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Research Papers

# Modeling of Moisture Transport Into an Electronic Enclosure Using the Resistor-Capacitor ApproachPUBLIC ACCESS

[+] Author and Article Information
Ž. Staliulionis

Process Modelling Group,
Department of Mechanical Engineering,
Technical University of Denmark,
Nils Koppels Allé,
Kgs. Lyngby 2800, Denmark
e-mail: zygsta@mek.dtu.dk

H. Conseil-Gudla, R. Ambat

Materials and Surface Engineering,
Department of Mechanical Engineering,
Technical University of Denmark,
Nils Koppels Allé,
Kgs. Lyngby 2800, Denmark

S. Mohanty, J. H. Hattel

Process Modelling Group,
Department of Mechanical Engineering,
Technical University of Denmark,
Nils Koppels Allé,
Kgs. Lyngby 2800, Denmark

M. Jabbari

Warwick Manufacturing Group (WMG),
University of Warwick,
Coventry CV4 7AL, UK

1Corresponding author.

Contributed by the Electronic and Photonic Packaging Division of ASME for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received April 30, 2017; final manuscript received March 16, 2018; published online May 10, 2018. Assoc. Editor: Amy Marconnet.

J. Electron. Packag 140(3), 031001 (May 10, 2018) (11 pages) Paper No: EP-17-1046; doi: 10.1115/1.4039790 History: Received April 30, 2017; Revised March 16, 2018

## Abstract

The aim of this paper is to model moisture ingress into a closed electronic enclosure under isothermal and non-isothermal conditions. As a consequence, an in-house code for moisture transport is developed using the Resistor-Capacitor (RC) method, which is efficient as regards computation time and resources. First, an in-house code is developed to model moisture transport through the enclosure walls driven by diffusion, which is based on the Fick's first and second law. Thus, the model couples a lumped analysis of moisture transport into the box interior with a modified one-dimensional (1D) analogy of Fick's second law for diffusion in the walls. Thereafter, under non-isothermal conditions, the moisture RC circuit is coupled with the same configuration of thermal RC circuit. The paper concerns the study of the impact of imperfections in the enclosure for the whole diffusion process. Moreover, a study of the impact of wall thickness, different diffusion coefficient, and initial conditions in the wall for the moisture transport is accomplished. Comparison of modeling and experimental results showed that the RC model is very applicable for simple and rough enclosure design. Furthermore, the experimental and modeling results indicate that the imperfections, with certain limits, do not have a significant effect on the moisture transport. The modeling of moisture transport under non-isothermal conditions shows that the internal moisture oscillations follow ambient temperature changes albeit with a delay. Although, moisture ingress is slightly dependent on ambient moisture oscillations; however, it is not so dominant until equilibrium is reached.

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## Introduction

Nowadays, application of outdoor electronics in daily life is constantly growing. Its deployment in climatically harsh environment creates a significant challenge for engineers to design reliable and durable electronic devices and systems. As a consequence, a harsh environment drives a range of moisture-related failure mechanisms: acceleration of corrosion, leakage currents, alternation of material properties, electrolytic metal migration causing short or open circuits [19]. Moreover, these moisture-related failures tend to increase due to ongoing miniaturization of electronic devices and lower power consumption [10]. Therefore, development of models for moisture prediction under known ambient conditions is highly important, especially, for usage in the early and final stages of electronics design. However, such models can also be used for materials selection for electronics packaging or estimating the lifetime under known operating (humidity and temperature) conditions.

The modeling time is one of the most important factors in the whole electronics design. Currently, the computational fluid dynamics or finite element method methods are mostly used; however, they are too time-consuming for the initial design phase due to the computational efforts in covering all configurations and complicated three-dimensional structures consisting of different materials [1116]. Thus, it is always preferable to have a fast method, which requires less spatial resolution and yet offers sufficient accuracy [17]. Consequently, the RC approach to hygrothermal modeling has been adopted in various studies in the scientific literature, but has mostly been limited to interior climate in buildings [1823]. Thus, the novelty of this paper lies in the simplicity that allows extensive use of RC model in early stage of electronics enclosure design.

Hence, the main objective of this paper is to build a model and apply it for the moisture ingress through a wall and a hole into an electronic enclosure under isothermal and nonisothermal conditions and to compare the obtained results with experiments.

## Moisture Ingress Modeling Via Resistor-Capacitor Approach

Many paths exist for the moisture ingress into electronic enclosures such as openings, cracks, permeable walls, imperfect seals in cable feedthroughs and between box and connecting covers, and porosities of solid materials (Fig. 1(a)) [10,16,24,25]. The moisture ingress through these different paths into any electronics system can be modeled using the RC approach [16,17,24,2634].

Basically, the RC method is the network of resistors and capacitors, which can be used to describe the moisture transport in 0D, one-dimensional (1D), two-dimensional, or three-dimensional dimensional analysis. Furthermore, the coupling between different dimensions allows having the RC circuit as simple as possible. Moreover, such RC models can be implemented in any electronic circuit simulators like ltspice,simetrix,pspice, and matlab [35].

This section outlines the isothermal case for development of RC models in order to explore the moisture response to constant ambient conditions, which would also help to understand the moisture loads [10]. Furthermore, a deeper understanding helps to develop reasonable RC models. To develop the RC moisture model under isothermal conditions, the following assumptions must be considered:

• The temperature is the same inside and outside of enclosure and remains constant during the moisture transport.

• Convection inside the air-filled volume of a box and in the surrounding environment distributes the water vapor evenly in both regions.

• The diffusive flux is the same everywhere inside the opening at a given time.

Then, the process of moisture diffusion is represented by the first and second Fick's laws [36] Display Formula

(1)$J=−D∂c∂x$
Display Formula
(2)$∂c∂t=D∂2c∂2x$

To model moisture distribution based on Fick's laws, the RC hygro (in other words, moisture) circuit is applied in this paper. Moreover, the modeling is performed in matlab where the RC hygro circuit couples lumped and 1D analysis.

###### Resistor-Capacitor Circuit Analogy for Lumped Analysis.

Let us take an impermeable enclosure whose interior volume is connected through an opening with the ambient environment (Fig. 1(b)) and apply the lumped analysis with the assumptions made above (Sec. 2).

To study the moisture transport based on the lumped analysis in electronics enclosure, the so-called quasi-steady-state (QSS) model is applied [24]. The solution of the QSS model for the interior moisture response (when the initial vapor concentration is not zero) is expressed by Display Formula

(3)$ci(t)=ca−(ca−c0i)e−tτ$
which can also be given in terms of relative humidity (RH) under isothermal conditions [31] Display Formula
(4)$RHi(t)=RHa−(RHa−RH0i)e−tτ$

where the time constant and diffusion resistance are Display Formula

(5)$τ=VairLAD=VairRopening$
Display Formula
(6)$Ropening=LDA$

This QSS model for diffusion can also be represented as electric circuit where capacitor is charged or discharged through resistor and has been illustrated in Fig. 2(a). The time constant is the response time for an RC circuit, which describes how quick a system accumulates the moisture inside defined volume.

###### Resistor-Capacitor Circuit Analogy for Walls.

One-dimensional analysis is based on Fick's second law (Eq. (2)) and RC circuit for 1D analysis of an enclosure wall is illustrated in Fig. 3. Here, the resistance of the discretized element in RC circuit is given Display Formula

(7)$Re=deDA$

Furthermore, the capacitor placed in the center of the element is equal to the volume of the element. Basically, the RC approach results in mathematically analogues relations as to using the resistance based control volume—finite difference method, because it uses a simple mesh of rectangular cells in which the capacity in the nodal point is lumped [37,38].

