The differential pressure model has the advantage that it can estimate the pressure variations at the DPF under various exhaust conditions when the parameters set on the model are accurate, as shown in
Table 5 and
Table 6. This model works based on Darcy’s law by considering the PM layer and filter. The simulation was performed using the BOOST program (AVL List GmbH, Graz, Austria), which is a 1D simulator. The parameters of the differential pressure model for the DPF with the specifications in
Table 1 were determined based on experimental data tested by considering the flow rate, temperature, and DPF loading condition of the exhaust gas [
20].
3.1. Differential Pressure Model
The modeled system of the BOOST program is illustrated in
Figure 5. The system consists of a defining component as follows: the inlet and outlet boundary that defines the exhaust gas flow rate and temperature for the test, the DPF component that simulates the differential pressure according to the test condition, the diesel oxidation catalyst (DOC) component to which the Forchheimer differential pressure model is applied, and pipe components that connect the DOC and DPF parts.
For the DPF differential pressure modeling based on the exhaust gas temperature, flow, and soot loading conditions, the differential pressure was measured under clean conditions, i.e., soot loading of between 0 and 4.27 g/L. The exhaust gas temperature at each soot loading condition and the flow rate were the same.
To create a DPF model that satisfies the differential pressure characteristics measured under the above conditions, both modeling and measurement comprise two steps. In the first step, a clean DPF differential pressure model is created for a soot loading of 0 g/L. In the second step, soot loading of 4.27 g/L was added to the DPF model completed in the first step, and the differential pressure characteristics caused by soot were modeled.
3.2. Clean DPF Modeling
The differential pressure of the DPF under clean conditions is shown in
Figure 6. No. 1 and No. 7 in
Figure 6 denote the exhaust gas flowing into the DPF inlet and exiting the outlet, respectively, resulting in a loss of inertia due to contraction/expansion at the DPF plug and inlet/outlet channels, as indicated by the numbers. It can be divided into pressure loss owing to frictional resistance, and that occurs while passing through the filter, as in No. 4.
The pressure loss caused by contraction/expansion at the DPF inlet and outlet is expressed in Equations (1) and (2). Here, ζinl and ζout are the friction loss coefficients, which are unknown variables for modeling the pressure loss of the DPF in this study. In Equations (1) and (2), the density and flow rate of the exhaust gas are determined using the temperature and flow rate, respectively, which are used as the test conditions in this study. As the density and flow rate increase, the pressure loss value increases. Similarly, the pressure loss value caused by contraction/expansion at the inlet and outlet of the DPF varies proportionally to the value of the friction loss coefficient, which is an unknown variable.
The pressure loss caused by the plug blocking the DPF inlet and outlet is expressed in Equations (3) and (4) [
3]. Because of the absence of soot accumulation in the plug, Equations (3) and (4) can describe the pressure changes accurately. With respect to the pressure loss, the longer the plug length and the greater the exhaust gas flow rate and viscosity, the greater the pressure loss; whereas, the larger the channel diameter, the smaller the pressure loss. In the equations,
φ denotes a shape factor of the channel for which a value of 1.05 is used as the shape of the DPF channel is square.
The pressure losses in the DPF inlet and outlet channels were calculated using Equations (5) and (6). Here, the pressure loss along the channel length is calculated using the steady-state Darcy model given in Equation (7); Equation (8) expresses the Darcy constant model. In this formula, DPF length is expressed as
z. As the channel length, exhaust gas flow rate, and density increased, the pressure loss increased. However, as the channel diameter increased, the pressure loss decreased. Similar to the plug, the shape factor value of 1.05 was used for the channel as the shape of the DPF channel is square. As shown in
Figure 2, when ash is accumulated in the inlet channel of the DPF, the pressure loss is calculated by subtracting the length of the ash from the length of the inlet and outlet channels. For the plug of the DPF outlet channel, the pressure loss is calculated using Equation (4), by adding the ash length [
21].
The pressure loss caused by the exhaust gas passing through the DPF wall was calculated using the modified model of Darcy’s law proposed by Konstandopoulos, as shown in Equation (9). For accurately predicting the pressure loss caused by the DPF wall, the model was developed using the measured data under the soot and ash loading conditions of 0 g/L. In Equation (9), Kw denotes the wall penetration area. The larger the Kw value, the larger the area through which the exhaust gas can pass; therefore, the calculated pressure loss is small. The Kw value depends on the filter material and production process. Because the DPF wall expands with the increase in temperature, independent Kw values must be used for each temperature. In this study, the differential pressure of the DPF was modeled using the Kw value, where various exhaust gas temperatures—200, 300, and 400 °C—were used as the parameter.
