Proposal, Robustness Analysis and Equivalent Implementation of Optimization Method for Row-by-Row Fin Distribution in Multi-Row Frosting Evaporator

The evaporator in a frost-free refrigerator typically has more tube rows, but frost deposition reduces along the airflow direction. Correspondingly, the evaporator fin distribution is thinner in the upstream rows but denser downstream, and a good match between frost and fin distribution is achieved to recover evaporator capacity loss. However, quantitative design principles of non-uniform fin distribution are lacking. A quasi-static frosting evaporator model is established and experimentally verified considering a three-dimensional (3D) evaporator, 1D frost growth and 1D non-uniform fin distribution. An optimization method for row-by-row fin distribution of a multi-row frosting evaporator is proposed based on the air pressure drop’s increase rate. When the increase rate in the air pressure drop of each row is almost equal, the smallest overall evaporator pressure drop is obtained, leading to the highest air flowrate and the greatest evaporator capacity. By applying the method, the air flowrate and the evaporator capacity increase by 5.5% and 4.6%, respectively, compared to the original fin distribution scheme. Moreover, the robustness of the optimization method is validated under wide temperature and humidity operating conditions. An equivalent implementation under an initial no-frost condition is also proposed to facilitate the optimization method without calculating the whole frosting process.


Introduction
The frost-free refrigerator is widely utilized nowadays because of its large capacity, automatic defrosting and multi-temperature compartments, i.e., refrigerating, variabletemperature and freezing ones. To meet the requirement of −18 • C temperature in the freezing compartment, the evaporation temperature should be even lower. Therefore, frost forms on the evaporator surface when the return air with higher temperature and humidity flows across. A frost layer will increase the thermal resistance and reduce the air flowrate of the evaporator, deteriorating the system's performance [1].
Much experimental work has been conducted on the frosting characteristics of evaporators and their influencing factors. These factors can be classified into three categories: (1) environment conditions such as the outdoor fan characteristics and the temperature & humidity of the frontal airflow. Kim et al. [2] used a slit fin bundle for the simulation of the heat exchanger in the air-source heat pumps (ASHPs) and experimentally explored the effects of cooling block temperature, air humidity and air velocity on its frosting characteristics. Zhang et al. [3] carried out the comparative study on the airside performance under different outdoor air fan control modes of ASHPs experimentally. (2) Geometrical parameters such as the fin types, the fin pitch and the surface treatment of fins. Huang et al. [4] experimentally studied effects of three fin types-flat, wavy and louver fins-on the periodic frosting and defrosting performance of evaporators in ASHPs. Wang and

Methods
The frosting model of the evaporator is described in detail in our previous work [21]. The frost growth process is assumed as a quasi-static phenomenon and adopts the Lewis analogy for calculations of mass transfer process. The water vapor is assumed to be saturated on the surface of the cold wall and the frost. The model is divided into three sub-models, namely, the geometric model, the heat and mass transfer model and the airflow model. The first divides the evaporator into many elements, the second calculates the heat and mass transfer both in the airside and inside the frost layer in every element and the third model connects each independent element by setting the rules of the air distribution and mixing.

Geometric Model
The evaporator studied in this research consists of 14 tubes spaced in 2 columns, as illustrated in Figure 1. The fins are separately mounted with free gaps between adjacent tube rows. The fin number generally increases from bottom to top tube rows (except for the top row) to mitigate the frost maldistribution caused by the leading-edge effect [20]. In this model, the evaporator is divided into elements based on the fin distribution. The internal element is composed of two adjacent fins and the flow channel between them, while the lateral element is only a half of the internal one.

Methods
The frosting model of the evaporator is described in detail in our previous The frost growth process is assumed as a quasi-static phenomenon and adopt analogy for calculations of mass transfer process. The water vapor is assume urated on the surface of the cold wall and the frost. The model is divided sub-models, namely, the geometric model, the heat and mass transfer model a flow model. The first divides the evaporator into many elements, the second the heat and mass transfer both in the airside and inside the frost layer in eve and the third model connects each independent element by setting the rule distribution and mixing.