Let us take the plastic closed box and assume it to be completely sealed (Fig. 1(b) without the opening). If possible, the RC circuit model is simplified as much as it can be under the assumptions made in the beginning of Sec. 2. Considering these assumptions, a closed enclosure can be modeled using a 1D dimensional analysis in combination with lumped components where the air-filled volume of the enclosure can be used as a lumped capacitor, Cair,m. The 1D description is used for the wall, which is discretized evenly along its thickness direction into three elements. Thus, the RC circuit for the closed enclosure is as shown in Fig. 3, but excluding the resistance of the opening (Ropening,m).

Usually, a real case of the box is not completely sealed and has some cracks or imperfect seals through a cable feedthrough and in between the box and its connecting covers. Thus, such imperfections can be presumed emulated as an equivalent opening, which is represented by a lumped resistor component (Ropening,m) in the RC circuit. In this case, the lumped resistor Ropening,m is connected in parallel to the circuit representing the wall, since there are two paths of moisture ingress (Fig. 3). The predominant path of moisture ingress depends on time constant of either opening or wall.

Thus, considering the imperfections, the RC circuit can also be used to study for the moisture ingress through a permeable wall and an opening (Fig. 1(b)). Furthermore, this RC circuit is a course model for the representation of simple electronic enclosures, which can always be improved until the simulation and experimental results are in good agreement, if needed.

The discussed RC models in this section are used for further interior moisture study; however, they have a drawback, which arises from inherent discontinuities of concentration at multimaterial interfaces and this is briefly discussed within Sec. 2.3.

###### Discontinuity at Multimaterial Interfaces.

In order to model the moisture transport using the equivalent RC circuit, one issue has to be solved, which arises from the fact that concentration is discontinuous at the environment–material or material–material interfaces [39]. Thus, a modification is needed to make the RC hygro circuit applicable for moisture modeling through different materials and this issue is broadly discussed in the papers [16,17]. Moreover, the diffusive mass transfer through the wall is mostly considered as Fickian in many polymeric materials used in electronics [17,40] where the discontinuity at the environment–material interface is normally described by the well-known Henry's law [4144].

The modification of the RC circuit is implemented through the transformed resistors and capacitors. The transformation of the resistors and capacitors is based on the usage of the reference material, which is usually air. Then, all investigated polymers and their segments are represented by equivalent air volumes, in order to have the same amount of water vapor as in the actual material. Such replacement with equivalent air enforces continuity at material interfaces where the RC circuit is described by one uniform material.

First, the actual volume (capacitor) of the wall segment is transformed to an equivalent air capacitor using the following equations [16,17]: Display Formula

(8)$Vequivalent=Mwater_in_polymer/ca=S(T)⋅pa⋅Vpolymerca$
Display Formula
(9)$ratioV=S(T)⋅pa/ca$
Display Formula
(10)$Cequivalent=Vequivalent=Vpolymer⋅ratioV$

From the equations, it is seen that ratioV defines the relation between actual and equivalent volumes. Basically, the dimensionless ratioV is the transformation factor, which relates the water vapor concentration in the polymer and water vapor concentration in air or equivalent volume of reference material to the actual material volume.

Second, the resistors representing diffusion needs to be transformed as well. Thus, the lumped component of each segment of different material is the diffusion resistance divided by ratioVDisplay Formula

(11)$Rdiff_equivalent=de/(A⋅D⋅ratioV)$

Since there was no experimental validation in the proposed modification, a comparison was made between two RC circuits in order to prove the validity of the model. Two RC circuits were analyzed with only different values of resistors and capacitors; however, the configuration of the RC circuit was the same. Now, let us take a plate with 3 mm length (Fig. 2(b)) and made of polycarbonate (PC) with solubility of 0.0016 kg/(m3·Pa) and diffusion coefficient of 6.5 × 10−12 m2/s at temperature of 25 °C. The moisture flow was modeled in 1D (x-direction) and assumed to be perpendicular to the plane of the plate.

The plate was discretized into three elements (Fig. 4(a)) and exposed to different environments on its two sides with the RH on the left side of the plate being 98% and on other side 40%. The maximum density of water vapor is 23 g/m3 at 25 °C and the initial value of RH in the wall was equal to 40%. Then, the resistors and capacitors of modified RC circuit were determined based on Eqs. (8)(11) discussed above. Here, the first RC circuit presents a PC wall (Fig. 4(a)), and the second represents diffusion through a wall replaced by equivalent air (Fig. 4(b)).

The comparison of curves is based on concentration values at each element node and performed by using the relation below: Display Formula

(12)$cn,solid=cn,air⋅ratioV$

Comparison of the results of these two RC circuits in Fig. 5 shows that concentrations at each element are even and therefore, such modification of RC hygro circuits can be used to model moisture response. Section 3 concerns the experimental setup used to analyze modeling results.

## Experiments

The aim of this section is to describe the experiments, which are used for the application of the RC model. Two cases are experimentally investigated, namely, a closed box and a box with the small opening.

The first experiment was accomplished using a closed box, more specifically an EKJB 130 T enclosure taken for a study from Fibox (Fig. 6(a)), which is made of polycarbonate with the dimensions of 280 mm × 190 mm × 130 mm and the ingress protection rating [45] of 66/67. Calibrated sensors (PT1000 and HIH4021, Honeywell) were placed inside the enclosure to monitor the temperature and relative humidity over time.

The sensors were connected by the cable to a data logging system (model 2700 Multimeter, Keithley Instruments); therefore, the feedthrough had to go through the enclosure wall (Fig. 6(b)). The enclosure placed in a climatic chamber (Espec, Escorp PL-3KPH) was exposed to a constant temperature of 25 °C and relative humidity of 98%. Before starting the experiment, the enclosure was kept inside the climatic chamber until the initial interior temperature and relative humidity reached equilibrium and was 25 °C of temperature and 34.63% of RH everywhere with the exterior of the enclosure. Some other experiments started at slightly different initial RH and therefore simulations were also adjusted accordingly (Sec. 4). Then, the RH in the climatic chamber was increased up to 98% within a few minutes until the equilibrium establishes. Such delay is considered negligible, since it is very short as compared to the whole mass transfer equilibrium time inside the enclosure.

The second experiment was made using the same enclosure, however, with a drilled opening in one of the walls (Fig. 6(c)). Two separate experiments were accomplished with 1 mm and 3 mm diameter opening while the length of the opening was 2.2 mm. This time, the experiment was not performed in the climatic chamber in order to eliminate impact of forced convection or, in other words, air circulations. As a consequence, the enclosure with a drilled opening was placed inside another bigger closed box with sufficient water to keep RH = 100% at 25 °C over the time of the experiment. When RH reached 100%, the enclosure was placed inside the box to run the experiment and record the results. Moreover, in the start of the experiments, the initial conditions of the RH inside the box with 1 mm opening was 38.09% and with 3 mm hole was 38.23%.

## Simulation Results

This section outlines the RC model application to an electronic enclosure for predicting moisture response based on the same boundary conditions as in the experiments. In order to model the moisture response, two RC moisture circuits discussed in Sec. 2.2 are used in this section. Furthermore, sensitivity analysis was accomplished based on different discretizations of the wall (Fig. 23). Since higher discretization did not change results noticeably, the RC circuit with three elements for the wall is used in this section. Then, simulation results based on these RC circuits were compared to experiments.

Having the final RC model for simulations, parametric study for moisture ingress into electronic enclosure was conducted in order to get a better agreement with experiments. Moreover, such modeling and analysis of different parameters, e.g., wall thickness, imperfections, initial conditions, can help to understand the whole diffusion process and other affecting factors occurring in the experiments. First, the effect of different wall thicknesses for the moisture response inside the enclosure was studied based on the experimentally found diffusion coefficient. Then, the imperfections of the closed enclosure were investigated, and the same modeling procedure was applied for the enclosure with an opening. Moreover, a temperature-dependent diffusion coefficient was studied as well. Finally, the modeling results from the aforementioned cases were compared with some experiments.