The analysis was performed under a soot loading of 0 g/L, as shown in
Figure 3. Flow rate and temperature were set to be 3 °C/min. When the exhaust gas temperature in front of the DPF reaches 200 °C, the differential pressure is measured while increasing the flow rate from 250 to 550 kg/h. The x-axis in
Figure 7 represents the test number, where 14 test conditions were used.
To create a DPF model that satisfies all 14 test conditions, shown in
Figure 7, the inlet and outlet friction loss coefficients and
Kw values for each temperature condition were determined as unknown variables, as listed in
Table 2. To obtain the optimal unknown variable value, an AVL optimization software (Design Explorer) was used.
Table 5 lists the unknown variables obtained using the optimization method.
The results in
Table 5 show that the higher the exhaust gas temperature, the higher the DPF temperature and the smaller the wall penetration area because the filter expands owing to the characteristics of the DPF material. Therefore, under the same flow rate conditions, the differential pressure increased as the temperature increased. As the exhaust gas temperature increases, the differential pressure increases proportionally as the exhaust gas flow rate increases.
Table 5 lists the unknown variables that account for these characteristics. Among the unknown variables, the value of wall permeability at 200 °C is significantly larger compared to that at other temperatures. When wall permeability values between 300 and 400 °C were used, the value of differential pressure was overestimated compared to that obtained via measurement. The unknown variables were optimized within a range of 1–10.
The results presented in
Figure 7 show that the simulation data are in good agreement with the measured data (dP of the DPF). The maximum error value obtained between the simulation and experimental data was less than 1 × 10
−3 bar; therefore, the model is effective and can make accurate predictions.
Soot Loading
The more the soot is filtered in the DPF, the greater the amount of soot loading; because the soot acts as a filter, the filtration efficiency of the DPF improves but the differential pressure increases. To model the differential pressure characteristics according to soot loading, an engine dynamometer test was conducted in this study until the soot loading was ≥4 g/L. During the soot loading, the same exhaust gas temperature and flow conditions used for the clean DPF test were employed, and the differential pressure was measured.
The differential pressure owing to the soot cake was modeled using Equation (10). The modification was carried out using the test conditions and measured data presented in
Figure 8. The unknown variables that were used for the modeling were the soot packing density (SPC) and
Ksc. The SPC indicates the mass per unit volume and is used to calculate the volume according to the soot mass. The thickness of the soot layer was determined from these values. It is important to determine the optimal value because the effective diameter of the DPF channel depends on the thickness of the soot layer, and this affects the pressure loss [
21].
In Equation (10), Ksc denotes the penetration area of the soot cake. As with Kw, the larger the Ksc value, the larger the area through which the exhaust gas can pass through the soot cake; thus, the calculated pressure loss is large. Further, the Ksc value varied depending on the engine.
In Equation (10), the DPF inlet channel diameter is the same as the ash cake thickness. In this study, as the differential pressure was measured in the absence of ash, it was not used in the differential pressure calculation. As the value of the soot cake thickness increases, the natural logarithmic value increases, and thus, the differential pressure increases. In addition, as the Ksc value also decreases, the calculated differential pressure is large.
Similar to
Figure 7, the AVL optimization software Design Explorer was used to obtain the soot packing density and
Ksc values that satisfy the differential pressure characteristics for conditions 1–14.
Table 5 lists the unknown variables obtained using the optimization method.
The Ksc value obtained using the optimization technique increased as the temperature increased. Unlike filters, soot cake does not have expansion or contraction characteristics that depend on the temperature; therefore, soot undergoes passive regeneration at the time of the test, and it has a significant influence on the values of differential pressure. The Ksc had the largest value at 400 °C where passive regeneration occurred relatively often.
The result is similar to that of the dP of DPF in
Figure 7; the measurement and analysis results were compared. The model showed an accuracy of within 1 mbar in all the conditions except for the 10th condition. As the test is completed in the 11–14th conditions, the decrease in soot is affected by the passive regeneration that occurred during the 11–14th tests. Therefore, in the 10th condition, the calculated analysis result was higher than the measurement result. In a future study, we plan to verify the results of this study by adding the passive regeneration model to the DPF model.