Geometric Model
The evaporator studied in this research consists of 14 tubes spaced in 2 c illustrated in Figure 1. The fins are separately mounted with free gaps betwe tube rows. The fin number generally increases from bottom to top tube rows the top row) to mitigate the frost maldistribution caused by the leading-edge In this model, the evaporator is divided into elements based on the fin distrib internal element is composed of two adjacent fins and the flow channel betw while the lateral element is only a half of the internal one.

Heat and Mass Transfer Model
The frosting process during humid air flowing over the cold surface is ill Figure 2 with heat and mass transfer indicated. The heat transfer includes tw sensible part ̇ and the latent part ̇. The former accounts for the air cooled by the cold evaporator or frost, while the latter comes from the desub water vapor (into frost) contained in the humid air. Moreover, the mass tr plays two roles in frost growth. One part of the water vapor ̇ enters the in frost layer and increases its density and the other part ̇ desublimates on th the frost layer to increase its height.

Heat and Mass Transfer Model
The frosting process during humid air flowing over the cold surface is illustrated in Figure 2 with heat and mass transfer indicated. The heat transfer includes two parts: the sensible part . Q sen and the latent part . Q lat . The former accounts for the airflow being cooled by the cold evaporator or frost, while the latter comes from the desublimation of water vapor (into frost) contained in the humid air. Moreover, the mass transfer also plays two roles in frost growth. One part of the water vapor . m ρ enters the inside of the frost layer and increases its density and the other part . m δ desublimates on the surface of the frost layer to increase its height. In the airside, convective heat and mass transfer betwe layer is mainly considered. The differential equations are as where index refers to humid air, refers to frost surfac the density, is the airflow rate, is the specific heat, side heat transfer coefficient, is the total fin efficiency, area, is humidity ratio and is the Lewis number. Within the frost layer, by contrast, the heat transfer pr conduction with an internal heat source, which is the latent The differential equations of energy and mass conservation a � � +̇= 0 In the airside, convective heat and mass transfer between the airstream and the frost layer is mainly considered. The differential equations are as following, respectively.
where index a refers to humid air, f s refers to frost surface, dry refers to dry air, ρ is the density, V a is the airflow rate, c p is the specific heat, T is temperature, h is the airside heat transfer coefficient, η is the total fin efficiency, A f s refers to the frost surface area, ω is humidity ratio and Le is the Lewis number. Within the frost layer, by contrast, the heat transfer process is regarded as the heat conduction with an internal heat source, which is the latent heat of vapor desublimation. The differential equations of energy and mass conservation are as following, respectively.
with the following boundary conditions: where the index w refers to the wall surface of evaporator tubes, i sv is the latent heat of vapor into frost, . ρ f r is the increase rate of frost density and λ f r is the thermal conductivity of frost. More information about the model and the physical properties of air and frost is provided in [21].

Air Flow Model
The heat and mass transfer process in each element is calculated individually first by the model above, and then connected together with the following rules of air distribution and mixing.
(1) The air distribution rule: the air pressure drop across all elements in the same row is equal to satisfy the conservation of momentum. This rule can be used to connect the elements situated in the same row. (2) The air mix rule: the element number varies in different rows and fins are separate along the air flow direction. Therefore, the air flow from different upstream elements mixes before entering the downstream ones. The air mixing process needs to meet the law of energy conservation and composition conservation. This rule establishes the coupling of rows along the air flow direction.
It is noteworthy that in many existing publications, the airflow rate through a frosted evaporator is considered constant, which hardly conforms to the practical truth. Frost deposition increases the flow resistance of the finned-tube evaporator, and will inevitably reduce the airflow rate of the axial fan at constant power input. Therefore, the flowratepressure curve is fitted based on the experimental tests of the axial fan in this article and is expressed as a 5th order polynomial fit equation to reproduce the actual air flowrate variation in the tested refrigerator.
The characteristic curves of the small axial fan are similar. With the increase of air pressure drop, airflow rate decreases first gradually and then sharply. Due to the long period of frosting in the refrigerator, most of fans will enter the stall area where the airflow rate decreases sharply with slight increase of air pressure drop.

