 9. The model predicts similar trend as the experimental results of Kondepudi and O'Neal (1993) and numerical results of Tso et al (2006), which also shows a decreasing trend. The decrease is due to the insulating layer of the frost.

[FIGURE 9 OMITTED]

Effects of Face Velocity (Air Mass Flow Rate)

Figure 10 shows the effects of air flow rate on heat transfer and pressure drop characteristics of heat exchanger. It is noted from Figure 10a that a higher air flow rate leads to a higher overall heat transfer coefficient as expected. The amount of frost formation increased as air flow rate decreased. This is because the surface of the heat exchanger becomes colder for a lower flow rate due to a lower heat transfer rate. The trend concerning the effect of air flow rate on frost formation is consistent with the experiments of Yasuda et al. (1990). However, it is contradictory to those of Rite and Crawford (1991). A decrease in air flow rate resulted in an increase in the frosting rate, thus the overall heat transfer coefficient degraded faster.

[FIGURE 10 OMITTED]

As shown in Figure 10b, the model data indicates that an increase flow rate results in a higher pressure drop initially. This is similar to the trends of dry heat exchangers. However, after 30 minute the pressure drop for [u.sub.a] = 0.381 m/s becomes the largest. This is because there is more frost formation blocking the flow passage and increases the pressure drop.

Effects of Relative Humidity

The effects of the air relative humidity on the performance of the heat exchanger are presented in Figure 11. Initially, the heat transfer rate and the overall heat transfer coefficient are very close to one another for 60%, 70% and 80% relative humidity. Air with a higher relative humidity has higher moisture content and leads to more frost formation. As a consequence, the overall heat transfer coefficient drop more quickly for higher relative humidity. The trend of increasing frost formation with humidity is consistent with that reported by Rite and Crawford (1991).

[FIGURE 11 OMITTED]

Figure 11b shows the effects of the relative humidity on the air side pressure drop. As the relative humidity increases, there is a higher pressure drop across the heat exchanger.

Effects of Working Fluid Inlet Temperature

Figure 12 shows the overall heat transfer coefficient, and pressure drop versus time for different working fluid temperatures. A lower working fluid temperature leaded to a lower surface temperature of the heat exchanger and a greater amount of frost; therefore, it had a larger heat transfer rate and a higher pressure drop. The trends are consistent with the results of Rite and Crawford (1991). It was also noted that for a lower working fluid temperature, the heat transfer rate declined and the pressure drop increased more rapidly. Because of a higher frost formation, the frost insulates and blocks the heat exchanger more quickly.

[FIGURE 12 OMITTED]

CONCLUSION

A comprehensive distributed model to evaluate the transient performance of fin-tube heat exchanger using single phase fluid i.e. glycol/water under frosting conditions has been presented. The model includes the thermal contact resistance between the fin and tube and a detailed frost growth model with a detailed distributed model that is able to predict the frost formation and heat exchanger performance on a practical fin tube heat-exchanger. The model is compared with existing reported experimental data and the results are well consistent with the experimental data. The following conclusions can be made.

a. Formation of frost degrades the performance of the heat exchanger.

b. The air and wall temperature vary along the tubes and coil which lead to a non-uniform frost growth along the coil.

c. The temperature along fin varies and this causes uneven frost growth. The frost height at fin base is higher than the fin tip.

d. The tube row where the working fluid inlet locates, which has lower wall temperature, has more frost accumulation and growth.

e. The frost growth propagates along the tube toward the coil exit.

f. The frost formation is greater for a lower air flow rate, and the rate of pressure drop increases.

g. The rate of pressure drop increases rapidly as the relative humidity increases.

NOMENCLATURE

A = area per meter, [m.sup.2]/m

[A.sub.T] = total heat transfer area

[A.sub.min] = minimum air flow area

[C.sub.p] = specific heat, kJ/kg*K

D = binary diffusion coefficient for water vapor in air at the frost temperature, [m.sup.2]/s

[D.sub.c] = collar diameter, m

[D.sub.h] = hydraulic diameter, m

[E.sub.0] = energy transfer coefficient, W/[m.sup.2]*K

f = friction factor

G = mass flux, kg/[m sup.2]*s

[G.sub.max] = maximum mass flux, kg/[m sup.2]*s

h = enthalpy, kJ/kg

[h.sub.sg] = enthalpy of sublimation, J/kg

[j.sub.a] = Colburn; factor

k = thermal conductivity, W/m*K

Le = Lewis number

LMED = logarithmic mean enthalpy difference

[M.sub.W] = mass per meter of tube wall, kg/m

m" = mass flux, kg/s*[m.sub.2]

m = mass flow rate, kg/s

Nu = Nusselt number

[N.sub.r] = number of rows

Pr = Prandtl number

[DELTA][P.sub.a] = pressure drop across coil, Pa

Q = total energy transferred, W

q = heat transfer rate per unit length, W/m

q" = heat flux, [W/[m.sup.2]]

r = distance in radial direction along fin, m

[R".sub.t,c] = thermal contact resistance between the collar fin and tube

Re = Reynolds number

S = heat transfer term, [W/[m.sup.3]]

[S.sub.fin] = fin spacing, m

T = temperature, K

t = time, s

u = velocity, m/a

[U.sub.m] = transfer coefficient of outside air, kg/s*[m.sup.2]

U = heat transfer coefficient, W/[m.sup.2]

[X.sub.l] = longitudinal tube pitch, m

[X.sub.t] = transverse tube pitch, m

x = distance in refrigerant flow direction, m

y = distance in coil depth (air flow) direction, m

z = distance perpendicular to cold surface, m

Greek Symbols

[delta] = thickness, m

[[epsilon].sub.fst] = absorption coefficient

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

[omega] = absolute humidity, kg/kg

[sigma] = minimum flow area/face area

Subscripts

a = air

fin = fin

free = free flow

fs = frost surface

fst = frost

i = inside

in = inlet condition

o = outside

r = refrigerant

S = surface

t = tube

[upsilon] = vapor

w = tube wall

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Yasuda, H, Senshu, T, Kuroda, S, Atsumi, iA, and Oguni, K. 1990. Heat pump performance under frosting conditions: Part II--Simulation of heat pump cycle characteristics under frosting conditions, ASHRAE Trans. 96 (1) 330-336.

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H. Shokouhmand, PhD

E. Esmaili

A. Veshkini

Y. Sarabi

H. Shokouhmand is a professor, A. Veshkini is a master's student, and E. Esmaili and Y. Sarabi are bachelor's students in the Department of Mechanical Engineering, University of Tehran, Tehran, Iran.
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Author:Shokouhmand, H; Esmaili, E; Veshkini, A; Sarabi, Y
Publication:ASHRAE Transactions
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Date:Jan 1, 2009
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