###### Impact of Wall Thickness.

The same closed PC box is used for the modeling as in the experiments (Figs. 6(a) and 6(b)) and the corresponding RC moisture circuit for this case is presented in Fig. 3; however, the opening resistance Ropening,m is not considered in the circuit. The enclosure was exposed to 98% of ambient relative humidity while initial RH inside the enclosure was 34.63% and the diffusion process was carried out at 25 °C of temperature.

The diffusion coefficient of 4.51 × 10−12 m2/s and the solubility coefficient of 0.0013 kg/(m3·Pa) (Table 2 [46]) for the PC material were found experimentally at 25 °C temperature. Moreover, the estimated solubility is used the same for all further simulations in the Sec. 4. As in reality, the thickness (L) of the enclosure wall varied from 2 mm to 3 mm; however, the range from 2 mm to 2.6 mm was considered for the moisture study and comparison with experimental results (Fig. 7). Such range was chosen in this case because it can represent the most realistic equivalent length of the wall to the investigated electronic enclosure.

Figure 7 shows that the interior moisture level up to almost 1 day remains the same as the initial RH. This shows that the wall behaves as a buffer at the beginning of the moisture transport process. Such behavior is explained by using the RC circuit in Fig. 3 without the opening resistance considered, that the interior moisture increases when first the wall absorbs a certain amount of moisture before the moisture is transported through the wall into interior volume Cair,m. In this case, the moisture transport process is mainly controlled by the thickness of the enclosure wall because other parameters like surface area, diffusion, and solubility coefficients are well defined and remain fixed.

Comparing the modeling and experimental results around day 14, the equivalent thickness of enclosure wall corresponding to the behavior of physical enclosure is somewhere between 2.2 and 2.4 mm. Since the thickness of actual wall was not even everywhere, such obtained thickness seems reasonable. Basically, all modeling results of moisture response followed the development of experimental curve over the time, however, completely mismatched the experimental results at the outset. Thus, one reason is that moisture ingress might be affected by another factor like imperfections or sealing of enclosure and this is thoroughly discussed in Sec. 4.2.

###### Impact of Imperfections and Hole.

Since the enclosure is never perfectly sealed in the real world, some imperfections or cracks might affect the moisture response inside the box. In order to consider the imperfections, the RC moisture circuit from Fig. 3 is used where the emulated opening of the imperfections is represented as a lumped resistor Ropening,m. Here, the 2.2 mm wall thickness is used in the model for moisture study, since it matched mostly with experimental results in Fig. 7. To investigate how the imperfections affect the total moisture transport process, the diameter (d) of the emulated opening representing the imperfections was considered in the range from 0.5 to 2 mm (Fig. 8).

Figure 8 shows that the moisture is mainly transported by diffusion through the wall and the imperfections do not have a significant impact for the overall moisture behavior over time. However, they still play an important role in the moisture transport at the beginning of the process until the wall absorbs a certain amount of water vapor and then a larger portion of the moisture flows through the wall into the box than through the imperfections.

In order to confirm findings in the modeling (Fig. 8), the additional experimental test (Fig. 9) was conducted when enclosure is sealed as much as possible on the most likely leaking parts, e.g., gasket of the enclosure and cable feedthrough (Figs. 6(b) and 6(c)). The experimental results were compared with modeling as well in Fig. 10.

When a proper sealing is ensured on the enclosure, the moisture behavior from experiments is almost identical as moisture transport was explained in Sec. 4.1 using RC circuit (Fig. 3 without the opening resistance). Thus, Fig. 10 shows that modeling results converges with experiments and here the best match with experiments was achieved with 2.4 mm wall thickness, which was even a little thicker as compared to Fig. 7. Moreover, the RC circuit model confirms and explains very well the mechanism occurring in experiments. This also shows that the RC circuit used for modeling of moisture transport into electronics enclosure is sufficient and can be used as a basis for further modeling. Furthermore, the experimental results in Fig. 9 also confirm the fact found in the modeling that imperfections are not predominant in the overall moisture diffusion process as well.

Next simulation was performed using the enclosure with drilled different size of opening where the diameter of opening was 1 mm and 3 mm (Fig. 6(c)). Moreover, the length of the opening was 2.2 mm. The same RC circuit is used as in the modeling of the imperfections, since it represents the same mechanism of the moisture transport. The modeling results were compared with the experimental results as well in Fig. 11.

In this case, the moisture transport occurs through both paths, namely walls and opening. Moreover, the moisture flowing through the opening into the box charges not only the interior air cavity, but also the wall through the inner surface of the box until the moisture flow increases in the wall more than in the beginning. Dependent on the resistance of the opening, the moisture transport might be predominant through either the opening or the wall. Moreover, it can also be evenly transported through both the opening and the wall. For instance, the moisture flow through 1 mm opening remains smaller than through the wall (Fig. 11) while it is opposite in case of 3 mm opening shown in Fig. 12.

Comparing experiments and modeling results in Fig. 11, the moisture behavior from the modeling is very similar in case shown in Fig. 7, since the moisture diffusion is mainly predominant through the wall. The mismatch of the results in the beginning of the moisture transport (Fig. 7) can also be affected by other factor as initial conditions in the wall. Since a certain time was needed to install the experimental setup in the investigated environment, during this period, the initial conditions in the wall might vary and change before experiment is started. In case of larger opening (Fig. 12), the modeling results are in good agreement with the experiments, because the moisture is more predominant through the opening and other factors including the imperfections and possible initial conditions can be neglected. However, the initial conditions remain interesting part for the study in Sec. 4.3 and how the moisture behavior is affected in both cases of 1 mm and 3 mm opening.

###### Impact of Initial Conditions.

Since the experiment with open enclosure was different, the initial conditions might be slightly different, because the enclosure was placed in the bigger box for a while until the setup was ready for monitoring the moisture. Then, the enclosure was kept until RH was reached up to 38.09% in case of 1 mm hole and 38.23% in case of 3 mm hole. In general, the readings were taken after one day, which means that initial conditions were different, especially, on the outer surface. Moreover, the modeling gives insights that the relative humidity increases from 38% up to 50% on outer element of wall (initial condition was RH = 38%), in about 5 h based on this enclosure case. This means that initial conditions must be taken into account in the modeling in order to investigate the behavior of interior moisture response.

Here, the modeling was accomplished for the enclosure with 1 mm and 3 mm opening and compared with the experiments. Additionally, small imperfections of emulated opening of 0.5 mm were considered in one of the curves in Fig. 13.

Figures 13 and 14 show that the initial conditions are more important in case of the enclosure with 1 mm opening, since moisture flow is predominant by the diffusion process through the wall. In case of the enclosure with 3 mm, the moisture response is also affected by the initial conditions especially in the beginning; however, the major portion of the moisture is still transported through the opening and therefore it is not so significant as in the case with 1 mm opening.

Summarizing the results, imperfections and initial conditions are more significant in moisture transport for a closed enclosure or enclosure with the small opening while these factors are negligible for enclosure with larger opening e.g., >3 mm. Moreover, the impact of initial conditions can also be significant in cyclic ambient conditions, if a cycle is relatively short; thus, the interior moisture behavior can be affected and can vary considerably.

###### Impact of Diffusion Coefficient.

When it comes to the selection of materials, the diffusion coefficient is one of the most important factors in the entire moisture diffusion process. Thus, this section studies the moisture behavior under different diffusion coefficient (D) of polycarbonate material.

The same enclosure and all boundary conditions from the previous Sec. 4.1 are used for the modeling. The wall thickness of the enclosure was 2.6 mm in the modeling and different diffusion coefficients of polycarbonate material were used for the wall (Fig. 15). These temperature-dependent diffusion coefficients are taken from the literature investigating polycarbonate material for the moisture transport [4749].