Initial Conditions and Boundary Conditions
In the real household refrigerator, the return air from different compartments is supposed to have different temperature and humidity conditions. The air stream from the refrigerating compartment is warm and humid, flowing through the left part of the evaporator, while that from the variable-temperature compartment flows through the right part of the evaporator as shown in Figure 1. By contrast, the return air from the freezing compartment is cold and dry, covering the whole frontal area of the evaporator. In this study, the temperature and humidity of the return air at the beginning of a stable on-cycle of refrigerators are taken as reference values as shown in Table 1. The main objective of this article is to investigate the influence of row-by-row fin distribution on the performance of the evaporator, so the effect of non-uniform return air is scarcely considered. Therefore, it is assumed that the return air is uniformly mixed at the bottom of the evaporator and the proportion is 1:1:2 for refrigerating, variable temperature and freezing return airflow, respectively, informed by the manufacturers. The parameters of return air after mixing are displayed in Table 2. The evaporating temperature varies in a frosting period and the value at the beginning of a stable on-cycle of refrigerators is taken as the simulated setting condition. The initial transient phase of the evaporating temperature is neglected due to the short time and small capacity. The initial frost thickness and frost density is set as 2 × 10 −5 m and 25 kg·m −3 , respectively. The initial frost surface temperature is equal to the evaporating temperature. The initial air flowrate is set as 0.0225 m 3 ·s −1 according to the fan characteristic curve. The time step is set as 1 s.

Solution Scheme
A flow chart describing the implementation of the frosting evaporator model in this paper is shown in Figure 3.
The initial frost thickness and frost density is s tively. The initial frost surface temperature is equa initial air flowrate is set as 0.0225 m 3 ·s −1 according to step is set as 1 s.

Solution Scheme
A flow chart describing the implementation of paper is shown in Figure 3.

Model Validation
The model is validated by comparing simulat

Model Validation
The model is validated by comparing simulation results with experimental data in open literature in our previous work [21] as shown in Figure 4. The test conditions are displayed in Table 3. The maximum deviation is less than 15%. Smaller air velocity in Test 1 leads to greater deviation, but the air velocity is close to 1.2 m/s in the following simulation process. The maximum deviation of Test 2 is less than 4% and the model is valid.
14, x FOR PEER REVIEW    Then, the model is utilized to study the effect of row-by-row fin distribution on frosting evaporator performance and a fin distribution design method of frost-free refrigerator evaporators is proposed based on the numerical results.