Thus, the moisture transport was studied for the calculated 0.35 × 10−12 m2/s, 4.51 × 10−12 m2/s, and 7.52 × 10−12 m2/s diffusion coefficients at 25 °C temperature (Table 1) [4749]. However, the diffusion coefficient is not considered from Ref. [48] as the value is very low.

Diffusion coefficient was calculated using the equation [50,51] Display Formula

(13)$D(T)=D0e(−EDRT)$

Then, the response of moisture inside box is presented in Fig. 16.

As can be seen, the moisture response changes significantly as diffusion coefficient increases. Figure 16 shows that in the beginning, the response under 4.51 × 10−12 m2/s is slower and has bigger error compared to experiments while in the end of the process, it gets closer. The larger diffusion coefficient causes faster diffusion process into the enclosure because of a faster absorption process of water vapor.

## Modeling Under Nonisothermal Conditions

The dependence of diffusion coefficient on temperature leads to the study of nonisothermal conditions. To model nonisothermal conditions, an additional RC thermal circuit is used to solve temperature field (Fig. 17).

Then, the two RC circuits (thermal and moisture) are coupled in the in-house code wherein the thermal field and concentration distribution are coupled via the diffusion (Eq. (13)) and solubility equation shown below [5053]: Display Formula

(14)$S(T)=S0e(ESRT)$

Here, the material's thermal conductivity is not dependent on the concentration in this study. Thus, the temperature is also represented as the voltage and the thermal resistance of discretized elements is expressed below [38]: Display Formula

(15)$Relement,th=de/kA$

and the thermal capacity is given by the equation as follows: Display Formula

(16)$Celement,th=ρcpdeA$

Here, thermal RC circuit uses the same configuration circuit as it is for the moisture study, except that resistance and capacitors values are different. Moreover, the implicit (Backward Euler) method [38] is applied to solve the coupled thermo-hygro-circuit in 4 to use an unconditionally stable scheme allowing for higher time increments. Such an in-house code can now solve a nonisothermal case.

###### Relative Humidity Modeling Under Oscillating Ambient Conditions.

First of all, the moisture transport is simulated when the enclosure is exposed to the cyclic ambient RH and constant ambient temperature (isothermal conditions). Here, the average ambient temperature and cyclic RH conditions of one month are taken from Copenhagen hourly climatic data according to the global humidity index2 and are shown in Fig. 18. The average temperature of a month was calculated to be 5.12 °C and was kept the same in the enclosure over simulation time; however, the initial relative humidity was 40% of RH. The same enclosure with 2.6 mm wall thickness from previous Sec. 4.1 is used. The modeling was accomplished when the pre-exponential diffusion factor D0 is 2.882 × 10−5 m2/s and diffusion activation energy ED − 38.844 kJ/mol (Table 1). Moreover, the pre-exponential factor of solubility S0 and solubility activation energy ES for the polycarbonate material are 5.37 × 10−10 kg/(m3·Pa) and 36.2 kJ/mol, respectively [48].

Figure 19 shows that for the time interval (t1), first the moisture absorption occurs in the wall and therefore there is no increase in interior RH. In the interval t2, the wall has certain absorbed moisture in the wall to increase the interior RH, which starts increasing exponentially until it reaches equilibrium. This shows that ambient oscillations do not have an impact to fluctuate the interior RH, because of a large moisture time constant of the wall which behaves as a buffer. If the time constant of a system is relatively small compared to frequency of concentration cycling, then the interior RH would start to fluctuate as it is discussed in literature [54].

The next simulation is accomplished for a year when both temperature and RH are changing (Figs. 20 and 21). Evaluation over a longer period allows the enclosure walls to be appropriately saturated over a time such that effects of sudden change in ambient temperature and moisture conditions, which are observed in time interval t1 in Fig. 19, have a more realistic level.

As the results show that under isothermal conditions, the ambient RH cycling does not have cycling effect on interior RH and therefore the RH increases exponentially until the equilibrium occurs. When a cyclic ambient temperature is applied to the enclosure, the interior RH is controlled significantly by the temperature, since the thermal time constant is much smaller than moisture time constant.

## Discussion

The RC model shows a quite good agreement with experimental results; therefore, it is also reasonable to use it for any other polymeric sealed box under isothermal and nonisothermal conditions. This developed model is also very simple and fast to implement in the modeling. Moreover, coupling of different dimensions like 0D, 1D in one model allows to optimize a model and to shorten computational time, however, still having a good accuracy of results. Sensitivity analysis of discretization showed that 3 elements discretization for enclosure wall is sufficient for the modeling.

In case of the complex shape of a box, the proper thickness of a wall in the modeling is always a sensitive part. Though 1D model with 2.4 mm wall thickness converged with experiments in Fig. 10, however, the average or the lowest value in the range of different wall thicknesses should be considered in the modeling which corresponds to the worst scenario. The issue is that the thinnest parts of wall are largest contributor for moisture transport while thicker ones might behave more as moisture buffers. The same procedure is with the surface area of the box having a complex shape where it should always be considered slightly larger than it is declared by the box dimensions. On the other hand, the two-dimensional modeling might be more accurate; however, the modeling is more time-consuming and would not give a lot of achievements since the 1D model was sufficient. Thus, such procedure of modeling box with complex shape could prevent of having critical areas for electronics reliability.

Moreover, the sensitivity of the RC model for a closed box of different dimensions has also been performed in a prior study [46]. Here, the opposite analysis was made, wherein the moisture was released from the saturated walls into the exterior environment and interior volume of a box. It is important to emphasize that the sensitivity of the model is also dependent on the imperfections in the corresponding physical enclosures being studied (and which are not included in the model). Assuming a proportional relation between the amount of imperfections and the dimensions of the electronic enclosure, the sensitivity of the modeled humidity prediction is higher for smaller boxes when compared to the experimental results. One method of increasing the sensitivity of the current model would, thus, comprise of including the effect of imperfections e.g., as considered in this paper. The accuracy may also be increased by including other phenomena such as natural convection into the model by corresponding equivalent RC elements.

Summarizing the modeling results, the imperfections (cracks through sealing and cable feedthrough) are more significant in the beginning of the moisture transport process and it was also confirmed by the experiments. Basically, the imperfections and the initial conditions are the most significant when a closed box with very small openings is investigated.

Under cyclic ambient conditions, the interior RH at constant temperature increases exponentially and similarly as considering constant ambient concentration. This shows that interior RH does not follow fluctuations of ambient RH because of relatively large moisture time constant and very frequent ambient oscillations. These two factors result that the wall behaves as a buffer and attenuates the oscillations of the ambient concentration. Moreover, Fig. 22 clearly shows that the interior RH is very controlled by the ambient temperature. This is caused by the quick response of the interior temperature due to the small thermal time constant, which also leads to a rapid interior moisture changes and therefore it almost identically follows the ambient temperature.

Comparison of modeling and experiments helps to create more accurate and optimized models. Furthermore, parameters can be optimized in the modeling using the experiments. However, the modeling can also help to evaluate some parameters in the experiments, especially when is difficult to attach some sensors or evaluate the total volume of material, for example, volume of enclosure wall when the shape is complex and difficult to measure.

## Conclusions

• The developed RC circuit is sufficient to predict in enough detail the moisture response under isothermal and nonisothermal conditions. Such RC model can be also developed for any more complex systems.

• The advantage of RC approach is that different dimensional analyses (0D, 1D) can be coupled together, which allow optimization of RC model.

• Sensitivity analysis of wall discretization shows that three elements for the wall give good results and can be applied for the future modeling.

• Investigation of imperfections in the modeling and experiments showed that the imperfections do not have a significant impact for the overall moisture behavior over a time and can be negligible, especially when box with an opening is studied.

• Under isothermal conditions, the ambient RH cycling does not cause cycling effect on interior RH and therefore the RH increases exponentially until the equilibrium occurs. However, the interior RH is controlled significantly by temperature fluctuations, since the thermal time constant is much smaller than a moisture time constant. Moreover, large moisture time constant through a wall gives a buffer effect under constant temperature.

## Acknowledgements

The authors would like to acknowledge the financial support from the Research Council for Technology and Production Sciences, Denmark (1335-00228A) for the ICCI project and Innovations-Fonden, Denmark for the IN-SPE project.

## Nomenclature

• A =

surface area of element perpendicular to the flow of heat or moisture (m2)

• c =

water vapor concentration (kg/m3)

• ca =

ambient water vapor concentration (kg/m3)

• cp =

specific heat capacity (J/(kg·K))

• cs =

water vapor concentration in solid (kg/m3)

• c0i =

initial water vapor concentration (kg/m3)

• Cequivalent =

equivalent air capacity for the transformed/normalized material (m3)

• de =

length of discretized element (m)

• D =

diffusion coefficient (m2/s)

• D0 =

pre-exponential factor/coefficient (m2/s)

• ED =

activation energy for diffusivity (J/mol)

• ES =

activation energy for solubility (J/mol)

• J =

moisture flow per unit area (kg/(m2·s))

• k =

thermal conductivity (W/(m·K))

• L =

length of a wall (m)

• mdry =

mass of dry sample (kg)

• mmoist =

mass of wet sample (kg)

• mwet =

mass of water (kg)

• Mwater_vapor_in_polymer =

water vapor mass in wall material (kg)

• n =

node number

• pa =

ambient water vapor pressure (Pa)

• ratioV =

the dimensionless transformation factor relating the water vapor concentration in the polymer and water vapor concentration in air or equivalent volume of reference material to the actual material volume

• R =

universal gas constant (J/(mol·K))

• Re =

resistance of discretized element (s/m3)

• Ropening,m =

resistance of opening (s/m3)

• RHa =

ambient relative humidity (kg/m3)

• RH0i =

initial relative humidity (kg/m3)

• S =

Solubility coefficient (kg/(m3·Pa))

• S0 =

pre-exponential factor/coefficient (kg/(m3·Pa))

• t =

time (s)

• T =

temperature (K)

• Vair =

volume of air (m3)

• Vequivalent =

equivalent air volume for the transformed/normalized material (m3)

• Vpolymer =

geometric volume of wall made of any polymeric material (m3)

• Vwall =

volume of wall (m3)

• u =

moisture content of material (kg/kg)

• ρ =

density (kg/m3)

• τ =

time constant (s)