Setup of Different Fin Distribution Schemes
To match the reduced frost formation in the vertical direction due to the leading-edge effect [20], the fin density is generally thinner in the upstream rows but denser downstream on the evaporator (except Row 7, considering the refrigerant migration during the starting stage of the compressor). However, too many fins in downstream tube rows can also cause severe frost blockage there, as shown in Figure 5.  Based on the original fin distribution, other are proposed for the evaporator, as shown in T schemes are the same so that the heat transfer are For the original scheme, the fin numbers of the there is a high possibility that frost will block th of the fins on downstream tube rows are move schemes to tackle this problem. Particularly, Sch of keeping the almost equal increase rate of the r Based on the original fin distribution, other three different fin distribution schemes are proposed for the evaporator, as shown in Table 4. The total fin numbers for these schemes are the same so that the heat transfer area can be the same when there is no frost. For the original scheme, the fin numbers of the Row 5 and Row 6 are the greatest and there is a high possibility that frost will block the airflow channel there. Therefore, some of the fins on downstream tube rows are moved to upstream tube rows in other three schemes to tackle this problem. Particularly, Scheme 2 is optimized based on the method of keeping the almost equal increase rate of the row-by-row air pressure drop. The heat transfer performance of the frosting evaporator is numerically studied under the four schemes for the time duration of 400 min, and the results are comparatively analyzed. Figure 6 comparatively illustrates the variations of the evaporator capacity with frosting time among the four schemes. The evaporator capacity for all schemes decreases first gradually and then rapidly. The turning points of the decrease rate occurs in the sequence of the original scheme, Scheme 1, Scheme 3, and Scheme 2 at 286 min, 321 min, 323 min and 337 min, respectively. Moreover, the optimized Scheme 2 has the highest evaporator capacity, followed by the Scheme 3, Scheme 1 and the original scheme. The difference between the highest and the smallest evaporator capacity gradually increases with frosting time. The maximum difference occurs at the end of frosting process when the evaporator capacity of the optimized Scheme 2 is 248 W, 24% higher than 200 W of the original scheme.  The fact that the evaporator capacity for the optimized sc other three schemes is mainly owing to its greater air flowrate. S capacity curves in Figure 6, the air flowrate through the evapor at first, and rapidly then as shown in Figure 7. The air flowrate o is greatest, which accords with that of the evaporator capacity process, the air flowrate of the optimized Scheme 2 is 0.0146 m 3 than that of the original scheme. The fact that the evaporator capacity for the optimized scheme is higher than for other three schemes is mainly owing to its greater air flowrate. Similar to the evaporator capacity curves in Figure 6, the air flowrate through the evaporator decreases gradually at first, and rapidly then as shown in Figure 7. The air flowrate of the optimized Scheme 2 is greatest, which accords with that of the evaporator capacity. At the end of frosting process, the air flowrate of the optimized Scheme 2 is 0.0146 m 3 ·s −1 , which is 35% higher than that of the original scheme. Energies 2021, 14,6069 For all cases, both evaporator capacity and air flowrate exhibit a gradual decrease period at first and a sharp decrease one then. The turning point occurs when the axial fan enters the stall area where the airflow rate decreases sharply with slight increase of air pressure drop. In general, the optimized Scheme 2 has the greatest air flowrate and the latest stall point of the axial fan, leading to its highest evaporator capacity and the latest turning point. Therefore, the optimized Scheme 2 shows the highest time-averaged evaporator capacity, 4.6% higher than the original fin scheme.

2021, 14, x FOR PEER REVIEW
The difference in the air flowrate among four schemes is induced by that in the overall pressure drop of the evaporator. The increase of pressure drop will decrease fan air flowrate according to the flow characteristics of the fan. When the pressure drop reaches 16 Pa, the fan stall occurs. As shown in Figure 8, the pressure drop of the optimized Scheme 2 is smallest among the three cases. At 400 min, the air pressure drop of the optimized Scheme 2 is 17.3 Pa, which is 9.6% lower than 19.1 Pa of the original scheme. pressure drop. In general, the optimized Scheme 2 has th latest stall point of the axial fan, leading to its highest eva turning point. Therefore, the optimized Scheme 2 show evaporator capacity, 4.6% higher than the original fin sche The difference in the air flowrate among four sche overall pressure drop of the evaporator. The increase of p air flowrate according to the flow characteristics of the reaches 16 Pa, the fan stall occurs. As shown in Figure 8, mized Scheme 2 is smallest among the three cases. At 400 the optimized Scheme 2 is 17.3 Pa, which is 9.6% lowe scheme. The original scheme and Scheme 2 are specially inves their largest difference in overall pressure drop. The hydr ing evaporator under these two fin distribution schemes is The row-by-row pressure drop of the evaporator bet pared in Figure 9. Figure 9 shows that the initial value of th