## References

Tencer, M. , and Moss, J. S. , 2002, “Humidity Management of Outdoor Electronic Equipment: Methods, Pitfalls, and Recommendations,” J. Comp. Packag. Technol., 25(1), pp. 66–72.
Ambat, R. , Jensen, S. G. , and Møller, P. , 2008, “Corrosion Reliability of Electronic Systems,” ECS Trans., 6(24), pp. 17–28.
Hoge, C. E. , 1990, “Corrosion Criteria for Electronic Packaging—Part I: A Framework for Corrosion of Integrated Circuits,” IEEE Trans. Comp. Hybrids Manuf. Technol., 13(4), pp. 1090–1097.
Klinger, D. J. , 1991, “Humidity Acceleration Factor for Plastic Packaged Electronic Devices,” Qual. Reliab. Eng. Int., 7(5), pp. 365–370.
Adams, J. , Salvador, M. , Lucera, L. , Langner, S. , Spyropoulos, G. D. , Fecher, F. W. , Voigt, M. M. , Dowland, S. A. , Osvet, A. , Egelhaaf, H.-J. , and Brabec, C. J. , 2015, “Water Ingress in Encapsulated Inverted Organic Solar Cells: Correlating Infrared Imaging and Photovoltaic Performance,” Adv. Energy Mater., 5(20), p. 1501065.
Asadpour, R. , Chavali, R. V. K. , and Alam, M. A. , 2016, “Physics-Based Computational Modeling of Moisture Ingress in Solar Modules: Location-Specific Corrosion and Delamination,” IEEE 43rd Photovoltaic Specialists Conference (PVSC), Portland, OR, June 5–10, pp. 840–843.
Liu, Y. , Ji, Z. , Chen, P. , Wang1, X. , Ye, N. , and Chiu, C.-T. , 2016, “Moisture Induced Interface Delamination for EMI Shielding Package,” IEEE CPMT Symposium Japan (ICSJ), Kyoto, Japan, Nov. 7–9, pp. 217–220.
Electronics Cooling, 2013, “Electronic Performance Impact of Elevated Humidity Environments—Implications for Free Air Cooling of Data Centers,” Electronics Cooling, Plymouth Meeting, PA, accessed Jan. 5, 2016,
Punch, J. , Grimes, R. , Heaslip, G. , Galkin, T. , Vakevainen, K. , Kyyhkynen, V. , and Elonen, E. , 2005, “Transient Hygrothermal Behavior of Portable Electronics,” Sixth International Conference on Thermal, Mechanical and Multiphysics Simulation and Experiments in Micro-Electrons and Micro-Systems (EUROSIME), Berlin, Apr. 18–20, pp. 398–405.
Jacobsen, J. B. , Peter Krog, J. , Hjarbæk Holm, A. , Rimestad, L. , and Riis, A. , 2014, “Climate-Protective Packaging: Using Basic Physics to Solve Climatic Challenges for Electronics in Demanding Applications,” J. Industr. Electr. Mag., 8(3), pp. 51–59.
Peter, N. , Toth, P. , Kovacs ., and Kristof, B. G. , 2016, “Implementation of Moisture Diffusion Model in Multi-Material System Including Air Cavities,” 22nd International Workshop on Thermal Investigations of ICs and Systems (THERMINIC), Budapest, Hungary, Sept. 21–23, pp. 224–229.
Barink, M. , Mavinkurve, A. , and Janssen, J. , 2016, “Predicting Non-Fickian Moisture Diffusion in EMCs for Application in Micro-Electronic Devices,” Microelectron. Reliab., 62, pp. 45–49.
Han, B. , and Kim, D.-S. , 2017, “Moisture Ingress, Behavior, and Prediction Inside Semiconductor Packaging: A Review,” ASME J. Electron. Packag., 139(1), p. 010802.
Liu, D. , Wang, J. , Liu, R. , and Park, S. B. , 2016, “An Examination on the Direct Concentration Approach to Simulating Moisture Diffusion in a Multi-Material System,” Microelectron. Reliab., 60, pp. 109–115.
Luiten, W. , and Kadijk, S. , 2009, “The Better Box Model: An Analytical Estimation of Temperature and Flow in a Free Convection Air Cooled Electronics Enclosure,” 25th Semiconductor Thermal Measurement and Management Symposium (SEMI-THERM), San Jose, CA, Mar. 15–19, pp. 70–75.
Bayerer, R. , Lassmann, M. , and Kremp, S. , 2014, “Transient Hygro-Thermal-Response of Power Modules in Inverters—Mission Profiling for Climate and Power Loading,” Eighth International Conference on Integrated Power Systems (CIPS), Nuremberg, Germany, Feb. 25–27, pp. 1–8.
Bayerer, R. , Lassmann, M. , and Kremp, S. , 2016, “Transient Hygrothermal-Response of Power Modules in Inverters—The Basis for Mission Profiling Under Climate and Power Loading,” IEEE Trans. Power Electr., 31(1), pp. 613–620.
Bacher, P. , and Madsen, H. , 2011, “Identifying Suitable Models for the Heat Dynamics of Buildings,” Energy Build., 43(7), pp. 1511–1522.
Barclay, M. , Holcroft, N. , and Shea, A. D. , 2014, “Methods to Determine Whole Building Hygrothermal Performance of Hempelime Buildings,” Build. Environ., 80, pp. 204–212.
Gudum, C. , 2003, “Moisture Transport and Convection in Building Envelopes,” Ph.D. thesis, Technical University of Denmark, Kgs. Lyngby, Denmark.
Weitzmann, P. , 2004, “Modelling Building Integrated Heating and Cooling Systems,” Technical University of Denmark, Kgs. Lyngby, Denmark, Report No. DTU R-091.
Zhong, Z. , and Braun, J. E. , 2008, “Combined Heat and Moisture Transport Modeling for Residential Buildings,” Ph.D. dissertation, Purdue University, West Lafayette, IN, p. 82.
Rode, C. , and Sørensen, K. G. , 2010, “Whole Building Hygrothermal Simulation Model,” American Society of Heating, Refrigeration and Air-Conditioning Engineers, Recent Advances in Energy Simulation: Building Loads, Symposium, Chicago, IL, Paper No. CH-03-09.
Tencer, M. , 1994, “Moisture Ingress Into Nonhermetic Enclosures and Packages. A Quasi-Steady State Model for Diffusion and Attenuation of Ambient Humidity Variations,” IEEE 44th International Conference on Electronic Components and Technology, Washington, DC, May 1–4, pp. 196–209.
Greenhouse, H. , 2000, Hermeticity of Electronic Packages, Elsevier, New York.
Tsividis, Y. , and Milios, J. , 2013, “A Detailed Look at Electrical Equivalents of Uniform Electrochemical Diffusion Using Nonuniform Resistance–Capacitance Ladders,” J. Electroanal. Chem., 707, pp. 156–165.
Sharafian, A. , and Bahrami, M. , 2013, “Adsorbate Uptake and Mass Diffusivity of Working Pairs in Adsorption Cooling Systems,” Int. J. Heat Mass Transfer, 59, pp. 262–271.
Kang, Y. S. , Hong, J.-M. , Jang, J. , and Kim, U. Y. , 1996, “Analysis of Facilitated Transport in Solid Membranes With Fixed Site Carriers 1. Single RC Circuit Model,” J. Membr. Sci., 109(2), pp. 149–157.
Hong, J.-M. , Kang, Y. S. , Jang, J. , and Kim, U. Y. , 1996, “Analysis of Facilitated Transport in Polymeric Membrane With Fixed Site Carrier 2. Series RC Circuit Model,” J. Membr. Sci., 109(2), pp. 159–163.
Hong, S. U. , Won, J. , Park, H. C. , and Kang, Y. S. , 1999, “Estimation of Penetrant Transport Properties Through Fixed Site Carrier Membranes Using the RC Circuit Model and Sensitivity Analysis,” J. Membr. Sci., 163(1), pp. 103–108.
Dahan, N. , Vanhoestenberghe, A. , and Donaldson, N. , 2012, “Moisture Ingress Into Packages With Walls of Varying Thickness and/or Properties: A Simple Calculation Method,” IEEE Trans. Compon. Packag. Manuf. Technol., 2(11), pp. 1796–1801.
Guenin, B., 2012, “Application of Transient Thermal Methods to Moisture Diffusion Calculations, Part I,” Electronics Cooling, Plymouth Meeting, PA, accessed Jan. 5, 2016,
Guenin, B., 2013, “Calculation Corner—Application of Transient Thermal Methods to Moisture Diffusion Calculations, Part 2,” Electronics Cooling, Plymouth Meeting, PA, accessed Jan. 5, 2016,
Wilson, J., 2007, “Moisture Permeation in Electronics,” Electronics Cooling, Plymouth Meeting, PA, accessed Jan. 5, 2016,
Nagel, L. W. , and Pederson, D. O. , 1973, SPICE (Simulation Program With Integrated Circuit Emphasis, University of California, CA, Berkeley.
Crank, J. , 1975, The Mathematics of Diffusion, 2nd ed., Oxford University Press, Oxford, UK, p. 2.
Drofenik, U. , and Kolar, J. W. , 2003, “Teaching Thermal Design of Power Electronic Systems With Web-Based Interactive Educational Software,” Applied Power Electronics Conference and Exposition (APEC), Miami Beach, FL, Feb. 9–13, pp. 1029–1036.
Hattel, J. , 2005, Fundamentals of Numerical Modelling of Casting Processes, 1st ed., Polyteknisk Forlag, Lyngby, Denmark, Chap. 3.
Comin, J. , 1985, Polymer Permeability, Chapman&Hall, London, Chaps. 2, 4, 8.
MarkusSchmidt, D. , Lunz, M. , and Becker, K. U. , 2016, “A New Method to Model Transient Multi-Material Moisture Transfer in Automotive Electronics Applications,” 17th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems (EUROSIME), Montpellier, France, Apr. 18–20, p. 7463322.
Wong, E. H. , and Park, S. B. , 2016, “Moisture Diffusion Modeling—A Critical Review,” Microelectronics Reliability, Vol. 65, pp. 318–326.
Carroll, J. , 1991, “What is Henry's Law,” Chem. Eng. Prog., 87(9), pp. 48–52.
Gibbs, J. , 1961, “The Scientific Papers of J Willard Gibbs,” Thermodynamics, Vol. I, Dover, New York Mineola, NY.
Job, G. , and Herrmann, F. , 2006, “Chemical Potential—A Quantity in Search of Recognition,” Eur. J. Phys., 27(2), p. 353.
CENELEC, 2000, “Degree of Protection Provided by Enclosures,” CENELEC, Brussels, Belgium, Standard No. IEC 60529:1989/A1:1999.
Conseil-Gudla, H. , Staliulionis, Z. , Jellesen, M. S. , Jabbari, M. , Hattel, J. H. , and Ambat, R. , 2017, “Humidity Buildup in Electronic Enclosures Exposed to Constant Conditions,” IEEE Trans. Compon., Packag. Manuf. Technol., 7(3), pp. 412–423.
Mao, Z. , Luo, X. , Yang, J. , and Liu, S. , 2010, “Moisture Diffusivity Analysis of Polycarbonate for LED Lens,” 11th International Conference on Electronic Packaging Technology and High Density Packaging (ICEPT-HDP), Xi'an, China, Aug. 16–19, pp. 1080–1087.
Moon, S. I. , and Extrand, C. W. , 2009, “Water Vapor Permeation Resistance of Polycarbonate at Various Temperatures,” Ind. Eng. Chem. Res., 48(19), pp. 8961–8965.
Golovoy, A. , and Zinbo, M. , 1989, “Water Sorption and Hydrolytic Stability of Polycarbonates,” Polym. Eng. Sci., 29(24), pp. 1733–1737.
Staliulionis, Z. , Jabbari, M. , and Hattel, J. H. , 2016, “Mathematical Modelling of Coupled Heat and Mass Transport Into an Electronic Enclosure,” 22nd International Workshop on Thermal Investigation of ICs and Systems (THERMINIC), Budapest, Hungary, Sept. 21–23, pp. 323–326.
Staliulionis, Z. , Joshy, S. , Jabbari, M. , Mohanty, S. , Ambat, R. , and Hattel, J. H. , 2016, “Analysis of Moisture Transport Between Connected Enclosures Under a Forced Thermal Gradient,” 18th Electronics Packaging Technology Conference (EPTC), Singapore, Nov. 30–Dec. 3, pp. 320–324.
Yoon, S. , Han, B. , and Wang, Z. , 2007, “On Moisture Diffusion Modeling Using Thermal-Moisture Analogy,” ASME J. Electron. Packag., 129(4), pp. 421–426.
Jang, C. , Park, S. , Han, B. , and Yoon, S. , 2008, “Advanced Thermal-Moisture Analogy Scheme for Anisothermal Moisture Diffusion Problem,” ASME J. Electron. Packag., 130(1), pp. 749–755.
Staliulionis, Z. , Jabbari, M. , and Hattel, J. H. , 2016, “Moisture Ingress Into Electronics Enclosure Under Isothermal Conditions,” AIP Conf. Proc., 1738, p. 030041.
View article in PDF format.

## References

Tencer, M. , and Moss, J. S. , 2002, “Humidity Management of Outdoor Electronic Equipment: Methods, Pitfalls, and Recommendations,” J. Comp. Packag. Technol., 25(1), pp. 66–72.
Ambat, R. , Jensen, S. G. , and Møller, P. , 2008, “Corrosion Reliability of Electronic Systems,” ECS Trans., 6(24), pp. 17–28.
Hoge, C. E. , 1990, “Corrosion Criteria for Electronic Packaging—Part I: A Framework for Corrosion of Integrated Circuits,” IEEE Trans. Comp. Hybrids Manuf. Technol., 13(4), pp. 1090–1097.
Klinger, D. J. , 1991, “Humidity Acceleration Factor for Plastic Packaged Electronic Devices,” Qual. Reliab. Eng. Int., 7(5), pp. 365–370.
Adams, J. , Salvador, M. , Lucera, L. , Langner, S. , Spyropoulos, G. D. , Fecher, F. W. , Voigt, M. M. , Dowland, S. A. , Osvet, A. , Egelhaaf, H.-J. , and Brabec, C. J. , 2015, “Water Ingress in Encapsulated Inverted Organic Solar Cells: Correlating Infrared Imaging and Photovoltaic Performance,” Adv. Energy Mater., 5(20), p. 1501065.
Asadpour, R. , Chavali, R. V. K. , and Alam, M. A. , 2016, “Physics-Based Computational Modeling of Moisture Ingress in Solar Modules: Location-Specific Corrosion and Delamination,” IEEE 43rd Photovoltaic Specialists Conference (PVSC), Portland, OR, June 5–10, pp. 840–843.
Liu, Y. , Ji, Z. , Chen, P. , Wang1, X. , Ye, N. , and Chiu, C.-T. , 2016, “Moisture Induced Interface Delamination for EMI Shielding Package,” IEEE CPMT Symposium Japan (ICSJ), Kyoto, Japan, Nov. 7–9, pp. 217–220.
Electronics Cooling, 2013, “Electronic Performance Impact of Elevated Humidity Environments—Implications for Free Air Cooling of Data Centers,” Electronics Cooling, Plymouth Meeting, PA, accessed Jan. 5, 2016,
Punch, J. , Grimes, R. , Heaslip, G. , Galkin, T. , Vakevainen, K. , Kyyhkynen, V. , and Elonen, E. , 2005, “Transient Hygrothermal Behavior of Portable Electronics,” Sixth International Conference on Thermal, Mechanical and Multiphysics Simulation and Experiments in Micro-Electrons and Micro-Systems (EUROSIME), Berlin, Apr. 18–20, pp. 398–405.
Jacobsen, J. B. , Peter Krog, J. , Hjarbæk Holm, A. , Rimestad, L. , and Riis, A. , 2014, “Climate-Protective Packaging: Using Basic Physics to Solve Climatic Challenges for Electronics in Demanding Applications,” J. Industr. Electr. Mag., 8(3), pp. 51–59.
Peter, N. , Toth, P. , Kovacs ., and Kristof, B. G. , 2016, “Implementation of Moisture Diffusion Model in Multi-Material System Including Air Cavities,” 22nd International Workshop on Thermal Investigations of ICs and Systems (THERMINIC), Budapest, Hungary, Sept. 21–23, pp. 224–229.
Barink, M. , Mavinkurve, A. , and Janssen, J. , 2016, “Predicting Non-Fickian Moisture Diffusion in EMCs for Application in Micro-Electronic Devices,” Microelectron. Reliab., 62, pp. 45–49.
Han, B. , and Kim, D.-S. , 2017, “Moisture Ingress, Behavior, and Prediction Inside Semiconductor Packaging: A Review,” ASME J. Electron. Packag., 139(1), p. 010802.
Liu, D. , Wang, J. , Liu, R. , and Park, S. B. , 2016, “An Examination on the Direct Concentration Approach to Simulating Moisture Diffusion in a Multi-Material System,” Microelectron. Reliab., 60, pp. 109–115.
Luiten, W. , and Kadijk, S. , 2009, “The Better Box Model: An Analytical Estimation of Temperature and Flow in a Free Convection Air Cooled Electronics Enclosure,” 25th Semiconductor Thermal Measurement and Management Symposium (SEMI-THERM), San Jose, CA, Mar. 15–19, pp. 70–75.
Bayerer, R. , Lassmann, M. , and Kremp, S. , 2014, “Transient Hygro-Thermal-Response of Power Modules in Inverters—Mission Profiling for Climate and Power Loading,” Eighth International Conference on Integrated Power Systems (CIPS), Nuremberg, Germany, Feb. 25–27, pp. 1–8.
Bayerer, R. , Lassmann, M. , and Kremp, S. , 2016, “Transient Hygrothermal-Response of Power Modules in Inverters—The Basis for Mission Profiling Under Climate and Power Loading,” IEEE Trans. Power Electr., 31(1), pp. 613–620.
Bacher, P. , and Madsen, H. , 2011, “Identifying Suitable Models for the Heat Dynamics of Buildings,” Energy Build., 43(7), pp. 1511–1522.
Barclay, M. , Holcroft, N. , and Shea, A. D. , 2014, “Methods to Determine Whole Building Hygrothermal Performance of Hempelime Buildings,” Build. Environ., 80, pp. 204–212.
Gudum, C. , 2003, “Moisture Transport and Convection in Building Envelopes,” Ph.D. thesis, Technical University of Denmark, Kgs. Lyngby, Denmark.
Weitzmann, P. , 2004, “Modelling Building Integrated Heating and Cooling Systems,” Technical University of Denmark, Kgs. Lyngby, Denmark, Report No. DTU R-091.
Zhong, Z. , and Braun, J. E. , 2008, “Combined Heat and Moisture Transport Modeling for Residential Buildings,” Ph.D. dissertation, Purdue University, West Lafayette, IN, p. 82.
Rode, C. , and Sørensen, K. G. , 2010, “Whole Building Hygrothermal Simulation Model,” American Society of Heating, Refrigeration and Air-Conditioning Engineers, Recent Advances in Energy Simulation: Building Loads, Symposium, Chicago, IL, Paper No. CH-03-09.
Tencer, M. , 1994, “Moisture Ingress Into Nonhermetic Enclosures and Packages. A Quasi-Steady State Model for Diffusion and Attenuation of Ambient Humidity Variations,” IEEE 44th International Conference on Electronic Components and Technology, Washington, DC, May 1–4, pp. 196–209.
Greenhouse, H. , 2000, Hermeticity of Electronic Packages, Elsevier, New York.
Tsividis, Y. , and Milios, J. , 2013, “A Detailed Look at Electrical Equivalents of Uniform Electrochemical Diffusion Using Nonuniform Resistance–Capacitance Ladders,” J. Electroanal. Chem., 707, pp. 156–165.
Sharafian, A. , and Bahrami, M. , 2013, “Adsorbate Uptake and Mass Diffusivity of Working Pairs in Adsorption Cooling Systems,” Int. J. Heat Mass Transfer, 59, pp. 262–271.
Kang, Y. S. , Hong, J.-M. , Jang, J. , and Kim, U. Y. , 1996, “Analysis of Facilitated Transport in Solid Membranes With Fixed Site Carriers 1. Single RC Circuit Model,” J. Membr. Sci., 109(2), pp. 149–157.
Hong, J.-M. , Kang, Y. S. , Jang, J. , and Kim, U. Y. , 1996, “Analysis of Facilitated Transport in Polymeric Membrane With Fixed Site Carrier 2. Series RC Circuit Model,” J. Membr. Sci., 109(2), pp. 159–163.
Hong, S. U. , Won, J. , Park, H. C. , and Kang, Y. S. , 1999, “Estimation of Penetrant Transport Properties Through Fixed Site Carrier Membranes Using the RC Circuit Model and Sensitivity Analysis,” J. Membr. Sci., 163(1), pp. 103–108.
Dahan, N. , Vanhoestenberghe, A. , and Donaldson, N. , 2012, “Moisture Ingress Into Packages With Walls of Varying Thickness and/or Properties: A Simple Calculation Method,” IEEE Trans. Compon. Packag. Manuf. Technol., 2(11), pp. 1796–1801.
Guenin, B., 2012, “Application of Transient Thermal Methods to Moisture Diffusion Calculations, Part I,” Electronics Cooling, Plymouth Meeting, PA, accessed Jan. 5, 2016,
Guenin, B., 2013, “Calculation Corner—Application of Transient Thermal Methods to Moisture Diffusion Calculations, Part 2,” Electronics Cooling, Plymouth Meeting, PA, accessed Jan. 5, 2016,
Wilson, J., 2007, “Moisture Permeation in Electronics,” Electronics Cooling, Plymouth Meeting, PA, accessed Jan. 5, 2016,
Nagel, L. W. , and Pederson, D. O. , 1973, SPICE (Simulation Program With Integrated Circuit Emphasis, University of California, CA, Berkeley.
Crank, J. , 1975, The Mathematics of Diffusion, 2nd ed., Oxford University Press, Oxford, UK, p. 2.
Drofenik, U. , and Kolar, J. W. , 2003, “Teaching Thermal Design of Power Electronic Systems With Web-Based Interactive Educational Software,” Applied Power Electronics Conference and Exposition (APEC), Miami Beach, FL, Feb. 9–13, pp. 1029–1036.
Hattel, J. , 2005, Fundamentals of Numerical Modelling of Casting Processes, 1st ed., Polyteknisk Forlag, Lyngby, Denmark, Chap. 3.
Comin, J. , 1985, Polymer Permeability, Chapman&Hall, London, Chaps. 2, 4, 8.
MarkusSchmidt, D. , Lunz, M. , and Becker, K. U. , 2016, “A New Method to Model Transient Multi-Material Moisture Transfer in Automotive Electronics Applications,” 17th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems (EUROSIME), Montpellier, France, Apr. 18–20, p. 7463322.
Wong, E. H. , and Park, S. B. , 2016, “Moisture Diffusion Modeling—A Critical Review,” Microelectronics Reliability, Vol. 65, pp. 318–326.
Carroll, J. , 1991, “What is Henry's Law,” Chem. Eng. Prog., 87(9), pp. 48–52.
Gibbs, J. , 1961, “The Scientific Papers of J Willard Gibbs,” Thermodynamics, Vol. I, Dover, New York Mineola, NY.
Job, G. , and Herrmann, F. , 2006, “Chemical Potential—A Quantity in Search of Recognition,” Eur. J. Phys., 27(2), p. 353.
CENELEC, 2000, “Degree of Protection Provided by Enclosures,” CENELEC, Brussels, Belgium, Standard No. IEC 60529:1989/A1:1999.
Conseil-Gudla, H. , Staliulionis, Z. , Jellesen, M. S. , Jabbari, M. , Hattel, J. H. , and Ambat, R. , 2017, “Humidity Buildup in Electronic Enclosures Exposed to Constant Conditions,” IEEE Trans. Compon., Packag. Manuf. Technol., 7(3), pp. 412–423.
Mao, Z. , Luo, X. , Yang, J. , and Liu, S. , 2010, “Moisture Diffusivity Analysis of Polycarbonate for LED Lens,” 11th International Conference on Electronic Packaging Technology and High Density Packaging (ICEPT-HDP), Xi'an, China, Aug. 16–19, pp. 1080–1087.
Moon, S. I. , and Extrand, C. W. , 2009, “Water Vapor Permeation Resistance of Polycarbonate at Various Temperatures,” Ind. Eng. Chem. Res., 48(19), pp. 8961–8965.
Golovoy, A. , and Zinbo, M. , 1989, “Water Sorption and Hydrolytic Stability of Polycarbonates,” Polym. Eng. Sci., 29(24), pp. 1733–1737.
Staliulionis, Z. , Jabbari, M. , and Hattel, J. H. , 2016, “Mathematical Modelling of Coupled Heat and Mass Transport Into an Electronic Enclosure,” 22nd International Workshop on Thermal Investigation of ICs and Systems (THERMINIC), Budapest, Hungary, Sept. 21–23, pp. 323–326.
Staliulionis, Z. , Joshy, S. , Jabbari, M. , Mohanty, S. , Ambat, R. , and Hattel, J. H. , 2016, “Analysis of Moisture Transport Between Connected Enclosures Under a Forced Thermal Gradient,” 18th Electronics Packaging Technology Conference (EPTC), Singapore, Nov. 30–Dec. 3, pp. 320–324.
Yoon, S. , Han, B. , and Wang, Z. , 2007, “On Moisture Diffusion Modeling Using Thermal-Moisture Analogy,” ASME J. Electron. Packag., 129(4), pp. 421–426.
Jang, C. , Park, S. , Han, B. , and Yoon, S. , 2008, “Advanced Thermal-Moisture Analogy Scheme for Anisothermal Moisture Diffusion Problem,” ASME J. Electron. Packag., 130(1), pp. 749–755.
Staliulionis, Z. , Jabbari, M. , and Hattel, J. H. , 2016, “Moisture Ingress Into Electronics Enclosure Under Isothermal Conditions,” AIP Conf. Proc., 1738, p. 030041.