Comparison of Hydraulic Characteristics among Different Fin Schemes
The original scheme and Scheme 2 are specially investigated for further study due to their largest difference in overall pressure drop. The hydraulic performance of the frosting evaporator under these two fin distribution schemes is comparatively studied.
The row-by-row pressure drop of the evaporator between the two schemes is compared in Figure 9. Figure 9 shows that the initial value of the pressure drop in the 5th and 6th rows in the original scheme is relatively large and increases rapidly due to the larger number of fins. At the end of the frosting process, the pressure drop of these two rows is 10.3 Pa and 4.85 Pa respectively, accounting for 80% of the overall pressure drop through the evaporator. However, the pressure drop for the 1st row is 0.34 Pa, which is only 3% of that for the 5th row. This extremely uneven distribution of row-by-row pressure drop for the original scheme leads to its highest overall pressure drop. By contrast, the single-row pressure drop in the optimized scheme increases gradually and uniformly. At 400 min, the difference between the highest and the smallest pressure drop is only 0.3 Pa. Therefore, it can be inferred that the total pressure drop will be the smallest when pressure drop of each row increases uniformly. number of fins. At the end of the frosting process, the pressure drop of these two rows is 10.3 Pa and 4.85 Pa respectively, accounting for 80% of the overall pressure drop through the evaporator. However, the pressure drop for the 1st row is 0.34 Pa, which is only 3% of that for the 5th row. This extremely uneven distribution of row-by-row pressure drop for the original scheme leads to its highest overall pressure drop. By contrast, the single-row pressure drop in the optimized scheme increases gradually and uniformly. At 400 min, the difference between the highest and the smallest pressure drop is only 0.3 Pa. Therefore, it can be inferred that the total pressure drop will be the smallest when pressure drop of each row increases uniformly. As frosting progresses, the frost layer gradually blocks the flow channel between adjacent fins, and leads to the increase in local pressure drop. Hence, the single-row pressure drop depends mainly on the blockage ratio defined by Equation (8).
where is the length of elements, is the width of t evaporator, is the height of fin, is the thickness of fin, is the thickness of frost, is the external tube diameter.
As shown in Figure 10, the blockage ratio of each tube row differs remarkably in the original scheme. The 5th and 6th rows have the highest blockage ratio, which explains their maximum single-row pressure drop. The blockage ratio of the 5th row is the largest, 29.5% higher than that of the 7th row which is the smallest. In the optimized scheme, however, the blockage ratio is uniformly distributed among the seven tube rows with very small difference. The blockage ratio of the 1st row is the largest and that of the 7th row is the smallest, which is 62.8% and 55.8% at 400 min, respectively, with a difference of only 7%. This leads to a more uniform distribution of single-row pressure drop in Figure 9.  The original scheme The optimized scheme As frosting progresses, the frost layer gradually blocks the flow channel between adjacent fins, and leads to the increase in local pressure drop. Hence, the single-row pressure drop depends mainly on the blockage ratio defined by Equation (8).
where l ele is the length of elements, W eva is the width of t evaporator, H f is the height of fin, δ f is the thickness of fin, δ f r is the thickness of frost, d is the external tube diameter. As shown in Figure 10, the blockage ratio of each tube row differs remarkably in the original scheme. The 5th and 6th rows have the highest blockage ratio, which explains their maximum single-row pressure drop. The blockage ratio of the 5th row is the largest, 29.5% higher than that of the 7th row which is the smallest. In the optimized scheme, however, the blockage ratio is uniformly distributed among the seven tube rows with very small difference. The blockage ratio of the 1st row is the largest and that of the 7th row is the smallest, which is 62.8% and 55.8% at 400 min, respectively, with a difference of only 7%. This leads to a more uniform distribution of single-row pressure drop in Figure 9. The blockage ratio between adjacent fins is influenced mainly by two factors: fin spacing and frost thickness. The frost layer thickness of each row for the original scheme and the optimized scheme is compared in Figure 11. Results show that the frost layer thickness of both two schemes decreases along the direction of air flow due to the lead- The blockage ratio between adjacent fins is influenced mainly by two factors: fin spacing and frost thickness. The frost layer thickness of each row for the original scheme and the optimized scheme is compared in Figure 11. Results show that the frost layer thickness of both two schemes decreases along the direction of air flow due to the leadingedge effect [18]. Hence, in order to prevent frost layer from completely blocking the airflow passage of upstream rows, the fin density is generally thinner in the upstream rows but denser in the downstream rows on the evaporator. In the original scheme, however, fins in the 5th and 6th tube rows seem to be over-dense and cause local blockage there. In the optimized scheme, by contrast, the fin spacing and frost thickness reached the condition that makes the blockage ratio of all tube rows similar. The blockage ratio between adjacent fins is influenced mainly by two factors: fin spacing and frost thickness. The frost layer thickness of each row for the original scheme and the optimized scheme is compared in Figure 11. Results show that the frost layer thickness of both two schemes decreases along the direction of air flow due to the leading-edge effect [18]. Hence, in order to prevent frost layer from completely blocking the airflow passage of upstream rows, the fin density is generally thinner in the upstream rows but denser in the downstream rows on the evaporator. In the original scheme, however, fins in the 5th and 6th tube rows seem to be over-dense and cause local blockage there. In the optimized scheme, by contrast, the fin spacing and frost thickness reached the condition that makes the blockage ratio of all tube rows similar. From the above discussion, it can be concluded that the optimization method of equal increase rate in row-by-row air pressure drop works by reaching a good match between the frost and fin distribution. Here the increase rate means the evolution in time of the pressure drop during a frosting process. Consequently, the overall air pressure drop of the evaporator can be smaller, leading to higher air flowrate and evaporator capacity.

Proposal of the Optimization Method
In Section 3.1.2, by comparing the heat transfer performance among different fin schemes, it can be concluded that when the air pressure drop of each row increases in From the above discussion, it can be concluded that the optimization method of equal increase rate in row-by-row air pressure drop works by reaching a good match between the frost and fin distribution. Here the increase rate means the evolution in time of the pressure drop during a frosting process. Consequently, the overall air pressure drop of the evaporator can be smaller, leading to higher air flowrate and evaporator capacity.

Proposal of the Optimization Method
In Section 3.1.2, by comparing the heat transfer performance among different fin schemes, it can be concluded that when the air pressure drop of each row increases in equal rate over time, the overall air pressure drop of the evaporator can be smaller, leading to higher air flowrate and evaporator capacity.
In Section 3.1.3, by comparing the hydraulic characteristics among different fin schemes, it can be concluded that the optimization method of equal increase rate in rowby-row air pressure drop works by reaching a good match between the airflow humidity and fin distribution.
The logical frame is displayed in Figure 12. The fin scheme with an equal increase rate in the air pressure drop of each row is the optimized scheme with best evaporator capacity.

Robustness Analysis of the Optimization Method under Wide Operating Conditions
In a real household refrigerator, the temperature and humidity of the airflow vary with the operating conditions. Thus, it is necessary to test the robustness of the optimization method under wide operating conditions.
The refrigerator studied in this research consists of refrigerating, variable-temperature and freezing compartments with a single evaporator. The return air from the freezing compartment always flows across the evaporator during the operation of the compressor and covers the whole frontal area of the evaporator. On the other hand, the return air from the refrigerating and variable-temperature compartments must first pass through the corresponding dampers which are controlled by the temperature sensor in the compartments. When the temperature of the compartments reaches the set value, the damper closes. Thus, based on the system control strategies above and port locations of the return air shown in Figure 1, the possible return air composition conditions are listed in Table 5. equal rate over time, the overall air pressure drop of the evaporator ca leading to higher air flowrate and evaporator capacity.
In Section 3.1.3, by comparing the hydraulic characteristics among schemes, it can be concluded that the optimization method of equal in row-by-row air pressure drop works by reaching a good match between t midity and fin distribution.
The logical frame is displayed in Figure 12. The fin scheme with an rate in the air pressure drop of each row is the optimized scheme with be capacity.