## Figures

Fig. 1

(a) Paths of moisture ingress into an enclosure (b) enclosure with the opening

Fig. 2

(a) Electric circuit for the enclosure (b) a plate for the test case

Fig. 3

Moisture ingress into electronic box through imperfections and wall

Fig. 4

(a) Modeling of RH inside enclosure using PC material for wall (resistance value Rn,m is equal to 76923076 s/m3 and capacitance value Cn,m = 0.001 m3, n – is the number of element) and (b) RC circuit when the PC material for wall is replaced with the air (resistance value Rn,m is equal to 353848 s/m3 and capacitance value Cn,m = 0.217 m3)

Fig. 5

Comparison of water vapor concentrations at elements node in the wall

Fig. 6

(a) Typical enclosure used in experiments from Fibox, (b) experimental setup, and (c) experimental setup with opening [39]

Fig. 7

Moisture response inside the enclosure dependent on wall thickness (L) when the diffusion coefficient is 4.51 × 10−12 m2/s

Fig. 8

Interior moisture response dependent on size of the opening (wall thickness is 2.2 mm)

Fig. 9

Experimental results when imperfections are analyzed

Fig. 10

Experimental results when imperfections are analyzed and compared to the modeling

Fig. 11

Interior moisture response for 1 mm opening and under different wall thickness

Fig. 12

Interior moisture response for 3 mm opening under different wall thickness

Fig. 13

Moisture response inside enclosure with 1 mm opening and 2.6 mm wall thickness

Fig. 14

Moisture response inside enclosure with 3 mm opening and 2.6 mm wall thickness

Fig. 15

Temperature dependent diffusion coefficient

Fig. 16

Moisture response inside enclosure under different diffusion coefficient (wall thickness −2.6 mm)

Fig. 17

Resistor-Capacitor circuit for modeling of temperature inside box through the wall

Fig. 18

Ambient temperature and RH

Fig. 19

Response of RH

Fig. 20

Ambient temperature and RH (Copenhagen climatic data) [47]

Fig. 21

Response of temperature

Fig. 22

Response of RH

Fig. 23

Comparison of wall discretization for moisture response

## Tables

Table 1 Properties of PC material

## Discussions

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