Robustness Analysis of the Optimization Method under Wide Operating Con
In a real household refrigerator, the temperature and humidity of th with the operating conditions. Thus, it is necessary to test the robustness zation method under wide operating conditions.
The refrigerator studied in this research consists of refrige ble-temperature and freezing compartments with a single evaporator. T from the freezing compartment always flows across the evaporator during of the compressor and covers the whole frontal area of the evaporator. On t the return air from the refrigerating and variable-temperature compartme pass through the corresponding dampers which are controlled by the temp in the compartments. When the temperature of the compartments reaches the damper closes. Thus, based on the system control strategies above and of the return air shown in Figure 1, the possible return air composition listed in Table 5.   The optimized fin distribution based on the method of keeping the almost equal increase rate of the row-by-row air pressure drop is calculated under different temperature & humidity conditions and the results are displayed in Figure 13. The difference among optimized fin distributions under different conditions is small. The maximum deviation of fin numbers in each row is less than 2. This shows the optimized fin distribution is adaptable to wide operating conditions and the optimization method proposed in this research has good robustness. By comparing condition #1 and #4, it is also proved that whether the return air is evenly mixed or not has little influence on the optimized fin distribution.

Equivalent Implementation of the Optimization Method under Initial No-Frost Condition
The optimization method proposed in this paper needs to obtain the row-by-row pressure drop data after the refrigerator has been operating for a while and then make fin distribution adjustment accordingly. However, it is difficult to do these operation experiments and adjustments in the actual evaporator design process. Thus, this paper also proposes an equivalent implementation method to provide predictive reference for evaporator design.
From the above discussion in Section 3.1, it can be concluded that the optimization method works by reaching a good match between the frost and fin distribution. Frost blockage ratio can be an index to evaluate the matching degree. Figure 14 comparatively illustrates the frost blockage ratio of three fin distribution at 0 min and 400 min. adaptable to wide operating conditions and the optimiz research has good robustness. By comparing condition # whether the return air is evenly mixed or not has little distribution.

Equivalent Implementation of the Optimization Method un
The optimization method proposed in this paper n pressure drop data after the refrigerator has been operatin distribution adjustment accordingly. However, it is diff periments and adjustments in the actual evaporator desig proposes an equivalent implementation method to pro evaporator design.
From the above discussion in Section 3.1, it can be c method works by reaching a good match between the f blockage ratio can be an index to evaluate the matching d illustrates the frost blockage ratio of three fin distribution

Equivalent Implementation of the Optimization Method under Initial No-Frost Condition
The optimization method proposed in this paper needs to obtain the row-by-row pressure drop data after the refrigerator has been operating for a while and then make fin distribution adjustment accordingly. However, it is difficult to do these operation experiments and adjustments in the actual evaporator design process. Thus, this paper also proposes an equivalent implementation method to provide predictive reference for evaporator design.
From the above discussion in Section 3.1, it can be concluded that the optimization method works by reaching a good match between the frost and fin distribution. Frost blockage ratio can be an index to evaluate the matching degree. Figure 14 comparatively illustrates the frost blockage ratio of three fin distribution at 0 min and 400 min. For the equal fin distribution, the frost blockage ratio is only influenced by the airflow humidity distribution (i.e., the leading-edge effect) and is the same in all tube rows at 0 min. With the frost growth, the frost blockage ratio gradually increases and at 400 min, it reduces along the airflow direction. On the other hand, the blockage ratio of each row is almost the same for the optimized fin distribution at 400 min. This is because the For the equal fin distribution, the frost blockage ratio is only influenced by the airflow humidity distribution (i.e., the leading-edge effect) and is the same in all tube rows at 0 min. With the frost growth, the frost blockage ratio gradually increases and at 400 min, it reduces along the airflow direction. On the other hand, the blockage ratio of each row is almost the same for the optimized fin distribution at 400 min. This is because the fin density in the optimized scheme is well coupled with the airflow humidity distribution.
The effects of airflow humidity distribution and the fin distribution are analyzed separately below. At 400 min, the relative slope factor of the blockage ratio for the equal fin distribution k ω is calculated by the Equation (9) below and it can be taken as the influence factor of the humidity distribution (i.e., the leading-edge effect). The relative slope factor of the blockage ratio for the optimized fin distribution k f in is also calculated and this factor is the required parameter in engineering applications. Results show that it meets